The general theory of nuclear magnetic resonance (NMR) imaging of large electromagnetically active systems is considered. We emphasize particularly noninvasive geophysical applications such as the imaging of subsurface water content. We derive a general formula for the NMR response voltage, valid for arbitrary transmitter and receiver loop geometry and arbitrary conductivity structure of the medium in which the nuclear spins reside. It is shown that in cases where the conductivity is large enough such that the electromagnetic skin depth at the Larmor frequency is of the same order or smaller than the measurement depth, there are diffusive retardation time effects that significantly alter the standard NMR response formula used in the literature. The formula now includes the full complex response, the imaginary part of which has previously been observed but not modeled. These differences are quantified via numerical investigation of various effectively one-dimensional model inverse problems with a horizontally stratified nuclear spin and conductivity distribution. It is found that inclusion of the imaginary part of the response significantly stabilizes the inversion. Large quantitative differences are found between conducting and insulating cases in physically relevant situations. It is shown also that the diffusive long time tail of the signal may be used to infer the distribution of time constants T1, normally not measurable in geophysical applications. Although in present applications the signal due to this tail is immeasurably small, this relationship may become useful in the future.

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In this paper we derive a theory, based on quasi-degenerate perturbation theory, that governs the effect of global-scale, steady-state convection and associated static asphericities in the elastic-gravitational variables (adiabatic bulk modulus κ, density ρ, and gravitational potential ∅) on helioseismic eigenfrequencies and eigenfunctions and present a formalism with which this theory can be applied computationally. The theory rests on three formal assumptions: (1) that convection is temporally steady in a frame corotating with the Sun, (2) that accurate eigenfrequencies and eigenfunctions can be determined by retaining terms in the seismically perturbed equations of motion only to first order in p -mode displacement, and (3) that we are justified in retaining terms only to first order in convective velocity (this is tantamount to assuming that the convective flow is anelastic). The most physically unrealistic assumption is (1), and we view the results of this paper as the first step toward a more general theory governing the seismic effects of time-varying fields. Although the theory does not govern the seismic effects of non-stationary flows, it can be used to approximate the effects of unsteady flows on the acoustic wavefield if the flow is varying smoothly in time. The theory does not attempt to model seismic modal amplitudes since these are governed, in part, by the exchange of energy between convection and acoustic motions which is not a part of this theory. However, we show how theoretical wavefields can be computed given a description of the stress field produced by a source process such as turbulent convection. The basic reference model that will be perturbed by rotation, convection, structural asphericities, and acoustic oscillations is a spherically symmetric, nonrotating, non-magnetic, isotropic, static solar model that, when subject to acoustic oscillations, oscillates adiabatically. We call this the SNRNMAIS model. An acoustic mode of the SNRNMAIS model is denoted by k = ( n, l, m ), where n is the radial order, l is the harmonic degree, and m is the azimuthal order of the mode. The main result of the paper is the general matrix element H m'm n'n,l'l for steady-state convection satisfying the anelastic condition with static structural asphericities. It is written in terms of the radial, scalar eigenfunctions of the snrnmais model, resulting in equations (90)—(110). We prove Rayleigh’s principle in our derivation of quasi-degenerate perturbation theory which, as a by-product, yields the general matrix element. Within this perturbative method, modes need not be exactly degenerate in the SNRNMAIS solar model to couple, only nearly so. General matrix elements compose the hermitian supermatrix Z . The eigenvalues of the supermatrix are the eigenfrequency perturbations of the convecting, aspherical model and the eigenvector components of Z are the expansion coefficients in the linear combination forming the eigenfunctions in which the eigenfunctions of the SNRNMAIS solar model act as basis functions. The properties of the Wigner 3 j symbols and the reduced matrix elements composing H m'm n'n,l'l produce selection rules governing the coupling of SNRNMAIS modes that hold even for time-varying flows. We state selection rules for both quasidegenerate and degenerate perturbation theories. For example, within degenerate perturbation theory, only odd-degree s toroidal flows and even degree structural asphericities, both with s ≤ 2 l , will couple and/or split acoustic modes with harmonic degree l . In addition, the frequency perturbations caused by a toroidal flow display odd symmetry with respect to the degenerate frequency when ordered from the minimum to the maximum frequency perturbation. We consider the special case of differential rotation, the odd-degree, axisymmetric, toroidal component of general convection, and present the general matrix element and selection rules under quasi-degenerate perturbation theory. We argue that due to the spacing of modes that satisfy the selection rules, quasi-degenerate coupling can, for all practical purposes, be neglected in modelling the effect of low-degree differential rotation on helioseismic data. In effect, modes that can couple through low-degree differential rotation are too far separated in frequency to couple strongly. This is not the case for non-axisymmetric flows and asphericities where near degeneracies will regularly occur, and couplings can be relatively strong especially among SNRNMAIS modes within the same multiplet. All derivations are performed and all solutions are presented in a frame corotating with the mean solar angular rotation rate. Equation (18) shows how to transform the eigenfrequencies and eigenfunctions in the corotating frame into an inertial frame. The transformation has the effect that each eigenfunction in the inertial frame is itself time varying. That is, a mode of oscillation, which is defined to have a single frequency in the corotating frame, becomes multiply periodic in the inertial frame.

Accurate models of the distribution of elastic heterogeneity in the Earth's mantle are important in many areas of geophysics. The purpose of this paper is to characterize and compare quantitatively a set of recent three‐dimensional models of the elastic structure of the Earth, to assess their similarities and differences, and to analyze their fit to one class of data in order to highlight fruitful directions for future research. The aspherical models considered are the following: M84C (Woodhouse and Dziewonski, 1984), LO2.56 (Dziewonski, 1984), MDLSH (Tanimoto, 1990a), SH.10c.17 (Masters et al., 1992), and S12_WM13 (Su et al., 1994). Through much of the discussion, M84C and LO2.56 are combined into a single whole mantle model, M84C + LO2.56. The fit of each model to previously tabulated even degree normal mode structure coefficients taken from Smith and Masters (1989a) and Ritzwoller et al. (1988) for multiplets along the normal mode fundamental and first, second, and fifth overtone branches is also presented. Rather than concentrating on detailed comparisons of specific features of the models, analyses of these models are general and statistical in nature. In particular, we focus on a comparison of the amplitude and the radial and geographical distribution of heterogeneity in each model and how variations in each affect the fit to the normal mode observations. In general, the results of the comparisons between the models are encouraging, especially with respect to the geographical distribution of heterogeneity and in the fit to the normal mode data sensitive to the upper mantle and lowermost lower mantle. There remain, however, significant discrepancies in amplitude and in the radial distribution of heterogeneity, especially near the top of the upper mantle and near the top of the lower mantle. The confident use of these models to constrain compositional and dynamical information about the mantle will await the resolution of these discrepancies. The factors that may be responsible for the differences in the models and/or for the misfit between the observed and predicted normal mode data are divided into two types: intrinsic (or procedural) and extrinsic (or structural). We discuss only three extrinsic factors at length here, including errors in the reference crustal models, unmodeled topography on discontinuities in the interior of the mantle, and errors in the assumed relationships between shear (υs) and compressional (υp) heterogeneity.