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1.
Ray tracing has recently been expressed for anisotropy specified in a local Cartesian coordinate system, which may vary continuously in a model specified by elastic parameters. It takes advantage of the fact that anisotropy is often of a simpler nature locally (and is thus specified by a smaller number of elastic parameters) and that the orientation of its symmetry elements may vary. Here we extend this approach by replacing the local Cartesian coordinate system with a curvilinear coordinate system of global extent and by applying the new approach to ray tracing and inhomogeneous dynamic ray tracing. The curvilinear coordinate system is orthogonal and is constructed so that the coordinate axes are consistent with the considered anisotropy of the medium. Our formulation allows for computation of ray attributes (e.g. ray velocity vector and paraxial ray attributes) in the curvilinear coordinate system, while rays are computed in global Cartesian coordinates. Compared to the classic formulation in terms of 21 elastic moduli in global Cartesian coordinates, the main advantages are improved efficiency, lower computer-memory requirements, and conservation of anisotropic symmetry throughout the model.  相似文献   

2.
Summary . Synthetic seismograms represented by integrals generally display signals associated with the limits of integration. Sometimes these 'end-point' contributions are spurious (e.g. in the WKBJ seismogram) and sometimes they are the main physical interest (e.g. the Kirchhoff integral for an edge). The end-point contributions may be asymptotically approximated using integration by parts or Laplace's method and it may then be possible to reduce them if desired. We describe examples in the WKBJ seism ogram for reflected or transmitted waves in homogeneous layers and for turning waves. We also study signals due to discontinuities in reflection coefficients, by partitioning the real slowness integral so that the discontinuities lie at end points. Examples are the head wave, which is a physically correct signal, and spurious diffractions caused by using plane-wave coefficients for grazing rays in the WKBJ seismogram.  相似文献   

3.
Summary. We show that Maslov's extension of the WKBJ method allows an extension of the dynamic ray tracing to wavefields involving caustics of arbitrary form. If the receiver lies off the caustics, then the synthetic seismogram can be obtained by integrating the DRT system along a single ray joining the receiver to the source which may touch caustics. If the receiver-lies in the vicinity of a caustic then DRT has to be carried out along a bunch of rays covering a neighbourhood of the receiver. Our approach encompasses pre-stressed and/or anisotropic media. Initial boundary conditions for a point source embedded in an anisotropic elastic medium are also presented.  相似文献   

4.
The eikonal equation is the equation of the phase slowness surface for isotropic and anisotropic media. In general anisotropic media, there is no simple explicit expression for the phase slowness surface. An approximate expression of the eikonal equation may be obtained in weakly anisotropic media. In orthorhombic media, the approximate eikonal equation of the qP wave is the sum of an ellipsoidal form and a more complicated term. The ellipsoidal form corresponds to what we call ellipsoidal anisotropy. Ray equations written in the Hamiltonian formulation are characteristics of the eikonal equation. Ray perturbation theory may be used to compute changes in ray paths and physical attributes (traveltime, polarization, amplitude) due to changes in the medium with respect to a reference medium. Examples obtained in homogeneous orthorhombic media show that a reference medium with ellipsoidal anisotropy is a better choice to develop the perturbation approach than an isotropic reference medium. Models with strong anisotropy can be considered. The comparison with results obtained by an exact ray program shows a relative traveltime error of less than 0.5 per cent for a model with relatively strong anisotropy. We propose a finite element approach in which the medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, using a perturbation approach, analytical expressions for rays and traveltimes are obtained Ray tracing reduces to connecting these analytical solutions at the vertices of the cells.  相似文献   

