van der Linden, Arjan (Shell International Exploration & Production) | Myers, Michael (Shell International Exploration & Production) | Dudley, John (Shell International Exploration & Production) | Love, Keith (Shell International Exploration & Production) | Addis, Tony (Shell International Exploration & Production) | Lehr, Christian (Shell International Exploration & Production) | Korndorffer, Frans (Shell International Exploration & Production) | Bauer, Andreas (Shell International Exploration & Production)
Basak, P.S. (Oil & Natural Gas Corporation Limited) | Deb, Abhijit (Oil & Natural Gas Corporation Limited) | Dotiwala, Sucheta (Oil & Natural Gas Corporation Limited) | Sanyal, A. (Oil & Natural Gas Corporation Limited) | Pokhriyal, S.K. (Oil & Natural Gas Corporation Limited)
We present a practical formulation for forward modeling and inverting time domain data arising from multiple transmitters. The underpinning of our procedure is the ability to factor the forward modeling matrix and then solve our system using direct methods. We first formulate Maxwell’s equations in terms of the magnetic field, H to provide a symmetric forward modeling operator. The problem is discretized using a finite volume technique in space and a backward Euler in time. The MUMPS software package is used to carry out a Cholesky decomposition of the forward operator, with the work distributed over an array of processors. The forward modeling is then quickly carried out using the factored operator. The time savings are considerable and they make the simulations of largeground or airborne data sets feasible and greatly increase our abilities to solve the 3D electromagnetic inverse problem in a reasonable time. The ability to use direct methods also greatly enhances our ability to carry out survey design problems.
In previous research (Haber, Oldenburg, & Shekhtman, 2006) we developed an inversion algorithm that allowed us to invert data from a single, or a very few, transmitters. Unfortunately the computational demands of that algorithm were too large to invert typical ground or airborne surveys acquired from many source locations. The principal difficulty is the time required to solve the forward problem. Simulating data that arise from multi-sources can be computationally onerous because each transmitter requires that Maxwell''s equations be solved. Usually this is done with iterative (eg. CG-type) algorithms and hence the computation time increases linearly with the number of transmitter locations. However, for surveys with a large number of sources, significant increases in efficiency can be had if the forward modeling matrix is factored. Factorization involves large computations and significant memory requirements. However, once this is accomplished, solving the factored system with a different right hand side proceeds very quickly. The idea of decomposing the matrix system and solving many right hand sides for different sources is not new (Dey & Morrison, 1979), and small problems have been solved in this manner. However, the matrices for 3D TEM problems have generally been considered to be too large to contemplate this approach. Over the last decade however, advances in mathematics and computing science have resulted in factorization algorithms that can be implemented on large scale computing systems (Amestoy, Guermouche, LExcellent, & Pralet, 2006). The efficacy of this approach depends upon the time required to factor the matrix compared to the time required to solve the matrix system using iterative solvers. We use the MUMPS codes and distribute the computation over many different processors.
MAXWELL’S EQUATIONS IN THE TIME DOMAIN
The forward model consists of Maxwell’s equations in time where the permeability is fixed but electrical conductivity can be highly discontinuous.
Solving the forward problem
As previously discussed in (Haber & Ascher, 2001; Haber, Ascher, Aruliah, & Oldenburg, 2000) consistent discretization of Maxwell''s equation leads to a near-singular system.