CN110895350B - GPU-based two-way wave Fourier finite difference wave field propagation method - Google Patents
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Abstract
The invention provides a GPU-based method for determining phase shift coefficients, a two-way wave Fourier finite difference wave field propagation method and a computer-readable storage medium. The method determines the phase shift coefficient by using the symmetry of the wavenumber domain, and improves the calculation efficiency of the Fourier finite difference wave field propagation algorithm by means of the FFT function of the GPU and the multithread parallel calculation of the GPU.
Description
Technical Field
The disclosure relates to the field of seismic wave numerical simulation, and in particular to a method for determining a phase shift coefficient based on a GPU (graphics processing unit) and a two-way wave Fourier finite difference wave field propagation algorithm based on the GPU, and further relates to a computer-readable storage medium.
Background
The two-pass wave seismic numerical simulation is an important means for researching the propagation characteristics of seismic waves in an underground medium, and is also an important step in the implementation process of seismic imaging (including reverse time migration, least square migration, full waveform inversion and the like). At present, the most common finite difference wave field simulation technical algorithm has a serious numerical dispersion problem and influences the frequency bandwidth of seismic simulation data. To solve the problem, scholars at home and abroad propose an optimization coefficient finite difference method, a rapid amplification method, a pseudo-analytic method and a Fourier finite difference method. The Fourier finite difference method combines Fast Fourier Transform (FFT) and finite difference algorithm, which can effectively solve the problem of numerical dispersion in seismic wave simulation and well process the problem of underground medium speed change, thereby having higher precision in seismic data numerical simulation. However, the fourier finite difference method requires one positive FFT and one inverse FFT at each step of time extension, and performs point-by-point calculation for the phase shift of each point in the three-dimensional space. The calculation amount of the three-dimensional FFT and the point-by-point phase shift calculation is large, so that the calculation efficiency of the Fourier finite difference method is low, and the popularization and the application of the method technology are restricted.
Therefore, a method for improving the computation efficiency of the finite difference fourier algorithm is needed.
Disclosure of Invention
The technical problem solved by the invention is as follows: in the prior art, the calculation efficiency of the two-way wave Fourier finite difference algorithm is low. To solve the problem, the present invention provides a method of determining a phase shift coefficient based on a GPU (graphics Processing Unit), a two-way wave fourier finite difference wave field propagation method, and a computer-readable storage medium.
According to a first aspect of the present invention, there is provided a method for determining a phase shift coefficient based on a GPU, comprising:
transforming the time-space domain wave field of each point in a first three-dimensional space preset in a time-space domain by using the FFT function of the GPU to obtain the time-wave number domain wave field of each point in a second three-dimensional space, wherein the second three-dimensional space is a corresponding space of the first three-dimensional space in the time-wave number domain;
determining a parallel domain of the second three-dimensional space;
for each point in the parallel domain, executing the following steps by utilizing a thread corresponding to the point in the GPU in parallel:
extracting all symmetrical points symmetrical to the point in the second three-dimensional space;
and determining the phase shift coefficient corresponding to the point, and taking the phase shift coefficient as the phase shift coefficient of each symmetrical point of the point.
Preferably, the phase shift coefficient is related to the average velocity of all points in the first three-dimensional space.
Preferably, expressions are used
Determining a phase shift coefficient corresponding to a point in the parallel domain, wherein,
representing the wave number, v, of the point0To representThe average speed and Δ t represent a preset time step.
Preferably, the coordinate range of the point in the parallel domain in the first direction is 0-nk1A coordinate range of 0-nk in the second direction2A coordinate range of 0-nk in the third direction3/2, wherein, nk1Representing the number of grid points, nk, of said second three-dimensional space in a first direction2Representing the number of grid points, nk, of said second three-dimensional space in a second direction3Representing a number of grid points of the second three-dimensional space in a third direction.
Preferably, the time-space domain wavefield for each point in the first three-dimensional space is FFT transformed using multiple threads in the GPU.
Preferably, the FFT of the time-space domain wavefield for each point in the first three-dimensional space is performed using a single thread in the GPU.
