Finite-frequency sensitivity kernels for head waves

Head waves are extremely important in determining the structure of the predominantly layered Earth. While several recent studies have shown the diffractive nature and the 3-D Fréchet kernels of finite-frequency turning waves, analogues of head waves in a continuous velocity structure, the finite-frequency effects and sensitivity kernels of head waves are yet to be carefully examined. We present the results of a numerical study focusing on the finite-frequency effects of head waves. Our model has a low-velocity layer over a high-velocity half-space and a cylindrical-shaped velocity perturbation placed beneath the interface at different locations. A 3-D finite-difference method is used to calculate synthetic waveforms. Traveltime and amplitude anomalies are measured by the cross-correlation of synthetic seismograms from models with and without the velocity perturbation and are compared to the 3-D sensitivity kernels constructed from full waveform simulations. The results show that the head wave arrival-time and amplitude are influenced by the velocity structure surrounding the ray path in a pattern that is consistent with the Fresnel zones. Unlike the 'banana–doughnut' traveltime sensitivity kernels of turning waves, the traveltime sensitivity of the head wave along the ray path below the interface is weak, but non-zero. Below the ray path, the traveltime sensitivity reaches the maximum (absolute value) at a depth that depends on the wavelength and propagation distance. The sensitivity kernels vary with the vertical velocity gradient in the lower layer, but the variation is relatively small at short propagation distances when the vertical velocity gradient is within the range of the commonly accepted values. Finally, the depression or shoaling of the interface results in increased or decreased sensitivities, respectively, beneath the interface topography.     

Quasi-isotropic approximation of ray theory for anisotropic media

Wave propagation in weakly anisotropic inhomogeneous media is studied by the quasi-isotropic approximation of ray theory. The approach is based on the ray-tracing and dynamic ray-tracing differential equations for an isotropic background medium. In addition, it requires the integration of a system of two complex coupled differential equations along the isotropic ray.
The interference of the qS waves is described by traveltime and polarization corrections of interacting isotropic S waves. For qP waves the approach leads to a correction of the traveltime of the P wave in the isotropic background medium.
Seismograms and particle-motion diagrams obtained from numerical computations are presented for models with different strengths of anisotropy.
The equivalence of the quasi-isotropic approximation and the quasi-shear-wave coupling theory is demonstrated. The quasi-isotropic approximation allows for a consideration of the limit from weak anisotropy to isotropy, especially in the case of qS waves, where the usual ray theory for anisotropic media fails.     

Wavepath traveltime tomography

The elastic-wave equation is used to construct sensitivity kernels relating perturbations in elastic parameters to traveltime deviations. Computation of the functions requires a correlation of the forward-propagating seismic wavefield with a backward propagation of the residual wavefield. The computation of the wavefields is accomplished using a finite difference algorithm and is efficiently executed on a CM-2 parallel processor. The source and receiver locations have maximum sensitivity to velocity structure. The sensitivity kernels or wavepaths are well suited for transmission traveltime inversion such as cross-borehole tomography and vertical seismic profiling. Conventional ray tomography and wavepath tomography are applied to a set of P -wave arrival times, from a cross-borehole experiment at Kesterson, California. Because the wavepaths have increased sensitivity near the source and receiver there are differences in resolution of the velocity structure. Both techniques recover the same relative variations in velocity where the coverage is adequate. The wavepath solution is more laterally continuous and the dominant variation is vertical, as is expected for the layered sediments in this region.     

Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods

We investigate the sensitivity of finite-frequency body-wave observables to mantle anisotropy based upon kernels calculated by combining adjoint methods and spectral-element modelling of seismic wave propagation. Anisotropy is described by 21 density-normalized elastic parameters naturally involved in asymptotic wave propagation in weakly anisotropic media. In a 1-D reference model, body-wave sensitivity to anisotropy is characterized by 'banana–doughnut' kernels which exhibit large, path-dependent variations and even sign changes. P -wave traveltimes appear much more sensitive to certain azimuthally anisotropic parameters than to the usual isotropic parameters, suggesting that isotropic P -wave tomography could be significantly biased by coherent anisotropic structures, such as slabs. Because of shear-wave splitting, the common cross-correlation traveltime anomaly is not an appropriate observable for S waves propagating in anisotropic media. We propose two new observables for shear waves. The first observable is a generalized cross-correlation traveltime anomaly, and the second a generalized 'splitting intensity'. Like P waves, S waves analysed based upon these observables are generally sensitive to a large number of the 21 anisotropic parameters and show significant path-dependent variations. The specific path-geometry of SKS waves results in favourable properties for imaging based upon the splitting intensity, because it is sensitive to a smaller number of anisotropic parameters, and the region which is sampled is mainly limited to the upper mantle beneath the receiver.     

