Refraction Traveltime Research Articles

Three‐Dimensional Seismic Transmission Tomography of the Deep Earthquake Faults in Yinchuan Basin

AbstractIn order to prove the validity and application value of three‐dimensional seismic transmission technology in active fault detection, a seismic array was deployed in the middle and north of Yinchuan basin. The basement and upper crust structure under the seismic array were reconstructed based on first‐arrival refraction traveltimes and reflection traveltimes from Moho at critical distances by using finite difference method, time‐term method, and the inversion method without model blocks. The results show that the basement of the studied area is shallow in the east and west and deep in the centre and it is steeper in the west than in the east. The greatest depth of the basement occurs from Luhuatai to Xidatan, and its depth reaches 7 km there. The Luhuatai fault, Yinchuan‐Pingluo fault, and Huanghe fault all appear as NNE‐trending velocity difference strips and there is significant variation in the depth of sediment interface across the faults. The east‐dipping, high‐angle Luhuatai fault extends under the basement. The west‐dipping Yinchuan‐Pingluo fault is a buried fault penetrating the basement. Huanghe fault dips toward the west and penetrates the basement as well. These exploration results prove that the three‐dimensional seismic transmission technology, which combines first‐arrival refraction wave exploration with Moho reflection wave exploration, is applicable in the reconstruction of fine upper crustal image of the studied area. With this technology, we can not only reveal the distribution and shallow features of the major faults, but also their extension under the basement.

Determination and sensitivity analysis of the seismic velocity of a shallow layer from refraction traveltimes measures

In this paper, we are interested in determining the seismic velocity of a shallow under-ground layer from refraction traveltimes measures. We present a study case taken from an experimental seismic survey. The study case is a wide-angle seismic inversion using experimental traveltimes measures and based on ray tracing technique and genetic algorithms. The hypothesis on the velocity distribution, coming from the seismic experiment, makes the computation of some seismic rays expensive in time. We propose to reduce the computations time by introducing a formulation of the inverse problem that avoids such costly rays, hence the inversion becomes feasible. Also we present a sensitivity analysis based on a singular value decomposition of the jacobian of the traveltimes with respect to velocity. We give the relationship between the traveltimes measure errors and the velocity estimation error. We discuss the advantages of this method over the classical one based on the resolution matrix.

VS and VP vertical profiling via joint inversion of Rayleigh waves and refraction travel times by means of bi-objective evolutionary algorithm

Rayleigh wave dispersion curves and refraction travel times are jointly inverted through a procedure based on a Multi-Objective Evolutionary Algorithm (MOEA) technique. The proposed procedure aims at improving the reconstruction of subsurface structure by exploiting the complementary information attainable by refraction seismics and surface-wave dispersion and by overcoming in this way the problems related to non-uniqueness of the solution (surface waves and refraction seismics) and hidden layers (refraction). The proposed scheme allows the joint inversion of the data and the validation of the provisional interpretation. In fact, Pareto front symmetry proves to be a valuable tool to verify the coherency of the adopted interpretation as an incorrect number of layers, refractor attribution or assumed Poisson values reflect in non-symmetric Pareto front as well as in wider model distribution in the objective space. Methodology is initially tested using synthetic data and successfully applied to a field dataset resulting from a single standard seismic survey with vertical geophones and vertically-incident seismic source (sledgehammer).

Crustal stucture of the Barents-Kara Region based on detailed deep seismic sounding

In 1995–1998 and 2003–2005, detailed deep seismic soundings were undertaken in the Barents-Kara Region along geotraverses 1-AR, 2-AR, 3-AR with a total length of over 3000 km. Seismic cross-sections, up to 50 km deep as an average, were obtained using the software package GODOGRAPH designed at the Department of Seismometry and Geoacoustics of the Lomonosov Moscow State University. The study was based on refraction traveltime curves with approximately 100 curves per profile. The sections obtained along the 1-AR and 2-AR traverses were geologically interpreted. The main crustal boundaries, fold-thrust structural features of the lower crust and a suture zone between the North Barents Basin and the Caledonian Orogenic Belt were distinguished. Based on our data, the structure of the suture can be interpreted as an ancient subduction zone. The possible pattern of tectonic movements of the Barents Plate is characterized.

