Multi-source inversion of TEM data: with field applications to Mt. Milligan D. Oldenburg, E. Haber, D. Yang Geophysical Inversion Facility, University of British Columbia, Vancouver, BC, Canada Summary 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 large ground or airborne data sets feasible and greatly increase our ability to solve the 3D electromagnetic inverse problem in a reasonable time. We illustrate the capability, and need, for 3D inversion by inverting airborne VTEM data from Mt. Milligan, a Cu-Au porphyry deposit in British Columbia. Introduction In previous research (Haber, Oldenburg, & Shekhtman, 2007) 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. CGtype) 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, 979), 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. We have shown preliminary results about the use of the decomposition method and inversion in a previous presentation (SEG 2008). Here we extend that analysis and apply it to field data. Maxwell s Equations in the Time Domain We first write the discrete form of Maxwell s equations. Using a backward Euler discretization in time with step size we obtain Hi+ Hi E i+ + μ = 0, () EGM 200 International Workshop Capri, Italy, April -4, 200
Ei+ Ei Hi+ σe i+ ε = si. (2) δ t E i and H i are the electric and magnetic fields that correspond to the source s i, μ is the permeability, σ is the conductivity, ε is the permittivity and s is a source. Eliminating the electric field E i+ from Maxwell s equations, we obtain an equation for the magnetic field H i+ μ ( σ + ε) Hi+ + Hi+ = rh s i + (3) The above system can be near-singular when the conductivity is small and the time steps are large. To stabilize it, we recall that we also have μh = 0 and therefore we can rewrite the system as μ ( σ + ε) Hi+ μ ( σ + ε) ( μh i+ ) + Hi+ = rhsi+ (4) Upon discretization in space, we obtain the linear system A( σ ) H : = ( C( σδ, t) + M) H = rhs i+ i+ where the matrix C( σ, δ t) is a discretization of the differential operator in (5). The matrix is SPD and depends on the conductivity and the time step. The matrix M depends on μ and. We note that by using the same time steps, the linear system (5) is identical for all times and all sources. Matrix factorization is an expensive computational process, however, when the same forward modeling matrix needs to be solved for many times and sources, the decomposition will be greatly superior to iterative techniques. The benefits of this are exacerbated when one proceeds to the inverse problem since the same factorization can be used for the computation of the gradient as well as for the solution of the linear system which arises at each Gauss-Newton iteration. In recent years there has been a growing effort to obtain scalable matrix factorizations on parallel machines. We use the package Multi-frontal Massively Parallel Solver (MUMPS) by the CERFACS group (Amestoy et al., 2006). MUMPS is a package for solving systems of linear equations of the form Ax = b, where the matrix A is sparse and can be either unsymmetric, symmetric positive definite, or general symmetric. Implementation of the Forward Modeling To obtain insight regarding the implementation of the decomposition we consider a representative time domain airborne survey using the VTEM system. The transmitter is a horizontal circular loop, of 26 meters diameter, flown at 50 meters elevation. Data, db/dt, are acquired in the center of the loop. The ideal waveform for this system is shown in Figure. Data are acquired in 27 time channels, equispaced in logarithmic time between 0-5 and 0-2 seconds after the turnoff time. Our equations are to be time stepped so that we capture the input waveform during the on-time and also capture the logarithmically spaced times after the turnoff (off-time). Also, we need to begin our time stepping a decade earlier than the first time channel and hence a = 6.6 0-7 seconds is required. Unfortunately, we can t use that step for the entire interval since it would then require 0 4 time steps and the cumulative time for all of these solves is approximately 6 seconds 0 4 8 hours. This difficulty can be circumvented by dividing our total time interval into subintervals, each of which has a constant. One additional factorization is required for each subinterval. The off-time part here can be solved using 3 factorizations and 58 time steps. The segments which are subsequently uniformly time-stepped are shown in Figure b. Each segment is divided into 5 ~ 25 steps and hence the total number of time steps and the total number of factorizations for a complete waveform are 88 and 5 respectively. With the factored system the time required to solve a forward problem, t s, is 6 seconds for this example. The total number of forward solutions is equal to (5) EGM 200 International Workshop Capri, Italy, April -4, 200
