Construction of the maximal solution of Backus’ problem in geodesy and geomagnetism

Abstract

The (simplified) Backus' Problem (BP) consists in finding a harmonic function u on the domain exterior to the three dimensional unit sphere S, such that u tends to zero at infinity and the norm of the gradient of u takes prescribed values g on S. Except for a change of sign, the solution is not unique in general. However, there is uniqueness of solutions in the class of functions with the additional property that the radial component of the gradient of u on S is nonpositive. This is the geodetically relevant case. If a solutionu with this property exists, then u is the maximal solution of the problem (and −u the minimal one). In this paper we propose a method of successive approximations to get this particular solution of BP and prove the convergence for functions g close to a constant function.K e y w o r d s : harmonic functions, fully nonlinear boundary problem, geodesy, geomagnetism

Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces

“…A solution of (1.1) is therefore obtained by means of (1.2), where w is the solution of (1.4) corresponding to f * . We conclude our review of known results with a couple of papers, [7,8], which provide a genuinely nonlinear approach to problem (1.1). In [7], (1.1) is converted into a boundary value problem in the unit ball B:…”

“…is tangential to S and is obtained by projecting e 3 = (0, 0, 1) on the tangent plane of S at x ∈ S. Notice that ∇d(x) has intensity |∇d(x)| = 1 + 3x 2 3 for x ∈ S, and points outward to the Earth's surface on the south hemisphere, becomes tangential on the equator E = {x ∈ S : x 3 = 0}, and points inward on the north hemisphere. This behavior of ∇d tells us that neither d nor any solution of (1.1) sufficiently close to d falls within the class of solutions studied in [7,8].…”

“…This property is then instrumental to the definition of maximal and minimal solutions of (1.1) such that u ν ≤ 0 on S. This fact allows the construction of a suitably defined viscosity solution of (1.6). In [8], a numerical scheme to construct a maximal solution is proposed. The aim of this paper is to study the local resolution of the geomagnetic case, i.e.…”

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

“…(3) with BC (4) represents the so-called nonlinear geodetic BVP for the actual gravity potential G. The existence, uniqueness and other properties to the solution of this problem, and its variants, were studied extensively in physical geodesy community, see e.g. [1,18,24,3,17,30,20,10,11]. In Earth gravity field modelling, the actual gravity field G is usually expressed as a sum of the selected model field U and the remainder T , i.e.…”

“…where γ(x) = V (x) · ∇U (x) = ∇U (x) |∇U (x)| · ∇U (x) = |∇U (x)| is the so-called normal gravity. Since all quantities depending on U are given analytically, the equation (11) represents a linear oblique derivative boundary condition. Together with equations (1a) and (1c), they are called the fixed gravimetric boundary value problem in the geodetic community [1,24,22,23,8,16,29] and give a basis for determining the Earth gravity field when gravity measurements are known on the Earth surface.…”