A fast, approximate 3D TEM inversion scheme has been developed, based on EM modeling at the resistive limit. The TEM data are converted to magnetic ...
Downloaded 07/19/18 to 134.7.152.114. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
Rapid, approximate 3D inversion of transient electromagnetic (TEM) data. Ralf Schaa*, ARC Centre of Excellence in Ore Deposits and Peter K. Fullagar, Fullagar Geophysics Pty Ltd
Summary A fast, approximate 3D TEM inversion scheme has been developed, based on EM modeling at the resistive limit. The TEM data are converted to magnetic moments via time integration. In effect, the moment transformation converts the nonlinear 3D TEM inversion problem into a linear 3D magnetic inversion problem. The resistive limit response is realized as a linear combination of a discretized 3D target response and a continuous host response. A starting model is constructed from conductivity depth images of the TEM profiles. In addition, the inversion is constrained by geological information and by standard potential field inversion devices such as depth weights. The underlying model is both geological and petrophysical. The inverse problem is solved using a fast steepest-descent algorithm. The approximate inversion scheme has been successfully tested on synthetic fixed-loop TEM examples and on real fixed-loop TEM field data. Typically, for ground-TEM, the approximate 3D inversion completes in minutes on a Pentium 4 2.8-Ghz (from 2006). Introduction
The 3D inversion algorithm was originally developed for potential fields data (Fullagar and Pears, 2007; Fullagar et al., 2008). The starting model is a geological model mapped onto the cubic 3D grid. Each cubic cell is assigned a rock type as well as a time constant. The geological significance of boundaries is preserved throughout the inversion process. Geological constraints ensure that the solution is consistent with all available geological information. The approximate inversion scheme is illustrated below, applied to both synthetic and real fixed loop TEM data. Approximate forward modeling The forward modeling scheme consists of superposition of the resistive limits of an anomalous volume, divided into cubic cells, together with a continuous host or background. Each cell is assigned a time constant, τk, proportional to its conductivity. At a particular location (x,y,z), the net TEM moment, M(x,y,z), is given by (
A fast approximate 3D TEM modeling and inversion algorithm has been developed, based on the concept of TEM moments. The ground is divided into cubic cells, and electromagnetic interaction between volume elements is ignored. Linear combination of the moments from the individual cubic cells is an acceptable approximation at sufficiently late times, i.e. in the resistive limit (Macnae, et al., 1999). For step-current shut-off, the resistive limit response ( ) is calculated as ( )
∫
( )
(1)
where B is the magnetic flux density. This is the first order TEM moment transform (Smith and Lee, 2002a). Application of the moment transform effectively converts the TEM inversion problem into a linear pseudo-magnetic inversion problem. The time-integration transforms each TEM decay into a single parameter. The time-depth resolution lost in the process is restored in three main ways: (i) a 3D starting model is constructed via interpolation of conductivity-depth imaging (CDI) sections; (ii) geological constraints are imposed if available; and (iii) the inversion is conditioned via weighting if desired, e.g. depth weighting (Li & Oldenburg, 1996).
© 2010 SEG SEG Denver 2010 Annual Meeting
)(
)
(
)(
)
∑
( )
(2)
where M0 denotes the background response, K is the number of cells; rk is the position vector to the centre of the k-th cell with time constant τk, and Gk represents the response of the k-th cell. After excitation by the transmitter, a magnetic dipole is induced at the centre of each cell. Therefore, Gk takes the form ̂ ̂ )̂ (̂ (3) ( ) where B0,k is the magnitude of the primary B-field at the centre of the k-th cell; Vk is the cell volume; ̂ is the unit vector parallel to rk and ̂ is the unit vector parallel to the primary field vector B0. Only large rectangular transmitter loops are considered here, but other inductive sources, e.g. magnetic dipoles for AEM, can easily be employed (Smith and Lee, 2002b). Each induced magnetic dipole inherits the orientation of the primary field. Consequently, the induced vortex currents are static and current diffusion cannot be simulated. For this reason a continuous background is introduced. The resistive limit response of a continuous conducting half space is adopted as the background contribution. For a receiver on the surface of the half-space the Z-component TEM moment for a linear current segment is given by (Schaa, 2010)
650
Approximate 3D inversion of TEM data *
( )+
(4)
where the wire extends from (x, y1) to (x, y2) with respect to the receiver at the origin. Summing the contributions from all four sides of the Tx-loop gives the net resistive limit response. The background response captures early-time and near-transmitter responses, as well as the late time response of an extensive unbounded medium. Slab-in-Host (Z-component) Subcelled (10m) Half space (1 mS/m)
Strike 400m 800m 1S/m
0.10
1mS/m
800m
RMS=0.0016
500E
300m
-500E
0.12
Resistive Limit (pTs/A)
Downloaded 07/19/18 to 134.7.152.114. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
( )
0.08
0.06
0.04
0.02
-600
-400
-200
0 Easting (m)
200
400
600
Figure 1: Resitive limit response of a 300m thick 1S/m slab in a 1mS/m half space (see inset). Open squares mark resitive limit responses from exact 3D-TEM computed by MARCO for 45 channels between 0.1-1400 ms and subsequently extrapolated at the first and last channel using the half space conductivity. Red curve is the approximate resistive limit response computed as per Eqn.(2). A square Tx-loop with side lenghts of 500 m is centered on top of the slab. Recordings are obtained along a line traversing the centre of the loop. Black triangles indicate the Tx-loop wires, and the grey shaded area marks the extent of the slab.
