3-D parallel shot-gather prestack depth migration

Transcription

1 3-D parallel shot-gather prestack depth migration Wensheng Zhang 1, Guanquan Zhang 1 and Xingfu Cui 2 1 LSSC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, CAS, China 2 Institute of Geophysics, Academy of Oil Prospecting and Exploitation, China. Abstract 3-D shot-gather prestack depth migration based on wave equation is an important method because of its good capability of imaging complex subsurface structures. In this paper, 3-D parallel shot-gather prestack depth migration with hybrid is developed and implemented. The shot-gather migration has high natural parallel feature because different processor can process shot-gather independently. Such spatial parallelism with message passing interface(mpi) can improve computational efficiency much more. After the basic formulae of wavefield extrapolation of the hybrid method are derived, two numerical examples are completed. One is as the test of our migration program with the SEG/EAEG benchmark model, and the excellent imaging result is offered. The other is the migration example for a 3-D real data. Both the imaging results demonstrate the correctness and effectiveness of our parallel program of 3-D shot-gather migration. It has shown that our parallel program has good potential values to process real data on PC cluster. keywords: 3-D prestack depth migration, shot-gather, hybrid method, MPI parallel, PC cluster. 1

2 1 Introduction 3-D prestack depth migration is an important and effective tool in identifying oil and gas structures. There are Kirchhoff and non non-kirchhoff two category methods. Kirchhoff approach is the most commonly used method for prestack depth migration due to its high efficiency and flexibility in handing 3-D data geometry. Kirchhoff method can be employed to efficiently migrate data sets with uneven spatial sampling and data sets that are subsets of the complete prestack data, such as common-offset cubes and common-azimuth cubes. In principle, downward continuation of 3-D prestack data should be carried out in the 5-D space of full 3-D prestack geometry (recording time, source surface location, and receiver surface location). For recursive methods, most of the computations are wasted on propagating components of the wavefield that either are equal to zero or do not contribute to the final image. These potential limitations of recursive methods have led the industry to adapt almost exclusively Kirchhoff method for 3-D prestack migration. The migration accuracy of this approach, however, relies on high-frequency asymptotic ray approximation. More accurate ray-tracing algorithm that takes into multi-pathing and the correct amplitudes of each arrival is computationally expensive. The non-kirchhoff migration, based on full (two-way) wave equation, is capable of obtaining wave solutions. All reflections including multiples and arbitrary steep reflectors can be imaged, but its imaging precision is inferior to that of one-way wave equation in the case of complex velocity media. The methods based on one-way wave extrapolation which are derived from the full wave equation and not an asymptotic solution based on a high-frequency assumption, can handle large lateral velocity variation and steep dipping events. Many people have developed numerical algorithms of this kind of non-kirchhoff method. Typically, there are the phase-shift method (Gazdag, 1978), the method of phase-shift plus interpolation (Gazdag and Sguazzero, 1984), the split-step method (Stoffa, 1990), the Fourier finitedifference method (Ristow and Rühl, 1995), the phase-screen method (Wu and Jin, 2

3 1998). All these methods can be used in the 3-D prestack depth migration. Methods implemented in frequency-wavenumber domain (Stolt, 1978) can handle steep dips up to 90 in principle and be very efficient because of high efficiency of fast Fourier transform. However, it can not handle lateral velocity variations. Phase-shift method can adapt the media with velocity only varies depth. The split-step Fourier migration can handle lateral migration dip. In order to improve migration dip, several extensions of the split-step Fourier (SSF) method have been proposed, such as the SSF method with multiple reference velocities and Fourier finite-difference (FFD) method, to improve the accuracy under strong velocity contrast. The FFD method is the method of cascading SSF method and finite-difference method for downward continuation to be applied to large lateral velocity variations and high steep reflectors. Plane wave migration or slant stack migration is another approximate wave equation method. Ottolini and Claerbout (1984) presented a migration method for common midpoint slant stack seismic data in constant velocity media. The common midpoint data are processed as plane wave section and the migration is performed on the individual common midpoint plane wave sections. However, the plane wave migration differs from conventional slant stack migration in the sense that the controlled illumination approach can control the complexity of the incident wavefield at the target level leads to a high image(rietveld and Berkhout, 1994). Migration by summing several shots together and imaging with a single migration is another migration algorithm with wavefield synthesizing method. This needs a phase-encoding process firstly proposed by Louis et al. (Louis A.R., Dennis C.G, et al., 2000). And in the reference of Zhang (Zhang w.,2004), the results of 3-D plane wave prestack depth migration for the SEG/EAEG models are presented. The double square-root equations provide an alternative domain for the development of migration methods in midpoint-offset coordinates. For these methods of wave equation migration, the wavefield extrapolation can be done with the single-square root or the double-square root(dsr). The DSR prestack migration equation, can be used 3

