CN108051854A - A kind of multiple dimensioned pseudo- bending ray tracing method - Google Patents
A kind of multiple dimensioned pseudo- bending ray tracing method Download PDFInfo
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Abstract
The invention discloses a kind of multiple dimensioned pseudo- bending ray tracing methods, are related to technical field of geophysical exploration.This method includes：By the ray path length between all two adjacent sections points by initial ray path one by one compared with the length of side of the cube grid of the three-dimensional velocity structure component of the correspondence scale of extraction；When ray path length between the length of side of the cube grid of the rate pattern component of the correspondence scale of the extraction is less than two adjacent sections point, using the intermediate point of the two adjacent sections point as new node；Modified using pseudo- bending ray tracing method to the new node, by the line between two nodes in initial ray path and amended node as an optimization after ray path, the ray path after this is optimized is as initial ray path.By the present invention method pseudo- bending method is made technically to be improved, make it is suitable for including in the arbitrary velocity distribution of velocity jump, with give full play to this method in itself possessed high-efficiency high-precision the advantages of.
Description
Technical field
The present invention relates to technical field of geophysical exploration, more particularly relate to a kind of multiple dimensioned pseudo- bending ray tracing side
Method.
Background technology
Ray tracing as a kind of quickly and effectively forward simulation technology, in the wild Data Acquisition Design, Wave field analysis with
Identification, static correction, tomography and migration imaging etc. have important application.Ray tracing is known media and observation
System, when asking for path that ripple between focus and receiving point propagates, walking, amplitude the problems such as.
Traditional ray tracing technique can be divided into：Shooting method (or shooting method), bending method and wave-front reconstruction method etc..It is numerous to penetrate
Bending method is fast with iteration speed in line method for tracing, and precision is also higher, but is unsuitable for continuous media, and pseudo- bending method ray chases after
Track considers velocity gradient, thus can be suitably used in the medium of continuous velocity variation, being but not suitable for Jie there are velocity jump
In matter.
The content of the invention
The embodiment of the present invention provides a kind of multiple dimensioned pseudo- bending ray tracing method, to solve pseudo- bending in the prior art
Not the problem of method ray tracing is not suitable in velocity jump medium.
The embodiment of the present invention provides a kind of multiple dimensioned pseudo- bending ray tracing method, including：
The first predetermined three-dimensional velocity structure is pressed cube mesh generation by step 1, carries out spatial discretization；
Step 2 is determined according to predetermined shot point coordinate and receiving in the first three-dimensional velocity structure of the point coordinates after subdivision
Initial ray path；
Step 3 sets initial gauges factor k=1；
Step 4, according to formula (1), the three-dimensional of corresponding scale is extracted from the first three-dimensional velocity structure after the subdivision
Rate pattern component Vk；
Wherein, in formula (1), L be Gaussian filter length, (x0, y0,z0) be Gaussian window central point, k for scale because
Son, W (k) be Gaussian filter, VkFor the convolution results of Gaussian filter and three-dimensional velocity structure, V is three-dimensional velocity structure；
It is step 5, the ray path length between all two adjacent sections points in initial ray path is corresponding with extraction one by one
The length of side of the cube grid of the three-dimensional velocity structure component of scale is compared；
Step 6, when the extraction correspondence scale rate pattern component cube grid the length of side be less than adjacent two
Between node during ray path length, using the intermediate point of the two adjacent sections point as new node；
Step 7 modifies to the new node using pseudo- bending ray tracing method, by two sections in initial ray path
Point amended node between line as an optimization after ray path, using the ray path after the optimization as initially
Ray path；
Step 8 makes scale factor k add 1, and three-dimensional velocity structure after scale factor k corresponding griddings is with making a reservation for
The second three-dimensional velocity structure when mismatching, repeat step 4-7；
The final ray path of step 9, output, and calculate ray traveltime；Wherein, the final ray path is to work as ruler
Initial ray road when three-dimensional velocity structure after the corresponding griddings of degree factor k is matched with the second predetermined three-dimensional velocity structure
Footpath.
It is preferred that described modified using pseudo- bending ray tracing method to the new node is specifically included：
S1, assume two nodes in initial ray path for Pk-1And Pk+1, Pk-1And Pk+1Line center at be inserted into it is new
Node is Pmid；
S2, by following iterative formula group to intermediate node PmidIt modifies, amended node is Pk, that is, after optimizing
Path node be Pk-1、PkAnd Pk+1；
Wherein, VmidFor midpoint PmidThe velocity amplitude at place,For point P after modificationkThe velocity gradient vector at place,For
Midpoint PmidThe velocity gradient vector at place, xk-1For node Pk-1Abscissa, xk+1For node Pk+1Abscissa, Γ be direction arrow
Amount, n are unit direction vector, and R is midpoint PmidPoint P after to modificationkDistance,C is point Pk-1With point Pk+1
Place's slowness is averaged, L=| xk+1-xmid|, L is node Pk+1With node PmidThe distance between, xmidFor node PmidAbscissa,
xkFor node pkAbscissa.
