In Sect. 2, automatic and semiautomatic cooperative inversion strategies were described. This was followed by description of the application of semiautomatic/automatic cooperative inversion with the aid of a field example from Nevada, USA. In Sect. 4, we will describe, analyse and compare the outcomes from application of nine MT inversion strategies.
Note that multiple strategies were run simultaneously on the Magnus supercomputer located at the Pawsey Supercomputing Centre in Perth, Western Australia. Run times for each strategy were of the order 24 h on the Cray Cascade system. Pethick and Harris (2016) provide a detailed account of applied parallel computing for EM methods. They include analysis of speed up and wall time for standard Intel processors versus the optimised Cray Cascade system that was also used for the current research.
There are limitless combinations of seismic attributes and MT inversion methods that could be attempted. We elected to systematically work through nine strategies we label S1–S9. These strategies were selected to facilitate comparison of misfit and final inverted conductivity distribution for: conventional unconstrained inversion (S1–S2), automatic cooperative inversion (S4–S6) and semiautomatic cooperative inversion (S3, S7–S9).
The “base case” unconstrained inversions are named S1 and S2. The seven cooperative inversion strategies, S3–S9, use seismic information to aid development of the prior model conductivity distribution, (m
) and/or covariance matrix values, (C
) needed for input to subsequent rounds of 3D MT inversion (see Eq. 1). We have organised the main points that differentiate these nine inversion strategies into summary in Table 1 and provide more complete descriptions below.
Unconstrained Inversion (S1–S2)
For the first two strategies, S1 and S2, the half-space prior model resistivity is set to 100 and 40 Ω m, respectively (see Table 1). The value 100 Ω m (S1) is selected to be consistent with the average wireline log resistivity measurements immediately below cover (i.e. in basement). The value 40 Ω m (S2) is selected to be consistent with the gross resistivity of the younger cover rocks above basement. Strategies S1 and S2 use a uniform covariance coefficient of 0.3.
Semiautomatic Inversion (S3)
Strategy S3 is identical to the unconstrained inversion strategy S2, except that for strategy S3 the covariance coefficient is reduced from 0.3 to 0.25 for cells that cross key high-impedance contrast interfaces picked from the seismic volume. An example of a high acoustic impedance interface is the boundary between cover and basement.
Automatic Cooperative Inversion (S4–S6)
Automatic cooperative inversion strategies can, in principle, be automated. For the Nevada field example, the structural seismic attribute, dip angle, is used to create the geometric framework (see Sect. 3.2). However, the ideas presented can certainly be adapted to take advantage of other seismic attributes or combinations of attributes.
Strategy S4 uses “direct mapping” of the seismic dip angle attribute to electrical conductivity distribution for the prior model (see schematic in Fig. 9). Direct mapping is in essence a mapping of some characteristic extracted from the seismic data to an electrical conductivity. This mapping is usually based either on a presumed or empirically determined petrophysical relationship possibly established by cross-plot of velocity and conductivity from wireline logs.
Given enough wells and good quality wireline log data a neural network approach could, in principle, be adopted to build nomogram connecting parameters derived from seismic and electromagnetic data (Bauer et al. 2012). However, as with the Nevada example, there is rarely sufficient log data to build the relationships.
Strategies S5 and S6 use the distribution of dip angle to divide the complete seismic volume into smaller subvolumes with a common seismic character. Each subvolume is subsequently assigned an electrical resistivity based on analysis of the distribution of conductivity derived from unconstrained inversions S1 and S2, respectively (see Fig. 14). We call this type of mapping from unconstrained inversion into a framework defined by seismic information “geometric mapping”.
For the Nevada example, five classes of subvolumes are created based on k-means cluster analysis of the distribution of the dip angle seismic attribute (see Fig. 11c). A resistivity value is then assigned to each subvolume by statistical analysis of the conductivity distribution derived from unconstrained inversion using half-space models of 100 Ω m (S1) and 40 Ω m (S2). In this way the seismic information provides a geo-electrical framework and the unconstrained inversion is used to generate a discrete conductivity that is assigned to each subvolume. It should be remembered that the prior model needs only to have a simple relatively gross and broadly correct 3D conductivity distribution for the subsequent rounds of inversion to yield a vastly improved final outcome.