4 to image media with strong velocity variations using a phase-shift plus interpolation or split-step correction. Popovici (Popovici A.M., 1996) proposed a simple split-step modification to the double-square root prestack migration, which leads to a powerful migration algorithm that can handle strong lateral velocity variations and produces very good images for Marmousi data. However, 3-D DSR prestack depth migration is not computationally efficient because of too many Fourier transforms in computations. Biondi (Biondi A. and Palacharla G., 1997) proposed the 3-D common-azimuthal DSR migration to save computational cost because the computations to the be carried out in the original 4-D space instead of the 5-D space that the application of the conventional full 3-D prestack downward-continuation operator would require. Wave equation migration currently become practical thanks to the advent of massively parallel computers based on commodity PC hardware and advancement of accurate approximation for one-way wavefield extrapolation. In this paper, 3-D shot-gather parallel migration based on wavefield extrapolation of the hybrid method is developed and implemented. The parallel implementation uses the MPI library for platform portability and a spatial decomposition for efficiency. The MPI computation is improved focusing on real data processing based on our serial 3-D migration program before. Numerical calculations have proven the good ability of our parallel code. It can be expected that 3-D parallel shot-gather prestack depth migration on PC cluster is a very practical imaging method. 2 Wavefield extrapolation equation The 3-D acoustic wave equation can be represented by the following equation 1 2 p v 2 t 2 p 2 x 2 p 2 y 2 p 2 z = 0 (1) 2 where t denotes time, (x, y, z) are space variables, z is the privileged direction, p(t, x, y, z) is the acoustic wavefields, v(x, y, z) is the velocity of medium. Introduce the following 4

5 Fourier transform of t, x and y for p P (ω, k x, k y, z) = 1 (2π) 3 e i(ωt kxx kyy) p(t, x, y, z)dxdydt (2) and make Fourier transform of these variables for p in equation (1), wet get the following equation in the frequency-wavenumber domain where ( 2 z 2 + k2 z)p = ( z + ik z)( z ik z)p = 0 (3) k z = ω v 1 v2 ω 2 (k2 x + k 2 y) (4) is the square-root operator, k x and k y are the wavenumber, ω is the angular frequency. Define the downgoing wavefield D(k x, k y, z, ω) = ( z ik z)p (5) and upcoming wavefield U(k x, k y, z, ω) = ( z + ik z)p (6) then, we get the one-way wave equation of downgoing wave D z = ik zd = ω 1 v2 v ω 2 (k2 x + ky)d 2 (7) and upcoming wave U z = +ik zu = + ω 1 v2 v ω 2 (k2 x + ky)u 2 (8) respectively. Transform equations (7) and (8) into the frequency-space domain, we get their corresponding expressions P z = ±iω v 1 v2 ω ( 2 2 x + 2 )P (9) 2 y2 where the + symbol before the square-root operator represents the upcoming wave P = U, and symbol represents the downgoing wave P = D. The derivation above 5

