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When comparing solutions for the propagation of SH waves in plane parallel layered elastic and viscoelastic (anelastic) media, one of the first things that becomes apparent is that in the elastic case the location of the saddle points required to obtain a high frequency approximation are located on the real p axis. This is true of the branch points also. In a viscoelastic medium this is not typical. The saddle point corresponding to an arrival lies in the first quadrant of the complex p-plane as do the branch points. Additionally, in the elastic case the saddle point and branch points lie on a straight line drawn through the origin (the positive real axis in the complex p-plane), while in the viscoelastic case this is generally not the case and the saddle point and branch points lie in such a manner as to indicate the degree of their complex values.In this paper simple SH reflected and transmitted particle displacement arrivals due to a point torque source at the surface in a viscoelastic medium composed of a layer over a half space will be considered. The path of steepest descent defining the saddle point in the first quadrant will be parameterized in terms of a real variable and the high frequency solutions and intermediate analytic results obtained will be used to formulate more specific constraints and observations regarding saddle point location relative to branch point locations in the complex p-plane.As saddle point determination for an arrival is, in general, the solution of a non-linear equation in two unknowns (the real and imaginary parts of the complex saddle point p
0), which must be solved numerically, the use of analytical methods for investigating this problem type is somewhat limited.Numerical experimentation using well documented solution methods, such as Newton's method, was undertaken and some observations were made. Although fairly basic, they did provide for the design of algorithms for the computation of synthetic traces that displayed more efficient convergence and accuracy than those previously employed. This was the primary motivation for this work and the results from the SH problem may be used with minimal modifications to address the more complicated subject of coupled P-SV wave propagation in viscoelastic media.Another reason for revisiting a problem that has received some attention in the literature was to approach it in a fairly comprehensive manner so that a number of specific observations may be made regarding the location of the saddle point in the complex p-plane and to incorporate these into computer software. These have been found to result in more efficient algorithms for the SH wave propagation and a significant enhancement of the comparable software in the P-SV problem. 相似文献
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Bingzhu Wang Stephen S. Gao Kelly H. Liu Edward S. Krebes 《Geophysical Prospecting》2012,60(5):1001-1016
Potential, potential field and potential‐field gradient data are supplemental to each other for resolving sources of interest in both exploration and solid Earth studies. We propose flexible high‐accuracy practical techniques to perform 3D and 2D integral transformations from potential field components to potential and from potential‐field gradient components to potential field components in the space domain using cubic B‐splines. The spline techniques are applicable to either uniform or non‐uniform rectangular grids for the 3D case, and applicable to either regular or irregular grids for the 2D case. The spline‐based indefinite integrations can be computed at any point in the computational domain. In our synthetic 3D gravity and magnetic transformation examples, we show that the spline techniques are substantially more accurate than the Fourier transform techniques, and demonstrate that harmonicity is confirmed substantially better for the spline method than the Fourier transform method and that spline‐based integration and differentiation are invertible. The cost of the increase in accuracy is an increase in computing time. Our real data examples of 3D transformations show that the spline‐based results agree substantially better or better with the observed data than do the Fourier‐based results. The spline techniques would therefore be very useful for data quality control through comparisons of the computed and observed components. If certain desired components of the potential field or gradient data are not measured, they can be obtained using the spline‐based transformations as alternatives to the Fourier transform techniques. 相似文献
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Two different approaches to finite-difference modeling of the elastodynamic equations have been used: the heterogeneous and the homogeneous. In the heterogeneous approach, boundary conditions at interfaces are treated implicitly; in the homogeneous, they are explicitly discretized. We present a homogeneous finite-difference scheme for the 2-D P-SV-wave case. This scheme represents a generalization of earlier such schemes, being able to model media with arbitrary non-uniformities, provided only that all interfaces are aligned with the numerical grid. We perform a detailed comparison of the generalized homogeneous scheme with the analogous heterogeneous scheme, and show the two schemes to be identical for media with a spatially constynt Poisson's ratio. For media where Poisson's ratio is spatially varying, the schemes differ by terms first-order in the spatial step size. However, a comparison of the numerical results produced by the two schemes shows that the resulting differences are negligible for a wide range of values of the Poisson's ratio contrast. 相似文献
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E. S. Krebes 《Geophysical Journal International》1983,75(3):839-846
Summary. Two apparently equivalent methods used in the literature for calculating the mean energy flux and other properties of harmonic plane waves in a viscoelastic medium are shown to give slightly different results when applied to inhomogeneous P/SV -waves. The discrepant results of the two methods have appeared in recent papers and textbooks. Hence, it is important to point out and explain them. It is also shown that the discrepancies occur even for perfect elasticity. The first method makes use of the familiar classical formulae for the energy flux and other quantities, which can be derived from first principles. However, in the second method, the formulae for these quantities are inferred from a specifically derived conservation relation_ The validity of the results of the first method cannot be questioned. It is shown that the errors in the results of the second method are due to the presence of a certain mathematical non-uniqueness. 相似文献