Simultaneous velocity and interface tomography of normal-incidence and wide-aperture seismic traveltime data

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.     

Elastodynamic and elastostatic Green tensors for homogeneous weak transversely isotropic media

An explicit analytical formula for the complete elastodynamic Green tensor for homogeneous unbounded weak transversely isotropic media is presented. The formula was derived by analytical calculations of higher-order approximations of the ray series. The ray series is finite and consists of seven non-zero terms. The formula for the Green tensor is complete and correct for the whole frequency range, thus it describes correctly the wavefield at all distances and at all directions including the shear-wave singularity direction. The Green tensor consists of P, SV and SH far-field waves and four coupling waves. Three of them couple P and SV waves, and the fourth wave couples the SV and SH waves. The P-SV coupling waves behave similarly to the near-field waves in isotropy. However, the SV-SH coupling wave, which is called 'shear-wave coupling', behaves exceptionally and it has no analogy in the Green tensor for isotropy. The formula for the elastostatic Green tensor is also derived.     

On crustal corrections in surface wave tomography

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.     

On crosswell diffusive time-domain electromagnetic tomography

We investigate the reconstruction of a conductive target using crosswell time-domain electromagnetic tomography in the diffusive limit. The work is a natural extension of our ongoing research in the modification of time-domain methods for the rugged marine mid-ocean-ridge environment, an environment characterized by extreme topography and pronounced variations in crustal conductivity on all scales. We have proved both in theory and in practice that 'traveltime', the time taken for an electromagnetic signal to be identified at a receiver following a change of current in the transmitter, is an excellent, robust estimator of average conductivity on a path between transmitter and receiver. A simple estimate of the traveltime for a parallel electric dipole-dipole system is the time at which the derivative of the electric field with respect to logarithmic time at the receiver reaches its maximum. We have derived the fundamental relationship between the traveltime and the conductivity of the medium for a uniform whole-space. We have applied the concept of the traveltime inversion to the related crosswell problem and demonstrated reconstructions of finite targets based on tomographic analyses. Results show that the crosswell time-domain electromagnetic tomography can supply useful information, such as the location and shape of a conductive target.     

Finite-frequency vectorial tomography: a new method for high-resolution imaging of upper mantle anisotropy

Amplitude measurements of the transverse component of SKS waves, the so-called splitting intensity, can be used to formulate a non-linear inverse problem to image the 3-D variations of upper mantle anisotropy. Assuming transverse isotropy (or hexagonal symmetry), one can parametrize anisotropy by two anisotropic parameters and two angles describing the orientation of the symmetry axis. These can also be written as two collinear pseudo-vectors. The tomographic process consists of retrieving the spatial distribution of these pseudo-vectors, and thus resembles surface wave vectorial tomography. Spatial resolution results from the sensitivity of low-frequency SKS waves to seismic anisotropy off the ray path. The expressions for the 3-D sensitivity kernels for splitting intensity are derived, including the near-field contributions, and validated by comparison with a full wave equation solution based upon the finite element method. These sensitivity kernels are valid for any orientation of the symmetry axis, and thus generalize previous results that were only valid for a horizontal symmetry axis. It is shown that both lateral and vertical subwavelength variations of anisotropy can be retrieved with a dense array of broad-band stations, even in the case of vertically propagating SKS waves.     

