George E. Backus

PROPAGATOR MATRICES IN ELASTIC WAVE AND VIBRATION 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 and 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.

Inference from Inadequate and Inaccurate Data, III

Having measured D numerical properties of a physical object E which requires many more than D parameters for its complete specification, an observer seeks to estimate P other numerical properties of E. This paper describes how he can proceed when E is adequately described by one member m(E) of a Hilbert space [unk] of possible models of E, when he believes that the Hilbert norm of m(E) is very likely rather smaller than some known number M, and (except for section 6) when all the observed and sought-after properties of E are continuous linear functionals on [unk]. Section 6 treats Frechet-differentiable non-linear functionals. A later paper will reduce unbounded functionals on arbitrary topological linear spaces to the present case.

Steady flows at the top of the core from geomagnetic field models: The steady motions theorem

It is demonstrated that the steady tangential velocity vs at the closed surface δK of a perfect fluid conductor bounded by a rigid, impenetrable exterior can be uniquely determined from knowledge o...

The electric field produced in the mantle by the dynamo in the core

Abstract 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 Earths surface ∂V, or (b) at a radius b inside the mantle, or (c) at the core-mantle boundary ∂K. In all three cases, the region where σ is no more than e times its minimum value constitutes a thin critical layer; in case (a), the radial electric field Er ≈ 0 there, while in cases (b) and (c), Er is very large there. Outside the critical layer, Er ≈ 0 in all three cases. In no case is the tangential electric field ES small, nearly toroidal, or nearly calculable from the magnetic vector potential A as −∂tAS. The defects in Muths (1979) argument which led him to contrary conclusions are identified. Benton (1979) cited Muths work to argue that the core-fluid velocity u just below ∂K can be estimated from measurements on ∂V of the magnetic field B and its time derivative ∂ t B . A simple model for westward drift is discussed which shows that Bentons conclusion is also wrong. In case (a), it is shown that knowledge of σ in M is unnecessary for estimating ES on ∂K with a relative error |r ∂r 1n σ|−1from measurements of ES on ∂V and knowledge of ∂tB in M (calculable from ∂tB on ∂V if σ is small). Then, in case (a), u just below ∂K can be estimated as − r × E S /B r . The method is impractical unless the contribution to ES on ∂V from ocean currents can be removed. The perturbation theory appropriate when σ in M is small is considered briefly; smallness of σ and of |r ∂r ln σ|−1 a independent questions. It is found that as σ → 0, B approaches the vacuum field in M but E does not; the explanation lies in the hydromagnetic approximation, which is certainly valid in M but fails as σ → 0. It is also found that the singular perturbation theory for |r ∂r ln σ|−1 is a useful tool in the perturbation calculations for σ when both σ and |r ∂r ln σ|−1 are small.

Potentials for tangent tensor fields on spheroids

AbstractIn three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let

Trimming and procrastination as inversion techniques

\hat n

Le Mouel receives Fleming Medal

be the outward unit normal to S, ▽Sthe surface gradient on S, IS the metric tensor on S, gij the four covariant components of IS (i,j = 1, 2), hijthe four covariant components of -