# On Picard criterion and the well-posed nature of harmonic downward continuation

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## Abstract

Harmonic downward continuation (HDC) is in the focus of this thesis. The main question we try to answer is whether the Helmert gravity anomaly reduction from the Earth’s topography to the Helmert co-geoid through the Poisson surface integral is a well-posed problem or not. From the mathematical point of view, the Poisson surface integral is nothing else but a linear Fredholm integral equation of the first kind. Using the Helmert 2[nd subscript] condensation technique, a transformation of the Stokes (geodetic) boundary value problem (GBVP) into the Helmert space is performed. Following that, the 3-D solution of the Helmert disturbing potential on and above the Helmert co-geoid is obtained by solving the (exterior) Dirichlet boundary value problem (DBVP) for the Laplace equation.
It is shown that the HDC, for the 5’ x 5’ grid-size, is a well-posed problem in Hadamard’s sense. Small variations of the Helmert gravity anomaly in the Earth’s topography produce reasonably small variations of the Helmert gravity anomaly on the Helmert co-geoid when the first kind Fredholm integral equation is used.
In order to measure the “magnification” by the HDC, we provide an effective tool for analyzing the exact nature of this problem in the form of the Picard criterion. The Picard criterion is a technique that indicates whether the sequence of the ratio becomes convergent or divergent. In the core of this thesis, a technical question will be asked:
“How do the properties of compact or discrete Picard criterion relate to the DBPV for the HDC problem?”
The answer to this question is important from both fundamental and practical points of view, because the criterion shows how the existence and stability estimate of the HDC problem can be used. The principles of the criterion are illustrated and the results of two applications are presented. For real-life application, we restrict ourselves to demonstrating a discrete case. Hence, we will be dealing with the discrete Picard technique (DPT). We concentrate on the finite-dimensional non-symmetric matrix of the HDC arising from the discretization of the integral equation, which are related to an eigenvalue system, referred to as the quasi-eigenvalue decomposition (QEVD).
Numerical results are presented throughout the thesis to illustrate the applications of a DPT studied, with emphasis on stability and existence of the HDC problem.