CN107479091A - A kind of method for extracting reverse-time migration angle gathers - Google Patents

A kind of method for extracting reverse-time migration angle gathers Download PDF

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CN107479091A
CN107479091A CN201710448348.0A CN201710448348A CN107479091A CN 107479091 A CN107479091 A CN 107479091A CN 201710448348 A CN201710448348 A CN 201710448348A CN 107479091 A CN107479091 A CN 107479091A
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mrow
msub
omega
angle
msup
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CN107479091B (en
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田坤
于海铖
王常波
张学涛
步长城
李国磊
王彦军
张敬仁
揭景荣
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China Petroleum and Chemical Corp
Geophysical Research Institute of Sinopec Shengli Oilfield Co
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China Petroleum and Chemical Corp
Geophysical Research Institute of Sinopec Shengli Oilfield Co
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01VGEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/28Processing seismic data, e.g. analysis, for interpretation, for correction
    • G01V1/282Application of seismic models, synthetic seismograms
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01VGEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/28Processing seismic data, e.g. analysis, for interpretation, for correction
    • G01V1/34Displaying seismic recordings or visualisation of seismic data or attributes
    • G01V1/345Visualisation of seismic data or attributes, e.g. in 3D cubes
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01VGEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V2210/00Details of seismic processing or analysis
    • G01V2210/70Other details related to processing
    • G01V2210/74Visualisation of seismic data

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  • Acoustics & Sound (AREA)
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  • General Life Sciences & Earth Sciences (AREA)
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Abstract

The present invention relates to oil-gas exploration seism processing field, more particularly to a kind of method for extracting reverse-time migration angle gathers, the extracting method improves the extraction accuracy of complex region angle gathers, and solves the problems, such as lineups intersection well, it is good to the imaging effect of complex region, be advantageous to carry out migration velocity analysis.The extracting method includes to be obtained inputting big gun record to default sound wave model progress limited number of time difference forward modelling;Extrapolate to obtain shot point wave field from focus point forward;Self-test wave point back-extrapolate obtains geophone station wave field;Fourier transformation obtains shot point part slowness domain, geophone station part slowness domain, and decomposition obtains shot point local angle component, geophone station local angle component;Determine that angle domain image-forming condition extracts to obtain the steps such as reverse-time migration angle gathers.

Description

A kind of method for extracting reverse-time migration angle gathers
Technical field
The present invention relates to oil-gas exploration seism processing field, more particularly to a kind of extraction reverse-time migration angle gathers Method.
Background technology
Migration velocity modeling is the important step of seism processing, and high-precision Migration velocity model is to obtain high quality The key factor of migration imaging result.Wherein, reverse-time migration shows its superiority in terms of complex dielectrics image quality is improved. , must in order to ensure reverse-time migration imaging has higher precision but response of the reverse-time migration to velocity accuracy is very sensitive The precision of velocity inversion must be improved.And output item of the angle domain common image gathers as reverse-time migration, can be by angle domain The inverting of common imaging gather obtains the information such as high-precision migration velocity field and amplitude, phase.Meanwhile precision highest imaging road Collection can be also used for AVO (amplitude variation with Offset analysis), AVA (amplitude is analyzed with angle change) analysis.But invention human hair Existing, existing angle domain common image gathers extracting method is normally based on the extracting method of poynting vector classes, and this kind carries Take method although there is the advantages such as efficient and high angular resolution, but it handles institute for handling complex region It is relatively low to obtain angle gathers precision, can not meet that inverting obtains the requirement of high-precision reverse-time migration.
The content of the invention
The invention provides a kind of method for extracting reverse-time migration angle gathers, the extracting method improves complex region angle road The extraction accuracy of collection, and solve the problems, such as lineups intersection well, it is good to the imaging effect of complex region, be advantageous into Line displacement velocity analysis.
