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.     

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.     

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.     

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.     

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.     

On the resolving power of tomographic images in the Aegean area

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The imaging of upper mantle heterogeneity by seismic tomography is strongly limited by the uneven global distribution both of seismic recording stations and earthquake sources. This can result in a loss of resolution and significance in the final image, particularly when a sparse data set contains few ray paths which intersect at sufficiently high angles in the volume of interest. In order to investigate the theoretical resolving power of a previously published tomographic image of the Aegean area, synthetic tests of the inversion procedure using a ray-path matrix obtained in this previous study for local and teleseismic P -waves were carried out. The aim was to examine the extent to which the shape of a synthetic lithospheric slab penetrating to different depths is inherently distorted by the tomographic imaging procedure, and to compare the synthetic tomographic images with the results from the actual inversion. The distortion is found to take the form of an artificial stretching of the lithospheric slab. The maximum 'stretching factor', as indicated by the downdip displacement of the peak amplitude of the synthetic high-velocity anomaly, is found to be a factor of 2 or so, though the distortion is usually less than this. The peak amplitude of the tomographic image of a lithospheric slab is found from the inversion of traveltime data to be at depths at or below 400 km. This indicates that the high-velocity lithospheric slab in the Aegean penetrates deeper than the Benioff zone seismicity of about 200 km. However, no constraints of the maximum depth of penetration could be established with the data set used in the present work.     

Computation of high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

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.     

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.     

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.     

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.     

Inversion of controlled-source seismic tomography and gravity data with the self-adaptive wavelet parametrization of velocities and interfaces

A self-adaptive automated parametrization approach is suggested for the sequential inversion of controlled-source seismic tomography and gravity data. The velocities and interfaces are parametrized by their Haar wavelet expansion coefficients. Only those coefficients that are well constrained by the data, as measured by the number of rays that cross the corresponding wavelet function support area and their angular coverage, are inverted for, others are set to zero. This approach results in a reasonable distribution of resolution throughout the model even in cases of irregular ray coverage and does overcome the trade-off between different types of model parameters. A modified sequential inversion approach is suggested to join the traveltimes and gravity anomalies inversion. An algorithm is developed that inverts for smooth velocity and density variations inside the seismic layer, the position of its bottom interface as well as for optimal values of the velocity-to-density regression coefficients. The algorithm makes use of direct (diving), reflected and head (critically refracted) wave traveltimes. The algorithm workflow is demonstrated on a synthetic data example.     

Joint traveltime inversion of wide-angle seismic data and a deep reflection profile from the central North Sea

We report results from the Seismic Wide-Angle and Broadband Survey carried out over the Mid North Sea High. This paper focuses on integrating the information from a conventional deep multichannel reflection profile and a coincident wide-angle profile obtained by recording the same shots on a set of ocean bottom hydrophones (OBH). To achieve this integration, a new traveltime inversion scheme was developed (reported elsewhere) that was used to invert traveltime information from both the wide-angle OBH records and the reflection profile simultaneously. Results from the inversion were evaluated by producing synthetic seismograms from the final inversion model and comparing them with the observed wide-angle data, and an excellent match was obtained. It was possible to fine-tune velocities in less well-resolved parts of the model by considering the critical distance for the Moho reflection. The seismic velocity model was checked for compatibility with the gravity field, and used to migrate and depth-convert the reflection profile. The unreflective upper crust is characterized by a high velocity gradient, whilst the highly reflective lower crust is associated with a low velocity gradient. At the base of the crust there are several subhorizontal reflectors, a few kilometres apart in depth, and correlatable laterally for several tens of kilometres. These reflectors are interpreted as representing a strike section through northward-dipping reflectors at the base of the crust, identified on orthogonal profiles by Freeman et al. (1988) as being slivers of subducted and imbricated oceanic crust, relics of the mid-Palaeozoic Iapetus Ocean.     

