Theoretical and Computational Acoustics 2005: Hangzhou, China, 19-22 September 2005

) are the surface spherical harmonics. The index £=0,1,2,.. with m=-l,..,l, but it is understood that UQQ = u ^ = 0. The notation u ° ^ , u ° ^ f , u ^ a n ( ^ uhn'uhrf >uhn 1S u s e c ^ f° r t n e t w o basic cases with ft as the spherical Bessel function jj and fi as the spherical Hankel function h ; + , respectively.

92 For an incident plane wave as in (3), the total scattered field u s c can be written u S c(r) = £ ( H? £ e l k " R • O Plm V R

- R) ) /

,

P = L,M,N.

(10)

The incoming field on the scatterer at the origin has two parts: the incident plane wave of type (3) and the scattered field from all the other scatterers. Both parts can be expanded in terms of u ^ , P = L,M,N, with expansion coefficients denoted a ^ and b^, respectively. It follows by a T-matrix (transition matrix) argument that hm

=

Z.^ Tlm;l'm' ' (al'm' + h'm') P'l'm'

(H)

with explicit expressions for a ^ and the T-matrix T^Vm, for a spherical scatterer. 6,7 A second equation system is derived by translating each wave bf • u t , (r — R ) to the origin: 7 ,'p _ sr^ c,pp' E^w^:, °lm

(12)

where the computable matrix fi^(/m/ depends on ku, the lattice, and on to/a and to/(3. Inserting Eq. (12) in Eq. (11), a linear equation system for bfm is obtained. In order to obtain the R / T matrices, the expansion (10) must be transformed to plane waves of the type (3). The following relation is crucial for this purpose: 5 ' 7 £

e i k " ' R h + M r - R | / C j ) Y, m (r - R) = £

R

g

2

~^=^

>T(K± ) e ^ V

.

(13)

%j*

Here, Kl"- should be used for z > 0 while K~- is needed for z < 0. A caret indicates the angular variables of the indicated quantity, and K^- is the z component of Kl_-. Using Eq. (13), a plane-wave representation of (10) is easily obtained 6 by expressing derivatives of Y";m in terms of itself and y ; m ± 1 . As anticipated from Sec. 2.1, it is the reciprocal lattice (2) that provides the changes of the lateral wavenumber vectors. 3. Basic Example A computer implementation has been made, with an existing program for photonic crystals 11 (the electromagnetic case) as a useful starting point. A basic example of the type in the left panel of Fig. 1 is now considered, with a 4 mm thick steel plate covered with a 3.5 mm rubber coating immersed in water with sound velocity c = 1480 m/s. In the middle of the rubber layer, spherical cavities with diameter 2 mm appear in a doubly periodic quadratic pattern with period d = 10 mm. The steel parameters are 5850 and 3230 m/s for the compressionaland shear-wave velocities, respectively, and 7.7 kg/dm 3 for the density. Only the rubber is anelastic, a viscoelastic solid with shear-wave velocity and absorption given by 100 m/s and 17.5 dB/wavelength, respectively. The corresponding compressional-wave parameters are 1500 m/s and 0.1 dB/wavelength, respectively, while the rubber density is 1.1 kg/dm 3 .

93 Curve (c) in Fig. 2 shows the frequency dependence of the corresponding reflectance. As compared to curves (a) and (b) for an uncoated and a homogeneously coated reference case, respectively, significantly reduced reflectance appears in the 10-60 kHz interval. At very low frequencies, the plate is thin compared to the wavelength and the reflectance drops. According to (2)-(4), with k|| = 0, only the normal beam is propagating in the water below c/d = 148 kHz. Nonnormal beam quartets become propagating at 148 kHz for (m,n) = (±1,0) and (0,±1) in (2), and at c^/2/d = 209.3 kHz for (m,n) = ( ± 1 , ± 1 ) . The corresponding small contributions to curve (c) are shown in curves (d) and (e), respectively.

Fig. 2. Variation with frequency of time- and space-averaged reflected energy flux, in dB relative to the timeaveraged normally incident plane-wave flux. The almost coinciding curves (a) and (b) show such reflectancies for two reference cases, an uncoated steel plate and a plate with a homogeneous rubber coating without cavities, respectively. Curves (c)-(e) concern the basic example as specified in the text. Curve (c) shows total reflectance, and curves (d) and (e) show the contributions from the two first nonnormal beam quartets.

4. Nonnormally Reflected B e a m s Pulse measurements in a water tank were designed to verify the existence of the nonnormal beam quartets from Fig. 2. A hydrophone at a distance of about 1 dm from a coated plate registered direct and reflected waves from a distant source. The hydrophone was moved laterally in x steps of 2.5 mm, covering 1.5 d = 15 mm for a constant y.

x=—d

x=0

x=+d/2

x=—d

x=0

x=+d/2

Fig. 3. Measurements (left panel) and modeling results (right panel) for pulse insonification centered at 177.5 kHz. The indicated tick-mark times are relative to a somewhat arbitrarily chosen reference time (zero, at the upper horizontal line). The seven traces in each panel correspond to the different lateral (x) hydrophone positions covering 1.5d = 15 mm. The direct arrival is denoted 'dir', the normally reflected beam 'rflO', and the first reflected beam quartet 'rfll'.

Figure 3 shows experimental data and modeling results for a source pulse centered at

94 177.5 kHz. The direct arrival 'dir' is followed by the normally reflected beam 'rflO', after about 0.135 ms. The pulse frequencies are higher than c/d = 148 kHz, allowing the existence of a propagating beam quartet corresponding to (m,n) = (±1,0) and (0, ±1) in (2)-(4). Indeed, a late arrival 'rfll' can be observed in both panels of Fig. 3. The geometrical and material parameters for the modeling, performed by Fourier synthesis, are exactly as in the basic example of Sec. 3, except that the cavities are adjacent to the steel to better match the actual coating. Since the water pressure is proportional to div(u), it follows from (2)-(4) that the lateral xy dependence of the pressure contribution by the beam quartet is given by ,.2nx. .2irx. ,.2ny. . 27ry. exp(i—) + exp(-i—) + exp(i—) + exp(-i—)

2-rrx 2iry\ •_ + « » - * ) .

, . (14)

For a constant y, varying constructive and destructive interference with x period d appears, as also observed for the 'rfll' arrival in the right panel of Fig. 3, computed for a particular y. The normal wavenumber of the beam quartet is kz = ^/w2/c2 — (2n/d)2, as obtained from (2)-(4), corresponding to a separated late arrival with normal group velocity du dkz

2it

C

(15)

d w

The expected lateral variations for the nonnormal beam quartet 'rfll' are not clearly seen in the measurements in the left panel. Contributing factors could be imperfections in the cavity lattice geometry, and that it was difficult keep the y value and achieve good accuracy during the desired 2.5 mm x translations of the hydrophone.

£=—d

x=0

x=+d/2

x=—d

x=0

x=+d/2

Fig. 4. As Fig. 3 but with a pulse centered at 250 kHz and two nonnormal beam quartets, 'rfll' and 'rfl2'.

Figure 4 is similar, but for a pulse center frequency of 250 kHz. Two nonnormal beam quartets are propagating in the water in this case, the previous 'rfll' quartet and an 'rf!2' quartet. By (15), the 'rfll' arrival gets an increased normal group velocity when the frequency is increased, and it now appears as a tail to the normally reflected beam 'rlfO'. The later beam quartet 'rfl2' consists of the four plane waves with representation (m,n) = ( ± 1 , ± 1 ) , according to (2)-(4). Results corresponding to (14)-(15) can easily be derived. The period in x for a fixed y is now halved to d/2. Both nonnormal beam quartets are weak in this case, as seen in both panels of Fig. 4. Noting that the spatial averaging in Fig. 2 causes cancellation of lateral energy flux, relative amplitudes in Figs. 3 and 4 are consistent with curves (d) and (e) in Fig. 2.

95 5. Design of Anechoic Coatings At lower frequencies, with only the normally reflected beam propagating in the water, Fig. 2 shows that an Alberich coating can provide significant echo reduction. Results of the same character have been given by Cederholm, 4 who computed reflection coefficients as functions of frequency based on parameter matching to certain experimental data. Unfortunately, direct measurements of the anechoic properties cannot be presented in an open publication. Anechoic coatings can be designed by numerical methods. The results obtained by two such methods, allowing certain variations to the basic example in Sec. 3, are shown as curves (d) and (e) in the left panel of Fig. 5. Curve (d) was obtained by differential evolution (DE) minimization. Simulating annealing and genetic algorithms have been popular global optimization methods during the last decade. DE is related to genetic algorithms, but the parameters are not encoded in bit strings, and genetic operators such as crossover and mutation are replaced by algebraic operations. For applications to underwater acoustics, DE has recently been claimed to be much more efficient than genetic algorithms 12 and comparable in efficiency to a modern adaptive simplex simulated annealing algorithm. 13 m s

/ > P3

HR

^ % r \^L^^^

VI

l.Or

LOU

-

10 20

«

\(e)/ 20

0.5 100

\i&f 10

'Jc^c'

. 30

.

Tf\

kHz

1450

m / s , j>2

1500

1550

O.Q[ 1.25 2

mn

3

?'.?6.' 4 5

Fig. 5. Left: Reflectancies as functions of frequency. Curves (a)-(c) are exactly as in Fig. 2, but for a restricted frequency interval. Curve (d) was obtained with DE to minimize the maximum reflectance in the band 15-30 kHz. Curve (e), obtained with the analytic design method of Sec. 5.1, exhibits a reflectance null at 22.5 kHz. Middle and right: ^-function characterizations of coating models with maximum 15-30 kHz band reflectance below -17 dB, jointly in terms of j>2,P3 (middle panel) and J>6,J>7 (right panel). The £^ level-curve values are 1,5,10,20,30,.., reaching 60 in the middle panel and 50 in the right panel. The two dashed arcs in the right panel represent cavity diameters of 2.7 mm (lower dashed arc) and 3.6 mm (upper dashed arc).

The objective function for the DE minimization was specified as the maximum reflectance in the frequency band 15-30 kHz. Starting from the basic example of Sec. 3, eight parameters, denoted Pi,P2,--,P8, were varied within a reasonable search space: rubber density [pi, 0.9-1.3 kg/dm 3 ] and compressional-wave velocity [p2, 1450-1550 m/s], rubber shear-wave velocity [j?3, 70-150 m/s] and absorption [p4, 7-27 dB/wavelength], and lattice period \p$ = d, 7-20 mm], coating thickness \pe, 1.25-5 mm], cavity diameter [0.5mm+p7 • (p^—1.25mm)], outer coating thickness between water and cavities [0.75mm+P8-(P6—1.25mm) ]. The parameters P7,p$ were defined as fractions, with nonnegative values such that pr + ps < 1. An echo reduction of at least 17.5 dB can be achieved throughout the band 15-30 kHz, as seen to the left in Fig. 5, curve (d). The corresponding rubber parameters are p i = 0.90 kg/dm 3 , p2 = 1455.3 m/s, ps = 149.8 m/s, p 4 = 26.9 dB/wavelength. The optimal geometrical parameters are p$ = d = 14.9 mm, pe = 4.98 mm, and p~t = 0.76, p% = 0.01. Improved echo

96 reduction could be obtained by also varying the rubber compressional-wave absorption. Some 40000 coating models were tested at this DE optimization. More information is contained in the search ensemble than just the optimal model. For example, let A be the set in the search space corresponding to coatings with maximum reflectance below -16.8 dB in the band 15-30 kHz, with characteristic function XA{PI,P2, --TPS)- Estimation is possible of certain dimensionless functions, generically denoted £4, of various parameters, such as . /

x

{nidP2dPz) IIIIIIIS

• IIJIIIxA(pi,P2,-,P8)dpidpidp5dp6dp7dp8 XA(PI,P2, -,PS) dpidp2dp3dpidp5dp6dp7 dp&

for the parameters P2 and P3, where each integral involves the whole corresponding searchspace cross-section. Estimates directly based on the DE search ensemble may be misleading, however, since the DE sampling is typically biased with an unknown sampling distribution. For Bayesian inverse problems, Sambridge 10 has proposed a resampling algorithm to estimate a posteriori probability density (PPD) function marginals. A neighborhood approximation 10 to the PPD, from a DE search ensemble, for example, can easily be evaluated along lines in parameter space. The new ensemble is constructed by random walks in directions parallel to the axes (Gibb's sampling), without further objective function calls. Here, a neighborhood approximation to \A is specified, and the Sambridge algorithm is adapted to produce an accordingly resampled ensemble with some 200000 models. The function £A{P2,P3) from (16), with average 1, is subsequently estimated in a straightforward way. The result is shown in the middle panel of Fig. 5. Most of the favorable coatings have rubber compressional-wave velocities below 1500 m / s , reasonably close to the velocity in the water, and rubber shear-wave velocities above 120 m / s . Without the resampling, the ^-function diagram would have appeared differently with higher values up to the left, indicating a DE-sampling bias as compared to the desired sampling here, controlled by XACharacterization in terms of a similarly produced £4 function of pe and p7 is made in the right panel of Fig. 5. It is natural that thick coatings (large p$) are preferred, but additional low-reflectance coatings appear within a large part of the region of p6iP7-space with cavity diameters, 0.5mm+p7-(p6 — 1.25mm), between 2.7 and 3.6 mm. Diameter-dependent singlecavity resonances, as modulated by multiple-scattering effects, appear to be essential for the loss mechanism and the frequency dependence of the reflectance. 5.1.

Designing

vanishing

reflectance

at an isolated

frequency

The second numerical design method is based on analytic function theory. It was used to produce curve (e) in the left panel of Fig. 5, with vanishing reflectance at 22.5 kHz. For a constant rubber density p and a varying complex rubber shear-wave velocity (3, the normal plane-wave reflection coefficient for waves from the water, now denoted 1Z, is an analytic function of the shear modulus fi = pf32 of the rubber material. The analyticity allows zeroes of lZ{n) to be identified by numerical winding-integral techniques, whereby the argument variation of 1Z is determined around search rectangles in the ji plane. Adaptive splitting of these search rectangles is applied until exactly one zero is enclosed. The secant method is finally used to refine the estimate of an isolated zero.

97 With carefully implemented error control, the existence of zeroes can actually be proved. The argument variation of H{n) around a closed path in the fj, plane is an integral multiple of 27r. The exact value is of course not obtained numerically, but a value close to 2w, for example, implies that one zero is enclosed. For the example in curve (e) in Fig. 5, exactly vanishing reflectance at 22.5 kHz is obtained at a rubber fi corresponding to a shear-wave velocity of about 98.3 m / s and a shear-wave absorption of about 26.7 dB/wavelength. All remaining parameters are kept at their values from the basic example in curve (c) of Fig. 2. Figure 6 shows corresponding time domain results obtained by Fourier synthesis, for a pulse with spectrum in the band 18-27 kHz. The reflected pulse as viewed at the water/rubber interface (left panel) is weak except in close connection to a spherical cavity at (x, y) = (0, 0) (the central trace). The corresponding energy is built up by evanescent waves with an exponential drop-off in the normal direction. At a distance of 1 m into the water, right panel, such waves are no longer discernible.

3

0.0ms-

— 0.0ms - 0.5ms

0.5ms -

- 1.0ms

-d/2

x=+d/2

=0

=+d/2

=0

x=-d/2

Fig. 6. Results of pulse computations by frequency synthesis, corresponding to curve (e) in Fig. 5. The reflected pulse is shown at the water/rubber interface (left panel) and 1 m into the water (right panel). There is a horizontal line for a common reference time (zero), where the center of the incident pulse has reached the water/rubber interface. Seven traces are drawn in each case, covering the overall period rf=10 mm along the x axis. The incident pulse is actually very similar in shape to the central trace in the left panel, but its amplitude is more than four times as large.

An apparent splitting of the pulse can be noted in the right panel of Fig. 6. To explain this effect, consider a general function g(t) of time t, with Fourier transform G(u>) = / g(t) exp(iwi) dt. It is the input to a linear filter with real-valued impulse function h(t) and transfer function H(tv) = J h(t) exp(iwi) dt. Thus, the output is given by the convolution (h * g){t) with Fourier transform H{LS)G(LO),

and H{—UJ) =

H*(LS).

For the notch filter from curve (e) of Fig. 5, there are real constants a,b such that H(w)

«

(17)

a (\ui\ — LOQ)+ib(u> — LUQsgn(uj))

in the vicinity of ±wo, where u>o = 2TT • 22500 Hz. For a function g(t) with spectrum concentrated to neighborhoods of ±WQ and Hilbert transform (TLg){t), it follows that

(h*g)(t)

-a ([Hg)'{t)+u0g(t))-b(g'{t)-uQ{Hg){t))

.

(18)

For a particular g(t) with G(u>) — $(o> — wo) + $(w + LOQ), where <E>(o;) is a real-valued, nonnegative, symmetrical function that vanishes for |w| > LJQ, g(t) = 2 cos(u>ot) tp(t)

and

(Tig)(t) = — 2 sin(uiot) ip(t)

(19)

98 w h e r e tp(t) is t h e inverse Fourier t r a n s f o r m of $ ( w ) . It follows by (18) t h a t (h*g)(t)

«

-2(p'(t)

(-a

sm(uJ0t) + b cos(uj0t))

.

(20)

T h e m o d u l a t i n g factor tp'(t) is small for small t, since tp(t) is a s y m m e t r i c a l function implying t h a t
Acknowledgments T h e m e a s u r e m e n t s were performed by Stefan B a n a n d T o r b j o r n S t a h l s t e n , a n d Alex Cederh o l m a n d G u n n a r S u n d i n shared their experience. Malcolm S a m b r i d g e provided his resampling code, a n d H a n n a G o t h a l l a n d R u n e Westin i m p l e m e n t e d t h e o p t i m i z a t i o n codes.

References 1. H. Oberst, Resonant sound absorbers, in Technical Aspects of Sound, ed. E.G. Richardson (1957), pp. 421-439. 2. G. Gaunaurd and J. Barlow, Matrix viscosity and cavity-size distribution effects on the dynamic effective properties of perforated elastomers, J. Acoust. Soc. Amer. 75 (1984) 23-34. 3. H. Strifors and G. Gaunaurd, Selective reflectivity of viscoelastically coated plates in water, J. Acoust. Soc. Amer. 88 (1990) 901-910. 4. A. Cederholm, Acoustic properties of Alberich rubber coatings subjected to an applied ambient pressure, in Proc. Tenth International Congress on Sound and Vibration (Stockholm, 2003), pp. 2515-2522. 5. J. B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). 6. Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen and J. H. Page, Elastic wave scattering by periodic structures of spherical objects: theory and experiment, Phys. Rev. B 62 (2000) 2446-2457. 7. I. E. Psarobas, N. Stefanou and A. Modinos, Scattering of elastic waves by periodic arrays of spherical bodies, Phys. Rev. B 62 (2000) 278-291. 8. B. L. N. Kennett, Seismic Wave Propagation in Stratified Media (Cambridge Univ. Press, Cambridge, 1983). 9. A.-C. Hladky-Hennion and J.-N. Decarpigny, Analysis of the scattering of a plane wave by a doubly periodic structure using the finite element method: application to Alberich anechoic coatings, J. Acoust. Soc. Amer. 90 (1991) 3356-3367. 10. M. Sambridge, Geophysical inversion with a neighbourhood algorithm - 2 appraising the ensemble, Geophys. J. Int. 138 (1999) 727-746. 11. N. Stefanou, V. Yannopapas and A. Modinos, Multem2: a new version of the program for transmission and band-structure calculations of photonic crystals, Comput. Phys. Coram. 132 (2000) 189-196. 12. C. van Moll and D. G. Simons, Improved performance of global optimisation methods for inversion problems in underwater acoustics, in Proceedings of the Seventh European Conference on Underwater Acoustics, ed. D. G. Simons (Delft, 2004), pp. 715-720. 13. H. Gothall and R. Westin, Evaluation of four global optimisation techniques (ASSA, DE, NA, Tabu Search) as applied to anechoic coating design and inverse problem uncertainty estimation, M.Sc. thesis, Swedish Defence Research Agency (2005). 14. S. Ivansson and I. Karasalo, Computation of modal wavenumbers using an adaptive windingnumber integral method with error control, J. Sound Vib. 161 (1993) 173-180.

SEISMIC CHARACTERIZATION AND MONITORING OF THIN-LAYER RESERVOIR

LONG JIN*, XIAOHONG CHEN, JINGYE LI China Petroleum University (Beijing), China jinlong@cup. edu. en

Thin-layer reservoir has great significance for oil exploration and development. Seismic characterization and monitoring of thin-layer reservoir has spatial advantage. New seismic attributes and attributes combination analysis are proposed, including attributes versus incidence angle, attributes versus scale, reflection coefficient spectrum and timefrequency analysis for detailed thin-layer reservoir characterization.

1 Introduction Thin-layer reservoirs are large in north china and other areas in the world. It is also an important research topic for geophysicists. With the development of oil exploration, detailed analysis of thin-layer reservoir is needed. Many researchers have done important work in this area. Widess studied amplitude character of thin-layer using normal pulse reflections[l]. Lange, Rafipour and Marfurt studied seismic attributes for thin-layer and fluid discrimination^][3][4]. Christopher and James analyzed the effect of the converted wave and multiple on thin-bed and AVO modeling [5] [6]. Chung analyzed the precision of different approximation[7]. Liu studied the amplitude attributes for thin-layer using acoustic wave equation modeling method[8]. Ellison studied the modeling and analysis method for thin-layer reservoir monitoring [9]. We studied new attributes and attributes combinations for thin-layer reservoir characterization and monitoring. 99

2 Thin-layer seismic modeling Many author using convolution based modeling method in thin-layer studies[l][5]. Reflectivity method can also be used[6]. We use a method similar to that used by Liu[8], but our method is based on elastic equation. There is no analytical expression for these method. Detailed derivation is in appendix. Figure 1 is the modeling result for a simple model with only three layers.

J 0

I_J—j_J 500

J

1

,

J

1

,

1000 OffsetCm)

L_i_i.

l__l 1500

L

1 J

L_ 2000

Figure 1 Seismic modeling result for a simple model with three layers

3 Seismic attributes versus incidence angle for thin-layer AVO has been widely used in oil exploration. Mazzotti also proposed the combined amplitude, phase and frequency versus offset analysis for layered bed[10]. We use this method in thin-layer reservoir analysis. Three layer model is studied. The upper and lower layer are both shale and the middle layer is sand. The rock properties for the model is in table 1, which is excerpted from [11]. The thickness of the middle layer can be changed. Later experiment is also based on this data. Figure2-Figure5 is amplitude and phase versus incidence angle for the three models when bed

101

thickness change. Conventional analysis used only amplitude. Our studies show that phase can help separate different thickness of the bed. Figure6 is amplitude and phase versus incidence angle for velocity change. There are good correlation between amplitude, phase versus incidence angle and velocity change. Table 1 rock properties used for the synthetic seismograms Sw

Vp(m/s)

Vs(m/s)

P (g /cc )

Shale

3900

2086

2.300

Sandl

3855

2202

2.320

1

Sand2

3597

2217

2.288

0.8

Sand3

3755

2254

2.192

0.2

0

S

10

15

20

23

30

incidence angle (a)

36

40

45

50

102

IS

30

25

30

35

incidence angle (b) Figure 2 Amplitude versus incidence angle for different thickness (a) and phase versus incidence angle for different thickness (b) for saturation=l

1S

30

25

30

incidence angle (a)

35

103

incidence angle

(b) Figure 3 Amplitude versus incidence angle for different thickness (a) and phase versus incidence angle for different thickness (b) for saturation=0.8

0

0

s

i

1

i

;

s

5

10

1S

20

25

30

incidence angle (a)

j

35

i

j .

j

40

45

50

104

>

S

i

!

i

k

1

J

1

3

"0

5

10

15

20

25

30

35

40

45

SO

incidence angle

(b) Figure 4 Amplitude versus incidence angle for different thickness (a) and phase versus incidence angle for different thickness (b) for saturation=0.2

0

5

10

15

20

25

30

incidence angle

(a)

35

40

45

50

15

20

25

30

35

incidence angle

(b) Figure 5 Amplitude versus incidence angle for different velocity change (a) and phase versus incidence angle for different velocity change (b) for saturation =1 and thickness=0.25 wavelength

• 0% velocity ' l^vetocfly • 2% velocity 3% velocity > 4%vsJoc%

15

JO

25

30

incidence angle

(a)

35

change change change change change

—— 0 % velocity change — 1 % velocity change - • - 2 % velocity change -*>~ 3 % veloctty change -**_4%vetocaychange

--1.S5-

00

-1.65

«

15

30

25

30

35

40

45

50

incidence angle (b) Figure 6 Amplitude versus incidence angle for different velocity change (a) and phase versus incidence angle for different velocity change (b) for saturation =1 and thickness=0.125wavelength

4 Seismic attributes versus scale for thin-layer Wavelet transform is helpful in analyzing energy and frequency difference[12]. Seismic attributes versus scale is proposed and tested in thin-layer analysis. The theory of wavelet is not discussed here. We use continuous wavelet transform and morlet wavelet is chosen. Figure7 is amplitude versus scale for different bed thickness. Both amplitude maximum and corresponding scale are different for bed thickness change. So the new attributes can better delineate thin-layer bed thickness. Figure8 is amplitude versus scale for different incidence angle. When incidence angle increases, amplitude increases and scale decreases. Figure0- is amplitude versus scale for different velocity change. Velocity change mainly affects amplitude.

Scale (a)

(b)

Figure7 Amplitude versus scale for different bed thickness when incidence angle is 0 (a) and Amplitude versus scale for different bed thickness when incidence angle is 30(b).

Scale (b)

Figure8 Amplitude versus scale for different incidence angle when bed thickness is 1/4 wavelength (a) and Amplitude versus scale for different incidence angle when bed thickness is l/8wavelength (b).

Scale

(a)

0

10

20

30

40

ao

60

TO

Scale

(b)

Figure9 Amplitude versus scale for different velocity change when bed thickness is 1/4 wavelength (a) and Amplitude versus scale for different velocity change when bed thickness is l/8wavelength (b).

5 Reflection coefficient spectrum for thin-layer thickness and velocity change Spectral decomposition has been successfully used in bed thickness estimation and fluid discrimination[13][14]. The basis of spectral decomposition is reflection coefficient spectrum dependence on thickness and velocity change. Using reflection coefficient spectrum, thickness and velocity change can be separated in thin-layer. Figure 10-11 is reflection coefficient spectrum for different bed thickness and different velocity change. Bed thickness mainly affect frequency of reflection coefficient spectrum maximum. Velocity change mainly affect amplitude of reflection coefficient spectrum maximum. Using these two attributes, the bed thickness and velocity change can be discriminated.

