**REVIEW ON STUDIES
OF SEISMIC WAVE**

*ZHANG
Haiming *and *CHEN
Xiaofei **

Department of Geophysics,
Peking University, Beijing 100871, China

Study on seismic wave propagation and excitation has been an important part of seismology at all times. It is the base of study on inverting the structure of the crust and upper mantle, investigating the rupture nature of seismic sources, and exploring the structure and geophysical characteristics of the inner part of the Earth. With more and more digital data, significant progress has been made on the research of seismic waves. In the past four years, studies of seismic have progressed remarkably in many aspects in China.

I. SEISMIC WAVE PROPAGATION IN LATERALLY HOMOGENEOUS MEDIAAs one of the classical problems in seismology, seismic wave propagation in laterally homogeneous media still drew much attention of seismologists. Improvements have been made on the conventional reflectivity method. Besides, some researchers have attempted to solve the problem from a bran-new point of view. Chen (1999a) presented a systematic and self-contained theory of elastic waves in multi-layered half-space media based on the development on the study of synthesizing seismograms by using the reflectivity method in the past two decades. In order to overcome the difficulties in numerical computation of synthetic seismograms in the multi-layered half-space with reflectivity method, Chen and Zhang (2001) and Zhang and Chen (2001) proposed a numerical integration method, the Self-Adaptive Filon's Integration Method, to calculate the synthetic seismograms with larger epicentral distances, which greatly enhanced the efficiency. Moreover, Zhang et al. (2001) modified the repeated averaging method to the so-called “peak-trough averaging method” to efficiently calculate the synthetic seismograms with sources and receivers at close or same depth. Shu et al. (2002) proposed a new method, based on the propagation matrix method, to get surface synthetic seismograms from the S wave input beneath a station. Applying the new method to S wave data recorded at two short-period stations in Hebei Province, they obtained the S wave velocity structure beneath the stations. By using synthetic seismogram, Zhang et al. (2002) analyzed that the dynamic characteristics of head wave in multi-layered half-space media models with high-velocity layer or low-velocity layer, and the model with a continuous transition-zone between the crust and the mantle.

Based on the analogy of structural mechanics and optimal control, Sun and Liu (2002) introduced the theory of Hamiltonian system into the elastic wave-guide problems. They built a semi-analytical finite element method (FEM) in Hamiltonian system. Luo et al. (2001a,b) proposed the representation of seismic wave propagation in the Hamiltonian frame and some symplectic schemes. Numerical results indicated that symplectic schemes are applicable to model wave field. The work of the above authors demonstrated a broad prospect of the Hamiltonian system and symplectic schemes in seismic wave propagation study.

II. SEISMIC WAVE PROPAGATION IN LATERALLY HETEROGENEOUS MEDIAThe laterally homogeneous media model is not a proper approximate model, if the lateral variation of the structure cannot be neglected. In such cases, a laterally heterogeneous model must be adopted. Accordingly, methods for studying seismic wave propagation in laterally heterogeneous media must be developed. By using the Fourier-Bessel series expansion technique, Liang et al. (2001a, b) derived an analytical solution of scattering of plane P and SV waves by circular-arc layered alluvial valleys, and utilized the solution to analyze the effects of the geometry of the model on the scattering of incident waves. Liu et al. (1999) extended the amplitude preserving common offset continuation equations in constant velocity medium to the case in laterally linearly varied velocity media. Due to the complexity of the problem, analytic solution can only be obtained in few cases in which the geometry of the model is extremely simple. For most cases, numerical technique must be introduced.

One of the important ways to solve the problem is to extend the reflectivity method for laterally homogeneous media to laterally heterogeneous media. Based on his earlier studies (1990, 1995,1996), Chen (1999b) introduced a unified and efficient method, the global generalized R/T matrices method, for synthetic seismograms in multi-layered media with irregular interfaces. By using this method, Chen (1999c) and Chen and Ge (1999) derived a new and more general formulation and the corresponding excitation formulation of Love waves in arbitrarily irregular multi-layered media. They also derived the characteristic frequencies and the corresponding distorted modes of Love wave in such media.

