What Is Seismic Tomography?

Geoscience tomography is a geophysical technology that uses the theory of medical X-ray CT to investigate the distribution of underground physical property parameters in detail. It is divided into seismic tomography, electromagnetic tomography, and resistivity tomography. Seismic tomography is a technique that uses seismic data to invert the physical properties of underground structures, and analyzes and draws the image layer by layer. Seismic tomography can be divided into global tomography, regional tomography, and local tomography according to the scale of the study area; according to the source of the used data, it can be divided into natural seismic tomography (large-scale deep lateral heterogeneity research) 2. Artificial seismic sounding (mainly studying the distribution of shallow interfaces). According to the theoretical basis, it is generally divided into tomography based on ray equation and tomography based on wave equation.

Geoscience tomography is a geophysical technology that uses the theory of medical X-ray CT to investigate the distribution of underground physical property parameters in detail. Divided into seismic tomography, electromagnetic wave tomography and resistivity tomography. Seismic tomography is a technique that uses seismic data to invert the physical properties of underground structures, and analyzes and draws the image layer by layer. Its main purpose is to determine the fine structure and local heterogeneity inside the earth. Relatively speaking, seismic tomography is more widely used than the other two methods. This is because the velocity of seismic waves has a relatively stable correlation with the properties of rocks. The degree of attenuation of seismic waves is less than that of electromagnetic waves, and the velocity of electromagnetic waves is fast and difficult to measure. ^[1]

Seismic tomography involves three aspects: data acquisition, data processing (data inversion calculation and image reconstruction), and interpretation of imaging results. Seismic tomography is the main purpose of data acquisition, the basis of data interpretation, and the main part of data processing. Seismic tomography mainly includes the following parts: parameterization of the model, theoretical calculation of the properties of the underground medium (ray tracing, waveform fitting), inversion and image reconstruction, evaluation of the inversion results (resolution analysis) ^{[ 3]}

The tomographic results are obtained by iterative inversion based on the initial model. Therefore, the closeness between the initial model and the real underground structure is directly related to whether the imaging results can accurately reflect the objective material properties. How to describe the initial model reasonably and accurately is very important. Early research generally assumed that the model was a homogeneous layered horizontally isotropic medium model. This is only a rough model, which is far from meeting the needs of practical applications. With the deepening of research, the model gradually transitions to a three-dimensional non-uniform anisotropic arbitrary interface medium model. Some international standard models include the Marmousi model in two dimensions, the salt dome model in three dimensions, and the inverse mask type.

In seismic tomography, the final inversion of the properties of the underground medium is based on dividing the study area into multiple non-overlapping pixels and mapping based on the gray levels of the pixels (the properties of the underground medium obtained by the inversion). Therefore, in the seismic tomography, the grid method is used to parameterize the model. The gridding method has also been developed from the initial uniform velocity distribution model in the grid to the subsequent node speed values. Interpolation is used to obtain the velocity of each point in the grid; from a regular uniform grid to a dynamically variable scale Regular grid.

Before the forward numerical simulation, another important work that needs to be done is data preprocessing. The quality of seismic tomography results depends not only on the selection of the initial model, but also on the completeness of the data space. Such as the amount of data, the accuracy of the data, the uniformity and density of the ray distribution. For artificial seismic data, shot points and receiving points can be artificially selected, so the above requirements can be met. However, for natural seismic data, the accuracy of imaging can only be improved as much as possible through data preprocessing, such as focal depth correction, earthquake relocation, time difference correction, remote earthquake height correction, and earth ellipse flatness correction ^[1]

The forward calculation plays an extremely important role in tomography. The accuracy and speed of the forward calculation directly determine the resolution and reliability of the imaging. The forward digital simulation technology is divided into numerical simulations of wave equations for solving partial differential equations and ray tracing numerical simulations based on integral equations to solve wave field propagation and travel time.