Refraction traveltime and amplitude corrections for very near‐ surface inhomogeneities

ABSTRACTThe performance of refraction inversion methods that employ the principle of refraction migration, whereby traveltimes are laterally migrated by the offset distance (which is the horizontal separation between the point of refraction and the point of detection on the surface), can be adversely affected by very near‐surface inhomogeneities. Even inhomogeneities at single receivers can limit the lateral resolution of detailed seismic velocities in the refractor.The generalized reciprocal method ‘statics’ smoothing method (GRM SSM) is a smoothing rather than a deterministic method for correcting very near‐surface inhomogeneities of limited lateral extent. It is based on the observation that there are only relatively minor differences in the time‐depths to the target refractor computed for a range of XY distances, which is the separation between the reverse and forward traveltimes used to compute the time‐depth. However, any traveltime anomalies, which originate in the near‐surface, migrate laterally with increasing XY distance. Therefore, an average of the time‐depths over a range of XY values preserves the architecture of the refractor, but significantly minimizes the traveltime anomalies originating in the near‐surface. The GRM statics smoothing corrections are obtained by subtracting the average time‐depth values from those computed with a zero XY value. In turn, the corrections are subtracted from the traveltimes, and the GRM algorithms are then re‐applied to the corrected data. Although a single application is generally adequate for most sets of field data, model studies have indicated that several applications of the GRM SSM can be required with severe topographic features, such as escarpments.In addition, very near‐surface inhomogeneities produce anomalous head‐wave amplitudes. An analogous process, using geometric means, can largely correct amplitude anomalies. Furthermore, the coincidence of traveltime and amplitude anomalies indicates that variations in the near‐surface geology, rather than variations in the coupling of the receivers, are a more likely source of the anomalies.The application of the GRM SSM, together with the averaging of the refractor velocity analysis function over a range of XY values, significantly minimizes the generation of artefacts, and facilitates the computation of detailed seismic velocities in the refractor at each receiver. These detailed seismic velocities, together with the GRM SSM‐corrected amplitude products, can facilitate the computation of the ratio of the density in the bedrock to that in the weathered layer. The accuracy of the computed density ratio improves where lateral variations in the seismic velocities in the weathered layer are known.

Structure of the San Andreas fault zone at SAFOD from a seismic refraction survey

Refraction traveltimes from a 46‐km long seismic survey across the San Andreas Fault were inverted to obtain two‐dimensional velocity structure of the upper crust near the SAFOD drilling project. The model contains strong vertical and lateral velocity variations from <2 km/s to ∼6 km/s. The Salinian terrane west of the San Andreas Fault has much higher velocity than the Franciscan terrane east of the fault. Salinian basement deepens from 0.8 km subsurface at SAFOD to ∼2.5 km subsurface 20 km to the southwest. A strong reflection and subtle velocity contrast suggest a steeply dipping fault separating the Franciscan terrane from the Great Valley Sequence. A low‐velocity wedge of Cenozoic sedimentary rocks lies immediately southwest of the San Andreas Fault. This body is bounded by a steep fault just northeast of SAFOD and approaches the depth of the shallowest earthquakes. Multiple active and inactive fault strands complicate structure near SAFOD.

Refraction traveltime tomography using damped monochromatic wavefield

For complicated earth models, wave-equation–based refraction-traveltime tomography is more accurate than ray-based tomography but requires more computational effort. Most of the computational effort in traveltime tomography comes from computing traveltimes and their Fréchet derivatives, which for ray-based methods can be computed directly. However, in most wave-equation traveltime-tomography algorithms, the steepest descent direction of the objective function is computed by the backprojection algorithm, without computing a Fréchet derivative directly. We propose a new wave-based refraction-traveltime–tomography procedure that computes Fréchet derivatives directly and efficiently. Our method involves solving a damped-wave equation using a frequency-domain, finite-element modeling algorithm at a single frequency and invoking the reciprocity theorem. A damping factor, which is commonly used to suppress wraparound effects in frequency-domain modeling, plays the role of suppressing multievent wavefields. By limiting the wavefield to a single first arrival, we are able to extract the first-arrival traveltime from the phase term without applying a time window. Computing the partial derivative of the damped wave-equation solution using the reciprocity theorem enables us to compute the Fréchet derivative of amplitude, as well as that of traveltime, with respect to subsurface parameters. Using the Marmousi-2 model, we demonstrate numerically that refraction traveltime tomography with large-offset data can be used to provide the smooth initial velocity model necessary for prestack depth migration.