(a)

Ill

(b)

Figure 10 Reflection coefficient spectrum for different bed thickness when incidence angle is 0(a) and Reflection coefficient spectrum for different bed thickness when incidence angle is 30(b).

Frequency{Hz)

(a)

112

0,3 -

20% velocity artf densfty cjiange }.

(b)

Figure 11 Reflection coefficient spectrum for different velocity change when bed thickness is l/4wavelength(a) and Reflection coefficient spectrum for velocity change when bed thickness is l/8wavelength (b). 6 Time-frequency analysis for thin-layer Time-frequency analysis can remove the tuning effect. The generalized S transform is used in the analysis. Figure 12 is the reflection coefficient and seismic trace. The seismic trace is affected by tuning. Figure 13 is the generalized S transform of the seismic trace. When the frequency increases, the spectrum has better correlation with reflection coefficient. Figure 14 is the comparison of one frequency spectrum and reflection coefficient. It can be shown that the position of maximum of spectrum can indicate the position of reflection coefficient. Figure 15 is the recovered reflection coefficient using time-frequency analysis. Time-frequency analysis is used to delineate the structure of seismic trace and combined with amplitude of the trace to form the recovered refection coefficient.

113 4)

!

"

o:

c

oi—fL

g-0.5 0) 0= -if9) 0

0.02

T~^7 0,04

O.06

0.08

0,1

0.12

0,14

0.16

0.18

0.2

time

Figure 12 Simple reflection coefficient and seismic trace

100

ISO

frequency

Figure 13 Spectrum for the modeled seismic trace

i;

1

S05i

T

•S-osi-

^

i

!f= I <5) I •--1.5 O

0.02

0.04

0.06

0,08

0.1

0,1!

0.14

0.16

0.18

0.2

time

Figure 14 Comparison of reflection coefficient and one high frequency spectrum

2

<§j" o

H-^Tb.oT~~oo4"™"o.o6

ooe

0.1

o,u

0.14

0.16

0.18

0.2

Figure 15 comparison of recovered reflection coefficient and true reflection coefficient

7 Conclusions and discussions The thin-bed seismic signature is affected by both bed thickness and reservoir property change. Using the proposed seismic attribute or

combination analysis method, thin-bed reservoir can be characterized and monitoring more precisely. Acknowledgement Thank my friend Chen tiansheng for his help in thin-layer seismic modeling. Reference 1 Widess, M.B., How thin is a thin bed?: Geophysics, 1973, 38:1176-1180. 2 James N. Lange, H. A. Almoghrabi., Lithology discrimination for thin layer using wavelet signal parameters.:Geophysics, 1988, 53:1512-1519. 3 B. Rafipour, E. Herrin. Phase offset indicator(POI): A study of phase shift versus offset and fluid content.:Geophysics, 1986, 51:679-688. 4 K. J. Marfurt, R. L. Kirlin. Narrow-band spectral analysis and thin-bed tuning. Geophysics, 2001, 66:1274-1283. 5 Christopher Juhlin, Roger Young. Implication of thin layers for amplitude variation with offset(AVO)

studies.: Geophysics, 1993,

58:1200-1204. 6 James L. Simmmons, Jr, Milo M. Backus. AVO modeling and the locally converted shear wave. Geophysics, 1994, 59:1237-1248. 7 Chung, H. M., Lawton, D. C. Amplitude responses of thin beds: Sinusoidal approximation versus Ricker approximation: Geophysics, 1995, 223-230. 8 Yinbin Liu and Douglas R.Schmitt. Amplitude and AVO responses of single thin bed.: Geophysics, 2003, 68(4): 1161 -1168. 9 Shelley J. Ellison, Matthias G. Imhof, Cahit Coruh, etc. Modeling offsetdependent reflectivity for time-lapse monitoring of water-floor production in thin-layer reservoir.: Geophysics, 2004, 69:25-36.

lOA.Mazzotti. Amplitude, phase and frequency versus offset applications.: Geophysics Prospecting, 1991, 39:863-886. 1 lFuping Zhu, Richard,L. Gibson JR., Joel S. Watakins, Sung Hwan Yuh. Distinguishing

fizz

gas

from

commercial

gas

reservoir

using

multicomponent seismic data.:TLE, 2000, 11, 1238-1245. 12 Yue Wenzheng, Tao guo. A method for recognition of fluid property in reservoirs using wavelet transformation.: Chinese Journal of Geophysics, 2003, 11,863-869 13 Partyka, Gridley, J., Lopez, J. Interpretation application of spectral decomposition in reservoir characterization,: TLE, 1999, 18, 353-355 14 K.R. Sandhya Devi, A.J. Cohen. Wavelet transforms and hybrid neural nets for improved pore fluid prediction and reservoir

properties

estimation.2004, 74rd SEG meeting extended abstracts.

Appendix For three layers model, the forum (1) can be derived using displacement and stress continuous conditions. " M (1)

V3)

w (,)

w(3)

=4^

°T 1 ,-»

2

r«> zx

1 /-I

ro, ZX

_ M

u ' =-i — smidAl -i—smidA2-i ax ax w(> =-i — cos idA{-i— ax ax aM = -P{0)2

— cosisB2 (3X

cosidA2-i—smisB2 f3x

cos(2r')4 1 - px0)2 cos{2is)A[ + A » 2 sin(2/])^

1

CO1 .

m

2ju{

T(£ =

(3)

,

CO2 .

., ,,

,

., ,,

CO

sin i\ cos i\A\ + — s i n i\ cos i\A\ + —— cos(2/s1 )B\ or, or, 2p,

• &>

.

.3 ,3

. #?

.3

n

3

- P ' - - ? — s i n / Ja 4 1 - z — cos/^5, 0

(3)

«"3

/"3

•
.3 .. &> 3 -3 j^3 w COSJ, A -I

.

.3

n

3

ww = -i — c o s Ja ^ ,1 - z — /5 sini s B x "3 ^3 «3

3)

A

°"i = - A ' ' cos(2zs )4 + p3
-^

n

2// 3 Where or,

9

m

3

CO

.

3

.3

.3 , 3

CO

= - ^ s i n / 3 cos/ 3 4 3 + - _ c o s ( 2 / 3 ) 5 1 3 a3 2p3

(l)

/?, and or3 > /?3 is the p and s wave velocity. is* id% /J% Zrf are angle

%

of refection s wave, reflection p wave, transmission s wave, transmission p wave. A^

A2s B2S Ax^ flj is the displacement amplitude of incidence p wave, reflection p

wave, reflection s wave, transmission p wave, transmission s wave. The ratio of displacement amplitude can be defined as,

HPP Where, R

4 A

R

B\ or,

=-f—

" 4A

/'pp

A] or,

A' a3

is reflection coefficient in frequency domain.

T

=^L^L

(2 )

THE ENERGY-CONSERVING PROPERTY OF T H E S T A N D A R D P E DING LEE Naval Undersea Warfare Center, Newport, RI,

USA

ER-CHANG SHANG CIRES,

University of Colorado, Boulder, Colorado,

USA

December 19, 2005

In 1974 a model was introduced by Frederick D. Tappert for predicting long-range wave propagation in a range-dependent environment. He applied the parabolic Equation approximation to transform the Helmholtz equation into a parabolic equation, the very first Parabolic Equation (PE). A pressure-release surface boundary is considered along with an artificial bottom boundary treatment. This paper proved that the Tappert model is energy-conserving.

1. Introduction Over the past quarter century, the authors had continuous technical interactions with Frederick D. Tappert who, in 1974, introduced a model which is to apply the parabolic equation approximation to transform the Helmholtz equation into a parabolic equation, the very first parabolic equation (PE). In 1984 the first author invited Tappert to spend a summer together to do research in relation to PE developments. The author raised a question to Tappert: You made a big contribution of PE to the acoustic community; in order to honor your contribution, should the very first PE be named after you? Tappert said: No, but suggested naming it the Standard PE. From that time on the Standard PE was recognized by the acoustics community. The Standard PE is a 2-dimensional (range and depth) representative wave equation which defines an initial-boundary value problem. Associated with the Standard PE, the surface boundary condition is considered pressure-release; the bottom boundary condition is treated by a special technique, introduced by Tappert, called "artifical bottom". The artifical bottom technique is to extend the field vertically down to the bottom deep enough such that u(r, z) = 0 at the bottom. The Standard PE to go with the assumed boundary conditions is regarded as the Tappert model. 119

120 Since the early 1980's, the authors and Tappert continued their technical discussions including the issues, contributions, and new results with reference to the PE-related developments. Various topics were among their discussions; Standard PE was one of the topics we discussed, but the energy-conserving issue entered the discussions but we did not pursue to prove that the Tappert model is energyconserving. This paper is to prove that the Tappert model is energy conserving. 2. Basic Development This section consists of the outline of two parts: The theoretical development of the standard PE and the associated surface and bottom boundary conditions. Theoretical details can be found in references [1, 2]. 2.1. The standard PE Let r be the range variable, z be the depth variable, u(r, z) the 2-dimensional wave field, n(r, z) be the index of refraction which is a real-valued function, and ko is the reference wavenumber which is a real-valued scalar. The very first Parabolic Equation introduced by Tappert [1] takes the form: iko , 0,

1 d2u

H.

-{n*(r,z)-l)*+—-^.

Ur =

2.2. Associated

,

boundary

(2.1)

conditions

Two types of boundary conditions are considered: the surface and the bottom boundaries. Let zs indicate the surface boundary and z&, the bottom boundary. 2.2.1. The surface boundary condition The assumed pressure-release surface condition indicates that u(r,zs) indication implies that the prescribed surface boundary conditions are u(r.zs) = 0,

u(r,z,) = 0,

du 7H*. az

= 0

-

du 7T^ oz

=

°-

= 0. The

(2'2)

2.2.2. The bottom boundary condition A technique was introduced by Tappert to generate the bottom boundary condition. This technique is known as the "artificial bottom" which is to extend the wave field vertically deep enough such that u(r, zb) = 0 there. Therefore; the prescribed bottom boundary conditions are u(r,zb)=0,

u(r,zb) = 0,

uu ^L=<>>

ou ^ U = °'•

(2-3)

121 3. Energy-Conserving Property Writing Eq. (2.1) in the form ur = a(n2(r, z) — \)u + buz

(3.1)

where ik0

a = —i

b = „ ,

2

(3.2)

.

2ik0

From Eq. (3.1), d2u ur = a(n2(r.z) — l)u + b

(3.3)

we have uru = a(n2(r, z) — l)uu + b

(3.4)

dz*

and —

-,

2/

\

T fd2U

-.s-

.

(3.5)

iru = a{n (r.z) — \)uu + b I —— J u Then

uru + uru =

d2u

d2i

u dz2

d2

£ W = 1(\U\>).

u

(3.6)

We want to examine whether or not \u\zdz = 0.

dr

(3.7)

Making use of expressions in Eq. (3.6), we have fZb

d2u\_

fz

r)

fd2u\

'

dr

dz.

(3.8)

Then, saving of the writing of ^ and the constant b, the first integral of the righthand-side of Eq. (3.8) can be evaluated by means of integration by parts; i.e. Zb / 2

(dus

d u\_1

du\

.

r-dudu,

dz

Tzn--LTzd-z -

/

,

x

(3 9)

'

Similarly, the second integral of the right-hand-side of Eq. (3.8) becomes czb /oa( ^ )

du\ UdZ

.

fdu\

dz~)uU+{dz')uL+

. /I

fZbdudu, dz -~^T~ dz dz

(3-10)

122 The term (§^)w|Zs in Eq. (3.9) and the term ( | j ) u | 2 s in Eq. (3.10) all go to zero due to the surface boundary condition, (2.2). Similarly, the term {%)u\Zb in Eq. (3.9) and the term ( f f H 2 „ in Eq. (3.10) all go to zero due to the bottom boundary condition, (2.3), therefore; d

fz\

,2,

d

A [Zb f

du9a\

,

fZbdudUl]

n

. „,

Then, the energy-conserving property of the standard PE with the prescribed boundary conditions, (2.2) and (2.3), is proved.

4. Remarks The Standard PE is a two-dimensional model with a narrow angle capability. These days the three-dimensional models have become more realistic in real applications. Not many users in the scientific community are using the two-dimensional model. Why bother to study the energy-conserving property for the Standard PE? There are a few answers for this question: 1. Because of the interest in three-dimensional problems, the Standard PE may not be used often in the acoustics community, to report this theoretical result to the public, we believe, may still interest the readers. 2. We selected the Tappert model to show it is energy-conserving, on the other hand, is to remember the late Tappert for what he did for the scientific community. 3. The technique, we used to examine the energy-conserving property, can be used to examine the energy-property of other PE models.

5. Conclusion The PE influence to the acoustic community is huge. All further-developed PE's are in use widely in the acoustifics community; they were all derived from the standard PE which benefited the scientific community a great deal. The Standard PE, even now-a-days is having limited use, it must not be forgotten; interestingly, the energy-conserving property of the Tappert model should not be unmentioned. This procedure may be applied to investigate the energy-conserving property for all other PE's, PE-like models, or other types of wave propagation models.

5.1.

Dedication

The impact of the PE to the scientific community is huge. In recognition of the PE contribution to the acoustic community, we cannot forget the Standard PE; and Frederick D. Tappert must be remembered. This paper is written in memory of our long time colleague Frederick D. Tappert.

123

Acknowledgments This research of the first author was supported by the U.S. Naval Undersea Warfare Center (Newport) Independent Research project.

References 1. Tappert, F. D., The Parabolic Equation Approximation Method, in Wave Propagation and Underwater Acoustics, ed. J. B. Keller and J. S. Papadakis, Lecture Notes in Physics 70, Springer-Verlag, Heidelberg, 1977, 224-287. 2. Lee, D. and S. T. McDaniel, BOOK. 3. Lee, D., A. D. Pierce, and E. C. Shang, Parabolic Equation Development in the Twentieth Century, J. Corny. Acoist. 8(4), 2000, 527-628.

A Dedication to Professor Tappert

Professor Frederick D. Tappert, who introduced the parabolic equation approximation to the acoustic community, passed away in 2001. Professor Michael I. Tarodakis and Dr. Finn B. Jensen organized a special memorial session for Prof. Tappert at the 6th International Conference on Theoretical and Computational Acoustics (ICTCA) at Hawaii, Honolulu, U.S.A. August 11-15, 2003. Professor Tarodakis and Dr. Jensen further encouraged the session speakers to contribute their articles to be included in the Proceedings of Theoretical and Computational Acoustics 2003. Their efforts in organizing this memorial session is appreciated by all of us. In January, 2000,1 visited Prof. Tappert in Miami, Florida, U.S.A. He expressed interest in contributing a paper to the 6th ICTCA. At that time, I started writing a paper on Revolutionary Influence of the Parabolic Equation Approximation to honor him. I continued to make progress on this article. At that stage, it was an article but, I left room for expansion. After the shocking news regarding Prof. Tappert, I immediately started writing another article entitled The Energy-Conserving Property of the Standard PE and dedicated it in memory of Prof. Tappert. Suddenly I was diagnosed with age-related Macular Degeneration. I had difficulty reading and writing. I was forced to stop writing this article, which I had planned to submit at the 2003 Hawaii ICTCA in the memorial session for Prof. Tappert, organized jointly by Prof. Tarodakis and Dr. Jensen. I was unable to submit the Energy Conserving paper on time. I felt very guilty for not being able to present this paper. After the conference, I was determined to complete writing this article, if possible. Prof. Er-Chang Shang came to help. With his help, this article has been completed. I thank Prof. Shang for his help and thank the committee chairs for giving me the opportunity to present this article at the 2005 Hangzhou ICTCA. Professor Frederick D. Tappert has gone; his PE contribution will be remembered. This article is dedicated in memory of my long-time colleague, Frederick D. Tappert. Ding Lee

124

Fredrick D. Tappert April 21, 1940 - January 9, 2001

Frederick D. Tappert was born April 21, 1940 in Philadelphia, Pennsylvania. His parents, the Reverend Dr. Theodore G. Tappert and Helen Louise Carson Tappert, raised their family of four children in the Lutheran Theological Seminary in Philadelphia, where the Reverend Dr. Tappert was a noted theologian. Fred showed an early penchant toward mathematics and science and attended Central High School in Philadelphia, which recognized outstanding young men in this area. From there he went on to study engineering at Penn State University, funded by the Ford Foundation, where he graduated with a B.S. in Engineering Science with honors in 1962. Fred went on to pursue his Ph.D. in theoretical physics from Princeton University with a full scholarship from the National Science Foundation. He earned his Ph.D. in 1967. Upon graduation Dr. Tappert was hired to the Technical Staff at Bell Laboratories in Whippany, New Jersey from 1967 - 1974, where he worked on plasma physics and high altitude nuclear effects, UHF radar propagation, solitons in optical fiber, and ocean acoustic surveillance systems. He left Bell Labs and became a Senior Research Scientist at the Courant Institute, at New York University from 1974 - 1978, where he performed research on controlled fusion and nonlinear waves, as well as ocean acoustics. It was at the Courant Institute that Fred first realized the impact that he could have upon students and thus his future took on even more meaning as the great professor and advisor began to emerge in Fred. Fred realized his potential as an educator and scientist when he left the Courant Institute and joined the faculty at the University of Miami's Rosenstiel School for Marine and Atmospheric Science in August 1978. At RSMAS he taught graduate courses in ocean acoustics, occasional undergraduate courses in physics, and supervised the research of more than 25 awardees of M.S. and Ph.D. degrees. In addition, Professor Tappert carried out a vigorous program of sponsored research in the areas of ocean acoustics, and wave propagation theory and numerical modeling. Dr. Tappert was a major participant in the ONR-sponsored initiative on "Chaos and Predictability in Long Range Ocean Acoustics Propagation." In this research he applied a recently developed 4-D (three space dimensional plus time) full-wave fully rangedependent parabolic equation (PE) ocean acoustic model to determine the limits of predictability of sound propagation and scattering. Since Dr. Tappert's most cited research was the original development of the PE numerical model, and he was also one of the originators of the concept of "ray chaos" in ocean acoustic propagation, this was a natural evolution for his research. Previously, Professor Tappert was a major participant in the ONR-sponsored "Acoustic Reverberation Special Research Project," the goal of which was to gain a scientific understanding of long-range low-frequency ocean surface and bottom reverberation by comparing numerical model predictions to measured 125

126 acoustic data, taking into account high resolution environmental data. In that research Professor Tappert developed a PE model of bistatic reverberation, the predictions of which compared favorably with measurements. In addition to his university research, Dr. Tappert was a consultant to many organizations involved in applied projects related to wave propagation theory and numerical modeling. This includes the DANTES project, sponsored by DARPA, in which he developed a novel technique call Broadband Matched Field Processing (BMFP) that localizes sources of acoustic transient signals using a back-propagation method. In October 2001, Fred was awarded the Superior Public Service Award from the Office of Naval Research. It was at this time that he was undergoing the rigors of chemotherapy in hopes that he would have more time in his fight against pancreatic cancer. This recognition brought tremendous joy to Fred. Unfortunately, he succumbed to the cancer only three months later on January 9, 2002. In November 2002, he was also posthumously awarded the Pioneers in Underwater Acoustics Award by the Acoustical Society of America. His wife, Sally, and two sons, Andrew and Peter, were present in Cancun, Mexico, to receive this award in his honor. Sally Tappert

ESTIMATION OF ANISOTROPIC PROPERTIES FROM A SURFACE SEISMIC SURVEY AND LOG DATA RUIPING LI, MILOVAN UROSEVIC Department of Exploration

Geophysics, Curtin University of Technology,

Australia.

[email protected]

Routine P-wave seismic data processing is tailored for isotropic rocks. Such assumption typically works well for small incidence angles and weak anisotropy. However, in the last decade it has become clear that seismic anisotropy is commonplace. Moreover, its magnitude often severely violates the presumptions taken for routine processing. Consequently reservoir characterization may be significantly distorted by anisotropic effects. In particular the intrinsic shale (often sealing rock) anisotropy often has first order effect on AVO gradient. Hence an assessment of the shale properties from surface seismic data may be of the primary importance for quantitative interpretation. There are several inversion approaches which require full set of geological information. In reality we expect to have at least the log and surface seismic data available for such a task. We present here a newly developed hybrid inversion method which is suitable for the recovery of anisotropic parameters of sealing rocks under such conditions. The effectiveness of this approach was successfully tested on seismic data recorded in the North West Shelf, Australia.

1

Introduction

Inversion of surface seismic data for the elastic properties of sealing rocks can impact on the accuracy of the reservoir characterisation. Since shales, which are intrinsically anisotropic, comprise often sealing rocks, an inversion has to at least incorporate recovery of the full set of anisotropic parameters for a transversely anisotropic medium. The shale anisotropy and its variation across an oil or gas field could have first order effect on Amplitude Versus Offset-and-azimuth analysis (AVOaz) [6; 1]. An example incorporating weak shale anisotropy is shown in Figure 1. Shale anisotropy in this case affects reflectivity curve on moderate to far angles. This "deviation" of the reflectivity curve could potentially impact onto our ability to accurately predict fluid type and its distribution across the field. Thus it is clear that before attempting detailed analysis for reservoir properties it is highly desirable to analyze and determine the magnitude of the seal anisotropy. Consequently an assessment of the shale properties from surface seismic data may be of the primary importance for quantitative interpretation of reservoir rocks. Thomsen [7] derived a convenient five-parameter model to describe seismic wave propagation in a transversely isotropic medium. There are many methods proposed to recover these elastic parameters, for example, the slowness surfaces method [2], the ray velocity field method from VSP surveys [4], the anisotropic moveout method from reflection events [8; 5]. Each of the above inversion method has been tested on field data sets separately provided enough information was available. However, we often have only surface seismic data and log data available for such inversion. In such case the existing methods fail to recover the elastic parameter accurately. For example the slowness method recovers the elastic parameters for an interval layer. The existence of a heterogeneous layer between successive receivers may produce errors in slowness surface determination. Deviation of the borehole, near surface inhomogeneities or topography of the surface also makes calculation of the slowness surfaces more

127

128 difficult. Because errors in slowness are in inverse proportion to the layer's thickness, errors for a thin interval layer will be larger due to the small time differences involved [3]. Using anisotropic NMO analysis, we may obtain information about overall anisotropy. We still need more constraints to determine the individual layer parameter values. For the ray velocity field method, the elastic parameters for an overall or interval layer may be estimated when the exact values for reflector depths are measured beforehand. Such method uses large number of observations, thereby statistically reducing the errors in the inverted parameters from measurement errors. However, any errors in the depth determination may produce inaccurate velocity field, which result in accumulated errors for the recovered parameters.

Incidence angles

Figure 1. Reflectivity curves for b-j\'A VTI and isotropic shale sealing an isotropic reservoir rock.

In the absence of suitable information, a new inversion approach which combines positive merits of different methods may be required. We present here a newly developed hybrid inversion which is suitable for the recovery of anisotropic parameters of sealing rocks (shales). The effectiveness of this approach was tested on seismic data recorded in the North West Shelf, Australia. 2

Recovery of elastic parameters using joint inversion method

We first discuss the inversion for the parameters for an overall layer, and then we will show how to recover the interval layer parameters. 2.1

Parameter for an overall layer to a reflector

For a reflection event, we use the anisotropic moveout velocity approximation [8] as below: 1 t2(x) = :

tl+^\

+4At^

(1)

Here, a represents the horizontal velocity. A is a newly defined parameter and its approximate value to the second order expressed in terms of Thomsen's anisotropy parameters eand £is [5]:

129 A*2-(e-8)-{ 2 /

Here,

/=

i-K

- i y + ( 3 + — )s2 -(4-y)«y. If'

(2)

with a0, /?<) are the vertical velocities for P and S-waves. To the first order

approximation, the A value is the double difference between the anisotropic parameters s and S, and A can be called as a dimensionless non-ellipticity parameter. The anisotropic velocity analysis which employs two-parameter (a and A) anisotropic semblance analysis is then implemented. For any set of parameter value of a and A, equation (1) is used to perform moveout corrections. The semblance coefficient Sc is then calculated. The values of the parameter a and A are determined for a specific reflection event when the semblance coefficients Sc achieves its maximum value Scmax. When we have the exact values for the reflector depth and the vertical velocities, the anisotropic parameters e and Scan be determined from the recovered parameters a and A using equation (2). From the log data tied with the surface seismic data, the reflector depth and the vertical velocities may be estimated. However, due to the sensitivity of the anisotropy parameters, the accuracy for the inverted parameters sand Sis inadequate. For the surface seismic survey, the two-way-travel times (TWTs) with different offsets for a specific reflector can also be picked. When the depth of this reflector is known, the velocity values at different travel angles can be inferred. Hence, the ray velocity field method could be applied for the parameter recovery. Combining the above two methods, a hybrid inversion is developed by best-fitting the TWT field with constraints of the parameter a and A values from the anisotropic semblance analysis. Figure 2 gives the program flow for the hybrid inversion. f Anise-tropic 3emblonces\ \ analysis Q, A /

J2!!giKfiSS*2LEffiSJyi*E*^Li_J

__

}_»iCoimjutetrial.obsraved velocity fields]*"

/

Reflection two way

travel time

/—

'Compare the observed velocity' fields w ith cilciilated field-,

~L__ N

,

Output parameters

e.6 Figure 2. The program flow for the hybrid inversion technique using surface seismic and log data.

2.2

Parameter for an interval layer between two reflectors

This approach can be used to obtain apparent elastic parameters for several interfaces such as top and bottom of the shale layer or the top and bottom reservoir interfaces. The parameters e and S for the interval layer between these two interfaces are then determined from the measured TWTs for different offsets and the depth values for the interfaces by the ray velocity field method. Subsequently, the

130 slowness surface for this interval layer is built from the measured TWTs for different offsets and the layer thickness. The inverted interval parameters e and 8 from the ray velocity field method are then validated by best-fitting the slowness curves. Figure 3 shows the flowchart for the interval layer inversion program.

Inversion from the velocity fields above and below the layer of interest

Output Barometers

I Inversion from the slowness curves ;

\

J

Figure 3. The flowchart for the hybrid inversion technique for the interval layer properties.

