Gaussian beams for surface waves in laterally slowly-varying media

Summary. Asymptotic ray theory is applied to surface waves in a medium where the lateral variations of structure are very smooth. Using ray-centred coordinates, parabolic equations are obtained for lateral variations while vertical structural variations at a given point are specified by eigenfunctions of normal mode theory as for the laterally homogeneous case. Final results on wavefields close to a ray can be expressed by formulations similar to those for elastic body waves in 2-D laterally heterogeneous media, except that the vertical dependence is described by eigenfunctions of 'local' Love or Rayleigh waves. The transport equation is written in terms of geometrical-ray spreading, group velocity and an energy integral. For the horizontal components there are both principal and additional components to describe the curvature of rays along the surface, as in the case of elastic body waves. The vertical component is decoupled from the horizontal components. With complex parameters the solutions for the dynamic ray tracing system correspond to Gaussian beams: the amplitude distribution is bell-shaped along the direction perpendicular to the ray and the solution is regular everywhere, even at caustics. Most of the characteristics of Gaussian beams for 2-D elastic body waves are also applicable to the surface wave case. At each frequency the solution may be regarded as a set of eigenfunctions propagating over a 2-D surface according to the phase velocity mapping.     

Pseudo-bending method for three-dimensional seismic ray tracing in a spherical earth with discontinuities

We present a 'pseudo-bending' approach to 3-D ray tracing in a spherical earth with discontinuities. This method is based on a three-point perturbation associated with a first-order approximation, while Snell's law in curvilinear coordinates is applied at the discontinuities. We demonstrate the computational accuracy and efficiency of the pseudo-bending method in tracing rays for various velocity models by comparing results with analytical solutions and with results from the bending method. The improvement of efficiency is significant, but is reduced as the number of discontinuities increases. Since the bending approach may be computationally unstable in some situations, even though it is exact, the pseudo-bending approach is preferable for automatic calculation of rays.     

3-D magnetotelluric inversion and model validation with gravity data for the investigation of flood basalts and associated volcanic rifted margins

20 magnetotelluric (MT) soundings were collected on the Isle of Skye, Scotland to provide a high-resolution three-dimensional (3-D) electrical resistivity model of a volcanic province within the framework of a project jointly interpreting gravity, seismic, geological and MT data. The full 3-D inversion of the MT data jointly interpreted with gravity data reveals upper crustal structure. The main features of the model are interpreted in conjunction with previous geological mapping and borehole data. Our model extends to 13 km depth, several kilometres below the top of the Lewisian basement. The top of the Lewisian basement is at approximately 7–8 km depth and the topography of its surface was controlled by Precambrian rifting, during which a 4.5 km thick sequence of Torridonian sediments was deposited. The Mesozoic sediments above, which can reach up to 2.2 km thick, have small-scale depocentres and are covered by up to 600 m of Tertiary lava flows. The interpretation of the resistivity model shows that 3-D MT inversion is an appropriate tool to image sedimentary structures beneath extrusive basalt units, where conventional seismic reflection methods may fail.     

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.     

Three-dimensional geometrical ray theory and modelling of transmitted seismic energy of data from the Nevada Test Site

This paper presents a geometrically based algorithm for computing synthetic seismograms for energy transmitted through a 3-D velocity distribution. 3-D ray tracing is performed to compute the traveltimes and geometrical spreading (amplitude). The formulations of both kinematic and dynamic ray-tracing systems are presented. The two-point ray-tracing problem is solved by systematically updating the initial conditions and adjusting the ray direction until the ray intersects the specified endpoint. The amount of adjustment required depends on the derivatives of the position with respect to the given starting angles between consecutive rays. The algorithm uses derivatives to define the steepest-descent direction and to update the initial directions. The convergence rate depends on the complexity of the model.
Test seismograms compare favourably with those from a 2-D asymptotic ray theory algorithm and a 3-D Gaussian-beam algorithm. The algorithm is flexible in modelling arbitrary source and recorder geometries for various smoothly varying 3-D velocity distributions. The algorithm is further tested by simulating surface-to-tunnel vibroseis field data. Shear waves as well as compressional waves may be approximately included. Application of the algorithm to a data set from the Rainier Mesa of the Nevada Test Site produced a good fit to the transmitted (first arrival) traveltimes and amplitudes, with approximately 15 per cent variation in the local 3-D velocity.     

