Reproducing-kernel-based splines for the regularization of the inverse ellipsoidal gravimetric problem

Abstract

To solve the inverse gravimetric problem, i.e. to reconstruct the Earth's mass density distribution by using the gravitational potential, we introduce a spline interpolation method for the ellipsoidal Earth model, where the ellipsoid has a rotational symmetry. This problem is ill-posed in the sense of Hadamard as the solution may not exist, it is not unique and it is not stable. Since the anharmonic part (orthogonal complement) of the density function produces a zero potential, we restrict our attention only to reconstruct the harmonic part of the density function by using the gravitational potential. This spline interpolation method gives the existence and uniqueness of the unknown solution. Moreover, this method represents a regularization, i.e. every spline continuously depends on the given gravitational potential. These splines are also combined with a multiresolution concept, i.e. we get closer and closer to the unknown solution by increasing the scale and adding more and more data at each step.

Structural damage identification with limited modal measurements and ultra‐sparse Bayesian regression

“…Structural damage detection by solving the linear regression model is actually a discrete inverse problem, which is usually ill-posed in Hadamard's original meaning. 15 The solution may not be unique or existed, or it does not depend on data (including observations and the responses) continuously. As a result of the ill-posedness, the influence of measurement noise is very significant.…”

“…Therefore, truncation errors may be an inherent shortcoming of the first order sensitivity‐based methods for structural identification. Structural damage detection by solving the linear regression model is actually a discrete inverse problem, which is usually ill‐posed in Hadamard's original meaning 15 . The solution may not be unique or existed, or it does not depend on data (including observations and the responses) continuously.…”

Spherical approximating and interpolating moving least squares in geodesy and geophysics: a case study for deriving gravity acceleration at sea surface in the Persian Gulf

“…Second is the outer, three-dimensional spline, de ned based on integral kernels such as Abel-Poisson. A number of di erent works have been devoted to the spherical and ellipsoidal case of the spline functions, including (Akhtar, et al, 2012), (Freeden, et al, 2018a), (Freeden, et al, 1998), and (Freeden, 2009). Spline functions have powerful characteristics that make them one of the most ideal classes of interpolants and approximants.…”

Spheroidal Spline Interpolation and Its Application in Geodesy

“…Although many works have been done on the spherical (spline) interpolation, there is little done on the spheroidal case. An important work for outer spheroidal spline, namely, Abel-Poisson kernel spline, has been done in Akhtar and Michel (2012); However, the case where data are on the surface of spheroid is fundamentally different and is investigated in the present paper.…”