According to a second aspect of the present invention, there is provided a GPU-based two-way wave fourier finite difference wave field propagation method, comprising:
acquiring the spatial point speed of each point in a first three-dimensional space preset in a time-space domain at the current moment, and calculating the average speed of all points in the first three-dimensional space at the current moment according to the spatial point speed;
acquiring a time-space domain wave field of each point in a first preset three-dimensional space in a time-space domain at the current moment, and accordingly determining phase shift coefficients of all points in the first three-dimensional space by the method for determining the phase shift coefficients based on the GPU;
for each point in the first three-dimensional space, respectively performing the following steps:
obtaining a first coefficient and a second coefficient according to the average speed, the grid intervals of the first three-dimensional space in a fourth direction, a fifth direction and a sixth direction respectively, the spatial point speed of the point and a preset time step;
phase shifting the time wavenumber domain wave field of the point at the current moment by using the phase shift coefficient of the point;
performing three-dimensional space inverse Fourier transform on the time wave number domain wave field of the point after phase shift at the current moment by using the IFFT function of the GPU to obtain the time space domain wave field of the point after phase shift at the current moment;
performing space finite difference calculation on the time-space domain wave field of the point after phase shift at the current moment by using the first coefficient and the second coefficient to obtain the time-space domain wave field of the point after phase shift at the next moment, wherein the next moment is separated from the current moment by the time step length;
and obtaining the time-space domain wave field of the point at the next moment according to the time-space domain wave field of the point after phase shift at the next moment and the time-space domain wave fields of the point at the previous moment and the current moment.
Preferably, the time-wave-number-domain wavefield of each point in the first three-dimensional space is phase shifted by multiplying the phase shift coefficient and the time-wave-number-domain wavefield.
Preferably, for each point in said first three-dimensional space
By using
To obtain the point
Time-space domain wavefield at the next time instant t +. DELTA.t
Wherein the content of the first and second substances,
representing the spatio-temporal wavefield of the point phase-shifted at the next time instant t +. DELTA.t,
representing the spatio-temporal wavefield of the point at the current time instant t,
the spatio-temporal wavefield of the point at the last time instant t- Δ t, Δ t being the time step.
According to a third aspect of the present invention, there is provided a computer-readable storage medium storing a program for causing a computer to execute the above-described method of determining a phase shift coefficient based on a GPU.
Compared with the prior art, one or more embodiments in the above scheme can have the following advantages or beneficial effects:
the method for determining the phase shift coefficient based on the GPU provided by the invention utilizes the symmetry of a wave number domain to reduce the calculated amount of the phase shift coefficient to 1/8 of the existing point-by-point calculation method, and carries out wave field propagation calculation on each point in a three-dimensional space in parallel by means of an FFT function of the GPU and multiple threads of the GPU through the method for propagating the finite difference wave field of the two-way wave Fourier based on the GPU provided by the invention. Therefore, by applying the method for determining the phase shift coefficient based on the GPU and the method for transmitting the double-pass wave Fourier finite difference wave field, the time for calculating the wave field phase shift and executing positive and negative FFT (fast Fourier transform) can be effectively reduced, and the calculation efficiency of the Fourier finite difference wave field transmission algorithm is improved.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The objectives and other advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
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The scope of the present disclosure may be better understood by reading the following detailed description of exemplary embodiments in conjunction with the accompanying drawings. Wherein the included drawings are:
FIG. 1 is a flow diagram illustrating a method for GPU-based determination of phase shift coefficients in accordance with an embodiment of the present invention;
FIG. 2 is a flow chart diagram illustrating a GPU-based two-way Fourier finite difference wavefield propagation method according to an embodiment of the present invention;
FIG. 3 shows the symmetry of the frequency domain data obtained after one-dimensional FFT of the time domain data;
fig. 4 shows a second three-dimensional space partitioned according to wavenumber domain symmetry and parallel domains therein.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention clearer, embodiments of the present invention will be described in detail below with reference to the accompanying drawings and examples, so that how to apply technical means to solve technical problems and achieve a technical effect can be fully understood and implemented.
In the existing finite difference wave field simulation technology algorithm, the fourier finite difference method needs to perform one positive FFT and one negative FFT in each step of time prolongation, and performs point-by-point calculation on the phase shift of each point in a three-dimensional space. The calculation amount of the three-dimensional FFT and the point-by-point phase shift calculation is large, so that the calculation efficiency of the Fourier finite difference method is low, and the popularization and the application of the method technology are restricted.