6 is accurate only for the case of constant velocity, however, it is well known that the resulting one-way wave equation (9) are used for the variable velocity. In fact, for the case of variable velocity, the wavefield splitting procedure above is only correct strictly in case of neglecting multiple reflections (Zhang G., 1993). A major difficulty with equation (9) is that it corresponds to a non local pseudodifferential equation and therefore is not very tractable from a computational point of view. There are several ways to approximate the square-root operator. The most familiar ones are the rational fraction approximations and Padé expansion. In 2001, Wnag (Wang Y., 2001) proposed an accurate finite solution of the 3-D paraxial wave equation. Here, we adopt a new scheme. With the identity formulae (Zhang G., 1993) 1 t2 = 1 1 π 1 the square-root operator k z can be written as k z = ω v [1 1 π S 2 Then by use of Gauss integral formulae t 1 S 2 ds (10) 1 t 2 S2 (v x )2 + (v y )2 ]ds. (11) ω 2 (Sv x )2 (Sv )2 y 1 π S2 f(s)ds 1 2 m C m,l f(s m,l ) (12) l=1 where lπ S m,l = cos( m + 1 ), C m,l = l lπ m + 1 sin2 ( m + 1 ) (13) when m = 1, S 1,1 = 0, C 1,1 = 1, insert the expression (11)-(13) into (9), we get the 15 one-way wave equation. And when m = 2, S 2,1 = S 2,2 = 1, C 2 2,1 = C 2,2 = 1, we get 2 the 45 equation. Because of large lateral velocity variations, we introduce a reference velocity v 0 (z) to handle lateral variable velocity like that in the reference of Ristow and Rühl in 1995, then square-root k z operator can be approximated as k z A 1 + A 2 + A 3 (14) 6

7 Therefore, the one-way wavefield extrapolation equation for downgoing wave D and upgoing wave U is P z = ±i[a 1 + A 2 + A 3 ]P (15) where A 1, A 2 and A 3 are the following expressions respectively ω A 1 = 2 kx 2 ky, 2 A 2 = ω v (1 v ), v 0 v 2 0 A 3 = b v ( ) ω x 2 y a v2 ( ) ω 2 x 2 y 2 One notes that A 1 has been written as its form in the frequency-wavenumber domain, A 2 and A 3 in the frequency-space domain. The parameters a and b are optimal coefficients. One of the typical selections can be found in the reference of Ristow and Rühl, i.e, a = 1(1 v 0 2 v ), b = 1[( v 0 4 v ) 2 + v 0 v + 1]. The wavefield extrapolation are three steps. The first step is the phase-shift correction in the frequency-wavenumber domain. The (16) second step is the time-shift computation in the frequency-space domain. And the third step is the finite-difference migration also in the frequency-space domain, which can improve migration dips. The one-way equation corresponds to the finite-difference migration operator A 3 is P z = ±i b v ω ( 2 x y 2 ) 1 + a v2 ω 2 ( 2 x x y 2 ) P (17) Equation (7) yields the following alternatively directional finite-difference scheme [1 + (α 1 iβ 1 )δ 2 x]p n+1/2 kl [1 + (α 2 iβ 2 )δ 2 y]p n+1 kl = [1 + (α 1 + iβ 1 )δ 2 x]p n kl = [1 + (α 2 + iβ 2 )δ 2 y]p n+1/2 kl (18) where P n kl represents P (ω, k x, l y, n z), δ2 x and δ 2 y are the second-order central differences with respect to x and y respectively. And x, y and z are the spatial steps of x, y and z respectively. The coefficients α 1, α 2, β 1 and β 2 can be written as α 1 = av2 ω 2 x, α 2 2 = av2 ω 2 y, 2 β 1 = b zv 2ω x 2, β 2 = b zv 2ω y 2. (19) The ADI schemes will cause numerical anisotropic errors. Fortunately, several authors proposed the multi-way methods to decrease the azimuthal errors (Collina and Joly, 7

