ALBERT

Notes: Seismic traveltime data are used to determine and assess a weighted least-squares model of the 3-D P-velocity structure of the Earth's mantle. A total of 682090 summary rays constructed from over three million ISC traveltime observations are used to constrain velocity perturbations in 12496 6° by 6° cells. An iterative block LU decomposition procedure, applied on the massively parallel Connection Machine 2 (CM-2), is used to compute a formal weighted least-squares model of P-velocity variations as well as the full resolution and covariance matrices. It is found that the shallow mantle beneath the continents of the northern hemisphere is well resolved. However, the majority of mid-oceanic features, such as spreading ridges and hot spots, are not well constrained. As a function of depth, the resolution is highest in the upper to mid-mantle, due to variations in ray coverage. Primarily because of ray geometry, single blocks are not resolved below 1870 km. The estimates of spatial resolution, constructed from the complete resolution matrix, are useful in judging whether subducting lithosphere extends into the lower mantle. The results indicate that possible extensions of subduction zones in the northern hemisphere are imaged reliably down to at least 1470 km. However, areas beneath most subduction zones in the southern hemisphere are averaged over scale lengths of 800 km or more below a depth of 1070 km. Velocity estimates for the mid to lower mantle beneath Hawaii are the result of averaging over 1900 km or more. Generally, vertical and lateral averaging of 600 to 1000 km is occurring in the depth range 2270 to 2670 km. P-wave velocity values in the region just above the Core-Mantle Boundary (CMB) are large-scale spatial averages (1500 km or more) and individual cells are poorly resolved. The model parameter standard error remains moderate throughout the lower mantle due to smaller traveltime errors associated with rays that bottom in this region. These standard errors reach no more than 0.3 per cent of the Jeffreys-Bullen average velocity. The greatest standard errors, 0.9 per cent of the average velocity, are found in the upper mantle underlying the Pacific basin. These large parameter errors are due primarily to the poor ray coverage in the Pacific coupled with the large arrival-time uncertainties for P recordings in the epicentral distance range 0° to 20°.

Notes: Non-invasive studies of the shallow Earth are an essential element in a broad range of earth science disciplines. Such studies are usually accomplished using geophysical techniques, which means that a geophysical inverse problem must be solved. Unfortunately, non-uniqueness is a basic property of almost all solutions to geophysical inverse problems. In this paper we describe a method of dealing with this non-uniqueness which we believe is more direct, more complete, and more informative than previous attempts in this area. Instead of using regularization methods to constrain the non-uniqueness of geophysical inverse problems, we confront directly the non-uniqueness and attempt to describe it. The basic technique is to generate and describe a collection of models that fit the data within acceptable limits based upon observational errors. This collection of models, called an ensemble, is analysed with statistical methods in order to characterize the uncertainty in the solution and also make inferences about properties that are shared by all acceptable models. A by-product of this approach is that it produces basic information about the degree of linearity contained in the problem and maps trade-offs between various parameters. The method proposed is quite general and should be applicable to a broad range of geophysical inverse problems. We illustrate the basic method by applying it to two typical geophysical problems. The first application is to a set of cross-borehole traveltime observations gathered in Kesterson, California. The small deviations of the observed traveltime residuals from those predicted by a uniform half-space indicates that the velocity contrasts are not large. The set of acceptable models appears to form a non-degenerate hyperellipsoid in the model space. The mean of the velocity models converges quite rapidly to a smoothed version of a linearized singular-value decomposition solution. The velocity variations agree very well with the results of a hydrological tracer test conducted prior to the seismic experiment. The second application is to the problem of simultaneously estimating the shape and density of a constant density body using surface gravity measurements. In this case a linear analysis is found to be inadequate. A quadratic formulation is necessary to accurately represent the trade-off between the magnitude of the density perturbation and the depth extent of the body.

Notes: A set of coordinate transformations is used to linearize a general geophysical inverse problem. Statistical and analytic techniques are employed to estimate the parameters of such linearization transformations. In the transformed space, techniques from linear inverse theory may be utilized. Consequently, important concepts, such as model parameter covariance, model parameter resolution and averaging kernels, may be carried over to non-linear inverse problems. I apply the approach to a set of seismic cross-borehole traveltimes gathered at the Conoco Borehole Test Facility. the seismic survey was conducted within the Fort Riley formation, a limestone with thin interbedded shales. Between the boreholes, the velocity structure of the Fort Riley formation consists of a high-velocity region overlying a section of lower velocity. It is found that model parameter resolution is poorest and spatial averaging lengths are greatest in the underlying low-velocity region.

