CN112882093B - Method and system for calculating internal co-vibration deformation of elastic earth - Google Patents
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Abstract
The application discloses a method and a system for calculating the internal co-vibration deformation of an elastic earth, wherein the internal co-vibration deformation of the earth caused by an independent point source under a homogeneous sphere model is calculated through a semi-analytic semi-numerical method. The method can calculate to an ultra-high order, avoids the problem of high-order oscillation in a pure numerical method, ensures the calculation accuracy, does not occupy too much calculation time, and has important significance for deep movement of the earth and analysis of the inoculation process and mechanism of the next earthquake.
Description
Technical Field
The present application relates to the field of geophysics technologies, and in particular, to a method and system for calculating internal seismological deformation of an elastic earth.
Background
The occurrence of earthquake generally accompanies the processes of earthquake pregnancy, earthquake initiation, stress adjustment and the like, and new earthquake inoculation is started, and research on the mechanism of earthquake inoculation, the cracking process, post-earthquake adjustment and the like is always the most important problem of scientists, so that the earthquake inoculation device can help us to know the earthquake more deeply and provide scientific basis for reducing the damage of the earthquake. Scientists began to study the deformation calculation of the same earthquake displacement and strain change concentrated on the earth surface caused by earthquake in 1958, and the knowledge of the earth model underwent the process of 'homogeneous semi-infinite space model-homogeneous sphere model-layered semi-infinite space model/layered sphere model', and the earth surface same earthquake deformation calculation method caused by earthquake was improved along with the evolution of the earth model.
With the development of seismology and the deepening of research on the characteristics of pregnant earthquakes, the internal displacement and strain change of the earth generated by earthquakes also draw attention, the research on the characteristics of the internal co-vibration deformation of the earth has important significance on the deep movement of the earth and the analysis on the inoculation process and mechanism of the next earthquake, however, the method for calculating the internal co-vibration displacement and strain change of the earth is quite few, and the analytical solution calculation method under a homogeneous semi-infinite space model proposed by Japanese seismologist Okada is quite simple, convenient and quick at present, and the internal co-vibration change of the earth caused by any point source and limited fault can be calculated quickly by directly adopting the analytical solution calculation method, but the earth model adopted by the method is not spherical. The numerical calculation method under the homogeneous sphere model proposed by Japanese scholars Takagi and Okada is that the surface displacement and strain calculation formula expressed in the form of spherical harmonic function is deduced through stress-strain relation, poisson equation and balance equation and by introducing spherical harmonic function and Association Legend function, the numerical calculation method is a numerical solution calculation method, the order required to be calculated to meet the precision requirement is especially high, the calculated amount is large, the time is relatively long, and numerical oscillation easily occurs when the highest order reaches a certain degree. Both methods have certain drawbacks.
Disclosure of Invention
Purpose of (one) application
Based on the method, in order to truly calculate the co-vibration deformation of any position in the earth caused by any point source, the high-order vibration problem of a pure numerical calculation method is avoided, the calculation time is saved while the calculation precision is ensured, the earthquake is recognized more deeply, a scientific basis is provided for reducing the earthquake injury, and the following technical scheme is disclosed.
(II) technical scheme
As a first aspect of the present application, the present application discloses a method for calculating an elastic earth internal seismostatic deformation, including:
the stress field ball function and the displacement field ball function are brought into a balance equation and a poisson equation which are formed by the stress field and the displacement field, a spherical differential equation set and a ring differential equation set are obtained through simplified calculation, and then a spherical basic solution and a ring basic solution are calculated;
the spherical basic solution and the annular basic solution are brought into a ground surface boundary condition and a seismic source function of an independent point source to obtain a non-homogeneous equation set, and then a spherical solution coefficient to be determined and an annular solution coefficient to be determined are calculated;
according to the spherical solution coefficient, the annular solution coefficient and the size relation between the depth of the calculated surface and the depth of the seismic source, calculating spherical deformation values and annular deformation values at different positions in the earth;
the spherical deformation value and the annular deformation value are brought into a non-homogeneous equation set, and the analysis solutions of dislocation Love numbers of different independent seismic sources are calculated;
carrying out analysis solution of dislocation Love number into a sphere function integral summation formula to obtain a displacement lattice function and a strain lattice function;
defining the magnitude factor of independent point source as UdS/R 2 The actual magnitude of vibration is brought into the displacement green function and the strain green function to obtain the internal co-vibration deformation of the elastic earth of the vibration source point to be calculated, wherein the co-vibration deformation comprises co-vibration displacement and co-vibration strain, U is the displacement, S is the area, and R is the radius;
the four independent point sources comprise a walk sliding point source, a tilt sliding point source, a horizontal tension point source and a vertical tension point source.