Because Eqn.(1) is defined for a time range from 0 to ∞, an extrapolation scheme is employed for early and late times, based on the apparent conductivities for the first and last channels respectively. Fig.1 shows an example which compares approximate TEM moment responses as calculated with Eqn.(2) with ‘exact’ responses computed with program MARCO, a fully 3D integral equation solver (Xiong and Tripp, 1995). Constrained potential field inversion Geological models are comprised of litho-stratigraphic domains. Litho-inversion schemes operating on geological models allow for property changes during inversion to be constrained by property bounds, specific to each geological unit (Fullagar and Pears, 2007). The starting model is based on 3D interpolation of CDI sections. Inversion can be conditioned using weights, e.g. (1D) depth weights or (3D) conductivity weights. Conductivity weights favor solutions close to the CDI
© 2010 SEG SEG Denver 2010 Annual Meeting
models. Depth weighting is a general device to penalize shallow solutions, based here on an empirical tanhfunction. During inversion a 3D time constant model is sought that adequately explains the set of ‘measured’ resistive limit responses. A starting model τ0 comprises an initial time constant distribution. Inversion proceeds via iterative improvement of the starting model τ0. At each iteration, a perturbation δτ is sought such that (Jupp and Vozoff, 1975) ( ) (5) where data vector contains the N observations; represents the forward model calculation (Eqn.2); is the derivative matrix or Jacobian, and denotes the time constant perturbation for K model cells. Usually there will be far more cells than there are observations so that an under-determined problem is subject to inversion (K > N). The inversion attempts to minimize error-weighted residuals between observed and calculated data by finding a new perturbation vector δτi at successive iterations i. Multiplicative weights are incorporated to inject depth resolution or to promote desired features. The inverse problem actually solved is as follows (
)
̃
(6)
where is a diagonal matrix which contains the data uncertainties qn; is a diagonal matrix which contains the multiplicative weights, e.g. depth weights, applied to the derivatives; ( ) denotes the calculated responses and where ̃ ( ) denotes the weighted vector of unknowns. A new time constant model is obtained upon ‘unweighting’ the solved perturbation, viz. (7)
̃ The updated model response vector is calculated via
(8) The perturbation vector ̃ at each iteration is calculated with the steepest descent method so that matrix inversion is not required. A solution anti-parallel to the gradient of χ2, the chi-squared misfit, is sought. Thus, ̃ where α is a scalar to be determined, and where
(11)
i
∑
(
)
(12)
where qn is the uncertainty associated with the n-th data point. At each iteration the optimal value of αi is found, minimizing the misfit. The recovered model τi is regarded as acceptable if the χ2-misfit of the model is 1 or less.
651
Approximate 3D inversion of TEM data
Downloaded 07/19/18 to 134.7.152.114. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
Synthetic inversion examples Fig.2 shows an inversion result for 231 fixed loop TEM (Zcomponent) data. Synthetic B-field decays were computed by MARCO for 45 channels between 0.1-1400 msec and integrated. The model comprises a 1 S/m horizontal slab in a resistive host (1 mS/m). Fig.1 shows the model configuration and slab dimension. 5% random Gaussian noise was added to the responses, which translates to a data uncertainty of about 0.004 pTs/A in the moment data. Model dimensions are 3km×2km×2km for east, north and depth, respectively. The model is discretized into a total of 768,000 25m cubic cells. All cell time constants were initialized with a time constant of zero and the inversion was constrained using conductivity weights based on interpolated CDIs (Fig.2A). Host conductivity was held fixed to the true value of 1 mS/m during inversion. The inversion succeeded after 4 iterations with a runtime of ~9 min. Initial χ2-misfit was ~51. As compared with the CDIs, where the base of the conductor is not well defined, the resulting time-constants are mostly confined to the true slab model.
