# Dr. Colin G. Farquharson

I'm interested in all aspects of forward-modelling and inversion of geophysical electromagnetic data. Below are some specific topics I'm working on now or have done so recently.

**Integral equation solution for large contrasts:** Traditional implementations of the integral equation solution to the geophysical electromagnetic forward-modelling problem fail for large conductivity contrasts. I have developed an implementation that does not have this limitation, and tested it against results from physical scale modelling performed by Ken Duckworth at the University of Calgary.

**Program EM1DTM:** This is the sister program to EM1DFM, which I wrote a few years ago while at UBC-GIF. (EM1DFM is available for commercial and academic licensing: see the relevant UBC-GIF page.) Program EM1DTM is for time-domain EM data, particularly lines or grids of many soundings (that is, what one gets from airborne surveys). Like EM1DFM, nifty things such as the GCV or L-curve criteria can be used to automatically estimate the trade-off parameter. Unlike EM1DFM though, EM1DTM includes general measures, and so can produce blocky, piecewise-constant models if desired. More below.

**3-D magnetotelluric inversion:** In conjunction with Jim Craven (GSC, Ottawa), and Placer Dome Inc., I'm testing the 3-D MT inversion program MT3Dinv on a number of field data-sets (all relevant to deep exploration). This program came out of the IMAGE consortium research project at UBC-GIF, and was a joint effort of Eldad Haber, Roman Shekhtman, Doug Oldenburg and myself. Impressions so far: have our nice pictures from 2-D inversions of MT data been more misleading than we would've suspected? More below.

## Integral equation solution for large contrasts

Traditional numerical solutions of the electric field integral equation for geophysical applications used pulse basis functions to represent the fields within the anomalous region, and used integrations over equivalent-volume spheres to approximate the integrations of both the current density within cells and surface charge density between cells. Such solutions give incorrect results for contrasts in conductivity between the anomalous region and the background that are greater than roughly 300:1. If the electric field within the anomalous region is instead represented by edge-element basis functions, and if the integration of surface charge density on interfaces across which there is a change in conductivity are explicitly included, a numerical implementation is possible which can successfully model the fields even when the conductivity contrast is high.

Results computed by this new integral equation formulation have been compared with the results of physical scale modelling experiments carried out by Ken Duckworth at his Electromagnetic Modelling Laboratory in the Department of Geology and Geophysics, University of Calgary. The figure below illustrates this for a graphite cube immersed in brine. The surface of the brine was 2cm above the top of the cube, and the transmitter-receiver pair was 2cm above the surface of the brine. (The conductivity of the graphite cube was 6.3E+04S/m, and that of the brine was 7.3S/m. The side-length of the cube was 14cm. The separation between transmitter and receiver was 20cm.) The heavy lines represent the scale modelling measurements; the narrow lines the numerical modelling values.

References:

Farquharson, C.G., K. Duckworth and D.W. Oldenburg. Comparison of integral equation and physical scale modelling of the electromagnetic responses of models with large conductivity contrasts, accepted for publication in GEOPHYSICS.

Farquharson, C.G., and D.W. Oldenburg, 2002. An integral-equation solution to the geophysical electromagnetic forward-modelling problem, in Three-Dimensional Electromagnetics: Proceedings of the Second International Symposium, M.S. Zhdanov and P.E. Wannamaker (eds).

Back to top.

.

## Program EM1DTM

This program is designed specifically to handle lots and lots of time-domain EM soundings, which is what results from airborne surveys. The voltage (or magnetic field) measurements for the set of time channels for a particular spatial location are inverted to give a horizontally layered conductivity model. All the models can then be pasted side-by-side to give an image of the subsurface (see picture below).

Data from an airborne time-domain EM survey (top panel; dots indicate observations, lines show the predicted data), and the 2-D conductivity image of the subsurface constructed from the 1-D inversions of the soundings.

EM1DTM is the sister program to EM1DFM, and has many of the same features (under-determined, minimum structure inversions; four possible ways of determining the trade-off parameter: fixed or cooling schedule, discrepancy principle, GCV, L-curve), doesn't have others (just does conductivity, not susceptibility), and has some features EM1DFM doesn't (general measures: for constructing blocky, piecewise-constant models and using robust measures of data misfit).