Notes: The relationship between the parameters describing a geophysical experiment and the number of solutions to the inverse problem is explored. It is found that degenerate situations, such as an inherent symmetry in an experimental set up, can give rise to multiple or even an infinite number of solutions. Techniques are developed to determine the simplest multivariable polynomial which represents the essential properties of the misfit functional. From the polynomial, the stability of a given solution to an inverse problem may be evaluated. It is shown how to calculate additional terms necessary to represent all the possible perturbations of a given minima due to changes in the description of the geophysical experiment. These terms indicate the number and types of solutions to the inverse problem which result from such changes. An example, the earthquake location problem, is considered. For differering station configurations the nature of the solutions to the location problem behave dramatically different with respect to changes in station position.

Notes: A path-following technique is presented as an alternative to the direct minimization of a misfit functional. the algorithm is based upon the iterative adjustment of an Earth model to fit a set of generated ‘data’. These values vary from a purely artificial set, produced from some prior model, to the observed data. the changes in the model that are required to satisfy the conditions for minimum misfit are calculated at each iteration. Multiple solutions appear when the path-following equations become singular and linear approximations fail. Using techniques from singularity theory in Hilbert space, the number, type and stability of solutions appearing at a singular point can be calculated. It is possible to project beyond a singular point and follow the additional solutions that emerge. Thus the algorithm allows one to track multiple solutions to a given inverse problem. the technique is used to infer velocity variations between two boreholes at the Conoco Borehole Test Facility. From a set of 471 arrival times, three velocity models are deduced. All of the models satisfy the conditions for minimum misfit. One model is very similar to the solution produced by a conjugate gradient algorithm. the three models all contain high velocities overlying low velocities. the path-following algorithm is also applied to a set of airborne gravity gradiometry data gathered in SW Oklahoma. an upthrown block of high-density basement material is found to coincide with a positive vertical gravity gradient anomaly. This conclusion agrees with the depths to basement observed in numerous oil wells in the region.

Notes: The moment tensor density is expanded in a series of orthogonal basis functions. The expansion coefficients represent integrated averages of the moment tensor density, a symmetric tensor of rank 2. A wide range of sources may be represented in this manner, from variable slip over a curved fault surface to the conventional point source expansion in moments. With this general formulation it is simple to change basis functions and to derive expressions for waveform inversion. Higher order terms in the expansion may be determined by a stable numerical integration over the basis elements.

Notes: The elastic-wave equation is used to construct sensitivity kernels relating perturbations in elastic parameters to traveltime deviations. Computation of the functions requires a correlation of the forward-propagating seismic wavefield with a backward propagation of the residual wavefield. The computation of the wavefields is accomplished using a finite difference algorithm and is efficiently executed on a CM-2 parallel processor. The source and receiver locations have maximum sensitivity to velocity structure. The sensitivity kernels or wavepaths are well suited for transmission traveltime inversion such as cross-borehole tomography and vertical seismic profiling. Conventional ray tomography and wavepath tomography are applied to a set of P-wave arrival times, from a cross-borehole experiment at Kesterson, California. Because the wavepaths have increased sensitivity near the source and receiver there are differences in resolution of the velocity structure. Both techniques recover the same relative variations in velocity where the coverage is adequate. The wavepath solution is more laterally continuous and the dominant variation is vertical, as is expected for the layered sediments in this region.

Notes: We have developed a new approach to the inversion of waveform data for the time-varying moment tensor. the method produces the source model which minimizes the modulus squared of any linear combination of moment tensor components, subject to the constraint that the data are satisfied within specified confidence intervals. This method allows the determination of possible source models other than the least-squares solution, enabling one to determine the significance of certain moment tensor properties; for example, the presence or absence of a volume change (isotropic component) in the source. Synthetic tests were used to examine the effect of microseismic noise and lateral heterogeneity on the extreme models of the isotropic component. Lateral heterogeneity is found to have a strong effect on the estimation of the isotropic component of the moment tensor. the method was tested by using long-period waveforms from the Global Digital Seismic Network to estimate the isotropic part of the moment tensor of a deep Bonin Islands earthquake. Modelling indicates that more than 10 per cent of the mechanism might have to be isotropic for detection of volume change in the presence of 10 per cent random noise and only 2 per cent lateral heterogeneity. the least-squares solution indicates that a relatively large change in volume was involved in the source mechanism. However, the minimum extreme solution shows that this volume change is not actually required by the data and thus may not be significant. the method was also tested on near-source data from the nuclear explosion Harzer. In this case, in spite of fairly large error bounds, it can be concluded that the source has a clear explosive component.