Semiautomatic Inversion (S7–S9)
Cooperative inversion strategies S7–S9 are designated semiautomatic because the key seismic boundaries that form the geo-electrical framework were picked by a partly manual and subjective process. For these strategies seismic attributes like amplitude, energy and energy texture were highly useful in identifying key horizons. After identifying key horizons, an auto-picking algorithm (Huck 2012) was used to recover them through the seismic volume. The auto-picking tended to be successful for recovery of the main high acoustic impedance contrast horizons. The result is that the seismic volume was readily partitioned into large subvolumes that ultimately formed a prior geo-electrical model framework for S7, S8 and S9.
The prior model framework for semiautomatic cooperative inversion strategies S7, S8 and S9 included just four interpreted horizons (see Fig. 6a, b) and five large subvolumes. The largest subvolume was the lower basement half-space. A resistivity was assigned to each of the other subvolumes by “geometric mapping” (see Fig. 9) based on statistical analysis of resistivity distribution derived from unconstrained inversion using a 40-Ω m half-space model (see strategy S2). The four subvolume resistivity values assigned to the cover rocks were 40, 30, 20 and 35 Ω m. A resistivity of 100 Ω m was mapped into the full basement subvolume because it is broadly consistent with outcome from unconstrained inversion (S1) and wireline log resistivity (see Sect. 3.1.2) immediately below cover.
Figure 14 provides an image representation of the original seismic reflectivity, the dip angle seismic attribute distribution and prior models conductivity distributions for cooperative inversion strategies S4–S9. The nine strategies are included to compare advantages and disadvantages of inversion strategies.
We immediately see that automatic cooperative inversion strategies S4–S6 permit prior model conductivity detail within basement that is not permitted for semiautomatic strategies S7–S9. For strategies S4–S6, the geo-electrical framework comes directly from the volumetric distribution of a seismic attribute whereas for S7–S9 the framework is based explicitly on selected seismic horizons. One approach is likely to provide accurate 3D geo-electrical boundaries, while the other provides an objective, mapping conductivities onto subvolumes with common seismic character.
The Final Conductivity Distribution for Inversion Strategies S1–S9
Next we provide final conductivity distributions for inversion strategies S1–S9 and analyse these outcomes with respect to fitting errors, the original seismic reflectivity and drill-hole data. A representation of final conductivity distributions from inversion strategies S1–S9 are provided in Fig. 15. All nine final inverted conductivity models broadly represent the geometry of the more conductive cover. However, as anticipated there are considerable differences in the distribution of conductivity in cover and below cover between the nine different strategies.
We will return to the analysis and comparison of these conductivity distributions after considering model convergence and data misfit for the nine inversion strategies.
Assessing Inversion Strategies S1–S9: Convergence And Misfit
For each 3D MT inversion strategy, a suitable error floor needs to be specified. The error floors for off-diagonal elements (Zxy and Zyx) are set to 5 % of the root mean square (RMS) absolute of their complex multiplication. Diagonal elements (Zxx and Zyy) share the same error floor with the off-diagonal elements because both are small and tend to be noisy. The error floor of tipper values is set to 0.02.
Assessing the fit between field and synthetic data is not straightforward. The rate of convergence for inversion strategies S1–S9 is shown in Fig. 16, and details are included in Table 2. At a basic level we could consider rate of convergence and global RMS error as indicators of the relative success for each inversion. However, this would be misleading and further analysis is required.
The global RMS error (Table 2) alone is a blunt instrument for analysing fit, which can be assessed for each sounding. Figure 17 shows a complete representation of field data and final inversion outcomes from all strategies for tensor MT station 320. The location of station 320 is provided in Fig. 2. The inversion strategies S1, S2 and S3 using half-space prior models generally do not achieve the rate of change observed in the field MT apparent resistivity curves. This is likely because unconstrained inversion strategies S1 and S2 do not contain the information necessary to accommodate high geo-electrical contrasts in their prior models. These high geo-electrical contrast interfaces are accurately located in the seismic volume, and this information is transferred to the MT inversion by different methods in strategies S4–S9.
A different way to assess the nine inversion strategies is to compare apparent resistivity and tipper values of the real field and synthetic data at each period for all MT stations that were included in inversion strategies S1–S9. In Fig. 18 we show a map of apparent resistivity and tipper at period of 0.092367 s for the nine inversion strategies. Figure 18 shows that the model apparent effective resistivity for all strategies is similar to the field values. However, Vozoff tipper directions at the measurement area edge tend to diverge from the tipper derived from the field data.
Figure 19 shows residuals between synthetic and field data at the same period for the nine cooperative inversion strategies. This provides a better diagnostic of the quality of the fit between modelled and real data than the global RMS value. Localised, high misfit areas can be readily identified.