5.
We use the Direct Solution Method (DSM) together with the modified operators derived by Geller & Takeuchi (1995) and Takeuchi, Geller & Cummins (1996) to compute complete synthetic seismograms and their partial derivatives for laterally heterogeneous models in spherical coordinates. The methods presented in this paper are well suited to conducting waveform inversion for 3-D Earth structure. No assumptions of weak perturbation are necessary, although such approximations greatly improve computational efficiency when their use is appropriate.
An example calculation is presented in which the toroidal wavefield is calculated for an axisymmetric model for which velocity is dependent on depth and latitude but not longitude. The wavefield calculated using the DSM agrees well with wavefronts calculated by tracing rays. To demonstrate that our algorithm is not limited to weak, aspherical perturbations to a spherically symmetric structure, we consider a model for which the latitude-dependent part of the velocity structure is very strong.  相似文献   

6.
Summary. Several approaches to computing body wave seismograms in 2–D and 3–D laterally inhomogeneous layered structures are suggested. They are based on the Gaussian beam method, which has been recently applied to the evaluation of time-harmonic high-frequency wavefields in inhomogeneous media. Three variants are discussed in some detail: the spectral method, the convolutory method and the wave-packet method. The most promising seems to be the wave-packet approach. In this approach, the wavefield, generated by a source, is expanded into a system of wave packets, which propagate along rays from the source in all directions. The wave packets change their properties due to diffusion, spreading, reflections/transmissions, etc. The resulting seismogram at any point of the medium is then obtained as a superposition of those packets which propagate close to the point. The final expressions in all the three methods are regular even in regions, in which the ray method fails, e.g. in the vicinity of caustics, in the critical region, at boundaries between shadow and illuminated regions, etc. Moreover, they are not as sensitive to the minor details of the medium as the ray method and, what is more, they remove the time-consuming two-point ray tracing from computations. Numerical examples of synthetic seismograms computed by the wave-packet approach are presented.  相似文献   

7.
The parameter that defines the ray tracing equations in the direct geometrical approach is the product of the radius of curvature of the wave front by the velocity on the wave front ( RV ). To show this, we derive motion equations for the centre and the radius of curvature of an expanding wave front. The continuity of RV along rays implies Snell's Law. For constant velocities the equation for the radius of curvature reduces to the original Huygens' Principle. The variable RV can be computed during ray tracing and used to determine the local radius of curvature, which in turn can be used in geometrical spreading, amplitude corrections and structure interpretation.  相似文献   

8.
Summary. Some relations between Gaussian beams, complex rays and the analytic extension of the Green's function in smoothly inhomogeneous media are shown in this paper. It is found that: (1) a single Gaussian beam is a paraxial approximation of the analytical extension of the ray-approximated Green's function in smoothly inhomogeneous media by putting the source point into a complex space. The Gaussian beam approximation of the Green's function has an advantage in computational efficiency and stability and can avoid the singularity problems at caustics, but also introduces a parabolic approximation to the wavefront and an angle-dependent amplitude damping. Therefore the validity of the Gaussian beam approximation should be checked using other methods. (2) Complex-ray tracing, which does not involve the paraxial approximation, can also avoid the singularity problemsm though without the computational efficiency. Therefore, it should be used to verify the Gaussian beam approximation, whenever possible. (3) The decomposition of a plane wave into an ensemble of Gaussian beams is equivalent to approximating the Green's function (the kernel of the ray-Kirchhoff method) with a single Gaussian beam. This introduces a parabolic approximation to the wavefront and a Gaussian windowing for arrival angles which may cause some problems in modelling wave propagation and scattering and has no advantages over other methods. (4) The representation of a point source field by a superposition of Gaussian beams, on the other hand, is equivalent to approximating the Green's function with a bundle of overlapped Gaussian beams. This representation is similar to a Maslov uniform asymptotic representation. It has no caustics and has improved accuracies at the caustics for quasi-plane waves compared to the extended WKBJ method.  相似文献   

