A new approach is presented to calculate the dispersion curves and normal mode solutions of multilayered solid earth model with low velocity layers based on the generalized reflection-transmission coefficient method. It is efficient to tackle these two problems by utilizing the single top-layer secular function proposed by Chen in 1993. However, it is difficult to obtain the lower modes by merely using the single top-layer secular function due to numerical singularity while the earth model contains low velocity layers. In this paper, we proposed using the secular function set, instead of the single secular function, to solve this problem. With secular function set, different modes can be calculated respectively through corresponding secular function. Numerical experiments show that the new approach is efficient, stable and reliable, thus it makes the generalized reflection-transmission coefficients method applicable to computation of normal mode in various stratified earth models. INTRODUCTIONComputing dispersion curves and their corresponding normal mode solutions in stratified earth model has been very useful in studying surface waves [1,2] , modelling regional Lg and Rg wave [3,4] , synthesizing seismogram by normal mode summation approach [5] , and exploring shallow velocity structure of the crust [6∼9] . Most of the researches concerning forward modelling of surface wave focus on the computation of dispersion curves. The classic Thomson-Haskell Matrix method [10] is the first efficient method for calculating the dispersion curve and corresponding displacement and traction eigenfunctions for a stratified earth model. This matrix method is perfect in theory, but inappropriate at high frequencies due to the 'precision-loss' problem. Numerous techniques were developed to tackle this problem including Schwab-Knopoff method [11] , delta matrix method [12] , and Abo-Zena method [13] , etc. These methods did extend the original Haskell matrix method to some higher frequencies, but didn't thoroughly solve the problem. Moreover, all these methods are time-consuming and more complicated for numerical implementation. Based on an alternative version of the reflection and transmission coefficients method, with a subtle decomposition of the wave field, Chen presented a systematic and efficient algorithm [14] to compute dispersion curve and normal modes for multi-layered solid half space. The new method eliminates the exponential growth component with respect to frequencies and depth which is numerically unstable in high frequencies, thus it is more efficient and stable than other methods, and is applicable for any stratified earth model in principle. Despite Chen's method has been successful for many cases, it becomes inefficient in the case that the stratified earth model contains low velocity layers (LVL). In this paper, by introducing a set of secular functions, we improve Chen's method hence make it possible to accurately and efficiently compute the dispersion curves and normal mode solutions of surface waves for any ...
A 2D boundary-element numerical simulation approach and a local slowness analysis method for an embedded array are used to quantify effects of topographic scattering on near-source energy partitioning for simple underground explosion sources. Various parameters, including free-surface models with different root mean square (rms) random topographic fluctuations and correlation lengths, source depths from 0.25 to 3 km, and Q values from 50 to infinity are included in the numerical simulations, with energy responses of different phases being determined as functions of frequency. The results reveal that for a crustal model with a relatively high surface velocity, near-source free-surface scattering provides an important coupling mechanism that can impart additional explosion energy to the Lg wave. At relatively low frequencies, and for a moderately rugged free-surface, the Rg-to-Lg transfer is quite efficient, while at higher frequencies or for a very rugged free surface, the body wave to Lg transfer may dominate the process. The Rg excitation functions, source depth, and topographic correlation length all contribute frequency dependence to the Lg excitation function. The presence of a low Q value within the uppermost crust severely attenuates the high-frequency energy transferred to the Lg wave.
It was widely accepted that the near-source scattering of explosion-generated Rg into S is the primary contributor to the low-frequency Lg from nuclear explosions, because the prominent low-frequency spectral null in Lg coincides with that in Rg from a compensated linear vector dipole (CLVD) source. In this study, the mechanism of the excitation of the spectral null of Rg generated by CLVD source is analyzed in detail with normal mode method. The most important result in this study is that the spectral null is due to a combination of displacement eigenfunction and its derivative of the fundamental-mode Rayleigh wave controlled by CLVD source, the zero-crossing alone can not fully explain the excitation of the spectral null.
S U M M A R YThe far-field approximation is commonly assumed in the derivation of Green's function retrieval by wavefield correlation using reciprocity theorems. This approximation can result in erroneous amplitude of the retrieved seismic phases and generate artificial arrivals. In an open system, we show that the far-field approximation can be removed by the exact integral equation method and this leads to the crosscorrelation kernel. In previous work, kernels in 2-D space had been explicitly constructed. With the kernel, we can construct the Green's function by crosscorrelation even when the sources are not in the far field. Here we extended the 2-D case to 3-D for acoustic waves for point sources on a plane and on a sphere. For the spherical case, the kernel is expressed by spherical harmonics and it degenerates to the Dirac Delta distribution under the high-frequency case; however, for a fixed frequency, if we increase the radius of the sphere, it reduces to that of the plane boundary.
Sensitivity kernels for wave equation based migration velocity analysis in local angle-domain are formulated. These sensitivity kernels serve as operators to backproject the residual moveout in local angle-domain common image gathers to migration models for velocity updating. To validate the theoretical formulation, we directly measure sensitivity kernels using a perturb-measuring approach. For single-shot data, the theoretical sensitivity kernels are consistent with those directly measured. For multi-shot data, the local angle domain sensitivity kernels are approximated by stacking the corresponding single-shot sensitivity kernels.
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