High-frequency-approximation based on ray tracing is another powerful tool for the study of seismic wave propagation in laterally heterogeneous media, especially the property of reflection waves on the irregular interfaces. By using the Maslov asymptotic theory, Chen and Liu (1999) proposed a new method to calculate the lithospheric receiver function in 3-D laterally inhomogeneous media, in which not only the accuracy of ray tracing is improved, but also the efficiency is enhanced. Moreover, the artificial ray coding in traditional ray methods is avoided. Based on this method, they (Chen and Liu, 2000) computed the synthetic receiver functions for the case in which dipping interfaces exist under a single station. In order to overcome the drawbacks in shortest ray path method, Wang and Chang (2000) introduced the Bresenham line algorithm into the travel-time calculation of nodes to improve the accuracy of travel-time and ray path results. Numerical tests showed that the modified ray tracing method is accurate and efficient. Based on Huygens' principle and Fermat's principle, Zhao et al. (2000) presented a new method to calculate seismic travel-time and ray-paths, which can be easily programmed and applied to heterogeneous media. Gao et al. (2000) and Qin and Chen (2000) introduced the symplectic algorithm to obtain travel-time and path of seismic ray, which not only enhanced the computation efficiency, but also overcame the caustic problem in seismic tomography.

The finite difference method and the finite element method have been used more and more widely in modeling seismic wave propagation in heterogeneous media. Zhang and Liu (1999, 2002) presented the 2D and 3D grid method to model P- and SV-wave propagation in 2D and 3D heterogeneous media, respectively. Based on the flux-corrected transport (FCT) technique, Yang et al. (2002) presented a fast finite-difference scheme that is based on the application of vectors and matrices in 2D anisotropic media, and obtained the stability equations. Compared with the results of reflectivity method and the conventional FD method without FCT technique and any absorbing boundary treatments, FCT based on FDM is very accurate and efficient in computing synthetic seismograms in general heterogeneous and anisotropic media. Dong et al. (2000) and Dong et al. (2000) used high-order finite difference method, combined with the staggered-grid technique, to numerically solve the one-order elastic wave equations expressed with velocity and stress, and derived a unified stability condition. Compared with conventional difference method, simulated results showed that the new method is more accurate and can depress the grid dispersion, meanwhile, bigger grid space can be used to enhance computational efficiency. Sun et al. (2000) used a hybrid method, based on mode summation and finite differences, to simulate the ground motion of SH wave in the Xiji-Langfu area of Beijing induced by the 1976 Tangshan earthquake. They also estimated the maximum acceleration amplitude and total energy of ground motion for engineering purposes. Di et al. (2000) constructed models of deposit sand body traps and conducted a complex seismic wave modeling study using the finite element method. Liao et al. (2002) systematically discussed the numerical instability, the high-frequency oscillation and the zero-frequency drift, resulted from the transmitting boundary in numerical simulation of near-field wave motion, and suggested a practical and complete scheme for stable implementation of the transmitting boundary. Zhang et al. (2000) and Zhang et al. (2001) presented new algorithms for multi-valued travel-time wave-field and precise images for complex structures with large lateral velocity variations, respectively.

Other than the above studies based on conventional analytic and numerical algorithms, some authors attempted to introduce new theories and techniques originated in other fields of science and technology to the study of seismic waves. Wang and Li (1999) applied the parallel algorithm for simulating seismic wave propagation by lattice gas automata based on the domain decomposition technique to the calculation of plane wave reflection seismic response in actual geologic model, and the computation efficiency was enhanced significantly. Liu et al. (1999) presented a new cellular automata model, Phononic Lattice Solid with Various Grids Length, for modeling seismic waves. Luo et al. (2001a,b) proposed the representation of seismic wave propagation in the Hamiltonian frame and some symplectic schemes. Numerical results indicated that symplectic schemes are applicable to model wave field. Based on Kaiser's acoustic wavelet theory, Chen et al. (2001) gave two kinds of construction methods of acoustic wavelet through introducing complex time function and imaginary time coordinate of point sources, respectively. The acoustic wavelet transform in time-space domain is applied both to the synthetic seismograms of point sources and to the seismic data produced by the complicated SEG-EAEG salt model.

The attenuation property is also an important aspect in seismology. Wang et al. (1999) studied the attenuation property of horizontal peak acceleration of near-field strong ground motion according to the strong ground motion record of Tangshan strong aftershocks. Shi et al. (2000) analyzed two methods for measuring Q value: frequency-amplitude ratio method and waveform inversion method, and proposed a new method of using the wave energy to calculate the Q value. Zhao et al. (2000) proposed a new method in which the sampling depth is used to describe the effect of lapse time on the Q-value, and investigated the dependence of coda wave of earthquakes on frequency and sampling depth.

