Inversion for interface structure using teleseismic traveltime residuals

An inversion method is presented for the reconstruction of interface geometry between two or more crustal layers from teleseismic traveltime residuals. The method is applied to 2-D models consisting of continuous interfaces separating constant-velocity layers. The forward problem of determining ray paths and traveltimes between incident wave fronts below the structure and receivers located on the Earth's surface is solved by an efficient and robust shooting method. A conjugate gradient method is employed to solve the inverse problem of minimizing a least-squares type objective function based on the difference between observed and calculated traveltimes. Teleseismic data do not accurately constrain average vertical structure, so a priori information in the form of layer velocities and average layer thicknesses is required. Synthetic tests show that the method can be used to reconstruct interface geometry accurately, even in the presence of data noise. Tests also show that, if layer velocities and initial interface positions are poorly chosen, lateral structure is still recoverable. The inversion method was applied to previously published teleseismic data recorded by an in-line array of portable seismographs that traversed the northern margin of the Musgrave Block, central Australia. The solution based on interface parametrization is consistent with models given by other studies that used the same data but different methods, most notably the standard tomographic approach that inverts for velocity rather than interface structure.     

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.     

Inversion of seismic attributes for velocity and attenuation structure

We have developed an inversion formuialion for velocity and attenuation structure using seismic attributes, including envelope amplitude, instantaneous frequency and arrival times of selected seismic phases. We refer to this approach as AFT inversion for amplitude, (instantaneous) frequency and time. Complex trace analysis is used to extract the different seismic attributes. The instantaneous frequency data are converted to t * using a matching procedure that approximately removes the effects of the source spectra. To invert for structure, ray-perturbation methods are used to compute the sensitivity of the seismic attributes to variations in the model. An iterative inversion procedure is then performed from smooth to less smooth models that progressively incorporates the shorter-wavelength components of the model. To illustrate the method, seismic attributes are extracted from seismic-refraction data of the Ouachita PASSCAL experiment and used to invert for shallow crustal velocity and attenuation structure. Although amplitude data are sensitive to model roughness, the inverted velocity and attenuation models were required by the data to maintain a relatively smooth character. The amplitude and t * data were needed, along with the traveltimes, at each step of the inversion in order to fit all the seismic attributes at the final iteration.     

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.     

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.     

Non-linear inversion using general measures of data misfit and model structure

We investigate the use of general, non- l ₂ measures of data misfit and model structure in the solution of the non-linear inverse problem. Of particular interest are robust measures of data misfit, and measures of model structure which enable piecewise-constant models to be constructed. General measures can be incorporated into traditional linearized, iterative solutions to the non-linear problem through the use of an iteratively reweighted least-squares (IRLS) algorithm. We show how such an algorithm can be used to solve the linear inverse problem when general measures of misfit and structure are considered. The magnetic stripe example of Parker (1994 ) is used as an illustration. This example also emphasizes the benefits of using a robust measure of misfit when outliers are present in the data. We then show how the IRLS algorithm can be used within a linearized, iterative solution to the non-linear problem. The relevant procedure contains two iterative loops which can be combined in a number of ways. We present two possibilities. The first involves a line search to determine the most appropriate value of the trade-off parameter and the complete solution, via the IRLS algorithm, of the linearized inverse problem for each value of the trade-off parameter. In the second approach, a schedule of prescribed values for the trade-off parameter is used and the iterations required by the IRLS algorithm are combined with those for the linearized, iterative inversion procedure. These two variations are then applied to the 1-D inversion of both synthetic and field time-domain electromagnetic data.     

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.     

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.     

Three-dimensional magnetotelluric inversion using non-linear conjugate gradients

We have formulated a 3-D inverse solution for the magnetotelluric (MT) problem using the non-linear conjugate gradient method. Finite difference methods are used to compute predicted data efficiently and objective functional gradients. Only six forward modelling applications per frequency are typically required to produce the model update at each iteration. This efficiency is achieved by incorporating a simple line search procedure that calls for a sufficient reduction in the objective functional, instead of an exact determination of its minimum along a given descent direction. Additional efficiencies in the scheme are sought by incorporating preconditioning to accelerate solution convergence. Even with these efficiencies, the solution's realism and complexity are still limited by the speed and memory of serial processors. To overcome this barrier, the scheme has been implemented on a parallel computing platform where tens to thousands of processors operate on the problem simultaneously. The inversion scheme is tested by inverting data produced with a forward modelling code algorithmically different from that employed in the inversion algorithm. This check provides independent verification of the scheme since the two forward modelling algorithms are prone to different types of numerical error.     