Numerical modelling and inversion of travel times of seismic body waves in inhomogeneous anisotropic media

Summary. Two approaches to travel-time computations in laterally inhomogeneous anisotropic media are presented. The first method is based on ray tracing in an anisotropic inhomogeneous medium, the second on the linearization procedure. The linearization procedure, which can be applied to inhomogeneous, slightly anisotropic media, does not require ray tracing in an anisotropic medium. Applications of linearized equations to the solutions of direct and inverse kinematic problems are discussed. A program package to perform the linearized computations for rather general 2-D laterally inhomogeneous layered structures is described and a numerical example is presented.     

On the computation of shear waves by the ray method

Summary. The ray series solution of the elastodynamic equation of motion for shear waves propagating through a laterally inhomogeneous three-dimensional medium can be simplified by the use of a particular coordinate system that accompanies the wave front along the ray of investigation. The system is entirely determined by parameters that are obtainable from the ray. The transport equations for the principal shear wave components are then no longer coupled, but reduce to the same type of equation which determines the principal compressional wave component.     

Chaotic ray behaviour in regional seismology

Rays propagating through strongly laterally varying media exhibit chaotic behaviour. This means that initially close rays diverge exponentially, rather than according to a power law. This chaotic behaviour is especially pronounced if the medium contains laterally varying interfaces. By studying simple 2-D and 3-D versions of models with laterally varying interfaces, the importance of chaotic ray behaviour is determined. A model of the Moho below Germany produces sharp variations with epicentral distance of the number of arrivals. In addition, the number of caustics grows dramatically: up to 1200 caustics are present between a distance of 0 and 800 km. Using the theory of Hamiltonian systems, a more in-depth study of the chaotic character of the ray equations is obtained. It is found that for realistic heterogeneous models most of the relevant rays will exhibit chaotic behaviour. The degree of chaos is quantified in terms of predictability horizons. Beyond the predictability horizons ray tracing cannot be carried out accurately. For the models under consideration, the length from the source to the predictability horizon has an order of magnitude of 1000 km. The chaotic behaviour of the rays makes it necessary to use extensions of asymptotic ray theory, such as Maslov theory, to compute seismic waveforms. It is shown that pseudo-caustics, an important obstacle in computing Maslov synthetics, are a generic feature of the 2-D laterally varying models that are studied. Eventually, the use of asymptotic methods is restricted because of the inaccuracy in the computation of the ray paths.     

Backscattering from spherical elastic inclusions and accuracy of the Kirchhoff approximation for curved interfaces

The Kirchhoff (or tangent plane) approximation, derived from the theoretically complete Kirchhoff–Helmholtz integral representation for the seismic wavefield, has been used extensively for the analysis of seismic-wave scattering from irregular interfaces; however, the accuracy of this method for curved interfaces has not been rigorously established. This paper describes an efficient Kirchhoff algorithm to simulate scattered waves from an arbitrarily curved interface in an elastic medium. Synthetic seismograms computed using this algorithm are compared with exact synthetics computed using analytical formulae for scattering of plane P waves by a spherical elastic inclusion. A windowing technique is used to remove strong internal reverberations from the analytical solution. Although the Kirchhoff method tends to underestimate the total scattering intensity, the accuracy of the approximation improves with increasing value of the wavenumber-radius product, kR . The arrival times and pulse shapes of primary reflections from the sphere are well approximated using the Kirchhoff approach regardless of curvature of the scattering surface, but the amplitudes are significantly underestimated for kR ≤ 5. The results of this work provide some new guidelines to assess the accuracy of Kirchhoff-synthetic seismograms for curved interfaces.     

