A Comparison of direct and indirect regularization methods for downward continuation problem of geoid computations without applying Stokes formula

The problem of downward continuation of the gravity field from the Earth’s surface to the reference ellipsoid arises from the fact that the solution to the boundary value problem for geoid determination without applying Stokes formula is sought in terms of the disturbing potential on the ellipsoid but the disturbing gravity observations are only available on the Earth’s surface. Downward continuation is achieved via Abel-Poisson integral and its derivatives. Using discrete observations, the Abel-Poisson integral has to be transformed into a summation form:
(1)
Where the matrix is the design matrix and stands for the disturbing
gravity observations vector. The downward continuation problem is an inverse
problem. Inverse problems are ill-posed, like any ill-posed problem it must be regularized. The objective of this paper is the comparison between direct and
iterative methods for solving downward continuation of the gravity field from the Earth’s surface to the reference ellipsoid for geoid determination without applying Stokes formula.
Direct regularization methods are methods where the solution is directly derived. In this contribution truncated method, standard Tikhonov method and generalized Tikhonov method using discretized norms at Sobolov subspaces , and Sobolov semi norms and are implemented. Based on SVD, in truncated methods, the solution can be obtained as:
(2)
Where and are the right and the left singular vectors, respectively. is rank of matrix that is a L2 norm approximation for matrix . In the case of TGSVD the solution is obtained as
(3)
In standard Tikhonov method, the minimizing function can be written as:
(4)
In this method, filter coefficients and solution become:
(5)
In standard Tikhonov method, the matrix was . In generalized Tikhonov method, we select the matrix as follows
(6)
Where the is obtained from discretization of derivative operators up to order s and coefficients are weight coefficients.
In contrast to direct methods, in iterative methods, normal equations are solved via construction of a sequence of the solutions that converge to the pseudo-inverse solution of the equations. In this contribution classical iterative method, Landweber-Fridman method, Tikhonov iterative method, Algebraic Reconstruction Technique (ART), conjugate gradient method and LSQR method are implemented.
Classical iterative methods are based on construction of sequences of solutions . For the matrix equation , The following relationship holds between solution and solution :
(7)
In Landweber-Fridman method the matrix is equal to diagonal matrix . Ergo, in this method, iterative relation between the solutions is defined as:
(8)
In Tikhonov iterative method, iterative relation between the solutions is defined as:
(9)
The idea of Algebraic Reconstruction Technique iteration to solve the matrix equation is to partition the system row wise, either into single rows or into blocks of rows. Each of these rows defines a hyper plane of dimension . The idea of the
ART iteration is to project the current approximate solution successively onto each one these hyper planes. It turns out that such a procedure converges to the solution of the system.
A best known method for solving large scale equations system is conjugate gradient. Conjugate gradient is a type of Krylov subspace method. Conjugate gradient method is suitable for positive definite operators.
In LSQR method, solution vector is def