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.     

A wide-angle seismic traverse through the Variscan of southwest Ireland

A wide-angle seismic profile across the western peninsulas of SW Ireland was performed. This region corresponds to the northernmost Variscan thrust and fold deformation. The dense set of 13 shots and 109 stations along the 120  km long profile provides a detailed velocity model of the crust.
  The seismic velocity model, obtained by forward and inverse modelling, defines a five-layer crust. A sedimentary layer, 5–8  km thick, is underlain by an upper-crustal layer of variable thickness, with a base generally at a depth of 10–12  km. Two mid-crustal layers are defined, and a lower-crustal layer below 22  km. The Moho lies at a depth of 30–32  km. A low-velocity zone, which coincides with a well-defined gravity low, is observed in the central part of the region and is modelled as a Caledonian granite which intruded upper-crustal basement. The granite may have acted as a buffer to northward-directed Variscan thrusting. The Dingle–Dungarvan Line (DDL) marks a major change in sedimentary and crustal velocity and structure. It lies immediately to the north of the velocity and gravity low, and shows thickness and velocity differences in many of the underlying crustal layers and even in the Moho. This suggests a deep, pre-Variscan control of the structural development of this area. The model is compatible with thin-skinned tectonics, which terminated at the DDL and which incorporated thrusts involving the sedimentary and upper-crustal layers.     

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.     

Multichannel appraisal deconvolution

Summary The single channel scalar deconvolution method presented by Oldenburg has been extended to include N channels of data and vector models of the form     ( t ) = ( m ₁( t ), m ₂( t ), …, m _α( t ))^T. The solution has its foundation in the linear inverse theory of Backus & Gilbert and is effected by computing a set of N filters, which, when convolved with the data, yield unique averages of one of the scalar functions of the model. Those averages are the summation of the scalar model convolved with a primary averaging function plus contamination from secondary averaging functions convolved with other model components. It is shown how a set of suitably selected weights can annihilate these secondary averaging functions and thereby greatly simplify the interpretation. The computations are efficiently carried out in the frequency domain and require the inversion of an N × N Hermitian matrix at each frequency. As a type example, we have shown how the time varying elements of a seismic moment tensor might be computed from a set of seismograms.     

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.     

A crustal seismic velocity model for the UK, Ireland and surrounding seas

A regional model of the 3-D variation in seismic P -wave velocity structure in the crust of NW Europe has been compiled from wide-angle reflection/refraction profiles. Along each 2-D profile a velocity–depth function has been digitised at 5 km intervals. These 1-D velocity functions were mapped into three dimensions using ordinary kriging with weights determined to minimise the difference between digitised and interpolated values. An analysis of variograms of the digitised data suggested a radial isotropic weighting scheme was most appropriate. Horizontal dimensions of the model cells are optimised at 40 × 40 km and the vertical dimension at 1 km. The resulting model provides a higher resolution image of the 3-D variation in seismic velocity structure of the UK, Ireland and surrounding areas than existing models. The construction of the model through kriging allows the uncertainty in the velocity structure to be assessed. This uncertainty indicates the high density of data required to confidently interpolate the crustal velocity structure, and shows that for this region the velocity is poorly constrained for large areas away from the input data.     

Resolution analysis for discrete systems

summary . Treatments of geophysical inverse problems have tended to polarize into approaches intended to generate models either described by piecewise continuous functions or with some prior discretization. The two approaches are here developed in parallel, and the ideas of a trade-off between the anticipated error and the attainable level of detail in the model estimate are extended to the discrete case, either with even or uneven discretization.
An alternative approach to specifying the potential resolution of a model is to establish upper and lower bounds on parameter values. Linear programming methods are extended to determine bounds which allow for subjective limits on parameter values.
For a non-linear system the possible resolution may be investigated by estimation procedures based on the full set of successful solutions obtained by Monte-Carlo inversion.     