Slowness-backazimuth weighted migration: A new array approach to a high-resolution image

We have developed a new array method combining conventional migration with a slowness-backazimuth deviation weighting scheme. All seismic traces are shifted based on the theoretical traveltime of the scattered wave from specific gridpoints in a 3-D volume. Observed slowness and backazimuth are calculated for each raypath and compared with theoretical values in order to estimate slowness and backazimuth deviations. Subsequently, stacked energy calculated by a conventional migration method is weighted by the slowness and backazimuth deviations to suppress any arrival energy whose slowness and backazimuth are inconsistent with the expected theoretical values. This new method was applied to two P- wave data sets which comprise (1) underside reflections at the 410 and 660 km mantle discontinuities and (2) D" reflections as well as their corresponding synthetic data sets. The results show that the weighting scheme dramatically increases the resolution of the migrated images and enables us to obtain well-constrained, focused images, making upper-mantle discontinuities and D" reflections more distinct by reducing their surrounding energy.     

Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.     

Phase-array decomposition of a seismic section

A seismic re fraction/wide-angle reflection profile is analysed for the presence of correlated events ('phases'). The correlation problem is formulated in terms of temporally, spatially and frequency-local complex covariances. For robustness, the method concentrates on phase rather than amplitude information. This allows a computationally efficient algorithm that can make allowance for signal correlation length and can model curved wavefronts. A statistical test based on residual phase misfit across the analysed subarray is used to assess the probability that a detected event represents a real correlated signal.
With our chosen analysis parameters and confidence level (over 99.9 per cent). 1222 events were detected in the data. Using simple techniques based on 1-D earth models, detected events are associated with a small number of particular wave types. In this way, we have succeeded in classifying almost 95 per cent of the detected events. Those that remain describe those components of the data that are inconsistent with our simple ray paths in the 1-D assumption and with our prescribed tolerance. These include reverberations, near-surface guided waves and reflected waves from strongly laterally inhomogeneous structures. According to our modelling, about 25 per cent of the detected events are consistent with simple P -wave reflected energy, and these are to a very large extent (over 85 per cent) distinct from all the other wave-type models we have used. A direct mapping of the detected events into the offset-depth domain reveals dear internal and external consistencies among the detections for the various wave types. Estimated earth structure is consistent with models from previous analyses based on much larger data sets.
We have thus succeeded in extracting correlated events from the data and decomposing these, approximately but meaningfully, into distinct classes (ray paths)     

Ray-reflectivity method for SH-waves in stacks of thin and thick layers

Summary Reflectivity and ray theories are united to produce a hybrid technique of computing synthetic seismograms for a plane layered medium in subcritical regions. Numerical experiments have indicated that this technique is useful when the depth structure is one composed of thick layers separated by finely layered zones. As the theory for wave propagation in a plane layered medium is well known, the simple SH case is investigated so that the basic idea of the method may be conveyed without an excess of mathematics that would be necessitated if the P-SV problem were considered.
In computing the ray-reflectivity seismogram, the thick layers are treated using asymptotic ray theory while the thin-layered zones are treated as quasiinterfaces where analogues of reflection and transmission coefficients called reflectivities and transmittivities are calculated utilizing a Thomson-Haskell formulation. A stationary phase approximation is employed when evaluating the integral which gives the displacement due to an arbitrary ray propagating in the thick layers of the above-mentioned medium, and the validity of this approximation is discussed.
A comparison of ray, numerical integration (reflectivity) and ray-reflectivity synthetic sections indicates that this method yields quite acceptable results for subcritical reflection work and is suitable for application in seismic interpretation as individual arrivals associated with ray-paths in the thick layers may be identified. Furthermore, the method is quite cost efficient and may be extended to a medium where the thick layers are non-planar using asymptotic ray theory in these layers.     

An approach to reserve estimation enhanced with 3-D seismic data

Reserve estimation for hydrocarbon reservoirs can be improved by incorporating values extracted from three-dimensional (3-D) seismic data with those obtained from more conventional data sources of data, such as drill-core and well-log data. An example of this improved method is illustrated by an application to the QW pool located in the Buohaiwan Basin in eastern China. Parameter values extracted from 3-D seismic data extend the knowledge about the spatial distributions of such reservoir parameters as net thickness, porosity, and oil saturation. To assist in the extraction of these values, different pattern-recognition techniques can be applied. The results that are obtained by this method offer a more reliable and more credible approach to reserve estimation and can be applied at every stage of resource extraction from exploration to development.     

