Large Scale Joint Inversion of Geophysical Data using the Finite Element Method in escript

Abstract

Inversion is finding the minimum value of a cost function measuring data defect and physical property variation subject to constraints defined by a partial differential equation (PDE). In general, this problem is equivalent to the solution of a system for coupled PDEs for three unknowns -- the physical property, the observation and a Lagrangean multiplier. Therefore it is appropriate to use established PDE solution methods such as the finite element method (FEM) and solver software systems such as escript (see https://launchpad.net/escript-finley) to tackle inversion problems. Besides the ability to handle non-linearity in the physical model (e.g. required for high susceptibility) as well as in the regularisation term or in the cross-gradient terms the PDE approach provides a number of computational advantages in comparison to traditional, linear algebra based solution approaches. The method is data sparse by nature and does not require compression or sparsification on the sensitivity matrix when solving large scale inversion problems. Moreover, domain decomposition can be applied to run the inversion across processors in a parallel computer. In contrast to traditional tiling, domain decomposition is not imposed on the inversion from the outside but applied on the lowest level which makes the approach more computaional more efficient. In the presentation we will introduce the concept of PDE based, joint inversion of gravity and magnetic data. We will outline the appropriate solution methods when using the finite element method, outline the implementation strategy using the PDE solver escript and show results from joint inversion runs of field data using massive parallel computer.