8 1995; Ristow and Rühl, 1997). And in Wang s method (Wang Y., 2001), the improved non-anisotropic solution is obtained at each extrapolation level by interpolation between the approximated solution from the ADI extrapolation and the wavefield before extrapolation, which can save computational cost. Usually, the Crank-Nicolson implicit difference scheme (18) is used because of its unconditional stability. For the stable of explicit depth extrapolation, the details, for example, can be found in the reference of Hale (Hale D., 1999). The following are the wavefield extrapolation procedures of the hybrid method above. 1. Make fast Fourier transform (FFT) with respect to t for known the shot-gather data p(t, x, y, z), i.e., p 1 (ω, x, y, 0)] = 1 2π p(t, x, y, 0)e iωt dt. (20) 2. Make FFT with respect to x and y for the data p 1 (ω, x, y, z) at the extrapolation depth z, i.e, F F T x,y [p(ω, x, y, z)] = 1 (2π) 2 p 1 (ω, x, y, z)e i(kxx+kyy) dxdy. (21) 3. Compute the phase-shift correction P (ω, k x, k y, z + z) = F F T x,y [p 1 (ω, x, y, z)]e A 1 z (22) 4. Transform the inverse FFT with respect to k x and k y for the resulting data P (ω, k x, k y, z+ z), i.e., IF F T kk,k y [P (ω, k x, k y, z + z)]. 5. Compute the time-shift correction p 1 (ω, x, y, z + z) = e iω( 1 v 1 v 0 ) IF F T kk,k y [P (ω, k x, k y, z + z)] (23) 6. Finite-difference migration with the data p 1 (ω, x, y, z + z) in the frequency-space domain with equation (18) and (19). Then return the step 2 till the maximum depth. The subsurface image can be obtained by extrapolating the shot wavefield D and receiver wavefield U simultaneously, and then applying the imaging condition I(x, y, z) = ω UD (24) 8

9 at each image point. Another imaging condition is for the reflection coefficient R which can be written as R(x, y, z) = ω UD ε + DD (25) where D is the complex conjugate of D, and a small positive number ε is added to the denominator to prevent unstability. However, the imaging condition (25) may produce a little noise in images than the condition (24). So the imaging condition (24) is preferred. The final whole images Ω are obtained by summing all the partial images and all the spatial subdomain Ω i Ω(x, y, z) = Ω i I(x, y, z) (26) shots 3 MPI implementation The most efficient parallel programs are ones which attempt to minimize the communication between processors while still requiring each processor to accomplish basically the same amount of work. Spatial parallelism rather than frequency parallelism in 3-D shot-gather migration tends to scale well. In this parallelism, the whole space domain is split among the processors so that each processor solves its problem in its own subdomain. Each subdomain has its own shot-gather data, and the wavefield extrapolation for downgoing wave D and upcoming wave U can be accomplished independently and the images of this subdomain can be obtained. The communications between processors are set at the begin of and the end of the computation. At the begin, the velocity model for migration of each subdomain is sent to its corresponding processor from the main node and then every processor does the same calculations. After images of each subdomain is yielded, they are sent back to the main node and produces the whole imaging results. One convenient spatial splitting scheme is according to the scope of each shot line. The flow chart of the parallelism is shown in figure 1 and figure 2. The procedure is as follows: after entering MPI system, every processor will receive its corresponding shot-gather data and velocity data, then do migration process and 9

10 yield partial images. These partial images are stored on all processors and finally are gathered together to yield finial image on the main processor. Figure 1. The flow chart of MPI parallelism for the shot-gather prestack migration. 10

11 Figure 2. The Flow chart of shot-gather prestack depth migration. 4 Numerical calculations The first calculation is for the SEG/EAEG subsalt model. The date used here is the shot-gather data. It has 50 shot lines and each line has 68 6 seismic traces. The line space is 160m and the shot space is 80m. The spatial steps of x, y and z are 40m, 40m and 20m respectively. The record length is 4992s with 8ms time sampling. And no any pre-processing such as noise suppressing or multiples removing is done before 11

12 migration. In the spatial parallelism, each processor is set to process partial or whole shot line. We present one most typical slice of 3-D shot-profile imaging result. Figure 3 is a vertical slice of the known 3-D velocity model at x = 5500m, and figure 4 is the migration slice at the same position. In both figures, the x direction is chosen as the inline direction, and the y direction is chosen as the crossline direction. From figure 4, the images of the salt model is very clear. The second numerical example is for a real 3-D data. There are shots and receivers totally. The maximal offset is m and the minimal offset is 270.4m. The trace space is 50m. The trace length is 6s with 1ms time sampling rate. The line space is 250m with 100m shot space. The stacking number is 96(16 6). Figure 5 is the builded velocity model before migration. Figures 7, 8, 9, 10 are the slices of 3-D shot-gather migration for inline No.27, No.32, No.41, No.123 and No.127 respectively. From these figures we can know the main subsurface structures in this area clearly. 5 Conclusions As the PC cluster becomes an economical and practical parallel tool, 3-D parallel prestack depth migration based on wave equation is playing an important role on seismic subsurface structures imaging. The spatial parallelism of 3-D shot-gather prestack depth migration is developed and implemented. Many ways can be adopted to parallelize the serial code. We choose to parallelize prestack depth migration code at the shots loop in order to minimize inter-processor communication and fully utilize the capability of every processor. Such spatial parallelism has little communications between processors and high parallel speedup and parallel efficiency can be expected. It is also an economical and practical algorithm because it divides large three dimensional space and dada volume into small one and so only needs relative small computer memory. The new expression of the hybrid migration is derived and two numerical examples are 12

