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CN116148925B - VTI medium spherical longitudinal wave reflection coefficient analysis method - Google Patents

VTI medium spherical longitudinal wave reflection coefficient analysis method Download PDF

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CN116148925B
CN116148925B CN202310408819.0A CN202310408819A CN116148925B CN 116148925 B CN116148925 B CN 116148925B CN 202310408819 A CN202310408819 A CN 202310408819A CN 116148925 B CN116148925 B CN 116148925B
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CN116148925A (en
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程广森
何传林
梁展源
张晓语
李冬
郑轶
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Institute of Oceanographic Instrumentation Shandong Academy of Sciences
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Abstract

The invention belongs to the technical field of geophysics, and particularly relates to a method for analyzing a spherical longitudinal wave reflection coefficient of a VTI medium. A method for analyzing the reflection coefficient of a spherical longitudinal wave of a VTI medium comprises the following steps: solving a VTI medium spherical wave displacement integral equation; solving displacement fields respectively excited by longitudinal and transverse wave point seismic sources by using a moment method; simplifying a spherical wave displacement equation of the VTI medium; solving the displacement of the spherical reflected wave at the receiving point; defining the spherical wave reflection coefficient of the VTI medium as the ratio of the displacement component of the spherical reflection wave to the displacement component of the incident wave in the polarization direction; and (3) deducing a frequency and depth dependent VTI medium spherical wave reflection coefficient analysis equation directly characterized by the vertical longitudinal wave speed, the density and the anisotropic parameters. The method can accurately describe the spherical wave amplitude and phase frequency-dependent reflection characteristics of the VTI medium, and quantify errors of geometric earthquake approximation under different frequencies, different reflection depths and different incidence angles.

Description

VTI medium spherical longitudinal wave reflection coefficient analysis method
Technical Field
The invention belongs to the technical field of geophysics, and particularly relates to a method for analyzing a spherical longitudinal wave reflection coefficient of a VTI medium.
Background
Transversely isotropic media (VTI media) with a vertical symmetry axis are widely available in the real world. Approximation to VTI-oriented media (ruger, 1997; stovas & Ursin, 2003; zhang & Li, 2013) and exact (Graebner, 1992) plane wave reflection coefficients (PRCs) have been widely used in AVO (amplitude variation with offset) inversion to estimate elasticity and anisotropy parameters in VTI media. However, for spherical wavefields excited by a point source, the VTI medium plane wave reflection coefficient suffers from fundamental problems, especially at low frequencies, near fields, and near or post critical angles of incidence where the spherical wave effect is significant.
In view of the complexity of solving the spherical wave reflection coefficient of the VTI medium, a technical scheme for directly solving the analysis expression of the spherical wave reflection coefficient of the VTI medium is not realized in the prior art. Therefore, the analysis method of the longitudinal wave reflection coefficient of the spherical surface of the VTI medium can fully consider the wavefront curvature of the spherical surface of the VTI medium and accurately characterize the amplitude and the phase frequency variation characteristics of the spherical surface reflection wave of the VTI medium.
Disclosure of Invention
The invention aims to provide a method for analyzing the reflection coefficient of a spherical longitudinal wave of a VTI medium. In order to achieve the above purpose, the invention adopts the following technical scheme: a method for analyzing the reflection coefficient of a spherical longitudinal wave of a VTI medium comprises the following steps:
step 1, solving a VTI medium spherical wave displacement integral equation;
step 2, solving displacement fields respectively excited by longitudinal and transverse wave point seismic sources by using a moment method;
step 3, simplifying a spherical wave displacement equation of the VTI medium;
step 4, calculating the displacement of the spherical reflection wave at the receiving point;
step 5, defining the spherical wave reflection coefficient of the VTI medium as the ratio of the displacement components of the spherical reflection wave and the incident wave in the polarization direction;
and 6, deducing a frequency and depth dependent VTI medium spherical wave reflection coefficient analysis equation directly characterized by the vertical longitudinal wave speed, the density and the anisotropic parameters.
Further, in the step 1, starting from the VTI medium wave equation of the coupling point source term, a VTI medium spherical wave displacement integral equation expressed by a stiffness tensor is constructed in a frequency wave number domain by performing time Fourier transform and space Fourier transform on the wave equation.
