Scientific Journal of Earth Science June 2014, Volume 4, Issue 2, PP.67-74
A Generalized Optimal 17-point Scheme for Frequency-domain Scalar Wave Equation Xiangde Tang 1, 2#, Hong Liu 1 1. Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China 2. University of Chinese Academy of Sciences, Beijing 100029, China #
Email: tangxiangde@mail.iggcas.ac.cn
Abstract Frequency-domain modeling is the basis of frequency-domain full waveform inversion. The rotated optimal 9-point scheme is an efficient algorithm for frequency-domain wave equation simulation, but this scheme fails when directional sampling intervals are different, and is only of second-order accuracy. To overcome the restriction on directional sampling intervals and low-accuracy seismic imaging of the rotated optimal 9-point, we introduce a new finite-difference algorithm, namely generalized optimal 17point scheme. Based on an average-derivative technique, the new algorithm uses a 17-point operator to approximate spatial derivatives and mass acceleration term. The coefficients can be determined by minimizing phase-velocity dispersion errors. This generalized optimal 17-point scheme applies to equal and unequal directional sampling intervals, and can be regarded as a generalization of the rotated 17-point scheme. The number of grid points per smallest wavelength is reduced to 2.4 by this scheme for equal and unequal directional sampling intervals. In order to suppress the reflection from the boundary, we apply a perfectly matched layer boundary condition. Numerical tests on complex model further confirm the feasibility of the generalized optimal 17-point scheme. Keywords: Scalar Wave Equation; Different Directional Intervals; 17-point Scheme; Frequency Domain
1 INTRODUCTION The frequency-domain approach of FWI has been shown to be efficient for several reasons (Pratt et al., 1996, 1998; Pratt, 1999; Brenders and Pratt, 2006). However, the frequency-domain finite-difference modeling technique has not gained popularity. The reason is that acceptable accuracy requires solving a large sparse system of linear equations and more grid points per wavelength than working in other domains. To overcome these serious limitations, improving the design of the finite difference scheme is one of the main ways. Pratt and Worthington (1990) developed the classical 5-point scheme for 2D frequency-domain scalar wave equation. Their scheme required 13 grid points per shortest wavelength for accurate modeling, with errors no larger than 1%. Jo et al. (1996) reduced the number of grid points per shortest wavelength to about 4 by including additional surrounding grid points for the derivative approximations, with consequent reductions of computer memory and CPU time. However, this scheme fails when directional sampling intervals are different. Chen (2012) introduced a new 9-point scheme applying to equal and unequal directional sampling intervals, and this scheme can be regarded a generalization of the rotated optimal nine-point scheme. Štekl and Pratt (1998) extended the method of Jo et al. (1996) to the elastic wave equations by introducing a 45°rotated operator within a conventional second-order scheme (Min et al., 2000). Elastodynamic finite-difference time-domain techniques moved from second-order approximations of spatial derivatives (Virieux 1984, 1986) to higher-order approximations (Dablain, 1986; Levander, 1988) with a good tradeoff between modelling accuracy and computational efficiency for the fourth-order spatial approach. Constructions of frequency-domain finite-difference schemes have followed a similar path (Hustedt et al., 2004). Shin and Sohn - 67 http://www.j-es.org
(1998) designed a 25-point operator that approximated a Laplacian operator and the number of grid points per wavelength can be reduced to 2.5. Based on a rotated coordinate system, Cao and Chen (2012) proposed one 17point scheme. This scheme is of fourth-order and reduces the number of grid points to 2.56. One drawback of the 25point and 17-point scheme is that equal directional sampling intervals are required, and in practice directional sampling intervals usually are different. Min et al. (2000) developed a 25-point optimal scheme for frequencydomain elastic modeling which reduces the number of grid points to 3.3 per shear wavelength, but their dispersion analysis was not carried out in the case of unequal directional sampling intervals. To overcome the disadvantage of requiring equal directional sampling intervals and meet the high-accuracy seismic imaging, a new scheme is introduced, called generalized optimal 17-point scheme. This new scheme imposes no restriction of equal directional sampling intervals and is of fourth-order accuracy. The resulting generalized optimal 17-point scheme reduces the number of grid points per wavelength to less than 2.4. This document is a template. An electronic copy can be downloaded from the journal website. For questions on paper guidelines, please contact the publications committee as indicated on the journal website. Information about final paper submission is available from the journal website.