It is preferred that the midpoint PmidThe velocity amplitude V at placemidIt can be calculated according to equation below (2)；
Wherein, in formula (2) formula, Vmid(x, y, z) is midpoint PmidSpeed at (x, y, z), Vmid((i+l,j+m,k+n)
Represent midpoint PmidVelocity amplitude at 8 grid nodes of surrounding；(x, y, z) is midpoint PmidCoordinate, (xi,yj,zk) it is net respectively
The x coordinate of lattice node, y-coordinate and z coordinate.
In the embodiment of the present invention, whether the sizing grid of the three-dimensional velocity structure component of the correspondence scale by judging extraction
Less than initial ray path；When the sizing grid of the rate pattern component of the correspondence scale of the extraction is less than initial ray path
When, new node will be inserted at the center in initial ray path；The new node is repaiied using pseudo- bending ray tracing method
Change, by the line between two nodes in initial ray path and amended node as an optimization after ray path, will described in
Ray path after optimization is as initial ray path, that is, by extracting step by step, and the correspondence ruler relatively per onestep extraction
After the sizing grid of the three-dimensional velocity structure component of degree and initial ray path, initial ray path is encrypted and optimized,
Until path meets the end condition of setting, so that pseudo- bending method is technically improved, make that it is suitable for include speed
Spend mutation arbitrary velocity distribution in, with give full play to this method in itself possessed high-efficiency high-precision the advantages of.
Description of the drawings
Fig. 1 is a kind of flow diagram of multiple dimensioned pseudo- bending ray tracing method provided in an embodiment of the present invention；
Fig. 2 is mesh generation schematic diagram of the present invention；
Fig. 3 is 3 iteration schematic diagrames of pseudo- bending method
Fig. 4 is the multiple dimensioned pseudo- bending method ray tracing process of complicated rate pattern.
Specific embodiment
Below in conjunction with the attached drawing in the embodiment of the present invention, the technical solution in the embodiment of the present invention is carried out clear, complete
Site preparation describes, it is clear that described embodiment is only part of the embodiment of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other without making creative work
Embodiment belongs to the scope of protection of the invention.
Multiple dimensioned puppet bending ray tracing method belongs to the forward problem in geophysical exploration, that is, passes through given three-dimensional speed
Degree model and shot point, the position of receiving point arrangement obtain the ray path of correct seimic wave propagation, and then obtain correctly
Earthquake record.To achieve the above object, the flow of a kind of multiple dimensioned pseudo- bending ray tracing method provided in an embodiment of the present invention
Schematic diagram, as shown in Figure 1, this method includes：
The first predetermined three-dimensional velocity structure is pressed cube mesh generation by step 1, carries out spatial discretization.
Wherein, subdivision is carried out to the first three-dimensional velocity structure according to cube grid, as shown in Figure 1.
Step 2 is determined according to predetermined shot point coordinate and receiving in the first three-dimensional velocity structure of the point coordinates after subdivision
Initial ray path.
Step 3 sets initial gauges factor k=1.
Step 4, according to formula (1), the three-dimensional speed of corresponding scale is extracted from the first three-dimensional velocity structure after the subdivision
Spend model component Vk。
Wherein, in formula (1), L be Gaussian filter length, (x0, y0,z0) be Gaussian window central point, k for scale because
Son, W (k) be Gaussian filter, VkFor the convolution results of Gaussian filter and three-dimensional velocity structure, V is three-dimensional velocity structure.
It is step 5, the ray path length between all two adjacent sections points in initial ray path is corresponding with extraction one by one
The length of side of the cube grid of the three-dimensional velocity structure component of scale is compared.
Step 6, when the extraction correspondence scale rate pattern component cube grid the length of side be less than two adjacent sections
Between point during ray path length, using the intermediate point of the two adjacent sections point as new node.
Step 7 modifies to the new node using pseudo- bending ray tracing method, by two nodes in initial ray path
Line between amended node as an optimization after ray path, the ray path after this is optimized is as initial ray
Path.
Wherein, the pseudo- bending ray tracing method of the use, which modifies to the new node, specifically includes：
S1, assume two nodes in initial ray path for Pk-1And Pk+1, Pk-1And Pk+1Line center at be inserted into it is new
Node is Pmid。
S2, by following iterative formula group to intermediate node PmidIt modifies, amended node is Pk, that is, after optimizing
Path node be Pk-1、PkAnd Pk+1。
Wherein, VmidFor midpoint PmidThe velocity amplitude at place,For point P after modificationkThe velocity gradient vector at place,For
Midpoint PmidThe velocity gradient vector at place, xk-1For node Pk-1Abscissa, xk+1For node Pk+1Abscissa, Γ be direction arrow
Amount, n are unit direction vector, and R is midpoint PmidPoint P after to modificationkDistance,C is point Pk-1With point Pk+1
Place's slowness is averaged, L=| xk+1-xmid|, L is node Pk+1With node PmidThe distance between, xmidFor node PmidAbscissa,
xkFor node PkAbscissa.