Besides, Wang et al. (2001) integrated the seismic velocity tomography, seismic amplitude attenuation tomography, and rise-time attenuation tomography into a series procedure, and applied it to field data obtained in a mine. Ding et al. (1999) resolved the coupling P1 and P2 wave by dividing the force potency field through two-phase saturated Biot's equation transform, and obtained the Green's function and wave field in two-phase saturated medium. Liu (1999) studied the propagation of the elastic waves in discrete medium, and constructed the solutions of 1D elastic waves in heterogeneous discrete medium by linearly combined the Bessel functions.

III. SEISMIC WAVE IN ANISOTROPIC AND POROUS MEDIAShear wave splitting is a special phenomenon of seismic waves in anisotropic media, through which, we can understand the properties of complex structures. From the polarization patterns of fast shear wave in Tangshan region, Gao et al. (1999) recognized that it is possible that the polarization directions of fast shear waves in a complicated tectonic zone change quickly, therefore it may be meaningless to simply overlap fast shear wave polarization observed in different stations. Three methods for analyzing azimuth anisotropy are examined with the cross-dipole logging data by Tao et al. (1999). These techniques are based on the phenomena of flexural wave splitting in anisotropic materials. Zhang and He (1999) studied crosshole tomography by using qP- and qSV-wave travel-time, based on the anisotropy travel-time disturbance theory. By resampling technique, Wang et al. (2000) found the variation of shear wave splitting directly from seismograms of each pair in a cluster. Lai et al. (2002) investigated the S-wave splitting at each station nearby the Jiashi strong earthquake sequence from the local event data. According to their study, they suggested that the Jiashi earthquake sequence is caused probably due to the release of the localized tensile strain-energy by lateral inhomogeneous deformation at the north edge of the Tarim basin. Liu et al. (2001) applied Butterworth band-pass filter to S-wave data recorded at 8 stations in China mainland, and S-wave splitting at different frequency bands is analyzed. The results showed that the delay time and the fast polarization directions of S-wave splitting depend upon the frequency bands.

In the aspect of properties and numerical simulations of seismic waves in anisotropic media, Zhang et al. (1999) developed a new algorithm, the Snell law for anisotropic media, to compute the angles of incident wave, reflection wave and transmission wave on the interface in an anisotropic media. Liu et al. (2000) derived a method for calculating reflection and transmission coefficients of elastic wave based on elastic wave equations and four continuity conditions of an interface between dissimilar two-phase, transversely isotropic media. Liu and Li (2000) derived the pseudo-spectral method to solve wave equations of two-phase anisotropic media. From the simulation results, they observed four types of waves in the snapshots, i.e., fast P wave and slow P wave, fast S wave and slow S wave. Yang (2002) presented the finite element equation in two-phase anisotropic media, which is used for solving the porous elastic wave equation under artificial absorbing conditions. He applied the new method to the numerical simulation of elastic wave propagation in the two-phase PTL and two-phase isotropic media. Liu and Li (1999) derived the Christoffel equation for elastic waves in arbitrary two-phase anisotropic media; they analyzed and computed the effects of frequency on the phase velocity, attenuation, amplitude ratio and so on. By adopting absorbing boundary conditions and FCT technique, Zhang et al. (1999) presented an FDM for the simulation of multi-component wave-field in viscous extensively dilatancy anisotropic (EDA) media.

The seismic wave propagation in porous media is also a problem to which researchers paid attention. Shao and Lan (2000) studied the wave propagation in fluid-saturated porous media, and developed a new kind of non-reflecting boundary condition on artificial boundaries. Zhu et al. (2001) studied the non-specular acoustic reflected field on a Water-FSPM (fluid-saturated porous media) interface by using the optical method and the transmitter-receiver method. Yang et al. (2002) performed the wave-field simulation based on the BISQ equation in the isotropic medium by using the FCT finite difference method. Li et al. (2002) determined the permeability for two-dimensional porous medium from the arrival time. Based on a first order hyperbolic system that is equivalent to Biot's equations, Zhang (1999) developed a quadrange-grid velocity-stress FDM for the simulation of wave propagation in 2D heterogeneous porous media.