9.
b
A new method is presented by means of which one can compute finite frequency synthetic seismograms for media whose velocity and density are continuous functions of two or three spatial variables. Basically, the method is a generalization of the familiar phase integral method, to which it reduces in a stratified medium. For a given source location the travel-time and distance functions needed to compute synthetics are obtained by numerically tracing rays through the model. This information is then used to evaluate a double integral over frequency and take-off angle at the source. The solution obtained reduces to the geometrical optics solution wherever that is valid but it also works in shadows and at caustics without knowing explicitly where these may be located. The method can be used as a spectral method, in which the integral over take-off angle is evaluated first, or as a slowness method, in which the frequency integral is evaluated first.  相似文献   

10.
We have developed a new array method combining conventional migration with a slowness-backazimuth deviation weighting scheme. All seismic traces are shifted based on the theoretical traveltime of the scattered wave from specific gridpoints in a 3-D volume. Observed slowness and backazimuth are calculated for each raypath and compared with theoretical values in order to estimate slowness and backazimuth deviations. Subsequently, stacked energy calculated by a conventional migration method is weighted by the slowness and backazimuth deviations to suppress any arrival energy whose slowness and backazimuth are inconsistent with the expected theoretical values. This new method was applied to two P- wave data sets which comprise (1) underside reflections at the 410 and 660 km mantle discontinuities and (2) D" reflections as well as their corresponding synthetic data sets. The results show that the weighting scheme dramatically increases the resolution of the migrated images and enables us to obtain well-constrained, focused images, making upper-mantle discontinuities and D" reflections more distinct by reducing their surrounding energy.  相似文献   

11.
Summary. An algorithm for the computation of travel times, ray amplitudes and ray synthetic seismograms in 3-D laterally inhomogeneous media composed of isotropic and anisotropic layers is described. All 21 independent elastic parameters may vary within the anisotropic layers. Rays and travel times are evaluated by numerical solution of the ray tracing equations. Ray amplitudes are determined by evaluating reflection/ transmission coefficients and the geometrical spreading along individual rays. The geometrical spreading is computed approximately by numerical measurement of the cross-sectional area of the ray tube formed by three neighbouring rays. A similar approximate procedure is used for the determination of the coefficients of the paraxial ray approximation. The ray paraxial approximation makes computation of synthetic seismograms on the surface of the model very efficient. Examples of ray synthetic seismograms computed with a program package based on the described algorithm are presented.  相似文献   

12.
Summary. Several important applications of the paraxial ray approximation (PRA) to numerical modelling of high-frequency seismic body wavefields are discussed. The PRA can be used to evaluate the displacement vector not only directly on the ray, as in the standard ray method. but also approximately in the vicinity of this ray. The PRA also offers simple ways of approximate evaluation of paraxial rays, situated in the vicinity of the central ray, and of two-point ray tracing. A very important application of the PRA consists in a simple, fast and effective Computation of body-wave synthetic seismograms in general, 3-D, laterally inhomogeneous, layered structures. Examples of synthetic seismograms for 3-D structures, computed using the PRA, are presented.  相似文献   

13.
Land-use allocation is of great importance for rapid urban planning and natural resource management. This article presents an improved artificial bee colony (ABC) algorithm to solve the spatial optimization problem. The new approach consists of a heuristic information-based pseudorandom initialization (HIPI) method for initial solutions and pseudorandom search strategy based on a long-chain (LC) mechanism for neighborhood searches; together, these methods substantially improve the search efficiency and quality when handling spatial data in large areas. We evaluated the approach via a series of land-use allocation experiments and compared it with particle swarm optimization (PSO) and genetic algorithm (GA) methods. The experimental results show that the new approach outperforms the current methods in both computing efficiency and optimization quality.  相似文献   

14.
Summary. The study of the asymptotic behaviour of eigenfrequencies of torsional modes of the Earth is of interest in the problem of the existence of discontinuities in the Earth's interior and the determination of their depth and scale, since the solotone effect, which is a persistent oscillatory cohponent in the asymptotic overtone structure, is very sensitive to the presence of discontinuities. The asymptotic behaviour of torsional eigenfrequencies of the second order differential equation for the Earth's free oscillations can be compared with that obtained from eigenfrequencies evaluated from synthetic SH-wave seismograms by Brune's phase correlation method, using various earth models. The solotone effect that appears in the former for discontinuous models can be explained in terms of multiple reflections fram the discontinuities, and can be reconstructed from synthetic SH -wave pulses arising from these reflections. Its properties vary systematically with the depth and the scale of discontinuities and can be superposed for several discontinuities.  相似文献   