Figure 2: Horizontal slab model. Model configuration as in Fig.1. Panel A shows the conductivity weights, obtained from CDIs. Cells with zero weights are effectively turned off. Panel B shows the recovered time constants after successful inversion. The empirical slab time constant from TEM decays is ~2.6 msec.
Fig.3 shows a total B-field CDI section for a large vertical dyke (50 S/m) in a fairly conductive host (50 mS/m). Elevated apparent conductivities extend underneath the Txloop. Model dimensions are 3km×3km×2.5km for east, north and depth, respectively, enclosing 180,000 50 m cubic cells. Z-component TEM B-field responses were calculated by MARCO at 231 stations and converted to resistive limits as previously described. After addition of 5% Gaussian random noise, these synthetic moments were subject to inversion. As before, cell time constants were initialized with zero values. The inversion was constraint using depth weights (Fig.3B) and a lower zero-bound. Host conductivity was held fixed at its true value (50 mS/m). The inversion succeeded after 91 iterations with a runtime of ~6½ min (Fig.4). Initial χ2-misfit was ~4.7.
© 2010 SEG SEG Denver 2010 Annual Meeting
Figure 3: Vertical dyke model. Panel A shows an eastwest section through the CDI, based on total B-field amplitudes (Schaa, et al., 2006). Dyke dimensions are 100m×1400m×1000m, centered at (750E,0N) with depth-to-top of -400m. The fixed Tx-loop is centered at (0E,0N) and has side lengths 500m×1000m. Exact 3DTEM were computed for 45 channels between 0.1-1400 ms using MARCO. Depth weights are on display in panel B. Smaller weights penalize shallow solutions.
Figure 4: Recovered time constants after inversion, conditioned with depth weights (vertical dyke model). The recovered time constants approximately coincide with the upper third of the vertical dyke. Estimated dyke time constant was ~475 msec
Fig.4A shows an eastwest section through the recovered time constant model; Fig.4B displays an iso-surface for time constants larger than 30 msec. Elevated values of the recovered time constants are found at depths broadly coinciding with the upper half of the vertical dyke model. Field data inversion example Fixed-loop TEM data were acquired over a resistive Proterozoic metavolcanic sedimentary sequence hosting Pb-Zn-Cu-Ag mineralization. TEM data had been recorded along nine NNE-SSW survey lines at 260 locations for 41 channel centre times ranging from 0.1 to 710.31 ms. The transmitter loop had dimensions 700m×400m, and the waveform was a bipolar square wave (0.2778 Hz fundamental, 18 Amp, turnoff ramp 0.4 ms). An interpolated CDI cross-section, calculated from total Bfield amplitudes, is on display in Fig.5. A conductive feature is evident at depth.
652
Downloaded 07/19/18 to 134.7.152.114. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
Approximate 3D inversion of TEM data was ~62, which reflects the misfit to the host response (1mS/m half space). The inversion results are consistent with drilled mineralization at depths of 500-800m. Because of non-uniqueness, the existence of conductive material at much greater depths cannot be unambiguously supported, but neither do the TEM inversion results preclude the occurrence of mineralization at those depths.
Figure 5: North-south cross-section through interpolated CDI model of the field data. Superimposed is the interpreted geometry, based on drilling, of an Iron Formation which hosts mineralization. Also shown are receiver locations (black) and the Tx-loop (red).
A 3D geological model, based on interpretation of aeromagnetic data, CDIs, geological mapping, and drill core logs was available for geologically constrained inversion. The 3D-model is a simplified representation of the local geology at the survey site and consists of three units as shown in Table 1. The structure of the prospect is such that the Gneiss is effectively enclosed by the folded Iron Formation (Fig. 5); Quartzite occupies the exterior volume. Model extents are 1480m East, 1720m North and 2320m depth extent, enclosing a total of 738,224 20m cubic cells. The Iron Formation contains 33,886 cells.