2 THEORY 2.1 17-point Scheme and Its Limitations In a Cartesian coordinate system with the x-axis horizontal and positive to the right and the z-axis positive downward, the 2-D scalar wave equation with no damping in the frequency domain is given by
2 P 2 P 2 P 0, x 2 z 2 v 2 where P is the pressure wavefield, is circular frequency, and v is the velocity.
(1)
The rotated optimal 9-point scheme (Jo et al., 1996) is an efficient algorithm for frequency-domain wave equation simulation, but this scheme is only of second-order accuracy. Cao and Chen (2012) designed a 17-point scheme which is of fourth-order accuracy. 1 4 3 Pm 1, n Pm 1, n Pm, n 1 Pm, n 1 12 Pm 2, n Pm 2,n Pm ,n 2 Pm ,n 2 5Pm, n 1 a 4 1 P Pm 1, n 1 Pm 1, n 1 Pm 1, n 1 Pm 2, n 2 Pm 2, n 2 Pm 2, n 2 Pm 2, n 2 5Pm, n 2 m 1, n 1 2 3 12 a 2
(2)
2
(bPm, n c( Pm 1, n Pm 1, n Pm , n 1 Pm, n 1 ) d ( Pm 1, n 1 Pm 1, n 1 Pm 1, n 1 Pm 1, n 1 ) v2 e( Pm 2, n Pm 2, n Pm, n 2 Pm, n 2 ) f ( Pm 2, n 2 Pm 2, n 2 Pm 2, n 2 Pm 2, n 2 )) 0，
The constants a, b, c, d, e and f are weighted coefficients and b+4c+4d+4e+4f=1. However, this method has a requirement of x z , which is not always fulfilled. For example, the horizontal and vertical sampling intervals of the Marmousi model are dx=12.5 m and dz=4 m, respectively. For such a model, the 17-point scheme fails. As an effort towards improvement, we introduce a generalization of Equation (2). The generalization applies to the case of x z and x z .
2.2 A New 17-point Finite-difference Scheme Based on an average-derivative technique (Chen, 2001, 2008), we introduce a generalized optimal 17-point scheme for Equation (1)
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4 Pm1,n Pm1,n 121 Pm 2,n Pm2,n 52 Pm,n 43 Pm,n1 Pm,n1 121 Pm,n 2 Pm,n2 52 Pm,n 3 x 2 z 2
2
(bPm, n c( Pm 1, n Pm 1, n Pm, n 1 Pm, n 1 ) d ( Pm 1, n 1 Pm 1, n 1 Pm 1, n 1 Pm 1, n 1 ) v2 e( Pm 2, n Pm 2, n Pm, n 2 Pm, n 2 ) f ( Pm 2, n 2 Pm 2, n 2 Pm 2, n 2 Pm 2, n 2 )) 0,
(3)
where 1 2 ( Pm 1, n 1 Pm 1, n 1 ), 2 1 2 Pm 1, n 2 Pm 1, n ( Pm 1, n 1 Pm 1, n 1 ), 2 1 1 Pm 2, n 1 Pm 2, n ( Pm 2,n 2 Pm 2,n 2 ), 2 1 1 Pm 2, n 1 Pm 2, n ( Pm 2, n 2 Pm 2, n 2 ), 2 1 2 4 3 Pm, n 3 Pm, n 4 ( Pm, n 1 Pm, n 1 ) ( Pm, n 2 Pm, n 2 ), 2 Pm 1, n 2 Pm 1, n
(4)
and 1 2 ( Pm 1, n 1 Pm 1, n 1 ), 2 1 2 Pm, n 1 2 Pm, n 1 ( Pm 1, n 1 Pm 1, n 1 ), 2 1 1 Pm, n 2 1 Pm, n 2 ( Pm 2, n 2 Pm 2, n 2 ), 2 1 1 Pm, n 2 1 Pm, n 2 ( Pm 2, n 2 Pm 2, n 2 ), 2 1 2 4 3 Pm, n 3 Pm, n 4 ( Pm 1, n Pm 1, n ) ( Pm 2, n Pm 2, n ), 2 where i , i , b, c, d , e and f are weighted coefficients and b+4c+4d+4e+4f=1. For details, see Figures 1a and 1b. Pm, n 1 2 Pm, n 1
(a)
(5)
(b)
