Published in Vadose Zone Journal 3:1215-1229 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION: HYDROGEOPHYSICS
Geostatistical Reconstruction of Gaps in Near-Surface Electrical Resistivity Data
Daniel Cornacchiuloa and
Amvrossios C. Bagtzogloub,*
a Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027
b Department of Civil and Environmental Engineering, University of Connecticut, Storrs, CT 06269-2037
* Corresponding author (acb{at}engr.uconn.edu)
Received 30 January 2004.
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ABSTRACT
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This study was motivated by the need to reduce the effects of various types of noise observed in geophysical field data. We focused on assessing the impact of noise and data gaps on electrical resistivity data and on evaluating whether geostatistical methods (in this case kriging) can be successfully used for restoring missing data before inversion. We used electrical resistivity forward and inverse modeling with a simple fault earth model to produce and invert synthetic datasets. We examined the effects of random background noise, data density deletion, and data gap and noise structure scenarios to study the influence of these factors on the inversion of resistivity data and the subsequent interpretability of the geologic structure. Our results suggest that geostatistical methods are potentially very useful for restoring data points deleted from noisy resistivity field data. Clearly, the efficacy of kriging depends on the level of noise and the amount of data deleted. The inversion RMSE of the kriged files is less than that of the original random background noise files containing all data. The magnitude of the improvement increases as random background noise increases. Even for cases where 80% of the original data were randomly eliminated and there was 10% random background noise, the kriging procedure resulted in significant improvement in the ability to resolve the basement and overburden structure, correctly place the orientation and location of the fault, and identify the downthrown block. At random background noise levels of 20 and 30%, kriging was effective at recovering the major geological features but to a lesser degree. The efficacy of the kriging procedure performed on the noisiest data appears to be a function of the location and magnitude of data gaps induced by editing or missing strings. Finally, the effect that coherent noise has on the efficacy of our approach was studied and contrasted to random deletion. Our study suggested that the geostatistical restoration approach improved the interpretability of electrical resistivity data that had been degraded by noise or data loss problems.
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INTRODUCTION
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EARTH RESISTIVITY PROFILING and sounding methods have been used to image earth structure since the first successful experiments by Conrad Schlumberger in 1920 (Van Nostrand and Cook, 1966). The electrical resistivity of earth materials in the subsurface is measured using a surface "array" of electrodes. For a given array, the voltage is the difference between the measured electrical potential at the potential electrodes resulting from the current injected into the ground through the current electrodes. The voltage values along with the array geometry provide information on the electrical resistivity of earth materials within the zone of influence of the array. As the offset between electrodes increases (and the array gets larger), the volume of earth investigated by the array increases, providing information about the earth resistivity distribution at progressively greater depths. Two-dimensional resistivity surveys are acquired by profiling along a section of the surface with successively larger linear arrays. The two-dimensional DC-resistivity profiling data can be interpreted to create a model or "image" of subsurface resistivity along the profiled section. Two common types of linear arrays used for two-dimensional resistivity imaging surveys are the: (i) dipole–dipole and (ii) Wenner–Schlumberger. Both arrays have useful specific attributes. For example, the dipole–dipole array has good horizontal resolution whereas the Wenner–Schlumberger array has good vertical resolution (Loke, 1997; Ward and Hohmann, 1987).
While applicable to a wide range of problems in the mining, petroleum, groundwater, and geotechnical engineering arenas, until recently the method has been overshadowed by the more rapid, noncontact, electromagnetic imaging methods. However, starting in the early 1990s, advances in data-acquisition hardware, computers, and data-interpretation software led to a rebirth in the use of electrical resistivity methods (Dahlin, 1996). Advances in data-acquisition systems increased the number of data that could be acquired in a typical field day by several orders of magnitude. The rapid measurement of subsurface electrical resistivity along profiles (rather than the classic one-dimensional sounding approach) led to the development of robust two-dimensional (and higher) data analysis algorithms that could rapidly invert electrical resistivity data using personal computers. Inversion of two-dimensional electrical resistivity data provides models of the lateral and vertical distribution of earth electrical properties that can be viewed as electrical images of the subsurface. For near-surface earth and environmental engineering applications, the images can provide insight into subsurface structure of interest to hydrogeologists and engineers (Loke and Barker, 1995; LaBrecque et al., 1996, 1998, 1999; Meads et al., 2003) including identification of the depth to bedrock, thickness of saturated materials, variation in grain size, moisture content differences, aquifer testing, location of clay layers and lenses or fracture zones, and the presence of high salinity groundwater (Urish, 1983; Rodrigues, 1984; Ross et al., 1990; Yeh and Liu, 2000; Yeh et al., 2002; Liu et al., 2002).