The degree of the sealing rock anisotropy has significant effect on the AVOaz reservoir signature. Hence, the inverted anisotropic parameters for the top sealing rocks should play an important part in the reservoir characterization. In the following section, we apply our hybrid inversion to real field data. 3

Field data application

The hydrocarbon field analyzed is located in Exmouth Sub-basin, offshore North-West Shelf, Western Australia. High-quality cross-dipole sonic logs showed significant shear wave splitting (1015%) over the reservoir interval. To utilize this information for reservoir rock characterization it was first necessary to estimate the anisotropy of the sealing shale and its effect on AVOaz signature. For a CMP (common mid-point), the anisotropic semblance analysis [9] is applied along the time axis (t0). For each t0, the Sc values are computed for a set of a and A values. The parameters a and A are then determined when the semblance coefficient Sc hold its maximum value. Figure 4 shows an example for the reflection event at t0=2005 ms. The corresponding parameters are determined as: a=2452.5 m/s and A=0.16. Along the seismic line, the parameter a and A are then determined using the above anisotropic semblance analysis. Figure 5 shows the parameter a and A values for different CMP for the same t0 value. The stability of the recovered parameters is demonstrated from the figures. At different t0, the Scmax values are then compared with the surface seismic section and the log data, as shown in figure 6. The top shale layer is identified as the interval layer between two strong reflection events with local maximum Scmax values. The corresponding parameters a, A for the reflection events on the top and the bottom of the shale layer are also obtained through the anisotropic semblance analysis.

131 -Visrtropffi Semblancs ^otysis:2-3W ? C P M * «

Figure 4. Anisotropic semblance analysis for CMP=4185 at t(i=2005 ms.

Horizontal velocities for different CMPs

2520 • 2510 2500 2490 2480 2470 2460 2450 2440 2430'4180

0.3 •

0.25 Parameter A

Jf E. " « o > S 0 •c 1

Parameter A for different CMPs

** *

0.2 0.15

.".

-

«

•

0.1 0.05

r 4185

4190

'•

~~ 4195

4200

4205

4210

0 -. 4180 4185

4190

4195 CMP

CMP

Figure 5. Anisotropy parameter ,4 and horizontal velocity a change along the seismic line (fixed to).

Surface seismic section

Figure 6. The surface seismic data, log data and the anisotropic semblance analysis.

Log data

4200

4205

4210

132 Even we have the analytical relation between anisotropic parameter A and s, 8 [5], it is still hard to obtain the anisotropic parameters e, 8directly from parameter^ because we lack enough information for the depth or vertical velocity. Making an assumption may cause big errors due to the sensitivity of the anisotropic parameters. The hybrid inversion which combines the ray velocity field method [4] and the anisotropic moveout method is then employed. We first pick the TWTs for different offsets for a reflection event. Then the hybrid inversion program is executed with the input of the TWTs and the recovered parameters a, A as a constraint. For the overall layer above the top of shale, we have £]=0.175, 8i=0.086. The reflector depth and the vertical velocity are also inverted. For the overall layer above the top of reservoir, we have s2=0.192, S2=0.081. Figure 7 shows the two-way-travel times from the measurements in circles (o) for the top layer. The asterisk (*) denotes the TWTs calculated using the inversion results. Both data sets match very well and the inversion results for the overall layer are quite satisfactory.

2.2

t 26 measurements calculated from the inversion results

2.7 28,

500

100D

ISOCI 2000 OUset (m)

2500

3000

3500

Figure 7. Comparison of the TWTs from the measurements and calculated from the inversion results. Very good agreement between these two sets of data indicates that the inversion is successful.

Subsequently, for the interval shale property, we apply the ray velocity field method based on a twolayer's model [4]. The anisotropic parameters obtained for the shale above the reservoir are: s=0.224, 8=0.108. Such results are also verified by the slowness surface plot in Figure 8. Notice that the thickness will affect the inversion so that for very thin shale layer at this CMP, the measured slowness surface in figure 8a is of low quality. Figure 8b shows another example with a thicker shale layer in another CMP position. The anisotropic parameters s and 8 for the overall layer to the top and the bottom of the shale are inverted first. Subsequently, the interval parameters e and 8 for the interval shale layer are then successfully recovered. The inverted anisotropic parameters can then be used in the AVOaz analysis aimed at the reservoir characterization.

133

(a) Slowness surface for a thin layer

(b) Slowness surface for a thick layer

Figure 8. The comparison of the slowness surfaces from the measurements and calculated from the inversion results.

4

Conclusions

From the log data and anisotropic semblance analysis, the reflection events at different two way travel times are analysed, as well as the horizontal velocities a and the anisotropic parameter A. From a seismic section, the two-way travel times for different offsets for a CMP location are manually picked. With the constraint of the parameter^ and horizontal velocity a values, a new hybrid inversion method is developed to recover anisotropic parameters e, S, reflector depth and the vertical velocity from the observations of two way travel times for different offsets. As the velocity field at different ray angles can be converted using the inverted reflector depth, verification procedure is carried out. The calculated values of TWT for different offsets using the recovered parameter values should coincide with the log measurements. Apparent differences between the measured and estimated values may suggest misfit of the seismic section with the log data. After obtaining the apparent average parameter for the top and the bottom sealing layer or reservoir, the interval anisotropy parameters are obtained from the velocity field data using two-layer model approach [4]. From the travel time picks, the slowness surface for the interval layer is also constructed which allows us again to recover the interval anisotropy parameters. These two estimates should match each other. The application of our new hybrid inversion methods to the field petroleum data suggests that the method is robust and should consequently result in reliable parameter estimates. 5

Acknowledgments

This is a project supported by the Curtin Reservoir Geophysics Consortium (CRGC). We thank CRGC for providing the field data. Thanks also go to Mr. Said Amiri Besheli for his help with the filed seismic data and log data.

134 6

References 1. Banik, N. C, An effective anisotropy parameter in transversely isotropic media: Geophysics, Soc. of Expl. Geophys., 52 (1987) pp. 1654-1664. 2. Hsu, K., Schoenberg, M. and Walsh, J. J., Anisotropy from polarization and moveout: 61st Ann. Internat. Mtg., Soc. of Expl. Geophys., (1991) pp. 1526-1529. 3. Kebaili, A., Le, L. H. and Schmitt, D. R., Slowness surface determination from slant stack curves, in Rathore, J. S., Ed., Seismic anisotropy: Soc. of Expl. Geophys., (1996) pp. 518-555. 4. Li R., Uren N. F., McDonald J. A. and Urosevic M., Recovery of elastic parameters for a multilayered transversely isotropic medium: J. Geophys. Eng., 1 (2004) pp. 327-335. 5. Li R. and Urosevic M., Analytical relationship between the non-elliptical parameter and anisotropic parameters from moveout analysis: (2005) being prepared for publication. 6. Ruger, A., Variation of P-wave reflectivity with offset and azimuth in anisotropic media, 66th Ann. Internat. Mtg: Soc. of Expl. Geophys., (1996) pp.1810-1813. 7. Thomsen, L., Weak elastic anisotropy. Geophysics 51 (1986) pp. 1954- 1966. 8. Zhang, F. and Uren, N, Approximate explicit ray velocity functions and travel times for p-waves in TI media: 71th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, (2001) pp. 106-109. 9. Zhang, F., Uren, N., and Urosevic, M., Anisotropic NMO corrections for long offset P-wave data from multi-layered isotropic and transversely isotropic media: 73rd Ann. Internat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, (2003) pp. 133-136.

USING GAUSSIAN BEAM MODEL IN OCEANS WITH PENETRATING SLOPE BOTTOMS Y I N G - T S O N G LIN*, CHI-FANG CHEN, Y U A N - Y I N G CHANG, WEI-SHIEN H W A N G Department

of Engineering

Science

and Ocean Engineering,

E-mail:

National

Taiwan

University

[email protected]

A numerical code using Gaussian Beam Model (NTUGBM) is developed for underwater acoustic propagation at high frequency (larger than 1 kHz) in oceans with penetrating slope bottom. Several test cases are used to benchmark NTUGBM. Cases include continental shelf and continental slope. The results of NTUGBM are compared with results using EFEPE and FOR3D (Nx2D version). Results of NTUGBM agree well with those of both codes.

1

Introduction

In order to accurately and efficiently simulate the acoustic field, some sorts of numerically methods have been developed. In this paper, a numerical model called NTURAY, which is developed using the Gaussian Beams Method, is illustrated [1]. The propagation models deduced from the Helmholtz Equation are classified in Fig. 1, which are divided into the range-dependent and the range-independent models. Our goal is to establish a high-frequency, range-dependent numerical model with the capability to accomplish the long-range ray tracing and transmission loss calculations in the laterally varying multi-layered ocean environment. According to the requirements, only the Ray method is efficient enough to handle the high-frequency and ray tracing computation. f=-

v$-

e

1 f--el)

Wave Equation

c- (Jt

Ranjrc ln
Humicmc \

!

d> + kld

NM

= 0

RdllJtC llL'pilull'Il! Hflniholl/. liquation

0 - /•'(.-Jf/li )

Ray o~Flx.y,:)i' I": Amrtifji- .•''-r.--'i;;i

!?

i I

I . • M I - ' U 1 :.«.-..!• : - H I -

PE 4-nr,0a)O(r)\ t f parabolic Equation i G. Bess^/H^iitsl Function. \ "iT-'p"

FD/FE.

•.'.. : i y j »••:

Pirate Eianenf.

Figure 1. The propagation models deduced from the Helmholtz Equation.

* Current position: Post-Doc fellow of Woods Hole Oceanographic Institute 135

136 A serious drawback is using the Ray Method in the vicinity of caustics, and Gaussian Beams Method can overcome this problem and effectively calculate the transmission loss in caustics and shadow zone as well. Cerveny et al. [2-4] first applied the Gaussian Beams Method in geophysics, and then this method is used in underwater acoustics application by Porter and Bucker [5] and Weinberg and Keenan [6]. All the applications introduced above dealt with a flat bottom, so the contribution of this paper is to apply the Gaussian Beans Method in cases of slope bottom and laterally varying layered bottom. In section 2, we will introduce the theory of Gaussian Beams Method. The verification of the NTURAY model is discussed in section 3, and section 4 will talk about the calculation of the layered bottom. Finally, section 5 will give brief discussions and conclusions. 2

Gaussian Beam Method

The linear acoustic wave equation is written as V

1 d2P 1 = l- ^ T '~c dt

'2

_

P

(1)

For a harmonic wave, the solution to the linear acoustic wave equation is

P(x,t)

= A{x)eim[t~T(s)\

(2)

where W\t — T\x))is the constant phase surface, G7 is the frequency and A(x)is amplitude. Substitute Eq. (2) into Eq. (1), we can obtain the following equation, UV2A-(O2A\VT\2

+^A)

+ i(26)VA-VT + coAVh)ieia('-T) =0

the

(3)

The real part and imaginary part are equal to zero as following,

y2A

,

.

-A\VT\

,2

A + ~T = 0

(4)

C

CO

2V^-Vr + ^V2r

= 0

(5)

Thus, if the amplitude changes slightly with the space and if the frequency is high enough, Eq. (4) becomes Eq. (6) |Vr|2=^ c

(6)

Because the directional vector of the ray is

dx

Vr

combining this directional vector with Eq. (6), the Eikonal Equation can be deduced to be Eq. (8). r

^dx}

ds c ds

(8) •

>

Thus we can obtain the geometry of acoustic rays by solving the functions in cylindrical coordinate (Nx2D calculation, eliminate the 0 coupling)

d(\dr\

j_dc_

ds ye ds_

c2 dr

d (\_dz^

]_dc_ c2 dz

ds c ds j 1 dr

(9)

„ 1 dz q and = Q , Eq. (9) becomes

Giving that

c ds dr(s)

c ds

=c(S)-m

ds dz(s) ds

am

1

dc(s)

ds

c2(s)

dr

d£(s)

1

dc(s)

2

ds

(10)

c (s)

dz

Thus we can solve the equation system simultaneously with the initial conditions to obtain the ray traces. If we rewrite the solution of the standard linear wave equation as P(x,t)=

^ ( x > ' r a [ ' - r ( i ) ] = u{x)eim<

,

(11)

then the solution in ray-centered coordinate system can be represented as

u(s,n)=A0

( \

•. \c(s,0) • e i W e x p

q(s)r

V

where

L=

CO

- — Im 2

K = c(s,0) • Re

K(s) 2c(5,0)

2

(12)

)

138 Therefore, we will obtain the sound field, u(x), by calculating two system parameters p(s) and q(s) ,

q,s = CP (13)

'"" \n=0

P,*=—rr^-q and the transmission loss is

TL = 20 log \u\

3

(14)

Verification of the NUTRAY Model

We verify the accuracy of the NTUTAY model by comparing the solutions of the NTURAY model and the analytic solution [6] and that of the EFEPE model. Case 1 is a free space case, the NTURAY solution is compared with the analytical solution and the result shows that they match very well, as shown in Fig. 2. Similar to case 1, case 2 and case 3 are the comparisons between the NTURAY solutions and the analytical solutions of the half-space case and the shallow water, hard-bottom waveguide case (See Fig. 3 and Fig. 4). These cases show that the NTURAY model is accurate and acceptable in the simple, basic condition. Case 4 ~6 are the comparisons between the NTURAY solutions and the EFEPE solutions. Case 4 is a shallow water waveguide case with penetrable bottom (as shown in Fig. 5), and both case 5 and case 6 are cases of continental shelf (slope of 1/500, see Fig. 6) and continental slope (slope of 1/20, see Fig. 7), respectively. Fig. 8-10 represent the results of case 4 ~6. — — - -

EXACT SOLUTION SO H i GAUSSIAN SEAM SOLUTION 3,5 Hz GAUSSIAN SEAM SOLUTION

Figure 2. The results of free space case. The NTURAY solution is compared with the analytical solution.

139 EXACT SOLUTION GAUSSIAN B E A U SOLUTION

Figure 3. The results of half space case. The NTURAY solution is compared with the analytical solution.

a

10 Range (km)

12

Figure 4. The results of the shallow water, hard-bottom waveguide. The NTURAY solution is compared with the analytical solution (the normal mode solution). All cases show that the NTURAY solutions match well with others except case 6. Two solutions in case 6 begin to deviate from 5 km and further away from the source. This may be due to the EFEPE model is not developed to calculate high-frequency sound field. Compare the EFEPE solution with another PE model, the FOR3D model [8], the discrepancy still exists between two PE models. Thus the NTURAY model still needs to be compared with the other well-developed, high-frequency model.

140 Air

100 m

Source

Receiver 10 Km

. *

c=1500 m/s p=1000kg/m 3

Penetrating Bottom c=1550m/s p=1200kg/m 3 a = l dB/A Figure 5. Case 4: the shallow water waveguide with a penetrable bottom.

Air Receiver Source

Penetrating Bottom Slope (1/500)

c= 1500 m/s , p=1000kg/m 3

c=1550m/s ,0=1200 kg/m 3 a=\ dB/A

Figure 6. Case 5: the case of continental shelf (slope is equal to 1/500) with penetrable bottom.

Air c=1500m/s ,0=1000 kg/m 3

Penetrating Bottom Slope (1/20)

c=1550m/s ,0=1200 kg/m 3 a =1 dB/A

Figure 7. Case 6: the case of continental slope (slope is equal to 1/20) with penetrable bottom.

141

Range
Figure 8. The results of shallow water waveguide with penetrable bottom. The NTURAY solution is compared with the EFEPE solution, and they math quite well.

4

5 Range (fan)

6

Figure 9. The results of continental shelf case (slope is equal to 1/500) with penetrable bottom. The NTURAY solution is compared with the EFEPE solution and match very well.

142

4

5 6 Range (km)

Figure 10. The results of the continental slope case (slope is equal to 1/20) with penetrable bottom. The NTURAY solution is compared with the EFEPE solution; two solutions start to diverge from about 5 km far from the source. 4

Two-layered Bottom Model

According to Frisk [7], the summarized reflection effect of the two layer bottom (as shown in Fig. 11) can be represented by the Rayleigh Reflection Coefficient R R =

o\ +

R e

n

\ + R01Rne

(15)

2ik,h cos 9

Air

Po.Co

Half Space Figure 11. The two layer bottom.

Pl,Cl

The results of the two-layered NTURAY model and the EFEPE model are presented m Fig. 12. It shows that the two-layered NTURAY model is accurate in this case.

• E F E P E solution • G B M 2 solution

1000

2000

3000

4000

5000 6000 Range (m)

7000

8000

9000

10000

Figure 12. The results of the two-layered bottom case. The NTURAY solution is compared with the EFEPE solution.

5

Conclusion

In this paper, the NTURAY model is proposed. The NTURAY is a range-dependent model which uses the Gaussian Beams Method to calculation the high-frequency, long range acoustic field. It can deal with the laterally varying multi-layered ocean environment and calculate the traces and the transmission loss. Several cases are used to verify the accuracy of the NTURAY, and the results show that the comparisons between analytic solutions or EFEPE solutions and NTURAY solutions are satisfactory. Reference 1. C. F. Chen, W. S. Hwang, L. W. Hsieh, and Y. T. Lin, "Verification and evaluation of the advanced sonar range prediction system (ASORPS)," National Taiwan University, report of Engineering Science and Ocean Engineering, Underwater Acoustic Laboratory, UAL-NTU TR 0101, 2002. 2. V. Cerveny, M. M. Popov and I. Psencik, "Computation of wave fields in inhomogeneous media - Gaussian beam approach," Geophys. J. astr. Soc. 70, pp.109-128, 1982. 3. V. Cerveny and I. Psencik, "Gaussian beams in two-dimensional elastic inhomogeneous media," Geophys. J. astr. Soc. 72, pp. 417-433, 1983. 4. V. Cerveny and I. Psencik, "Gaussian beams in 2-D laterally varying layered structures," Geophys. J. astr. Soc. 78, pp. 65-91, 1984.

5. M. B. Porter and H. P. Bucker, "Gaussian beam tracing for computing ocean acoustic fields," J. Acoust. Soc. Am. 82(4), pp. 1349-1359, 1987. 6. H. Weinberg and R. Keenan, "Gaussian ray bundles for modeling high-frequency propagation loss under shallow water conditions," J. Acoust. Soc. Am. 100(3), 1996. 7. G. V. Frisk, Ocean and Seabed Acoustics: a theory of wave propagation, PrenticeHall, NJ, 1994. 8. D. Lee and M. H. Schultz, "Numerical Ocean Acoustic Propagation in Three Dimensions, " Singapore: World Scientific, pp. 138-144, 1995.

APPLICATION NICHE GENETIC ALGORITHMS TO AVOA INVERSION IN O R T H O R H O M B I C M E D I A

MING-HUI LU, HUI-ZHU YANG Department

of Engineering Mechanics, Tsinghua University, Beijing 100084, China lmh02 @ mails, tsinghua. edu. en

A forward modeling of P-wave propagation in a bi-layer model of an isotropic layer overlying an orthorhombic

layer

is

performed.

The

observation

data

of

four

differently

oriented

common-midpoint (CMP) lines show that P-wave amplitude exhibit strong azimuthal anisotropy. A formula is deduced to obtain the azimuth angle by using the amplitude variation of four differently oriented lines. Thomsen anisotropic parameters and the ratio of SV- wave and P-wave vertical velocity can be inverted from Amplitude Versus Offset and Azimuth (AVOA) by using the Niche Genetic Algorithms (NGA).The numerical simulation shows that the inversion method has enough stabilization and precision.

1. Introduction Seismic detection of subsurface fracture plays an important role in making decisions on drilling locations and determining fluid flow during production1"2. Natural fractures in reservoirs tend to be vertical to the minimum horizontal in-situ stress, so the horizontal transverse isotropy (HTI) model is commonly used to describe a system of parallel vertical penny-shaped cracks embedded in an isotropic host rock. Some scholars have done many researches on fracture prediction by using the properties of azimuthal anisotropy of seismic wave velocities and reflection amplitudes in HTI media, and have obtained some theoretical achievements and oilfield data processing experiences3"7. However, the orthorhombic model (ORT) is believed to be more realistic than HTI model to describe the naturally-fractured reservoirs. The approximate reflection coefficient for a bi-layer model of an isotropic layer overlying an orthorhombic layer has been derived by Corrigan in 1990. In terms of the similar form of the symmetry planes of orthorhombic media and transverse isotropy media, Tsvankin9 introduced Thomsen-style anisotropic parameters10 of transverse isotropy model into P-wave kinematic study on ORT model to deduce a series of simplified velocity formulae. In the same way, Ruger' presented a modified P-wave reflection coefficient formula in ORT media, which is the basis for our study on AVOA inversion. It is known that the inversion algorithm is an important factor to affect the results of inversion. In respect that most of the optimization problems in geophysical prospecting are nonlinear, the conventional Newton's method or gradient method are prone to trapping in local minima. To overcome the problems above, Genetic Algorithm (GA) or simulated annealing algorithm etc. are often applied to nonlinear inversion. As a modified GA, the Niche Genetic Algorithm (NGA) can maintain the population various, meanwhile owns the properties of preventing premature 145

146 convergence11. Therefore, we apply the Niche Genetic Algorithm to the AVOA inversion of fracture parameters in ORT media and obtain a highly precise inversion results. 2. AVOA for ORT Media The P-wave reflection coefficient for ORT medium has been given by Ruger1 as „,

^

1 AZ

1 I Aa

2/3 2 AG

2 Z

2[a

a

G

A^+2(^-)2Ar

a

cos^ + A^'sin 2

(1)

where i denotes the incidence phase angle,

line one

ISO

ORT

Fig. 1. Orientation of four survey lines for an isotropic layer overlying an ORT medium.

In a 2-D survey, reflection amplitudes are recorded at different offsets and azimuths and strong AVOZ is observed in ORT media (shown in Fig.2). Therefore, it is possible to invert fracture orientation and density from AVOZ information. Generally speaking, the azimuth angle is unknown ahead of inversion. In this paper, we define the first survey line oriented eastward as the baseline, then put the second line at the direction of 45degree east-north, and put the third line oriented northward, finally put the fourth line at the direction of 135degree east-north (shown in Fig.l). When we get the observation

147 data of the four lines, we can deduce the azimuth angle in terms of the amplitudes variations among the four lines. From equation (1), we can deduce

1 R(i, + 90) A8^ = - +2^-)2Ay-AS^

cos 20 sin / ,

a

(2)

and

R(i,0 +45)- R(i,0 +135) =

1 AS^-AS^-2(^)2Ay

a

2 sin 2(f) sin / .

(3)

According to the equations above, we can obtain the azimuthal angle by r R(i, = arctan V R(i,0)-R(i,0+ 90)

(4)

where R(i,&), R(i,&+45), R(i,&+90), R(i,&+135) are the reflection coefficients for the first, the second, the third and the fourth line. When the azimuthal angle <£> is know, Thomsen anisotropic parameters A£ (1) , AS(2),

Ay and the ratio of SV- wave and P-wave vertical velocity (to simplified

P as g = = ) can be obtained by performing some nonlinear optimizing inversion

a according to equation (2) or equation (3). -U.04 • -0.06 • |

-0.08-

-azimuth=30 • azimuth=75 azimuth=120 • azimuth=165

fc o -0.10 •

8 c o

-0.12-

t5 jjS -0.14

"S 01

-0.16-0.18

o -0.20

10

20

30

Incidence angle(degree)

Fig. 2. Reflection coefficient for an isotropic layer overlying an isotropic layer overlying an ORT medium for azimuth of 30(solid),75(dash),120(dot),165(dash dot). Table 1 lists the model parameters.

148 3. NGA Inversion Algorithm For complex nonlinear optimization problem, the conventional GA is prone to trapping in local minima owing to its searching the extreme point in a population, which is also named premature convergence. To overcome the problem above, the population variety should be maintained during the evolutionary process. In some other words, the optimum solution and the extreme solution must be co-existed during the search process to ensure the global optimum solution be obtained by comparing some peaks of every population. The NGA is one of the modified algorithms to overcome the problem above, whose basic idea concludes": the individuals of many populations are relatively independent in propagation for extending the search space, and the population variety is maintained by controlling the fitness of the individual. The NGA in this paper combines the benefits of the distance isolation model and the panmixia model, which are applied to the optimizing inversion of multi-parameters. The basic idea of the distance isolation model is to divide a population into many smaller populations (islands), and make the individuals propagate independently in a small population, and commute individuals among the populations. The idea of the panmixia model is embodied in the improved standard fitness sharing method. The sharing function Sh is defined as Sh(du)

Jl-dy/a1

0

dv
(5)

where a is the given niche radius, dtj is the distance from the individual i and the individual/ Then the sharing fitness is

f\*i) = n*i)/llshVv)-

<6>

In every population, when one individual is close to another one, its sharing fitness will reduce largely to make it easy to be eliminated, whereas its fitness will decrease little. Therefore, the rare individuals can be maintained and propagated and the population variety is preserved very well. This algorithm can prevent premature convergence and hold the properties of global optimum and fast convergence, especially for the complex multi-mode optimization problem. The essence of inversing Thomsen parameters by NGA is an optimization problem of seeking the adequate model parameters p(ph p2,--pn) to make the theoretical predicted

149 results match with the observation data. The objective function E(p) is defined as

E(p) = j^-fJ[RP(pJ)-K(i)]2 (7) where Rp(P,i)

is the ith predicted amplitude of P-wave, R (i)is the ith observation

data, C is the amplified number, and N is the number of sample. 4. Numerical Simulation In order to test the effectivity of our inversion method, we perform a numerical simulation inversion process. First, we substitute the amplitudes calculated in equation (1) for the observation data required before inversion. The numerical simulation model is a bi-layer one of an isotropic layer overlying an ORT media. The elastic tensor of ORT media is as follows (referred to literature 12):

30.779 3.163 3.551 0 0 0 0 3.163 23.611 2.801 0 0 3.551 2.801 22.941 0 0 0 C= 0 7.903 0 0 0 0 9.386 0 0 0 0 0 0 0 10.558 0 0 0 The parameters of velocities and Thomsen parameters in ORT media are calculated according to the equations in reference [3,9], and the results are listed in Table 1 with the parameters of the isotropic layer. Next, substitute the values of these parameters in Equation (1) and obtain four sets of amplitudes when the azimuthal angle is 30°, 75°, 120°and 165° respectively as observation data of four survey lines. Table 1. Parameters of the bi-layer model. £0)

£<2)

P (kg/m3)

a(m/s)

ISO media

2600

4000

2200

0

0

0

0

ORT media

2200

3229

1895

-0.16

-0.026

0.168

30

parameter

/?(m/s)

r

value

fin

150 Second, we substitute four sets of amplitudes in Equation (5), and obtain a curve of illustrating the relationship between the azimuthal angle and the incidence angle. Fig.3 shows that the azimuthal angle inverted is approximated as 30°and it is highly accurate compared with the real value.

i n c i d e n c e angle(d e g r e e )

Fig.3. The azimuthal angle inverted.