Further, in the step 2, the moment method is used to consider the action of the longitudinal wave point source and the transverse wave point source on the spherical wave field so as to solve the spherical wave displacement field of the VTI medium excited by the longitudinal wave point source and the transverse wave point source respectively.
Further, in the step 3, the spherical wave displacement equation of the VTI medium is converted from a cartesian coordinate system to a polar coordinate system, the spherical wave displacement equation is expressed as an integral of horizontal slowness by using a residue theorem, and the simplification of the spherical wave displacement equation is further realized by means of a bessel function.
Further, in the step 4, the displacement of the spherical reflection wave at the receiving point is solved by using the boundary conditions of continuous displacement and continuous stress.
Further, in the step 5, the VTI medium spherical wave reflection coefficient is defined as a ratio of a spherical reflection wave to a displacement component of an incident wave in a polarization direction, and a VTI medium spherical wave reflection coefficient equation represented by a stiffness tensor is derived in the form of integration of a VTI medium plane wave reflection coefficient.
Further, in the step 6, the quantitative relation between the stiffness tensor and the model parameters is utilized, the vertical longitudinal wave speed, the vertical transverse wave speed, the density and the anisotropic parameters on two sides of the reflecting interface are substituted into the VTI medium spherical wave reflection coefficient equation represented by the stiffness tensor, and the frequency and depth dependent VTI medium spherical wave reflection coefficient analysis equation directly represented by the vertical longitudinal transverse wave speed, the density and the anisotropic parameters is deduced.
Compared with the prior art, the invention has the following beneficial effects:
(1) The method for directly solving the analysis expression of the spherical wave reflection coefficient of the VTI medium is provided for the first time;
(2) The method can accurately describe the spherical wave amplitude and phase frequency-dependent reflection characteristics of the VTI medium, quantify errors of geometric seismic approximation (plane wavefront approximation) under the conditions of different frequencies, different reflection depths and different incidence angles, and is beneficial to improving the pre-stack seismic elasticity and anisotropic inversion stability of the VTI medium.
Drawings
FIG. 1 is a flow chart of a method for resolving a spherical longitudinal wave reflection coefficient of a VTI medium according to an embodiment of the present invention;
FIG. 2 is a graph showing the amplitude of the spherical wave reflection coefficient of the VTI medium calculated in example 1;
FIG. 3 is a graph showing the calculated amplitude of spherical wave reflection coefficients of VTI media in example 2;
FIG. 4 is a graph showing the calculated amplitude of spherical wave reflection coefficients of VTI media in example 3;
FIG. 5 is a graph showing the calculated amplitude of spherical wave reflection coefficient of VTI medium in example 4.
Detailed Description
In order that the invention may be readily understood, a more particular description thereof will be rendered by reference to specific embodiments that are illustrated in the appended drawings. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete.
The invention provides a method for analyzing the reflection coefficient of a spherical longitudinal wave of a VTI medium, which is shown in a figure 1, and specifically comprises the following steps:
step 1, solving a VTI medium spherical wave displacement integral equation
Starting from a VTI medium wave equation of a coupling point seismic source item, constructing a VTI medium spherical wave displacement integral equation expressed by a stiffness tensor in a frequency wave number domain by carrying out time Fourier transform and space Fourier transform on the wave equation.
The VTI medium wave equation for the coupled point source term is expressed as:
Figure SMS_1
；
wherein,,
Figure SMS_11
for a single point force at the origin of coordinates>
Figure SMS_4
Component of->
Figure SMS_7
Representing the strength of force, ++>
Figure SMS_12
Density (I)>
Figure SMS_16
Is a spatial dirac function, +.>
Figure SMS_14
Is a time dirac function and is used to describe the source,/->
Figure SMS_18
Is time, & lt>
Figure SMS_13
Is the 4 th order stiffness tensor,>
Figure SMS_15
and->
Figure SMS_2
Is the subscript of the 4 th order stiffness tensor and takes the values 1, 2 and 3 respectively; />
Figure SMS_9
And->
Figure SMS_5
Respectively is a displacement->
Figure SMS_8
One of the components (+)>
Figure SMS_17
），/>
Figure SMS_19
And->
Figure SMS_3
Position vector +.>
Figure SMS_6
One of the components (+)>
Figure SMS_10
）。
Using
Figure SMS_21
Symbol (S)>
Figure SMS_26
Can be expressed as->
Figure SMS_30
，/>
Figure SMS_22
，/>
Figure SMS_27
. Above->
Figure SMS_31
(or->
Figure SMS_33
) Is a Croneck function, when->
Figure SMS_20
Time->
Figure SMS_24
Equal to 1. In VTI medium, < >>
Figure SMS_28
Comprises only->
Figure SMS_32
、/>
Figure SMS_23
、/>
Figure SMS_25
、
Figure SMS_29
、/>
Figure SMS_34
There are a total of 5 independent stiffness parameters.
