Nonlinear Optimization For Velocities And Structures From Reflection Picks

We present the use of a nonlinear optimization scheme called generalized simulated annealing to invert seismic reflection times for velocities, reflector depths, and lengths. A finite-difference solution of the eikonal equation computes reflection traveltimes through the velocity model and avoids raytracing. We test the optimization scheme on synthetic models and compare it with results from a linearized inversion. The synthetic tests illustrate that, unlike linear inversion schemes, the results obtained by the optimization scheme are independent of the initial model. The annealing method has the ability to produce a suite of models which satisfy the data equally well. We make use of this property to determine the uncertainties associated with the obtained model parameters. Synthetic examples demonstrate that allowing the reflector length to vary, along with its position, helps the optimization process obtain a better solution. This we put to use in imaging the Garlock fault, whose geometry at depth is poorly known. We use reflection times picked from shot gathers recorded along COCORP Mojave Line 5 to invert for the Garlock fault and velocities within the Cantil Basin below Fremont Valley, California. INTRODUCTION Reflection times depend on both the velocity and the depth of the reflectors, making traveltime inversion a nonlinear process. The last decade has seen the development of numerous inversion schemes to obtain velocity and reflector depth from reflection traveltimes. One wa.y is to parametrize both velocity and reflector depth and perform a joint inversion (e.g., Bishop et al., 1985; Stork and Clayton, 1986; Farra and Madariaga., 1988; Williamson, 1990; van Trier, 1990). Most of these met hods involve local linearization of the problem, and require the starting model to be close to the desired final solution. In this paper we examine the use of a nonlinear optimization scheme, namely simulated annealing, to invert reflection traveltimes. We must rapidly compute traveltimes through models by a met hod avoiding ray tracing. It employs a fast finite-difference scheme based on a, solution to the eikonal equation (Vidale, 1988). This accounts for curved rays and all types of primary arrivals, i.e., the fastest arrival, be it a direct arrival or a, diffraction. We test the optimization process on synthetic models and compare its performance with linear inversion schemes that use curved rays. Synthetic examples and optimizations of data demonstrate that the method is not dependent on the choice of the initial model. Another advantage of the simulated-annealing algorithm is that it produces a suite of final models with comparable least-square error. This enables us to choose the model most likely to represent the geology of the region. We also use this property to determine the uncertainties associated with the model parameters obtained. Lastly, we use the scheme to invert reflection times to image the Garlock fault and the velocity within the Cantil Basin below Fremont Valley, California,. The traveline picks are made from the shot records of COCORP Mojave Line 5. Strong lateral velocity variations across the basin make it difficult to image the Garlock fault by standard seismic velocity analysis techniques. This makes it important to use an inversion scheme that will image the velocities with minimum a priori constraints. In addition, the geometry of the Garlock fault at depth is not unequivocally known. The non1inea.r optimization scheme we use can reconstruct velocities, as well as recover estimates of dip or length of the fault. We need not specify the length or dip prior to the inversion. METHODS Simulated annealing avoids local linearization and does not require the calculation of partial derivatives. Ideally? it has the ability to test a series of local minima in search of the global minimum (Kirkpatrick et al., 1983). We make use of a variation of this optimization process called generalized simulated annealing (Bohachevsky et al., 1986). Its basis is a Monte Carlo technique due to Metropolis et al. (1953). The algorithm essentially is a two step process: ( 1) a random guess for the model is made, (2j a decision is made either to accept or reject this new guess. The medium velocity and reflector depth are perturbed simultaneously in order to preserve the nonlinearity of the problem. We perturb the velocity by adding random sized boxes, followed by smoothing with averaging over four adjacent, cells. The boxes can vary between one cell size and the entire model size. To perturb the reflector we add random length lines that are smoothed by averaging adjacent nodes. Again, the added lines can be as small as one grid spacing or as long as the length of the model. The objecive of the inversion is to minimize the least-square error between the observed times and times calculated through the model. New models are accepted when the error is less than that of the previous iteration. Conditional acceptance of models with a larger least-square error helps the inversion escape out of local minima. The generalized annealing schedule is dependent largely on two empirical parameters. They are the initial tern perature and the rate of temperature decrease. These are determined by a process based on the procedure developed by Basu and Frazer (1990). RESULTS We compare the performance of annealing with a linearized inversion scheme. In the linearized inversion. we linearize the nonlinear relationship between traveltime and slowness and the reflector depth. Generally, this linear scheme is cast: in a matrix form and the solution obtained using a standard least-squares technique (Jackson, 1972). We use a singular value decomposition method to do this. We compute the traveltimes by the same method we used in the annealing. To speed up convergence and stabilize the inversion we smooth the velocity perturbations using a Laplacian operator and the reflector depth