Figure 6: Left: Recovered time constant model for geologically constrained inversion, restricted to the Iron Formation. Inversion succeeded after 3 min. Right: Depth-weighted inversion result, without geological constraints. Recovered time-constants straddle the Iron Formation and extend into all three units of the geological model. Volume dispersion of time constants results in much lower values. Drill markers designate locations where drill holes have intersected mineralization.
Conclusion Category
Start τ
Min τ
Max τ
Gneiss
0
0
0
Iron Formation
100
0
2000
Quartzite
0
0
0
Table 1: Time constant starting values and bounds for field site. Each cell of the geological model belongs to a predefined geological category. A large upper bound is assigned to the iron formation, where ore mineralization is expected. Rock types with identical lower and upper bounds play no part in inversion.
Fig.6 shows the result for two different inversions: on the left side the result for the geologically constrained inversion is shown, where parameter changes were confined to the Iron Formation. A ‘best fit’ plate is superimposed. Conductive mineralization has been intersected in the vicinity of the time constant high. The right panel shows the inversion result when all cells were allowed to change, except for a surficial layer of 200m. A depth weighting scheme was used to condition the inversion. Both models were initialized with time constants set to zero. For a data uncertainty of 0.04 pT/A, inversion succeeded after 3 min (164 iterations) and 32 min (107 iterations) for the geologically constrained and depthweighted inversion respectively. Initial χ2-misfit for both
© 2010 SEG SEG Denver 2010 Annual Meeting
Motivated by the demand for a fast method for constrained 3D TEM inversion, an approximate 3D inversion scheme has been developed. The presented inversion method effectively converts the TEM inversion problem into a magnetostatic inversion problem, by exploiting the quasilinear character of TEM in the resistive limit. Depth resolution is recovered by means of CDIs (used to create a starting model), by means of geological constraints, and by depth weighting. The inversion scheme was successfully tested on fixed-loop TEM examples for both synthetic and real data. Inversion consistently defined conductive regions that generally coincided with actual conductors (in synthetic cases) or with drilled mineralization (in field data case). Runtimes are measured in minutes for the examples shown. The approach is suitable for inversion of large airborne TEM data sets. Acknowledgements This research was funded by the ARC Centre of Excellence in Ore Deposits, with industry support from Fullagar Geophysics Pty Ltd, Geophysical Resources and Services Pty Ltd, and Anglo American Geoscience Resource Group (South Africa). Mira Geoscience Asia Pacific Pty Ltd constructed the geological model.
653
Downloaded 07/19/18 to 134.7.152.114. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/
EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2010 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES
Fullagar, P. K., and G. A. Pears, 2007, Towards geologically realistic inversion: Fifth Decennial International Conference on Mineral Exploration, Exploration 07. Fullagar, P. K., G. A. Pears, and B. McMonnies, 2008, Constrained inversion of geological surfaces pushing the boundaries: The Leading Edge, 27, 98–105, doi: 10.1190/1.2831686. Jupp, D. L. B., and K. Vozoff, 1975, Stable iterative methods for the inversion of geophysical data: Geophysical Journal, 42, 957–976. Li, Y., and D. W. Oldenburg, 1996, 3-D inversion of magnetic data: Geophysics, 61, 394–408, doi: 10.1190/1.1443968. Macnae, J., A. King, N. Stolz, and P. Klinkert, 1999, 3-D EM inversion to the limit, in M. Oristaglio and B. Spies, eds., Three-dimensional electromagnetic s: SEG, 489–501. Schaa, R., 2010, Rapid approximate 3D inversion of transient electromagnetic data: Ph. D thesis, University of Tasmania. Schaa, R., J. E. Reid, and P. K. Fullagar, 2006, Unambiguous apparent conductivities for fixed-loop transient electromagnetic data: Exploration Geophysics, 37, 348–354, doi: 10.1071/EG06348. Smith, R. S., and T. J. Lee, 2001, The impulse-response moments of a conductive sphere in a uniform field, a versatile and efficient electromagnetic model: Exploration Geophysics, 32, 113–118, doi: 10.1071/EG01113. ———, 2002, The moments of the impulse response: A new paradigm for the interpretation of transient electromagnetic data: Geophysics, 67, 1095–1103, doi: 10.1190/1.1500370. Xiong, Z., and A. C. Tripp, 1995, A block iterative algorithm for 3-D electromagnetic modeling using integral equations with symmetrized substructures: Geophysics, 60, 291–295, doi: 10.1190/1.1443757.
© 2010 SEG SEG Denver 2010 Annual Meeting
654