(1) Numerical simulation method of ray tracing

There are many types of ray tracing methods. The classical methods are the test firing method based on the initial value problem and the bending method based on the boundary value problem. The disadvantages of the classical method are: it is difficult to handle the strong speed change in the medium, it is difficult to find the global minimum travel time in the multi-value travel time, and the calculation efficiency is low. Moreover, the test firing method cannot track the ray path in the first wave and the shadow area (the ray theory is not valid); the bending method is less efficient for cases where the two points are far away. With the development of ray tracing methods, a large number of new algorithms have emerged that are different from traditional methods. The main feature of these methods is that they are no longer limited to the description of the ray path of seismic waves, but directly from the Huygens principle or Fermat principle, and use equivalent wavefronts to describe the characteristics of seismic wavefields.

(2) Waveform fit method

Wave equation-based tomography generally includes theoretical seismic map method and receiver function method. The numerical simulation of the wave equation is essentially to solve the seismic wave equation. Therefore, the simulated seismic wave field contains all the information of the seismic wave. However, since the tomographic imaging method based on the wave equation requires ultra-large-scale three-dimensional numerical calculations, the calculation speed is higher than that of the geometric ray method It is slow, and it is easy to introduce interference waves. At present, there are still many difficult issues that have not been resolved. However, the wave equation contains all the information of the seismic wave field, which can more objectively reflect the information of the underground structure than the ray tracing tomography, which only uses the travel time data and only used to simulate the kinematic characteristics of the wave. The wave field is the most effective and has broad development prospects.

The commonly used methods are: pseudospectral method, finite element method, and finite difference method. Pseudospectral method is flexible in processing the boundary, which is the limit when the approximate order of finite difference method approaches infinity. It uses fast Fourier transform to calculate the space derivative, and the calculation accuracy is higher than that of finite difference method. But like the finite difference method, it has a large amount of calculation and low efficiency. Due to the arbitrariness of the division and the variational principle on which it is based, the processing of multiple media and natural boundary conditions is very convenient and effective, and it has become a solution. An important method for numerical simulation of seismic wave propagation. It is by far the most accurate forward modeling method, but it is computationally intensive. The main advantage of the finite element method is that it is suitable for simulating any morphology of the geological body.

The fidelity of layer morphology simulation. The main disadvantage of the finite difference method and the finite element method is the limitation on high-frequency resolution. For typical speeds and frequencies in seismic exploration, a large number of grid points are required in the calculation, while the pseudospectral method is relatively more effective ^[1]

The inversion methods in tomography can be divided into linear methods and non-linear methods. The current non-linear inversion methods are: fast-pass algorithm, simulated annealing method and neural network method. In body tomography, there are many linear inversion methods, such as SVD (Singular Value Decomposition), Conjugate Gradients (CG), and Least Squares Quadrature (LSQR) Matrix Right-up-per-triangular-matrix Decomposition). The damped least squares method (DLSQR) is obtained by adding damping to the least squares method. The essence is to solve the overdetermined linear equations by using the triangular matrix decomposition method and the damped least squares method. Most geophysical problems are highly nonlinear, such as tomography is a typical nonlinear problem. The non-linear inversion method searches globally and does not depend on the initial model. It is suitable for studies that have little knowledge of the initial information of the studied area. The results of the inversion can be used as the initial model for local optimal image reconstruction. Although the calculation speed is slow, it has significant effect on complex nonlinear inversion problems. The linear inversion methods mostly artificially linearize non-linear problems, have large instabilities, and depend on the initial model, but their calculation speed is faster. Therefore, both the linear inversion method and the nonlinear inversion method have many applications in tomography.

The evaluation of tomographic solutions is an important part of tomographic research. Through the analysis of the solution, we can understand the important information such as the reliability, resolution, and error of the results. The commonly used methods are:

(1) Ray density method. By measuring the number of rays near each node as an evaluation of the reliability of the solution. However, this evaluation method only gives a preliminary measure of the reliability of the solution, and the evaluation of the resolution of the solution needs further analysis.

(2) Theoretical method of linear inversion. This method uses the classical Backus-Gilbert's generalized linear inversion theory, and uses the model resolution matrix, data resolution matrix, and covariance matrix to describe the evaluation method of the solution.