Solution of the inverse problem of seismology for laterally inhomogeneous media

Summary. The Backus-Gilbert method has been extended to the estimation of the seismic wave velocity distribution in 2-D or 3-D inhomogeneous media from a finite set of travel-time data. The method may be applied to the inversion of body wave as well as surface wave data. The problem of determining a local average of the unknown velocity corrections may be reduced to a choice of a suitable δ-ness criterion for the averaging kernel. For 2-D and 3-D inhomogeneous media the simplest criterion is to minimize a sum of 'spreads' over all the coordinates. The use of this criterion requires the solution (the averaged velocity corrections) to be represented as a sum of functions, each of which depends only on one coordinate. This is a basic restriction of the method. In practice it is possible to achieve good agreement between the solution and a real velocity distribution by a reasonable choice of the coordinate system.
Numerical tests demonstrate the efficiency of the method. Some examples of the application of the method to the inversion of real seismological data for body and surface waves are given.     

An algebraic formulation of geophysical inverse problems

Many geophysical inverse problems derive from governing partial differential equations with unknown coefficients. Alternatively, inverse problems often arise from integral equations associated with a Green's function solution to a governing differential equation. In their discrete form such equations reduce to systems of polynomial equations, known as algebraic equations. Using techniques from computational algebra one can address questions of the existence of solutions to such equations as well as the uniqueness of the solutions. The techniques are enumerative and exhaustive, requiring a finite number of computer operations. For example, calculating a bound to the total number of solutions reduces to computing the dimension of a linear vector space. The solution set itself may be constructed through the solution of an eigenvalue problem. The techniques are applied to a set of synthetic magnetotelluric values generated by conductivity variations within a layer. We find that the estimation of the conductivity and the electric field in the subsurface, based upon single-frequency magnetotelluric field values, is equivalent to a linear inverse problem. The techniques are also illustrated by an application to a magnetotelluric data set gathered at Battle Mountain, Nevada. Surface observations of the electric ( E _y ) and magnetic ( H _x ) fields are used to construct a model of subsurface electrical structure. Using techniques for algebraic equations it is shown that solutions exist, and that the set of solutions is finite. The total number of solutions is bounded above at 134 217 728. A numerical solution of the algebraic equations generates a conductivity structure in accordance with the current geological model for the area.     

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.     

The Backus–Gilbert approach to the 3-D structure in the upper mantle – II. SH and SV velocity

Summary. Phase velocity variations obtained in the previous paper are inverted by the Backus–Gilbert method for the velocity structure of the upper mantle. Spheroidal modes and toroidal modes in the period range of 125–260 s are used in the inversion. The data cannot constrain all six parameters in a transversely isotropic medium and we chose to perturb only two parameters, SH and SV velocities. SV velocities are resolved between the depths of about 200 and 400 km and SH velocities between 0 and 200 km. Resolution kernels have half-peak widths of about 200–300 km in depth, becoming broader for deeper target depths. SV velocity kernels show secondary peaks near the surface of the Earth, with widths varying from 50 to 100 km. The deeper the target depths, the wider the secondary peaks near the surface. SH velocity kernels do not possess such secondary peaks. The trade-off between SV and SH velocities is small. SV velocity is essentially determined by spheroidal modes and SH velocity by toroidal modes. Because of the broad width of the resolution kernels, the structure in the resolved region is difficult to detect from our data set; for example the differences in SV velocity structure between 250 and 350 km or the differences in SH velocity between 100 and 200 km are difficult to distinguish. Considering the horizontal resolution of about 2000 km, obtained in the previous paper, averaging kernels for 3-D structure are quite elongated in the horizontal dimension.     