In order to solve the above technical problems, present invention employs following technical scheme:
A kind of method for extracting reverse-time migration angle gathers, methods described include having the following steps:
Step 1:Limited number of time difference forward modelling is carried out to default sound wave model, obtains inputting big gun record;
Step 2:According to input big gun record, extrapolate to obtain shot point wave field from focus point forward;
Step 3:According to input big gun record, self-test wave point back-extrapolate obtains geophone station wave field;
Step 4:Fourier transformation is carried out to shot point wave field, geophone station wave field respectively and local angle decomposes, obtains big gun Point local angle component, geophone station local angle component;
Step 5:Angle domain image-forming condition is determined, and it is local according to step 4 gained shot point local angle component and geophone station Angle component, extraction obtain reverse-time migration angle gathers.
More preferable, the default sound wave model is the sound wave model of single order stress-velocity form.
Preferably, Fourier transformation satisfaction is carried out to shot point wave field:
Wherein, ugFor shot point wave field, Wg(x'-x) window function at focal point x is referred to, p is the modulus value of slowness vector,It is to pass Broadcast the unit vector on direction.
Preferably, ug(x, ω) meets:
Wherein,For slowness vector, For the average speed in sampling window;θ is plane wave propagation direction and list Bit vectorBetween angle;ω p are the Jacobians of two-dimentional slowness-angular transformation.
Preferably, Fourier transformation satisfaction is carried out to geophone station wave field:
Wherein, usIt is geophone station wave field, Ws(x'-x) window function at geophone station x is referred to, p is the modulus value of slowness vector,It is Unit vector on the direction of propagation.
Preferably, us(x, ω) meets:
Wherein,For slowness vector, For the average speed in sampling window;θ is plane wave propagation direction and list Bit vectorBetween angle;ω p are the Jacobians of two-dimentional slowness-angular transformation.
Preferably, it is determined that angle domain image-forming condition meet:
Wherein, ugIt is shot point wave field, usIt is geophone station wave field, θ is angle of reflection, meets θ=(θsg)/2。
The invention provides a kind of method for extracting reverse-time migration angle gathers, the method that this kind extracts reverse-time migration angle gathers Include and default sound wave model progress limited number of time difference forward modelling is obtained inputting big gun record;Extrapolate to obtain from focus point forward Shot point wave field;Self-test wave point back-extrapolate obtains geophone station wave field;Fourier transformation obtains shot point part slowness domain, geophone station office Portion's slowness domain, decomposition obtain shot point local angle component, geophone station local angle component;Determine that angle domain image-forming condition extracts To steps such as reverse-time migration angle gathers.Extracting method with above-mentioned steps is higher to the extraction accuracy of complex region angle gathers, The data accuracy that progress Tomography Velocity inverting obtains is also higher, and imaging effect is more preferable.
Brief description of the drawings
Fig. 1 is a kind of schematic flow sheet of method for extracting reverse-time migration angle gathers provided by the invention;
Fig. 2 is a kind of complex model schematic diagram;
Fig. 3 is that complex model shown in Fig. 2 is carried out using the inventive method to choose the imaging of proper angle domain imaging condition simultaneously Extract the schematic diagram of angle gathers;
Fig. 4 is the specific extraction position view of Fig. 3 angle gathers;
It using the inventive method is the migrated section figure that obtains after complex model extraction process shown in Fig. 2 that Fig. 5, which is,;
Fig. 6 is data model schematic diagram used in Shengli Oil Field;
Fig. 7 is to carry out selection proper angle to data model used in Shengli Oil Field shown in Fig. 6 using the inventive method Domain imaging condition is imaged and extracts the schematic diagram of angle gathers;
Fig. 8 is the specific extraction position view of Fig. 7 angle gathers;
Fig. 9 is to obtain after data model extraction process used in Shengli Oil Field shown in Fig. 6 using the inventive method Migrated section figure;
Figure 10 is the schematic diagram to certain section real data processing gained migration velocity field using the inventive method;
Figure 11 is to certain section real data processing gained part angle gathers schematic diagram using the inventive method;
Figure 12 is the migrated section figure for handling to obtain to certain section real data using the inventive method.