Adaptive traveltime tomography of densely sampled seismic data

We present a new 2-D traveltime tomography method for the inversion of densely sampled seismic streamer data. This method was especially designed for the efficient inversion of long-offset multichannel data. A layer–interface model is used to fit ray-traced traveltime data to observed seismic data. The solution of the forward problem is based on initial-value ray tracing in a triangulated grid with a linear interpolation of the squared slowness. We implement an adaptive model parametrization based on ray density, which allows for smaller velocity cells with subsequent iteration steps. We solve the inverse problem using an iterative linearized joint inversion of reflection and refraction data for interface and velocity structures. Adaptive smoothing regularization is implemented in the form of a priori model covariances. As the cell sizes decrease with increasing iteration numbers, the model covariance ranges are reduced, allowing for more detail to emerge in the model. We demonstrate the algorithm's ability to invert successfully a realistic crustal velocity structure in a synthetic model. Several adaptive and non-adaptive model parametrizations are tested. The joint interface and velocity inversion of real long-offset reflection and refraction traveltime data is presented as a second example. We demonstrate that our results are in good agreement with independently derived velocity models.

The Inverse Problem of Refraction Travel Times, Part II: Quantifying Refraction Nonuniqueness Using a Three-layer Model

This paper is the second of a set of two papers in which we study the inverse refraction problem. The first paper, “Types of Geophysical Nonuniqueness through Minimization,” studies and classifies the types of nonuniqueness that exist when solving inverse problems depending on the participation of a priori information required to obtain reliable solutions of inverse geophysical problems. In view of the classification developed, in this paper we study the type of nonuniqueness associated with the inverse refraction problem. An approach for obtaining a realistic solution to the inverse refraction problem is offered in a third paper that is in preparation. The nonuniqueness of the inverse refraction problem is examined by using a simple three-layer model. Like many other inverse geophysical problems, the inverse refraction problem does not have a unique solution. Conventionally, nonuniqueness is considered to be a result of insufficient data and/or error in the data, for any fixed number of model parameters. This study illustrates that even for overdetermined and error free data, nonlinear inverse refraction problems exhibit exact-data nonuniqueness, which further complicates the problem of nonuniqueness. By evaluating the nonuniqueness of the inverse refraction problem, this paper targets the improvement of refraction inversion algorithms, and as a result, the achievement of more realistic solutions. The nonuniqueness of the inverse refraction problem is examined initially by using a simple three-layer model. The observations and conclusions of the three-layer model nonuniqueness study are used to evaluate the nonuniqueness of more complicated n-layer models and multi-parameter cell models such as in refraction tomography. For any fixed number of model parameters, the inverse refraction problem exhibits continuous ranges of exact-data nonuniqueness. Such an unfavorable type of nonuniqueness can be uniquely solved only by providing abundant a priori information. Insufficient a priori information during the inversion is the reason why refraction methods often may not produce desired results or even fail. This work also demonstrates that the application of the smoothing constraints, typical when solving ill-posed inverse problems, has a dual and contradictory role when applied to the ill-posed inverse problem of refraction travel times. This observation indicates that smoothing constraints may play such a two-fold role when applied to other inverse problems. Other factors that contribute to inverse-refraction-problem nonuniqueness are also considered, including indeterminacy, statistical data-error distribution, numerical error and instability, finite data, and model parameters.

