Application of a Non-Linear Boundary-Value Problem for Laplace's Equation to Gravity and Geomagnetic Intensity Surveys

Backus, George E.

doi:10.1093/qjmam/21.2.195

Cited by 67 publications

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“…Backus 1968Backus , 1970. This problem has been prompted by the fact that total intensity can typically more easily and more cheaply be measured than directions.…”

Section: Introductionmentioning

confidence: 99%

A non-standard boundary value problem related to geomagnetism

Kaiser¹,

Neudert²

2004

Quart. Appl. Math.

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Abstract.Consider the following boundary value problem in the exterior space 5'd~1 = {x e : |x| > 1} of a sphere in two and three dimensions (d -2,3): Given a vector field D : Sd~1 -> we ask for all harmonic vector fields B : Sd~l -* IRd which decay at least as fast as a dipole field at infinity and are parallel to D on 5d_1;i.e. there is / : Sd~l -> R such that B = / D. For d = 3, this problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth's surface. The question for uniqueness or non-uniqueness is of particular interest here.In this paper we characterize the solution space of the boundary value problem as orthogonal complement of a certain set of functions determined by the vector field D in an appropriate Hilbert space. Based on the Hilbert space approach we determine and its dimension dim for certain classes of vector fields D. In particular, we find in d = 2 for those fields D^v which are obtained by restricting a 2jV pole field on S1, dim VpN -2(N -1) + 1. This result is robust in the sense that perturbations of Djy which are small in a certain norm do not change the dimension of the solution space. In d = 3 we consider only the axisymmetric situation.Here, we find in the case that D is given by polynomials of order not larger than N the upper bound dim < N and in the 2^-pole case dim = N. For N = 1 (dipole field) the result is proved to be robust, which implies uniqueness of the boundary value problem for all vector fields D close to Di. For Djv with N > 2 it is shown that uniqueness can be enforced if either the Hilbert space is truncated or if stronger decay conditions at infinity are imposed.

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“…Backus 1968Backus , 1970. This problem has been prompted by the fact that total intensity can typically more easily and more cheaply be measured than directions.…”

Section: Introductionmentioning

confidence: 99%

A non-standard boundary value problem related to geomagnetism

Kaiser¹,

Neudert²

2004

Quart. Appl. Math.

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“…Nonetheless, the band-limited case shows that the commonly used finite expansion of magnetizations in terms of spherical harmonics introduces some level of uniqueness that is not inherent to the original problem (similar to the case of reconstructing magnetic fields from intensity data; see, e.g. [13]). …”

Section: M(x) = Q(x)mentioning

confidence: 99%

On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere

Gerhards

2018

Inverse Problems in Science and Engineering

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Reconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e. the magnetization is described by a scalar susceptibility and the dipole direction). Here, we investigate the situation under the additional constraint that the susceptibility is either spatially localized in a subregion of the sphere or that it is band-limited. If the dipole direction is known, then the susceptibility is uniquely determined under the spatial localization constraint while it is only determined up to a constant under the assumption of bandlimitedness. If the dipole direction is not known, uniqueness is lost again. However, we show that all dipole directions that could possibly generate the measured magnetic potential need to be zeros of a certain polynomial which can be computed from the given potential. We provide examples of non-uniqueness of the dipole direction and examples on how to find admissible candidates for the dipole direction under the spatial localization constraint. ARTICLE HISTORY

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“…We begin by proving a Holder estimate for the gradient of a solution of (1). In order to make sense of the equations in (1), one should assume u to be fairly smooth; however, this smoothness can be relaxed provided the equations are interpreted in a suitable generalized way.…”

Section: Preliminariesmentioning

confidence: 99%

Two-Dimensional Nonlinear Boundary Value Problems for Elliptic Equations

Lieberman¹

1987

Transactions of the American Mathematical Society

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ABSTRACT. Boundary regularity of solutions of the fully nonlinear boundary value problem F(x,u,Du, D2u) = 0 inn, G(x,u, Du) = 0 on dO is discussed for two-dimensional domains Q. The function F is assumed uniformly elliptic and G is assumed to depend (in a nonvacuous manner) on Du. Continuity estimates are proved for first and second derivatives of u under weak hypotheses for smoothness of F, G, and 0.In [9] nonlinear boundary value problems for nonlinear, uniformly elliptic equations were studied, and several important existence and regularity results were proved when the boundary condition is oblique, i.e., it prescribes a nontangential directional derivative. Results were derived there for problems in any number of dimensions, but it was shown that the two-dimensional case is simpler than the higher-dimensional one. Here we examine the two-dimensional case in more detail using different arguments. By exploiting special features of the two-dimensional problem, we can weaken the regularity hypotheses in [9] and, more significantly, remove the obliqueness assumption. We refer the reader to [1 and 14] for existence results with nonoblique boundary condition; our main concern here is with the regularity of solutions.

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Application of a Non-Linear Boundary-Value Problem for Laplace's Equation to Gravity and Geomagnetic Intensity Surveys

Cited by 67 publications

References 0 publications

A non-standard boundary value problem related to geomagnetism

A non-standard boundary value problem related to geomagnetism

On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere

Two-Dimensional Nonlinear Boundary Value Problems for Elliptic Equations

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