In addition, the mode of insertion new node also referred to as encrypts ray node, as shown in Figure 3, it is assumed that initial ray path
Two nodes are Pk-1And Pk+1, in Pk-1And Pk+1Line center at be inserted into new node be Pmid, i.e., by increasing among ray
Node obtains encrypted ray path.
In addition, to continuous three points Pk-1、PmidAnd Pk+1, utilize Pk-1And Pk+1By iterative formula group to intermediate point Pmid
It modifies, amended node is Pk, that is, the path node after optimizing is Pk-1、PkAnd Pk+1, expression such as Fig. 3 institutes of the node
Show.
Specifically, midpoint PmidThe velocity amplitude V at placemidIt can be calculated according to equation below (2)；And midpoint PmidPosition
It puts as shown in Figure 2.
Wherein, in formula (2), Vmid(x, y, z) is midpoint PmidSpeed at (x, y, z), Vmid((i+l, j+m, k+n) table
Show midpoint PmidVelocity amplitude at 8 grid nodes of surrounding；(x, y, z) is midpoint PmidCoordinate, (xi,yj,zk) it is grid respectively
The x coordinate of node, y-coordinate and z coordinate.
Step 8 makes scale factor k add 1, and three-dimensional velocity structure after scale factor k corresponding griddings is with making a reservation for
The second three-dimensional velocity structure when mismatching, repeat step 4-7.
The final ray path of step 9, output, and calculate ray traveltime.
Wherein, which is three-dimensional velocity structure after scale factor k corresponding griddings with making a reservation for
The second three-dimensional velocity structure matching when initial ray path.
Fig. 4 is to utilize a kind of multiple dimensioned pseudo- bending ray tracing method pseudo- bending multiple dimensioned to complicated rate pattern of the present invention
Method ray tracing process, in Fig. 4, a is the line in initial ray path, i.e. shot point and receiving point；When b is scale factor k=1
Ray path is inserted into a node at this time between shot point and receiving point；Ray path ... when c is scale factor k=2, always
It is final ray path to j.
In the embodiment of the present invention, whether the sizing grid of the three-dimensional velocity structure component of the correspondence scale by judging extraction
Less than initial ray path；When the sizing grid of the rate pattern component of the correspondence scale of the extraction is less than initial ray path
When, new node will be inserted at the center in initial ray path；It is modified using pseudo- bending ray tracing method to the new node,
By the line between two nodes in initial ray path and amended node as an optimization after ray path, by the optimization
Ray path afterwards is as initial ray path, that is, by extracting step by step, and the correspondence scale relatively per onestep extraction
After the sizing grid of three-dimensional velocity structure component and initial ray path, initial ray path is encrypted and optimized, until
Path meets the end condition of setting, and obtained ray path is final result, so that pseudo- bending method ray tracing can
Suitable for velocity jump medium.
Disclosed above is only several specific embodiments of the present invention, and those skilled in the art can carry out the present invention
Various modification and variations without departing from the spirit and scope of the present invention, if these modifications and changes of the present invention belong to the present invention
Within the scope of claim and its equivalent technologies, then the present invention is also intended to comprising including these modification and variations.