Parametrizing surface wave tomographic models with harmonic spherical splines

We present a mathematical framework and a new methodology for the parametrization of surface wave phase-speed models, based on traveltime data. Our method is neither purely local, like block-based approaches, nor is it purely global, like those based on spherical harmonic basis functions. Rather, it combines the well-known theory and practical utility of the spherical harmonics with the spatial localization properties of spline basis functions. We derive the theoretical foundations for the application of harmonic spherical splines to surface wave tomography and summarize the results of numerous numerical tests illustrating the performance of a practical inversion scheme based upon them. Our presentation is based on the notion of reproducing-kernel Hilbert spaces, which lends itself to the parametrization of fully 3-D tomographic earth models that include body waves as well.     

Sequential integrated inversion of refraction and wide-angle reflection traveltimes and gravity data for two-dimensional velocity structures

A new algorithm is presented for the integrated 2-D inversion of seismic traveltime and gravity data. The algorithm adopts the 'maximum likelihood' regularization scheme. We construct a 'probability density function' which includes three kinds of information: information derived from gravity measurements; information derived from the seismic traveltime inversion procedure applied to the model; and information on the physical correlation among the density and the velocity parameters. We assume a linear relation between density and velocity, which can be node-dependent; that is, we can choose different relationships for different parts of the velocity–density grid. In addition, our procedure allows us to consider a covariance matrix related to the error propagation in linking density to velocity. We use seismic data to estimate starting velocity values and the position of boundary nodes. Subsequently, the sequential integrated inversion (SII) optimizes the layer velocities and densities for our models. The procedure is applicable, as an additional step, to any type of seismic tomographic inversion.
We illustrate the method by comparing the velocity models recovered from a standard seismic traveltime inversion with those retrieved using our algorithm. The inversion of synthetic data calculated for a 2-D isotropic, laterally inhomogeneous model shows the stability and accuracy of this procedure, demonstrates the improvements to the recovery of true velocity anomalies, and proves that this technique can efficiently overcome some of the limitations of both gravity and seismic traveltime inversions, when they are used independently.
An interpretation of field data from the 1994 Vesuvius test experiment is also presented. At depths down to 4.5 km, the model retrieved after a SII shows a more detailed structure than the model obtained from an interpretation of seismic traveltime only, and yields additional information for a further study of the area.     

Is there an observable lack of reciprocity in PKP(DF) traveltimes?

On average, traveltimes of PKP_DF for equatorial ray paths through the quasieastern hemisphere of the inner core are around 0.5 s faster than equivalent ray paths through its quasiwestern hemisphere. In these observations, the eastern hemisphere is sampled primarily by westward and the western hemisphere by eastwardpropagating waves. Noting that westward propagation is faster than eastward propagation inside a rotating earth, I estimate the expected traveltime difference from Coriolis splitting of the displacement eigenfunctions of the PKP_DF equivalent modes. It turns out that Coriolis effects are too small to give rise to residuals of the required magnitude. Thus, the observations must be primarily due to velocity heterogeneities.     

Imaging upper-mantle discontinuity topography usingunderside-reflection data

This paper presents a method to invert underside-reflection ( P _d P or S _d S arrivals) data for lateral depth variations of upper-mantle discontinuities, combining traveltime and amplitude data. The method greatly improves the resolution of small-scale undulations obtained by existing imaging methods and does not suffer from the long-wavelength biases that are likely to be present in currently available models. Existing inversion methods account for the large size of the Fresnel zone of underside reflections, but not for its complexity, arising from the mini-max traveltime nature of PP- and SS -related waves. This neglect results in long-wavelength artefacts from small-scale undulations of the discontinuities, obscuring true long-wavelength depth variations. The inversion method presented in this paper uses a complex-valued sensitivity kernel, derived from the representation of underside reflections through a Kirchhoff integral formulation. The sensitivity kernel accounts for the varying sensitivity of the waveforms to discontinuity structure over the Fresnel zone. The method is applied to a large, synthetic data set. The data set consists of P _d P amplitudes and traveltimes. The results show that the new inversion method resolves depth variations on a lateral scale that is smaller than the size of the Fresnel zone of individual underside reflections (but larger than the dominant wavelength), retaining the resolution of large-scale variations. The results presented here suggest that the discontinuity depth variations induced by slab penetration of the 670 discontinuity could be resolved by current broad-band P ₆₇₀ P data sets.     