Based on the above, the embodiment of the invention provides a method for determining a phase shift coefficient based on a GPU, a two-way wave Fourier finite difference wave field propagation method and a computer readable storage medium, aiming at improving the algorithm efficiency of a Fourier finite difference algorithm in three-dimensional calculation. Various embodiments of the present invention will be described in detail below.
Example one
The present embodiments relate to a method of determining a shift coefficient based on a GPU. Fig. 1 shows a flow diagram of a method of determining a shift coefficient based on a GPU. As shown in fig. 1, the method for determining a phase shift coefficient based on a GPU of the present embodiment mainly includes steps S101 to S103.
In step S101, the time-space domain wave field of each point in the first three-dimensional space is transformed by using the FFT function of the GPU, so as to obtain the time-wave-number domain wave field of each point in the second three-dimensional space. Here, the first three-dimensional space is a space of a time-space domain, and the second three-dimensional space is a space of a time-wavenumber domain, and the second three-dimensional space is a corresponding space of the first three-dimensional space in the time-wavenumber domain.
Specifically, when the FFT function of the GPU is used to transform the wave field, it may be preferable that one point corresponds to one thread, i.e., a multi-thread parallel execution transformation operation, or that multiple points correspond to one thread, i.e., a single thread sequential execution transformation operation.
In step S102, a parallel domain of the second three-dimensional space is determined.
Specifically, the method for determining the phase shift coefficient provided by the embodiment of the invention uses a method for transforming a time-space domain wave field into a time-wavenumber domain wave field by using three-dimensional FFT. As is well known, because the FFT has symmetry, the FFT-processed data has symmetry in the time-wavenumber domain, as shown in the waveform diagram of the time-wavenumber domain data obtained by performing the one-dimensional FFT on the time-space domain data in fig. 2. The three-dimensional space domain FFT is consistent with the one-dimensional FFT principle, only the one-dimensional time domain is changed into the three-dimensional space domain, and the three-dimensional space domain is expanded from one dimension to three dimensions, so that the three-dimensional FFT also has symmetry.
The symmetry of the three-dimensional space of the time-wavenumber domain is explained below with reference to fig. 3. As shown in fig. 3, the embodiment of the present invention divides the second three-dimensional space into 8 symmetrical portions according to symmetry, wherein 1 portion is shown as a gray portion in fig. 3. The grey part represents the parallel domain of the present embodiment. In this embodiment, the second three-dimensional space is in a first direction (i.e., k)1Direction) is nk1The second three-dimensional space being in a second direction (i.e., k)2Direction) is nk2The second three-dimensional space is in a third direction (i.e., k)3Direction) is nk3. In this case, the coordinate range of the point in the parallel domain in the first direction is 0-nk1A coordinate range of 0-nk in the second direction2A coordinate range of 0-nk in the third direction3/2. In the present embodiment, it is preferred that,
representing points in the spatio-temporal domain
The corresponding wavenumber in the temporal wavenumber domain. As shown by the solid black dots in figure 3,
representing a point in the parallel domain (which may be referred to as a fiducial point). There are 7 points in the remaining 7 portions of the second three-dimensional space, respectively, that are symmetrical to the reference point. These 7 points are referred to as symmetrical points of the reference points in the parallel domain, as indicated by the open black dots in fig. 3.
In step S103, the phase shift coefficients of all points in the second three-dimensional space are determined using the symmetry. The specific method comprises the following steps: for each point in the parallel domain, executing the following steps by utilizing a thread corresponding to the point in the GPU in parallel: extracting all symmetrical points of the middle points of the parallel domain in the second three-dimensional space; and determining the phase shift coefficient of the point in the parallel domain, and taking the phase shift coefficient as the phase shift coefficient of all symmetrical points of the point.
Specifically, in the FFT, since the phase shift coefficient of the reference point in the parallel domain is the same as the phase shift coefficient of the symmetric point corresponding thereto, that is, the reference point and the symmetric point share the phase shift coefficient, it is only necessary to calculate the phase shift coefficient of the reference point, and then share the phase shift coefficient to the remaining 7 symmetric points.