In one possible implementation, poisson and equilibrium equations, which are built up from stress fields, displacement fields, are as follows:
where f is the unit point force of the elastomeric sphere surface, I is the unit tensor, superscript T denotes the transpose, μ and λ are the elastic dielectric constants at the source, i.e., the lame constants. τ is the stress tensor, u is the displacement field, ρ is the density.
In one possible embodiment, the displacement field ball function, the strain field ball function, the Shan Weidian force ball function are as follows:
wherein,for the displacement field +.>Is stress tensor->Is under stressRadial component, spherical harmonic->And->Is an associative Legend function->Is the function of θ is the complementary weft, +.>Is longitude, r is earth radius, +.>Representing the position of Shan Weidian force f, "×" represents the complex conjugate, v is the unit vector, and t represents the annular deformation. y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress, +.>And->Is the horizontal displacement and stress component in the cyclic deformation factor.
In one possible embodiment, a spherical differential equation set and a toroidal differential equation set are obtained, and then a spherical basic solution and a toroidal basic solution are calculated, wherein the spherical differential equation set is obtained as follows:
wherein y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial direction of stress and waterA horizontal component; r is the earth radius, μ and λ are the elastic dielectric constants at the source, β=λ+2μ,delta represents the variation, n represents the order, r 0 Is the location of the single-point source,
the set of ring differential equations obtained is as follows:
wherein,and->Is the horizontal displacement and stress component in the cyclic deformation factor, the superscript t indicates the cyclic solution part,/and-> Is an associative Legend function->Function of->Representing the position of Shan Weidian force f, representing the complex conjugate, v is the unit vector,
the basic solution of the sphere is as follows:
the basic solution of the ring is as follows:
in one possible implementation, the surface boundary conditions and the source function of the independent point sources are as follows:
y 2 (r)| r＝R ＝y 4 (r)| r＝R ＝0
y(r)| r＝0 ＜+∞
where s represents the source function of 4 independent point sources, including: a walk-slide point source, a tilt-slide point source, a horizontal tension point source and a vertical tension point source.
In one possible implementation, the non-homogeneous system of equations is as follows:
wherein R is the earth radius, R s ＝(R-r 0 ) R represents regularized source depth, beta i (i=1, 2, …, 6) is a sphere solution coefficient;
wherein beta is i t (i=1, 2, 3) is a cyclic solution coefficient.
In one possible embodiment, the ball-type deformation value and the ring-type deformation value are as follows:
when the depth of the calculated surface is smaller than the depth of the seismic source, the spherical deformation value and the annular deformation value are as follows:
when the calculated surface depth is greater than the source depth, the spherical deformation value and the annular deformation value are as follows:
in one possible implementation, calculating dislocation Love number by using the spherical deformation value and the annular deformation value, and then obtaining a displacement green function and a strain green function displacement green function by using integral summation of the spherical function;
the calculated dislocation Love number of the independent seismic source is analyzed as follows:
where ij= 12,32,220,33 represents 4 independent sources;
the resulting displacement green function is as follows:
the resulting strain green function is as follows:
as a second aspect of the present application, the present application also discloses a system for calculating an internal seismostatic deformation of an elastic earth, including:
the basic solution calculation module is used for bringing the stress field ball function and the displacement field ball function into a balance equation and a poisson equation which are formed by the stress field and the displacement field, simplifying and calculating to obtain a spherical differential equation set and a ring differential equation set, and further calculating a spherical basic solution and a ring basic solution;
the undetermined coefficient calculation module is used for bringing the spherical basic solution and the annular basic solution into the surface boundary condition and the seismic source function of the independent point source to obtain a non-homogeneous equation set, and further calculating the spherical solution undetermined coefficient and the annular solution undetermined coefficient;
the deformation value calculation module is used for calculating spherical deformation values and annular deformation values at different positions in the earth according to the spherical solution coefficient, the annular solution coefficient and the size relation between the depth of the calculated surface and the depth of the seismic source;
the dislocation Love number calculation module is used for bringing the spherical deformation value and the annular deformation value into a non-homogeneous equation set and calculating dislocation Love number analysis solutions of different independent seismic sources;
the Green function calculation module is used for bringing the dislocation Love number analysis solution into a ball function integral summation formula to obtain a displacement green function and a strain green function,
the co-vibration deformation calculation module is used for defining that the magnitude factor of the independent point source is UdS/R 2 The actual magnitude of vibration is brought into the displacement green function and the strain green function to obtain the internal co-vibration deformation of the elastic earth of the vibration source point to be calculated, wherein the co-vibration deformation comprises co-vibration displacement and co-vibration strain, U is the displacement, S is the area, and R is the radius;
the four independent point sources comprise a walk sliding point source, a tilt sliding point source, a horizontal tension point source and a vertical tension point source.