Concurrent with advances in the theoretical and practical development of two-dimensional resistivity imaging methods is the need to develop methods to reduce the effects of noise observed in the field data or missing data strings due to, for example, malfunctioning electrodes. Sources of noise include natural noise (e.g., atmospheric noise from electrical storms and near-surface inhomogeneities), cultural or anthropogenic noise (e.g., electric and electromagnetic noise from cathodic protection systems and power lines), and noise induced by poor electrode contacts and hardware faults. One approach to identify and eliminate noise is by visual inspection. Because earth resistivity is generally conceptualized to vary "smoothly" within a single geologic unit, large differences in the apparent resistivity of adjacent data points ("spiky" data) are generally suspect and are often removed from field data before inversion through simple editing. Both approaches of eliminating data points create spatial gaps in the field data that affect the inversion process. In highly noisy environments, large amounts of data might be removed, which raises concern about the validity and utility of the resulting earth models. Therefore, approaches that may provide a rational way to restore the deleted data before data inversion are needed. Geostatistical methods have been successfully applied before in either filling the gaps or providing the necessary continuity for a variety of geophysical techniques (e.g., Parks and Bentley, 1996). In this study, we applied such approaches to degraded electrical resistivity data, and evaluate the efficacy of the reconstruction approach.
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OBJECTIVES AND APPROACH
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The purpose of this study was to (i) investigate the effect of random background noise and random and coherent data deletion density on the inversion of two-dimensional resistivity profiles and (ii) test the utility of the geostatistical technique of kriging to restore data eliminated from noisy two-dimensional electrical resistivity profiles.
The study was conducted following the general approach of Oldenburg and Li (1999) and employed a combination of electrical resistivity forward and inverse models and project specific software written in MATLAB. Synthetic data were generated using RES2DMOD in block-based finite difference mode (Loke, 1997). RES2DMOD solves the resistivity forward model problem over user-defined earth models to generate synthetic two-dimensional electrical resistivity data. Zhang et al. (1995) presented a detailed discussion of the complete forward modeling formulation and the effect of different boundary conditions on the solution. For this study, synthetic data were generated for the commonly used Wenner-
array (Van Nostrand and Cook, 1966) profiled over a three-layered earth model, which has horizontal strata that are offset by a vertical fault (Fig. 1) . In this figure, the electrodes are depicted as white crosses at the ground surface and are at equidistant separations of 5 m. Note, however, that we show every electrode for the first five electrodes only and every fourth beyond this point for clarity in visualization purposes. We recognize that dealing with real data sets presents challenges unaccounted for in our analysis. However, before significant effort is expended applying the geostatistical approach to data derived from field campaigns, analyses such as the one presented here can provide benchmarks for future investigations.
A MATLAB code was developed to generate the synthetic data sets to study the effects of random background noise addition, data deletion level and structure (i.e., coherent vs. random), and the use of kriging to restore data deleted from the synthetic data sets. The code allows the user to specify the level of random background noise (sampled from a uniform distribution; however, other distributions can be readily implemented) and percentage of data deletion from the raw synthetic files. Subroutines in our code permit the user to select different variogram models (exponential, Gaussian, linear, and spherical) to fit the data before the implementation of kriging. Variography was conducted on a layer-by-layer basis and the optimal variogram model and its corresponding parameters were then used in one-dimensional ordinary kriging to recover the deleted data points again on a layer-by-layer basis. Even though the addition of random background noise introduced a nugget, this effect was found to be negligible below the fourth layer (corresponding to a depth of 12 m) for all cases tested. Because the interface between the first and second strata and the top of the upthrown block are located below a depth of 15 m, we opted to not include the variogram nugget as a parameter to be optimized in the rest of our analyses.
The electrical resistivity files used in the inversion were created by superimposing random background noise on the apparent resistivity result from the forward model or by deleting data points obtained from the forward model in a random or coherent manner. The latter represents coherent or systematic noise that is commonly encountered during acquisition such as poor electrode connection or incorrect cable connection, which typically forces the analyst to ignore (delete) a portion of the data set related to the electrodes in question.
