Preliminary reference earth model

Most bulk elastic media are weakly anisotropic. -The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted 6) controls most anisotropic phenomena of importance in exploration geophysics. some of which are nonnegligible even when the anisotropy is weak. The critical parameter 6 is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.

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Weak elastic anisotropy

Scalp electric potentials (electroencephalograms) and extracranial magnetic fields (magnetoencephalograms) are due to the primary (impressed) current density distribution that arises from neuronal postsynaptic processes. A solution to the inverse problem--the computation of images of electric neuronal activity based on extracranial measurements--would provide important information on the time-course and localization of brain function. In general, there is no unique solution to this problem. In particular, an instantaneous, distributed, discrete, linear solution capable of exact localization of point sources is of great interest, since the principles of linearity and superposition would guarantee its trustworthiness as a functional imaging method, given that brain activity occurs in the form of a finite number of distributed hot spots. Despite all previous efforts, linear solutions, at best, produced images with systematic nonzero localization errors. A solution reported here yields images of standardized current density with zero localization error. The purpose of this paper is to present the technical details of the method, allowing researchers to test, check, reproduce and validate the new method.

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Standardized low-resolution brain electromagnetic tomography (sLORETA): technical details.

It is possible to use the waveform data not only to derive the source mechanism of an earthquake but also to establish the hypocentral coordinates of the ‘best point source’ (the centroid of the stress glut density) at a given frequency. Thus two classical problems of seismology are combined into a single procedure. Given an estimate of the origin time, epicentral coordinates and depth, an initial moment tensor is derived using one of the variations of the method described in detail by Gilbert and Dziewonski (1975). This set of parameters represents the starting values for an iterative procedure in which perturbations to the elements of the moment tensor are found simultaneously with changes in the hypocentral parameters. In general, the method is stable, and convergence rapid. Although the approach is a general one, we present it here in the context of the analysis of long-period body wave data recorded by the instruments of the SRO and ASRO digital network. It appears that the upper magnitude limit of earthquakes that can be processed using this particular approach is between 7.5 and 8.0; the lower limit is, at this time, approximately 5.5, but it could be extended by broadening the passband of the analysis to include energy with periods shorter that 45 s. As there are hundreds of earthquakes each year with magnitudes exceeding 5.5, the seismic source mechanism can now be studied in detail not only for major events but also, for example, for aftershock series. We have investigated the foreshock and several aftershocks of the Sumba earthquake of August 19, 1977; the results show temporal variation of the stress regime in the fault area of the main shock. An area some 150 km to the northwest of the epicenter of the main event became seismically active 49 days later. The sense of the strike-slip mechanism of these events is consistent with the relaxation of the compressive stress in the plate north of the Java trench. Another geophysically interesting result of our analysis is that for 5 out of 11 earthquakes of intermediate and great depth the intermediate principal value of the moment tensor is significant, while for the remaining 6 it is essentially zero, which means that their mechanisms are consistent with a simple double-couple representation. There is clear distinction between these two groups of earthquakes.

Determination of earthquake source parameters from waveform data for studies of global and regional seismicity

The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models.To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model.Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric-resistivity inversion, demonstrate the method and show it to have practical application.

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Occam's inversion; a practical algorithm for generating smooth models from electromagnetic sounding data

A horizontally layered inhomogeneous medium, isotropic or transversely isotropic, is considered, whose properties are constant or nearly so when averaged over some vertical height l′. For waves longer than l′ the medium is shown to behave like a homogeneous, or nearly homogeneous, transversely isotropic medium whose density is the average density and whose elastic coefficients are algebraic combinations of averages of algebraic combinations of the elastic coefficients of the original medium. The nearly homogeneous medium is said to be ‘long-wave equivalent’ to the original medium. Conditions on the five elastic coefficients of a homogeneous transversely isotropic medium are derived which are necessary and sufficient for the medium to be ‘long-wave equivalent’ to a horizontally layered isotropic medium. Further conditions are also derived which are necessary and sufficient for the homogeneous medium to be ‘long-wave equivalent’ to a horizontally layered isotropic medium consisting of only two different homogeneous isotropic materials. Except in singular cases, if the latter two-layered medium exists at all, its proportions and elastic coefficients are uniquely determined by the elastic coefficients of the homogeneous transversely isotropic medium. The observed variations in crustal P-wave velocity with depth, obtained from well logs, are shown to be large enough to explain some of the observed crustal anisotropies as due to layering of isotropic material.

Long-Wave Elastic Anisotropy Produced by Horizontal Layering

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of interia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail; and how to determine the shortest length scale which the given data can resolve at any particular depth. We apply the general theory to the linear problem of finding the depth-variation of a frequency-independent local Q from the observed quality factors Q of a finite number of normal modes. We also apply the theory to the non-linear problem of finding density vs depth from the total mass, moment, and normal-mode frequencies, in case the compressional and shear velocities are known.

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The Resolving Power of Gross Earth Data

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.

Uniqueness in the Inversion of Inaccurate Gross Earth Data

Summary
A gross datum of the Earth is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We prove that the collection of Earth models which yield the physically observed values of any independent set of gross Earth data is either empty or infinite dimensional. We exploit this very high degree of non-uniqueness in real geophysical inverse problems to generate computer programs which iteratively produce Earth models to fit given gross Earth data and satisfy other criteria. We describe techniques for exploring the collection of all Earth models which fit given gross Earth data. Finally, we apply the theory to the normal modes of elastic-gravitational oscillation of the Earth.

Numerical Applications of a Formalism for Geophysical Inverse Problems

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order, ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerical methods for their computation. When the dispersion relation is some mth order minor of the integral matrix it is possible to deal with mth minor propagators so that the dispersion relation is a single element of the mth minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations for SH and for P‐SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator e...

George E. Backus

Papers

Long-Wave Elastic Anisotropy Produced by Horizontal Layering

The Resolving Power of Gross Earth Data

Uniqueness in the Inversion of Inaccurate Gross Earth Data

Numerical Applications of a Formalism for Geophysical Inverse Problems

Propagator matrices in elastic wave and vibration problems