Third, we use the NGA to invert Thomsen parameters AS, A^and g from the amplitudes difference between line one and line three. The parameters of NGA are selected as: the population number is 4, the number of individuals in every population is 50, the crossover probability is Pc = 0.8, the mutation probability is Pm- 0.01, the ratio of selection from the parent population is 70%, and the generation number is 30. The ranges of inversion parameters are -0.2 ^ A
Table 2 parameters

Comparison between inversion results and true values. ASW

AS(2)

Ay

g

real value

-0.16

-0.026

0.168

0.566

inverted value

-0.18

-0.027

0.164

0.558

absolute error

0.02

0.001

0.004

0.008

relative error

12.5%

3.8%

2.3%

1.4%

151 From table 2, we can conclude that the results of inversion accord well with the real values of model parameters. The ratio of SV- wave and P-wave vertical velocity inverted has the smallest error, which can provide a way to estimate SV- wave vertical velocity accurately because it is difficult to acquire SV- wave vertical velocity directly from the oilfield observation. The relative error of Ay parameter is only 2.3%, so it is accurate enough to predict the density of fractures. On Fig.4, we can see that the convergence of the objective function tends to be stable after the 20th generation, which shows that our inversion algorithm has fast and stable convergence. 70605040-

^» >

\ \

302010-

v„„ Generation number Fig.4 The convergence of objective function

5. Conclusions In studies of naturally-fractured reservoirs, the orthorhombic model (ORT) is more realistic than horizontal transverse isotropy (HTI) model. In respect that P-wave amplitudes are very sensitive to azimuthal seismic anisotropy, the properties of AVOA can be applied to fracture detection. In this paper, a new method using the amplitudes variation of four differently oriented common-midpoint (CMP) lines to obtain a highly accurate azimuth angle is proposed; the procedure of AVOA inversion of Thomsen anisotropic parameters by using the Niche Genetic Algorithms is described in detail. The numerical simulation shows that the direction and density of fractures inverted are highly accurate and the Niche Genetic Algorithms has enough stabilization and precision. Further study will focus on extending this inversion method to the processing of the oilfield seismic data. Acknowledgments The authors would like to thank the China National Natural Science Foundation for supporting this work under Grant 10272064.

152 References 1. Ruger, A., 1998, Variation of P-wave reflectivity with offset and azimuth in anisotropic media: Geophysics, 63(3), 935-947. 2. Sayers, CM. and Dean, S., 2001, Azimuth-dependent AVO in reservoirs containing non-orthogonal fracture sets: Geophysical Prospecting, 49(1), 100-106. 3. Ruger, A., 1997, P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry: Geophysics, 62(3), 713-722. 4. Perez, M.A., Grechka, V., Michelena, R.J., 1999, Fracture detection in a carbonate reservoir using a variety of seismic methods: Geophysics, 64(4), 1266-1276. 5. Gray, D. and Head, K., 2000, Fracture detection in Manderson Field: A 3D AVAZ case history: The Leading Edge, 19(11), 1214-1221. 6. Li, X-Y., Liu, Y-J., and Liu, E. etc., 2003, Fracture detection using land 3D seismic data from the Yellow River Delta, China: The Leading Edge, 22(7), 680-683. 7. Bakulin, A., Grechka, V., and Tsvinkin, I., 2000, Estimation of fracture parameters from reflection seismic data-part I, II, and III: Geophysics, 65(6), 1788-1830. 8. Corrigan, D., 1990, The effect of azimuthal anisotropy on the variation of reflectivity with offset: Workshop on Seismic Anisotropy: Soc. Expl. Geophys. 4IWSA, 1645. 9 Tsvankin, I., 1997, Anisotropic parameters and P-wave velocity for orthorhombic media: Geophysics, 62(4), 1292-1309 10. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51(10), 1954-1966. 11. Nie, J-X., Yang, D-H. and Yang, H-Z., 2004, Inversion of reservoir parameters based on the BISQ model in partially saturated porous medium: Chinese J. Geophys. (in Chinese), 47(6), 1101-1105. 12. Xun, H., 1994, The seismic wave forward and analysis on AVO in anisotropic media: Ph.D. thesis (in Chinese), Univ. of Petroleum (Beijing), 14.

RECONSTRUCTION OF SEISMIC IMPEDANCE FROM MARINE SEISMIC DATA B. R. Mabuza Faculty of Applied and Computer Sciences, Vaal University of Technology, Private Bag X021, Vanderbijlpark, South Africa M. Braun and S. A. Sofianos Physics Department, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa

IRCCyN/CNRS,

J. Idier 1 rue de la Noe, BP92101, 44321 Nantes cedex 3, France (Dated: October 31, 2005)

In this paper we focus our attention on the Marchenko inversion method which requires as input the reflectivity sequence of the medium with the view to reconstructing the seismic impedance from seismic reflection data. The reflectivity sequence and the relevant seismic wavelet are extracted from marine reflection data by applying the statistical estimation procedure known as Markov Chain Monte Carlo method to the problem of blind deconvolution. In order to implement the inversion method, the assumption of pure spike trains that was used previously has been replaced by amplitudes having a narrow bell-shaped form to facilitate the numerical solution of the Marchenko integral equation from which the underlying profile of the medium is obtained. Various aspects of our inversion procedure are discussed. These include questions related to the handling of experimental data and the numerical solution of the Marchenko integral equation using piecewise polynomials. PACS numbers: 58.11.Ab

I.

INTRODUCTION

Various methods for seismic exploration have been employed in the past to extract information on subsurface properties of the Earth. T h e most commonly applied is the seismic reflection method in which both the source and receiver are spread out on the surface. The success of this method is mainly due to the multi-layered structures of sedimentary basins, which reflect the seismic wavelet back to the surface. In this work we will consider marine exploration only. In order to obtain quantitative information on subsurface properties, in particular, the seismic impedance, we employ the Marchenko integral equation (MIE) [1-3]. The method is closely connected to the inverse problem [4, 5] and its historical evolution can be found, for example, in Refs. [2, 6, 7]. A complete bibliography of pioneering papers dealing with the inverse problem can also be found in Faddeev's paper [5] and in references therein. As indicated by its name, the inverse scattering problem has a counterpart known as the direct scattering problem, in which one proceeds from the potential to the scattering data. Thus the methodology used for solving the inverse problem relies strongly on the formulation of the direct problem. For most practical situations in the seismic reflection method, the E a r t h can be considered as an elastic medium. The elastic wave equation which can be transformed into a Schrodinger-type equation is therefore adequate for the direct problem. This in turn allows treatment via the Marchenko inverse scattering method. The solution of the MIE requires as input the reflectivity sequence of the medium which can be

153

154 extracted from the marine reflection data. This can be achieved by applying the Markov Chain Monte Carlo (MCMC) method [8-11] based on the Gibbs sampler to iteratively generate random samples from the joint posterior distribution of the unknowns. The MCMC method is based on Bayesian analysis and provides a general mechanism to sample the parameter vector from its posterior distribution via the Monte Carlo method. In section II the blind deconvolution approach which uses the MCMC method as an alternative form for simultaneously deconvolving the seismic wavelet and reflectivity sequence from marine reflection d a t a is discussed. In section III the inverse reflection problem and the Marchenko inversion method are briefly described. Calculations and results are given in section IV while the conclusions are summarized in section V.

II.

BLIND DECONVOLUTION A.

Deconvolution process

Before discussing the deconvolution process we present a brief description of the convolution model. This model can be described schematically as [12] measured output = output + noise = wavelet * x + noise, where x is the reflectivity sequence. Mathematically, it can be written as min(iV,t)

zt=

Yl

hkxt.k+1+nt,

t = l,...,N + M-l,

(1)

fc=i

where z is an observed seismic trace of length N + M — 1, h represents the seismic wavelet of length N, x stands for the white reflectivity sequence of the medium of length M and n is a zero-mean white noise of Gaussian type. The noise sequence is characterized by its variance a2 [12]. Eq. (1) can be written in a convolutional form z = h* x + n .

(2)

Our objective is to seperate the reflectivity sequence and seismic wavelet from each other by applying the blind deconvolution procedure. In the literature the system's unit response is called the reflectivity sequence. In our model it will also include multiple reflections (only a finite number is needed) effected by the system provided the seismic wavelet is shorter t h a n the travel time distance between the consecutive interfaces. For our numerical computations we identify the reflectivity sequence up to a scaling factor with the unit response of the medium B(£), which is discussed in section III. Deconvolution of the seismic reflection d a t a series z when the source wavelet h is known, is a well-understood problem; however, in some investigations such as ours, only the marine seismic reflection d a t a have been provided and both reflectivity and seismic wavelet should be retrieved from them. In order to estimate these quantities, we apply the blind deconvolution method.

155 We assume the seafloorto consist of several homogeneous layers that are separated by interfaces. Such an assumption makes it possible to express the reflectivity sequence in terms of a BernoulliGaussian (BG) sequence [13, 14]. Thus, the reflectivity sequence t h a t defines the generalized BG sequence can be modeled by using two random sequences expressed as [12] Xk = rkQk ,

(3)

where r = (rk) denotes a zero-mean Gaussian white sequence with variance of and q = (qk) stands for the Bernoulli sequence with the probability parameter A being equal to its mean value [15]. For the probabilities associated with this sequence we have P(9 fc = l ) = A,

(4)

P(qk = 0) = l - \ ,

(5)

t h a t is, t h e random variable qk is one with probability A and zero with probability 1 — A. The probability of the whole sequence q reads

P(qW = Y[P(qk) = \n(i-x)M-n,

(6)

k

where n is the number of ones in the sequence.

B.

Markov Chain Monte Carlo method

We are concerned with the MCMC method in a pure Bayesian approach. Upon using the Bayes' rule, we can write the probability distribution in the form [16]

Pm=^m,

(7)

where 9 stands for all parameters of the problem [17], t h a t is, 8 = (h,x,\,a2),

(8)

and h, x, A, and a2 have the same meaning as above. P(9\z) is the posterior probability of the model conditional on the observed d a t a z, P(0), and P(z) describe the prior knowledge and seismic reflection d a t a respectively while P(z\0) describes the discrepancy between the model and observation. T h e complete joint probability distribution is expressed in the form [16] P(8\z) oc P(z\0)P(0),

(9)

since P(z) is in this case a normalizing constant. The MCMC algorithm is iterative and may require a number of iterations before the first sample from the correct distribution can be provided. These initial iterations are called burn-in iterations and should not be used in the statistical analysis. Thus, the estimation of reflectivity sequence and the seismic wavelet is determined by the simulation of random variables via the MCMC algorithms based on the Gibbs sampler [9], which is regarded

156 as the best known and most popular of the MCMC algorithms [10]. It is an algorithm in which the vector Q(k+l) is obtained from 0^ by updating the vector elements one at a time. T h e prior distribution in Eq. (9) can be written as P(0) = P(x\X)P(X)P(h)P(a2),

(10)

2

where P(h) = P(
pm_P(z\«

,x,X)P(x\X)P(h)

^

The assumption of a white Gaussian noise sequence n of variance a2 leads to P(z\a2,x,X)

h *•

(2-Ka2)^N+M-1^2 exp

=

2a2

(12)

For our purpose, we need to calculate the distributions P(h\z,x,a2, A), P(x\z,h, X,a2), P(a2\z,x,h,X), and P(X\z,x, h,a2). Thus, Eq. (11) can be employed to handle the relevant re-sampling processes.

1.

Re-sampling of the amplitude of the reflectivity sequence

The reflectivity sequence x contains information about the Earth's structure. In order to statistically separate it from the seismic wavelet we use the BG white sequence model [13]: P(x)=nmP(*m), With P(xm)

= 0) + AJV(0, B2),

- (1 - X)5(xm

(13)

where M is a Gaussian distribution with specified mean and variance and where A € (0,1) is the probability that xm = 1, and both A and g2 are unknown. It can be shown [13] t h a t the posterior probabilities involving single components of the vector x remain Gaussian mixtures with a structure comparable to that of Eq. (13). More precisely, P(xm\X,h,z,x^m,) where X-

~ (1 -

(ii,...,im-i,im+i,...,XM)

Xm)S{xv

0) +

(14)

and

aV

(15)

o-2 + e2\\h\\2

(16)

fih'en),

1 +

XmU(x*m,al

l-A

Q

V2CTm

(17)

and -h*x{m) m

,

(18)

where x< > is identical to x except for Xm = 0. Using Eqs. (14)-(17), the components xm of the reflectivity sequence can be re-sampled, one at a time.

157 2.

Resampling

of the seismic wavelet

In order t o re-sample the seismic wavelet h, we deduce from the Bayes rule t h a t P(h\a2,x,z) oc P(z\a2,x,h), given P(h) = 1, where P(z\a2,x, h) is given by Eq. (12). Moreover, it is easy t o check the following identity:

_\\z~h*xf

=_^{h_ii)TR_l{h_^

(19)

where » = {XTX)-1XTz,

(20)

R = (XTX)-1a2l,

(21)

and

where X is the Toeplitz matrix of size (N + M — 1, N) such that Xh = h*x.

(22)

This allows us to conclude t h a t the posterior probability of h is a multivariate Gaussian with mean vector n and with covariance matrix R. T h e latter probability is easy t o sample according t o h = fi + Qe, where e is a normalized Gaussian white noise and QT is a square root matrix of R (that is, such t h a t R = QQT), such as the one resulting from the Cholesky decomposition.

3.

Re-sampling of the hyperparameter a

Given P(<J 2 ) = 1, it is also true that P(a2\z,x, takes the form P(z\a2,x,\)

h, A) ex P{z\a2,x,

h). As a function of a 2 , Eq. (12)

= pjSTTexp(-/3/a2)

up to a multiplicative constant, with a = (N + M — l ) / 2 — 1 and f3 = \\z — h * x\\2/2, which means that the posterior probability of a2 follows an inverse gamma distribution of parameters (a, /3). T h e latter can be easily sampled by taking the inverse of a gamma random generator output with the same parameters.

4-

Re-sampling of the hyperparameter A

T h e reflectivity sequence x gathers all the information about A contained in (z,x,h, a2), t h a t is, P(X\z,x,h,a2) = P(\\x). Following [13], let us remark that the Bernoulli sequence q can b e retrieved from x with probability one according t o q^ = 1 if Xt ^ 0, q^ = 0 otherwise. Thus, P(\\x) = P(X\q), the latter being proportional t o P(q\X) since we assumed a flat prior P(A) = 1. Finally, according to Eq. (6), we get P(A|z,x,/i,a2)cxAn(l-A)M-n,

(23)

which belongs to the family of b e t a probability densities B(a,f3) with a = n+1 and (3 = M — n + 1.

158 III.

I N V E R S E REFLECTION P R O B L E M

The one-dimensional seismic wave equation for the elastic displacement u is given by [18] p

d2u

d (

2du

W-dz{pC8-z)=°'

W

where t is the time, z is t h e space coordinate along the direction of propagation, p = p{z) is the density of the medium, and c = c(z) is the speed of t h e seismic wave. We are considering here a longitudinal displacement in t h e ^-direction. T h e Marchenko integral equation is directly applicable to t h e case of inversion with a seismic wave normally incident on a planar stratified medium, provided t h a t t h e one-dimensional seismic wave equation is converted t o the Schrodinger equation. Thus, the coordinate variable z is changed t o the travel time £ defined by

dz= c^y •

(25)

When integrating Eq. (25) we obtain

*=La£)"'

(26)

which is the travel time for a pulse to move from the origin to position z. Upon using this relation we can rewrite t h e wave equation as , ,d2u

d ( ,^du\

, s (27)

"«><*= ae("«>ae)'

where r/(£) = pc is t h e seismic impedance of the medium. Defining rf> via ip = yjiju we obtain d2ip

d2i>

T

,

where V is given by

nO = - ^ r For an ansatz of the form il>(£,t) = exp(—ikt)f(£) ~

(29)

the Schrodinger-type equation

+ (k2-V(O)f = 0,

(30)

is obtained. From t h e definition of V in Eq. (29) we can write d2^j di2

V(OVV

= 0,

(31)

which is a reduced form of Eq. (30) with k = 0. For t h e inversion procedure we apply the Marchenko integral equation given as [2] K{£,t)

+ B{£ + t)+

dt'K(Z,t')B(t

+ t')=0,

|t| < f ,

(32)

159 for which K(£,t) = 0, for \t\ > £, and it denotes a non-causal function while the function B is causal and represents a reflectivity sequence. T h e function K satisfies the wave equation given by Eq. (28). Thus, K(C,-O

= 0,

(33)

V(6 = 2

^ . at.

(34)

and

The output kernel K(£,£,') can be determined by using the collocation method and piecewise polynomials, in our case Hermite splines. The Schrodinger-type equation, Eq. (30), is equivalent to the Marchenko integral equation via Eq. (34). The seismic impedance r\ can be calculated from t h e potential V(£) in Eq. (29) or directly from the relation [19]

v(0 = v(o) ( 1 + / + *K.£'K') •

(35)

This means that, given the ??(0), the seismic impedance rj(£) for £ > 0 can be recovered from the knowledge of t h e kernel K(£,t).

IV.

RESULTS

We illustrate the use of the blind deconvolution method on recorded marine seismic reflection d a t a derived from a seismic survey in a deep water location in the North sea. We use d a t a collected with a streamer containing 240 hydrophone groups. T h e group interval is 15 m. T h e sampling rate is 4 ms and the total length is 8 s. Each trace is composed of 2001 samples as shown in Figs. 1 and 2. Depicted in Fig. 3 are the seismic reflection data. We migrated the seismic reflection d a t a using the standard moveout correction method [20] (or any other standard textbook) with the result as shown in Fig. 4. T h e main modification as compared to Chen's version [13] is t h a t we assume a shape given by s = [0.1, 0.4, 1.0, 0.4, 0.1],

(36)

instead of pure spikes in order to have a narrow bell-shaped form to facilitate the numerical solution of the Marchenko inversion. This means t h a t the observation model, Eq. (2), now becomes z = h*s*x

+ n,

(37)

where s is a known shape. If the shape s is equal to unity then of course the original equation (2) is recovered. We use in our calculations the seismic dataset from Fig. 4 which only include seismic traces from groups 90 to 240 since the normal moveout correction method did not give satisfactory results for the other groups because their offsets are too large. In addition we used all migrated traces collectively as observed seismic reflection d a t a and modified the MCMC algorithm accordingly. We also note, t h a t the observed seismic reflection d a t a are not calibrated, that is, they only provide relative amplitudes. The details of the source of the signal, t h a t is, the airgun, are also

160 not known. Therefore, the reflectivity sequence t h a t we obtain from t h e statistical procedure will be related to the unit response of the medium by a suitable factor. This calibration problem can be solved by using additional information, such as the seismic impedance j u m p at the ocean bottom if known via other means. For our purpose we model the sea floor as fluid so t h a t only the compressional seismic wave can be supported. If we assume that the sea bottom consists of silt (fine sand or soil) and that the density changes much more t h a n the velocity, then we can write [21-23] — = 1.7, P\

(38)

where pi = 1000 kg m - 3 is the density of the sea water and P2 = 1700 kg m~ is the density of silt. Similarly, if we assume t h a t the velocity of sound does not change much, then we obtain the ratio — = 1.05, c\

(39)

where c\ = 1500 m s - 1 is the velocity of the seismic compressional wave in sea water and c-2 = 1575 m s - 1 is the velocity of the seismic compressional wave in silt. Thus, the seismic impedance is expressed in the form Z » = £22* = 1 . 7 8 5 , Vi

(40)

PiCi

where rji is the seismic impedance of the sea water and r]2 is the seismic impedance of silt. Assuming this value of t h e ratio 772/V1 w e proceed to re-scale t h e amplitude of t h e estimated reflectivity sequence by a suitable factor. This factor is obtained by scaling the amplitude of the input kernel into the Marchenko equation, such t h a t the inversion procedure yields a first j u m p approximately equal to r\il"t)\ = 1.785. Shown in Fig. 5 is the seismic wavelet extracted from the migrated seismic reflection d a t a in Fig. 4. Fig. 6 depicts the statistically retrieved reflection sequence corresponding to t h e seismic wavelet in Fig. 5. The estimated seismic impedance is shown in Fig. 7. Fig. 8 shows the reflectivity sequence scaled by a suitable factor and the corresponding estimated seismic impedance is shown in Fig. 9, while in Fig. 10 we observe the invariance of peak ranking and location of peaks between the estimated seismic impedances with and without a scaling factor. Thus a lot of information can be retrieved even without knowledge of the proper scaling factor.

V.

CONCLUSIONS

We have presented the blind deconvolution of the Marine seismic reflection d a t a wherein the Bernoulli-Gaussian white sequence model for the reflectivity sequence has been used. We presented an M C M C method for simultaneously estimating seismic wavelet and reflectivity sequence under the Bayesian approach. W i t h the estimated reflectivity sequence at hand, the seismic impedance of the E a r t h medium has been reliably estimated by applying the Marchenko inverse scattering method. However, since the marine seismic reflection d a t a are not calibrated and the details of the source signal are not known, we related the acquired reflectivity sequence to the unit response of the

161 E a r t h medium by a suitable scaling factor. Since the statistically acquired reflectivity sequence and seismic wavelet appear geophysically reasonable, the blind deconvolution of reflection d a t a is judged as successful. The results we have obtained indicate t h a t we have uncovered the information about the seismic impedances t h a t are coded into the measured seismic traces. Further work is under way to handle d a t a from other seismic surveys.

ACKNOWLEDGEMENTS

I would like to place on record my appreciation to chemistry at the University of Cape Town for his research by generously giving me access to seismic the North Sea which was used to test the model in 1

1

1

Mr George Smith of the Department of Geoenlightening discussions and support for this reflection d a t a from a deep water location in this thesis. 1

1

40

1 1 marine data —

30 20 •§

io

•a. s

o

l ,1

* -10 -20 -30 -40

1

i

i

i

i

i

i

i

FIG. 1: 90th seismic trace marine data -

150 100 50 0 —^M/Vv~—4

II«4*I* ,> *' II |I>*' 11 ''*'

-50 -100

FIG. 2: 240th seismic trace.

~ -

162 0 T

3 4 -

5 E-i

6 7 0

50

100

150

Number of traces FIG. 3: The seismic data without moveout correction.

0

-1 + -2

w
-3 .- c .^ipvv.

::.-MU,

-4 -5 -6 -7 90

1—

— i —

120

150

180

210

240

number of traces FIG. 4: The seismic data with moveout correction.

[1] Z. S. Agranovich and V. A. Marchenko The Inverse Problem of Scattering Theory. New York: Gordon and Breach, 1963 [2] K. Chadan and P.C. Sabatier, in Inverse Problems in Quantum Scattering Theory, 2nd edition, Springer, Berlin, Heidelberg, N.Y (1982 and 1989). [3] D. N. Gosh Roy, in Methods of inverse problems of scattering Theory, Gordon and Beach N.Y (1963) p.2, 4; D. N. Gosh Roy, in Methods of inverse problems in Physics, CRC Press, Boston, (1991). [4] I. Kay, The inverse Scattering Problem. New York: University Research Report, EM-74, 1955. [5] L.D. Faddeev, Uspekhi Matem. Nauk 14, II, 57; English translation: J. Math. Phys. 4, 72 (1963). [6] D. L. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: SpringerVerlag, 1992.

163

0.2

FIG. 5: Seismic wavelet statistically estimated from a dataset in Fig. 4. 1

1

1

Reflectivity

relative amplitude

4 2

_

IIIIJLUJUA wTTPTnr

0

4

j

^

f

Iff

-2 -4

0

0.5

1

1.5

2

2.5

J

L

3

3.5

t

FIG. 6: Statistically estimated reflectivity sequence corresponding to a seismic wavelet in Fig. 5. [7] A. G. Raima, Multidimensional Inverse Scattering Problems, Longman Scientific and Wiley, New York, 1992. [8] C. Andrieu, N. D. Freitas, A. Doucet and M. I. Gordan, An introduction to MCMC for machine learning Mach. Learning, 50, 2003, p. 5-43. [9] C. P. Robert, The Bayesian choice, New York: Springer-Verlag, 1994. [10] A. Buland and H. Omre, Bayesian wavelet estimation from seismic and well data, Geophysics, Vol. No. 6, November-December 2003, p. 2000-2009. [11] S. Geman and D. Geman, Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images, IEEE Trans. Pattern. Anal. Mach. Intell., vol. 6 1984, p721-741. [12] J. M. Mendel, Maximum-Likelihood deconvolution: A journey into model-based signal processing, Berlin, Germany: Springer-Verlag, New York, Berlin, 1990, p. 4-15. [13] Q. Cheng, R. Chen, and T. -H. Li, Simultaneous wavelet estimation and deconvolution of reflection seismic signals, IEEE Trans. Geosci. and Remote sensing, vol. 34, no. 2, March 1996, p. 377-384 [14] J. Kormylo and J. M. Mendel, Maximum-likelihood detection and estimation of Bernoulli-Gaussian processes, IEEE Trans. Inform. Theory, vol. IT-28, p.482-488, 1982. [15] A. Papoulis, Probability, random variables and stochastic processes. McGraw-Hill, New York, 1965 and 1984. [16] J. Wang and N. Zabaras, Hierarchical Bayesian models for inverse problems in heat conduction, Inverse Problems 21, 2005, p. 183-206. [17] S. V. Vaseghi, Advanced signal processing and digital noise reduction, John Wiley and Teubner, Chichester, New York, 1996 p. 66-67, 71-72. [18] R. Burridge, in Wave motion 2, North-Holland Publishing Company, 1980, p. 305-323.

164 1

1.14 a,

1

1

1

1

1 1 1 impedance! —

1.12 1.1

S

1.08

&

1.06

Z

1-04

| "o3

1.02

1

- 11 [Vi \

0.98 0.96

0

1 0.2

«

i

• u 1i r - 1 p, 1

1 0.4

1 0.6

0.8

1

v

1 1.4

1.2

_ -

1 1.6

1.8

? FIG. 7: Estimated seismic impedance corresponding to a statistically estimated reflectivity sequence in Fie. 6.

~

Reflectivity

r:

U ' U u h * J 1>JL IWKV'fl (( f p

I- 10 -

>">' ,

-20

i

0

0.5

i

1

i

1.5

i

2

l

2.5

l

3

i

3.5

FIG. 8: Scaled reflectivity sequence in Fig. 6.