FIG. 1 GRIDS OF GENERALIZED OPTIMAL 17-POINT SCHEME EQUATION (3) IS TO PROVIDE A FAMILY OF APPROXIMATIONS TO THE DERIVATIVES FROM WHICH THE OPTIMIZATION APPROXIMATION CAN BE CHOSEN TO MEET OUR NEEDS. THE SCHEME IS VALID FOR x z AND ... MOREOVER, EQUATION (3) IS A SPECIAL CASE OF THE GENERALIZED OPTIMAL 17-POINT SCHEME WHEN x z ,1 2 3 1 2 3 ,4 8 15(1 1 ), 4 8 15(1 1 ). .
2.3 Determination of Weighting Coefficients and Dispersion Analysis - 69 http://www.j-es.org
To minimize grid dispersion and numerical anisotropy, we must determine the weighting coefficients that make normalized phase velocity close to unity. The normalized phase velocity is obtained from dispersion relations. We assume a uniform and infinite medium that supports a plane wave, expressed as P( x, z ) P0ei ( k x k z ) in the frequency domain. x
z
When x z , the number of grid points per wavelength G is defined with respect to the larger sampling interval. Therefore, we first consider x z . In this case, G is defined as x The normalized phase velocity can be derived 12
G E r2F , v 2 b 2c( A B) 4dAB 2e(C D ) 4 fCD 2 cos( ) A cos( ), G 2 sin( ) B cos( ), rG 4 cos( ) ), . C cos( G 4 sin( ) D cos( ) rG 1 8 5 8 1 5 E 1 (C CD) 2 ( A AB) 3 (1 D) 5 4 ( B D) AB CD D, 6 3 2 3 6 2 1 8 5 8 1 5 F 1 ( D CD) 2 ( B AB) 3 (1 C ) 5 4 ( A C ) AB CD C , 6 3 2 3 6 2 2 2 r x z V ph
(6)
Where kx k cos( ), k z k sin( ) .
We obtain the coefficients i , i , b, c, d, e and f by minimizing the error defined by the difference between the normalized phase velocity and unity (see Jo et al. 1996). Optimized coefficients for different r x z are listed in Table 1. TABLE 1 Optimization coefficients for different x z when x z 1
2
3
4
1
2
3
r=1.0 r=3.125
1.150847 0.009972 -0.08050 0.137957
1.150847 0.009982 c 0 0.152780
-0.08050 0.704004 d 0.039990 -0.01569
1.150847 0.887209 e -0.02859 -0.02431
1.150847 0.763481 f 0.00110 0.002598
1.150847 0.737511
r=1.0 r=3.125
1.150847 0.999997 b 0.950015 0.538545
4
The coefficients 1 , 2 , 3 and 4 exchange of 1 , 2 , 3 and 4 separately when x z . In both cases, the coefficients b, c, d, e and f are the same. Figure 2 shows normalized phase velocity curves of the conventional fourth-order 9-point scheme and our scheme based on coefficients for different r x z when x z . If we require the normalized phase velocity to be less than ±1%, the conventional 9-point scheme in the frequency domain requires G =5. For a comparable degree of accuracy, our scheme requires G = 2.4. The new scheme offers a substantial reduction (50%) in the number of grid points while maintaining the same bandwidth of the complex impedance matrix.