Developing time and space Fourier transformation on the wave equation (1), and constructing a VTI medium spherical wave displacement integral equation expressed by a stiffness tensor in a frequency wave number domain, wherein the equation is expressed as follows:
Figure SMS_35
；
wherein,,
Figure SMS_46
is imaginary unit, ++>
Figure SMS_38
Is an exponential function, ++>
Figure SMS_41
Is angular frequency, +.>
Figure SMS_49
As a three-dimensional integral variable,
Figure SMS_53
、/>
Figure SMS_52
、/>
Figure SMS_54
is wave vector +.>
Figure SMS_47
Component of->
Figure SMS_51
Is a matrix->
Figure SMS_37
Component of->
Figure SMS_43
And->
Figure SMS_39
Slowness vectors respectively->
Figure SMS_44
（
Figure SMS_45
,/>
Figure SMS_50
,/>
Figure SMS_36
)（/>
Figure SMS_40
) One of the components (+)>
Figure SMS_42
)，/>
Figure SMS_48
Is constant.
Step 2, solving displacement fields respectively excited by longitudinal and transverse wave point seismic sources by utilizing moment method
And taking the action of the longitudinal wave point source and the transverse wave point source on the spherical wave field into consideration by using a moment method to solve the spherical wave displacement field of the VTI medium excited by the longitudinal wave point source and the transverse wave point source respectively. The VTI medium spherical wave displacement equation for longitudinal wave point source excitation is expressed as:
Figure SMS_55
；
wherein,,
Figure SMS_59
is a matrix->
Figure SMS_60
Is>
Figure SMS_67
And->
Figure SMS_58
Is->
Figure SMS_63
Root of (A)>
Figure SMS_66
Represents vertical slowness of longitudinal wave, P represents longitudinal wave,>
Figure SMS_69
vertical slowness representing transverse waves, SV representing transverse waves; />
Figure SMS_57
、
Figure SMS_62
、/>
Figure SMS_65
For matrix->
Figure SMS_68
Is marked +.>
Figure SMS_56
Representing the companion matrix; />
Figure SMS_61
Is->
Figure SMS_64
Representing the integral.
The displacement field of the spherical waves of the VTI medium excited by the transverse wave point source is expressed as follows:
Figure SMS_70
；
wherein,,
Figure SMS_71
，/>
Figure SMS_72
；/>
Figure SMS_73
and->
Figure SMS_74
Is polar coordinates; />
Figure SMS_75
Is level slowness, ->
Figure SMS_76
Is vertical slowness.
Step 3, simplifying the spherical wave displacement equation of the VTI medium
The spherical wave displacement equation excited by the VTI medium longitudinal and transverse wave point seismic source is converted into a polar coordinate system from a Cartesian coordinate system, the spherical longitudinal and transverse wave displacement equation is expressed as the integral of horizontal slowness by using a residue theorem, and the simplification of the spherical longitudinal and transverse wave displacement equation is further realized by means of a Bessel function.
(1) Spherical longitudinal wave displacement equation for longitudinal wave point source excitation
Figure SMS_77
，/>
Figure SMS_78
，
Figure SMS_79
，/>
Figure SMS_80
，/>
Figure SMS_81
Substituting the equation (3) to obtain a spherical longitudinal wave displacement equation under the polar coordinate system, which is expressed as:
Figure SMS_82
；
wherein,,
Figure SMS_83
and->
Figure SMS_84
Is the rotation angle under the polar coordinate system; />
Figure SMS_85
，
Figure SMS_86
；/>
Figure SMS_87
Is->
Figure SMS_88
Representing the integral.