[19] W. C. Chew, in Waves and fields in inhomogeneous media, Van Nostrand, Reinhold, New York, 1990, p.49-52, 532-547. [20] J. M. Reynolds, in An introduction to Applied and Environmental Geophysics, John Wiley and Sons, Singapore, 1997, p. 218, 226, 233, 360. [21] F. B. Jensen, W. A. Kuperman, M. B. Porter, H. Schmidt, in Computational ocean acoustics, American Institute of Physics, New York, 1994, p. 41-54. [22] E. L, Hamilton, Geoacoustic modeling of the sea flour, J. Acoust. Soc. Am. Vol. 68, 1980, p. 1313-1340. [23] E. L. Hamilton, Acoustic properties of sediments, in Acoustics and Ocean Bottom, edited by A. LaraSaenz, C. Ranz-Guerra and C. Carbio-Fite (C.S.I.C, Madrid, Spain, 1987), p. 3-58.

165

FIG. 9: Estimated seismic impedance corresponding to the scaled reflectivity sequence in Fig. 8.

FIG. 10: Comparison between the estimated seismic impedances in Figs. (9) and (7) with and without a scaling factor respectively.

Journal of Computational Acoustics © IMACS

C H A R A C T E R I Z A T I O N OF A N U N D E R W A T E R A C O U S T I C S I G N A L U S I N G T H E STATISTICS OF T H E WAVELET S U B B A N D COEFFICIENTS MICHAEL I. TAROTJDAKIS Department of Mathematics, University of Crete, Institute of Applied and Computational Mathematics, FORTH, P.O.Box 1385, 711 10 Heraklion, Crete, Greece taroud@iacm. forth, gr GEORGE TZAGKARAKIS and PANAGIOTIS TSAKALIDES Department of Computer Science, University of Crete, Institute of Computer Science, FORTH, P.O.Box 1385, 711 10 Heraklion, Crete, Greece {gtzag, tsakalid} @ics. forth.gr

A novel statistical scheme for the characterization of underwater acoustic signals is tested in a shallow water environment for the classification of the bottom properties. The scheme is using the statistics of the 1-D wavelet coefficients of the transformed signal. For geoacoustic inversions based on optimization procedures, an appropriate norm is defined, based on the Kullback-Leibler divergence (KLD), expressing the difference between two statistical distributions. Thus the similarity of two environments is determined by means of an appropriate norm expressing the difference between two acoustical signals. The performance of the proposed inversion method is studied using synthetic acoustic signals generated in a shallow water environment over a fluid bottom.

1. Introduction Recently, a new method for the classification of the underwater acoustic signals has been proposed by the authors, aiming at the definition of an alternative set of "observables" to be used for geoacoustic inversions l. The study was motivated by the fact that it is not always possible to obtain a set of identifiable and measurable properties of the acoustic signal to be used in the framework of an inversion process. As the efficiency of an inversion procedure is directly related to the character of the observables, a major task on a specific physical problem is to define observables which will be more sensitive to changes of the environmental parameters and easily identified in noisy conditions. In previous works *'2 it was shown that the modelling of the statistics of the wavelet subband coefficients of the measured signal, provides an alternative way for obtaining a set of observables which is easily calculated and has the necessary sensitivity in changes of the environmental parameters, so that its use for inversions to be well justified. Here, this method is tested in shallow water environments for the recovery of the bottom parameters. The inversion is based on an optimization scheme utilizing the Kullback-Leibler divergence to measure the similarity between the observed signal and a signal calculated using a candidate set of bottom parameters.

167

168 2. T h e classification scheme In the framework of the proposed approach, an acoustic signal is classified using the statistics of the subband coefficients of its 1-D wavelet transform. In particular, the measured signal is decomposed into several scales by employing a multilevel 1-D Discrete Wavelet Transform (DWT) 3 . This transform works as follows: starting from the given signal s(t), two sets of coefficients are computed at the first level of decomposition, (i) approximation coefficients Al and (ii) detail coefficients D l . These vectors are obtained by convolving s(t) with a low-pass filter for approximation and with a high-pass filter for detail, followed by dyadic decimation. At the second level of decomposition, the vector Al of the approximation coefficients, is decomposed in two sets of coefficients using the same approach replacing s(t) by A l and producing A2 and D2. This procedure continues in the same way, namely at the k-th level of decomposition we filter the vector of the approximation coefficients computed at the (k-l)-th level. 2.1. Derivation

of the statistics

of the wavelet

subband

coefficients

The Feature Extraction (FE) step is motivated by previous works on image processing 4 ' 5 , 6 . The signal is first decomposed into several scales by employing a 1-D DWT as described above. The next step is based on the accurate modelling of the tails of the marginal distribution of the wavelet coefficients at each subband by adaptively varying the parameters of a suitable density function. The extracted features of each subband are the estimated parameters of the corresponding model. For the acoustical signals studied, the wavelet subband coefficients are modelled as symmetric alpha-Stable (SaS) random variables. The SaS distribution is best defined by its characteristic function 7 ' 8 : (Kw)=exp(?<5w-7aMa),

(!)

where a is the characteristic exponent, taking values 0 < a < 2, 5 (—oo < S < oo) is the location parameter, and 7 (7 > 0) is the dispersion of the distribution. The characteristic exponent is a shape parameter, which controls the "thickness" of the tails of the density function. The smaller the value of a, the heavier the tails of the SaS density function. The dispersion parameter determines the spread of the distribution around its location parameter, similar to the variance of the Gaussian. In general, no closed-form expressions exist for the SaS density functions. Two important special cases of SaS densities with closed-form expressions are the Gaussian (a = 2) and the Cauchy (a = 1). Unlike the Gaussian density, which has exponential tails, stable densities have tails following an algebraic rate of decay (P(X > x) ~ Cx~a, as x —> 00, where C is a constant depending on the model parameters), hence random variables following SaS distributions with small a values are highly impulsive. 2.2. Feature

Extraction

After the implementation of the 1-D wavelet transform, the marginal statistics of the coefficients at each decomposition level are modelled via a SaS distribution. Then, to extract

169 the features, we simply estimate the (a, 7) pairs at each subband. Thus, a given acoustic signal S, decomposed in L levels, is associated with the set of the L + l pairs of the estimated parameters: S ^ { ( a i , 71), (a2, 72), • • •, («L+i, 7 i + i ) } .

(2)

where (o^, 7$) are the estimated model parameters of the i-th subband. Note that we follow the convention that i = 1 corresponds to the detail subband at the first decomposition level, while i = L + 1 corresponds to the approximation subband at the L-th level. The total size of the above set equals 2(_L + 1) which means that the content of an acoustic signal can be represented by only a few parameters, in contrast with the large number of the transform coefficients. As it has already been mentioned, the FE step becomes an estimator of the model parameters. The desired estimator in our case is the maximum likelihood (ML) estimator. The estimation of the SaS model parameters is performed using the consistent ML method described by Nolan 9 , which provides estimates with the most tight confidence intervals. 2.3. Similarity

Measurement

In the proposed classification scheme, the similarity measurement between two distinct acoustic signals was carried out by employing the Kullback-Leibler divergence (KLD) 10 . As there is no closed-form expression for the KLD between two general SaS densities which are not Cauchy or Gaussian, numerical methods should be employed for the computation of the KLD between two numerically approximated SaS densities. In order to avoid the increased computational complexity of a numerical scheme, we first transform the corresponding characteristic functions into valid probability density functions and then the KLD is applied on these normalized versions of the characteristic functions. Due to the one-to-one correspondence between a SaS density and its associated characteristic function, it is expected that the KLD between normalized characteristic functions will be a good similarity measure between the acoustic signals. If 4>{UJ) is a characteristic function corresponding to a SaS distribution, then the function

fa) = ^

(3)

is a valid density function when oo 4>(u>) duo.

/

-00

For the parameterization of the SaS characteristic function given by Eq. (1) and assuming that the densities are centered at zero, that is 5 = 0, which is true in the case of wavelet subband coefficients since the average value of a wavelet is zero, the normalization factor is given by

170 By employing the KLD between a pair of normalized SaS characteristic functions, the following closed form expression is obtained 4 :

^ll^) = l n ( ^ ) - ^ + ( j r . ^ l \CXJ

«i

V71/

(5)

r ( —)

where (on, %) are the estimated parameters of the characteristic function <j>i(-) and C; is its normalizing factor. It can be shown that D is the appropriate cost function for our application as DfyiWfa) > 0 with equality if and only if (on, 71) = («2> 72)Thus, the implementation of an L-level DWT on each underwater acoustic signal results in its representation by L + 1 subbands, (D\, D2, • • •, DL,AL), where Di, A{ denote the i-th level detail and approximation subband coefficients, respectively. Assuming that the wavelet coefficients belonging to different subbands are independent, Eq. (5) yields the following expression for the overall distance between two acoustic signals Si, S2: L+l

£(S1||S2) = ^£(
(6)

fc=i

3. S t u d y of t h e sensitivity of t h e K L D for geoacoustic inversions In order to validate the proposed classification scheme, first we need to study the sensitivity of the proposed cost function, which measures the similarity with respect to changes of the environmental parameters. Our previous efforts in this respect were mainly oriented towards the sensitivity of the KLD with respect to small changes of the sound speed profile in the water column. First we observed that the set of statistical parameters of the subband coefficients of a specific signal, propagated through the water column, change significantly when the sound speed profile varies and eventually that the KLD is a suitable tool for monitoring the corresponding model parameters' variation 1 . In the last paper, we have also shown that similar conclusions can be derived for the sensitivity of the KLD when applied to signals measured after interacting with different types of ocean beds. The classification parameters of the sea-bed are typically the compressional and shear velocities, the densities of the various layers, the attenuation coefficients and the thickness of the sedimentary layers. In order to simplify the study we had chosen to simulate experiments by assuming semi-infinite fluid bottoms, thus restricting our environmental parameters to compressional velocity and bottom density only. Here, we extend this study by adding the case of an ocean bottom consisting of two layers of fluid material. The properties of the substrate are considered constant, while we allow the compressional velocity and density of the sedimentary layer to vary within prescribed limits. We simulate the propagation of an acoustic pulse of central frequency / o = 200 Hz and bandwidth A / = 50 Hz over a distance of R = 5 km and we apply the proposed classification scheme to the simulated measurements for a source and receiver pair placed at middepth of the water column. The environmental parameters for the reference environment

171 appear in Table I. We estimated the statistics of the four subband coefficients of the 3-level 1-D wavelet transform applied to each one of the synthetic signals and the KLD between a reference signal and the set of simulated signals that are obtained by changing the sediment compressional velocity from the value of c& = 1550 m/sec to the value of Cj, = 1650 m/sec in steps of 5 m/sec and the density from pi, = 1180 kg/m3 to p& = 1220 kg/m3 in steps of 1 kg/m3. We have chosen to restrict our analysis to narrow limits of the two parameters in order to focus on small variations of the geoacoustic parameters. It should also be pointed out that we have chosen to study a two layered bottom in order to assess the performance of the KLD in cases where a small part of the bottom changes.

Table 1. The shallow water environment.

Water Depth (H) Range (R) Central Frequency (/o) Bandwidth ( A / ) Source/Receiver depth Sound speed profile in the water: cw{0) cw(50) cw{200) Sediment layer : Cb

Pb

200 m 5 km 200 Hz 50 Hz 100 m 1500 m/sec 1490 m/sec 1515 m/sec 1550 m/sec 1200 kg/m3

Semi-infinite substrate: Csb Psb

1800 m/sec 1500 kg/m3

The simulated data are calculated using the Normal-Mode program MODE1 developed at FO.R.T.H. These data are provided as input to the inverse discrete Fourier transform to yield the signals in the time domain. Each of the time-domain signals is decomposed by implementing a 3-level 1-D DWT using the db2 and db4 wavelets. The reference signal is that corresponding to the reference environment. Fig. 1 displays the KLD between the reference signal and each signal corresponding to the geoacoustic parameters indicated at the axes of the diagram. In order to be consistent with previous studies, we have included two plots in the figure, the first of which corresponds to the case where both the approximation and detail subbands are considered and the second corresponding to the case where only the detail subbands are taken into account. The star in the two plots corresponds to zero KLD, that is, its coordinates are equal to the geoacoustic parameters of the reference signal. It can be seen that the inclusion of the approximation subband only affects the discrim-

172 db2 , APPROX. + DETAILS

301 1191

' 1192

1 1193

' 1194

• 1195

> 1196

• 1197

> 1196

J 1199

P,b [kg/m3] db2 , DETAILS

p^IkgAn3]

Fig. 1. KLD between the reference signal and signals corresponding to different values of the sediment compressional velocity and density, decomposed with the db2 wavelet, using (a) all wavelet subbands and (b) only the details

db4 , APPROX. + DETAILS

i

1

,

1

1

1

1

1

uJ

1191

1192

1193

1194

1195

1196

1197

1198

1199

8 0

Psb [kg/m3] db4 , DETAILS

Fig. 2. KLD between the reference signal and signals corresponding to different values of the sediment compressional velocity and density, decomposed with the db4 wavelet, using (a) all wavelet subbands and (b) only the details

ination power of the KLD between the reference signal and the signals which are already "far" from it. Fig. 2 presents the KLD between the reference signal and each signal corresponding to different geoacoustic parameters when all the signals are decomposed using the db4 wavelet. As we can see, there is an improvement in comparison with the results provided by the db2 wavelet, which is a good indication that the db4 wavelet can be used with confidence for performing the proposed classification process. Fig. 3 presents the variation of the KLD between a reference signal corresponding to a

173 0.012 1186 -*-• 1192 1199 1205 1212 1220

0.01

0.008

\ \

Q

y/

'•

y

1190

1195

\

j- ^*-

-J 0.006

0.004

0.002 1"180

, ,,«_. -'<;.., 1185

1200

1205

1210

_, 1215

1220

3

P sb [kg/m ]

Fig. 3. KLD between each one of the signals corresponding to the specific densities of the sediment layer and signals corresponding to different values of the sediment density, decomposed with the db4 wavelet.

sediment density other than that of the reference environment and the signals corresponding to sediment layers of different densities within the limits adopted in the previous study. The compressional velocity is considered constant (=1600 m/sec) for each one of these signals. The purpose of the study illustrated in Fig. 3 is to assess the performance of the classification scheme for small variations of the bottom parameter which is known to be the less accurately estimated by any inversion scheme applied to acoustical data, namely the bottom density, for a class of different reference values. Although the reference values are chosen within the prescribed limits, they can be used for the derivation of more general conclusions with respect to the performance of the proposed classification scheme. We observe that, for each one of the reference signals, the correct value of the bottom density is obtained with confidence limits that are narrow enough to ensure a reliable estimation of the parameter. This observation is again consistent with that of the preliminary studies presented in l .

4. C o n c l u s i o n s The purpose of the present paper was to provide additional evidence of the reliability and good behavior of an acoustic signal classification scheme based on a SaS modelling of the coefficients of a 1-D wavelet decomposition. The scheme was originally developed for the classification of the acoustical signals to be used for tomographic applications and has illustrated its efficiency through simulations corresponding to shallow water environments. At the present stage, it is the performance of the KLD, being used so far for signal similarity measurements, which is systematically studied. This is considered to be the first necessary step before proceeding to the inversions. Here, the sensitivity of the KLD with respect to variations of the sediment parameters is studied as an additional step towards the full validation of the proposed method as the basic tool for geoacoustic inversions. The results presented here, based on simulated signals in shallow water environments over a

174 two-layered b o t t o m , s u p p o r t our s t a t e m e n t t h a t t h e p r o p o s e d t e c h n i q u e c a n classify, w i t h high probability, a n u n d e r w a t e r signal in t h e correct e n v i r o n m e n t w h e r e it w a s recorded. It is also i m p o r t a n t t o n o t e t h a t conclusions derived in p r e v i o u s studies w i t h respect t o t h e use of specific wavelets or t h e use of specific s u b b a n d s a r e also derived here, which is a n a d d i t i o n a l i n d i c a t i o n t h a t t h e p r o p o s e d classification s c h e m e is r o b u s t in its b e h a v i o r . T h e n e x t s t e p in o u r s t u d y is t h e a p p l i c a t i o n of t h e signal classification scheme t o a c t u a l inversion p r o c e d u r e s involving m u l t i d i m e n s i o n a l search spaces a n d , if possible, t o acoustical signals from real e x p e r i m e n t s .

References 1. M.I.Taroudakis, G. Tzagkarakis and P. Tsakalides, "Classification of shallow-water acoustic signals via alpha-Stable modeling of the 1-D wavelet coefficients," to appear in J. Acoust. Soc. Am., 2006. 2. M.I.Taroudakis and G. Tzagkarakis, "Acoustic signal representation by the statistical distribution of the wavelet subband coefficients for tomographic inversion," in Proceedings of the 7th European Conference on Underwater Acoustics, ECUA 2004, Delft, The Netherlands, 5-8 July, 2004, pp. 639-644. 3. S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, 1998. 4. G. Tzagkarakis and P. Tsakalides, "A statistical approach to texture image retrieval via alphastable modeling of wavelet decompositions," 5th International Workshop on Image Analysis for Multimedia Interactive Services, Lisboa, Portugal, April 21-23, 2004. 5. G. Tzagkarakis, B. Beferull-Lozano, and P. Tsakalides, "Sub-Gaussian Rotation-Invariant Features for Steerable Wavelet-based Image Retrieval," in Proceedings of the International Asilomar Conference, 2004. 6. G. Tzagkarakis, B. Beferull-Lozano, and P. Tsakalides, "Rotation-Invariant Texture Retrieval with Gaussianized Steerable Pyramids," to appear in IEEE Trans. Image Processing, 2006. 7. C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications. New York: John Wiley and Sons, 1995. 8. J. P. Nolan, "Parameterizations and modes of stable distributions," Statistics & Probability Letters, no. 38, pp. 187-195, 1998. 9. J. P. Nolan, "Numerical calculation of stable densities and distribution functions," Commun. Statist-Stochastic Models, vol. 13, pp. 759-774, 1997. 10. S. Kullback, Information Theory and Statistics. Dover, 1997.

SOME THEORETICAL ASPECTS FOR ELASTIC WAVE MODELING IN A RECENTLY DEVELOPED SPECTRAL ELEMENT METHOD

X.M.WANG1,2 1

CSIRO Petroleum, ARRC, POBOX 1130, Technology Park, Bentley, WA 6102, E-mail: xiuming. wang @ csiro. au

Australia

G. SERIANI Instituto Nazonnale di Oceangrafia e di Geofisica Sperimentale Borga Grotta Gigante 42/c - Sgonico, 1-34016, Trieste, Italy E-mail: [email protected]

W.J. LIN 2

2

Institute of Acoustics, Chinese Academy of Sciences, 21 Beisihuan Western Road Beijing 100080, China E-mail:[email protected]

A spectral element method has been recently developed for solving elastodynamic problems. The numerical solutions are obtained by using the weak formulation of the elastodynamic equation for heterogeneous media and by the Galerkin approach applied to a partition, in small subdomains, of the original physical domain under investigation. In the present work some mathematical aspects of the method and of the associated algorithm implementation are systematically investigated. Two kinds of orthogonal basis functions, constructed with Legendre and Chebyshev polynomials, and their related Gauss-Lobbatto collocation points, used in reference element quadrature, are introduced. The related analytical integration formulas are obtained. The standard error estimations and expansion convergence are discussed. In order to improve the computation accuracy and efficiency, an element-by-element pre-conditioned conjugate gradient linear solver in the space domain and a staggered predictor/multi-corrector algorithm in the time integration are used for strong heterogeneous elastic media. As a consequence neither the global matrices, nor the effective force vector is assembled. When analytical formula are used for the element quadrature, there is even no need for forming element matrix in order to further save memory without loosing much in computational efficiency. The element-by-element algorithm uses an optimal tensor product scheme which makes spectral element methods much more efficient than finite-element methods from the point of view of both memory storage and computational time requirements. This work is divided into two parts. The second part will give the algorithm implementation, numerical accuracy and efficiency analyses, and then the modelling example comparison of the proposed spectral element method with a conventional finite-element method and a staggered pseudo-spectral method that is to be reported in the other work.

1. Introduction Despite the ever-increasing power of conventional computers, the challenge to accurately simulate elastodynamic problem still exists in computational acoustics and wave propagation simulation. An example in oil exploration occurs when it is necessary to simulate seismic wave propagation in a complex large-scale three-dimensional structure incorporating an irregular stress-release boundary, and at the same time a realistic rheology 175

176 must be taken into account for understanding the wave propagation in real porous formation containing multiphase fluids. In all aspects of seismology and ultrasonics, experimental observation points are located at or near the stress-release surface which may have a strong effect on the received signals. Modelling and understanding this effect has been one of the major issues in seismic exploration (Tessmer, Kosloff, and Behle, 1992; Hestholm and Ruud, 2000; Robertsson, 1996), in earth seismology (Komatitsch and Vilotte, 1998), and non-destructive ultrasonic detection (Kishore, Sridhar, and Iyengar, 2000). Several approaches have been proposed for simulating wave propagation in heterogeneous media with a topographic stress-release boundary. These include finite-element methods (FEM), boundary element methods (BEM), finite-difference methods (FDM), pseudo-spectral methods (PSM), and spectral element methods (SEM). Finite-element methods handle complex models with curved stress-release boundaries occurring at the air-solid boundary. These methods, based on a weak variational formulation of the wave equation, allow a natural treatment of free-boundary conditions, and are suitable for heterogeneous elastic media with complex geometries (Mu, 1984; Teng, 1988). However these methods are computationally expensive compared to the explicit finite-difference methods (Graves, 1996). This probably is the main reason why FEM has attracted a little attention from geophysical modellers in oil exploration. Also, low-order finite-element methods exhibit poor dispersion properties (Marfurt, 1984), while high-order conventional finite-element methods unfortunately generate spurious waves (Komatitsch and Vilotte, 1998). As the finite-difference is concerned, the method works efficiently only for regular boundaries and for fairly strong velocity or density contrasts of the model of interest (Madariaga, 1976; Virieux, 1986; Levander, 1988). Various schemes for the treatment of the irregular free surface and with an improved numerical accuracy have been reported by several authors (Tessmer, Kosloff, and Behle, 1992; Robertsson, 1996; Hestholm, 2000; Wang and Zhang, 2004). Although easy to be implemented, it is argued that whatever order of the finite-difference one may use in a staggered high-order finite-difference, in practice the accuracy seems to be limited into no more than a second order, for the best case with flat free surfaces. A pseudo-spectral method, that initially was introduced for the fluid dynamics (Orszag, 1980), has been proposed also for solving elastodynamic problems (Gazdag, 1981; Kosloff and Baysal, 1982; Carcione, 1995). It based either on the FFT technique or on the Chebyshev transform, and it has been one of the most important numerical techniques because of its accuracy and the minimum number of grid points needed to represent the Nyquist wavelength for a non-dispersive propagation. The spatial derivatives are computed by using a FFT procedure which in turn allows for the use of coarse and more efficient computational grids with a minimum number of nodes per wavelength that theoretically can reach the value of two. The numerical accuracy can be further increased by using a Chebyshev expansion for representing the time domain operator (Tal-ezer et al, 1986; Carcione, 1992), which leads to the rapid expansion method (REM). This method cannot directly handle media with curved stress-release surfaces. To this end a set of algebraic polynomials, such as Chebyschev polynomials in space, must be used to replace the original Fourier series (Kosloff et al., 1990; Tessmer, Kosloff, and Behle, 1992). The spatial differencing in the horizontal direction is calculated by the FFT technique; while the vertical derivatives are performed by the Chebychev transform to

177 incorporate boundary conditions into the numerical scheme. The implementation of a curved free surface is done by mapping a rectangular grid onto a curved one. Because of this, it is computationally expensive. Boundary integral equations (BIE) and boundary element methods (BEM) are other alternative schemes to simulate wave propagation for a curved stress-release surface. These methods are based on an integral equation representation of the problem relating quantities on physical boundaries. Integral formulations employ fundamental solutions and Green's theorem to represent the wave field. Also, BIE techniques with the discrete wave-number Green's function representation have been used to study wave propagation in multi-layered media having irregular interfaces (Bouchon, Campillo, and Gaffet, 1989; Durand, Gaffet, and Virieux, 1999). Although efficient, methods of this kind are most often limited to linear and homogeneous problems. Seriani et al. (1992) was the first one to introduce a so called spectral-element method for solving forward elastic wave propagation problems, and the related work has been extended greatly in recent years (Seriani and Priolo, 1994; Seriani et al., 1997, and 1998; Dauksher and Emery, 1997; Komatitsch and Vilotte 1998; Komatitsch et al, 1999). This method, originally proposed by Patera (1984) for fluid dynamics, combines a finite-element scheme with the spectral expansion on each element, to greatly reduce computation time and memory access. The basic idea of the spectral element method is that the sought solution is based on an expansion of orthogonal polynomials. The related shape functions are similar to the sine and cosine terms in a Fourier series, which leads to a high rate of convergence for the series that represent the solution. The steps of the method are, a) decompose the computational domain into many sub-domains; b) express the sought solution as a truncated expansion of a product of Chebyshev polynomials (Seriani et al., 1992; Seriani, 1997, Dauksher and Emery, 1997; Seriani, 1998) or of Legendre polynomials (Komatitsch and Vilotte, 1998; Komatitsch et al, 1999) in each sub-domain; 3) compute the solution by solving the variational formulation of the orthogonal problem via the Galerkin approach. The spectral element method is a high-order variational method for the spatial approximation of the elastic wave equation. This method can reduce the total number of elements needed to discretize the physical domain which drastically reduces the computational cost. This is particularly true by using a preconditioned conjugate gradient solver based on an element by element approach as proposed by Seriani (1997, 1998). In this paper, first the spectral element theory is briefly introduced; then, two kinds of basis functions, i.e., Legendre and Chebyshev functions, used for function expansions in reference element quadrature are studied analytically and numerically; followed by derivation of analytical solutions of the element quadrature, standard error estimations and function expansion convergence are discussed; finally, the element by element associated with pre-conditioned conjugate gradient and staggered predictor/multi-corrector procedure for time domain update are summarized. In the Appendix, the detailed derivation of analytical quadrature solutions based on the Jacobian expansion is given. 2. Basic Principles of the Spectral Element Methods In a general sense, the spectral element method, stemming from finite-element and pseudo-spectral methods, adapts the advantages as well as the disadvantages of both the two

178 methods, and sets up a much fast and efficient numerical modeling method. In this section, we briefly introduce the basic concept of the spectral element method for elastodynamic problems starting from the initial statement of the problem up to the global matrix equations. In order to understand the method we follow the finite-element point of view, since the method has a similar approach. However, we will see how some important concepts, as the global matrices or the global force vector, or specific treatments, as the one-dimensional band storage, are no longer used in the spectral elements. On the contrary new concepts are introduced like the spectral concept, the tensor product, the element by element procedure, and so on. 2.1. Statement of the problems Suppose that we consider wave propagation on a nd -dimensional domain denoted by Q."d for a period of time (0,T) where nd = 1,2,3 . We define the domain Q."d x(0,T) d

that

d

belongs to a real space W , i. e., for a given time dependent physical variable on Q." , we say the variable denoted by u(x, y,z,t):£ln'' x(0,T) —>R . For the physical variables denoted by u(x,y,z,t) , constrained to T , the boundary of Q."" , and (0,7) , we sayu(x,y,z,t):rx(0,T)—>R , where T i s either the part of £2"" or the whole boundary 1 of CI"" . In order to avoid intensive mathematical difficulties, we assume that the functions of interest belong to the / / ' space, i.e., a classical Sobolev space that belongs to a Hilbert space ]} , or a space of functions, with square Integrabel generalized first derivatives. This assumption requires that, each physical variable, is a kind of function such that it is piece-wise continuous, and the integration of the square of its first derivative on the defined space domain is finite, which is generally true for our linear elastodynamic problem. With this in mind, the boundary value problem for elastic wave equation can be expressed as the followings Given ft, g.,h t , uoi, uoj, find w. : Q,"" X (0,T) —» M such that p(x,y,z)uii„(x,y,z,t) ui(x,y,z,t)

= (TUj(.x,y,z,t) + fi(x,y,z,t)

= gi(x,y,z,t)

on £>"" x(0,T),

on Tgj x(0,T),


on Thix{QJ),

(1.1)

d

a,.(x, y, z,0) = um(x, y, z)

x,y,ze Cl" ,

uit (x, y, z, 0) = um (x, y, z)

x, y, z e SI"'.