In one possible implementation, poisson and equilibrium equations, which are built up from stress fields, displacement fields, are as follows:
where f is the unit point force of the elastomeric sphere surface, I is the unit tensor, superscript T denotes the transpose, μ and λ are the elastic dielectric constants at the source, i.e., the lame constants. τ is the stress tensor, u is the displacement field, ρ is the density.
In one possible embodiment, the displacement field ball function, the strain field ball function, the Shan Weidian force ball function are as follows:
wherein,for the displacement field +.>Is stress tensor->Is the radial component of stress, spherical harmonic +.>And->Is an associative Legend function->Is the function of θ is the complementary weft, +.>Is longitude, r is earth radius, +.>Represents Shan Weidian force f position, "×" represents complex conjugate, v is unit vector, t represents ring type transformationShape. y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress, +.>And->Is the horizontal displacement and stress component in the cyclic deformation factor.
In one possible embodiment, a spherical differential equation set and a toroidal differential equation set are obtained, and then a spherical basic solution and a toroidal basic solution are calculated, wherein the spherical differential equation set is obtained as follows:
wherein y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress; r is the earth radius, μ and λ are the elastic dielectric constants at the source, β=λ+2μ,delta represents the variation, n represents the order, r 0 Is the location of the single-point source,
the set of ring differential equations obtained is as follows:
wherein,and->Is a ringHorizontal displacement and stress component in the deformation factor, superscript t stands for cyclic solution part,/-for-> Is an associative Legend function->Function of->Representing the position of Shan Weidian force f, "×" represents the complex conjugate, v is the unit vector,
the basic solution of the sphere is as follows:
the basic solution of the ring is as follows:
in one possible implementation, the surface boundary conditions and the source function of the independent point sources are as follows:
y 2 (r)| r＝R ＝y 4 (r)| r＝R ＝0
y(r)| r＝0 ＜+∞
where s represents the source function of 4 independent point sources, including: a walk-slide point source, a tilt-slide point source, a horizontal tension point source and a vertical tension point source.
In one possible implementation, the non-homogeneous system of equations is as follows:
wherein R is the earth radius, R s ＝(R-r 0 ) R represents regularized source depth, beta i (i=1, 2, …, 6) is a sphere solution coefficient;
wherein beta is i t (i=1, 2, 3) is a cyclic solution coefficient.
In one possible embodiment, the ball-type deformation value and the ring-type deformation value are as follows:
when the depth of the calculated surface is smaller than the depth of the seismic source, the spherical deformation value and the annular deformation value are as follows:
when the calculated surface depth is greater than the source depth, the spherical deformation value and the annular deformation value are as follows:
in one possible implementation, calculating dislocation Love number by using the spherical deformation value and the annular deformation value, and then obtaining a displacement green function and a strain green function displacement green function by using integral summation of the spherical function;
the calculated dislocation Love number of the independent seismic source is analyzed as follows:
where ij= 12,32,220,33 represents 4 independent sources;
the resulting displacement green function is as follows:
the resulting strain green function is as follows:
(III) beneficial effects
According to the method and the system for calculating the internal isoseism displacement and strain of the elastic earth, the isoseism deformation at any position in the earth caused by any point source can be truly calculated through the semi-analytic semi-numerical calculation method, the high-order concussion problem of the pure numerical calculation method is avoided, the calculation accuracy is ensured, the calculation time is saved, the earthquake is deeply recognized, and the scientific basis is provided for reducing the earthquake injury.
Drawings
The embodiments described below with reference to the drawings are exemplary and intended for the purpose of illustrating and explaining the present application and are not to be construed as limiting the scope of protection of the present application.
FIG. 1 is a flow chart of an embodiment of a method of calculating elastic earth internal co-vibration deformation disclosed herein.
Fig. 2 is a model diagram of 4 independent sources.
FIG. 3 is a flowchart of a method for calculating the internal seismic deformation of the elastic earth disclosed in the present application.
Fig. 4 is a graph of deformation values of a walk-Slip point source (Slip-Slip) at different depths inside the earth.
