George Backus

University of California, San Diego | UCSD

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. They satisfy Poincar\'e equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are therefore solutions of an ill-posed boundary-value problem. Using the Hilbert space $\underline{\bf\Lambda}$ of physically admissibl...

SUMMARYA six-parameter statistical model of the non-dipole geomagnetic field is fitted to 2597 harmonic coefficients determined by Cain, Holter & Sandee (1990) from MAGSAT data. The model includes sources in the core, sources in the crust, and instrument errors. External fields are included with instrument errors. The core and instrument statistics...

We show how to represent a vector field in a spherical shell in terms of three scalar fields, and a second-order tensor field in terms of nine scalar fields. We derive the scalar representations of the most common algebraic and differential operations on vector and second-order tensor fields, and apply the results to obtain scalar formulations of v...

In a very ingenious paper, Forsyth and Uyeda used observed plate shapes and speeds to compute the ratios of the eight rms torques which eight different types of plate force exert on 12 crustal plates. We describe one resolution of the conceptual problems arising in Forsyth and Uyeda's estimates for the values and uncertainties of those ratios. We c...

On the spherical surface S just below the core-mantle boundary layer, let u be the core-fluid velocity, Br the radial magnetic field, θ the colatitude, and φ=Br sec θ. In the approximation where the flux is frozen and u is tangentially geostrophic, u can be determined from Br and ∂tBr at all points of S which are connected to the geographical equat...

By examining the processes of truncating and approximating the model space (trimming it), and by committing to neither the objectivist nor the subjectivist interpretation of probability (procrastinating), we construct a formal scheme for solving linear and non-linear geophysical inverse problems. The necessary prior information about the correct mo...

Recent work on reversals of the earth's magnetic field [Clement, 1991; Constable, 1992; Laj et al., 1991; Runcorn, 1992] and work on low rates of change of the radial magnetic field, Br, in the Pacific hemisphere (see Jacobs [1994] page 8, for a summary) suggest that the statistics of Br differ between the Pacific and Atlantic hemispheres. The ques...

Statistical modelling of the Earth's magnetic field B has a long history. In particular, the spherical harmonic coefficients of scalar fields derived from B can be treated as Gaussian random variables. In this paper, we give examples of highly organized fields whose spherical harmonic coefficients pass tests for independent Gaussian random variable...

The main results of the grant were (1) finishing the manuscript of a proof of completeness of the Poincare modes in an incompressible nonviscous fluid corotating with a rigid ellipsoidal boundary, (2) partial completion of a manuscript describing a definition of helicity that resolved questions in the literature about calculating the helicities of...

Research in the following areas is summarized: (1) statistical modeling of main field geomagnetic secular variation on the longest time scales; and (2) constructing images of the geomagnetic field at the core mantle boundary.

Neyman's (1937) theory of confidence sets is developed as a replacement for Bayesian interference (BI) and stochastic inversion (SI) when the prior information is a hard quadratic bound. It is recommended that BI and SI be replaced by confidence set interference (CSI) only in certain circumstances. The geomagnetic problem is used to illustrate the...

In the uniqueness part of a geophysical inverse problem, the observer wants to predict all likely values of P unknown numerical properties z = (z sub 1,...,z sub p) of the earth from measurement of D other numerical properties y(0)=(y sub 1(0),...,y sub D(0)) knowledge of the statistical distribution of the random errors in y(0). The data space Y c...

In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about the correct earth model xE is essential if y is to supply useful limits for z. The one exception occurs when all the prediction functionals are linear combinations of the data functionals. We compare two forms of p...

The equations of Willis and Young (1987) for the field lines of an arbitrary axisymmetric multipole are generalized to an arbitrary linear combination of multipoles, i.e., to an arbitrary axisymmetric magnetic field B outside a sphere of radius a, S(a), centered on the origin, and containing all the sources of B. For this field, axisymmetric Stokes...

The inverse problem in empirical geomagnetic modeling is investigated, with critical examination of recently published studies. Particular attention is given to the use of Bayesian inference (BI) to select the damping parameter lambda in the uniqueness portion of the inverse problem. The mathematical bases of BI and stochastic inversion are explore...

The theory of surface operators is described and applied to four surface operators on spheres: the dimensionless surface gradient, ▽1=r▽-r∂r; the dimensionless surface curl, Λ1=rˆ×▽1; the dimensionless surface Laplacian, ▽1²=▽1 · ▽1; and the Funk-Hecke operators, integral operators with axisymmetric kernels. Three methods are given for solving ▽1²g...

Achache et al.'s averages over 10 west European observatories of the annual mean magnetic components X, Y, Z are studied from 1953 to 1979. A 1970 jerk is fitted to the data by least squares, and the fit is quantitatively compared with that of a polynomial having the same number of free parameters as the jerk (a quintic). A crude correction for the...

It is demonstrated that the steady tangential velocity at the closed surface of a perfect-fluid conductor bounded by a rigid impenetrable exterior can be uniquely determined from knowledge of the normal component of the time-varying magnetic-flux density on the surface. In the context of a simple earth model consisting of an electrically insulating...

Which features of a geomagnetic field model on the surface of the core are really necessary in order to fit, within observational error, the field observations at and above the Earth's surface? To approach this question, we define ‘roughness’ in various ways as a norm on an appropriate Hilbert space of field models which is small when the field is...

A rigorous singular perturbation theory is developed to estimate the electric field E produced in the mantle M by the core dynamo when the electrical conductivity σ in M depends only on radius r, and when |r ∂rln σ| ⪢ 1 in most of M. It is assumed that σ has only one local minimum in M, either (a) at the Earth's surface ∂V, or (b) at a radius b ins...

Suppose U is an arbitrary set, W is a subset of n-dimensional real Hilbert space $R^n$, $P:U \times R^n \to R$, and for each $u \in U$, $P(u, {\bf x})$ is a polynomial in ${\bf x}$ of degree p or less. We give an algorithm for calculating $(\sup P|U \times W)$ and $(\inf P|U \times W)$ if algorithms are available which calculate $\sup \{ P(u,{\bf x...

Contents: Constructing P-velocity models to fit restricted sets of travel-time data; Inversion of inaccurate gross earth data; Tidal to seismic frequency investigations with a quartz; Accelerometer of new geometry; Earth normal modes from a 6.5 magnitude earthquake; Microseism measurements on the deep ocean bottom; Ocean microseism measurements wit...

A scheme recently proposed by the authors for constructing Earth models which fit a given finite set of gross Earth data is applied to the problem of constructing a P-velocity structure which, within experimental error, fits the observed travel times in the range Δ = 25°(5°)95°. Three such models are obtained, all of which fit the observed travel t...

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 fi...

The motions in the Earth's electrically conducting fluid core which are the probable cause of the geomagnetic secular variations have time scales of the order of a few centuries or less. Seismic bounds on the kinematic molecular viscosity of the core and order-of-magnitude arguments about the eddy viscosity make plausible the hypothesis that at suc...

Computer applications in education are becoming more and more prevalent. Perhaps the most talked about use of computers in the schools is to control the educational material presented to students... the Computer Assisted Instruction (CAI) application. ...

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 and ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by...

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 th...