Elastic scattered waves from a continuous and heterogeneous layer

Elastic scattering from a continuous and laterally unbounded heterogeneous layer has been formulated using the Born approximation. A general solution of the scattered wave equation for the above-stated medium has been given in terms of a Fourier integral over plane waves. Far-field asymptotic expressions for weak elastic scattering by a finite, continuous and inhomogeneous layer have been presented which agree with earlier results. For perturbations of the two elastic parameters and the density having the same form of spatial variation, the spectrum of plane waves scattered from a heterogeneous layer is expressed as a product of an 'elastic scattering factor'and a 'distribution factor'. As in earlier results for small-scale heterogeneity, the scattering pattern depends on various combinations of perturbations of elastic parameters and density. In order to show the general characteristics of the elastic wave scattering, some scattering patterns have been given.     

Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.     

3-D linearized scattering of surface waves and a formalism for surface wave holography

Summary. Scattering of surface waves by lateral heterogeneities is analysed in the Born approximation. It is assumed that the background medium is either laterally homogeneous, or smoothly varying in the horizontal direction. A dyadic representation of the Green's function simplifies the theory tremendously. Several examples of the theory are presented. The scattering and mode conversion coefficients are shown for scattering of surface waves by the root of an Alpine-like crustal structure. Furthermore a 'great circle theorem'in a plane geometry is derived. A new proof of Snell's law is given for surface wave scattering by a quarter-space. It is shown how a stationary phase approximation can be used to simplify the Fourier synthesis of the scattered wave in the time domain. Finally a procedure is suggested to do 'surface wave holography'.     

Surface wave tomography: global membrane waves and adjoint methods

We implement the wave equation on a spherical membrane, with a finite-difference algorithm that accounts for finite-frequency effects in the smooth-Earth approximation, and use the resulting 'membrane waves' as an analogue for surface wave propagation in the Earth. In this formulation, we derive fully numerical 2-D sensitivity kernels for phase anomaly measurements, and employ them in a preliminary tomographic application. To speed up the computation of kernels, so that it is practical to formulate the inverse problem also with respect to a laterally heterogeneous starting model, we calculate them via the adjoint method, based on backpropagation, and parallelize our software on a Linux cluster. Our method is a step forward from ray theory, as it surpasses the inherent infinite-frequency approximation. It differs from analytical Born theory in that it does not involve a far-field approximation, and accounts, in principle, for non-linear effects like multiple scattering and wave front healing. It is much cheaper than the more accurate, fully 3-D numerical solution of the Earth's equations of motion, which has not yet been applied to large-scale tomography. Our tomographic results and trade-off analysis are compatible with those found in the ray- and analytical-Born-theory approaches.     

Time-lapse traveltime change of singly scattered acoustic waves

We present a technique based on the single-scattering approximation that relates time-lapse localized changes in the propagation velocity to changes in the traveltime of singly scattered waves. We describe wave propagation in a random medium with homogeneous statistical properties as a single-scattering process where the fluctuations of the velocity with respect to the background velocity are assumed to be weak. This corresponds to one of two end-member regimes of wave propagation in a random medium, the first being single scattering, and the second multiple scattering. We present a formulation that relates the change in the traveltime of the scattered waves to a localized change in the propagation velocity by means of the Born approximation for the scattered wavefield. We validate the methodology with synthetic seismograms calculated with finite differences for 2-D acoustic waves. Potential applications of this technique include non-destructive evaluation of heterogeneous materials and time-lapse monitoring of heterogeneous reservoirs.     

Time-domain computation of scattering from a heterogeneous layer

Adopting Born and ray approximations, time-domain synthetic seismograms for P-P and P-S scattering from a plane wave incident on a thin, laterally heterogeneous layer are presented in this paper. The time-domain P coda is a convolution between a structure function and the second-order derivative of the time function of the incident P wave. Examples of synthetic seismograms are given using a time function from a computed short-period seismogram for a point explosive source in a half-space. These show that it is impossible, with realistic values of the parameters involved, to generate significant codas when only single scattering is involved.     

Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

Born scattering of longperiod body waves

The Born approximation is applied to the modelling of the propagation of deeply turning longperiod body waves through heterogeneities in the lowermost mantle. We use an exact Green's function for a spherically symmetric earth model that also satisfies the appropriate boundary conditions at internal boundaries and the surface of the earth. The scattered displacement field is obtained by a numerical quadrature of the product of the Green's function, the exciting wavefield and structural perturbations. We study three examples: scattering of longperiod P waves from a plume rising from the coremantle boundary (CMB), generation of longperiod precursors to PKIKP by strong, localized scatterers at the CMB, and propagation of corediffracted P waves through largescale heterogeneities in D". The main results are as follows: (1) the signals scattered from a realistic plume are small with relative amplitudes of less than 2 per cent at a period of 20 s, rendering plume detection a fairly difficult task; (2) strong heterogeneities at the CMB of appropriate size may produce observable longperiod precursors to PKIKP in spite of the presence of a diffraction from the PKP B caustic; (3) corediffracted P  waves ( P _diff) are sensitive to structure in D" far off the geometrical ray path and also far beyond the entry and exit points of the ray into and out of D"; sensitivity kernels exhibit ringshaped patterns of alternating sign reminiscent of Fresnel zones; (4) P _diff also shows a nonnegligible sensitivity to shear wave velocity in D"; (5) down to periods of 40 s, the Born approximation is sufficiently accurate to allow waveform modelling of P _diff through largescale heterogeneities in D" of up to 5 per cent.     

Inverse scattering of surface waves: imaging of near-surface heterogeneities

An efficient inverse scattering method is developed for imaging near-surface heterogeneities using scattered surface waves. Three dimensional elastodynamic wave propagation and scattering in a laterally invariant embedding medium is considered. The Born Approximation is used and the scattered wavefield is expressed as a domain type integral representation. The computation time of Green's tensor elements is reduced by considering the radial symmetry of the medium. The method is validated by numerical tests. Ultrasonic laboratory data obtained from a scale model experiment are used for imaging the near-surface inhomogeneities caused by an epoxy-filled hole in the surface of an aluminum block. Both synthetic and the scale model tests show that the location, the actual density contrast and the depth of the inhomogeneities are reasonably well estimated.     

Elastic-wave scattering in a random medium characterized by the Von Karman correlation function and small-scale inhomogeneities in the lithosphere

A Gaussian correlation function characterizes smoothly heterogeneous media, while real heterogeneities in the Earth are often non-Gaussian in nature. Using the Born approximation, mean square amplitudes of the scattered waves have been derived for an elastic media characterized by the Von Karman correlation function. Heterogeneities with different power laws can be defined by the Von Karman correlation function. The sensitivity of fore- and backscattering to heterogeneities with different scales and properties (that is velocity and impedance) is discussed in this paper. The analytical expression for total scattered energy for the incident P waves is also derived for a random medium having the Von Karman correlation function. We find that at high frequencies, the scattered power of converted waves is a function of frequency. In the case of codawave excitation by local earthquakes, which must be handled by the full elastic-wave theory, we can define any type of inhomogeneity by the Von Karman correlation function. It also supports the idea that the lithosphere might have multiple-scale inhomogeneities.     

On seismic-wave scattering by a rough core—mantle boundary

summary . A formulation is given for the seismic-wave scattering by a rough solid—liquid interface, in analogy to results derived for a solid—solid interface and a heterogeneous volume. Using Kennett's approach and the reciprocity theorem, the scattering is formulated as the excitation by an equivalent dislocation. Using interface parameters relevant to the core—mantle boundary (CMB), computational results for several types of body-wave scattering are given and compared to scattering by a heterogeneous volume. In an application to the generation of PKP precursors it is concluded that, whereas some data groups point to heterogeneity (which may not be small) in the lower mantle above CMB, in other cases a rough CMB may be considered equally feasible. Scattering at the source or receiver side of the core by both a slightly rough CMB (radial variations up to a few hundred metres) and a slightly heterogeneous lower mantle (relative variations in physical parameters up to a few per cent) produces the energy level that is observed in most of the PKP precursors; also the relevant scale lengths of variation are about the same ie both mechanisms (10–20 km with possibly somewhat higher values at relatively long epicentral distances).     