The Inverse Problem of Refraction Travel Times, Part I:Types of Geophysical Nonuniqueness Through Minimization

In a set of two papers we study the inverse problem of refraction travel times. The purpose of this work is to use the study as a basis for development of more sophisticated methods for finding more reliable solutions to the inverse problem of refraction travel times, which is known to be nonunique. The first paper, “Types of Geophysical Nonuniqueness through Minimization,” emphasizes the existence of different forms of nonuniqueness in the realm of inverse geophysical problems. Each type of nonuniqueness requires a different type and amount of a priori information to acquire a reliable solution. Based on such coupling, a nonuniqueness classification is designed. Therefore, since most inverse geophysical problems are nonunique, each inverse problem must be studied to define what type of nonuniqueness it belongs to and thus determine what type of a priori information is necessary to find a realistic solution. The second paper, “Quantifying Refraction Nonuniqueness Using a Three-layer Model,” serves as an example of such an approach. However, its main purpose is to provide a better understanding of the inverse refraction problem by studying the type of nonuniqueness it possesses. An approach for obtaining a realistic solution to the inverse refraction problem is planned to be offered in a third paper that is in preparation. The main goal of this paper is to redefine the existing generalized notion of nonuniqueness and a priori information by offering a classified, discriminate structure. Nonuniqueness is often encountered when trying to solve inverse problems. However, possible nonuniqueness diversity is typically neglected and nonuniqueness is regarded as a whole, as an unpleasant “black box” and is approached in the same manner by applying smoothing constraints, damping constraints with respect to the solution increment and, rarely, damping constraints with respect to some sparse reference information about the true parameters. In practice, when solving geophysical problems different types of nonuniqueness exist, and thus there are different ways to solve the problems. Nonuniqueness is usually regarded as due to data error, assuming the true geology is acceptably approximated by simple mathematical models. Compounding the nonlinear problems, geophysical applications routinely exhibit exact-data nonuniqueness even for models with very few parameters adding to the nonuniqueness due to data error. While nonuniqueness variations have been defined earlier, they have not been linked to specific use of a priori information necessary to resolve each case. Four types of nonuniqueness, typical for minimization problems are defined with the corresponding methods for inclusion of a priori information to find a realistic solution without resorting to a non-discriminative approach. The above-developed stand-alone classification is expected to be helpful when solving any geophysical inverse problems.

Estimating the Correlation Function of a Self-affine Random Medium

The medium covariance function is of principal importance in refraction travel-time tomographic inversion, especially when estimating the accuracy of the seismic model, its relation to the geological structure, or the covariance matrix describing the statistics of synthetic travel times. The medium correlation function for the travel-time tomography should be obtained from travel times. Since a geological structure contains heterogeneities of all sizes, very similar on various scales, a self-affine random medium is a mathematical model very suitable for approximating the statistics of a geological structure. A particular class of self-affine random media, composed of a heterogeneous mean value and a stationary self-affine random function, is considered. The self-affine random function is assumed to be realized in terms of a white noise filtered by the power-law spectral filter of amplitude proportional to a reasonable power of the wavenumber. The corresponding power-law medium correlation function depends on two parameters: the Hurst exponent and the reference standard deviation. The corresponding geometrical travel-time covariances are derived. The geometrical travel-time variances are proportional to the power of ray lengths. A method designed to estimate the parameters of the medium correlation function using field travel times is proposed, and applied to data from the Western Bohemia region. The determination of the Hurst exponent from field travel times is very difficult and sensitive to numerical parameters selected for the inversion. The medium correlation functions with the values of the Hurst exponent like N = -0.1 or N = -0.2 are equally acceptable to statistically describe the travel times measured in the Western Bohemia region. On the other hand, for the fixed Hurst exponent, the determination of the reference standard deviation of slowness is easy and reliable.

Anisotropic inversion of refracted waves in vertical cable data in the presence of dip

Refracted arrivals are analysed to estimate the near-surface anisotropy of marine sediments using a vertical-cable (VC) configuration. In the presence of dip, the horizontal and vertical ray-slownesses are obtained from the observed apparent slownesses in the up- and downdip directions using a sum or difference at each azimuth. The multiple azimuths generated by a VC geometry permit the ray-slowness distribution of the marine sediments to be determined. An inversion procedure is developed to provide dip and anisotropy parameters for refractive layers from the measured refraction traveltimes in multilayered azimuthally isotropic and anisotropic media. Two sets of transversely isotropic models are used to analyse the azimuthal variations of apparent and ray slownesses. In the first set, we fix the anisotropic parameters of the models but vary the dip (0°, 5° and 10°) to test the effects of the presence of dip. In the second set, we vary the P-wave anisotropy strength (5.2%, 10.3%, 15.8% and 22.0%) to examine the sensitivity and accuracy of ray-slowness approximations which are independent of dip. We test this inversion procedure on synthetic P-wave VC data calculated for six different models by a finite-difference method. The results of applications to real VC data acquired from the North Sea are also presented.