(3) Spike test method. This method uses synthetic data to obtain the column vectors of the resolution matrix to test the distortion effect of the ill-conditioned equations on the solution. However, it can only estimate the resolution of a single parameter point, and cannot evaluate the reliability of the entire solution. The main purpose of this method is to study whether a certain shape anomaly of known data can be distinguished. This test can provide the imaging capability of short-wave anomaly images, which can help distinguish the advantages and disadvantages of vertical resolution and lateral resolution. It can also perform spike tests on abnormal bodies of different sizes and shapes to verify algorithms and data The ability to image this abnormal body.

(4) Checkerboard resolution test method. The basic principle of this method is that, first, an existing synthetic data set is used to replace the existing observation data set. The synthetic data set consists of rational travel time values calculated using a real ray distribution under a specific three-dimensional velocity model. The velocity distribution (ie, checkerboard) of this particular three-dimensional mesh model is based on the initial one-dimensional velocity model, plus a regular distribution of perturbation values (that is, the perturbation values of each node are the same, but the order is positive and negative Permutation, this is done for ease of analysis). Then, an inversion calculation is performed on the synthetic data set, and the similarity between the 3D velocity structure of the inversion result and the detection plate is used as an estimate of the reliability and resolution of the solution. As the checkerboard experimental method is intuitive and practical, it is easy to analyze and compare. Most of the results of tomography today are applied to evaluate.

(5) Recovery resolution experimental method. The basic principle is to use the inversion result as an artificial synthesis model, perform ray tracing in this model, calculate the travel time, and add random errors of the same order of magnitude as the real data errors in this data to obtain an artificially synthesized data set. Invert this data set and compare the obtained results with the real results to analyze the situation of image restoration. The perturbation of rules set in the checkerboard experiment cannot simulate the image restoration in the case of complex models. Relatively speaking, the restoration resolution experiment is closer to the real situation than the checkerboard experiment, but it is not easy to analyze and compare ^[2]

In the past studies of the three-dimensional structure of the crust and upper mantle in China and adjacent areas, people paid more attention to the study of the three-dimensional velocity structure of the crust and upper mantle. On the one hand, because the velocity structure can explain some aspects of the three-dimensional structure inside the earth, on the other hand, because the calculation of the velocity parameters involves relatively few factors. However, from a general point of view, the velocity parameter only uses the kinematic information of the seismic wave and ignores the characteristic information of the seismic wave dynamics. Seismic tomography research mostly inverts a single physical quantity based on a single observed value on the local part of the seismic record. The methods are independent, showing global and systematic deficiencies, which hinders the use of seismic tomography to a considerable extent. Therefore, a new research method, the geophysical multi-parameter simultaneous inversion seismic wave tomography method, will be the direction of future development. The so-called geophysical multi-parameter refers to the inversion of two or more rock physical parameters and multiple components by the method of seismic tomography. Synchronous inversion refers to the simultaneous solution of multiple parameters in the inversion algorithm, such as joint inversion of interface and velocity. Zhou Huawei (2002), Zhang Yuansheng (1998), and Li Songlin (1997) have all studied this. Seismic fluctuation is a comprehensive response of rock physical properties. There is an inevitable relationship between rock physical properties. The single parameter inversion method is very weak in analyzing the multi-parameter and multi-component comprehensive response. Therefore, the study of multi-parameter and multi-component synchronous inversion is very necessary. . In the previous seismic tomography methods, travel time information and amplitude information were used to invert the wave velocity and attenuation of rocks, respectively. Regardless of travel time and amplitude, it can only represent local characteristics of seismic waves, and the overall seismic waveform is the comprehensive response of underground geological conditions and rock physical properties. Multi-component wave tomography breaks through the past method of extracting only local information, using the overall information of seismic waves, and modifying the physical parameters in the elastic wave equation by quantifying the waveform residuals between the theoretical seismic wave data and the measured seismic wave data. , And then modify the rock physical parameters of the geological model. Through such research, the seismic tomography research method will undergo substantial changes, which will transform the research based on the local characteristics of seismic waves into the research based on the overall dynamic characteristics of seismic waves ^[4] .