Claims (3)
- A kind of 1. multiple dimensioned pseudo- bending ray tracing method, which is characterized in that including：The first predetermined three-dimensional velocity structure is pressed cube mesh generation by step 1, carries out spatial discretization；Step 2, determined according to predetermined shot point coordinate and receiving in the first three-dimensional velocity structure of the point coordinates after subdivision it is initial Ray path；Step 3 sets initial gauges factor k=1；Step 4, according to formula (1), the three-dimensional velocity of corresponding scale is extracted from the first three-dimensional velocity structure after the subdivision Model component Vk；<mrow> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>V</mi> <mi>k</mi> </msup> <mo>=</mo> <mi>W</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>&CircleTimes;</mo> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>k</mi> <mrow> <mn>3</mn> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <mi>L</mi> </mrow> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>L</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mfrac> <mi>k</mi> <mrow> <mn>3</mn> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <mi>L</mi> </mrow> </mfrac> <msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>L</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> </msub> <mfrac> <mi>k</mi> <mrow> <mn>3</mn> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <mi>L</mi> </mrow> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>L</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>Wherein, in formula (1), L be Gaussian filter length, (x0, y0,z0) it is Gaussian window central point, k is scale factor, W (k) it is Gaussian filter, VkFor the convolution results of Gaussian filter and three-dimensional velocity structure, V is three-dimensional velocity structure；Step 5, by the corresponding scale with extraction one by one of the ray path length between all two adjacent sections points in initial ray path The length of side of cube grid of three-dimensional velocity structure component be compared；Step 6, when the extraction correspondence scale rate pattern component cube grid the length of side be less than two adjacent sections point Between ray path length when, using the intermediate point of the two adjacent sections point as new node；Step 7 modifies to the new node using pseudo- bending ray tracing method, by two nodes in initial ray path and Line between amended node as an optimization after ray path, using the ray path after the optimization as initial ray Path；Step 8 makes scale factor k add 1, and three-dimensional velocity structure after scale factor k corresponding griddings and predetermined the When two three-dimensional velocity structures mismatch, step 4-7 is repeated；The final ray path of step 9, output, and calculate ray traveltime；Wherein, the final ray path be when scale because Initial ray path when three-dimensional velocity structure after the corresponding gridding of sub- k is matched with the second predetermined three-dimensional velocity structure.
- 2. multiple dimensioned pseudo- bending ray tracing method as described in claim 1, which is characterized in that described using pseudo- curved rays Method for tracing is modified to the new node and is specifically included：S1, assume two nodes in initial ray path for Pk-1And Pk+1, Pk-1And Pk+1Line center at be inserted into new node For Pmid；S2, by following iterative formula group to intermediate node PmidIt modifies, amended node is Pk, that is, the path after optimizing Node is Pk-1、PkAnd Pk+1；<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>&Gamma;</mi> <mo>=</mo> <msub> <mo>&dtri;</mo> <msub> <mi>p</mi> <mi>k</mi> </msub> </msub> <mi>V</mi> <mo>-</mo> <mo>&lsqb;</mo> <msub> <mo>&dtri;</mo> <msub> <mi>p</mi> <mi>k</mi> </msub> </msub> <mi>V</mi> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mfrac> <mrow> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>n</mi> <mo>=</mo> <mi>&Gamma;</mi> <mo>/</mo> <mo>|</mo> <mi>&Gamma;</mi> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>R</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mi>cV</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>c</mi> <mi>n</mi> <mo>&CenterDot;</mo> <msub> <mo>&dtri;</mo> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mi>u</mi> <mi>d</mi> </mrow> </msub> </msub> <mi>V</mi> </mrow> </mfrac> <mo>+</mo> <msqrt> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>cV</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <msup> <mrow> <mo>(</mo> <mn>4</mn> <mi>c</mi> <mi>n</mi> <mo>&CenterDot;</mo> <msub> <mo>&dtri;</mo> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mi>u</mi> <mi>d</mi> </mrow> </msub> </msub> <mi>V</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msub> <mi>cV</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mi>R</mi> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>Wherein, VmidFor midpoint PmidThe velocity amplitude at place,For point P after modificationkThe velocity gradient vector at place,For midpoint PmidThe velocity gradient vector at place, xk-1For node Pk-1Abscissa, xk+1For node Pk+1Abscissa, Γ is direction vector, n For unit direction vector, R is midpoint PmidPoint P after to modificationkDistance,C is point Pk-1With point Pk+1Place is slow Degree is averaged, L=| xk+1-xmid|, L is node Pk+1With node PmidThe distance between, xmidFor node PmidAbscissa, xkFor Node PkAbscissa.
- 3. multiple dimensioned pseudo- bending ray tracing method as claimed in claim 2, which is characterized in that the midpoint PmidThe speed at place Angle value VmidIt can be calculated according to equation below (2)；<mrow> <msub> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>1</mn> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>1</mn> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>1</mn> </munderover> <msub> <mi>V</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>d</mi> </mrow> </msub> <mo>(</mo> <mrow> <mi>i</mi> <mo>+</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> <mo>)</mo> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mo>|</mo> <mfrac> <mrow> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mi>l</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mi>l</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>|</mo> </mrow> <mo>)</mo> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mo>|</mo> <mfrac> <mrow> <mi>y</mi> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>y</mi> <mrow> <mi>j</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>|</mo> </mrow> <mo>)</mo> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mo>|</mo> <mfrac> <mrow> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo>|</mo> </mrow> <mo>)</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>Wherein, in formula (2) formula, Vmid(x, y, z) is midpoint PmidSpeed at (x, y, z), Vmid((i+l, j+m, k+n) is represented Midpoint PmidVelocity amplitude at 8 grid nodes of surrounding；(x, y, z) is midpoint PmidCoordinate, (xi,yj,zk) it is grid section respectively The x coordinate of point, y-coordinate and z coordinate.
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