IV. SURFACE WAVE AND SEISMIC WAVE INVERSIONSeismic surface wave inversion is an important approach to investigate the interior structure of the Earth. Many efforts have been made on the surface wave theory and its inversion in the past four years. Chen and Chen (2001) presented a new systematic and efficient algorithm to calculate the modal solutions of multi-layered ocean-Earth model. The algorithm distinguishes itself as terseness of formulation, efficiency in numerical computation, and stableness at high frequencies, thus, thoroughly solving the problem of loss-of-precision at high frequencies. Zhang et al. (2000, 2002) analyzed the mechanism of zigzag dispersion curves in the Rayleigh wave exploration and studied the propagation characteristics and interrelation of the guided waves. It is presented that the zigzag structure of Rayleigh wave dispersion curves is the result of many modes generated by the low velocity layers or fractures in multilayered. In the theoretical study on the inversion of surface waves, Zhu et al. (2001) transformed the inversion of surface-wave velocities into the linear inversion of spherical harmonic coefficients through conformal mapping. The new algorithm has a fast computing speed, smooth contours, and clear tectonic boundaries. Wu et al. (2001) introduced an indirect smooth constraint technique into the genetic inversion, and applied it to invert the phase velocity of Raleigh wave in the Tibetan Plateau, revealing the horizontal variation of S wave velocity structure near the center of the Tibetan Plateau. Many researchers (Hu and Su, 1999; Hu et al., 1999; Xu et al., 2000; He et al., 2000, 2001, 2002; Teng et al., 2001; Cao et al., 2001; Zhu et al., 2002; Zhu et al., 2002; Peng et al., 2002) used the dispersion curves extracted from surface wave data to invert the structures in different areas. These studies indicated that the lateral variation in Earth's structure exist widely.

Travel-time inversion is another classical way to obtain information on the Earth's structure. The results of teleseismic P-wave tomography study conducted by Shi et al. (1999) revealed that the Altyn fault is a low velocity stripe which extends downward vertically. Xu et al. (2001) reconstructed three-dimensional velocity images of the crust and upper mantle beneath orogenic belts and adjacent basins of the northwestern continent of China by seismic tomography, based on arrival data of P wave recorded in seismic networks in Xinjiang, Qinghai, Gansu of China and Kyrgyzstan. Liu et al. (2001) used a least-squares approach in conjunction with QR decomposition to reconstruct 3-D velocity model from the actual first-break times obtained from 3-D data, and applied the fractal algorithm to determine the first breaks, which overcomes the error caused by the differences of wavelet shapes and the leg-jump of refractions. By using travel time and path of seismic ray for inverting the velocity structure of medium, Li et al. (2000) proposed a set of methods for tomographic inversion of seismic first break in surface model aimed at complex surface problem in seismic prospecting.

In the aspect of inversion by using the far-field body wave receiver function, Liu et al. (2000) studied the 3D S-wave velocity structures at a depth of 0-100km in crust and upper mantle beneath the stations in Jiashi strong earthquake region and its neighborhood by using superposition offset analysis technique for far-field body wave receiver function and non-linear inversion technique. Wu et al. (2001) obtained the S wave velocity structure within the depth of 0—100km beneath digital seismic stations of Yunnan Province from teleseismic receiver function modeling. Qian et al. (2001) modified the inversion algorithm based on receiver function, and used it to investigate the structure of crust beneath the eastern part of the Tibetan Plateau. By receiver function method, Yang and Zhou (2001) determined the 410km and 460km discontinuities under the stations in China and adjacent areas. Zou and Chen (2003) used analyzed the advantages of SV component receiver function than the conventional radial component receiver function in mapping the crustal shear wave velocity structure. They found that the change of amplitude of SV-component receiver function against the change of epicentral distance is less than that of radial receiver function. Moreover, the waveform of SV-component receiver function is simpler than the radial receiver function and gives prominence to the PS converted phases that are the most sensitive to the shear wave velocity structure in the inversion.

In recent years, nonlinear inversion has been an active subject that more and more seismologists get interested. Some researchers introduced the new techniques in optimization theory to the nonlinear inversion problem in seismology. Based on wavelet transform technique, Meng and Liu (1999) proposed a multi-scale seismic waveform inversion method. Compared with direct iterative method and multi-grid method, this new method proved to be more efficient for a 1-D nonlinear seismic waveform inversion problem. Wang et al. (1999) modified the Genetic Algorithm according to the principles of simulated annealing and taboos search, making it possible to solve large-scale, non-linear, discrete and complex inversion problems. Yang et al. (1999) analyzed the dynamical characteristics of the seismic-trace nonlinear inversion, and presented a new control scheme for inversion based on the chaos controlling theory. The relative errors and stability appraisals for the three methods used in nonlinear inversion problem were given by Chang et al. (1999). This provided quantitative parameters for the analysis of results of seismic tomography. Yang and Nie (2000) proposed a novel inverse iteration method, variational Born iteration method, for the inversion and reconstruction of two-dimensional axisymmetic inhomogeneous media. Several examples showed that this method has faster convergence and higher inverse quality than Born iteration method.