Delineation of shallow structure and the Gondwana graben in the Mahanadi delta, India, using forward modeling of first arrival seismic data

We derived 2-D shallow velocity structure, in order to delineate low-velocity Gondwana sediments underlain by high-velocity volcanics in the north, to determine sediment thickness in the south and to map the basement configuration along the N–S trending Konark–Mukundpur profile, Mahanadi delta, India. We applied the ray-based 2-D forward modeling technique to the first arrival seismic refraction traveltime data. The shallow velocity structure has been derived to a depth of 4 km. The main features of the model are the Konark depression (0–15 km), the Bhubaneswar ridge (15–50 km), the Cuttack depression or the Gondwana graben (50–100 km) and the Chandikhol ridge (100–115 km) along the profile. The overall structures represent alternate graben and horst features. In the south, the Konark depression is composed of three sedimentary formations with velocities of 1.75, 2.4 and 4.0 km/s and attains a maximum depth of 2.9 km at 9 km profile distance. To the north, a low velocity (4.0 km/s) layer of basinal shape, believed to be the Gondwana sediments, is delineated in the Gondwana graben using the ‘skip’ phenomenon of travel time data. This layer with a maximum thickness of 1.75 km near Cuttack lies between a thin (∼100–300 m) cover of high-velocity volcanic (5.25 km/s) and underlying basement (6.0 km/s) rocks. The model indicates upwarping of basement on either side of the Gondwana graben.

2-D crustal structure of Changbaishan-Tianchi volcanic region determined by seismic traveltime inversion

2-D velocity structure and tectonics of the crust and upper mantle is revealed by inversion of seismic refraction and wide-angle reflection traveltimes acquired along the profile L1 in the Changbaishan-Tianchi volcanic region. It is used in this study that seismic traveltime inversion for simultaneous determination of 2-D velocity and interface structure of the crust and upper mantle. The result shows that, under Changbaishan-Tianchi crater, there exists a low-velocity body in the shape of an inverted triangle, and the crustal reflecting boundaries and Moho all become lower by a varying margin of 2–6 km, forming a crustal root which is assumed to be the Changbaishan-Tianchi volcanic system. Finally, we make a comparison between our 2-D velocity model and the result from the studies by using trial-and-error forward modeling with SEIS83.

New evidence of Tasmania's tectonic history from a novel seismic experiment

In March 1995, 44 land‐based recorders were deployed throughout Tasmania, SE Australia, to record seismic energy from an encircling array of marine normal‐incidence reflection shot lines. We invert refraction and wide‐angle reflection traveltimes for crustal structure, with the principal outcome being a map of the Tasmanian Moho. Key tectonic inferences from this map include: (1) the Arthur Lineament metamorphic belt in NW Tasmania overlies a major change in crustal thickness (over 5 km) and probably represents the NW limit of deformation in Tasmania during the Mid‐Late Cambrian Tyennan Orogeny, (2) thickening of the crust beneath central northern Tasmania may be associated with the juxtaposition of the Eastern and Western Tasmania Terranes during the Mid‐Devonian Tabberabberan Orogeny, and (3) the difference in crustal thickness between the east and west coasts reflects the presence of differing strain regimes during the Cretaceous break‐up of Gondwana.