Calculation of melt pond albedos on arctic sea ice

An analytical approximation of spectral albedo is derived for a melt pond with a Lambertian bottom assuming that Rayleigh scattering in the water is small compared to absorption. A Monte Carlo method is used to verify that scattering can he ignored in the water. This enables us to calculate pond albedos in the 400–700 nm wavelength hand using the analytical approximation. Model calculations and observations indicate that a step-decrease in albedo is likely t o occur when a melt pond initially forms, and melt pond albedos in the visible depend more on the structural and optical properties of the bottom than on the depth of the pond.     

Vertical profile of effective turbidity reconstructed from broadening of incoherent body-wave pulses—I. General approach and the inversion procedure

A method is developed for the reconstruction of a non-uniform distribution of scattering properties in the upper layers of the Earth using data on broadening of an incoherent body-wave group or pulse along a number of rays. The theoretical basis for this reconstruction is a linear integral formula after Bocharov (1985, 1988), which is employed to design a linear inversion procedure. The inversion is performed in terms of a single scalar parameter of effective turbidity. This parameter presents an adequate generalization of the common turbidity parameter used in the isotropic scattering case; it describes, simultaneously, scattering attenuation, pulse broadening and backscattering or coda formation. As a preliminary step, necessary conditions of applicability of the transport equation approach for the analysis of regional high-frequency seismic waves are verified. A new compact derivation of Bocharov's formula is then presented. A linear least-squares inversion procedure for recovering a layered turbidity structure is proposed and tested on synthetic data of onset-to-peak delays of incoherent body-wave pulses. A few practical aspects of the application of the general approach to seismological data are analysed, including the correctness of the low-angle approximation, the use of peak delay observations instead of pulse centroid, the effects of a realistic spatial spectrum of inhomogeneity field, the potential bias produced by intrinsic loss, and the distortions produced by a non-spherical (double dipole) source radiation pattern. The latter point is considered as critically important, as one can expect significant data contamination by nodal arrivals. An efficient robust estimation procedure is designed and tested that is capable of suppressing distortions from nodal and near-nodal data.     

Surface wave tomography for azimuthal anisotropy in a strongly reduced parameter space

Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy.     

Gaussian beams, complex rays, and the analytic extension of the Green's function in smoothly inhomogeneous media

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.     

On the relative scattering of P- and S-waves

Summary. Using a single scattering approximation, we derive equations for the scattering attenuation coefficients of P- and S -body waves. We discuss our results in the light of some recent energy renormalization approaches to seismic wave scattering. Practical methods for calculating the scattering attenuation coefficients for various earth models are emphasized. The conversions of P - to S -waves and S- to P -waves are included in the theory. The earth models are assumed to be randomly inhomogeneous, with their properties known only through their average wavenumber power spectra. We approximate the power spectra with piecewise constant functions, each segment of which contributes to the net, frequency-dependent, scattering attenuation coefficient. The smallest and largest wavenumbers of a segment can be plotted along with the wavevectors of the incident and scattered waves on a wavenumber diagram. This diagram gives a geometric interpretation for the frequency behaviour associated with each spectral segment, including a 'transition' peak that is due entirely to the wavenumber limits of the segment. For regions of the earth where the inhomogeneity spectra are concentrated in a band of wavenumbers, it should be possible to observed such a peak in the apparent attenuation of seismic waves. We give both the frequency and distance limits on the accuracy of the theoretical results.     