15.
Rays propagating through strongly laterally varying media exhibit chaotic behaviour. This means that initially close rays diverge exponentially, rather than according to a power law. This chaotic behaviour is especially pronounced if the medium contains laterally varying interfaces. By studying simple 2-D and 3-D versions of models with laterally varying interfaces, the importance of chaotic ray behaviour is determined. A model of the Moho below Germany produces sharp variations with epicentral distance of the number of arrivals. In addition, the number of caustics grows dramatically: up to 1200 caustics are present between a distance of 0 and 800 km. Using the theory of Hamiltonian systems, a more in-depth study of the chaotic character of the ray equations is obtained. It is found that for realistic heterogeneous models most of the relevant rays will exhibit chaotic behaviour. The degree of chaos is quantified in terms of predictability horizons. Beyond the predictability horizons ray tracing cannot be carried out accurately. For the models under consideration, the length from the source to the predictability horizon has an order of magnitude of 1000 km. The chaotic behaviour of the rays makes it necessary to use extensions of asymptotic ray theory, such as Maslov theory, to compute seismic waveforms. It is shown that pseudo-caustics, an important obstacle in computing Maslov synthetics, are a generic feature of the 2-D laterally varying models that are studied. Eventually, the use of asymptotic methods is restricted because of the inaccuracy in the computation of the ray paths.  相似文献   

16.
In case of a complex overburden, the seismic data can be greatly improved by applying a full wavefield redatuming procedure. In practice, the application of the redatuming process to 3-D data acquired by conventional acquisition designs is non-trivial. Because of the large amount of data involved in the 3-D redatuming process and because of the sparseness of these data, it is impossible to apply conventional wave equation datuming directly.
We present a data mapping approach to redatuming (DMR), which follows the concept of Kirchhoff data mapping. A simplified background medium where no ray bending occurs is assumed for the medium below the datum in order to map an input data set referenced to the acquisition surface to an output data set referenced to the new datum level. The DMR method can be interpreted as a simplified version of the Kirchhoff summation redatuming (KSR) method, where one of the 2-D integrals over the acquisition coordinates can be solved analytically. Consequently, in this approach fewer traces are involved in the computation of one time sample (a 2-D integral is computed instead of a 4-D integral), which makes it particularly attractive for the application to 3-D data sets.
In this paper the theory underlying data mapping redatuming is discussed and the proposed approach is tested on fully sampled 2-D and 3-D synthetic data from models with both simple and complex velocity distributions in the subsurface.
The tests clearly show that the objective of producing results that are comparable to the conventional KSR has been achieved. The redatumed traces are dynamically and kinematically correct. Furthermore, these results confirm that the dependency of the new approach on the assumed medium below the datum level is, indeed, weak because the assumption of a velocity medium where no ray bending occurs is already sufficient to produce correct results.  相似文献   

17.
Today's numerical methods like the Spectral Element Method (SEM) allow accurate simulation of the whole seismic field in complex 3-D geological media. However, the accuracy of such a method requires physical discontinuities to be matched by mesh interfaces. In many realistic earth models, the design of such a mesh is difficult and quite ineffective in terms of numerical cost. In this paper, we address a limited aspect of this problem: an earth model with a thin shallow layer below the free surface in which the elastic and density properties are different from the rest of the medium and in which rapid vertical variations are allowed. We only consider here smooth lateral variations of the thickness and elastic properties of the shallow layer. In the limit of a shallow layer thickness very small compared to the smallest wavelength of the wavefield, by resorting to a second order matching asymptotic approximation, the thin layer can be replaced by a vertically smooth effective medium without discontinuities together with a specific Dirichlet to Neumann (DtN) surface boundary condition. Such a formulation allows to accurately take into account complex thin shallow structures within the SEM without the classical mesh design and time step constraints. Corrections at receivers and source—when the source is located within the thin shallow layer—have been also derived. Accuracy and efficiency of this formulation are assessed on academic tests. The stability and limitations of this formulation are also discussed.  相似文献   