Anisotropic tomography of P-wave traveltimes

The ray path of a P -wave is specified in terms of the ray parameter and three Euler angles. the P -wave traveltime depends only on the ray parameter for a spherically symmetric earth. If we introduce an aspherical perturbation, including general ani-sotropy, the dependence on Euler angles can be expanded in terms of the rotation matrix for a fixed ray parameter. If the perturbation is isotropic, the expansion coefficients satisfy certain relations which may be used to obtain definite evidence for anisotropy rather than isotropic lateral heterogeneity.     

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.     

New advances in three-dimensional controlled-source electromagnetic inversion

New techniques for improving both the computational and imaging performance of the three-dimensional (3-D) electromagnetic inverse problem are presented. A non-linear conjugate gradient algorithm is the framework of the inversion scheme. Full wave equation modelling for controlled sources is utilized for data simulation along with an efficient gradient computation approach for the model update. Improving the modelling efficiency of the 3-D finite difference (FD) method involves the separation of the potentially large modelling mesh, defining the set of model parameters, from the computational FD meshes used for field simulation. Grid spacings and thus overall grid sizes can be reduced and optimized according to source frequencies and source–receiver offsets of a given input data set. Further computational efficiency is obtained by combining different levels of parallelization. While the parallel scheme allows for an arbitrarily large number of parallel tasks, the relative amount of message passing is kept constant. Image enhancement is achieved by model parameter transformation functions, which enforce bounded conductivity parameters and thus prevent parameter overshoots. Further, a remedy for treating distorted data within the inversion process is presented. Data distortions simulated here include positioning errors and a highly conductive overburden, hiding the desired target signal. The methods are demonstrated using both synthetic and field data.     

Modelling and inversion of single-ended refraction data from the shot gathers of multifold deep seismic reflection profiling—an approach for deriving the shallow velocity structure

A multifold crustal-scale deep seismic near-vertical reflection profile generates a large number of single-ended shot gathers, which provide redundant data sets because of overlapping coverage of the shallow refractors. We present an approach for deriving the shallow velocity structure by modelling and inversion of single-ended seismic refraction first arrival traveltime data. We apply this method to a data set acquired with a 12-km long spread with 100 m spacing of shots and receivers, of the Neoproterozoic Marwar basin in the NW Indian shield. The approach is shown to be quite successful for delineating the shallow refractor depths, steep dips and velocities, even in the absence of regular reverse refraction profiles. The study reveals two-layered sedimentary formations, Malani volcanics and a complicated basement configuration of the Marwar basin, and provides a measure of resolution and uncertainty of the estimated model parameters. A seismic section of the near-trace gather is found to be qualitatively consistent with the derived structural features of the basin. The relative highs and lows, observed in the Bouguer gravity profile, further corroborate the derived velocity model. The present approach can be especially useful in offshore areas and elsewhere, where the single-ended multifold seismic profiles are the only available data sets.     

Moment tensor inversion of complex earthquakes

Summary. Moment tensor inversion methods can be applied with success in the determination of source properties of simple earthquakes. However, these methods utilize the assumption of a point source, which is inadequate for modelling many complicated, shallow earthquakes. For complex earthquakes, an inversion using finite faulting models is desirable but the number of parameters involved requires that a good starting model be found or that independent constraints be placed on some of the parameters. A method is presented for low-pass filtering both the data and Green's functions, passing only signals with wavelengths greater than the dimension of the entire fault. The filter tends to smooth complications in the waveforms and allows application of the point source moment tensor inversion. This method is applied to body waves from the 1978 Thessaloniki, Greece, earthquake, the 1971 San Fernando earthquake and to a multiple-point source synthetic model of the San Fernando event. For the Thessaloniki event, although a multiple-source mechanism has been suggested, inversion results before and after filtering were essentially identical, indicating that a point source mechanism is sufficient in modelling the long-period, teleseismic body waves. In the case of the San Fernando earthquake, the point source Green's functions were incapable of simultaneously modelling the P - and SH -waves. Inversion of P -waves alone resulted in extreme parameter resolution problems, but allowed constraint in one axis of the moment tensor and suggested an overall source time function. Inversion of a synthetic San Fernando data set yielded similar results, but allowed an investigation of the shortcomings of the method under controlled circumstances. Although the results may require substantial interpretation, the method presented represents a simple first step in the analysis of complex earthquakes.     