Fig. 5 shows deformation values of a Dip-slide point source (Dip-slide) at different depths inside the earth.
Fig. 6 is a graph of deformation values of a horizontal stretching point source (Horizontal Tensile) at different depths inside the earth.
Fig. 7 is a graph of deformation values of a Vertical Tensile point source (Vertical tension) at different depths inside the earth.
FIG. 8 is a graph of the displacement and strain changes produced by 4 independent point sources at 0km-40km inside the earth.
FIG. 9 is a block diagram of an embodiment of a system for calculating elastic earth internal seismic deformation disclosed herein.
Detailed Description
In order to make the purposes, technical solutions and advantages of the implementation of the present application more clear, the technical solutions in the embodiments of the present application will be described in more detail below with reference to the accompanying drawings in the embodiments of the present application.
An embodiment of a method for calculating the internal seismic deformation of elastic earth disclosed in the present application is described in detail below with reference to fig. 1. As shown in fig. 1, the method disclosed in this embodiment mainly includes the following steps 100 to 600.
Step 100, bringing the stress field ball function and the displacement field ball function into a balance equation and a poisson equation which are formed by the stress field and the displacement field, simplifying and calculating to obtain a spherical differential equation set and a ring differential equation set, and further calculating a spherical basic solution and a ring basic solution.
Dislocation is generated in the elastic earth so as to cause vibration deformation on the earth surface and different depth planes in the earth, the unit point force on the surface of the elastic ball generates a same-vibration displacement field and a stress field, and the displacement field is generatedStress tensor->Writing into sphere function, taking it into poisson equation and balance equation in sphere function form,
poisson's equation is:
wherein Δ represents the Laplacian, anAnd f may be an equation of real or complex values on the manifold.
The equilibrium equation is a mathematical form of the equilibrium condition of the force system. The balance of any force system in space is that the principal vector and the principal moment of the force system are both equal to zero, i.e. r=0, m O ＝0
The equilibrium equation for any force system for a plane is:
∑F x ＝0，∑F y ＝0，∑M O (F)＝0
the equilibrium equation for the planar junction system is:
∑F x ＝0，∑F y ＝0
in at least one embodiment, poisson and equilibrium equations, which are built up from stress fields, displacement fields, are as follows:
where f is the unit point force of the elastomeric sphere surface, I is the unit tensor, superscript T denotes the transpose, μ and λ are the elastic dielectric constants at the source, i.e., the lame constants. τ is the stress tensor, u is the displacement field, ρ is the density.
In at least one embodiment, the displacement field ball function, the strain field ball function, shan Weidian force ball function are as follows:
/>
wherein,for the displacement field +.>Is stress tensor->Is the radial component of stress, spherical harmonic +.>And->Is an associative Legend function->Is the function of θ is the complementary weft, +.>Is longitude, r is earth radius, +.>Representing the position of Shan Weidian force f, "×" represents the complex conjugate, v is the unit vector, and t represents the annular deformation. y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress, +.>And->Is the horizontal displacement and stress component in the cyclic deformation factor.
In at least one embodiment, a spherical differential equation set and a circular differential equation set are obtained, and then a spherical basic solution and a circular basic solution are calculated, wherein the spherical differential equation set is obtained as follows:
wherein y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress; r is the earth radius, μ and λ are the elastic dielectric constants at the source, β=λ+2μ,delta represents the variation, n represents the order, r 0 Is the location of the single-point source,
the set of ring differential equations obtained is as follows:
wherein,and->Is the horizontal displacement and stress component in the cyclic deformation factor, the superscript t indicates the cyclic solution part,/and-> Is an associative Legend function->Function of->Representing the position of Shan Weidian force f, representing the complex conjugate, v is the unit vector,
the basic solution of the sphere is as follows:
the basic solution of the ring is as follows:
step 200, bringing the spherical basic solution and the annular basic solution into a seismic source function of an earth surface boundary condition and an independent point source to obtain a non-homogeneous equation set, and further calculating a spherical solution coefficient and an annular solution coefficient.
Boundary conditions refer to the law of variation of the variable or derivative thereof solved over time and place at the boundaries of the solution area. The boundary conditions are the preconditions for a definite solution of the control equation, given the boundary conditions are required for any problem. The processing of boundary conditions directly affects the accuracy of the calculation result. While solving differential equations with a definite solution, conditions must be introduced, these additional conditions being called definite solution conditions.