In the above equation, the initial conditions involve specifications of both displacement and velocity components. Thus K 0 ,(jc,3>,z,0):Jr'->R, «„,,,(*, y,z,0): are given functions for each i,\
(1.2a) ft*-»R,

The remaining prescribed variables are

(1.2b)

179 f(x,y,z,t):^x(0,T)^R, gi(x,y,z,t):rg,x(0,T)-*R,

(1.3a) (1.3b)

h, (x, y, z, t): rM x (0, T) -> K. (1.3c) They denote the given body force, displacement and surface stress components, respectively. The density, p: Q."" —» M, assumed to be positive, needs also to be specified in the present case. Note that the boundary T for the domain 0."" may be decomposed into two basic sets in accordance with the given boundary conditions, i.e.,r = r f t [ J r g .The stresses are given on Th ,while the displacements are given on r However, r ^ l j r ^ = 0 , which is an empty set .The constitutive equation, cr. = cjjklu(kl) , links stress and displacement

through

with

+ M

" ( U ) =("*./

*,/)

/ 2

geometric

equations

or strain-displacement

equations,

•

2.2. Weak formulation of elastodynamic equation In spectral element methods, the strong formulation in (1.1) is converted into weak formulation, and then it is converted into Galerkin weak formulation in which the weighted function and trail solutions are expressed by finite terms of orthogonal basis functions. These functions can be constructed with orthogonal either Chebyshev or Legendre polynomials. The corresponding weak formulation is Given f, g, h, u oi , iioi, find u e LI,,f G (0,T), such that for all test function w e PL (w,/ni) + a(w,u) = (w,f) + (w,h) r , (w,/7u|,=0) = (w,pu 0 ),

(2.1)

(w,pu| (=0 ) = (w,pu 0 ). In the above equations, the inner products are defined as (w,u) - f w^u.dQ.,

(2.2a)

(w,h)r=J|^,Vr,

(2.2b)

a(w u) =

' L wu.J)cmuiu)dQ' <2-2c) where w = wiei , u = ujei and h = hiel , and e. is the unit vector in x-, y-, and z-axis, respectively, and \ = xiel , and U and Yl denote for two function spaces, i.e.,U,={ueW 1 (ii""):a"" ->K" J ,u| rs =g} , and n = {w6ff'(fl"'):fl"' -»R"";w| r =0} . Some times we denote (r, ,?)„„,, = \ rsdQ. where r and s are two scalar functions with the same properties as the uv.

180 Note that the choice of the test function in such a way, as we may see, will bring a great convenience in the following mathematical and numerical treatments. It results in natural satisfactions of traction free boundary conditions if the displacements are prescribed on the whole boundary. If we take the whole domain as one element, the semi-discrete Galerkin formulation of the elastodynamic problem is (Hughes, 1987) given f,g,h,u 0 , and ii 0 , find uh = \h + g \ u * ( 0 e UhJe [0,7], such that for all w* e I T ,

(w\/7vVa(w%v*)Kw\f)+(w\h)r-(w,pg*)-a(w\g*), (w\/7v*(O)) = (w\ / 9u o )-(w\ / 0g' i (O)), (w\/>V*(0)) = ( W \/?U o )-(w\/?g''(0)). (2.3) In the above equations, w*, v* and g* can be expanded by finite numbers of basis functions denoted by NA(x) w.'•=

I NA(x)c,A(t), ten-v.,

v,"= £

(2.4a)

NA(x)diA(t),

(2.4b)

S,*=5>,(x)gM(0,

(2.4c)

where A belongs to the node point in the element Q"* . Note that 77 E {1,2,..., N} is the set of node numbers and 77g is the set of nodes at whichui = g.. Also, wh = whiei, vh - v'i'ei, and gh = g'.e!. In order to solve the weak formulations as shown in Equation (2.1), the space domain is discretized into a finite number of non overlaying elements Nd , the eth element is denoted by£2 e , and Q."J ={J&e whereee [l,2,...,Nel],

and Ne[ is the total

element number. In each element, there are a number of nodes to be used in order to interpolate the related function values. Equation (2.3) for all of the elements is converted into a matrix equation, that is Given F : [0.7] -^ K, find the d : [ 0 , T ] ^ R such that Md + Kd = F,
(2-6a)

181

(2.6b) and the associated stiffness matrixes K and k e K=A(kc),

(2.6c)

ke=[F

<

(2.6d)

2

*; = k"..* wv" ' while the effective force vector can be written as F(0 = F „ A + A ( n O ) ,

/; = LNj,da+ I

(2.6e)

NMF-^KX

+

where j , y ' e { l , 2 , . . . , « d } , /? = n d ( a - l ) + i , q = nd{b-\) + j ,a,be

<0> {l,2,...,nen}

,

nee = nm • nd, and F„ are the body force or gravity at nodal point nb. In Equation (2.6), nen

A standards for the matrix assembling function. It takes care of the contributions of each element matrix as well as of effective force to the correspondent global ones. In Equation (2.6d), when the isotropic media is taken into account, it is simplified into *« = M(4 i NaJtNhJtda + INaJNbJd£l)

+ AlNaJNhJdQ,

(2.7)

where p and q are determined by the given formulas before. Also, the four indices can be used for the stiffness matrix components, i.e., ke = k'ajb. If g and h are zeros, Equation (2.6f) can be simplified into

f'P = lNaf,da.

(2.8)

Also, the two indices can be used for the element fore vector, i.e., / J = f'p . As shown in the above equations, all of the computation are related to the element quadrature of f NamNbndQ.,

f NaNbdQ., and f NafjdQ,, where a and b are local element nodal

numbers, and i is the space degree of freedom in local element. The last term can also be expressed using the second term if the function f. is expanded using the basis functions of Nm . So there are only two quadrature terms in element matrix calculations, i.e., L NaNbd£l and f Na mNb nd£l. Although Equation (2.7) is restricted to isotropic media at the moment, the idea in calculating the element matrix with the help of the above two quadrature terms is also true for any other linear elastodynamic problems, including anisotropic and porous elastic media.

182 As shown, the above treatment follows the finite-element approach. However, unlike the finite element method, the spectral element does not need to form the global system matrices and computations can be done in an element by element (EBE) way. Moreover, since the spectral element method uses much more grid points per element (such as 16, 36, and 64) than the standard finite element method, memory and computational time requirements can be greatly reduced by using a pre-conditioned conjugate gradient solver based on an EBE approach. Furthermore, a "good" choice of basis functions based on some orthogonal polynomials can increase the overall efficiency. 3. Basis Functions and Element Quadrature In this section, we will focus our attention on the basis (shape) functions construction by using orthogonal Legendre and Chebshev polynomials in a reference (master) element. In fact, a detailed formulation of the elastic wave modeling can be easily given if we use a spectral expansion of the wave field on a master element and, furthermore, it can be directly exploited for the numerical algorithm implementation. One of the key points in spectral element is the choice of the shape functions and the related collocation points needed for interpolating physical quantities. Moreover, a great care must be used in that choice since different function expansions on the reference element result in different computational efficiency. First we need to define an appropriate transformation for mapping the master element space to each physical element space, isoparametric transformations (Hughes, 1987) are the most used. As a consequence, the transform for computing the derivatives of the basis functions can be written as (Hughes, 1987)

^=K. f ^)(y J 7 -yJ/J^rj), tf«.,=K, tfa,#)ta

-XJ/J&TJ),

(3.1a) 0.2b)

where the (x, y) and (£,7) are the coordinates of a nodal point in physical and reference elements, respectively, and J (£, Tj) is the Jacobian determinant, written in the form of, f

J(£»7) = det

*4 V

?4 V-

and /#,= X X , ? / ? Q , where

fie[x,y),

(3.2c)

y*,

££{£77},and

]3ae{xa,ya}.

In the particular case of 2-dimensional problems, the physical space is decomposed into quadrilateral curved elements, denoted by Q.e, and they will be mapped to a rectangular reference element defined as Of = [-1.+1]x[-l,+l]. The element matrix m'pq and kepq can be calculated numerically using the Gauss-Lobatto quadrature rule. However, in order to improve the overall accuracy, analytical solution would be rather used, if available, for this kind of computation. From the above equations, it is known that the analytical solutions are possible if proper basis functions are used. To derive analytical formulas for element mass

183 and stiffness matrix, one may seek the analytical solutions of (Na, Nb)Q, and (NaJ, Nby)a,, and J5, y& (x, y). Since there is a one to one map between physical and reference elements, we may focus our attention on reference element. In the next section, we will treat the map between a general physical quadrilateral element and the reference element. If the 2-D basis functions are constructed based on some 1-D basis function formed with orthogonal polynomials with the weight of 1, they can be written as Na&V) = M?>MV)> (3-3) It is easy to show that (Na,Nb)n,=(^t)]^^

(3-4)

where the inner product is defined as

In this case, the element mass matrix m =[mepq] is diagonal, which can be see from equation (2.6b), where <pa is the 1-D basis functions. 3.1. Basis functions constructed using Legendre polynomials When Legendre polynomials Pn (£) are chosen for constructing the basis functions with n = 0, 1,2, ... N, the element mass matrix is always diagonal, which reduces computation time in forming element mass matrix. For stiffness matrix, by checking(N a ^,N br ) Qr , we have Waj'Ni.Jv = MatfWMh-m&MfiXr):,; • Let P = ^, and y = t], the above equation will become

Waj'^a;

= ((hAZWMAifr&^K', •

(3-6a)

< 3 - 6b )

When we use $,(£) = c a P a (£) as a basis function, it is easy to show that (^./^A; = ^aP^Ah-ic.P^c^ .

(3.6c)

Since PB4Pb={a + mPaPb-PB+xP„)l{\-e), We may convert it into a sum of truncated series Pa A =(« + l)Z[
(3.7a)

"+1P„Pb]-

Some of integrations are derived, such as (Pa(^),Pb^))i =28abl{2a + V).

(€Pa&,Ph(&).

2(a + l) (2a + l)(2a + 3) 2a (2a-l)(2a + l)

(3-7b) (3.8a)

for b = a + l, (3.8b) for b = a-l,

184 2(a + l)(a + 2) (2a + l)(2a + 3)(2a + 5)

{fpa(apb(t)) =

for b = a + 2,

2(2a2+2a-l) for b = a, (2a-l)(2a + l)(2a + 3) 2a(a-l) for b = a-2. (2a-3)(2a-l)(2a + l)

(3.8c)

For the term associated with^ m ,m > 3, there are no general closed-form solutions and one may derive its analytical formula in accordance with each detailed term. In fact, n can not be too large, and usually it is taken to be not larger than 8 that corresponds to 64 nodal points in each element. It is applicable to use closed form solutions although complicated, so that high accuracy can be retained before time step iteration. If an element quadrature is chosen for numerical calculations, the basis function is taken to be

m=

=i

q-frt(fl,

(3.9a)

where ^ is zeros of P'N(%) for i = 1,2,3,..., JV — 1, and with special point of 4 = - 1 , and 4N = 1, which coincide with Legendre-Gauss-Lobatto collocation points. In this case, the element mass matrix can be calculated using Gauss-Lobatto quadrature rules. However, the accuracy becomes lower than the analytical solutions. Equation (3.5) holds in a numerical sense, and the element matrix is only calculated for the diagonal components, i.e., m' . Note that, if £ = <£ in Equation (3.9a), the De I'Hopital's rule is used for evaluating the value of the basis function that must be 1. Unlike the element mass matrix, calculations for the element stiffness matrix are made only with its symmetric properties. In this case, the derivative of $(£) is used. From equation (3.9a), we can not directly calculate the derivative of the basis function because there are singular points at £ = £ , apparently. However, indeed, only the values of the basis function and its first derivative at these points are required. For
=!

l i m

(i-f)p;^

=

2 ^ ^ . ) -

(

i - ^ ) ^ . ) ^

where the Legendre equation has been used. So the basis function, as expected, possesses the Lagrange interpolation properties. In order to obtain the first derivative of the basis function from equation (3.9a), we first suppose that £ ^ ^ , so that

*(£)(£-£)= ~ (1 ~^ ) P w ( ^ . MAS s.) N(N + l)pN^y also

(3-9c)

185 , m = 2£P^)-(l-f)P^£)-^)N(N + l)PN(^) } ™ NiN + DP^it-Z) Again, by using the Legendre equation, Equation (3.9d) can be simplified into , m = N(N + 1)PN(£)-^(£)N(N + 1)PN(£,)

(3 gd)

Therefore, for any collocation point £ = £} except for £} = £ , and j*i, <*'(£.) =

P (£ ~) H^il

.

(3.9f)

When j = i, and i, je {1,2,..., N - 1 } , it is easy to show that the following formula holds from Equation (3.9e) #(6) = 0, (3.9g) where, again, the De l'Hopital's rule has been used. For i = 0 , or N, we have ^(£0) = (N + l)N/4, (3.9h) &(&) = -#(&).

(3.9i)

3.2. Basis functions constructed using Chebyshev polynomials When the basis functions are constructed with Chebyshev polynomial, it can be written as &(£) = c?X(£). (3.10a) where c"J is a normalized coefficient, Ta is the Chebyshev polynomial and is determined by cl

=

1 / -yJ(Ta, Ta )^K in which the inner product is defined with a weight of 1 / ^/l - £ 2 .

According to the orthogonality of the Chebyshev polynomials, \\ln,for a=0, T (3 10b) i \2lJt, for a>\. Following the previous step, the element mass matrix components can be written as

<=P(fc.&) f •(&.&)„.

(3.H) 2

Since the Chebyshev polynomials are orthogonal with a weight o f l / ^ / l - ^ , and the related element matrix is not diagonal. However, the closed form solutions can be easily obtained from Equation (3.11). The related term can be written as f0 for a + b is odd, T T (hA)4=clcTb(Ta,Tb)s=\_T_ (3.12) \ca chQ for a + b is even, where Q = (Ta,Tb)?, which can be expressed in a closed form, i.e., Q = J^cos(a0)cos(b0)sin0d0

,

(3.13a)

where another form of the first kind of Chebyshev polynomials are used with £ = cos(0), that is, Ta(£) = cos(a0). (3.13b)

186 So Equation (3.13a) can be obtained analytically, i.e., 1 1 (3.13c) Q= \-(a + bf l-(a-b)2 Also, the major terms that contribute to the element stiffness matrix in Equations (2.6d), (2.7), and in Equation (3.6a) can be expressed analytically. There are four kinds of integrations associated with the calculations of k' in Equation (3.6a), and the treatment for their closed form solutions is similar to m

, for example,

= {cTam),cim))£

(Na4,N^)nt

•

{cXmc^irj))

(3.14)

In this equation, /_ r , _ r ? ^ = [ 0 \ca a>cb „){ j - r - r ^ , ^

for a + b is even, fora

+

(3.15)

bisodd

By using the properties of the Chebyshev Polynomials, we have K

1

m) = 2ajj-

(3.16)

*,(#),

n=0 *~a-\-ln

where K'a is the integer part of {a -1) / 2 , it follows that

(T:,Tb)(=2afj-^—[jMn^)Tb^)d^ n=0

(3.17a)

C

a-l-2n

Taking into account Equation (3.12) and (3.13), we have K' 1

fjXdt =2at— 1=0

C

a-\-2n

\-{a-\-2n

+ bf

\-{a-\-2n-bf

(3.17b)

where [2 if a-l-2n = 0, [l if a-\-2n>\.

(3.17c)

If we interchange a withb , the inner product of (Tb\Ta)^ is also obtained with the similar result. The followings are for the derivation of {T'a,Tb)^ . Using Equation (3.16),

T'M)T^) = AabYZ

0 n=0

1 C

a-l-2mCb

-Ta+lm{Z)TMn{Z)

•

(3.18a)

Thus, for a + b = odd,

0

(T:X\=

K K

i

**IZ—ra=0 n=0

-(Ta-i-2m>Tb+2n)i;fara + b = even.

(3.18b)

C

a-\-2mCb-\-2n

Again, by using Equation (3.12) and (3.13), we have, 1 (*a-l~2m>*b-\-2n)t;

~

}-{a + b-2m-2-2nf

\-(a-b-2m

+ 2nf

(3.19)

Until now, the four kinds of integrations are derived completely. Therefore, the element mass matrix and stiffness matrix can be calculated analytically.

187 When Gauss-Lobatto quadrature role is used for calculating element mass and stiffness matrix, the basis function is taken to be (Canuto, Hussaini, and Zang,1987)

with at =1 (r = 1 , 2 , . . . , N - l ) , a 0 - a N = 2 . Note that £ is the Gauss-Lobatto collocation points, i.e., £.=cos(i>r/AO. (3.20b) Now we show that the basis function set given by Equation (3.20) is orthogonal with respect to the grid collocation points. First, we show that $(£,.) = Sy. For i*j,

$($) = () since(l-^)7 , ;(^) = 0.

Fori = j , using the De l'Hopital's rule, we have

^(^)=^^(1"f-^(^=^[(1"ff)r;(j:')"2^^)]-

(3 2 C)

-°

By using the properties of'TN,T'N , and T"N at the special points, i.e., Y w (£) = (-1)'', <7*;(£) = 0, +1

/or i = 0,1,2,..., AT, /or* = 1,2,..., AM, 2

(3.21)

2

r ; ^ , ) = (-1)' JV / ( l - £ ) , /or i = 1,2,..., AM. It is easy to show that $(£.) = 1, and $ (£,•) = oV . Now let us derive the derivative of $(£•). When ;' =£ y , it is easy to show that a (-1)'+J #(£)= ,. >,•

0.22a)

When i = y, we have to follow the approach used previously for Legendre polynomials. After some derivations, we have ff(5)=

~jL

/ o r i = ./e(l,2,...,AM).

^ 0 '(4) = (2Af 2 +l)/6 fari = j = 0, 2

^ ( ^ ) = -(2Af + l)/6 fori = j = N.

(3.22b) (3.22c) (3.22d)

Based on the above knowledge, one may show that W - t f . W ^ . (3-23) with respect to the quadrature points. Note that because the orthogonal formula in Equation (3.23) holds only with a weight ofw = l / ^ / l - £ 2 , again, numerically the element mass matrix is not diagonal, i.e., iNm,Nn)a^Smn.

(3.24)

188 In principle it seems that Chebyshev polynomial expansion does not have an evident advantage over Lengendre one. However, Chebyshev expansion has closed form solutions for element quadrature, as shown in Equations (3.15)-(3.19) and as a consequence it is easy to calculate the element matrix terms. These feature, then, can be exploited in order to avoid to form the element matrices so that memory access can be drastically reduced. Also, from the analytical solutions we see that, not all of the matrix terms are calculated, and some of them are zeros. On the other hand, when implicit algorithms are used, that allows for using larger step in time integration, the diagonal features will loss its advantage. 4.

Element by Element Iterative Solver

In the spectral element method, the element matrices are big and dense. When the global system matrices are assembled they are sparse with a large bandwidth and they require huge computer memory storages. A remarkable improvement can be obtained by using element by element iterative solvers (Seriani, 1997; 1998) since this approach greatly increases the computation efficiency. The EBE iterative solver is part of the process used for computing the time evolution of the wave field. In the following section, we present the time integration scheme first with the staggered predictor/multi-corrector algorithm, and then with the preconditioned conjugate gradient algorithm based on EBE approach. 4.1.

Staggered predictor/multi-corrector algorithms

In order to find the solution we need to solve in time the matrix equation (2.5). Using the staggered predictor/multi-corrector algorithms (Hughes, 1987) we have n=0 (The time iteration index) (1) d = d„ + (1 + a)At\n + (1 + a)(\ - 2/3)At\ (1)

= v„ + (1 + a){\ - Y)AtAn

(1)

=0

v a

Loop i slove the matrix equation: M*Aa(l) = AF^j jpi> =d-(o +(i + ar)yS\/2Aa<') v ( / + 1 ) =v ( 0 +(l + a)?AfAa(/) a ( ' + , ) =a ( 0 +Aa ( ' ) a - a(,+1) End of Loop i v„ + l =(v„ + 1 -v„)/(l + tf) + v„ d„ + 1 =(d„ + 1 -d„)/(l + oO + d„ if time iteration is not over

12

189

n = n+l go for time step iteration or else stop In the above paragraph, a, v, and d standard for the particle-acceleration, velocity and displacement, respectively, and M* are global effect mass matrix and AFn(+, is the residual force vector. 4.2. Element by element iterative solver The major step in the time iteration is to solve the matrix equation M*Aac,) = AF", written in the form of SX = B , where the element by element (EBE) procedure and the preconditioned conjugate gradient (PCG) are used. The EBE-PCG algorithm (Seriani, 1997) is given by Initialize: m = 0,R0=B-SX0,Z0=P0=Q"'R0 Iterate unitl convergence (P„„SP,„)

X,„+1=X,„+«rPra R„+=Rm-«sP,„

tfl^H-il^lNI st°p Z,„+, = Q R m + I

P„,=Z

1

(R„.zj

'"

m+\ In the above algorithm, the key points for a fast convergence to the sought solution are, first, the choice of the preconditioning matrix Q such that Q~'S = I , and where Zm - Q~'Rm should be easily evaluated; second, the choice of the initial guess of X0 that must be as close as possible to the real solution X ; and third, the calculations of SX0 or using the EBE approach, which is one of the most important step in spectral element method in order to reduce the computational time and the memory access. In staggered predict/multi-corrector time iteration, the initial value of X0 can be taken as the particle acceleration of the previous time iteration. According to our calculation, usually 2-3 time iterations are used to converge if £ = 1 .e'4; while in the work by Seriani (1997), three options for the initial values are given.

190 As Q is concerned, following the work by Seriani (1997), we may use Q = diag(S). Since only the diagonal values are used, the matrix Q can be stored as a vector, rather than a matrix. This will greatly reduce memory usage and Q 'Rm can be calculated with a fast term to term vector product. Finally, let us look at the implementation to calculate SPm . This is the most computationally-intensive part of the spectral element method. From element point of view, the vector of SPm can be computed at the level of individual elements (Seriani, 1997), i.e.,

where s"e,pj;, and \'m are the element contributions to the matrix S and vector Pm respectively, and v"m = ¥ e p^. In the above formula the multiplication of the global matrix with the global vector is converted to the simpler and faster matrix-vector-multiply at element level. The detailed algorithm consist in: i) computation of the element matrices; ii) derivation of the element vectors using the global and local links (connectivity map), i.e., the vectors pjj, are gathered from the global vector p m ; iii) evaluation of the matrix-vector multiplications at element level, so that yem = ~sepem is obtained; iv) distribution of \'m , the computed element vectors, into the global vector, i.e., the data should be correctly scattered to form the global vector \m . With this approach the global matrix S is never needed and formed.. To further increases the computation efficiency, a novel EBE treatment was proposed by Seriani ( 1997) where all of the calculations are converted to matrix by matrix multiplications on local elements, and there is no need for storing the element matrices, neither is there the storage for se , which can greatly reduce memory sizes. Although element matrices are calculated in each iteration time, the matrix by matrix multiplications improve computation efficiency. This is especially true by suing analytical solutions for element quadrature. 5.

Error Analysis and Convergence Comparisons

The error estimates in spectral elements can be introduced using standard finite-elements with a little modification. In the approximate finite-element problems, we assume that, a. The discretized spaces of U* and FT belong to U and IT, respectively; b. The inner products defined as (•,•) and a ( v ) are symmetric and bilinear; c.

(•,•) and|-| are with equivalent norms on the defined space I I , i.e., we may write that ^Ja(w, w) = |w|| where |w| is the rth Sobolev norm of w .

