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Prestack depth migration for seismic reflection data is commonly used tool for imaging complex geological structures such as salt domes, faults, thrust belts, and stratigraphic structures. Phase shift plus interpolation (PSPI) algorithm is a useful tool to directly solve a wave equation and the results have natural properties of the wave equation. Amplitude and phase characteristics, in particular, are better preserved. The PSPI algorithm is widely used in hydrocarbon exploration because of its simplicity, efficiency, and reduced efforts for computation. However, meaningful depth image of 3D subsurface requires parallel computing to handle heavy computing time and great amount of input data. We implemented a parallelized version of 3D PSPI for prestack depth migration using Open-Multi-Processing (Open MP) library. We verified its performance through applications to 3D SEG/EAGE salt model with a small scale Linux cluster. Phase-shift was performed in the vertical and horizontal directions, respectively, and then interpolated at each node. This gave a single image gather according to shot gather. After summation of each single image gather, we got a 3D stacked image in the depth domain. The numerical model example shows good agree- ment with the original geological model.

Seismic migration is a process to turn the reflectors on the stack image into true geological interface. It can be divided ray- and wave equation-based method. Seismic migration based on ray methods such as Kirchhoff (Gray and May, 1994 [

The PSPI method in isotropy media, presented by Gazdag and Sguazzero (1984) [

where u = u(x, y, z, t) is the pressure field and v(x, y, z) is a velocity model of geological media. The double Fourier series of the pressure field is

After taking 2^{nd} partial derivative Equation (2) with respect to x, y and t then substitution of Equation (2) into Equation (1) give

where

Equation (3) shows the second-order ordinary differential equation in the wave number-frequency domain (k_{x}, k_{y}, ω) of homogeneous acoustic wave equation. Though Equation (3) has two characteristic solutions relating the field at the depth z, we can choose only one characteristic solution for the following downward displacement of the signals in reverse time

A solution for the one-way wave equation

which is the basic tool used by other migration methods to approximate one-way wave propagation in the space- frequency domain (x, y, ω). In order to map the solution

Equations (4), (5) and (7) are the basis of the phase shift migration. Since we cannot directly apply the phase shift to the imaging condition in the case of lateral velocity variations, the phase shift formula is divided into vertical and horizontal components. Then the formula is modified to handle wave propagation inside the layer z + Δz that has a laterally variable velocity field. The first phase shift for the vertically traveling waves is

which k = ω/v and v = v(x, y, z). The second phase shift for the horizontal components with a reference velocity v^{j}, one of

where

for all points (x, y, z) with

When we parallelize a problem, we must first decompose the problem into smaller problems and then assign these problems to processors to be worked on simultaneously. There are methods to decompose a problem, domain decomposition and functional decomposition. In the case of the domain decomposition, which entails data parallelization, data are divided into pieces of approximately the same size and then mapped to different processors. Each processor works on a portion of the assigned data and needs to communicate with the other processors to exchange data. The Finite Difference Method (FDM) is a good example of the domain decomposition method. Functional decomposition, which is also known as task parallelization, decomposes the problem into a large number of tasks and these tasks are assigned to all processors. The tasks are allocated to a group of slave processes by a master process. Matrix-vector multiplication is a good example of functional decomposition. The PSPI algorithm is intrinsically data parallel according to decoupling in the frequency domain, and depth extrapolation in each step consists of concurrent calculations. The imaging condition at each depth requires the sum over frequency of the results of the depth extrapolation. Since the PSPI has a data-parallel nature, PSPI code is naturally structured for efficiency on the Linux cluster. In the 3D PSPI, each node performs wavefield extrapolation and interpolation for lateral velocity variation for a single image and then sends this to the master node. The migrated image is the result of the summation of all single images. In

We conducted a numerical model test to verify prestack depth migration by the 3D PSPI. We applied it to the 3- D SEG/EAGE salt velocity model. The dimensions of the geological model are 10.8 km × 10.4 km × 4.2 km. The velocity range is 1500 m/s to 4482 m/s.

MPI_init MPI_Common_size MPI_Comm_rank MPI_BCat (shot gathers, velocity data) for all tasks simultaneously for every shot -1^{st} phase shift -2^{nd} phase shift -interpolation -single image gather MPI_Reduce (common image gathers) MPI_Finalize |
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shot interval is 80 m and there are 96 shots to each survey line. A shot gather thus consists of 544 traces. The total number of survey lines is 50, with 160 m spacing. The sampling interval is 8 ms for a recording time of 5 s and the source frequency is 20 Hz. A common shot gather at the 30^{th }shot point is shown in

The prestack imaging by 3D PSPI entails extrapolation of wavefields in the wave number-frequency domain which are recorded on the surface, followed by finding wavefields at t = 0. A phase-shift was performed in the vertical and horizontal directions, respectively, and then interpolated at each node. This gave a single image gather according to the shot gather. After summation of every single image gather, we got a stacked image in depth domain. The results of wavefields extrapolation are shown in ^{nd} image is the resulting subsurface image for inline 7760 m, which shows some portions of the reflectors, the saltbody, and the reflectors of the subsalt. The 3^{rd} and 4^{th} images are the result at inline 9360 m and 10360 m. ^{nd} and 3^{rd} images are the resulting subsurface image for xline 5320 m and 7540 m, which show some portions of the reflectors, the saltbody, and the reflectors of the subsalt. The 4^{th} is the result at inline 8640 m. ^{nd} and 3^{rd} results at 1000 m and 1040 m show the saltbody, the reflectors, and thefault zone, respectively. The bottom image is the result at zline 4000 m.

Since 3D PSPI migration is efficient and requires less computation than RTM, it is a suitable tool for prestack depth migration, which requires heavy data input-output and huge computing time. We implemented 3D prestack depth migration on PSPI algorithm using Open MP and verified its performance on the synthetic 3D SEG/EAGE salt velocity model. The numerical model test shows that the results are in good agreement with the original geological model. The outline of saltbody is well imaged, but some geological interfaces of the lower parts of saltbody are poor. Though the numerical test shows reasonable results, for further study we need to perform field data application to focus on developing a more elaborate velocity model. And, it also needs to develop an algorithm using GPU codes for fast more computation.

This work was supported by the Energy Efficiency & Resources Core Technology Program of the Korea Institute of Energy Technology Evaluation and Planning(KETEP), granted financial resource from by the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20132510100060).

Seonghyung Jang,Taeyoun Kim, (2016) Prestack Depth Migration by a Parallel 3D PSPI. International Journal of Geosciences,07,904-914. doi: 10.4236/ijg.2016.77067