Embodiment
The invention provides a kind of method for extracting reverse-time migration angle gathers, the extracting method improves complex region angle road The extraction accuracy of collection, and solve the problems, such as lineups intersection well, it is good to the imaging effect of complex region, be advantageous into Line displacement velocity analysis.
The embodiment of the present invention is described in detail with reference to following accompanying drawings.
The invention provides a kind of method for extracting reverse-time migration angle gathers, as shown in figure 1, the extracting method specifically includes Have:
Step 1:Limited number of time difference forward modelling is carried out to default sound wave model, obtains inputting big gun record;
A kind of more preferred embodiment wherein as the present invention, select the sound wave mould of single order stress-velocity form Type is as default sound wave model.And the sound wave model progress limited number of time difference forward modelling of single order stress-velocity form is obtained The wave field recorded to input big gun opens up the process prolonged can be with specific reference to following formula (1):
Wherein, v represents particle seimic wave velocity, and ρ is density, and u is stress value at particle.
Step 2:According to input big gun record, extrapolate to obtain shot point wave field from focus point forward;
Step 3:According to input big gun record, self-test wave point back-extrapolate obtains geophone station wave field;
It is anti-from the extrapolation of focus point forward and self-test wave point according to input big gun record respectively on the basis of step 1 is completed Push out, shot point wave field u can be obtainedgAnd geophone station wave field us
Step 4:Fourier transformation is carried out to shot point wave field, geophone station wave field respectively and local angle decomposes, obtains big gun Point local angle component, geophone station local angle component;
On the basis of step 2, step 3 is completed, further to shot point wave field ugAnd geophone station wave field usCarry out Fourier Conversion and local angle resolution process.First with to shot point wave field ugExemplified by progress Fourier transformation and local angle decompose It is introduced.
Wherein, shot point wave field ugFourier transformation meet following formula (3):
Specifically, ugFor shot point wave field, Wg(x'-x) window function at focal point x is referred to, p is the modulus value of slowness vector,It is Unit vector on the direction of propagation.
And u in formula (3)g(x, ω) is further decomposed by formula (4), formula (5):
Wherein,For slowness vector, For the average speed in sampling window;θ is plane wave propagation direction and list Bit vectorBetween angle;ω p are the Jacobians of two-dimentional slowness-angular transformation.
Likewise, geophone station wave field usFourier transformation meet following formula (6):
Wherein, usIt is geophone station wave field, Ws(x'-x) window function at geophone station x is referred to, p is the modulus value of slowness vector,It is Unit vector on the direction of propagation.
And further, u in formula (6)s(x, ω) can be decomposed into (7), (8) as the following formula:
Wherein,For slowness vector, For the average speed in sampling window;θ is plane wave propagation direction and list Bit vectorBetween angle;ω p are the Jacobians of two-dimentional slowness-angular transformation.
Step 5:Angle domain image-forming condition is determined, and it is local according to step 4 gained shot point local angle component and geophone station Angle component, extraction obtain reverse-time migration angle gathers.
On the basis of step 4 is completed, angle domain image-forming condition is further determined that.Specifically, the angle domain imaging determined Condition meets following formula (9), formula (10):
Wherein, ugIt is shot point wave field, usIt is geophone station wave field, θ is angle of reflection, meets θ=(θsg)/2。
Determine again after completing angle domain image-forming condition, shot point local angle component and geophone station office with reference to obtained by step 4 Portion's angle component, you can extraction obtains reverse-time migration angle gathers.