Setting polar coordinate axis
Figure SMS_89
The axis is downward, in->
Figure SMS_90
If the ratio is greater than 0, the downlink spherical longitudinal wave corresponds to the downlink spherical longitudinal wave. Will Bessel function
Figure SMS_91
、/>
Figure SMS_92
And->
Figure SMS_93
Substituting into equation (5) simplified by the remainder theorem, at +.>
Figure SMS_94
The coordinate system can obtain the downlink spherical longitudinal wave displacement equation:
Figure SMS_95
；
wherein,,
Figure SMS_96
is a Bessel function of order 0, +.>
Figure SMS_97
Is a Bessel function of order 1; />
Figure SMS_98
Representing the integral of the horizontal slowness.
At the position of
Figure SMS_99
If the ratio is less than 0, the spherical upward longitudinal wave is corresponded. Also using the Bessel function and the remainder theorem, the spherical upward longitudinal wave displacement equation is expressed as:
Figure SMS_100
；
(2) Spherical transverse wave displacement equation for transverse wave point focus excitation
Referring to the processing flow of the longitudinal wave displacement equation, the simplification of the VTI medium spherical transverse wave displacement equation (4)) of the transverse wave point source excitation is realized by means of a polar coordinate system, a residue theorem and a Bessel function.
Figure SMS_101
Corresponding to the spherical downlink transverse wave displacement equation, expressed as:
Figure SMS_102
；
Figure SMS_103
corresponding to the spherical upward transverse wave displacement equation, expressed as:
Figure SMS_104
。
step 4, solving the displacement of the spherical reflection wave at the receiving point under specific frequency and depth
And solving the displacement of the spherical reflection wave at the receiving point by utilizing the boundary conditions of continuous displacement and continuous stress. In combination with the spherical longitudinal and transverse wave displacement equation, the point focus is located at the coordinates
Figure SMS_105
Under the condition of (1), the spherical incident longitudinal wave displacement equation of the medium positioned on the upper layer is as follows:
Figure SMS_106
；
wherein,,
Figure SMS_107
；
the spherical reflection longitudinal wave displacement equation of the upper medium is as follows:
Figure SMS_108
；
the spherical reflection transverse wave displacement equation of the upper medium is as follows:
Figure SMS_109
；
the spherical transmission longitudinal wave displacement equation of the medium positioned at the lower layer is as follows:
Figure SMS_110
；
the spherical transmission transverse wave displacement equation of the medium positioned at the lower layer is as follows:
Figure SMS_111
；
wherein,,
Figure SMS_112
is a coefficient to be determined; />
Figure SMS_113
，/>
Figure SMS_114
，/>
Figure SMS_115
，/>
Figure SMS_116
Superscripts (1) and (2) represent an upper medium and a lower medium, respectively.
Spherical wave displacement vector of upper mediumrAndzthe components are respectively as follows:
Figure SMS_117
；
spherical wave displacement vector of lower mediumrAndzthe components are respectively as follows:
Figure SMS_118
；
by using the displacement continuous and stress continuous boundary condition, the method can obtain
Figure SMS_121
，/>
Figure SMS_123
，
Figure SMS_124
，/>
Figure SMS_119
，/>
Figure SMS_122
Is the longitudinal wave reflection coefficient of the plane of the VTI medium, < + >>
Figure SMS_125
Is the transverse wave reflection coefficient of the plane of the VTI medium, < >>
Figure SMS_126
And->
Figure SMS_120
Is the VTI medium plane longitudinal and transverse wave conversion coefficient.
Will be
Figure SMS_127
Substitution of displacement of spherical reflected wave at receiving point at specific frequency and depthrComponents andzthe components are respectively obtained:
Figure SMS_128
；
and
Figure SMS_129
；
wherein,,
Figure SMS_130
is the upper layerLongitudinal wave velocity of medium, < >>
Figure SMS_131
For transverse wave velocity of upper medium, +.>
Figure SMS_132
Is the incidence angle of the longitudinal wave.