An improved method for the determination of the tectonic stress field from focal mechanism data

All conventional stress inversion methods, when applied to earthquake focal mechanism data, suffer from uncertainty as to which plane is the true fault plane. This paper deals with several problems in stress inversion brought about by this uncertainty. Our analysis shows that the direction of shear stress on the auxiliary plane does not coincide with the hypothetical slip direction unless the B -axis is parallel to one of the three principal stress directions. Based on this simple fact, we propose a new algorithm dealing with the ambiguity in fault/auxiliary plane identification. We also propose a method to handle the inhomogeneity problem of data quality, which is common and unique for focal mechanism data. Different inversion methods and algorithms are applied to two sets of 'focal mechanism' data simulated from field fault-slip measurement data. The inversion results show that, among the four stress parameters inverted, the stress ratio suffers the most from the ambiguity in fault/auxiliary plane identity, whereas the solutions for the principal stress directions are surprisingly good. The errors in inversion solutions resulting from the fault/auxiliary plane ambiguity can be significantly reduced by controlling subjectively the sample variance of the measurement errors. Our results also suggest that the fault plane cannot be distinguished correctly from the auxiliary plane with a high probability on the basis of the stress inversion alone.     

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.     

Non-linear optimization for seismic traveltime tomography

This paper presents a non-linear algorithmic approach for seismic traveltime. It is based on large-scale optimization using non-linear least-squares and trust-region methods. These methods provide a natural way to stabilize algorithms based on Newton's iteration for non-linear minimization. They also correspond to an alternative (and often more efficient) view of the Levenberg-Marquardt method. Numerical experience on synthetic data and on real borehole-to-borehole problems are presented. In particular, results produced by the new algorithm are compared with those of Ivansson (1985) for the Kråkemåla experiment.     

Error propagation in non-linear delay-time tomography

Delay-time tomography can be either linearized or non-linear. In the case of linearized tomography, an error due to the linearization is introduced. If the tomography is performed in a non-linear fashion, the theory used is more accurate from the physical point of view, but if the data have a statistical error, a noise bias in the model is introduced due to the non-linear propagation of errors. We investigate the error propagation of a weakly non-linear delay-time tomography example using second-order perturbation theory. This enables us to compare the linearization error with the noise bias. We show explicitly that the question of whether a non-linear inversion methods leads to a better estimation of the model parameters than a linearized method is dependent on the signal-to-noisc ratio. We also show that, in cases of poor data quality, a linearized inversion method leads to a better estimation of the model parameters than a non-linear method.     

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.     

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.     

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.     

Groups, algebras, and the non-linearity of geophysical inverse problems

Mathematical methods from the theory of continuous groups are used to determine whether a non-linear inverse problem, in the form of a functional, can be transformed into a linear inverse problem. If such transformations exist they can be constructed from the solutions of a linear system of differential equations. An illustration of the methodology is given by the linearization of the functional relating basement topography to observed surface gravity. The linearized inversion of gravity data for basement topography is applied to observations from Yucca Mountain, Nevada. A 2.0 km step in the basement to the west of Yucca Mountain, corresponding to the Bare Mountain fault, matches the Bouguer gravity anomaly. The resolution and uncertainty associated with the estimates of basement topography indicate that the structure directly beneath the gravity line is well constrained.     

Sensitivities of seismic traveltimes and amplitudes in reflection tomography

Seismic traveltimes and amplitudes in reflection-seismic data show different dependences on the geometry of reflection interfaces, and on the variation of interval velocities. These dependences are revealed by eigenanalysis of the Hessian matrix, defined in terms of the Fréchet matrix and its adjoint associated with different norms chosen in the model space. The eigenvectors and eigenvalues of the Hessian clearly show that for reflection tomographic inversion, traveltime and amplitude data contain complementary information. Both for reflector-geometry and for interval-velocity variations, the traveltimes are sensitive to the model components with small wavenumbers, whereas the amplitudes are more sensitive to the components with high wavenumbers. The model resolution matrices, after the rejection of eigenvectors corresponding to small eigenvalues, give us some insight into how the addition of amplitude information could potentially contribute to the recovery of physical parameters.
In order to cooperatively invert seismic traveltimes and amplitudes simultaneously, we propose an empirical definition of the data covariance matrix which balances the relative sensitivities of different types of data. We investigate the cooperative use of both data types for, separately, interface-geometry and 2-D interval-velocity variations. In both cases we find that cooperative inversions can provide better solutions than those using traveltimes alone. The potential benefit of including amplitude-data constraints in seismic-reflection traveltime tomography is therefore that it may be possible to resolve the known ambiguity between the reflector-depth uncertainty and the interval-velocity uncertainty better.     

Three-dimensional massively parallel electromagnetic inversion—I. Theory

An iterative solution to the non-linear 3-D electromagnetic inverse problem is obtained by successive linearized model updates using the method of conjugate gradients. Full wave equation modelling for controlled sources is employed to compute model sensitivities and predicted data in the frequency domain with an efficient 3-D finite-difference algorithm. Necessity dictates that the inverse be underdetermined, since realistic reconstructions require the solution for tens of thousands of parameters. In addition, large-scale 3-D forward modelling is required and this can easily involve the solution of over several million electric field unknowns per solve. A massively parallel computing platform has therefore been utilized to obtain reasonable execution times, and results are given for the 1840-node Intel Paragon. The solution is demonstrated with a synthetic example with added Gaussian noise, where the data were produced from an integral equation forward-modelling code, and is different from the finite difference code embedded in the inversion algorithm     

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.