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1.
Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c o, where | c o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, co =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.  相似文献   

2.
Summary. The Radon transform or slant stack is becoming a widely used technique for analysing high-quality reflection and refraction data. The transform normally used is applicable to data from a line source in a plane model, that is, one Cartesian coordinate. The theoretical basis for the Radon transform pair for one Cartesian coordinate has appeared in the seismological literature. For a point source in plane or spherical geometry, or a line source in cylindrical geometry only the Radon transform for the direct problem (computation of synthetic seismograms) has been published. To analyse data an approximate inverse transform has been used. In this research note, the exact forms of the generalized Radon transform pairs are completed for a point source in plane or spherical geometry, and for a line source in cylindrical geometry. The differences will be important if the waveforms are being interpreted, and are most significant for near-vertical reflections—the type of data most commonly slant stacked.  相似文献   

3.
Summary. Seismic investigations using shear-wave and converted wave techniques show that very often reflected PS - and SS -waves have anomalous polarizations ( accessory components ). This phenomenon cannot be explained in terms of isotropic models with dipping boundaries. Computations of synthetic seismograms of reflected PS - and SS -waves were made for different models of transversely isotropic media with dipping anisotropic symmetry axes not normal to the boundaries. Synthetic seismograms were computed by ray techniques using an optimization algorithm to construct all rays arriving at a given receiver. These computations indicate that accessory components arise when the medium above the boundary is anisotropic, where they are caused by the constructive interference of qSV - and qSH -waves. If a low-velocity layer is present, displacement vectors of both waves have horizontal projections which are approximately orthogonal. The algorithm for wave separation is presented and some results of its use are given.  相似文献   

4.
Seismic waves in stratified anisotropic media   总被引：4，自引：0，他引：4  
Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmétry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmétry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.  相似文献   

5.
Summary. There are two ways to apply the Cagniard de-Hoop method when generating synthetic seismograms due to a source in a three-dimensional medium. One is the Hankel transform method (hereafter called 'the cylindrical wave representation method'), which utilizes the property of the modified Bessel function. The other is 'the plane wave representation method', which replaces the Bessel function by a superposition of plane waves. In extending a point source to a finite dimensional source, the latter method is extremely useful, because it enables the integrations on the fault surface to be performed analytically.
Using the latter method, expressions for displacements due to a Haskell type vertical fault in a uniform half-space are obtained. A solution is given as a sum of four quadrantal source contributions, which is similar to Madariaga's solution for a source in the whole space. Each contribution consists of a single finite range integration or a single integration plus a pole contribution, depending on the location of the observation point with respect to the source. The procedure can be extended to other fault types.  相似文献   

6.
P-SH conversion is commonly observed in teleseismic P waves, and is often attributed to dipping interfaces beneath the receiver. Our modelling suggests an alternative explanation in terms of flat-layered anisotropy. We use reflectivity techniques to compute three-component synthetic seismograms in a 1-D anisotropic layered medium. For each layer of the medium, we prescribe values of seismic velocities and hexagonally symmetric anisotropy about a common symmetry axis of arbitrary orientation. A compressional wave in an anisotropic velocity structure suffers conversion to both SV -and SH -polarized shear waves, unless the axis of symmetry is everywhere vertical or the wave travels parallel to all symmetry axes. The P-SV conversion forms the basis of the widely used 'receiver function' technique. The P-SH conversion occurs at interfaces where one or both layers are anisotropic. A tilted axis of symmetry and a dipping interface in isotropic media produce similar amplitudes of both direct ( P ) and converted ( Ps ) phases, leaving the backazimuth variation of the P-Ps delay as the main discriminant. Seismic anisotropy with a tilted symmetry axis leads to complex synthetic seismograms in velocity models composed of just a few flat homogeneous layers. It is possible therefore to model observations of P coda with prominent transverse components with relatively simple 1-D velocity structures. Successful retrieval of salient model characteristics appears possible using multiple realizations of a genetic-algorithm (GA) inversion of P coda from several backazimuths. Using GA inversion, we determine that six P coda recorded at station ARU in central Russia are consistent with models that possess strong (> 10 per cent) anisotropy in the top 5 km and between 30 and 43 km depth. The symmetry axes are tilted, and appear aligned with the seismic anisotropy orientation in the mantle under ARU suggested by SKS splitting.  相似文献   