Seismic tomographic imaging of P- and S-waves velocity perturbations in the upper mantle beneath Iran

The inverse tomography method has been used to study the P - and S -waves velocity structure of the crust and upper mantle underneath Iran. The method, based on the principle of source–receiver reciprocity, allows for tomographic studies of regions with sparse distribution of seismic stations if the region has sufficient seismicity. The arrival times of body waves from earthquakes in the study area as reported in the ISC catalogue (1964–1996) at all available epicentral distances are used for calculation of residual arrival times. Prior to inversion we have relocated hypocentres based on a 1-D spherical earth's model taking into account variable crustal thickness and surface topography. During the inversion seismic sources are further relocated simultaneously with the calculation of velocity perturbations. With a series of synthetic tests we demonstrate the power of the algorithm and the data to reconstruct introduced anomalies using the ray paths of the real data set and taking into account the measurement errors and outliers. The velocity anomalies show that the crust and upper mantle beneath the Iranian Plateau comprises a low velocity domain between the Arabian Plate and the Caspian Block. This is in agreement with global tomographic models, and also tectonic models, in which active Iranian plateau is trapped between the stable Turan plate in the north and the Arabian shield in the south. Our results show clear evidence of the mainly aseismic subduction of the oceanic crust of the Oman Sea underneath the Iranian Plateau. However, along the Zagros suture zone, the subduction pattern is more complex than at Makran where the collision of the two plates is highly seismic.     

Three-dimensional inversion without blocks

Summary. We propose a method for solving non-linear inverse problems in the case where the unknown is a function of the spatial coordinates and the data set is discrete and finite. The method is based on a generalized leastsquares criterion, it is defined directly for non-linear problems (without previous linearization of the forward problem), and in the particular linear case it gives the same results (although slightly more general) than the Backus & Gilbert approach. As an example, we apply the method to the three-dimensional seismic velocity inverse problem, using as data the arrival times of seismic waves. The following paper (Nercessian et al .) shows some esults obtained using the present method.     

Preliminary Results on the 3-dimensional Seismic Structure of the Lithosphere under the USGS Central California Seismic Array

About 1500 readings of teleseismic P -time residuals obtained from the US Geological Survey seismograph network in central California have been used to obtain a three-dimensional image of seismic velocity anomalies for this area by the method of Aki, Christoffersson & Husebye We found that the California network is less suitable than the LASA and NORSAR arrays for this kind of studies because of its greater proportion of peripheral blocks in which the resolution is very poor for the stochastic inverse solution and the random error effect is severe for the generalized inverse solution. Nevertheless, the resultant velocity anomalies show a remarkable correlation with the San Andreas fault zone to a depth of 75 km. The anomaly pattern changes drastically as the depth exceeds 75 km, suggesting that the asthenosphere has been reached.     

Application of stacking and inversion techniques to three-dimensional wide-angle reflection and refraction seismic data of the Eastern Alps

We present new methods for the interpretation of 3-D seismic wide-angle reflection and refraction data with application to data acquired during the experiments CELEBRATION, 2000 and ALP 2002 in the area of the Eastern Alps and their transition to the surrounding tectonic provinces (Bohemian Massif, Carpathians, Pannonian domain, Dinarides). Data was acquired on a net of arbitrarily oriented seismic lines by simultaneous recording on all lines of seismic waves from the shots, which allows 2-D and 3-D interpretations. Much (80%) of the data set consists of crossline traces. Low signal to noise (S/N) ratio in the area of the young orogens decreases the quality of travel time picks. In these seismically heterogeneous areas it is difficult to assign clearly defined arrivals to the seismic phases, in particular on crossline record sections.
In order to enhance the S/N ratio, signal detection and stacking techniques have been applied to enhance the Pg -, Pn - and PmP phases. Further, inversion methods have been developed for the interpretation of WAR/R-data, based on automated 1-D inversion ( Pg ) and the application of the delay time concept ( Pn ). The results include a 3-D velocity model of the crust based on Pg waves, time and depth maps of the Moho and a Pn -velocity map. The models based on stacked data are robust and provide a larger coverage, than models based on travel time picks from single-fold (unstacked) traces, but have relatively low resolution, especially near the surface. They were used as the basis for constructing models with improved resolution by the inversion of picks from single-fold data. The results correlate well with geological structures and show new prominent features in the Eastern Alps area and their surrounds. The velocity distribution in the crust has strong lateral variations and the Moho in the investigation area appears to be fragmented into three parts.     

