ALBERT

Notizen: The response of a seismic model to excitation by a source can be represented in terms of the action of reflection and transmission operators for portions of the structure. This approach provides a flexible framework for both modeling and processing problems.The operator development provides a physical description of the wave propagation process and, via the expansion of reverberation operators, gives a mechanism for assessing the accuracy of approximate developments. The representation suggests new ways of developing modeling algorithms by balancing the computational effort expended on minor and major features of the model.For processing problems, the operator representation shows the relation of processing stages to the seismic wave field and thereby indicates effective sequences of operations. For migration it is possible to specify an ideal pre-stack migration procedure in terms of the inverse of the propagation operators and to examine the problems which need to be overcome by practical algorithms.

Notizen: The presence of the water layer in marine seismic prospecting provides an effective waveguide for acoustic energy trapped between the sea-bed and the sea-surface. This energy persists to large ranges and can be the dominant early feature on far-offset traces. On airgun records, there is commonly a lower frequency set of arrivals following the water-trapped waves. These arrivals are not as obvious with higher frequency watergun sources. By using a combination of intercept-time/slowness (τ—p) mapping on observational data and theoretical modelling, we are able to identify the origin of the events. If a very rapid increase in a seismic wavespeed occurs beneath the sea-bed sediments, a new waveguide is formed bounded by the sea surface and this transition zone. The low frequency waves are principally guided within this thicker waveguide. Numerical filtering in the τ—p domain followed by trace reconstruction is very effective in removing the low frequency noise.

Notizen: A common example of a large-scale non-linear inverse problem is the inversion of seismic waveforms. Techniques used to solve this type of problem usually involve finding the minimum of some misfit function between observations and theoretical predictions. As the size of the problem increases, techniques requiring the inversion of large matrices become very cumbersome. Considerable storage and computational effort are required to perform the inversion and to avoid stability problems. Consequently methods which do not require any large-scale matrix inversion have proved to be very popular. Currently, descent type algorithms are in widespread use. Usually at each iteration a descent direction is derived from the gradient of the misfit function and an improvement is made to an existing model based on this, and perhaps previous descent directions.A common feature in nearly all geophysically relevant problems is the existence of separate parameter types in the inversion, i.e. unknowns of different dimension and character. However, this fundamental difference in parameter types is not reflected in the inversion algorithms used. Usually gradient methods either mix parameter types together and take little notice of the individual character or assume some knowledge of their relative importance within the inversion process.We propose a new strategy for the non-linear inversion of multi-offset reflection data. The paper is entirely theoretical and its aim is to show how a technique which has been applied in reflection tomography and to the inversion of arrival times for 3D structure, may be used in the waveform case. Specifically we show how to extend the algorithm presented by Tarantola to incorporate the subspace scheme. The proposed strategy involves no large-scale matrix inversion but pays particular attention to different parameter types in the inversion.We use the formulae of Tarantola to state the problem as one of optimization and derive the same descent vectors. The new technique splits the descent vector so that each part depends on a different parameter type, and proceeds to minimize the misfit function within the sub-space defined by these individual descent vectors. In this way, optimal use is made of the descent vector components, i.e. one finds the combination which produces the greatest reduction in the misfit function based on a local linearization of the problem within the subspace. This is not the case with other gradient methods. By solving a linearized problem in the chosen subspace, at each iteration one need only invert a small well-conditioned matrix (the projection of the full Hessian on to the subspace). The method is a hybrid between gradient and matrix inversion methods. The proposed algorithm requires the same gradient vectors to be determined as in the algorithm of Tarantola, although its primary aim is to make better use of those calculations in minimizing the objective function.

Notizen: A new approach is presented for the suppression of multiples reflected at the surface of a horizontally layered fluid or elastic medium, recorded at non-zero offsets from the source. The scheme used is to extract the effect of the free surface in the frequency-wavenumber domain and then to replace this surface by a non-reflecting boundary. The multiple suppression operator requires a detailed knowledge of the source time function and the elastic properties of the medium between the source and the surface.For a stratified fluid or a liquid layer overlying a stratified elastic medium, complete multiple suppression can be achieved with noise free data. If only the vertical component is available for an elastic medium an approximate approach may be used which removes most of the multiple energy. Good results may be achieved with this multiple suppression scheme in the presence of noise. The method is designed to be used before records are stacked in a CDP gather.

Notizen: Theoretical seismograms for an explosive source in a multilayered elastic medium are constructed by Fourier synthesis and plane wave superposition. The calculation scheme which builds up a reflection matrix layer by layer in the frequency and wave number domain allows the inclusion of attenuation and a choice of the level of internal multiples in each layer.Comparative calculations of theoretical seismograms for an elastic model and in the acoustic approximation, neglecting shear, show that the main differences arise at large offsets. The inclusion of shear waves leads to lower reflected P wave amplitudes at the end of the spread but only small amounts of converted phases.