RES2DINV (Loke, 1997) was used to obtain an earth model for each of these data sets. The iterative inversion routine implements (among several others) the L2-norm, smoothness-constrained least-squares method described by deGroot-Hedlin and Constable (1990) and Sasaki (1994). The blocky L1-norm constraint has also been found to perform favorably compared with the L2-norm (Claerbout and Muir, 1973; Zhou and Dahlin, 2003), especially in cases where there exist large contrasts across geological boundaries or with high noise levels (Loke et al., 2003). In our test cases we have not found this to be extremely critical as we have attained very similar performance values in both tests (e.g., RMSE of 6.7 and 5.5% for the L2- and L1-norm, respectively). It should also be noted that the L1-norm results we obtained provided a much better identification of the upthrown horizontal strata and subvertical fault, but a significantly poorer identification of the downthrown strata.
The inversions were first conducted on files with a range of random background noise and percentages of deleted points. The randomly deleted points for each data set were restored using our numerical code through a standard kriging procedure; after kriging the data sets were inverted using RES2DINV and the results compared with the original data sets. It is well recognized that resistivity field data suffer from coherent noise (e.g., electrical noise or a hardware fault that affects only one or two electrodes). Consequently, editing a data set containing such noise can induce significant nonrandom, highly patterned data gaps. This analysis is presented in the last section.
The effects of random background noise, data deletion, and the efficacy of kriging were determined by: (i) comparing the change in inversion RMSE as a function of the changing parameter (percentage noise or data deletion) and (ii) examination of the geoelectrical structure and comparison with inversions under different conditions to the baseline data set (no added noise or deleted data). Similarly, the kriged data sets were compared with inversions with the equivalent amount of random background noise and the prekriged data inversions. For the inversion of each data set, three model iterations were permitted. The software used for the inversion (RES2DINV) automatically calculates the RMSE, which is routinely used for characterizing the efficacy of the inversion and optimization method intercomparison (Loke and Dahlin, 1997, 2002). Therefore, the RMSE was used in this study as an indicator of success in the inversion process, despite that one could easily use the variance and/or bias in the residuals. However, for the sake of completeness, we tested several of the test cases presented in this work using both indicators and determined that our conclusions would not change if residual statistics were used instead of RMSE statistics.
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RESULTS
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Synthetic Data: Test of Random Background Noise Effects
Inversions of the synthetic fault model data were made after the addition of 10, 20, 30, 40, and 50% random background noise. It is recognized that most field data sets have random background noise of <10%, particularly with the Wenner- array, which is fairly robust. Very high noise levels of more than 20% are more common with arrays such as the dipole–dipole because of the very low potentials measured. However, we included these higher noise levels in our study to test our concepts and algorithms. The inversion RMSE generally increases linearly with the percentage of random background noise, as will be discussed in more detail in the last section.
In Fig. 2, 3, 4, and 5
inverted results from the noise-free, 20, 30, and 50% added random-noise profiles are shown, with Parts a and b in each figure corresponding to the measured apparent resistivity and inverse model resistivity, respectively. In general, the resistivity inversion procedure tolerates the addition of significant amounts of random background noise before the underlying essential geoelectric structure is obscured. The addition of 10 to 20% random background noise resulted in the development of spurious resistivity anomalies in the near-surface layers, but had a negligible effect on resolution of the geoelectric structure. As random background noise increases, the effect of the spurious resistivity anomalies becomes more severe and resolution of the basement structure is degraded. For the range of noise levels tested, the upper noise limit where the basement structure could be reasonably interpreted was 30% Moreover, Fig. 4 shows results of two realizations of 30% random background noise.
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Fig. 2. Results of inverting synthetic data generated from fault model (noise-free data set, no data deleted). Fault contact is located at about 140 m and downthrown block is to the right. (a) Measured apparent resistivity, (b) calculated apparent resistivity, and (c) inverse model result.
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Fig. 3. Results of inverting synthetic data generated from fault model (20% random background noise added, no data deleted). Fault contact and downthrown block are clearly recognizable. Spurious resistivity anomalies are beginning to obscure the continuity of the overburden layer. (a) Measured apparent resistivity, and (b) inverse model result.
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Fig. 4. Two realizations of synthetic data generated from fault model containing 30% random background noise (no data deleted). Basement structure and overburden layers are distorted, but the fault contact is correctly placed and downthrown block is present. (a) Measured apparent resistivity, and (b) inverse model result for the first realization. (c) Measured apparent resistivity, and (d) inverse model result for the second realization.
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Fig. 5. Results of inverting synthetic data generated from fault model (50% random background noise, no data deleted). Basement structure and overburden layers are severely distorted. Downthrown block is missing. (a) Measured apparent resistivity, and (b) inverse model result.
