The Resolving Power of Gross Earth Data

Abstract

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 frequencyindependent 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.

Inverse eigenvalue problems for layered media

“…Remark. This result substantiates a conjecture of Backus and Gilbert [4] in that information contained in spectra corresponding to different angular numbers can be used to determine the density and the rigidity uniquely. The theorem is not best possible, because, as we shall show in Section 6, the lowest eigenfrequency Go can be deleted.…”

Inverse eigenvalue problems for layered media

“…Remark. This result substantiates a conjecture of Backus and Gilbert [4] in that information contained in spectra corresponding to different angular numbers can be used to determine the density and the rigidity uniquely. The theorem is not best possible, because, as we shall show in Section 6, the lowest eigenfrequency Go can be deleted.…”

"The Ray Paths Distribution Significant Role in the Resolution of Tomographic Images"

“…This agrees with (6). That we cannot integrate down to r = 0 reflects the non-physical nature of a point source.…”

Deep Earth Seismology: An Introduction and Overview

“…All three studies agreed in mode identification and found that their periods were very close to those predicted by the existing Earth models (Pekeris et al, 1961a). Backus and Gilbert (1967), Backus and Gilbert (1968), and Backus and Gilbert (1970) provided the formal background for consideration of geophysical inverse problems, and even though their own experiments with inversion of normalmode periods for the velocities and density, using a subset of normal-mode data, were discouraging (two very different models were found fitting the data nearly exactly; Backus and Gilbert, 1968), the idea of resolving kernels and trade-offs became part of the geophysical terminology. Thus, normal-mode seismology was born.…”

Inverse eigenvalue problems for layered media

“…Remark. This result substantiates a conjecture of Backus and Gilbert [4] in that information contained in spectra corresponding to different angular numbers can be used to determine the density and the rigidity uniquely. The theorem is not best possible, because, as we shall show in Section 6, the lowest eigenfrequency Go can be deleted.…”

"The Ray Paths Distribution Significant Role in the Resolution of Tomographic Images"

“…This agrees with (6). That we cannot integrate down to r = 0 reflects the non-physical nature of a point source.…”

Deep Earth Seismology: An Introduction and Overview

“…All three studies agreed in mode identification and found that their periods were very close to those predicted by the existing Earth models (Pekeris et al, 1961a). Backus and Gilbert (1967), Backus and Gilbert (1968), and Backus and Gilbert (1970) provided the formal background for consideration of geophysical inverse problems, and even though their own experiments with inversion of normalmode periods for the velocities and density, using a subset of normal-mode data, were discouraging (two very different models were found fitting the data nearly exactly; Backus and Gilbert, 1968), the idea of resolving kernels and trade-offs became part of the geophysical terminology. Thus, normal-mode seismology was born.…”

Inverse eigenvalue problems for layered media

“…Remark. This result substantiates a conjecture of Backus and Gilbert [4] in that information contained in spectra corresponding to different angular numbers can be used to determine the density and the rigidity uniquely. The theorem is not best possible, because, as we shall show in Section 6, the lowest eigenfrequency Go can be deleted.…”

"The Ray Paths Distribution Significant Role in the Resolution of Tomographic Images"

“…This agrees with (6). That we cannot integrate down to r = 0 reflects the non-physical nature of a point source.…”

Deep Earth Seismology: An Introduction and Overview

“…All three studies agreed in mode identification and found that their periods were very close to those predicted by the existing Earth models (Pekeris et al, 1961a). Backus and Gilbert (1967), Backus and Gilbert (1968), and Backus and Gilbert (1970) provided the formal background for consideration of geophysical inverse problems, and even though their own experiments with inversion of normalmode periods for the velocities and density, using a subset of normal-mode data, were discouraging (two very different models were found fitting the data nearly exactly; Backus and Gilbert, 1968), the idea of resolving kernels and trade-offs became part of the geophysical terminology. Thus, normal-mode seismology was born.…”