2D inversion of refraction traveltime curves using homogeneous functions

A method using simple inversion of refraction traveltimes for the determination of 2D velocity and interface structure is presented. The method is applicable to data obtained from engineering seismics and from deep seismic investigations. The advantage of simple inversion, as opposed to ray‐tracing methods, is that it enables direct calculation of a 2D velocity distribution, including information about interfaces, thus eliminating the calculation of seismic rays at every step of the iteration process. The inversion method is based on a local approximation of the real velocity cross‐section by homogeneous functions of two coordinates. Homogeneous functions are very useful for the approximation of real geological media. Homogeneous velocity functions can include straight‐line seismic boundaries. The contour lines of homogeneous functions are arbitrary curves that are similar to one another. The traveltime curves recorded at the surface of media with homogeneous velocity functions are also similar to one another. This is true for both refraction and reflection traveltime curves. For two reverse traveltime curves, non‐linear transformations exist which continuously convert the direct traveltime curve to the reverse one and vice versa. This fact has enabled us to develop an automatic procedure for the identification of waves refracted at different seismic boundaries using reverse traveltime curves. Homogeneous functions of two coordinates can describe media where the velocity depends significantly on two coordinates. However, the rays and the traveltime fields corresponding to these velocity functions can be transformed to those for media where the velocity depends on one coordinate. The 2D inverse kinematic problem, i.e. the computation of an approximate homogeneous velocity function using the data from two reverse traveltime curves of the refracted first arrival, is thus resolved. Since the solution algorithm is stable, in the case of complex shooting geometry, the common‐velocity cross‐section can be constructed by applying a local approximation. This method enables the reconstruction of practically any arbitrary velocity function of two coordinates. The computer program, known as godograf, which is based on this theory, is a universal program for the interpretation of any system of refraction traveltime curves for any refraction method for both shallow and deep seismic studies of crust and mantle. Examples using synthetic data demonstrate the accuracy of the algorithm and its sensitivity to realistic noise levels. Inversions of the refraction traveltimes from the Salair ore deposit, the Moscow region and the Kamchatka volcano seismic profiles illustrate the methodology, practical considerations and capability of seismic imaging with the inversion method.

Two-dimensional inversion of refraction traveltimes by progressive model development

SUMMARY We develop a systematic approach for the analysis of 2-D refraction experiments using traveltimes that allows progressive improvement of velocity structure through a sequence from 1-D models to pseudo-2-D models, and then 2-D models. The approach consists of three steps. First, 1-D velocity models are constructed for each segment of the pro¢le using a genetic algorithm inversion, and then pseudo-2-D models are constructed using a turning point approximation. The purpose of this step is to provide an approximate image of 2-D velocity structure, and to infer the number and general location of layer boundaries. The second step uses 1-D layered modelling, again with a pseudo-2-D conversion, to generate a rough 2-D layered structure. The third step consists of smoothing the pseudo-2-D model in order to create initial models for use in 2-D inversion and the construction of a 2-D model using a Bayesian formulation of non-linear iterative inversion. All the steps exploit traveltime data for the ¢rst arrival and do not use any trial-and-error forward modelling. The progressive approach is e⁄cient because the results of each step are employed as the initial model for the next step. The method is applied to real data from an along-strike experiment and also to a related synthetic example so that the quality of the solution can be judged. The results indicate that the method is robust, and this is con¢rmed by a further synthetic example that represents a survey across a trench and dipping subduction zone. The systematic approach to the inversion of refraction data enables a complex inversion to be undertaken in 1 or 2 days. The sequential approach allows the incorporation of additional information if desired.

Nonlinear refraction and reflection travel time tomography

We develop a rapid nonlinear travel time tomography method that simultaneously inverts refraction and reflection travel times on a regular velocity grid. For travel time and ray path calculations, we apply a wave front method employing graph theory. The first‐arrival refraction travel times are calculated on the basis of cell velocities, and the later refraction and reflection travel times are computed using both cell velocities and given interfaces. We solve a regularized nonlinear inverse problem. A Laplacian operator is applied to regularize the model parameters (cell slownesses and reflector geometry) so that the inverse problem is valid for a continuum. The travel times are also regularized such that we invert travel time curves rather than travel time points. A conjugate gradient method is applied to minimize the nonlinear objective function. After obtaining a solution, we perform nonlinear Monte Carlo inversions for uncertainty analysis and compute the posterior model covariance. In numerical experiments, we demonstrate that combining the first arrival refraction travel times with later reflection travel times can better reconstruct the velocity field as well as the reflector geometry. This combination is particularly important for modeling crustal structures where large velocity variations occur in the upper crust. We apply this approach to model the crustal structure of the California Borderland using ocean bottom seismometer and land data collected during the Los Angeles Region Seismic Experiment along two marine survey lines. Details of our image include a high‐velocity zone under the Catalina Ridge, but a smooth gradient zone between Catalina Ridge and San Clemente Ridge. The Moho depth is about 22 km with lateral variations.

