A Robust and fast method for gravity data inversion

In most geophysical inverse problems there is a non-linear dependence of the observable quantities on the parameters describing the model. The most common way of solving a nonlinear inversion is to make a local linear expansion about a current model and then to conduct a linear inversion for perturbing of the model required to match the observed data. The updated model should be used as the basis for a further linear inversion and the iterations are continued until convergence is achieved, i.e. the model perturbation or the data misfit lies below a pre-assigned threshold. This type of inversion depends on the direct solution of a set of simultaneous linear equations which lead to a matrix inversion. The solution of a set of simultaneous linear equations is computationally very intensive when the number of data points and of model parameters becomes large. Thus for large- scale problems, linearized techniques involving inversion of a matrix rapidly become difficult to handle. It can therefore be computationally advantageous to use techniques which can achieve good results without the inversion of large matrices. To handle this problem, subspace methods can be applied. Subspace methods use a local minimization of an objective function in a subspace spanned by a limited number of vectors in the model space. The spanning vectors are called basis vectors. It is well known that the success or failure of a subspace method depends upon the selection and the number of basis vectors chosen for the subspace. The basis vectors chosen are orthonormal functions constructed from the positions of the data on the surface. The matrix inversion in the subspace of the model parameters is better conditioned due to the smaller dimensionality and the limited number of basis vectors used in the inversion. Since the most significant basis vectors are used in the inversion, those elements of the model which are likely to have less influence in the fitting the data or lead to local minima are eliminated. The solution of the inversion in the subspace of the model parameter, in this way has small variance. The method is tested by inverting two- dimension irregularly spaced data.