Lower mask body combination complex model carries out tentative calculation test to extracting method provided by the invention.As Figure 2-Figure 5, Wherein, Fig. 2 is a kind of complex model schematic diagram;Fig. 3 is imaged to choose proper angle domain imaging condition and extracts angle gathers;And have The extraction position of body angle gathers is as shown in figure 4, choose 333~336CDP opening positions.By choosing suitable rate pattern It is also relatively flat to be imaged trace gather, and because deep layer angle illumination scope is smaller than shallow-layer, hence in so that angular-trace gather deep layer is concentrated In small angle range.The above results show:Extraction reverse-time migration angle gathers method provided by the invention is to large inclination Adaptability is preferable, can preferably complete the inverting to Tomography Velocity using this method, obtain accurate velocity field.And then Fig. 5 For the migrated section figure obtained using the inventive method, from figure 5 it can be seen that obtained migrated section accurately reflects very much Go out the structural configuration of given speed model, it is basically identical with given speed model.Therefore extracting method provided by the invention is utilized Gained angle gathers are more conducive to carry out the analysis of migration velocity, also have larger help to improving imaging precision.
In addition, also further by taking data model used in Shengli Oil Field and certain section real data model as an example, it is right Extracting method provided by the invention carries out tentative calculation test.Wherein Fig. 6 is data model schematic diagram used in Shengli Oil Field;Fig. 7 It is imaged for selection proper angle domain imaging condition and extracts angle gathers;And the extraction position of specific angle gathers is as shown in Figure 8.Can be with It was found that angle gathers lineups are straight clear, especially middle part complicated structure, angle gathers imaging effect are very good.Fig. 9 is to Fig. 6 Shown data model uses the migrated section figure obtained after the inventive method.The migrated section chart is bright, is obtained using this method Angle gathers carry out that the obtained speed ratio of Tomography Velocity inverting is more accurate, and imaging effect is very good, and precision is very high.The section with The realistic model goodness of fit is very high, and image space is accurate, can reflect Construction of A Model with relatively good.Figure 10-11 are utilization The inventive method is to the treatment effect figure of certain section real data, and wherein Figure 10 is the schematic diagram of migration velocity field, and Figure 11 is portion Subangle trace gather schematic diagram.Those skilled in the art are it can be found that extraction gained angle gathers lineups are straight clear, especially deep Energy is stronger, shows that this method can be completely applied to real data.Figure 12 is the migrated section obtained using this method.From figure It can be found that the section can reflect the actual configuration of this area underground well in 12, to deep layer engineering construction system effect It is good.
The invention provides a kind of method for extracting reverse-time migration angle gathers, the method that this kind extracts reverse-time migration angle gathers Include and default sound wave model progress limited number of time difference forward modelling is obtained inputting big gun record;Extrapolate to obtain from focus point forward Shot point wave field;Self-test wave point back-extrapolate obtains geophone station wave field;Fourier transformation obtains shot point part slowness domain, geophone station office Portion's slowness domain, decomposition obtain shot point local angle component, geophone station local angle component;Determine that angle domain image-forming condition extracts To steps such as reverse-time migration angle gathers.Extracting method with above-mentioned steps is higher to the extraction accuracy of complex region angle gathers, The data accuracy that progress Tomography Velocity inverting obtains is also higher, and imaging effect is more preferable.
The foregoing is only a specific embodiment of the invention, but protection scope of the present invention is not limited thereto, any Those familiar with the art the invention discloses technical scope in, change or replacement can be readily occurred in, should all be contained Cover within protection scope of the present invention.Therefore, protection scope of the present invention should be based on the protection scope of the described claims.

Claims (7)

  1. A kind of 1. method for extracting reverse-time migration angle gathers, it is characterised in that methods described includes having the following steps:
    Step 1:Limited number of time difference forward modelling is carried out to default sound wave model, obtains inputting big gun record;
    Step 2:According to input big gun record, extrapolate to obtain shot point wave field from focus point forward;
    Step 3:According to input big gun record, self-test wave point back-extrapolate obtains geophone station wave field;
    Step 4:Fourier transformation is carried out to shot point wave field, geophone station wave field respectively and local angle decomposes, obtains shot point office Portion's angle component, geophone station local angle component;
    Step 5:Angle domain image-forming condition is determined, and according to step 4 gained shot point local angle component and geophone station local angle Component, extraction obtain reverse-time migration angle gathers.