13 presented. The numerical results have shown the good ability of our parallel program. Along with the good velocity building tool, it can image subsurface structures precisely. 6 Acknowledgements This research is supported financially by the Major State Basic Research Program of Peoples s Republic of China (No. G ), National Key Foundation Project (No ) and Director Foundation of ICMSEC. References [1] Alexander Mihai Popovici, 1996, Prestack migration by split-step DSR: Geophysics, 61(5), [2] Biondi, B., and Gopal Palacharla, 1996, 3-D prestack migration of commonazimuth data: Geophysics, 61(6), [3] Collina, F. and Joly, P. Splitting of operators, alternated directions, and paraxial approximations for the three-dimensional wave equation: SIAM J. Sci. Comput., 1995, 16: [4] Gazdag, J., 1978, Wave equation migration with the phase method: Geophysics, 43(1), [5] Gazdag, J., and Sguazzero, P., 1984, Migration for seismic data by phase shift plus interpolation: Geophysics, 49(1), [6] Hale, D., 1999, Stable explicit depth extrapolation of seismic wave fields: Geophysics, 56(6), [7] Louis A. Romero, Dennis C. Ghiglia, Curtis C. Ober and Scott A. Mortor., 2000, Phase encoding of shot records in prestack migration: Geophysics, 65(2),

14 [8] Ottolini, R., and Claerbout, J.F., 1984, The migration of common midpoint slant stack: Geophysics, 49, [9] Ristow, D. and Ruhl, T., 1997, 3-D implicit finite-difference migration by multiway splitting: Geophysics, 62(2): [10] Wang, Y., 2001, ADI plus interpolation: accurate finite-difference solution to 3-D paraxial wave equation: Geophysical Prospecting, 49, [11] Rietveld, W.E.A., Berkhout, A.J., 1994, Prestack depth migration by means of controlled illumination: Geophysics, 59(5), [12] Ristow, D., and Rühl, T., 1995, Fourier finite-difference migration: Geophysics, 59(2), [13] Wu, R., and Jin, S., 1998, Window GSP (Generalized Screen Propagators) migration applied to SEG-EAEG salt model data. 68th Ann. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstracts. [14] Stoffa, P. L., Forkema, J. T., de Luna Freire, et al., 1990, Split-step Fourier migration: Geophysics, 55(4), [15] Stolt, R. H., 1978, Migration by Fourier transform: Geophysics, 43(1), [16] Zhang, W., 2004, 3-D prestack depth migration with planewave synthesizing method. 74th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, October, Denver, USA. [17] Zhang, G., 1993, System of coupled equations for downgoing and upcoming waves: Acta Math. Appl. Sinica (in Cinese), 16(2),

15 0 y/km z/km Figure 3. The slice of 3-D SEG/EAEG salt velocity model. The position is at x = 5500m, a typical one. 0 y/km z/km Figure 4. The slice of 3-D shot-profile migration result at the same position with figure 1. The images of the salt body is very clear 15

16 Figure 5. The builded 3-D velocity model for a real data migration. Figure 6. The slice of 3-D shot-profile migration of inline No

17 Figure 7 The slice of 3-D shot-profile migration of inline No.32. Figure 8. The slice of 3-D shot-profile migration of inline No.41. Figure 9. The slice of 3-D shot-profile migration of inline No.123. Figure 10. The slice of 3-D shot-profile migration of inline No