It can be seen that, in the technical scheme provided by this embodiment, after the phase shift coefficient of the reference point in the parallel domain is determined, the phase shift coefficient of the reference point is shared with all the symmetric points of the reference point in the second three-dimensional space, so that the calculation step of the phase shift coefficients of the symmetric points is omitted, and the calculation time is saved. The method for sharing the phase shift coefficient is provided by the embodiment, the phase shift coefficient calculation is only performed once at every 8 points in the three-dimensional space, and the calculation amount of the phase shift coefficient can be reduced to 1/8 of the existing point-by-point calculation method, so that the algorithm efficiency of the Fourier finite difference algorithm in the three-dimensional calculation is greatly improved.
In a preferred embodiment of the invention, expressions are used
Determining in parallel domainA reference point
The corresponding phase shift coefficient. Where Δ t denotes a predetermined time step, v0The average speed of each point in the first three-dimensional space at the current moment is obtained. According to the space point velocity v of each point in the first three-dimensional space at the current moment1，v2，…，vnxyzV is obtained by calculation0. The specific expression is as follows:
wherein nxyz is assumed to represent the total number of grid points in the first three-dimensional space.
It can be seen that, the average speed of each spatial point speed is introduced as the background speed in the method of the present embodiment, and compared with a method in which the root mean square speed of each spatial point speed is used as the background speed in the conventional method, the error between the background speed and the spatial point speed is reduced, thereby improving the algorithm precision speed of the finite difference fourier algorithm during the three-dimensional calculation.
Example two
The embodiment relates to a GPU-based two-way wave Fourier finite difference wave field propagation method. Fig. 2 shows a flowchart of the GPU-based two-way wave fourier finite difference wavefield propagation method according to the present embodiment. As shown in fig. 2, the GPU-based two-way wave fourier finite difference wave field propagation method of this embodiment mainly includes steps S201 to S210.
In step S201, a three-dimensional mesh number nx of a first three-dimensional space preset in a time-space domain is acquired1,nx2,nx3Space grid spacing Δ x1,△x2,△x3And the space point velocity v of each point in the first three-dimensional space at the current moment1，v2，…，vnxyzAnd calculating the average velocity v according to the formula (1)0。
Representing coordinates of a point in a first three-dimensional space. nx1Representing the first three-dimensional space in a fourth direction (i.e., x)1Direction), nx2Representing the first three-dimensional space in a fifth direction (i.e., x)2Direction), nx3Representing said first three-dimensional space (i.e. x)3Direction) of the grid points in the sixth direction.
In step S202, phase shift coefficients for all points in the first three-dimensional space are determined.
Obtaining the time-space domain wave field of each point in the first three-dimensional space preset in the time-space domain at the current moment
And accordingly, according to steps S101 to S103 of the GPU-based method for determining phase shift coefficients according to the first embodiment of the present invention, the phase shift coefficients of all points in space are determined.
Next, the wave field of each point in the first three-dimensional space at each time is determined through steps S203 to S210. It should be noted that steps S203 to S206 are performed by using multiple threads of the GPU, where each thread corresponds to a point in the first three-dimensional space. Specifically, the method comprises the following steps:
in step S203, for each point in the first three-dimensional space, a temporal-spatial domain wavefield is generated for each point at the current time at the shot point using GPU multithreading
The source wavelet is loaded.
In step S204, multithreading according to v using GPU0、△x1、△x2、△x3Obtaining a first coefficient a and a second coefficient b of each point by the space point velocity v of each point and a preset time step length delta tn。
Specifically, a first coefficient a and a second coefficient b for each pointnTo average velocity v0Introducing the background velocity into a two-way wave equation corresponding to each point and performing Taylor expansionThus obtaining the product. According to the principle of the two-pass wave fourier finite difference algorithm, the two-pass wave equation can be expressed as:
where t represents the current instant of wavefield propagation,
representing the time-wavenumber domain wavefield.
Thereafter, v is introduced in expression (2)0As background velocity. The following expression can be obtained:
further, by performing Taylor expansion on the formula (3), a first coefficient a and a second coefficient b of each point can be obtainednExpression (c):
calculating a first coefficient a and a second coefficient b of each point according to the formula (5) and the formula (6)n。
In step S205, the phase shift coefficients of each point in the three-dimensional space are used to perform a time-wave-number domain wave field of each point at the current time by using multiple threads of the GPU
A phase shift is performed. In particular by phase shifting pointsAnd phase shifting the time wave number domain wave field of each point in the first three-dimensional space by multiplying the coefficient by the current time wave number domain wave field of each point. The specific expression is as follows:
wherein the content of the first and second substances,
the time-space domain wave field after phase shift of each point in the first three-dimensional space.