The per cent residual maps for strategies S1 and S2 show distinct localised high residual areas (i.e. localised high misfit) when compared to residuals from strategies S3–S9. Figure 20 shows residuals along a section mark as AB on the survey map. Again, per cent residuals for strategies S1 and S2 exhibit localised zone of exceedingly poor fit. The red and dark blue zones in Figs. 19 and 20 represent over 20 % residual error between synthetic and real data for the absolute (abs) value of Zyz. The light blue to light green on these images represents areas of good fit between synthetic and field data after inversion. In the section views provided in Fig. 20 localised misfit is substantially reduced for strategies S7 and S9. The simple message is that misfit can and should be assessed throughout the full data volume.
Assessing Inversion Strategies S1–S9: Conductivity Distribution
Figure 15 shows images of the final electrical conductivity distributions after inversion for strategies S1–S9. Each strategy uses a different prior model conductivity and/or covariance coefficient to seed the inversion (see Table 1; Fig. 14). In all cases (S1–S9) final inversion resulted in significant reduction in RMS error and a substantial change to the conductivity distribution compared to the prior model. Gross distribution of electrical conductivity is broadly similar for all strategies and clearly represents more conductive cover over basement. However, a closer review of the nine images leads to a salient conclusion: in detail, all strategies yield different 3D conductivity distributions and as usual a “good fit” is a necessary but not sufficient condition for the actual subsurface conductivity distribution to be recovered by inversion.
The unconstrained MT inversions using half-space prior models S1 and S2 result in a highly smoothed representation showing a usual “blurred” image of conductive cover against the higher-resistivity basement. There is little detail in either cover or basement rock volumes and certainly no sharp boundaries for these unconstrained inversions. These results are typical and often accepted outcomes from the application of the MT method. We know these conductivity distributions must be incorrect, as sharp geo-electrical boundaries certainly exist (i.e. from wireline logs, logged geology, core geochemistry and seismic reflectivity).
Semiautomatic inversion strategy S3 uses a prior root mean square covariance of 0.25 across key seismic boundaries along with a 40-Ω m half-space prior model. Lowering the covariance coefficient across the boundaries appears to play a minor role in directing the outcome of the inversion. While the lower covariance permits rapid change in electrical conductivity at the high-reflectivity boundary, it does yield a significantly different result to strategy S2, which has its covariance coefficient set to 0.3 everywhere.
Automatic cooperative inversion strategy S4 is interesting in that the prior model is initialised via direct mapping of a seismic attribute to electrical resistivity for all cells in the MT model domain (see Table 1). This type of “direct mapping” has the advantage of being straightforward to implement. However, a distinct risk in applying such a strategy is that it requires an inferred link between conductivity and the selected seismic attribute. Even if this is based on wireline log data, such a relationship may only apply locally.
For the Nevada example, the result for S4 is surprisingly effective. In particular, S4 was the only strategy to recover possible linear features that may be associated with faulting in basement. What we suspect is that faulted zones in basement have a recoverable dip angle attribute and so are mapped to slightly higher electrical conductivity in the prior model compared to basement rock. A close look at the far left of the image representation of strategy S4 in Fig. 14 reveals slightly higher conductivity along linear seismic features. The global RMS misfit and percentage residual images (Figs. 16, 19, 20) indicate good model fit to data. In the absence of information to the contrary, we must consider the conductivity distribution generated from automatic cooperative inversion strategy S4 as a possible and valid representation. We highlight this example because, for exploration, fault systems are often of high significance. Representing their existence within the prior geo-electrical model may permit full inversion to reveal important information concerning subsurface rock properties proximal to faults.
Strategies S5 and S6 use the dip angle seismic attribute to assist in construction of the geometric framework for the prior conductivity model. For automatic cooperative inversion strategies S5 and S6, we have used geostatistical methods to restrict the number of “index groups” to five, which ultimately means that the large-scale seismic structures are retained and represented in the prior geo-electrical model by a small set of discrete conductivities (see schematic Fig. 10). In contrast to discrete conductivity values filling each subvolume for strategies S5 and S6, the prior conductivity model derived from direct mapping used for strategy S4 permits continuous variations in conductivity that follow the seismic character of the model domain.