If we define r, = u'' - u , the error in the finite element approximation, then we havea(w\r e ) = 0 V w h e ] l * , a n d a(re,re) < a(V -u,U* - u ) VUh e U*. This means in

191

Galerkin approach, the obtained solution of u* is the best approximation foru. Also, this method always underestimates strain energy, i.e., a(u*,u*) < a(u,u). The previous discussion is based on no error time integration. If the time domain error is concerned, we will sue the followings to analyze the error estimates. Let£(u,u) = [(u,yCu) + a(u,u)]/2, i.e., the total energy, then

V ^ X ) * c{/."[|u(o)L+l|u«lL1]+n||u(o)|L+l+||u(o|L + J 0 'IML w

(5.1)

wherev-k + l-m , and fi-min{fc +1,2(k +1 - m ) } . In this case, ^JE(re,r,) that defines a norm on T\xL? is equivalent to the norm on// m xL 2 . Note that the integral in (5.1) is 0(t). Therefore, the rate of convergence is v for times no smaller than 0(h~m)3 (Strang and Fix, 1973) The above analysis assumes that we rigorously adhere to the Galerkin recipe, which means all of the integrals are calculated exactly. However, if numerical integration is used, we will break up the rule. That is why the analytical solution is important in improving the accuracy estimate or to give a predicted standard error. The error in space domain can be explained using spectral approximations since in each element the function is expressed into spectral expansions. For example, in one-dimensional problem, for analytical functions, exponential (or spectral) decay of the coefficients can be obtained for trial functions that are eigen-functions of singular Sturm-Liouville problems defined onQ.' = [ - l , + l ] . In general, polynomial solutions of singular Sturm-Liouvelle problems are Jacobi polynomials like Chebyshev and Legendre polynomials. The following theorem provides us the error of a truncated Chebyshev expansion (Mason and Handscomb, 2003), i.e., If a function f(x) e Hm+' (Q.1) , where H'n+I is a Sobolev space of degree m+1, then \\f(x)-ST„f(x)\\

= 0(n-m) ,

for all xe Q.', where Sj f(x) = VCT.(JC) , and c =

(5.2)

< f,T'•> ,

———.

The Proof of this theorem can be seen in Mason and Handscomb (2003). The theorem can also be satisfied with the other Jacobi Polynomials, such as Legendre Polynomials. More detailed work can be seen through Canuto et al. (1988). On the other hand, people argue that first-kind Chebyshev expansion is superior to expansions in other orthogonal polynomials. That means, the first kind Chebyshev expansion converges faster than any other orthogonal polynomial expansions. Light (1979) proved that, the first-kind Chebyshev expansion of a function f(x) converges faster than other ultraspherical expansions in the conventional sense, i.e.,

\\f(x)-STnf(x)\\

f(x)-^pr

(5.3)

192 for sufficiently large n at a > - 1 / 2 , where | ||^ defines a Holder norm of || | as p-»«> . In this case, Light's work assumed that the function f(x)

has a Chebyshev expansion

f(x) ='Y_lbkTk(x) with 2* \bk | —> A as k —> oo; where A is a constant. Although equation k=\

(5.3) holds for a > - 1 / 2, Light's analysis does not exclude the possibility that we could get faster convergence to such a function by taking into a < -1 / 2 . From the above discussions, it is known that the spectral expansion converges fast with an exponential convergence. Usually, a Chebyshev expansion is better than a Legendre expansion numerically in that, at collocation points there is no need for really calculating the Chebyschev polynomial, while in Legendre expansion, this is not true, which can be seen from Equations (3.9) and (3.22), respectively.

6. Conclusions and Discussions Theory and algorithm implementation techniques for the spectral element method in solving elastodynamic problems are systematically analysed. Theoretically, the orthogonal basis functions, constructed with Legendre and Chebyschev polynomials, are introduced. The related analytical formula for arbitrary quadrilateral element in the physical domain for Legendre and Chebyshev expansions are discussed and those for Chebyshev expansions have been derived completely. It is pointed out that, theoretically, the first-kind Chebyshev expansion is superior to the Legendre expansions in the spectral element method. In numerical algorithm analyses, some apparent singular points in basis functions are carefully treated at the Legendre- and Chebyshev- Gauss-Lobatto collocation points. Especially, the derivatives of Legendre and Chebyshev basis functions at the boundary collocation points, used in element quadrature, are discussed. The element by element procedure with a pre-conditioned conjugate gradient linear solver in space domains, and a staggered predictor/multi-corrector algorithm in time iteration, are also introduced. Thank to the present approach neither the global matrix, nor the effective force vector is assembled. There is even no need for forming element matrix in order to further save memory without loosing much computation efficiency. This is true especially when element quadratic analytical solutions are used. The detailed numerical analysis will be seen in the second part of the work. 7. Acknowledgments This work was jointly supported by the Abdus Salam International Centre for Theoretical Physics of UNESCO, the International Science Link Program by the Department of Education, Science and Technology of Australia, and the Young Research Fellowship of Chinese Academy of Sciences in China. The first author wishes to extend his thank to Prof. Hailan Zhang in Institute of Acoustics of Chinese Academy of Sciences for reviewing this paper and for his helpful comments.

193 APPENDIX Analytical solutions of the terms in element mass and stiffness matrices for Chebyshev Polynomial expansions for arbitrary quadrilateral in physical domain are derived in the Appendix. In general because of the mapping from a physical element to a reference element, the contributions of basis functions and derivatives to element matrices are __ N /__ .. \ dgdr/ {Na,.,Nb,X, = £<->/__ £ {N (A-l) a,y,-Na^){Nb^-Nb4X^

where J is the Jacobian determinant, and (x^,y^) and yx^,y^j

can be calculated using

Chebyshev polynomial expansions, i.e., xg=x(£,7]) = YJN,ixl,

(A-2a)

y.f = :y(£»7) = I X f y , ,

(A-2b)

wheregs {^,7]}, and (x^y,) are the nodal coordinates in the physical element. Also, the Jacobian determinant can be expanded into 1 (A-3) = Z-W(£»7). where Ji ={J~\ Nt)

. Insert Equations (A-2a) and (A-2b) into Equation (A-l), yielding l.Nj*yj j=0

1

\ i=0 ;_n

J

dQ..

{N^-NH)

ZX^*

Vm=0

\k=0

J

(A-4a) The integrand in the above equation can be written as

F&f) = £W

N^N^Nj^N^-N^N^Nj^jt^^ J=0

1=1

V

*=1

m=0

*=0

(A-4b) ;=0

It can be further simplified into i,;,;=o

+ £

i,j,m=0

ly^^N^N^N^N^-

£ i,M=0

(A-4c)

J.y^N^N^^N^.

194 The key point to evaluate (Nax,Nby)n,

analytically is to find solution of this type

integral IF , i.e.,

where p,q,r,s,te

[l,2,...,neJ,

and a,ji,y,g^ T

{^,rf\ . IF can also be written as

T

h=c pc qcJcJcJl), T

T

where ~c p ,~c

,~cTr ,cf,~cj

(A-5b)

are determined by equation (3.1 Ob), and

/ , = [jp{g)TqXg)TXg)T'{g)sTXg),dg

•

(A-5c)

According to Equation (3.16), the derivative of a Chebyshev Polynomial can be expressed into a sum of the Chebyshev polynomials. Eventually, Equation (A-5c) can be expressed by the major term (H-i fO for p+q+r+s+t is odd, J [Tp{g)Tq{g)Tr(g)Ts{g)Tl{g)dg =\ (A"6a) M \ M Mt • M • [QF for p + q + r + s + t is even, where QF = f cos(ad) cos(b&) cos(c0) cos(dO) cos(eO) sin 6d6. (A-6b) In the above equation, the variable replacement and Equation (3.13a) have been used. After some derivation, we have

ef4lnr' 8 m=i 1 -

(A 6c)

-

K

where Lm= p + qi + rj + sk + tl, and i,j,k,le {-1,1}, whichmeans i,j,k,l can take either 1 or -1 so that there are 16 permutations, and there are 16 values for Lm , where m e {1,2,...,16}. Note that, since p + q + r+s + t is an even number, it is easy to show that Lm is also even number. Therefore the denominator in Equation (A-13) is never zero. We take into account a simpler example for detailed formulation derivation with three Chebyshev polynomial multiplications. For random elastic media, a double grid Chebyshev expansion is used (Seriani, 2004) in a rectangular element, so that the variation in medium properties is taken into account by using an independent set of basis functions on a temporary local grid in such a way that small scale fluctuations are accurately handled. In this case, a variable material property, such as density, is also expanded into fXt,Tj) = %paNatf,Tl).

(A-7)

The element mass matrix and the stiffness matrix contain the term i _„ |0 for a + b + c is odd,

f Q%(g)T (g)T (g)dg 1

b

c

=\

. [Q Qabc for a + b + c is even,

where QT = ~cTa~cTh~cTc , and Qabc = Jf cos(a<9) cos(b0) cos(c0) sin 9d6.

(A-8a) (A-8b)

195 In the above equation, the variable replacement and Equation (3.13a) have been used. After some derivation, we have

_ 1 cos(l + a + b + c)0 cos(l + c-a-b)0 l+c-a-b \+a+b+c cos(l-c + a + b)0 cos(l-c-a-b)0 l-c-a-b l-c+a+l

cos(l + c + a-b)0 cos(l + c - a + b)0 l+c+a-b l+c-a+b cos{l-c + a-b)0 \-c+a-b

cos(l-c-a + b)0lX l-c—a+b (A-8c)

So the above formula can be simplified into 1 , 1

1

1

-+2 l-(a + b + cY \-{a + b-cf

& * = - [ •

1

l-(a-b + c)2

Note that, since a + b + c is an even number, a + b-c,

-].

l-(a-b-c)-

a-b + c, and a — b-c

(A-8d)

are also

even numbers. Therefore the denominator in the above equations is never zero. Also, the following term is needed during the element integral evaluation for stiffness matrix calculation: ffi

T

_.

,

T

,

fO

lcTacThcX^)T'b{g)TXg)dg

for a + b + c is odd,

= \_c T_c T_c

.

x

(A-9)

{ a b c Qabc for a + b + c is even,

where

QL^fjMWfcK'^dg.

(A-10a)

Also Q\bc can be simplified into

QL = YLAbcQlM,n)..

(A-10b)

where 1

S i (m,n) =

2cb-\-2m^

c-l-2n

1-m,

"+

1 1 1 r- + - +\ — m2 l - r a 3 1 —m4

(A-10c)

In the above equation, m, = a + b + c — 2m — 2 — 2n ,

(A-11 a)

m2 =a + b-2m-c

+ 2n ,

(A-lib)

m3 = a — b + 2m + c — 2n,

(A-lie)

m. =a—b + 2 + 2m-c

(A-lid)

+ 2n .

By using the above formulas, an element quadrature is calculated analytically and high accuracy will be retained.

196 References 1. Bouchon, M., Campillo, M., and Gaffet, S., 1989, A boundary integral equation-discrete wave number representation method to study wave propagation in multilayered media having irregular interfaces: Geophysics, 54, 1134-1140. 2. Canuto, C , Hussaini, M. Y., Quarteroni, A., and Zang, T.A., 1988, Spectral Methods in Fluid Dynamics: Springer Series in Computational Physics, Springer-verlag. 3. Carcione, J. M, Kosloff, D., Behle A., and Seriani G., 1992, A spectral Scheme for wave propagation simulation in 3-D elastic-anisotropic media: Geophysics, 57,1593-1607. 4. Dauksher W., and Emery A. F., 1997, Accuracy in modelling the acoustic wave equation with Chebyshev spectral finite elements: Finite Elements in Analysis and Design, 26, 115-128. 5. Durand, S., Gaffet, S., and Virieux, J., 1999, Seismic diffracted waves from topography using 3-D discrete wavenumber-boundary integral equation simulation: Geophysics, 64, 572-578. 6. Gazdag, J. 1981, Modelling of the acoustic wave equation with transform methods: Geophysics, 46, 854-859. 7. Graves, R., 1996, Simulation seismic wave propagation in 3D elastic media using staggered-grid finite differences: Bull. Seism. Soc. Am., 86, 1091-1106. 8. Hestholm, S. O., and Ruud, B. O., 2000, 2D finite-difference viscoelastic wave modelling including surface topography: Geophys. Prosp., 48, 341-373. 9. Hughes, T. J. R., 1987, The finite element method: linear static and dynamic finite element analysis: Prentice-Hall international, Inc. 10. Kishore, N. N., Sridhar, I., and Iyengar, N. G. R., 2000, Finite element modelling of the scattering of ultrasonic waves by isolated flaws: NDT & E International, 33, 297-305. 11. Komatitsch, D., and Vilotte J. P., 1998, The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures: Bull. Seism, Soc. Am., 88, 368-392. 12. Komatitsch, D., Vilotte, J.P., Vai, R., Castillo-Covarrubias, J.M., and Sanchez-Sesma, F.J., 1999, The spectral element method for elastic wave equations -application to 2-D and 3-D seismic problems: Int. J. Numer. Meth. Engng., 45, 1139-1164. 13. Komatitsch,D., and Tromp, J., 1999, Introduction to the spectral element method for three-dimensional seismic wave propagation: Geophys. J. Int., 139, 806-822. 14. Kosloff, R., and Baysal, E., 1982, Forward modelling by a Fourier method: Geophysics, 47, 1402-1412. 15. Kosloff, D., Kessler, D., Filho, A. Q., Tessmer, E, Behle, A., and Strahilevitz, R., 1990, Solution of the equations of dynamic elasticity by a Chebychev spectral method: Geophysics, 55, 734-748. 16. Levander, A. R., 1988, Fourth-order finite-difference P-SV seismograms: Geophysics, 53, 1425-1436. 17. Mason, J. C , and Handscomb, D. C , 2003, Chebyshev Polynomials: Chapman & Hall /CRC, 131-133. 18. Madariaga, R., 1976, Dynamics of an expanding circular fault: Bull. Seism. Soc. Am., 65, 163-182.

197 19. Marfurt, K., J., 1984, Accuracy of finite-difference and finite-element modelling of the scalar and elastic wave equations: Geophysics, 49, 533-549. 20. Mu, Y. G., 1984, Elastic wave migration with finite element method: Acta Geophysica Sinica, 27, 268-278. 21. Orszag, S. A., 1980, Spectral methods for problems in complex geometries: J. Comput. Phys., 37, 70-92. 22. Patera, A. T., 1984, A spectral element method for fluid dynamics: laminar flow in a channel expansion: J. of Computational Physics, 54, 468-488. 23. Robertsson, J. O. A, 1996, A numerical free-surface condition for elastic/viscoelastic finite-difference modelling in the presence of topography: Geophysics, 61, 1921-1934. 24. Seriani, G., Priolo, E., Carcione, J. M., and Padovani, E., 1992, High-order spectral element method for elastic wave modelling: Expanded Abstracts of 62nd SEG Annual Int. Mtg., 1285-1288. 25. Seriani, G., and Priolo, E., 1994, Spectral element method for acoustic wave simulation in heterogeneous media: Finite Element in Analysis and Design, 16, 337-348. 26. Seriani, G., 1997, A parallel spectral element method for acoustic wave modeling: Journal of Computational Acoustics, Vol.5 (1), 53-69. 27. Seriani, G., 1998, 3-D large-scale wave propagation modelling by spectral element method on Cray T3E: Comp. Meth. Appl. Mech. and Eng., 164, 235-247. 28. Strang, G., and Fix, G. J., 1973, An analysis of the finite element method: Englewood Cliffs, N. J., Prentice-Hall. 29. Tal-Ezer, H., 1986, Spectral methods in time for hyperbolic problems: SIAM J. Numer. Anal., 23, 12-26. 30. Teng, Y. C , 1988, Three-dimensional finite element analysis of waves in an acoustic media with inclusion: J. Acoust. Soc. Am., 86, 414-422. 31. Tessmer, E., Kosloff, D., and Behle, A., 1992, Elastic wave propagation simulation in the presence of surface topography: Geophys. J. Internat., 108, 621-632. 32. Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method: Geophysics, 51, 889-901. 33. Wang, X. M, Zhang, H. L., and Wang, D., 2003, Modeling of seismic wave propagation in Heterogeneous poroelastic media using a high-order staggered finite-difference method: Chinese J. of Geophysics, 46 (6), 842-849 34. Wang, X. M., and Zhang, H. L., 2004, Modeling of elastic wave propagation on a curved free surface using an improved finite-difference algorithm: Sciences in China, Series A , 47 (5), 633-648.

INVERSION OF B O T T O M BACK-SCATTERING MATRIX J. R. WU, T. F. GAO Institute ofAcoustics, Chinese Academy of Sciences, Beijing, China E-mail: [email protected] E. C. SHANG CIRES, University of Colorado, Boulder, CO80303, USA E-mail: [email protected] Abstract The normal mode model of reverberation in shallow-water waveguides has been presented based on Born approximation. The key component of this model is the modal back-scattering matrix. The characteristics of the modal back-scattering caused by (1) the roughness of the bottom interface and (2) the volume inhomogeneities under the interface are discussed. Approaches of inversion of the matrix from reverberation data are proposed. Examples of the inversed result are shown both for numerical simulation and experiment. 1. Introduction Reverberation process includes sound transmission and scattering. In shallow water the bottom scattering can be considered as the dominant scattering source, especially for a relatively smooth sea surface or for a water sound speed profile with a negative gradient[l]. Therefore, it is important to understand the characteristics and the mechanisms of bottom modal back-scattering in order to establish a predictable model. Bottom scattering, generally, can be attributed to two major mechanisms: (a) roughness of various interfaces; (b) volume inhomogeneities of the medium parameters - density and compressibility. Shallow-water reverberation due to roughness[2] and volume inhomogeneities[3] has been developed separately based on small perturbation theory[4]. Recently, a unified approach to volume and roughness scattering is proposed[5], which provides the possibility of solving the scattering field due to both of the two mechanisms. The key component for modeling reverberation in shallow-water is the bottom modal back-scattering matrix @m„. To extract the 0 m „ from the reverberation data has been a challenging topic for a long time. There are some work dealing with the extraction of ®mn from the reverberation data[6-8]. However, the inversion are based on either empirical law (Lambert' law) or an assumption that 0 m „ is separable: ®mn = ®m®n

(1)

Recently, Shang[9] has proposed a new approach to extract ®m„ from the reverberation data without any a priori assumption on the scattering. With this method, the bottom backscattering matrix ©m„ can be extracted by mode-filtering at the receiving vertical array and changing the point source depth to obtain different incident mode excitation. Some numerical simulations are conducted in [10]. In this paper, the normal mode model of reverberation in shallow-water waveguides has been presented based on Born approximation firstly. Then the characteristics of the modal back-scattering caused by (1) the roughness of the bottom interface and (2) the volume inhomogeneities under the interface are discussed. At last, approaches of inversion of the matrix from reverberation data are proposed. Examples of the inversed result are shown both for numerical simulation and experiment. 199

200 2. Normal Mode Model of Reverberation in Shallow-Water Waveguide The acoustic pressure due to a point source of unit strength at depth zs in a shallow-water waveguide can be written as M

p(r,z) = ix^m(zsWm(z)Hll)(Kmr)

(2)

where z is the depth coordinate(measured positive downward from the ocean surface), r is the range coordinate, M is the number of trapped modes, t) due to the source at frequency / = a>/2 a has been suppressed. By writing the asymptotic form of the Hankel function H?(Kmr) = pl{nkmr)

exp{i{Kmr -xl4)}

(3)

and using exp(i7i/4)=i(1/2>, the Eq.(2) can be expressed as [

M

0

kmr

-Smr

p(r,z) = (2mf1Yd^m{z,)4,m(z)-—^r

(4)

\kmr)

m=\

In [5], the exact expression of scattering can be presented as multiple scattering series, and the first-order solution is called the single-scattered field or Born approximation. In this paper, the Born approximation of backscattering is considered, and the reverberation field can be represented based on normal mode: JIT

p\r,z) * O X X ^ ( Z J 7 ^ „ ( z ) e - < ^ >

(5)

where Tmn is scattering kernel function, for roughness and medium volume inhomogeneities, it can be expressed as

""

\S™' lle(r,zyUM-"-"-''drdz

(7)

where y describe the interface roughness, and s describe the volume inhomogeneities. The expression of Tmn for roughness[2] is sin2 0m sin2 0„Gm„

TL = ©m„ = ("l*)°W

(8)

where a is mean square root of roughness, r0 is correlation length of the roughness, 8 is grazing angle of the normal mode. Gmn is proportional to the two-dimensional space spectrum of y. And the expression of Tmn for volume inhomogeneities[3] is :

"»«

—C

mn ' Lmn ' Jm„

{.")

Where

• (rf^YTX

«

Lm„*«)e

+W +U„ -PmPnW i-tr^,*u 2

(10) (11)

201 (12)

J.

where ac is the standard deviation of sound speed fluctuation, / i s the density ratio between the bottom and the water column at the source depth, rj is a constant, £ is the eigenvalue of the normal mode, J3 is the transmission coefficient of a plane wave, Lmn is spectrum of the horizontal correlation function of inhomogeneity, Jm„ is spectrum of the vertical correlation function of inhomogeneity 3. Characteristics of Modal Back-Scattering Matrix Modal back-scattering matrix is key component of normal reverberation model, and it describes the coupling relationship between incident normal mode and scattering normal mode in the shallow water. In this section, the theory and numerical analysis of back-scattering matrix was given in three aspects: 1) Separable approximation, 2) Sub-Matrix approximation, and 3) Mode-space. 3.1. Separability of back-scattering matrix The modal back-scattering matrix in the shallow water can be expressed as: 0, 0,

©,: 0,

©,, 0,

"Ml

"M2

0,

O1

(13)

If the matrix is separable, just like Eq.(l), Eq.(8) can become separable matrix

V©^

0„ 0 2 = V©22©U •\/®MM©ll

V®22©*

©22

V©^

(14)

22

The difference between the full back-scattering matrix(0 ) and the separable matrix(02) is the separability error. Whether the matrix is separable can be decided by quantity of separability error. As a numerical simulation example, we consider a Pekeris waveguide with water depth H = 50m, c0 = 1500m/s, cb = 1600m/s, pb = 1.77, bottom attenuation a = 0.23A. /dB. 0 po = 1 g/cm c 0 =1500m/s H=50m Pb = 1.77g/cm c 4 =1600m/s ZH

Fig. 1. Pekeris waveguide.

202

Firstly, considering roughness back-scattering. Taking 0.1m, r0 = 6m Then full back-scatteririg matrix is (dB) "-52.9126 - 48.1497 -46.2195 -48.1497 - 43.3745 -41.4262 ®\ = -46.2195 - 41.4262 -39.4511 -45.3982 - 40.5837 -38.5774 The separable matrix is (dB) "-52.9126 -48.1435 ®l = -46.1818 -45.2901

--48.1435 --43.3745 --41.4128 --40.5211

the roughness parameters as: a

-45.3982 -40.5837 -38.5774 -37.6677

(15)

-46.1818 -45.2901 -41.4128 -40.5211 -39.4511 -38.5594 -38.5594 -37.6677

(16)

Separability error matrix is (dB) 0 0.0061 0.0376 0.1081" 0 0.0061 0.0134 0.0626 ®\ = 0.0376 0.0134 0 0.0180 0.1081 0.0626 0.0180 0

(17)

Then, considering volume inhomogeneity back-scattering. Taking the volume inhomogeneity parameters as: ac = 0.025, rj = 5.2, lh = 2m, /„ = 0.5m Then full back-scattering matrix is (dB) -86.1092 - 80.9139 -78.2050 -78.2752 -80.9139 - 75.7099 -72.9776 -72.8896 (18) ®*,= -78.2050 - 72.9776 -70.1820 -69.6368 -78.2752 - 72.8896 -69.6368 -61.4255 The separable matrix is (dB) -86.1092 -80.9095 ®l = -78.1456 -73.7674

- 80.9095 - 75.7099 - 72.9459 - 68.5677

-78.1456 -73.7674" -72.9459 -68.5677 -70.1820 -65.8037 -65.8037 -61.4255

Separability error matrix is (dB) 0 0.0043 0.0594 4.5079n 0 0.0317 4.3220 0.0043 ®l = 0.0594 0.0317 0 3.8331 4.5079 4.3220 3.8331

(19)

(20)

0

From Eq.(17) and Eq.(20), we can decided that the back-scattering matrix due to roughness is quasi-separable; and the back-scattering matrix due to inhomogeneities is unseparable.

203 3.2. Sub-matrix of back-scattering matrix In Eq.(15) and Eq.(18), the elements of right-down is bigger than the elements of left-up of the matrices. In this section, the right-down submatrix is used to synthesize reverberation data in stead of full back-scattering matrix. As a numerical simulation example, we consider the same Pekeris waveguide as we used in section 3.1. Considering center frequency i s / = 450Hz, so there are 11 trapped normal modes in the waveguide. Taken the order of submatrix as 11, 10, 9 and 8 respectively, The synthesized reverberation data used four submatrices are in the Fig.2

8

10 12 Range (km)

Fig.2. Reverberation data using submatrices. The full modal back-scattering matrix can be replaced by submatrix sometimes judged from the Fig.2. 3.3. Characteristics of back-scattering matrix in mode space For convenience, we define the "effective matrix" ®m„eff as follows ©*=©m„exp{-2(^+^K}

(21)

In the numerical simulation, we consider the same Pekeris waveguide as section 3.1. Considering centre frequency / = 150Hz, so there are 4 trapped normal modes in the waveguide. We only consider the diagonal elements of the effective matrix because it is quasi-separable for roughness back-scattering. The ranges are 500m, 5km, 10km and 20km respectively.

204

Fie.3.r c = 500 m.

Fig.5. rc = 10 km.

Fig.4. rc = 5 km.

Fig.6. rc =20 km.