18.
The state of stress within a bending spherical shell has some special features that are caused by sphericity. While most lithospheres are more like spherical shells than flat plates, our ideas of the state of stress have been dominated by flat-plate models. As a consequence, we might be missing some important aspects of the state of stress within subducting lithospheres. In order to examine this problem, we analyse spherical-shell bending problems from basic equations. We present two approaches to solve spherical-shell bending problems: one by the variational approach, which is suitable for global-scale problems, and the other by the asymptotic equation, which is valid to first order in h/R , where h is the thickness of the lithosphere and R is its curvature radius (i.e. under the assumption of small curvature). The form of the equation for displacement shows that wavelengths of deformation are determined by the spherical (elastic) effect and the gravitational buoyancy effect, for which only the latter effect is included in the usual flat-plate formulations. In the case of the Earth, the buoyancy force is dominant and, consequently, spherical effects are suppressed to a large extent; this explains why flat-plate models have been successful for Earth's lithospheric problems. On the other hand, the state of stress shows interesting spherical effects: while bending (fibre) stress along the subduction zone is always important, bending stress along the trench-strike direction can also be important, in particular when the subduction zone arc is small. Numerical results also indicate that compressive normal stress along the trench-strike direction is important when a subduction zone arc is large. These two stresses, the bending stress and the compressive normal stress, both along the trench-strike direction, may have important implications for intraplate earthquakes at subduction zones.  相似文献   

19.
Understanding the interaction between groundwater and surface water in permafrost regions is essential to study flood frequencies and river water quality, especially in the high latitude/altitude basins. The application of heat tracing method,based on oscillating streambed temperature signals, is a promising geophysical method for identifying and quantifying the interaction between groundwater and surface water. Analytical analysis based on a one-dimensional convective-conductive heat transport equation combined with the fiber-optic distributed temperature sensing method was applied on a streambed of a mountainous permafrost region in the Yeniugou Basin, located in the upper Heihe River on the northern Tibetan Plateau. The results indicated that low connectivity existed between the stream and groundwater in permafrost regions.The interaction between surface water and groundwater increased with the thawing of the active layer. This study demonstrates that the heat tracing method can be applied to study surface water-groundwater interaction over temporal and spatial scales in permafrost regions.  相似文献   

20.
A tomographic inversion technique that inverts traveltimes to obtain a model of the subsurface in terms of velocities and interfaces is presented. It uses a combination of refraction, wide-angle reflection and normal-incidence data, it simultaneously inverts for velocities and interface depths, and it is able to quantify the errors and trade-offs in the final model. The technique uses an iterative linearized approach to the non-linear traveltime inversion problem. The subsurface is represented as a set of layers separated by interfaces, across which the velocity may be discontinuous. Within each layer the velocity varies in two dimensions and has a continuous first derivative. Rays are traced in this medium using a technique based on ray perturbation theory, and two-point ray tracing is avoided by interpolating the traveltimes to the receivers from a roughly equidistant fan of rays. The calculated traveltimes are inverted by simultaneously minimizing the misfit between the data and calculated traveltimes, and the roughness of the model. This 'smoothing regularization' stabilizes the solution of the inverse problem. In practice, the first iterations are performed with a high level of smoothing. As the inversion proceeds, the level of smoothing is gradually reduced until the traveltime residual is at the estimated level of noise in the data. At this point, a minimum-feature solution is obtained, which should contain only those features discernible over the noise.
The technique is tested on a synthetic data set, demonstrating its accuracy and stability and also illustrating the desirability of including a large number of different ray types in an inversion.  相似文献   

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