7.
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   

8.
Summary. The transformation of a set of seismograms to the delay time-slowness, τ—p, domain is presented as a sequence of Fourier and Bessel transforms, For a horizontally layered medium, this sequence gives an exact cylindrical wave decomposition of the response to a point source; correctly compensating for the phase shifting and geometrical spreading associated with transmission through the Earth. The resultant τ—p map or 'slant stack' contains true amplitude and phase information. The spatial aliasing properties of the transformation, when applied to a dataset, are greatly improved by the use of only outgoing waves in the Bessel transform. This is equivalent to using Hankel functions rather than Bessel functions, and is justified by the absence of incoming waves from most datasets. The WKBJ approximation to the medium response enables predictions to be made about the shape and amplitude variation with slowness of truncation effects. Theoretically the τ—p transformation is reversible, thus the τ—p domain is a suitable one in which to perform filtering operations before seismogram reconstruction.  相似文献   

9.
We present two equivalent algorithms for iterative linearized waveform inversion for 3-D Earth structure with respect to an arbitrary 3-D starting model; one is a matrix formulation, and the second is a wavefield formulation. Both algorithms require the computation of accurate synthetic seismograms, but neither requires that any particular method be used to compute the synthetics. The matrix formulation is equivalent to our previously published algorithm (Hara, Tsuboi & Geller 1991), but requires less than 10 per cent of the CPU time of the previous algorithm. The wavefield algorithm is equivalent to that of Tarantola (1986) and Mora (1987), but appears to be substantially more efficient.  相似文献   

10.
This paper presents a geometrically based algorithm for computing synthetic seismograms for energy transmitted through a 3-D velocity distribution. 3-D ray tracing is performed to compute the traveltimes and geometrical spreading (amplitude). The formulations of both kinematic and dynamic ray-tracing systems are presented. The two-point ray-tracing problem is solved by systematically updating the initial conditions and adjusting the ray direction until the ray intersects the specified endpoint. The amount of adjustment required depends on the derivatives of the position with respect to the given starting angles between consecutive rays. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The convergence rate depends on the complexity of the model.
Test seismograms compare favourably with those from a 2-D asymptotic ray theory algorithm and a 3-D Gaussian-beam algorithm. The algorithm is flexible in modelling arbitrary source and recorder geometries for various smoothly varying 3-D velocity distributions. The algorithm is further tested by simulating surface-to-tunnel vibroseis field data. Shear waves as well as compressional waves may be approximately included. Application of the algorithm to a data set from the Rainier Mesa of the Nevada Test Site produced a good fit to the transmitted (first arrival) traveltimes and amplitudes, with approximately 15 per cent variation in the local 3-D velocity.  相似文献   

11.
In contrast to previous work, which treats the Earth's lateral heterogeneity as an infinitesimal perturbation to a spherically symmetrical starting model, we conduct iterative linearized waveform inversion for the Earth's laterally heterogeneous structure. We use the Direct Solution Method (DSM) (Geller et al. 1990a) to calculate synthetic seismograms and their partial derivatives for a laterally heterogeneous earth model. We invert surface-wave data from the IDA and GEOSCOPE networks. We expand the lateral heterogeneity of rigidity in spherical harmonics up to angular order number 8 and use three parameters to specify the depth dependence of each harmonic, giving us a total of 240 unknowns. The short-wavelength lateral heterogeneity (s = 4, 6 and 8) in the deeper part of the upper mantle obtained by this study differs significantly from M84A. The relative improvement in the variance reduction as compared with model M84A is about 20 per cent for the IDA data and more than 100 per cent for the GEOSCOPE data.  相似文献   