In step S206, IFFT function pairs in multithreading with GPUs
Performing three-dimensional space inverse Fourier transform to obtain the time-space domain wave field of each point after phase shift at the current moment
In step S207, a first coefficient a and a second coefficient b are usednTo pair
Performing space finite difference calculation to obtain the time-space domain wave field of each point in the three-dimensional space after phase shift at the next moment
The next time is separated from the current time by Δ t. Specifically, the finite difference format thereof can be written according to equations (4), (5) and (6):
then, the time-space domain wave field of each point in the first three-dimensional space after phase shift at the next moment is obtained by calculation according to the formula (8)
In step S208, according to
Time-space domain wave field of each point in three-dimensional space at last moment t-delta t
And the time-space domain wave field at the current time t
For each point in the first three-dimensional space, utilizing
Obtaining the time-space domain wave field of each point at the next moment t plus delta t
In step S209, the wave field at each time of each point in the first three-dimensional space is calculated by repeating the above steps S203 to S208.
In step S210, the wavefield at each time of each point in the first three-dimensional space is saved to the disk, so that the GPU-based finite difference fourier wavefield propagation calculation is implemented.
For the problem that the efficiency of the two-way wave Fourier finite difference algorithm is low, the GPU is adopted to perform multi-thread parallel forward FFT, phase shift, inverse FFT and other calculations in each time wave field propagation process, the calculation time consumption of the Fourier finite difference algorithm is reduced, the seismic wave field propagation method which gives consideration to effects and efficiency is formed, and the calculation efficiency of the Fourier finite difference algorithm is improved. Compared with the traditional method in which the root mean square velocity of each spatial point velocity is used as the background velocity, the present embodiment introduces the average velocity of each spatial point velocity as the background velocity, and reduces the error between the background velocity and the spatial point velocity, thereby improving the algorithm precision of the fourier finite difference algorithm in the three-dimensional calculation.
Embodiments of the present invention also provide a computer-readable storage medium storing a program for causing a computer to execute the method for determining a phase shift coefficient based on a GPU.
In summary, the beneficial effects of the present invention can be summarized as follows:
1. the method of the invention utilizes the symmetry of the time wave number domain, shares the phase shift coefficients of each point to the phase shift coefficients of all the symmetrical points of each point in the second three-dimensional space after determining the phase shift coefficients of each point in the parallel domain, omits the calculation step of the phase shift coefficients of the symmetrical points, saves the calculation time, and thus improves the algorithm efficiency of the Fourier finite difference algorithm in the three-dimensional calculation.
2. Compared with the traditional method which uses the root mean square velocity of each space point velocity as the background velocity, the method reduces the error between the background velocity and the space point velocity, thereby improving the algorithm precision of the Fourier finite difference algorithm during three-dimensional calculation.
3. Aiming at the problem of low efficiency of the two-way wave Fourier finite difference algorithm, the GPU is adopted to carry out calculation such as multi-thread parallel positive FFT, phase shift, inverse FFT and the like in each time wave field propagation process, so that the calculation time consumption of the Fourier finite difference algorithm is reduced, the seismic wave field propagation method which gives consideration to both the effect and the efficiency is formed, and the technical effect of improving the calculation efficiency of the Fourier finite difference algorithm is achieved.
Technologies consistent with the present disclosure provide, among other features, systems and methods for distributing offers and offer documents based on redemption history, and systems and methods for fulfilling requests for documents. While various exemplary embodiments of the disclosed system and method have been described above, it should be understood that they have been presented by way of example only, and not limitation. The present disclosure is not intended to be exhaustive or to limit the precise forms disclosed. Modifications and variations are possible in light of the above teachings or may be acquired from practice of the disclosure without departing from the breadth or scope of the disclosure.