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In order to trace a ray between known source and receiver locations in a perfectly elastic medium, the take-off angle must be determined, or equialently, the ray parameter. In a viscoelastic medium, the initial value of a second angle, the attenuation angle (the angle between the normal to the plane wavefront and the direction of maximum attenuation), must also be determined. There seems to be no agreement in the literature as to how this should be done. In computing anelastic synthetic seismograms, some authors have simply chosen arbitrary numerical values for the initial attenuation angle, resulting in different raypaths for different choices. There exists, however, a procedure in which the arbitrariness is not present, i.e., in which the raypath is uniquely determined. It consists of computing the value of the anelastic ray parameter for which the phase function is stationary (Fermat's principle). This unique value of the ray parameter gives unique values for the take-off and attenuation angles. The coordinates of points on these stationary raypaths are complex numbers. Such rays are known as complex rays. They have been used to study electromagnetic wave propagation in lossy media. However, ray-synthetic seismograms can be computed by this procedure without concern for the details of complex raypath coordinates. To clarify the nature of complex rays, we study two examples involving a ray passing through a vertically inhomogeneous medium. In the first example, the medium consists of a sequence of discrete homogeneous layers. We find that the coordinates of points on the ray are generally complex (other than the source and receiver points which are usually assumed to lie in real space), except for a ray which is symmetric about an axis down its center, in which case the center point of the ray lies in real space. In the second example, the velocity varies continuously and linearly with depth. We show that, in geneneral, the turning point of the ray lies in complex space (unlike the symmetric ray in the discrete layer case), except if the ratio of the velocity gradient to the complex frequency-dependent velocity at the surface is a real number. We also present a numerical example which demonstrates that the differences between parameters, such as arrival time and raypath angles, for the stationary ray and for rays computed by the above-mentioned arbitrary approaches can be substantial. 相似文献
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Based on knowledge of a commutative group calculation of the rock stiffness and on some geophysical assumptions, the simplest fractured medium may be regarded as a fracture embedded in an isotropic background medium, and the fracture interface can be simulated as a linear slip interface that satisfies non‐welded contact boundary conditions: the kinematic displacements are discontinuous across the interface, whereas the dynamic stresses are continuous across the interface. The finite‐difference method with boundary conditions explicitly imposed is advantageous for modelling wave propagation in fractured discontinuous media that are described by the elastic equation of motion and non‐welded contact boundary conditions. In this paper, finite‐difference schemes for horizontally, vertically, and orthogonally fractured media are derived when the fracture interfaces are aligned with the boundaries of the finite‐difference grid. The new finite‐difference schemes explicitly have an additional part that is different from the conventional second‐order finite‐difference scheme and that directly describes the contributions of the fracture to the wave equation of motion in the fractured medium. The numerical seismograms presented, to first order, show that the new finite‐difference scheme is accurate and stable and agrees well with the results of previously published finite‐difference schemes (the Coates and Schoenberg method). The results of the new finite‐difference schemes show how the amplitude of the reflection produced by the fracture varies with the fracture compliances. Later, comparisons with the reflection coefficients indicate that the reflection coefficients of the fracture are frequency dependent, whereas the reflection coefficients of the impedance contrast interface are frequency independent. In addition, the numerical seismograms show that the reflections of the fractured medium are equal to the reflections of the background medium plus the reflections of the fracture in the elastic fractured medium. 相似文献
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Consider a plane homogeneous harmonic SH wave incident upon an interface between two anelastic half-spaces. Computing the plane wave displacement and energy-flux-based reflection and transmission coefficients correctly requires determining the proper signs of the vertical slowness components of all the reflected and transmitted waves, i.e., determining which of the two values of the square root for a given vertical slowness should be chosen. For anelastic media, this can be problematic, as unphysical results can arise. Previous research has led to a specific recommendation on how to choose the signs. However, when this recommendation is employed, it is found via numerical experimentation that for certain values of the medium parameters, the energy-flux-based transmission coefficient T can be negative for certain supercritical values of the incidence angle, whereas physical reasoning suggests it should be zero. To investigate this seemingly unphysical result, an analytical determination of the mathematical conditions under which it occurs would be useful. Such a determination is performed in this article. Letting V 1 and V 2 be the wave speeds of homogeneous SH plane waves in the incidence and transmission media respectively, and Q 1 and Q 2 be the corresponding quality factors, with Q 1, Q 2 ? 1, it is found that if V 1 < V 2 and Q 1 < Q 2 (a common situation in the Earth), then T will be negative for part of the supercritical incidence angle range if 1 < Q 2/Q 1 ?? 2 ? (V 1/V 2)2 and for all of it if 2 (V 1/V 2)2 < Q 2/Q 1. 相似文献
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