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Noisy Synthetic Data: Test of Data Deletion Effects
Synthetic data files containing 10 to 30% random background noise were used to assess the effects of data deletion density on resistivity inversions. Figure 6
shows the effects of data deletion on the resolution of the geologic structure for random data point deletions ranging from 20 to 60% The inversions shown in Fig. 6 contain 10% random background noise.
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Fig. 6. Results of inverting synthetic data generated from fault model containing 10% random background noise. The effect of random deletion of data points is shown for (a) 20%, (b) 40%, and (c) 60% deletion density.
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For the fault model tested, the basement, overburden structure, and downthrown block are observed in the inverted resistivity sections at the 20 and 40% data deletion levels. Resolution of vertical fault structure and location degrades somewhat after the 40% data deletion level is reached. The results of the data deletion for files containing 10% random background noise shown in Fig. 6 indicate (for the simple three-layered earth model used for the synthetic experiment) that two-dimensional resistivity profiles containing moderate amounts of noise are relatively robust and can tolerate very high levels of data loss while preserving essential geologic information necessary to interpret inverted resistivity profiles. Figure 7
depicts randomly deleted data points (circles) for one realization corresponding to 10% random background noise and 20% deletion density.
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Fig. 7. Measured apparent resistivity with 10% random background noise showing 20% of randomly deleted data points.
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Figures 8 and 9
show the effects of 20, 40, and 60% data deletion for synthetic data containing higher amounts of random background noise (20 and 30%, respectively). As the random background noise level increases, the level of data deletion that results in serious degradation of the resolution of basement and overburden structure and downthrown block decreases. For the synthetic data files containing 20% random background noise, inversion interpretability is lost when the 60% data deletion level is reached. For the synthetic data files containing 30% random background noise, inversion interpretability is lost when the 40% data deletion level is reached.
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Fig. 8. Results of inverting synthetic data generated from fault model containing 20% random background noise. The effect of random deletion of data points is shown for (a) 20%, (b) 40%, and (c) 60% deletion density.
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Fig. 9. Results of inverting synthetic data generated from fault model containing 30% random background noise. The effect of random deletion of data points is shown for (a) 20%, (b) 40%, and (c) 60% deletion density.
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These results indicate that a method capable of restoring data deleted from noisy resistivity field data might be useful in situations where noise-contaminated data require the deletion of certain field measurements or there are data strings missing; the method could be increasingly valuable as noise levels and the percentage of deleted data increases.
Noisy Synthetic Data: Test of Kriging Efficacy to Restore Randomly Deleted Data
On the basis of the results of the random background noise and data deletion tests, we chose synthetic data files characterized by 10, 20, and 30% random background noise with 10, 20, 40, and 60% data deletion levels for variography analysis and kriging. Our numerical code was used to perform variography and kriging by first fitting several different variogram models to a given data set and subsequently interpolating using kriging associated with the appropriate variogram model. For this study, exponential, Gaussian, linear, and spherical models were tested to see which model would best fit variograms produced from the synthetic data.
For the fault model used in this study, variography analysis was conducted for a range of noises and data deletion levels. The Gaussian model produced the best fit to the synthetic data variograms. After the completion of the variography, the kriging procedure implemented in our numerical code was used to restore data to files containing a range of noise and different levels of data deletion. Figures 10, 11, and 12
show the results of inversion after kriging is used to restore data files containing 10, 20, and 30% random background noise, respectively, for deletion percentages of 10, 20, 40, and 60%.
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Fig. 10. Results of inverting synthetic data generated from fault model containing 10% random background noise where kriging was used to restore (a) 10, (b) 20, (c) 40, and (d) 60% of randomly deleted data points. The last three panels can be compared directly with Fig. 6.
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Fig. 11. Results of inverting synthetic data generated from fault model containing 20% random background noise where kriging was used to restore (a) 10, (b) 20, (c) 40, and (d) 60% of randomly deleted data points. The last three panels can be compared directly with Fig. 8.
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Fig. 12. Results of inverting synthetic data generated from fault model containing 30% random background noise where kriging was used to restore (a) 20, (b) 40, and (c) 60% of randomly deleted data points.
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In Fig. 10 and 11, the effects of kriging on synthetic data containing different levels of random background noise and a range of data deletion levels differ depending on the level of noise and the amount of data deleted. For the lowest noise level test (10%) the kriging method resulted in a substantial restoration of the geologic structure observed in the original data files compared to the inversions performed on files where the data had been deleted (Fig. 10). Even in cases when 80% of the original data were eliminated (results not shown), the kriging procedure resulted in significant improvement in the ability to resolve the basement and overburden structure, correctly place the orientation and location of the fault, and identify the downthrown block.