Claims (10)
1. A method for determining a finite difference phase shift coefficient of a double-pass wave Fourier based on a GPU (graphics processing Unit), which is characterized by comprising the following steps:
transforming the time-space domain wave field of each point in a first three-dimensional space preset in a time-space domain by using the FFT function of the GPU to obtain the time-wave number domain wave field of each point in a second three-dimensional space, wherein the second three-dimensional space is a corresponding space of the first three-dimensional space in the time-wave number domain;
determining a parallel domain of the second three-dimensional space;
for each point in the parallel domain, executing the following steps by utilizing a thread corresponding to the point in the GPU in parallel:
extracting all symmetrical points symmetrical to the point in the second three-dimensional space;
and determining the double-pass wave Fourier finite difference phase shift coefficient corresponding to the point, and taking the double-pass wave Fourier finite difference phase shift coefficient as the double-pass wave Fourier finite difference phase shift coefficient of each symmetrical point of the point.
2. The method of claim 1, wherein the two-way wave Fourier finite difference phase shift coefficient is related to an average velocity of all points in the first three dimensional space.
3. The method of claim 2, wherein the expression is utilized
Determining a two-way wave Fourier finite difference phase shift coefficient corresponding to a point in the parallel domain, wherein,
representing the wave number, v, of the point0Represents the average speed and Δ t represents a preset time step.
4. A method according to any one of claims 1 to 3, wherein the points in the parallel domain have a coordinate range in the first direction of 0-nk1A coordinate range of 0-nk in the second direction2A coordinate range of 0-nk in the third direction3/2, wherein, nk1Representing the number of grid points, nk, of said second three-dimensional space in a first direction2Representing the number of grid points, nk, of said second three-dimensional space in a second direction3Representing a number of grid points of the second three-dimensional space in a third direction.
5. The method of claim 1, wherein the time-space domain wavefield for each point in the first three-dimensional space is FFT transformed using multiple threads in the GPU.
6. The method of claim 1, wherein a single thread in the GPU is used to perform an FFT of the spatio-temporal domain wavefield for each point in the first three-dimensional space.
7. A GPU-based two-way wave Fourier finite difference wave field propagation method is characterized by comprising the following steps:
acquiring the spatial point speed of each point in a first three-dimensional space preset in a time-space domain at the current moment, and calculating the average speed of all points in the first three-dimensional space at the current moment according to the spatial point speed;
acquiring a time-space domain wave field of each point in a preset first three-dimensional space in a time-space domain at the current moment, and determining a two-way wave Fourier finite difference phase shift coefficient of all the points in the first three-dimensional space according to the method of any one of claims 1 to 6;
for each point in the first three-dimensional space, respectively performing the following steps:
obtaining a first coefficient and a second coefficient according to the average speed, the grid intervals of the first three-dimensional space in a fourth direction, a fifth direction and a sixth direction respectively, the spatial point speed of the point and a preset time step;
phase shifting the time wavenumber domain wave field of the point at the current moment by using the double-pass wave Fourier finite difference phase shift coefficient of the point;
performing three-dimensional space inverse Fourier transform on the time wave number domain wave field of the point after phase shift at the current moment by using the IFFT function of the GPU to obtain the time space domain wave field of the point after phase shift at the current moment;
performing space finite difference calculation on the time-space domain wave field of the point after phase shift at the current moment by using the first coefficient and the second coefficient to obtain the time-space domain wave field of the point after phase shift at the next moment, wherein the next moment is separated from the current moment by the time step length;
and obtaining the time-space domain wave field of the point at the next moment according to the time-space domain wave field of the point after phase shift at the next moment and the time-space domain wave fields of the point at the previous moment and the current moment.
8. The method of claim 7, wherein the time-wave-domain wavefield is phase shifted for each point in the first three-dimensional space by multiplying the two-way wave Fourier finite difference phase shift coefficients and the time-wave-domain wavefield.
9. The method of claim 7, wherein for each point in the first three-dimensional space
By using
To obtain the point
At the next instant t + Δ tTime-space domain wavefield
Wherein the content of the first and second substances,
representing the spatio-temporal wavefield of the point phase-shifted at the next time instant t + at,
representing the spatio-temporal wavefield of the point at the current time instant t,
the spatio-temporal wavefield at this point at the last time instant t- Δ t, Δ t being the time step.
10. A computer-readable storage medium storing a program for causing a computer to execute the method of determining a two-way wave fourier finite difference phase shift coefficient based on a GPU according to any one of claims 1 to 6.
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