Use of Kirchhoff s formula for body wave calculations in the Earth

Summary. Kirchhoff's time-dependent surface integral representation of a scalar wavefield is applied to the problem of computing synthetic seismograms for P -waves in the Earth. By means of an appropriate parameterization, the Kirchhoff integral is transformed into a convolution of a weight function with the derivative of the source function in the time domain. The weight function is calculated using simple ray theory. The method extends the applicability of simple ray theory to caustics and other diffraction phenomena and allows certain kinds of departures from spherical symmetry to be taken into account. The method is illustrated in detail by application to the PKP -wavefield in the Earth.     

Simulating three-dimensional seismograms in 2.5-dimensional structures by combining two-dimensional finite difference modelling and ray tracing

Finite difference (FD) simulation of elastic wave propagation is an important tool in geophysical research. As large-scale 3-D simulations are only feasible on supercomputers or clusters, and even then the simulations are limited to long periods compared to the model size, 2-D FD simulations are widespread. Whereas in generally 3-D heterogeneous structures it is not possible to infer the correct amplitude and waveform from 2-D simulations, in 2.5-D heterogeneous structures some inferences are possible. In particular, Vidale & Helmberger developed an approach that simulates 3-D waveforms using 2-D FD experiments only. However, their method requires a special FD source implementation technique that is based on a source definition which is not any longer used in nowadays FD codes. In this paper, we derive a conversion between 2-D and 3-D Green tensors that allows us to simulate 3-D displacement seismograms using 2-D FD simulations and the actual ray path determined in the geometrical optic limit. We give the conversion for a source of a certain seismic moment that is implemented by incrementing the components of the stress tensor.
Therefore, we present a hybrid modelling procedure involving 2-D FD and kinematic ray-tracing techniques. The applicability is demonstrated by numerical experiments of elastic wave propagation for models of different complexity.     

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.     

Synthetic seismograms for the case of the receiver within the reflectivity zone

Synthetic seismograms are shown and discussed for the case of the receiver within the medium. Most of the discussion is on the reflectivity method with the receiver within the reflectivity zone, but results using the ray method are shown for comparison. Such synthetic seismograms can be used to interpret data from Oblique Seismic Experiments where shots generated on the surface up to large ranges are recorded in crustal boreholes.     

Inferring network paths from point observations

The use of geographically referenced point data, such as that obtained from global positioning systems (GPS), is rapidly increasing. However, due to error and uncertainty inherent in most geographic datasets, the ability to accurately associate these point locations with other layers of geographic data is still a challenge. One difficulty in particular is how to associate spatially and temporally referenced point-based observations of a network activity with a network topology such that a continuous network path can be best inferred. In this article, an optimization method for inferring a network path from a temporal sequence of point observations of location is presented. An application to GPS data is provided to highlight various characteristics of the proposed modeling approach relative to several other available techniques.     

A search for the ratio between gravity variation and vertical displacement due to a surface load

A data space approach to magnetotelluric (MT) inversion reduces the size of the system of equations that must be solved from M × M , as required for a model space approach, to only N × N , where M is the number of model parameter and N is the number of data. This reduction makes 3-D MT inversion on a personal computer possible for modest values of M and N . However, the need to store the N × M sensitivity matrix J remains a serious limitation. Here, we consider application of conjugate gradient (CG) methods to solve the system of data space Gauss–Newton equations. With this approach J is not explicitly formed and stored, but instead the product of J with an arbitrary vector is computed by solving one forward problem. As a test of this data space conjugate gradient (DCG) algorithm, we consider the 2-D MT inverse problem. Computational efficiency is assessed and compared to the data space Occam's (DASOCC) inversion by counting the number of forward modelling calls. Experiments with synthetic data show that although DCG requires significantly less memory, it generally requires more forward problem solutions than a scheme such as DASOCC, which is based on a full computation of J .