12.
Geophysical observables are generally related to earth structure and source parameters in a complicated non-linear way. Consequently, a large number of forward modelling processes are commonly necessary to obtain a satisfactory estimate of such parameters from observed data. The most time-consuming part of the forward modelling is the computation of the Green's functions of the different earth models to be tested. In this study, we present a fast converging algorithm: the differential transform method for the computation of Green's functions in terms of spherical or cylindrical harmonics. In this method, a deconvolutable high-pass filter is used to enhance the numerical significance of the far-field spectrum of Green's functions. Compared with existing fast converging algorithms such as the Kummer's transformation and the disc factor method, the differential transform method is more efficient except for the extremely near-source region. The new method can be used to suppress numerical phases (non-physical seismic signals) associated with the aliasing effect that may arise in synthetic seismograms when the latter are computed from a windowed wavenumber (or slowness) spectrum. The numerical efficiency of the new method is demonstrated via two representative tests.  相似文献   

13.
Summary. The Lanczos method of separating exponentials is applied to the Fourier transform of seismograms in order to separate the various modes that contribute to the given portion of the seismograms. Phase velocities and amplitudes are obtained as functions of the frequency. When applying the method to artificial seismograms, which are built as an exact superposition of a number of modes, the separation is very accurate. The method was also applied to the surface-wave portion of numerical seismograms for a vertical point force in a layered medium. The phase velocity and amplitude of the fundamental mode are obtained. These functions were taken as the first guess in the Backus—Gilbert generalized inverse procedure and the process converged very rapidly. When a perturbation of the phases and amplitudes is taken as the first guess the process converges to the true model when enough data are available.  相似文献   

14.
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   

15.
On crustal corrections in surface wave tomography   总被引：1，自引：0，他引：1  
Mantle models from surface waves rely on good crustal corrections. We investigated how far ray theoretical and finite frequency approximations can predict crustal corrections for fundamental mode surface waves. Using a spectral element method, we calculated synthetic seismograms in transversely isotropic PREM and in the 3-D crustal model Crust2.0 on top of PREM, and measured the corresponding time-shifts as a function of period. We then applied phase corrections to the PREM seismograms using ray theory and finite frequency theory with exact local phase velocity perturbations from Crust2.0 and looked at the residual time-shifts. After crustal corrections, residuals fall within the uncertainty of measured phase velocities for periods longer than 60 and 80 s for Rayleigh and Love waves, respectively. Rayleigh and Love waves are affected in a highly non-linear way by the crustal type. Oceanic crust affects Love waves stronger, while Rayleigh waves change most in continental crust. As a consequence, we find that the imperfect crustal corrections could have a large impact on our inferences of radial anisotropy. If we want to map anisotropy correctly, we should invert simultaneously for mantle and crust. The latter can only be achieved by using perturbation theory from a good 3-D starting model, or implementing full non-linearity from a 1-D starting model.  相似文献   

16.
Summary. The inverse gravity potential problem consists in the determination of the form and the density of the body by its exterior gravity potential. We describe two similar classes of bodies for which this problem has a unique constructive solution.
(1) The first class contains the cylindrical bodies with finite length, arbitrary form of section and ρ( R , ø, z) =ρ1( z )ρ2( R , ø) density distribution, where z is the cylindrical coordinate; R , ø are the polar coordinates in a section plane. This class is important for prospecting geophysics in that it allows us to determine in a unique and constructive way, the function ρ1( R , ø), the length, form and orientation of the cylinder if we know the function ρ1( z ) and the exterior potential. The classical moment problem of functions is the basis for the solution of this problem.
(2) The analogous problem for the class of the spherical cylinders, or bodies bounded by arbitrary similar sections of two different concentric spheres and the radial lateral surface, appears when bodies of planetary size are studied. (An example of these bodies would be the Moon mascons.) The density distribution of these cylinders is ρ(τ, θ, ø) =ρ1(τ)ρ2(θ, ø) where τ, θ, ø are the spherical coordinates. The function ρ1(θ, ø), length and form of spherical sections can be uniquely determined by exterior potential if we know the function ρ1(τ). We propose a new constructive method for harmonic continuation of the gravity potential into the region containing the perturbing masses for the solution of the problem.  相似文献   