In at least one embodiment, the surface boundary conditions and the source function of the independent point sources are as follows:
y 2 (r)| r＝R ＝y 4 (r)| r＝R ＝0
y(r)| r＝0 ＜+∞
where s represents the source function of 4 independent point sources, including: a walk-slide point source, a tilt-slide point source, a horizontal tension point source and a vertical tension point source.
The model diagrams of the 4 independent seismic sources are shown in fig. 2, and the model diagrams are a walk sliding point source, a tilt sliding point source, a horizontal stretching point source and a vertical stretching point source in sequence from left to right.
In at least one embodiment, the system of non-homogeneous equations is as follows:
wherein R is the earth radius, R s ＝(R-r 0 ) R represents regularized source depth, beta i (i=1, 2, …, 6) is a sphere solution coefficient;
wherein beta is i t (i=1, 2, 3) is a cyclic solution coefficient.
And 300, calculating spherical deformation values and annular deformation values at different positions in the earth according to the spherical solution coefficient, the annular solution coefficient and the size relation between the depth of the calculated surface and the depth of the seismic source.
The source depth refers to the vertical distance from the source to the ground (center of the epicenter).
The size relation between the depth of the calculated surface and the depth of the seismic source is divided into two types, wherein the first type is that the depth of the calculated surface is smaller than the depth of the seismic source, namely the calculated surface is above the seismic source and is positioned between the ground surface and the seismic source; the second is that the depth of the calculated surface is greater than the depth of the source and the calculated surface is below the source between the source and the boundary of the nuclear valance. The internal seismomorphism calculation expressions at different positions are different, and spherical and toroidal morphisms inside the earth need to be calculated in cases.
In at least one embodiment, the ball-type deformation value and the ring-type deformation value are as follows:
when the depth of the calculated surface is smaller than the depth of the seismic source, the spherical deformation value and the annular deformation value are as follows:
when the calculated surface depth is greater than the source depth, the spherical deformation value and the annular deformation value are as follows:
/>
step 400, the spherical deformation value and the annular deformation value are brought into a non-homogeneous equation set, and the dislocation Love number analytic solutions of different independent seismic sources are calculated.
Dislocation Love number: the dislocation Love number is a dimensionless parameter introduced by describing the elastic deformation of the earth under the action of a point seismic source, and comprises h (related to radial displacement), l (related to horizontal displacement) and k (related to gravitational potential).
And calculating dislocation Love number analysis solutions of different independent seismic sources according to the spherical deformation values and the annular deformation values corresponding to different relations between the depth of the calculated surface and the depth of the seismic sources.
In at least one embodiment, the calculated dislocation Love number of the independent seismic source is resolved as follows:
where ij= 12,32,220,33 represents 4 independent sources.
And 500, bringing the dislocation Love number analysis solution into a sphere function integral summation formula to obtain a displacement lattice function and a strain lattice function.
And (3) according to the dislocation Love number analytic solution obtained in the step 400, obtaining a displacement green's function and a strain green's function generated by 4 independent point sources in the earth on any layer by integrating and summing the ball function.
In at least one embodiment, the resulting displacement green function is as follows:
/>
the resulting strain green function is as follows:
step 600, defining the magnitude factor of the independent point source as UdS/R 2 =1, the actual magnitude is taken into the displacement green function and strainAnd (3) a Green function, namely obtaining the internal co-vibration deformation of the elastic earth of the vibration source point to be calculated, wherein the co-vibration deformation comprises co-vibration displacement and co-vibration strain, U is the displacement, S is the area, and R is the radius.
The set of green's function is directly caused by an independent point source, which we define the magnitude factor of the independent point source as UdS/R 2 =1 (i.e. displacement x area/radius 2 =1). And (5) carrying out actual magnitude according to the displacement green function and the strain green function obtained in the step 500, so as to obtain the internal co-vibration displacement deformation and the strain deformation of the elastic earth of the vibration source point to be calculated.
Specifically, referring to fig. 3, the method includes the following complete calculation steps:
and establishing a stress-strain relation, a poisson equation and a balance equation according to a co-vibration displacement field and a stress field generated by unit point force on the surface of the elastic ball, introducing a displacement field ball function and a stress field ball function to construct a differential equation set, and then solving the equation set to obtain a spherical basic solution (analytic solution) and a circular basic solution (analytic solution). The spherical basic solution and the annular basic solution are introduced into a seismic source function of an earth surface boundary condition and an independent point source, a non-homogeneous equation set is built, and unknown coefficients of the spherical solution and the annular solution are solved. And carrying the solved spherical solution and the annular solution back into a non-homogeneous equation set, calculating displacement components and stress components, judging the relation between the depth (h) of a calculated surface and the depth (ds) of a seismic source, respectively calculating internal common-seismic deformation values at different positions above the seismic source (h < ds) and below the seismic source (h > ds) in the earth by using different expressions, and obtaining dislocation Love number analysis solutions of 4 independent seismic sources according to calculation results. And finally, according to the obtained dislocation Love number analysis solution, utilizing sphere function integral summation to obtain displacement green's functions and strain green's functions generated by 4 independent point sources in the earth on any layer. And calculating the internal co-vibration displacement deformation and the strain deformation of the elastic earth of the vibration source point to be calculated by using the obtained displacement green function and the strain green function.