Notizen: A number of techniques which exploit the waveforms of seismic surface waves depend on simple approximations for the character of the propagation process from source to receiver based on the representation for a stratified medium. Commonly the propagation path is assumed to lie along a great circle and to be representable by a path-averaged structure. The influence of structure near the source and near the receiver is included by using local modal formulations. However, the terms that depend on source depth and receiver depth in the stratified medium results are not purely local in character, and so care has to be taken to ensure a simple mapping between the modal shapes for the different structures.For frequencies less than 0.03 Hz, different crustal structures can be used at the source, near the receiver, and along the propagation path, provided that the change in crustal thickness is not more than 10 km between contiguous structures. Furthermore, for frequencies up to 0.035 Hz, it should be possible to use a single modal set in non-linear waveform inversions for perturbations of up to 5 per cent in lithospheric velocities along the propagation path.For propagation paths of length from 1000 to 4000 km, typical of a continental scale, the path-averaged structure approximation should be suitable for waveform fitting for frequencies in the range 0.01-0.03 Hz. The lower limit depends on the use of asymptotic approximations and the upper on the influence of heterogeneity on the modal content of the seismograms.Where surface waves cross a major structural boundary such as the continent-ocean transition, some aspects of the wavefield can still be represented using the path-averaged approximation.

Notizen: This paper shows how the performance of a fully non-linear earthquake location scheme can be improved by taking advantage of problem-specific information in the location procedure. The genetic algorithm is best viewed as a method of parameter space sampling that can be used for optimization problems. It has been applied successfully in regional and teleseismic earthquake location when the network geometry is favourable. However, on a series of test events with unfavourable network geometries the performance of the genetic algorithm is found to be poor.We introduce a method to separate the spatial and temporal parameters in such a way that problems related to the strong trade-off between depth and origin time are avoided. Our modified algorithm has been applied to several test events. Performance over the unmodified algorithm is improved substantially and the computational cost is reduced. The algorithm is better suited to the determination of hypocentral location whether using arrival times, array information (slowness and azimuth) or a combination of both.A second type of modification is introduced which exploits the weak correlation between the epicentral parameters and depth. This algorithm also improves performance over the standard genetic algorithm search, except in circumstances where the depth and epicentre are not weakly correlated, which occurs when the azimuthal coverage is very poor, or when azimuth and slowness information are incorporated. On a shallow nuclear explosion with only teleseismic P arrivals available, the algorithm consistently converged to a depth very close to the true depth, indicating superior depth estimation for shallow earthquake locations over the unmodified algorithm.

Notizen: Many techniques for the solution of seismic-wave propagation problems depend on the representation of the seismic wavefield in terms of a linear combination of basis functions, as for example in Fourier or Gaussian Beam expansions. A common formal representation encompasses such methods when a preferred coordinate is isolated to track the propagation path. Different techniques can be classified by the dependence of the basis functions on this preferred coordinate. The common representation provides useful insight into the relation between apparently disparate methods and can guide the development of computational techniques. This common framework allows the development of generalized propagator methods and a compact formulation of reflection and transmission problems. A general perturbation approach can be used either to add heterogeneity to an existing structure or to restore features, such as coupling between P and S waves, which have been ignored in an approximate development.

Notizen: Over the last three years, a major international effort has been made by the Sub-Commission on Earthquake Algorithms of the International Association of Seismology and the Physics of the Earth's Interior (IASPEI) to generate new global traveltime tables for seismic phases to update the tables of Jeffreys & Bullen (1940). The new tables are specifically designed for convenient computational use, with high-accuracy interpolation in both depth and range. The new iasp91 traveltime tables are derived from a radially stratified velocity model which has been constructed so that the times for the major seismic phases are consistent with the reported times for events in the catalogue of the International Seismological Centre (ISC) for the period 1964–1987. The baseline for the P-wave traveltimes in the iasp91 model has been adjusted to provide only a small bias in origin time for well-constrained events at the main nuclear testing sites around the world.For P-waves at teleseismic distances, the new tables are about 0.7s slower than the 1968 P-tables (Herrin 1968) and on average about 1.8–1.9 s faster than the Jeffreys & Bullen (1940) tables. For S-waves the teleseismic times lie between those of the JB tables and the results of Randall (1971).Because the times for all phases are derived from the same velocity model, there is complete consistency between the traveltimes for different phases at different focal depths. The calculation scheme adopted for the new iasp91 tables is that proposed by Buland & Chapman (1983). Tables of delay time as a function of slowness are stored for each traveltime branch, and interpolated using a specially designed tau spline which takes care of square-root singularities in the derivative of the traveltime curve at certain critical slownesses. With this representation, once the source depth is specified, it is straightforward to find the traveltime explicitly for a given epicentral distance. The computational cost is no higher than a conventional look-up table, but there is increased accuracy in constructing the traveltimes for a source at arbitrary depth. A further advantage over standard tables is that exactly the same procedure can be used for each phase. For a given source depth, it is therefore possible to generate very rapidly a comprehensive list of traveltimes and associated derivatives for the main seismic phases which could be observed at a given epicentral distance.

Notizen: The effect of the free surface can be removed from three-component seismic recordings to recover the incident upgoing wavefield, if the slowness and azimuth of the current wavefront are known as a function of time. For a single three-component station it is usually possible to estimate an azimuth for an event from the first arriving P-waves, but slowness estimates are less reliable when more than one wavetype is presented in the seismic wavetrain. However, the free surface correction operators are generally slowly varying functions of slowness and so some error in slowness can be tolerated.Effective approximations for the removal of the free surface effects can be made for hard rock sites to cover slowness bands for the main regional phases Pn, Pg, Sn and Lg. By applying these operators in turn over group velocity windows appropriate to the particular phases, the relative amplitude of the P, SV and SH contributions to the wavefield can be estimated. Because the free surface amplification effects have been removed, the amplitudes can be compared directly and provide useful constraints on the radiation characteristics of the source. This procedure is therefore helpful for developing discrimination measures for different classes of sources.