About 3km's worth of the conductivity image produced by EM1DTM from a long line of GEOTEM data acquired during exploration for oil sands in Alberta. An L1-type norm was used for the measure of model structure, which produces this character of model with relatively sharp interfaces between units of different conductivities. Jamin Cristall did the inversions as part of his BASc thesis at UBC. (More information here.)

References:

Cristall, J., C.G. Farquharson and D.W. Oldenburg, Inversion of Airborne Electromagnetic Data: Application to Oil Sands Exploration, The 2004 Joint Assembly of the CGU, AGU, SEG and EEGS, Montréal, PQ, 17-21 May 2004.

Cristall, J., C.G. Farquharson and D.W. Oldenburg, Airborne Electromagnetic Inversion Applied to Oil Sands Exploration, 2004 CSEG National Convention, Calgary, AB, 10-13 May 2004.

Also, mainly for EM1DFM:

Farquharson, C.G., and D.W. Oldenburg, 2004. A comparison of automatic techniques for estimating the regularization parameter in nonlinear inverse problems, Geophysical Journal International, 156, 411-425.

Farquharson, C.G., D.W. Oldenburg and P.S. Routh, 2003. Simultaneous one-dimensional inversion of loop-loop electromagnetic data for magnetic susceptibility and electrical conductivity, Geophysics, 68, 1857-1869.

, and has many of the same features (under-determined, minimum structure inversions; four possible ways of determining the trade-off parameter: fixed or cooling schedule, discrepancy principle, GCV, L-curve), doesn't have others (just does conductivity, not susceptibility), and has some features EM1DFM doesn't (general measures: for constructing blocky, piecewise-constant models and using robust measures of data misfit).

Back to top.

## 3-D magnetotelluric inversion

One of the products of the UBC-GIF IMAGE consortium research project was the 3-D magnetotelluric inversion program MT3Dinv. This was a combined effort of Eldad Haber, Roman Shekhtman and Doug Oldenburg and myself. (Other products of the IMAGE project were the related frequency-domain controlled-source EM forward-modelling and inversion programs EH3D and EH3Dinv, and the time-domain forward-modelling program EH3DTD.) The forward modelling is done using a finite-volume discretization, a primary-secondary field separation, and the relevant 2-D boundary conditions. The inversion algorithm is a basic minimum-structure, Gauss-Newton one, with the system of equations at each iteration solved (to some extent!) using conjugate gradients. The requisite action of the Jacobian matrix, or its transpose, on a vector is done by solving essentially the same system of equations as for the forward problem, but with the appropriate right-hand side. All these solutions of course try to exploit the sparseness of the forward-modelling matrix as much as possible.

I've currently got inversions of a few field data-sets churning away. These are running in parallel: the obvious, straight-forward parallelization of one frequency per cpu. One example is a data-set collected at the Turquoise Ridge mine-site in Nevada by Quantec using their Titan-24 system. This test of the 3-D MT inversion is a collaboration with Placer Dome, Inc. The others are classic individual-site tensor data that I'm investigating in conjunction with Jim Craven at the GSC.

Section through a preliminary conductivity model generated from the 3-D inversion of a sub-set of the MT data acquired at the Turquoise Ridge mine site in Nevada.

References:

Farquharson, C.G., D.W. Oldenburg, and P. Kowalczyk. Three-dimensional inversion of magnetotelluric data from the Turquoise Ridge mine, Nevada, 74th Meeting of the SEG, Denver, Colorado, 10-15 October 2004.

Farquharson, C.G., D.W. Oldenburg, E. Haber and R. Shekhtman. An algorithm for the three-dimensional inversion of magnetotelluric data, 72nd Meeting of the SEG, Salt Lake City, Utah, 6-11 October 2002.

consortium research project was the 3-D magnetotelluric inversion program MT3Dinv. This was a combined effort of , Roman Shekhtman and Doug Oldenburg and myself. (Other products of the IMAGE project were the related frequency-domain controlled-source EM forward-modelling and inversion programs EH3D and EH3Dinv, and the time-domain forward-modelling program EH3DTD.) The forward modelling is done using a finite-volume discretization, a primary-secondary field separation, and the relevant 2-D boundary conditions. The inversion algorithm is a basic minimum-structure, Gauss-Newton one, with the system of equations at each iteration solved (to some extent!) using conjugate gradients. The requisite action of the Jacobian matrix, or its transpose, on a vector is done by solving essentially the same system of equations as for the forward problem, but with the appropriate right-hand side. All these solutions of course try to exploit the sparseness of the forward-modelling matrix as much as possible.