3 NUMERICAL EXAMPLES We tested the generalized optimal 17-point scheme with a homogeneous model whose velocity structure is shown in Figure 3a. The sampling intervals of the model are dx=5 m and dz=5 m. Horizontal and vertical samplings are - 70 http://www.j-es.org
nx=100 and nz=100. For this ratio of directional sampling intervals, the coefficients used are listed in Table 1 (r=1). And the source is located at (250 m, 20 m) with peak frequency of 20 Hz. The geophones are spread at Z=20 m horizontally. The time sampling interval is 2 ms and the recorded length is 1 s. Complex frequencies can be used to suppress the wraparound effect of the Fast Fourier transform (Mallick and Frazer, 1987). Based on this model, we implement forward modeling by our generalized optimal 17-point scheme with PML boundary (Figure 8).
FIG .2 Normalized phase velocity curves of the conventional fourth-order 9-point scheme and the generalized optimal 17point scheme for different r z x when x z
In order to verify the correctness of our scheme, we consider a more realistic model. Figure 4a shows part of the Marmousi model (nx=301, nz=301). The sampling intervals are dx=12.5 m and dz=4 m. A Ricker wavelet with peak frequency of 12 Hz is placed at (x=500 m, dz=16 m) as a source and the receivers are located at Z=16 m. The time sampling interval is 4 ms and the recorded length is 2 s. PML boundary conditions are used at four sides of the - 71 http://www.j-es.org
model. And we use compressed storage method to store the huge impendence matrix so that significantly decrease the memory consumption. The optimization coefficients are listed in Table 1 (r=3.125).
(a)
(b)
FIG . 3 TWO LAYER MODEL (A) VELOCITY STRUCTURE (B) SYNTHETIC SEISMOGRAMS COMPUTED WITH THE GENERALIZED OPTIMAL 17-POINT SCHEME.
(a)
(b)
(c)
(d)
FIG . 4 (A) PART OF THE MARMOUSI MODEL. (B) 25 HZ MONOCHROMATIC WAVEFIELD COMPUTED BY THE GENERALIZED OPTIMAL 17-POINT SCHEME. (C) T IME-DOMAIN SEISMOGRAMS COMPUTED WITH THE GENERALIZED OPTIMAL 17-POINT SCHEME. (D) SEISMOGRAMS OBTAINED BY TIME-DOMAIN METHOD.
Figure 4b is 25 Hz monochromatic wavefield computed by generalized optimal 17-point scheme with PML boundary. Figures 4c and 4d are the seismograms obtained by our method and time-domain method. Through the comparison, we can conclude that generalized optimal 17-point scheme has a good consistency with time-domain - 72 http://www.j-es.org
finite difference method on arrival time and lineups position. For the Marmousi model, the rotated-coordinate 17point scheme cannot be applied due to the fact of, but our method still is valid due to its flexibility.
4 CONCLUSIONS The generalized optimal 17-point scheme overcomes the disadvantage of the rotated 17-point scheme by removing the requirement of equal directional sampling intervals. Fourth-order accuracy of the scheme can meet the need of high-accuracy seismic imaging. And the number of grid points required per wavelength is reduced to 2.4. The generalized optimal 17-point scheme includes the rotated 17-point scheme as a special case, and can be regarded as a generalization of the rotated 17-point scheme to the case of general directional sampling intervals. Two examples demonstrate the theoretical.
ACKNOWLEDGMENT This research was supported by the Project of National 863 Plan of China (grant No. 2012AA061202) and the Important Specific Projects (grant No. 2011ZX05008-006-50).
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AUTHORS 1
2
University of Geosciences (Wuhan) in
Geology & Geophysics, Chinese Academy of Sciences.
Xiangde Tang Graduated from China
Hong Liu Geophysicist (Research Professor), Institute of
2009 and received bachelor’s degree in geophysics. I am reading a PhD degree in Institute of geology and geophysics, Chinese Academy of Sciences. My main researches include seismic data processing, seismic waveform inversion and GPU HPC.
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