N TX, where is the number of transmitters. Thus the total time taken to solve our problem N δ t N TX is T = N N t + t where tf is the time carrying out a factorization. total TX δ t s f (a) on-time discretized waveform (b) off-time waveform and time channels Figure. Transmitter current waveform of VTEM system Inversion The real benefit of using decomposition in our forward modeling becomes apparent when we treat the inverse problem. Our inversion algorithm (Haber et al., 2007), is based upon a Gauss-Newton procedure where a model m = lnσ is sought. The forward modeling is generically written as A(m)u = q where A is the forward modeling matrix, u are the fields, and q is the right hand side which contains the source terms. At every Gauss-Newton iteration, the equation ( ( ) T T Jm Jm ( ) + βwws ) = gm ( ) (8) is solved to obtain a perturbation s from a current model m. W is a sparse regularization matrix, β is a constant, and g(m) is the gradient of the objective function. The sensitivity matrix is available as J(m) = -QA(m) - G(m,u), where Q is an interpolation matrix that extracts the simulated data from the computed fields and G(m,u) is a known sparse matrix ( A( mu ) ) G( m, u) = (9) m We note that G requires the fields for all transmitters and for all times. These have been computed to evaluate the misfit and gradient, but they need to be stored or recomputed to carry out the effect of multiplying J or J T on a vector. Field Example Mt. Milligan is a Cu-Au porphyry deposit in central British Columbia and it has served as a test site for the inversion of many different types of geophysical surveys (Oldenburg et al., 997). The basic geologic and petrophysical structure consists of a resistive monzonite stock surrounded by a conductive alteration zone. The host rock is volcanic (Figure 2a). The area is covered with a variable thickness, moderately conductive overburden. EGM 200 International Workshop Capri, Italy, April -4, 200
(a) geologic model at Mt. Milligan (b) D inversion of VTEM data Figure 2. Comparison between geologic model and D inversion VTEM data were acquired along several lines in the area and a D inversion was carried out. One conductivity cross section of our D inversion is shown in Figure 2b. We note that soundings directly over the resistive stock (horizontal coordinate is 434260) indicate a conductive layer at depth which is contrary to the expected result. To test the hypothesis that the conductive layer was an artifact of using a D inversion model we selected a small subset of the airborne data and carried out a 3D inversion using only the 28 transmitter locations shown on the map in Figure 3a. (a) locations of soundings for 3D inversion (b) cross section of 3D inversion model Figure 3. 3D inversion with 28 VTEM soundings at Mt. Milligan In our 3D inversion, the model domain was discretized by a 50 50 48 mesh (20, 000 cells), in which the finest cells are with a dimension of 75 75 20 meters. The inversion was carried out on 3 dual quadcore machines. The initial and reference models were set to 500Ωm. After 2 iterations, the inversion achieved a data misfit equivalent to chifact =.6. One cross section of the 3D inversion model, shown in Figure 3b, highlights a massive resistive rock unit corresponding to the drillholeproven monzonite stock. This coincides spatially whered inversion found relatively conductive material. Conclusions We have demonstrated the benefits of solving multi-source time domain electromagnetic problems by decomposing the matrix and using direct solvers. This methodology has been proven efficient and practical from the computational point of view if the transmitter waveform and mesh are appropriately designed. Our field data example of VTEM at Mt. Milligan showcases the advantage of 3D multisource inversion over D inversion in revealing complicated geologic structure. References Amestoy, P.R., Guermouche, A., L Excellent, J.-Y., Pralet, S. 2006. Hybrid scheduling for the parallel solution of linear systems. Parallel Computing, 32, 36-56. Dey, A., Morrison, H.F.979. Resistivity modeling for arbitrarily shaped three-dimensional structures. Geophysics, 44, 753-780. EGM 200 International Workshop Capri, Italy, April -4, 200
Haber, E., Oldenburg, D., & Shekhtman, R. 2007. Inversion of time domain 3D electromagnetic data. Geoph. J. Int., 7, 550-564. Oldenburg, D., Li, Y., Ellis, R. 997. Inversion of geophysical data over a copper gold porphyry deposit: A case history for Mt. Milligan. Geophysics, 62, 49-43. Oldenburg, D., Haber, E., Shekhtman, R., 2008. Forward modeling and inversion of multi-source TEM data. SEG Annual Meeting, Las Vegas. EGM 200 International Workshop Capri, Italy, April -4, 200