  2. A kind of 2. method for extracting reverse-time migration angle gathers according to claim 1, it is characterised in that the default sound Wave pattern is the sound wave model of single order stress-velocity form.
  3. 3. a kind of method for extracting reverse-time migration angle gathers according to claim 1, it is characterised in that enter to shot point wave field Row Fourier transformation meets:
    <mrow> <msub> <mi>u</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;Integral;</mo> <msub> <mi>W</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mover> <mi>e</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>u</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mover> <mi>e</mi> <mo>^</mo> </mover> <mo>&amp;rsqb;</mo> <mo>&amp;times;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>&amp;omega;</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>p</mi> </mrow> </msup> <mo>&amp;CenterDot;</mo> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> </mrow>
    Wherein, ugFor shot point wave field, Wg(x'-x) window function at focal point x is referred to, p is the modulus value of slowness vector,It is propagation side Upward unit vector.
  4. A kind of 4. method for extracting reverse-time migration angle gathers according to claim 3, it is characterised in that ug(x, ω) meets:
    <mrow> <msub> <mi>u</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mi>&amp;theta;</mi> </munder> <msub> <mi>u</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>u</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;omega;pu</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mover> <mi>p</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein,For slowness vector, For the average speed in sampling window;θ be plane wave propagation direction with unit to AmountBetween angle;ω p are the Jacobians of two-dimentional slowness-angular transformation.
  5. 5. a kind of method for extracting reverse-time migration angle gathers according to claim 1, it is characterised in that to geophone station wave field Carry out Fourier transformation satisfaction:
    <mrow> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;Integral;</mo> <msub> <mi>W</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mover> <mi>e</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mover> <mi>e</mi> <mo>^</mo> </mover> <mo>&amp;rsqb;</mo> <mo>&amp;times;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>i</mi> <mi>&amp;omega;</mi> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>p</mi> </mrow> </msup> <mo>&amp;CenterDot;</mo> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> </mrow>
    Wherein, usIt is geophone station wave field, Ws(x'-x) window function at geophone station x is referred to, p is the modulus value of slowness vector,It is to propagate Unit vector on direction.
  6. A kind of 6. method for extracting reverse-time migration angle gathers according to claim 5, it is characterised in that us(x, ω) meets:
    <mrow> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <mi>&amp;theta;</mi> </munder> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;omega;pu</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mover> <mi>p</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> </mrow>
    Wherein,For slowness vector, For the average speed in sampling window;θ be plane wave propagation direction with unit to AmountBetween angle;ω p are the Jacobians of two-dimentional slowness-angular transformation.
  7. 7. it is according to claim 1 it is a kind of extract reverse-time migration angle gathers method, it is characterised in that it is determined that angle domain Image-forming condition meets:
    <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <msub> <mi>&amp;theta;</mi> <mi>s</mi> </msub> </munder> <munder> <mo>&amp;Sigma;</mo> <msub> <mi>&amp;theta;</mi> <mi>g</mi> </msub> </munder> <mi>I</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>s</mi> </msub> <mo>,</mo> <msub> <mi>&amp;theta;</mi> <mi>g</mi> </msub> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mi>&amp;theta;</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>&amp;Sigma;</mo> <mi>&amp;omega;</mi> </munder> <mo>&amp;lsqb;</mo> <msub> <mover> <mi>e</mi> <mo>^</mo> </mover> <mi>s</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>u</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>*</mo> <mo>&amp;lsqb;</mo> <msub> <mover> <mi>e</mi> <mo>^</mo> </mover> <mi>g</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>u</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>g</mi> </msub> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>&amp;omega;</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>;</mo> </mrow>
    Wherein, ugIt is shot point wave field, usIt is geophone station wave field, θ is angle of reflection, meets θ=(θsg)/2。
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