Will be
Figure SMS_133
Substitution of displacement of spherical incident wave at receiving point at specific frequency and depthrComponents andzthe components are respectively obtained:
Figure SMS_134
；
and
Figure SMS_135
。
and 5, defining the spherical wave reflection coefficient of the VTI medium as the ratio of the displacement components of the spherical reflection wave and the incident wave in the polarization direction, and deducing a spherical wave reflection coefficient equation of the VTI medium represented by the stiffness tensor, wherein the equation is expressed as follows:
Figure SMS_136
；
in the form of integration of the plane wave reflection coefficient of the VTI medium, where
Figure SMS_137
Is the polarization angle of the upper medium longitudinal wave.
Step 6, deducing a frequency and depth dependent VTI medium spherical wave reflection coefficient analysis equation directly characterized by vertical longitudinal and transverse wave speed, density and anisotropic parameters
Will vertical longitudinal wave velocity
Figure SMS_138
Vertical transverse wave speed +.>
Figure SMS_139
，/>
Figure SMS_140
，
Figure SMS_141
Substituting the stiffness tensor-characterized VTI medium spherical wave reflection coefficient equation (23) to derive a frequency and depth-dependent VTI medium spherical wave reflection coefficient analysis equation directly characterized by vertical longitudinal and transverse wave speed, density and anisotropic parameters:
Figure SMS_142
；
wherein,,
Figure SMS_143
representing vertical longitudinal wave velocity, +.>
Figure SMS_144
Representing vertical transverse wave velocity, +.>
Figure SMS_145
Representing density; />
Figure SMS_146
And->
Figure SMS_147
Representing an anisotropic parameter; />
Figure SMS_148
Representing the nonlinear relationship between the parameter and the reflection coefficient.
Examples 1-4 are the calculation of spherical wave reflection coefficients of VTI media using formation model parameters using the methods provided by the present invention. Upper layer formation model parameters: vertical longitudinal wave velocity 2000 (m/s), vertical transverse wave velocity 1200 (m/s), density 2150 (kg/m) 3 ) The method comprises the steps of carrying out a first treatment on the surface of the Lower layer formation model parameters: vertical longitudinal wave velocity 2400 (m/s), vertical transverse wave velocity 1400 (m/s), density 2350 (kg/m) 3 ）。
Implementation of the embodimentsThe upper layer stratum model anisotropy parameters in example 1 are:
Figure SMS_149
、/>
Figure SMS_150
the method comprises the steps of carrying out a first treatment on the surface of the The anisotropic parameters of the lower stratum model are as follows: />
Figure SMS_151
、/>
Figure SMS_152
. The frequencies are set as: 5 Hz, 30 Hz and 90 Hz; the interface depth was set to 500 m.
FIG. 2 is a graph showing the calculated amplitude of spherical wave reflection coefficients of VTI media at frequencies of 30 Hz and reflection depths of 250 m, 1000 m and 4000 m, respectively, for example 1, wherein the amplitude is a dimensionless number. It can be seen that the spherical wave reflection coefficient of the VTI medium has a depth dependence, and that as the depth increases, the plane wave approximation error becomes smaller.
Upper layer formation model anisotropy parameters in example 2:
Figure SMS_153
、/>
Figure SMS_154
the method comprises the steps of carrying out a first treatment on the surface of the Lower layer formation model anisotropy parameters: />
Figure SMS_155
、/>
Figure SMS_156
. The frequency was set to 30 Hz; the interface depths are set as follows: 250 m, 1000 m and 4000 m.
FIG. 3 is a graph showing the calculated amplitude of the spherical wave reflection coefficient of the VTI medium for example 2 at reflection depths of 500 m and frequencies of 5 Hz, 30 Hz and 90 Hz, respectively. It can be seen that the spherical wave reflection coefficient of the VTI medium has a frequency dependence, and the plane wave approximation error becomes smaller as the frequency increases. It is also disclosed that elasticity and anisotropy parameter predictions can be implemented from seismic data of different frequency components.
Upper layer formation model anisotropy parameters in example 3:
Figure SMS_157
、/>
Figure SMS_158
the method comprises the steps of carrying out a first treatment on the surface of the Lower layer formation model anisotropy parameters:
Figure SMS_159
、/>
Figure SMS_160
0, 0.1, 0.2 respectively; the frequency is set as follows: 30 Hz; the interface depth was set to 1500 m.