Besides, Liu et al. (2001) used broadband three-component seismic data recorded by Beijing station (BJI) of CDSN to calculate P-wave polarization of teleseismic events. These polarization data were then used in the inversion for the underground structure around the Beijing station, especially for the details of velocity discontinuities.

V. SEISMIC WAVE STUDY IN PROSPECTING AND LOGGING PROBLEMS AND OTHERIn the aspects of seismic prospecting and logging, many authors proposed new methods aiming at problems occurring in practical work. Yin et al. (1999) proposed a new method of common offset continuation. A new common offset continuation equation is built based on the Hamilton equation. Liu et al. (1999) introduced Hamilton method to describe CRP trajectory for the first time, and calculated the trajectories of the points on several typical reflective surfaces. Shen and Zhang (2000) studied the 3D acoustic field generated by a source close to the borehole wall, and calculated waveforms, including direct wave, reflected wave, longitudinal wave, shear wave and interface wave, at different positions on the wall. Zhao (1999) imaged refractor data by using an automated reconstruction wave-front method, and extrapolated wave field, thus thoroughly removed the difficulties existed in traditional method. Zhang and Wang (2000) presented an effective analytical perturbation method and applied it to multi-pole acoustic logging in borehole model, which can be regarded as the perturbations added to the relevant isotropic two-phase medium. Cheng et al. (2001) introduced a prestack depth migration operator with coefficients-optimized paraxial wave equation on the common shot gathers. Under the imaging condition based on refection coefficients estimating, the prestack depth imaging can be realized. Chen et al. (2001) proposed a new prestack depth migration method based on quasi-linear Born approximation, which enlarged the range of quasi-linear Born approximation, and made it more applicable to strong lateral velocity variation. Based on the finite element method and the interface point method of shortest path ray tracing, Zhang and Wang (20001) realized reverse-time migration for anisotropic elastic wave. Chen et al. (2002) developed a robust method to effectively set up velocity-depth models. The best migration velocity can be approached by cascade migration based on the sensibility of seismic imaging to velocity variation. Gu et al. (2002) derived an algorithm for decomposition of multi-wave and multi-component data into up-going and down-going P-wave and S-wave in frequency-wavenumber domain, and calculated the reflection-coefficient-versus-incident-angle (RVA). Jin et al. (2002) proposed an efficient and accurate offset domain prestack depth migration method for 2-D common offset and 3-D common azimuth seismic data. Wang et al. (2002) conducted 3-D migration of seismic data for an irregular acquisition geometry. Zhang et al. (2002) provided a 5-parameter stacking formula to transform 2D prestack data into a particular common-offset section. This method can handle P-P, P-S, and S-S reflections.

Other than the above studies, Shen and Zheng (1999) studied the transient spectra of underground nuclear explorations and natural earthquakes by using the non-stationary theory of Winger Distribution. Liu et al. (1999) applied the stacked spectral ratio method to the measurement of Lg Q

_{Lg}^{c}at regional epicentral distances (<500km). Zhang et al. (1999) proposed a new algorithm of seismic signal deconvolution based on dyadic wavelet transform, which is called multi-resolution seismic signal deconvolution. Liu et al. (1999) found that simultaneously measured P-and S-wave velocities as well as their ratioVp/Vsmay give an estimate of crack densities in rock samples. Huang (2001) gave some new imaging formulas for seismic reflection wave and their theoretical basis. Zhang et al. (1999) presented a new method for filtering or wave-field separation based on the representation of seismic data, τ-p spectrum, by combining wavelet transform and τ-p transform. Wang et al. (2002) present a τ-p domain scheme to separate P- and SV-wave-fields from multi-component seismic data observed at free surface and ocean bottom. Gao et al. (2002) presented a new earthquake location method (SAMS), in which the object function is the absolute values of the remnants of travel time together with the arrival time, and the fast simulated annealing method is introduced.

Acknowledgement:The authors are grateful to the supports of the National Natural Science Foundation of China (grant No. 40134010).

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