Semiautomatic cooperative inversion strategies S7, S8 and S9 result in considerable vertical and lateral detail in conductivity distribution in cover. What is particularly exciting about these results is that the final inversions (S8 and S9) created detailed conductivity distributions that follow geometries clearly in the seismic reflectivity that were not included in the prior model. That is the inversions have “found” clear geo-electrical units within the framework of the prior model that appear to match the detail of the reflectivity image. S7 results in the smoothest image due to the higher, default covariance boundary value of 0.3 and the smooth prior conductivity model. Strategy S9 provides the sharpest image and the greatest contrast in conductivity between key subvolumes. It includes a prior model with sharp conductivity changes at boundaries and boundary-defined covariance that will enable the inversion process to maintain these sharp boundaries (see Table 1).
Figure 21 provides a conductivity slice at 466 m below average ground level with an intensity overlay from the 3D seismic volume at the same depth. The prior conductivity model for strategies S8 and S9 are identical and are presented at depth 466 m in Fig. 21a. At this depth slice the prior model conductivity distribution consists of just 3 conductivities. Small arrows in each image point to selected prominent seismic reflections that are not incorporated in the prior model for S8 or S9. We observe that the application of semiautomatic cooperative inversion strategies S8 and S9 reveal detailed conductivity distributions that correlate with seismic information not provided to the prior model.
Semiautomatic cooperative inversion strategy S9 differs from S8 in that the boundary or covariance coefficient is reduced at the prior model boundaries. What this does is to allow more rapid changes of conductivity across the boundaries. The outcome is that several features close to high-contrast boundaries may be better resolved in S9 compared to S8. Such features are represented by yellow and tan arrows in Fig. 21c. Certainly, the relatively small feature highlighted by Fig. 21c matches the seismic and appears to be accurately resolved by strategy S9. It is difficult to fully appreciate the detail resulting from strategy S9 in 2D images so 3D representation of conductivity distribution of S9 is provided in the “Appendix”.
We suspect that strategy S9 is the most likely to represent the true subsurface conductivity distribution in cover because the results are consistent with (1) the wireline log data, (2) seismic data, (3) geochemical information and lithological information. We would also suggest that recovery and resolution of subtle conductivity features proximal to the large faults is substantially enhanced by changing smoothness constraints across the fault, which is accurately located in the seismic reflection data.
Next we will compare automatic cooperative inversion strategy S6 and semiautomatic cooperative inversion strategy S9 with available drill-hole information. Figure 22 shows a comparison between “smooth” strategy S6 and “sharp” inversion strategy S9. The 3D image representation includes a depth slice of 560 m below average surface level. The final conductivity distribution after application of strategies S6 and S9 is overlaid on the 3D seismic image. Detailed information at drill hole 1 is provided in the left-hand panel.
Within cover, conductivity distributions derived from strategies S6 and S9 broadly match with lithology and geochemistry information (see Sect. 3.3). However, in detail semiautomatic inversion strategy S9 results in discrete clear conductivity zones that appear to give a superior match to the lithological and geochemical rock groupings. Also, strategy S9 recovers the correct resistivity in basement according to the wireline log. Strategy S9, which includes definition of a specific covariance value across key boundaries (dashed lines), also provides the closest agreement with average basement resistivity as recovered from induction logging (approximately 100 Ω m).
A key observation from the lithological and geochemical groupings for hole 1 (Fig. 22) is that at several locations they simply do not match. Further, we should not expect that the distribution of pore water chemistry is necessarily matched to lithological or geochemical groupings. As noted earlier the distribution of solute concentration will be determined by both local effects and large-scale basin hydrodynamics. These are some of the reasons why strict forms of joint inversion (e.g., cross-gradient methods) or cooperative inversion-based direct mapping of seismic to electromagnetic properties may fail to generate a sensible outcome.
Figures 23 and 24 show a comparison of drill holes’ 1 and 2 information with conductivities extracted from final inverted results from all strategies. These are included to highlight the range of possible solutions generated by inversion strategies S1–S9 at the two well locations. Table 3 provides the key observations from these strategies.
All strategies other than S7, S8 and S9 fail to recover conductivities that match wireline log conductivity in the basement with S2, S3, S4 and S6 being the main offenders. However, S4 did provide a level of detail proximal and below the basement not present in other strategies and in this sense represented valuable information about conductivity distribution now not present in other outcomes. A message from our analysis is that many possible inversion outcomes (i.e. strategies) must be considered when interpreting MT data. The information that is desired may not exist because of the inversion strategy selected.
For cooperative strategies S4–S9, outcomes show details consistent with the seismic image. The inversion process refined the conductivity distribution to match remarkably well within the detailed seismic reflectivity image not included in the prior model.