FIG. 4 is a graph of the calculated anisotropy of example 3 at a frequency of 30 Hz and an interface depth of 1500 m
Figure SMS_161
The amplitude of the spherical wave reflection coefficient of the VTI medium is shown. It can be seen that the spherical wave reflection coefficient of the VTI medium versus the anisotropy parameter +.>
Figure SMS_162
Is enhanced. Critical angle is along->
Figure SMS_163
The spherical wave reflection coefficient of the VTI medium is horizontally shifted while being increased.
Upper layer formation model anisotropy parameters in example 4:
Figure SMS_164
、/>
Figure SMS_165
the method comprises the steps of carrying out a first treatment on the surface of the Lower layer formation model anisotropy parameters:
Figure SMS_166
、/>
Figure SMS_167
0, 0.1, 0.2 respectively; the frequency is set as follows: 30 Hz; the interface depth was set to 1500 m.
FIG. 5 shows calculated example 4 at a frequency of 30 Hz and an interface depth of 1500 m, different anisotropy parameters
Figure SMS_168
The amplitude of the spherical wave reflection coefficient of the VTI medium is shown. It can be seen that the spherical wave reflection coefficient of the VTI medium at near-critical angle (around 43 DEG) versus the anisotropy parameter +.>
Figure SMS_169
And->
Figure SMS_170
Is more sensitive.
The invention provides a method for analyzing the spherical longitudinal wave reflection coefficient of the VTI medium by considering the excitation condition of a point focus and the spherical wave front effect in the VTI medium, and deduces a spherical longitudinal wave reflection coefficient equation of the VTI medium. As shown in fig. 2 and 3 (black solid lines in fig. 2 and 3), compared with the prior art, the method considers the frequency and depth dependence of the seismic reflection wave, can accurately describe the spherical wave amplitude and phase frequency-dependent reflection characteristics of the VTI medium, can quantify errors of geometrical seismic approximation under different frequencies, different reflection depths and different incidence angles, and further is beneficial to improving the inversion prediction precision of the underground stratum parameters by utilizing the seismic data of different frequency components.

Claims (4)

1. A method for analyzing the reflection coefficient of a spherical longitudinal wave of a VTI medium is characterized by comprising the following steps:
step 1, solving a VTI medium spherical wave displacement integral equation;
step 2, solving displacement fields respectively excited by longitudinal and transverse wave point seismic sources by using a moment method;
step 3, simplifying a spherical wave displacement equation of the VTI medium: converting a VTI medium spherical wave displacement equation from a Cartesian coordinate system into a polar coordinate system, expressing the spherical wave displacement equation into an integral of horizontal slowness by using a residue theorem, and further simplifying the spherical wave displacement equation by using a Bessel function;
step 4, solving the displacement of the spherical reflection wave at the receiving point;
step 5, defining the spherical wave reflection coefficient of the VTI medium as the ratio of the displacement components of the spherical reflection wave and the incident wave in the polarization direction; deriving a VTI medium spherical wave reflection coefficient equation characterized by the stiffness tensor;
and 6, substituting the vertical longitudinal wave speed, the vertical transverse wave speed, the density and the anisotropic parameters on two sides of the reflecting interface into a VTI medium spherical wave reflection coefficient equation represented by the stiffness tensor by utilizing the quantitative relation between the stiffness tensor and the model parameters, and deducing a frequency and depth dependent VTI medium spherical wave reflection coefficient analysis equation directly represented by the vertical longitudinal and transverse wave speed, the density and the anisotropic parameters.
2. The method for resolving a spherical longitudinal wave reflection coefficient of a VTI medium according to claim 1, wherein in the step 1, starting from a VTI medium wave equation of a coupling point source term, a VTI medium spherical wave displacement integral equation represented by a stiffness tensor is constructed in a frequency wavenumber domain by performing a time fourier transform and a space fourier transform on the wave equation.
3. The method for resolving a spherical longitudinal wave reflection coefficient of a VTI medium according to claim 1, wherein in the step 2, the moment method is used to consider the effects of a longitudinal wave point source and a transverse wave point source on a spherical wave field so as to solve the spherical wave displacement fields of the VTI medium excited by the longitudinal wave point source and the transverse wave point source respectively.
4. The method for resolving a spherical longitudinal wave reflection coefficient of a VTI medium according to claim 1, wherein in the step 4, the displacement of the spherical reflected wave at the receiving point is resolved using a continuous displacement and continuous stress boundary condition.
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