Taking a sliding point seismic source at 20km in the earth as an example for calculation:
to show the deformation of 4 independent point sources in the earth generated by different depths in the earth, we draw the calculation results of the displacement and strain in the range of different earthquake angular distances (θ), and calculate their internal deformation values at the ground surface (h 0), the underground 2km (h 2), 12km (h 12), 28km (h 28) and 40km (h 40), respectively, as shown in fig. 4-7;
to show the results of the deformation of the earth with the depth, we give the results of the Displacement (Displacement) and Strain (Strain) component calculation of 4 independent sources, respectively, the deformation value of the earth decays with the increase of the center distance of the seismic, so we only give the calculation results of the depth from the earth surface (0 km) to 40km, as shown in figure 8,
an embodiment of a system for calculating elastic earth internal seismic deformation disclosed herein is described in detail below with reference to fig. 9. The embodiment is used for implementing the embodiment of the method for calculating the internal seismic deformation of the elastic earth. As shown in fig. 9, the system disclosed in this embodiment includes:
the basic solution calculation module is used for bringing the stress field ball function and the displacement field ball function into a balance equation and a poisson equation which are formed by the stress field and the displacement field, simplifying and calculating to obtain a spherical differential equation set and a ring differential equation set, and further calculating a spherical basic solution and a ring basic solution;
the undetermined coefficient calculation module is used for bringing the spherical basic solution and the annular basic solution into the surface boundary condition and the seismic source function of the independent point source to obtain a non-homogeneous equation set, and further calculating the spherical solution undetermined coefficient and the annular solution undetermined coefficient;
the deformation value calculation module is used for calculating spherical deformation values and annular deformation values at different positions in the earth according to the spherical solution coefficient, the annular solution coefficient and the size relation between the depth of the calculated surface and the depth of the seismic source;
the dislocation Love number calculation module is used for bringing the spherical deformation value and the annular deformation value into a non-homogeneous equation set and calculating dislocation Love number analysis solutions of different independent seismic sources;
the green's function calculation module is used for bringing the dislocation Love number analysis solution into a ball function integral summation formula to obtain a displacement green's function and a strain green's function;
the co-vibration deformation calculation module defines that the magnitude factor of the independent point source is UdS/R 2 =1, willThe actual magnitude of the vibration is brought into the displacement green function and the strain green function to obtain the internal co-vibration deformation of the elastic earth of the vibration source point to be calculated, wherein the co-vibration deformation comprises co-vibration displacement and co-vibration strain, U is the displacement, S is the area, and R is the radius;
the four independent point sources comprise a walk sliding point source, a tilt sliding point source, a horizontal tension point source and a vertical tension point source.
In at least one embodiment, poisson and equilibrium equations, which are built up from stress fields, displacement fields, are as follows:
where f is the unit point force of the elastomeric sphere surface, I is the unit tensor, superscript T denotes the transpose, μ and λ are the elastic dielectric constants at the source, i.e., the lame constants. τ is the stress tensor, u is the displacement field, ρ is the density.
In at least one embodiment, the displacement field ball function, the strain field ball function, shan Weidian force ball function are as follows:
wherein,for the displacement field +.>Is stress tensor->Is the radial component of stress, spherical harmonic +.>And->Is an associative Legend function->Is the function of θ is the complementary weft, +.>Is longitude, r is earth radius, +.>Represents the position of Shan Weidian force f, x represents the complex conjugate, v is the unit vector, and t represents the annular deformation. y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress, +.>And->Is the horizontal displacement and stress component in the cyclic deformation factor.