17.
Summary. We report the initial results of our attempts to obtain theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram, including both body waves and surface waves, can be generated for a spherical, anelastic earth by simple inverse Fourier transformation of the sum of the propagating fundamental and higher-mode surface waves. The key to success in reproducing the WWSSN records involves the number of modes, and the minimum period used in these computations; here we use eight modes and a minimum period of 2 s. Efficient computational algorithms make it possible to handle up to 2000 frequency points for each mode; approximately 200 layers are used to model the radial heterogeneity of the earth; attenuation is treated exactly. Examples are given of the SH theoretical seismograms resulting from dislocation sources buried at various depths in the Earth.  相似文献   

18.
From basic Fourier theory, a one-component signal can be expressed as a superposition of sinusoidal oscillations in time, with the Fourier amplitude and phase spectra describing the contribution of each sinusoid to the total signal. By extension, three-component signals can be thought of as superpositions of sinusoids oscillating in the x -, y -, and z -directions, which, when considered one frequency at a time, trace out elliptical motion in three-space. Thus the total three-component signal can be thought of as a superposition of ellipses. The information contained in the Fourier spectra of the x -, y -, and z -components of the signal can then be re-expressed as Fourier spectra of the elements of these ellipses, namely: the lengths of their semi-major and semi-minor axes, the strike and dip of each ellipse plane, the pitch of the major axis, and the phase of the particle motion at each frequency. The same type of reasoning can be used with windowed Fourier transforms (such as the S transform), to give time-varying spectra of the elliptical elements. These can be used to design signal-adaptive polarization filters that reject signal components with specific polarization properties. Filters of this type are not restricted to reducing the whole amplitude of any particular ellipse; for example, the 'linear' part of the ellipse can be retained while the 'circular' part is rejected. This paper describes the mathematics behind this technique, and presents three examples: an earthquake seismogram that is first separated into linear and circular parts, and is later filtered specifically to remove the Rayleigh wave; and two shot gathers, to which similar Rayleigh-wave filters have been applied on a trace-by-trace basis.  相似文献   

19.
A seismogram that is several times the length of the source-receiver wavelet is windowed into two parts—these may overlap—to obtain two seismograms with approximately the same source function but different Green's functions. A similarly windowed synthetic seismogram gives two corresponding synthetic seismograms. The spectral product of the window 1 data with the window 2 synthetic is equal to the spectral product of the window 1 synthetic with the window 2 data only if the correct earth model is used to compute the synthetic. This partition principle is applied to well-log sonic waveform data from Ocean Drilling Project hole 806B, a slow formation, and used there to estimate Poisson's ratio from a single seismogram whose transmitter and receiver functions are unknown. A multichannel extension of the algorithm gives even better results. The effective borehole radius R b, was included in the inversion procedure, because of waveform sensitivity to R b. Inversion results for R b agreed with the sonic caliper, but not the mechanical caliper; thus if R b is not included in the inversion its value should be taken from the sonic caliper.  相似文献   

20.
Summary. Body wave synthetic siesmograms for laterally varying media are computed by means of a slowness implementation of the extended WKBJ (EWKBJ) theory of Frazer & Phinney. An EWKBJ seismogram is computed by first tracing rays through a particular model to obtain conventional ray information (travel time, ray end point, ray slowness) and then using these data in the finite frequency integral expression for the EWKBJ seismogram. The EWKBJ seismograms compare favourably to geometrical ray theory (GRT) seismograms but are significantly better because of the finite frequency nature of the EWKBJ calculation. More realistic behaviour is obtained with EWKBJ seismograms at normal seismic frequencies near caustics, where the GRT amplitude is infinite, and within geometrical shadow zones where GRT predicts zero amplitudes. In addition the EWKBJ calculation is more sensitive than GRT to focuses and defocuses in the ray field. The major disadvantage of the EWKBJ calculation is the additional computer time over that of GRT, necessary to calculate one seismogram although an EWKBJ seismogram costs much less to compute than a reflectivity seismogram. Another disadvantage of EWKBJ theory is the generation of spurious, non-geometrical phases that are associated with rapidly varying lateral inhomogeneities. Fortunately the amplitudes of these spurious phases are usually much lower than that of neighbouring geometrical phases so that the spurious phases can usually be ignored. When this observation is combined with the moderately increased computational time of the EWKBJ calculation then the gain in finite frequency character significantly outweighs any disadvantages.  相似文献   

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