Earthquake location in strongly heterogeneous media

Traveltime computation methods for strongly heterogeneous 3-D media developed during recent years are well suited for earthquake location. We present here a new method based on the traveltime algorithm of Podvin-Lecomte, related to the inverse problem formulation of Tarantola & Valette. The Podvin-Lecomte method, based on the Huygens principle, is very robust and allows arbitrary surface topography and station placement even for borehole instruments. First arrival traveltimes are computed for each of the recording stations using a fine 3-D velocity mesh (up to 10⁶ cells on a workstation). The traveltime grid allows the use of the Tarantola & Valette formulation, which enables a full non-linear approach. The solution is given as a 3-D probability density function of hypocentre coordinates, which accounts for the arrival time measurements as well as a priori information for the location, the accuracy of both the arrival time readings and the computation of the theoretical traveltimes. This powerful method called 3DGRIDLOC gives the location of the induced seismicity of the gas field of Lacq (France) using 443 520 cells of a 3-D velocity mesh and the observations from nine recording stations, one of which is located at the bottom of a 3880 m deep borehole. Location of synthetic foci as well as more than 500 actual earthquakes shows the real advantages of this new method versus the classical HYPO71. A new insight into the induced seismicity is now possible: induced seismicity may occur as far away as 10 km from the gas reservoir and involve a much greater volume of rock than expected using earlier locations.     

Anisotropy in multi-offset deep-crustal seismic experiments

Modelling of deep-seismic wide-angle data commonly assumes that the Earth is heterogeneous and isotropic. It is important to know the magnitudes of errors that may be introduced by isotropic-based wide-angle models when the Earth is anisotropic. It is equally important to find ways of detecting anisotropy and determining its properties.
  This paper explores the errors introduced by interpreting anisotropic seismic data with isotropic models. Errors in P -wave reflector depths are dependent on the magnitude of the velocity anisotropy and the direction of the fast axis. The interpreted, isotropic, model velocity function is found to correspond closely to the horizontal velocity of the anisotropic medium. An additional observed parameter is the time mismatch , which we define to be the difference between the vertical two-way traveltime to a reflector and the time-converted wide-angle position of the reflector. The magnitude of the time mismatch is typically <1.0  s (when the whole crust is anisotropic) and is found to be closely related to the magnitude and sign of the anisotropic anellipticity. The relationships are extendible to more complicated models, including those with vertical velocity gradients, crustal zonation, and lower symmetry orders.
  A time mismatch may be symptomatic of the presence of anisotropy. We illustrate the observation of a time mismatch for a real multi-offset seismic data set collected north of Scotland and discuss the implications for crustal anisotropy in that region.     

Formalism for traveltime inversion with finite frequency effects

A simple modification of the waveform inversion formula, based on the normal mode perturbation theory, is shown to lead to a formula for traveltime anomalies. The kernel which is derived can be used for traveltime inversion with automatic inclusion of finite frequency effects. Inversion for Earth structure with such kernels will lead to better resolution estimates than ray-theoretical traveltime inversion. Examples of kernels for transverse component seismograms are shown for direct S waves, ScS , Love waves and diffracted S waves. A measure of finite frequency effects is also proposed by comparing our formula with the one from ray theory. A quantity which should be 1 in the case of ray theory is computed for the finite frequency kernels and is shown to have deviations up to about 30 per cent from 1. Therefore, the use of ray theory for long-period body waves applies incorrect weight along a ray path and may introduce a small bias to an earth model.