In at least one embodiment, a spherical differential equation set and a circular differential equation set are obtained, and then a spherical basic solution and a circular basic solution are calculated, wherein the spherical differential equation set is obtained as follows:
wherein y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress; r is the earth radius, μ and λ are the elastic dielectric constants at the source, β=λ+2μ,delta represents the variation, n represents the order, r 0 Is the location of the single-point source,
the set of ring differential equations obtained is as follows:
wherein,and->Is the horizontal displacement and stress component in the cyclic deformation factor, the superscript t indicates the cyclic solution part,/and-> Is an associative Legend function->Function of->Representing the position of Shan Weidian force f, representing the complex conjugate, v is the unit vector,
the basic solution of the sphere is as follows:
the basic solution of the ring is as follows:
in at least one embodiment, the surface boundary conditions and the source function of the independent point sources are as follows:
y 2 (r)| r＝R ＝y 4 (r)| r＝R ＝0
y(r)| r＝0 ＜+∞
where s represents the source function of 4 independent point sources, including: a walk-slide point source, a tilt-slide point source, a horizontal tension point source and a vertical tension point source.
In at least one embodiment, the system of non-homogeneous equations is as follows:
wherein R is the earth radius, R s ＝(R-r 0 ) R represents regularized source depth, beta i (i=1, 2, …, 6) is a sphere solution coefficient;
wherein beta is i t (i=1, 2, 3) is a cyclic solution coefficient.
In at least one embodiment, the ball-type deformation value and the ring-type deformation value are as follows:
when the depth of the calculated surface is smaller than the depth of the seismic source, the spherical deformation value and the annular deformation value are as follows:
when the calculated surface depth is greater than the source depth, the spherical deformation value and the annular deformation value are as follows:
/>
in at least one embodiment, the spherical deformation value and the annular deformation value are utilized to calculate dislocation Love number, then the spherical function integral summation is utilized to obtain a displacement green function and a strain green function,
the calculated dislocation Love number of the independent seismic source is analyzed as follows:
where ij= 12,32,220,33 represents 4 independent sources.
The resulting displacement green function is as follows:
the resulting strain green function is as follows:
the division of modules herein is merely a division of logic functions, and other manners of division are possible in actual implementation, for example, multiple modules may be combined or integrated in another system. The modules illustrated as separate components may or may not be physically separate.
The foregoing is merely specific embodiments of the present application, but the scope of the present application is not limited thereto, and any changes or substitutions easily conceivable by those skilled in the art within the technical scope of the present application should be covered in the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.
Claims (10)
1. A method of calculating the internal seismological deformation of an elastic earth, comprising:
the stress field ball function and the displacement field ball function are brought into a balance equation and a poisson equation which are formed by the stress field and the displacement field, a spherical differential equation set and a ring differential equation set are obtained through calculation, and then a spherical basic solution and a ring basic solution are calculated;
bringing the spherical basic solution and the annular basic solution into a ground surface boundary condition and a seismic source function of an independent point source to obtain a non-homogeneous equation set, and further calculating a spherical solution coefficient to be determined and an annular solution coefficient to be determined;
according to the spherical solution coefficient, the annular solution coefficient and the size relation between the depth of the calculated surface and the depth of the seismic source, calculating spherical deformation values and annular deformation values at different positions in the earth;
carrying the spherical deformation value and the annular deformation value into the non-homogeneous equation set, and calculating dislocation Love number analysis solutions of different independent seismic sources;
carrying the dislocation Love number analysis solution into a sphere function integral summation formula to obtain a displacement lattice function and a strain lattice function;
defining the magnitude factor of independent point source as UdS/R 2 The actual magnitude of the vibration is brought into the displacement green 'S function and the strain green' S function to obtain the internal co-vibration deformation of the elastic earth of the vibration source point to be calculated, including the co-vibration displacement and the co-vibration strain, wherein U is the displacement and S is the areaR is a radius;
the four independent point sources comprise a walk sliding point source, a tilt sliding point source, a horizontal tension point source and a vertical tension point source.
2. The method of claim 1, wherein the spherical differential equation set and the annular differential equation set are obtained, and further a spherical basic solution and an annular basic solution are calculated, wherein the spherical differential equation set is obtained as follows:
wherein y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress; r is the earth radius, μ and λ are the elastic dielectric constants at the source, β=λ+2μ,delta represents the variation, n represents the order, r 0 Is the location of the unit point source;
the set of ring differential equations obtained is as follows:
wherein,and->Is the horizontal displacement and stress component in the cyclic deformation factor, the superscript t indicates the cyclic solution part, is an associative Legend function->Function of->Representing the position of Shan Weidian force f, representing the complex conjugate, v being the unit vector;
the basic solution of the sphere is as follows:
the basic solution of the ring is as follows:
3. the method of claim 2, wherein the system of non-homogeneous equations is as follows:
wherein R is the earth radius, R s ＝(R-r 0 ) R represents regularized source depth, beta i (i=1, 2, …, 6) is a sphere solution coefficient;
wherein beta is i t (i=1, 2, 3) is a cyclic solution coefficient.
