Preliminary reference earth model

The nearly coincident forms of the relations between seismic moment M_0 and the magnitudes M_L, M_S, and M_w imply a moment magnitude scale M = ⅔ log M_0 - 10.7 which is uniformly valid for 3 ≲ M_L ≲ 7, 5 ≲ M_s ≲ 7½, and M_w ≳ 7½.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JB084iB05p02348%4010.1002/%28ISSN%292169-9356.FAULT1

A moment magnitude scale

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.

/pdf/standardized-low-resolution-brain-electromagnetic-tomography-nyto1m0b7x.pdf

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

Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

/pdf/an-overview-of-full-waveform-inversion-in-exploration-59sei7fzvh.pdf

An overview of full-waveform inversion in exploration geophysics

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

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.

/pdf/the-resolving-power-of-gross-earth-data-31h1wjivgj.pdf

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

A cyclic process of refining models of the mechanical structure of the Earth and models of the mechanism of one or more earthquakes is developed. The theory of the elastic-gravitational free oscillations of the Earth is used to derive procedures for resolving nearly degenerate multiplets of normal modes. We show that a global network of seismographs (W.W.S.S.N.) permits resolution for angular orders l ≤ 76 and for frequencies a) w ≤ 0.090 s -1 . The peak or centre frequency of each nearly degenerate multiplet is interpreted to be a gross Earth datum. Together, the data are used to refine models of the mechanical structure of the Earth. The theory of free oscillations is used further to derive procedures for retrieving the mechanism, or moment tensor, of an earthquake point source. We show that a globa network of seismographs permits retrieval for frequencies 0.0125 s-1 ≤ w ≤ 0.0825 s-1 . We show that refined models of structure and mechanism lead to improved resolution and retrieval, and that an array of sources further complements the resolution of multiplets. We present a ‘standardized dataset’ of 1064 distinct, observed eigenfrequencies ol the elastic-gravitational free oscillations of the Earth. These gross-Earth data are compiled from 1461 modes reported in five studies: 2 modes reported by Derr (1969), 159 modes observed by Brune & Gilbert (1974), 240 modes observed by Mendiguren ( 1973), 248 modes observed by Dziewonski & Gilbert (1972,1973) and 812 modes reported here. It is our opinion that the establishment of a standardized dataset should precede the establishment of a standardized model of the Earth. Two new Earth models are presented that are compatible with the modal data. One is derived from model 508 (Gilbert & Dziewonski 1973) and the other from model B1 (Jordan & Anderson 1974). In the outer core and in the lower mantle, below a depth of about 950 km, the differences between the two models are negligibly small. In the inner core there are minor differences and in the upper mantle there are major differences in detail. The two models and the modal data are compatible with traditional ray data, provided that appropriate baseline corrections are made to the latter. The source mechanisms, or moment tensors, of two deep earthquakes, Colombia (1970 July 31) and Peru-Bolivia (1963 August 15), have been retrieved from the seismic spectra. In both cases the moment tensor possesses a compressive (implosive) isotropic part. There is good evidence that isotropic stress release begins gradually, over 80s before the origin time derived from the onset of short-period P and S waves. During the process of stress release the principal axes of the moment rate tensor migrate. The axis of compression is relatively stable, the compressive stress rate is dominant, and the other two axes rotate about the axis of compression. We speculate that earthquakes, occurring deep within descending lithospheric plates, are not sudden shearing movements alone but do exhibit compressive changes in volume such as would be associated with a phase change. We further speculate that compressive changes in volume may occur without sudden shearing movements, that there may be 9 silent earthquakes’.

An Application of Normal Mode Theory to the Retrieval of Structural Parameters and Source Mechanisms from Seismic Spectra

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...

Propagator matrices in elastic wave and vibration problems

Summary

Some century-old results, due to Rayleigh and Routh, are used to derive a very compact and simple representation for the transient response of the Earth to earthquakes. In particular, it is shown how the residual static displacement field is naturally represented in terms of the normal mode eigenfunctions.

/pdf/excitation-of-the-normal-modes-of-the-earth-by-earthquake-2omojboy7l.pdf

Freeman Gilbert

Papers

The Resolving Power of Gross Earth Data

Uniqueness in the Inversion of Inaccurate Gross Earth Data

An Application of Normal Mode Theory to the Retrieval of Structural Parameters and Source Mechanisms from Seismic Spectra

Propagator matrices in elastic wave and vibration problems

Excitation of the Normal Modes of the Earth by Earthquake Sources