ALBERT

Notes: 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.

Notes: The problem of constraining 3-D seismic anomalies using arrival times from a regional network is examined. The non-linear dependence of arrival times on the hypocentral parameters of the earthquakes and the 3-D velocity field leads to a multiparameter-type non-linear inverse problem, and the distribution of sources and receivers from a typical regional network results in an enormous 3-D variation in data constraint. To ensure computational feasibility, authors have tended to neglect the non-linearity of the problem by linearizing about some best-guess discretized earth model. One must be careful in interpreting 3-D structure from linearized inversions because the inadequacy of the data window may combine with non-linear effects to produce artificial or phantom ‘structure’.To avoid the generation of artificial velocity gradients we must determine only those velocity variations which are necessary to fit the data rather than merely estimating local velocities in different parts of the model, which is the more common practice. We present a series of inversion algorithms which seek to inhibit the generation of unnecessary structure while performing efficiently within the framework of a large-scale inversion. This is achieved by extending the subspace method of Kennett, Sambridge & Williamson (1988) and incorporating the smoothing strategy proposed by Constable, Parker & Constable (1987). A flexible model parametrization involving Cardinal spline functions is used, and full 3-D ray tracing performed. A comparison between linear and non-linear inversions shows that if a breakdown in the linearizing approximation occurs spurious velocity models may be obtained which would appear acceptable in a linear inversion. Application of the techniques to a SE Australian data set show that unnecessary structure can be suppressed. As the smoothing power of the algorithm is improved a robust low-velocity anomaly dipping to the north becomes the most dominant feature of the P-wave model and much of the complex structure of pure data-fitting models is removed.

Notes: Traveltime calculations in 3-D velocity models have become more commonplace during the past decade or so. Many schemes have been developed to deal with the initial value problem, which consists of tracing rays from a known source position and trajectory usually towards some distant surface. Less attention has been given to the more difficult problem of boundary value ray tracing in 3-D. In this case, source and receiver positions are known and one, or more, minimum time paths are sought between fixed endpoints.A new technique for boundary value ray tracing is proposed. The scheme uses a common numerical integration technique for solving the initial value problem and iteratively updates the take-off angles until the ray passes through the receiver. This type of ‘shooting’ technique is made efficient by using expressions describing the geometrical spreading of the wavefront to determine the relationship between the ray position at any time and the take-off angles from the source. The use of numerical integration allows the method to be compatible with a wide variety of structures. These include models with velocity varying smoothly as a function of position and those with arbitrarily orientated surfaces of discontinuity. An examination of traveltime accuracy is given as well as a discussion of efficiency for a few classes of velocity model.To improve upon the first guess pair of take-off angles, a small-scale non-linear inverse problem must be solved. The difference between the receiver position and the arrival point of a ray, on a plane through the receiver, describe a mis-match surface as a function of the two take-off angles of the ray. The shape of this surface can possess local minima and multiple ‘global’ minima even for relatively simple 1-D velocity models. Its study provides some insight into the non-linearities of a small-scale geophysical inverse problem.

Notes: 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.