4. A method according to claim 3, wherein the ball-type deformation value and the ring-type deformation value are as follows:
when the depth of the calculated surface is smaller than the depth of the seismic source, the spherical deformation value and the annular deformation value are as follows:
when the calculated surface depth is greater than the source depth, the spherical deformation value and the annular deformation value are as follows:
5. the method of claim 4, wherein the displacement green's function and the strain green's function are obtained by integrating and summing ball functions after calculating dislocation Love numbers by using the ball-type deformation values and ring-type deformation values;
the calculated dislocation Love number of the independent seismic source is analyzed as follows:
where ij= 12,32,220,33 represents 4 independent sources;
the resulting displacement green function is as follows:
the resulting strain green function is as follows:
6. a system for calculating the internal seismic deformation of an elastic earth, comprising:
and a basic solution calculation module: the spherical differential equation set and the annular differential equation set are obtained by simplified calculation, so that a spherical basic solution and an annular basic solution are calculated;
and the undetermined coefficient calculating module is used for: the spherical basic solution and the annular basic solution are brought into a ground surface boundary condition and a seismic source function of an independent point source to obtain a non-homogeneous equation set, and then a spherical solution coefficient to be determined and an annular solution coefficient to be determined are calculated;
deformation value calculation module: the spherical deformation value and the annular deformation value at different positions in the earth are calculated according to the spherical solution coefficient, the annular solution coefficient and the size relation between the depth of the calculated surface and the depth of the seismic source;
dislocation Love number calculation module: the method is used for bringing the spherical deformation value and the annular deformation value into the non-homogeneous equation set and calculating dislocation Love number analysis solutions of different independent seismic sources;
the green function calculation module: the dislocation Love number analysis solution is brought into a sphere function integral summation formula to obtain a displacement lattice function and a strain lattice function,
the co-vibration deformation calculation module: the magnitude factor used to define the independent point source is UdS/R 2 The actual magnitude of vibration is brought into the displacement green function and the strain green function to obtain the internal co-vibration deformation of the elastic earth of the vibration source point to be calculated, wherein the co-vibration deformation comprises co-vibration displacement and co-vibration strain, U is the displacement, S is the area, and R is the radius;
the four independent point sources comprise a walk sliding point source, a tilt sliding point source, a horizontal tension point source and a vertical tension point source.
7. The system of claim 6, wherein the system of spherical differential equations and the system of annular differential equations are derived to calculate a spherical base solution and an annular base solution, wherein the system of spherical differential equations is derived as follows:
wherein y is 1 To y 4 Is a spherical deformation factor, y 1 And y 3 Is the radial and horizontal component of displacement, y 2 And y 4 Is the radial and horizontal component of stress; r is the earth radius, μ and λ are the elastic dielectric constants at the source, β=λ+2μ,delta represents the variation, n represents the order, r 0 Is the location of the unit point source;
the set of ring differential equations obtained is as follows:
wherein,and->Is the horizontal displacement and stress component in the cyclic deformation factor, the superscript t indicates the cyclic solution part, is an associative Legend function->Function of->Representing the position of Shan Weidian force f, representing the complex conjugate, v being the unit vector;
the basic solution of the sphere is as follows:
the basic solution of the ring is as follows:
8. the system of claim 7, wherein the system of non-homogeneous equations is as follows:
wherein R is the earth radius, R s ＝(R-r 0 ) R represents regularized source depth, beta i (i=1, 2, …, 6) is a sphere solution coefficient;
wherein beta is i t (i=1, 2, 3) is a cyclic solution coefficient.
9. The system of claim 8, wherein the ball-type deformation value and the ring-type deformation value are as follows:
when the depth of the calculated surface is smaller than the depth of the seismic source, the spherical deformation value and the annular deformation value are as follows:
when the calculated surface depth is greater than the source depth, the spherical deformation value and the annular deformation value are as follows:
10. the system of claim 9, wherein the displacement green's function and the strain green's function are obtained by integrating and summing the ball function after calculating the dislocation Love number using the ball-type deformation value and the ring-type deformation value;
the calculated dislocation Love number of the independent seismic source is analyzed as follows:
where ij= 12,32,220,33 represents 4 independent sources;
the resulting displacement green function is as follows:
the resulting strain green function is as follows:
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