Computation of qP-wave rays, traveltimes and slowness vectors in orthorhombic media

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.     

Properties of S waves near a kiss singularity: a comparison of exact and ray solutions

We have studied the properties of S waves generated by a point source in a homogeneous, transversely isotropic, elastic medium, propagating in directions close to that of a kiss singularity, which coincides with the symmetry axis of the medium. We have proved analytically as well as numerically that the ray solution can describe the S waves correctly far from the source in all directions, including that of the kiss singularity. We have found that, in contrast to the far-field P wave, which can be reproduced satisfactorily by the zeroth-order ray approximation in all directions from the source, the far-field S waves can be reproduced satisfactorily by the zeroth-order ray approximation only in directions far from the kiss singularity. In directions near the kiss singularity, the first-order ray approximation must also be considered, because the zeroth- order ray approximation yields distorted results. The first-order ray approximation can be of high frequency and can be detected in the far field.     

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.     

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.     

Refining seismic structure models by the systematic phase identification of late-arriving groups

We develop a systematic approach to the phase identification of late-arriving groups in 2-D seismic data. Waveforms in the same traveltime branch are grouped, and synthetic traveltimes for all phases are calculated using an initial approximation to the 2-D structure. For each group, we identify the two synthetic phases providing the smallest RMS residuals. If their ratio is less than some predetermined threshold, then the group's phase is ambiguous and both assignments must be tested by traveltime inversion. If there are n unidentified groups, we construct 2 ⁿ phase tables and perform a traveltime inversion on every plausible phase assignment. The phase table that provides the highest value of the posterior probability density is taken as correct, and a 2-D velocity model is constructed from the data. This approach is shown to be effective and efficient on both simulated and real data. In addition, the residuals associated with late-arriving groups provide a means of identifying deficiencies in the initial model.     

Observations and origin of Rayleigh-wave amplitude anomalies

This is a report of observations of amplitude anomalies of fundamental-mode Rayleigh waves ( R 1) between periods of 17 and 100  s. The anomalies are with respect to amplitudes predicted by Rayleigh-wave excitation for a reference earth model and catalogued centroid earthquake source parameters, such as are used in large-scale waveform inversions. The observations indicate that the amplitude anomalies are consistent for nearby recordings of the same event, while there is no obvious relation between the observed anomalies and the paths travelled by the waves. This is in contrast to Rayleigh-wave phase anomalies, which are consistent for similar propagation paths, and hence form the input in many inversions for along-path structure. The observations in this paper show that a similar inversion of intermediate-period amplitude anomalies for along- and near-path structure is not warranted without eliminating source effects, since the amplitude anomalies are dominated by scattering off near-source earth structure and by possible uncertainties in the source parameters. Sensitivity kernels that take the coupling between the moment tensor and displacement field into account demonstrate that Rayleigh-wave amplitude sensitivity is largest near the source. This report argues that the interaction between source-radiated Rayleigh waves and near-source earth structure may not be ignored in amplitude inversion procedures.     

P-wave traveltime and polarization tomography of VSP data

The presence of anisotropy requires that tomographic methods be generalized to account for anisotropy. This generalization allows geological structure to be correctly imaged and allows the anisotropic parameters to be estimated. Use of isotropic inversion for imaging anisotropic structures gives systematic trends in the traveltime and polarization residuals. However, due to the limited directional coverage, the traveltimes along may not be sufficient to study the anisotropic properties of the structure. Polarizations can provide independent information on the structure. Traveltime and polarization inversion are applied to synthetic examples simulating VSP experiments. Transverse isotropy and 1-D structure are assumed. Plots of traveltime and polarization residuals are an important tool to detect the anomalies due to the presence of anisotropy. For receivers located in anisotropic layers, polarization residuals display consistent anomalies of several degrees. The synthetic examples show that even the simple 1-D problem is difficult, when using direct arrivals only. Large a posteriori errors in anisotropic parameters are obtained by traveltime inversion in layers where available incidence angles are less than 45°. Resolution of the tomographic image of VSP data is greatly improved by a combination of traveltime and polarization information. In order to obtain accurate inversion results, the measurement error of polarization data should be kept to within a few degrees.