Lateral velocity resolution from three-dimensional seismic refraction data

SUMMARY A method for estimating the lateral velocity resolution from 3-D seismic refraction traveltimes is presented. The method is non-linear in that synthetic data calculated from ‘checkerboard’ models are inverted using an iterative tomographic approach incorporating smoothness constraints and updated ray paths at each iteration. Two applications are presented: (1) real data from the Faeroe Basin experiment using the ray coverage and noise level of approximately 50 000 picks, and (2) ideal data from the same experiment corresponding to 100 per cent data recovery, equivalent to about three times as many ray paths. The refraction data constrain the sedimentary and basement structure to approximately 12 km depth. A comparison of the results from the two data sets provides a qualitative check on the resolution method, illustrates the pitfalls of using ray coverage as an indicator of resolution, and demonstrates the potential resolution of currently feasible 3-D refraction experiments. 32 checkerboard models consisting of alternating positive and negative velocity anomalies superimposed on the preferred final model of the basin were tested, each with a diVerent cell size, position or orientation. The problem of ray bending associated with the 5 per cent checkerboard anomalies is shown to be small. Using models with four cell sizes allowed for resolution estimates from 3 to 15 km, this range being imposed by the frequency content of the data and the areal dimensions of the study area (44◊26 km). An operator centred on each model node measures the local semblance between the known and recovered checkerboard models. A determination of the smallest well-resolved cell size at each model node is then made using a 0.7 semblance threshold to provide a spatially dependent lateral resolution estimate. Resolution for the real data set varies from an average of just over 3 km at 2 km depth, to 10 km at 10 km depth, where there is a local maximum in ray coverage due to a concentration of ray-bottoming points. For the ideal data set, the average lateral resolution drops oV roughly linearly from better than 3 km at 2 km depth, to 6.3 km at 10 km depth. The largest diVerence between the resolution associated with the ideal and real data occurs at 7.2 km depth, where it averages 4.6 km for the former and 10.8 km for the latter due to the near absence of ray-bottoming points between 7 and 9 km. The results show that the final model for the Faeroe Basin obtained by regularized inversion with smoothness constraints is consistent with the resolution estimates in the sense that the model does not contain structure with a wavelength smaller than the resolution estimate at any point. The results for the ideal data show that good-quality 3-D refraction experiments can provide a lateral resolution of no worse than the receiver spacing (shot spacing for land data) throughout most of the sampled volume.

Nonlinear refraction traveltime tomography

A few important issues for performing nonlinear refraction traveltime tomography have been identified. They include the accuracy of the traveltime and raypath calculations for refraction, the physical information in the refraction traveltime curves, and the characteristics of the refraction traveltime errors. Consequently, we develop a shortest path ray‐tracing method with an optimized node distribution that can calculate refraction traveltimes and raypaths accurately in any velocity model. We find that structure ambiguity caused by short and long rays in the seismic refraction method may influence the inversion solution significantly. Therefore, we pose a nonlinear inverse problem that explicitly minimizes the misfits of the average slownesses (ratios of traveltimes to the corresponding ray lengths) and the apparent slownesses (derivatives of traveltimes with respect to distance). As a result, we enhance the resolution as well as the convergence speed. To regularize our inverse problem, we use the Tikhonov method to avoid solving an ill‐posed inverse problem. Errors in refraction traveltimes are characterized in terms of a common‐shot error, a constant deviation for recordings from the same shot, and a relative traveltime‐gradient error with zero mean with respect to the true gradient of the traveltime curve. Therefore, we measure the uncertainty of our tomography solution using a nonlinear Monte Carlo approach and compute the posterior model covariance associated with two different types of random data vectors and one random model vector. The nonlinear uncertainty analysis indicates that the resolution of a tomography solution may not correspond to the ray coverage. We apply this tomography technique to image the shallow velocity structure at a coastal site near Boston, Massachusetts. The results are consistent with a subsequent drilling survey.