Transient and impulse responses of a one-dimensional linearly attenuating medium — I. Analytical results

Summary. The transient and impulse responses (Green's function) for onedimensional wave propagation in a standard linear solid are calculated using a Laplace Transform method. The spectrum of relaxation times is chosen so as to model a constant Q medium within an absorption band covering a broad frequency range which may be chosen so as to include the seismic frequencies. The inverse transform may be evaluated asymptotically in the limit of very long propagation times using the saddle point method. For shorter propagation times the method of steepest descent may be modified so as to yield an accurate first motion approximation. The character of the small amplitude precursor to the large amplitude Visible' signal is investigated analytically. It is shown that the signal velocity is intermediate between the high-frequency ('unrelaxed') and the low-frequency ('relaxed') limits of the phase velocity.     

Mode-Converted (P-SV) Waves for Imaging Thin Intercalated Basaltic Layers and Their Possible Geologic Implications

The subsurface imaging using conventional seismic reflection technique is challenging in areas where high velocity rocks such as basalts are underlain by low velocity rocks. The seismic image quality worsens in the presence of intercalated sediments within the basaltic layers. In the recent years, the multicomponent seismic exploration technique has drawn great attention because it reduces the ambiguity in seismic imaging, enlarges the S-wave information, and improves the prediction and identification of reservoir fluids. Improvements in sub-basalt imaging techniques could hold highly significant geologic implications such as resource exploration and identifying permanent geochemical trapping potential (such as for carbon sequestration studies). In this article, we examine the possibility of utilizing mode-converted (P-SV) waves for sub-basalt imaging as well as likely complicacies one may expect in such processes.     

Inferring the relative measure of principal stress components

The phase velocity and the attenuation coefficient of compressional seismic waves, propagating in poroelastic, fluid-saturated, laminated sediments, are computed analytically from first principles. The wavefield is found to be strongly affected by the medium heterogeneity. Impedance fluctuations lead to poroelastic scattering; variations of the layer compressibilities cause inter-layer flow (a 1-D macroscopic local flow). These effects result in significant attenuation and dispersion of the seismic wavefield, even in the surface seismic frequency range, 10–100 Hz. The various attenuation mechanisms are found to be approximately additive, dominated by inter-layer flow at very low frequencies. Elastic scattering is important over a broad frequency range from seismic to sonic frequencies. Biot's global flow (the relative displacement of solid frame and fluid) contributes mainly in the range of ultrasonic frequencies. From the seismic frequency range up to ultrasonic frequencies, attenuation due to heterogeneity is strongly enhanced compared to homogeneous Biot models. Simple analytical expressions for the P -wave phase velocity and attenuation coefficient are presented as functions of frequency and of statistical medium parameters (correlation lengths, variances). These results automatically include different asymptotic approximations, such as poroelastic Backus averaging in the quasi-static and the no-flow limits, geometrical optics, and intermediate frequency ranges.     

Optimized and robust experimental design: a non-linear application to EM sounding

Pragmatic experimental design requires objective consideration of several classes of information including the survey goals, the range of expected Earth responses, acquisition costs, instrumental capabilities, experimental conditions and logistics. In this study we consider the ramifications of maximizing model parameter resolution through non-linear experimental design. Global optimization theory is employed to examine and rank different EM sounding survey designs in terms of model resolution as defined by linearized inverse theory. By studying both theoretically optimal and heuristic experimental survey configurations for various quantities of data, it is shown that design optimization is critical for minimizing model variance estimates, and is particularly important when the inverse problem becomes nearly underdetermined. We introduce the concept of robustness so that survey designs are relatively immune to the presence of potential bias errors in important data. Bias may arise during practical measurement, or from designing a survey using an appropriate model.