The results(Fig.3-Fig.6) show that: 1) in near distance, the value of back- scattering matrix elements inclines towards high modes. The matrix is similar to highpass filter; 2) in middle distance, the value of back-scattering matrix elements inclines towards middle modes. The matrix is similar to bandpass filter; 3) in far distance, the value of back-scattering matrix elements inclines towards low modes. The matrix is similar to lowpass filter. This phenomenon is caused by mode attenuation in shallow water waveguide. 4. Inversion of Modal Back-Scattering Matrix Extracting the bottom back-scattering information from reverberation data in shallow-water waveguide is an attractive but difficult issue. In this section, two kinds of methods were proposed. One has a priori assumption; the other has no a priori assumption. They were called separable inversion and unseparable inversion of modal back-scattering matrix[ll]. 4.1 Numerical simulation of inversion of modal back-scattering matrix First, make an assumption that the modal back-scattering matrix ®m„ is separable (Eq.l) Then, make mode-filtering of the reverberation field. When the receiving vertical array is weighted by thej'-th mode function, we have

205

Psj(z0,rc) =

lps(z,z0,rc)^(z)dzr = f ^ f ^ o M , ^ ) • Snj • ^(zH)jArj(x)e,(^')xdX

(22)

Taking only the non-interference term of the reverberation intensity as the averaged intersity j

M

k(jrc

n

Ij(z0,rc) = 7 ^ 2 > " ( z o ) © > , exp{-2(<5„ + Sj)re)

(23)

Letj = l,2 (24)

72 exp(-2£2rc)@2 Eq.(24) can be transformed to the following equation Q2=(/2_)(exp(-2^,rc))Qi /, exp(-2£2rc)

(25)

And 0

3 = (

^)(^=^^)© /, exp(-2£3rc)

(26)

1

Mwexp(-2yc))0

(27)

©M=(^L)(

Ix exp(-2£ M r c ) Integrated (25),(26),(27) and (23), we can get ©,, 02> 03> back-scattering matrix 0m„ using the separability(Eq.l). Neglecting the dispersion effect, we have j

and 0 M . At last, get the modal

M

(28) Ij(z0,rc) = -^-^(zoy®„ exp{-2(Sm + Sj)re} Krc m In Eq.(28), we have assumed that the unperturbed stratified waveguide is known, which means that ®mJ is the only unknown. If, for each filtered modal reverberation intensity 7,, we change the source depth zs M times: ZS\^S2,"'^SM, then we will have MxM equations for solving the MxM unknown ®mJ In the numerical simulation, we consider the same Pekeris waveguide as section 3.1. And take the roughness back-scattering as an example, its roughness parameters is: a = 0.1m, xo = 6m The back-scattering matrix used to synthesize reverberation data is Eq.(15). And the inversed back-scattering matrix using separable method is -52.9640 -48.1760 -46.2087 -45.3440"

-48.1760 -43.3880 -41.4207 -40.5560 -46.2087 -41.4207 -39.4533 -38.5886 -45.3440 -40.5560 -38.5886 -37.7239 The inversed back-scattering matrix using unseparable method is Qsep

(29)

206

s^\ unsep

•52.9126 -48.1497 -46.2195 -45.3982

-48.1497 -46.2195 -45.3982" -43.3745 -41.4262 -40.5837 -41.4262 -39.4511 -38.5774 -40.5837 -38.5774 -37.6677

At last, we get the separable inversion error separable inversion error is (dB) 0.0515 0.0264 0.0264 0.0135 i©< • © 0.0108 0.0055 0.0542 0.0277 The unseparable inversion error is (dB) "0.0 0.0

107^ • • 0

(30)

and unseparable inversion error. The 0.0108 0.0055 0.0022 0.0112

0.0542 0.0277 0.0112 0.0562

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

(31)

(32)

4.2. Inversion of modal back-scattering matrix from experiment data Reverberation data were collected in a reasonably flat shallow-water area in South China Sea. A vertical line array was deployed to record monostatic reverberation from explosive charges. The explosive charges denoted at a depth of 7m. The vertical array contained 32 hydrophones, which were spaced from 7m to 69m. The sound speed profile measured during experiment is shown in Fig.7. The depth of the experiment sea area is 88.84m. o First Time Second Time

10 20 30

1 40 |

50 60 70

1529

1529.5 1530 Sound Speed (m/s)

1530.5

Fig.7. Sound speed profile. Firstly, the separable inversion of the back-scattering matrix was presented. The separable inversion include following six steps: The first step: make frequency filtering of the reverberation data, the center frequency

207

i s / = 200Hz The second step: make mode-filtering of the reverberation data The third step: get the I/j=l,2,---,M) The fourth step: get the ratio of ®m(m=l,2,---,M) The fifth step: calculate the ©i The sixth step: get the back-scattering matrix -82.9903 -82.6730 -82.3185 -76.6115 -75.0499 -82.6730 -82.3557 -82.0012 - 76.2942 -74.7326 -82.3185 -82.0012 -81.6466 -75.9397 -74.3781 &sep = -76.6115 -76.2942 -75.9397 -70.2327 -68.6712 -75.0499 -74.7326 -74.3781 -68.6712 -67.1096 -65.9585 -65.6412 -65.2867 -59.5797 -58.0182

-65.9585 -65.6412 -65.2867 -59.5797 -58.0182 -48.9268

(33)

Then, the unseparable inversion of the back-scattering matrix was discussed. The reverberation intensity can expressed as: T

I(z0,z,rJ

MM

= -^^fi(z0)®mifi(z)vq>{-2(Sm+Sn)rc} k

Drc

m

(34)

n

The number of trapped mode in the waveguide is M=6, the unkown parameters the back-matrix is M-(\I2)(M-M)=2\ x, xg

0:

x12 x13

(35)

x14 x15

Using 21 channels of receiver array, we can get the unseparable back-scattering matrix -84.9205 -78.9090 -81.4483 -80.5997 -77.7279 -67.0755 -78.9090 -76.6619 -77.6056 -73.0682 -67.7421 -56.3842 -81.4483 -77.6056 -74.9338 -73.0582 -80.4350 -60.9133 ©" (36) -80.5997 -73.0682 -73.0582 -72.9733 -71.1268 -53.2428 -77.7279 -67.7421 -80.4350 -71.1268 -71.2151 -51.7836 -67.0755 -56.3842 -60.9133 -53.2428 -51.7836 -46.2848 5. Conclusion In this paper, the normal mode model of reverberation in shallow-water waveguides has been presented based on Born approximation. Then characteristics of the modal back-scattering caused by (1) the roughness of the bottom interface and (2) the volume inhomogeneities under the interface are discussed. The back-scattering matrix due to roughness is quasi-separable; and the back-scattering matrix due to inhomogeneities is unseparable. In the certain condition, the full back-scattering matrix can be replaced by its submatrix. In mode-space, 1) in near distance, the value of back- scattering matrix elements inclines towards high modes. The matrix is similar to highpass filter; 2) in middle distance,

208 the value of back-scattering matrix elements inclines towards middle modes. The matrix is similar to bandpass filter; 3) in far distance, the value of back-scattering matrix elements inclines towards low modes. The matrix is similar to lowpass filter. This phenomenon is caused by mode attenuation in shallow water waveguide. Two approaches of inversion of the matrix from reverberation data are proposed. One is separable inversion method, the other is unseparable method. Examples of the inversed result of the two inversion methods are shown both for numerical simulation and experiment. Acknowledgements This work supported by the National Science Foundation of China under Grant No 10474111 and by Funds of Header of Institude of Acoustics(CAS) under Grant No S2004-10

References [I] G. L. Jin, R. H. Zhang and X. F. Qiu, "Characteristics of shallow water reverberation and inversion for bottom properties", Proceedings of SWAC, Ed. Zhang and Zhou, 303-308, 1997 [2] T. F. Gao, "Relation between waveguide and non-wave guide scattering from a rough interface", Acta Acust. 14, 126-132(1989) [3] D. J. Tang, "Shallow-water reverberation due to sediment volume inhomogeneities" (to be published) [4] F. G. Bass and I. M. Fuks, Wave Scattering from Statistical Rough Surface, Pergamon Press, 1979 [5] A. N. Ivakin, "A unified approach to volume and roughness scattering", J. Acoust. Soc. Am. 103, 827-837(1998) [6] D.D. Ellis and P. Gerstoft, "Using inversion technique to extract bottom scattering strength and sound speed from shallow-water reverberation data", Proceedings of 3rd ECUA, Ed. By J. Pappadakis, Vol.1, 320-325,1999 [7] V. M. Kurdryashov, "Low-frequency reverberation in shallow-water Arctic Seas", Acoustical Physics, 45, 320-325, 1999 [8] Ji-Xun Zhou and Xue-Zhen Zhang, "Shallow-water acoustic reverberation and small grazing angle bottom scattering", Proceedings of SWAC, Ed. Zhang and Zhou, pp.315-322, 1997 [9] E.C.Shang, T.F.Gao, and D.J.Tang, "Extraction of Modal Back-Scattering Matrix from Reverberation Data in Shallow-water Waveguide. Part I — Theory", Theoretical and Computational Acoustics 2001, pp.67-74, Ed. E. C. Shang, Qihu Li and T. F. Gao, 2001, Beijing [10] L. Brekhovskikh, Ocean Acoustics, Moscow, HAYKA, 1974, Ch.4 [II] J.R.Wu. "Doctoral Disertation" (2005, IOA, Beijing)

NEW METHODS OF SCATTERING COEFFICIENTS COMPUTATION FOR THE PREDICTION OF ROOM ACOUSTIC PARAMETERS

Institute of Environmental

XIANGYANG ZENG Engineering, PB58, Northwestern Polytechnical XVan, 710072, China E-mail: zenggxy @nwpu. edu. en

University,

CLAUS LYNGE CHRISTENSEN ODEON A/S c/o Acoustic Technology, Technical University of Denmark, DK-2800, Denmark JENS HOLGER RINDEL Department of Acoustics Technology, Technical University of Denmark, DK-2800, Denmark To include the sound scattering caused by limited size of surfaces in room acoustic computer simulations, some model for scattering must be included in room acoustics computer models. A large concert hall usually contains a variety of small and complex surfaces and it is not practical to obtain accurate scattering coefficients of all these surfaces. Even if these frequency dependent coefficients could be obtained in the design phase, the modeling process would become more time consumed and increase the cost of design. In such a case, the appropriate simplification of the model and the definition of scattering coefficients by experience will become important. But in some other cases, calculation of a detailed model is necessary and possible. For these different cases, practical methods to define or calculate scattering coefficients, which include a new approach of modeling surface scattering and scattering caused by edge diffraction, have been presented. The predicted and measured acoustic parameters have been compared in order to testify the practical approaches recommended in the paper.

1.

Introduction

Scattering has been validated to be one of the most important properties of sound in enclosed spaces [1-3]. Scattered reflection can improve the uniformity of a reverberant field and reduce the risk of areas of poor acoustics within a room. Surface scattering has found a role in dispersing reflections which are causing echoes or coloration [4]. The first international round robin test of computer modeling software has clearly indicated that inclusion of scattered reflections is an important factor in achieving good simulation results [5]. The investigations carried out by Hodgson [1], Dalenback [2] and Lam [3] have shown that the scattered reflections will not only affect the accuracy of the calculation of acoustical parameters, but also have influence on the quality of auralization. Scattering coefficient is usually used to describe the scattering property of walls. It has been found that the coefficient is not only dependent on surface material, frequency of sound source, but also dependent on the geometry of the computer model [6]. It is possible for different computer models to get different calculation results on condition that the same scattering coefficients are used. This makes how to obtain the scattering coefficient a key problem. One way is direct measurement [7]. Another is to define it based on experience [8]. It can be found that the direct measurement is not practical for an arbitrary surface. This requests the users to be room acoustics experts, thus will limit the application of the program. 209

210 In this paper, rooms are divided into two groups: large and complicated rooms; small and simple rooms. For both cases, a method to consider both scattering due to surface property and scattering caused by edge diffraction has been applied. Practical methods to define the model and the scattering coefficients when using the program ODEON and other similar packages in both room cases have been given. 2.

Current Methods for Modeling Surface Scattering

The basic idea to consider sound scattering in program ODEON is that the reflected energy can be divided into two parts at a surface: specular and scattered. Their relation can be denoted by the absorption coefficient CC and the scattering coefficient S

(l-s)(l-a) + a + s{\-a) = \

(l)

The randomized ray propagations and secondary sources have been combined to simulate the scattering. Instead of separating the ray tracing process into two parts, the model uses only a single ray tracing process for each ray. Furthermore, using the secondary sources to model radiation from the surface reflections to the receiver means that it is no longer necessary to check the validity and visibility of the image sources, thus reducing the computation time. However, the image sources in the model may not be the purely specular images. Even when the scattering coefficient is set to be zero, there is still some scattering that has been modeled. And it will reduce the effect of specular sound. To solve the problem, a factor named transition order (TO) has been defined, which can limit the scattering calculation only to those reflections having orders higher than TO. This scattering model has been applied into ODEON from version 2.5 to version7.0, and has been validated to be an efficient model. The scattering coefficient used in the current model mainly considers the surface scattering due to material property and the TO can take into consideration the shape or structure of the acoustic room [6]. However, it has been found out that sound scattering is also dependent on the distance from the receiver to the edge of some small surfaces where diffraction usually occurs. This means it is not enough to take into account only the scattering due to surface property. Especially at low frequency and in the case where there are many varied small surfaces, the scattering caused by edge diffraction becomes more important and need to be calculated separately. In the following section, the method to define a scattering coefficient combining both parts of scattering will be described at first. And then practical recommendations to define scattering coefficient for both kinds of rooms will be given. 3.

Practical Methods to Define Scattering Coefficient

3.1. A new method for the calculation of scattering coefficient To consider surface edge diffraction, we take a small panel as an example, which is shown in Fig.l. S,S' are the original sound source and image source, R is the receiver. It can be derived that the limiting frequency is [9]

211

/ „ = •

c-d 2Acos<9

(2)

where c is the sound speed, A is the area of the small surface and d is the characteristic distance, which can be calculated from 2dj • d2

d =

(3)

dx +d2

S"

R

Figure 1. Sound reflection from a small surface.

Above the limiting frequency, the diffraction losses can be considered negligible, while below the limiting frequency, it is

AL = 201og10

/

(4)

/.

This means at frequency higher than the limiting frequency, the sound energy can be thought totally specular and below the limiting frequency the scattering energy due to diffraction increases rapidly (6dB per octave band). This part of scattered energy can be represented by a factor sD, which can be calculated from

=i-(A 2 =i-( 2 / - A ; o s V

(5)

Therefore, the total scattering coefficient s can be calculated from s = \-(l-sD)i\-ss) (6) Where sD is the factor related with distance from receiver to the edge of a surface and ss\s the scattering coefficient due to the surface property (defined in ISO/ FDIS17497-1). Then the new direction of a reflected ray can be determined according to the value of s. If 5=1, the reflected ray will propagate in a scattered way which can be calculated according to Lambert's law; if 5=0,the reflected ray will propagate in a specular direction and can be easily obtained from Snell's law; if s is between 0 and 1, a new reflected direction can be determined by using s a s a weighting between the pure specular direction and scattered direction [10].

212 The above algorithm has combined the scattering due to edge diffraction and scattering due to surface property, therefore, it can reduce the influence of the scattering coefficient due to surface, which usually has to be defined according to the subjective experience of users. 3.2. Methods to define scattering coefficients for different rooms For the case there are a variety of small and complex surfaces the computation will become more time consuming and it is also impossible to obtain the frequency dependent scattering coefficients due to surface property for all the surfaces. But in other cases, especially when the low frequency sound is more important, the detailed room model has to be considered. Therefore, we suggest two different ways to deal with small surfaces in large concert halls and small rooms. A. Small and simple rooms For such kind of rooms, the number of main walls is usually small and the diffuser arrays may distribute in a few walls to achieve special acoustics effect. For instance, a studio room may need better acoustic behaviors at low frequency bands. It is required that some walls have to be equipped with special diffusers to counterbalance the weak scattering due to the simple structure of the room. On the other hand, as the sources or receivers are close to the reflector, which will produce strong reflections, considering the detailed structure of the model is necessary. In this case, one scattering coefficient due to surface ss can be assigned to all these small surfaces. The recommended value is between 0.01 and 0.05. The scattering coefficient due to surface then will be combined with the part representing the scattering caused by surface edge diffraction. In offices or classrooms, there is furniture such as tables and shelves. If a table plate is close to a source or receiver point, it likely to produce a strong reflection at the receiver, so it also should be included in the model. B. Large and complicated rooms For a large artistic room, the shape and its interior structure are usually complicated. It is likely to contain too many small surfaces and to establish a model with such a degree of detail is likely to be a waste of time. It is recommended to simplify the real room when turning it into a visible computer model. That means some detailed parts of the walls may be deleted. But for such kind of walls the comparatively bigger scattering coefficients should be defined. The value is usually defined bigger than 0.3 and smaller than 0.8. There are some guidelines for the simplification: (1) Curved surface. Curved surfaces have to be approximated by dividing them into plane sections. How finely to subdivide depends on the type of curved surface and how important the surface is. Using many surfaces in the model will make the model visually complex, and increase the probability of errors in the model, typically small leaks becomes a problem. Subdivisions about every 10° to 30° will probably be adequate to reproduce focusing trends, without excessive numbers of surfaces. (2) Audience area. Modeling each step between the rows in an audience area is not recommended. The audience area can be simplified a lot without compromising the quality of the results. However, this rule does not apply to open-air theatres.

213 (3) Podium on stage. The guideline is the same as that for the audience area. Rather than modeling each step of the podium, the podium can be simplified into a few sloped surfaces. 4.

Prediction of Various Models

4.1. Conditions of experiments A. PTB studio The PTB studio, which was used to test different computer models in international round robin, has been chosen as an example in this paper. Two computer models for the studio have been designed for ODEON, named "simple" and "detailed" respectively, see Fig.2. There are 70 surfaces in the simple model and the total surface area is 421 m2. For the detailed model, there are 268 surfaces and the total surface area is 450 m2. In the simple model the small diffusers on the ceiling and one wall have been neglected. The omni-directional point source is located at (x,y,z) = (1.5,3.5,1.5) and three receivers are: Rl (-2.00, 3.00, 1.20), R2 (2.00, 6.00, 1.20), R3 (0.00, 7.50, 1.20). The total ray number is 10000 and the transition order is 0. The scattering coefficients of various surfaces are listed in table 1. The measurement results are the mean value of 18 participants.

(a)

(b)

Figure 2. PTB studio (a) simple model (b) detailed model.

To validate the scattering method presented in the paper, it has been compared with the old method applied in the ODEON ver7.0. Three cases have been studied: (1) simple model, old scattering method; (2) simple model, new scattering method; (3) detailed model, new scattering method. The acoustic parameters C80, T30, Ts, EDT, LF8o, D50 and G have been modeled and compared with the measured results.

214 Table 1. Scattering coefficients for PTB studio model.

Case

Simple model Old method

Simple model New method Detailed model New method

Surface Parquet Wilhelmi Curtain(open) Studio wall Window glass Wood absorber Ceiling Ceiling Wood absorber Other surfaces All other surfaces

Scattering coefficient 0.20 0.30 0.48 0.20 0.10 0.95 0.95 0.85 0.85 0.02 0.02

B. Elmia concert hall Two kinds of models of the Swedish concert hall Elmia were set up in ODEON. One is simplified and another is in detail. See Fig. 3. There are 94 surfaces in the simple model and the total surface area is 4409 m2. For the detailed model, there are 470 surfaces and the total surface area is 4932 m2. In the simple model the small diffusers on the side faces have been simplified. The omni-directional point source is located at (x,y,z) = (8.5, 0.0, 25.5) and six receivers are: Rl (13.8, 0.0, 24.9), R2 (12.9, 10.5, 28.7), R3 (19.9, 5.1, 26.1), R4 (25.5, -4.9, 27.5), R5 (24.8, 11.9, 29.1), R6 (37.80, 6.40, 131.85). 10000 rays have been used to calculate the acoustics parameters C80, T30, Ts, EDT, LF80, D50 and G. The transition order is set to be 4 and the scattering coefficients of various surfaces are listed in table 2. The predicted acoustic parameters have been compared with those of measurements.

(a)

(b)

Figure 3. Elmia concert hall (a) simplified model (b) detailed model.

Table 2. Scattering coefficients for Elmia model.

model Simplified model Detailed model

Surface Audience area Simplified surfaces Other surfaces Audience area Side reflectors Ceiling Other surfaces

Scattering coefficient 0.60 0.30 0.02 0.60 0.35 0.30 0.02

4.2. Results and discussion A. Accuracy of different scattering models For the PTB studio, the mean errors of the three cases at 6 frequency bands are listed in table 3. For the Elmia concert hall, the mean errors of the two cases at 6 frequency bands are listed in table 4. Table 3. Average errors of at three receiving positions in PTB studio.

Parameter Simple modelold method Simple modelnew method Detailed modelnew method

Qo(dB) G(dB) T30(s) C80(dB) G(dB) Ts(s) C80(dB) G(dB) T30(s)

125 1.8 2.5 0.17 1.6 2.0 0.16 3.0 0.1 0.16

250 1.1 2.2 0.27 1.1 1.8 0.20 1.2 0.7 0.09

Frequency (Hz) 500 1000 0.2 0.1 1.5 1.3 0.13 0.06 0.1 0.2 1.1 0.8 0.09 0.04 0.4 0.4 1.0 0.7 0.03 0.07

2000 0.2 1.3 0.12 0.1 0.8 0.06 0.3 0.6 0.04

4000 0.1 1.0 0.06 0.1 0.8 0.12 0.4 0.7 0.03

Table 4. Average errors of six receiving positions in Elmia concert hall.

Parameter Simple modelnew method Detailed modelnew method

C80(dB) G(dB) T30(s) Qo(dB) G(dB) T30(s)

125 2.9 1.0 0.32 3.7 1.6 0.42

250 0.8 2.2 0.13 1.3 1.8 0.33

Frequency band(Hz) 500 1000 2000 1.4 1.2 1.3 1.1 0.8 0.7 0.18 0.15 0.09 0.8 0.9 0.9 0.9 0.4 0.6 0.23 0.20 0.27

4000 1.7 1.3 0.23 0.8 1.8 0.28

216 Table 3 shows that when using the new method for the simple model, the results are better than those of the old method. And when using the new method for the detailed model, the results are the best except for C80. The predicted C80 is bigger than the measured ones and the difference is much bigger at low frequency bands when using the new method for the detailed model. This may indicate more early sound energy is collected because of the reflection and diffraction from those small surfaces that have not been considered in the simple model. Table 4 has shown that the prediction accuracy of C80 and G of the two models is comparable. Both of them can get acceptable results at different frequency bands. But for the parameter T30, the simple model can obtain better results at all frequency bands. This means the simplified model can also achieve satisfied results. According to the results of some other parameters like LF80, Ts and D50, it also can be concluded that the accuracy of these two models is approximate. B. Influence of TO on the prediction accuracy As in the ODEON model, the value of TO is also a important factor which can affect the scattering modeling, different TO cases have also been calculated. C8o and EDT are investigated in this paper. Table 5 and 6 have given the results. It can be found that when the TO is set to be zero or 1 for the PTB studio model, the accuracy is better. From TO=2 to TO=5, the average error of the six frequency bands will exceed ldB. From table 6, it can be found that when TO=4 or TO=5, the mean errors at all frequency bands are smaller. These results may indicate that for the typical concert hall the TO should be higher than 3, while for approximately proportionate rooms zero or 1 are the best choice for TO. Table 5. TO and predicted Cso(dB) in PTB detailed model.

f(Hz) TO=0 TO=l TO=2 TO=3 TO=4 TO=5 Measured

125 8.7 8.8 9.1 9.5 9.3 8.8 5.7

250 4.6 4.8 5.4 6.0 6.3 6.1 3.5

500 3.4 3.8 4.3 4.8 5.1 5.0 3.1

1000

2000

4000

4.3 4.6 4.9 5.2 5.3 5.1 3.9

3.5 3.7 4.1 4.2 4.3 4.2 3.3

4.7 4.8 4.9 4.8 4.7 4.6 4.4

Table 6. TO and predicted EDT(s) in Elmia simple model.

f(Hz) TO=0 TO=l TO=2 TO=3 TO=4 TO=5 Measured

125

250

500

1.47 1.86 1.88 2.06 1.91 2.02 2.10

1.86 1.66 1.88 2.09 2.34 2.23 2.33

1.92 1.70 1.98 1.98 2.15 2.08 2.19

1000 1.89 1.70 1.84 1.75 2.44 2.10 2.22

2000 1.76 1.74 1.90 1.72 2.25 1.93 2.04

4000 1.50 1.14 1.40 1.58 1.86 1.96 1.70

217 C. Computation time and model complexity From table 7, it can be found that with the same number of rays, the computation time of the detailed and the simple PTB studio model is very close. But as concluded in section "A", there is an obvious increase of accuracy. It means the new method for the detailed model is more practical than that for the simple model. For the Elmia concert hall this is more obvious. The computation time has increased to 525% if the detailed model is used, but the accuracy has almost no increase. Table 7. Comparison of different models.

Total surfaces Total area(m2) Total rays Source number Receiver number Computation time(s)

PTB simple 70 421 20000 1 3 34

PTB detailed 268 450 20000 1 3 33

Elmia simple 94 4409 20000 1 6 40

Elmia detailed 470 4932 20000 1 6 211

*the CPU frequency of the computer is 2.0GHz.

5.

Concluding Remarks

A scattering model has been presented and practical methods for the consideration of surface scattering when using room acoustics computer model ODEON. These methods are better than the conventional methods that only consider the scattering coefficient due to surface property. For acoustic consultants or computer model users, it is also an important problem to realize the balance between the accuracy and design difficulty. Two different ways have been given for the two groups of rooms, and some rules have been recommended on the inclusion or simplification of small surfaces in rooms. The discussion of the paper has indicated that for large complicated rooms, it is practical to simplify model geometry, while for small rooms, it will be benefited from the calculation of detailed models. The recommendation of scattering coefficient definition in various cases is as follows. If the geometry of a model has been simplified, the coefficient of the substituted surface should be a bit smaller than that determined by experience and usually between 0.3 and 0.8. If all the detail of a room is considered, each surface can be thought as smooth except for some special ones, therefore, the same scattering coefficient (ss) for all of these smooth surfaces can be set to be a low value from 0.01 to 0.05. As for transition order, in a general way, it should be smaller than 3 in small and simple shaped rooms and in large concert hall, it is better to set the transition order around 4.

218 6.

Acknowledgements

This work is partially supported by National Natural Science Foundation of China [no. 10404021] and the Fund for Outstanding Persons in Northwestern Poly technical University of China. References 1. Hodgson M.R., Evidence of diffuse surface reflection in rooms, J. Acoust. Soc. Am. 89(1991)765-771. 2. Dalenback B.I.,Kleiner M. and Svensson P., A macroscopic view of diffuse reflection, J. Audio Eng. Soc. 42 (1994) 793-806. 3. Lam Y.W., On the parameters controlling diffusions calculation in a hybrid computer model for room acoustics prediction, Proc. IOA. 16 (1994) 537-544. 4. Antonio P.D. and Cox T.J., Diffusor application in rooms, Applied Acoustics 60 (2000) 112-143. 5. Vorlander M., International round robin on room acoustical computer simulations, ICA'95 (1995), pp. 689. 6. Lam Y.W., The dependence of diffusion parameters in a room acoustics prediction model on auditorium size and shapes, J. Acoust. Soc. Am.100 (1996) 2193-2203. 7. Mommertz E. and Vorlander M., Measurement of scattering coefficients of surfaces in the reverberation chamber and in the free field, ICA '95 (1995) 577-590. 8. Christensen C.L., Manual of ODEON room acoustics program version 7.0, 2003. 9. Rindel J.H., Acoustic design of reflectors in auditoria, Proc. I.O.A. 14 (1992) 119128. 10. Christensen C.L. and Rindel J.H., Predicting acoustics in classrooms, Inter-noise 2005 (2005) 1782.

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