Backscattering from spherical elastic inclusions and accuracy of the Kirchhoff approximation for curved interfaces

The Kirchhoff (or tangent plane) approximation, derived from the theoretically complete Kirchhoff–Helmholtz integral representation for the seismic wavefield, has been used extensively for the analysis of seismic-wave scattering from irregular interfaces; however, the accuracy of this method for curved interfaces has not been rigorously established. This paper describes an efficient Kirchhoff algorithm to simulate scattered waves from an arbitrarily curved interface in an elastic medium. Synthetic seismograms computed using this algorithm are compared with exact synthetics computed using analytical formulae for scattering of plane P waves by a spherical elastic inclusion. A windowing technique is used to remove strong internal reverberations from the analytical solution. Although the Kirchhoff method tends to underestimate the total scattering intensity, the accuracy of the approximation improves with increasing value of the wavenumber-radius product, kR . The arrival times and pulse shapes of primary reflections from the sphere are well approximated using the Kirchhoff approach regardless of curvature of the scattering surface, but the amplitudes are significantly underestimated for kR ≤ 5. The results of this work provide some new guidelines to assess the accuracy of Kirchhoff-synthetic seismograms for curved interfaces.     

Global multiresolution models of surface wave propagation: comparing equivalently regularized Born and ray theoretical solutions

I invert a large set of teleseismic phase-anomaly observations, to derive tomographic maps of fundamental-mode surface wave phase velocity, first via ray theory, then accounting for finite-frequency effects through scattering theory, in the far-field approximation and neglecting mode coupling. I make use of a multiple-resolution pixel parametrization which, in the assumption of sufficient data coverage, should be adequate to represent strongly oscillatory Fréchet kernels. The parametrization is finer over North America, a region particularly well covered by the data. For each surface-wave mode where phase-anomaly observations are available, I derive a wide spectrum of plausible, differently damped solutions; I then conduct a trade-off analysis, and select as optimal solution model the one associated with the point of maximum curvature on the trade-off curve. I repeat this exercise in both theoretical frameworks, to find that selected scattering and ray theoretical phase-velocity maps are coincident in pattern, and differ only slightly in amplitude.     

Hierarchical traveltime tomography

A new traveltime tomographic method was developed with hierarchical shape functions of the finite element method as slowness or velocity interpolation functions. The degree of the approximation of velocity modelling is adjusted by selecting a set of hierarchical shape functions in each element. The ray density parameter of each element controls the selection to make the approximation fine or coarse in the high- or low-ray-density area. The proposed method is applied to both synthetic traveltime data and actual data. The AIC is used to determine the number of model parameters. The result of the synthetic data shows that low-resolution model parameters can be eliminated by the ray density parameter. The result of the actual data shows that the velocity pattern is approximately the same in the fine approximation area and that the velocity fluctuation is suppressed in the coarse approximation area, compared with that obtained from a full set of hierarchical shape functions. The number of model parameters is drastically reduced. The resolution can be estimated by the checkerboard restoration test. The result of the real data set was compared with that of the linear velocity grid model.     

Regional tomographic inversion of the amplitude and phase of Rayleigh waves with 2-D sensitivity kernels

In this study, we test the adequacy of 2-D sensitivity kernels for fundamental-mode Rayleigh waves based on the single-scattering (Born) approximation to account for the effects of heterogeneous structure on the wavefield in a regional surface wave study. The calculated phase and amplitude data using the 2-D sensitivity kernels are compared to phase and amplitude data obtained from seismic waveforms synthesized by the pseudo-spectral method for plane Rayleigh waves propagating through heterogeneous structure. We find that the kernels can accurately predict the perturbation of the wavefield even when the size of anomaly is larger than one wavelength. The only exception is a systematic bias in the amplitude within the anomaly itself due to a site response.
An inversion method of surface wave tomography based on the sensitivity kernels is developed and applied to synthesized data obtained from a numerical simulation modelling Rayleigh wave propagation over checkerboard structure. By comparing recovered images to input structure, we illustrate that the method can almost completely recover anomalies within an array of stations when the size of the anomalies is larger than or close to one wavelength of the surface waves. Surface wave amplitude contains important information about Earth structure and should be inverted together with phase data in surface wave tomography.