As the random background noise level increased to 20 and 30% kriging remained effective but to a lesser degree. For example, in Fig. 11 application of kriging to the data set where 60% of the data was eliminated resulted in an inversion that restored the location and orientation of the fault, but distorted the depth and structure of the downthrown block. In Fig. 12, the kriging procedure failed to significantly improve the inversion of the data set where 40% of the data points were deleted. The inversion bears little resemblance to the modeled geologic structure. Conversely, in the same figure, application of the kriging procedure to the data file where 60% of the data points were deleted resulted in an inversion where the fault location and orientation was improved yet the structure of the downthrown block was not.
The efficacy of the kriging procedure appears to be a function of the location and magnitude of the data gap. Qualitatively, where noise levels are high, large gaps of data that occur where the gradient of the field data is high are not restored effectively.
Noisy Synthetic Data: Test of Kriging Efficacy to Restore Coherently Deleted Data
Coherent or systematic noise, which may be the result of electrical noise or a hardware fault that affects only one or two electrodes, is quite common and can distort all of the measurements that involve a given electrode. The effect of such noise can differ depending on whether the electrode is being used to inject current or measure electrical potential. Consequently, editing a data set containing such noise can induce significant nonrandom, highly patterned data gaps. For the following tests we selected a minimum of four and a maximum of eight electrodes near the left boundary of the model (upthrown block) to "suffer" from coherent noise. These scenarios are equivalent to 10 and 20% of the data points been deleted, and they are comparable with some of the random deletion tests conducted earlier. The only difference is that in the coherent noise the deletion is concentrated near the left boundary of the model, while the random deletion is distributed throughout the model. Figure 13
shows an example of the measured apparent resistivity with the spikes in resistivity due to coherent noise. The following tests were conducted for two different scenarios of electrode deletion. The first scenario had four electrodes deleted (Electrodes 3, 7, 11, and 15), which corresponded to 10% of the data points being eliminated. The second scenario had a 20% deletion density, through deletion of Electrodes 3, 7, 8, 11, 12, 15, 16, and 17.
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Fig. 13. Measured apparent resistivity results with coherent noise from Electrodes 3, 7, 11, and 15 resulting in a 10% data deletion density.
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Figure 14
shows that very little difference exists between the inferred resistivity model under coherent and random deletion schemes for moderate background noise and data deletion density levels (both equal to 10%). The RMSE is only slightly higher (5.8% as opposed to 5.5% for the coherent noise case). However, increasing the amount of deleted points from 10 to 20% while also increasing the random background noise from 10 to 20% indeed has a significant effect on the inversion model. The stratum on the left side of the fault is not as well defined for the coherent noise (Fig. 15a)
as it is for the random deletion case (Fig. 15b). This is attributed to the high density of deleted data points in this region for the coherent noise case. Moreover, the upthrown block appears to be subhorizontal, and the fault is clearly subvertical. Similarly, the coherent deletion scheme affects the downthrown block by distorting it and making it appear subhorizontal. The RMSE metric also increases from 10.7% for the random deletion to 10.9% for the coherent noise deletion. However, this difference is considered minimal, especially given the fact that there exists variability from one realization of deleted points to the next that is unaccounted for. This will be addressed in the following section.
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Fig. 14. Inversion results when kriging was used to recover 10% of the data points deleted with 10% random background noise. Data points deleted are either (a) affected by Electrodes 3, 7, 11, and 15 (coherent noise) or (b) randomly selected. Note that Fig. 14b is identical to Figure 10a and is presented here for ease of comparison.
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Fig. 15. Inversion results when kriging was used to recover 20% of the data points deleted with 20% random background noise. Data points deleted are either (a) affected by Electrodes 3, 7, 8, 11, 12, 15, 16, and 17 (coherent noise) or (b) randomly selected. Note that Fig. 15b is identical to Fig. 11b and is presented here for ease of comparison.
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Effect of Random Background Noise on Kriged Inversion RMS for Random and Coherent Deletion Structures
One of the goals of the study was to determine if kriging would be useful for restoring data edited from noisy field data sets. The previous sections discuss the effects of random background noise, data deletion density and structure, and kriging on the resolution and interpretability of the geologic structure. Another way to examine the effects of kriging is shown in Fig. 16
, which provides a comparison of inversion RMSE of the noisy and kriged files for the case of coherent and random deletion schemes. There are results from 33 simulations presented in each panel of Fig. 16. Six runs correspond to coherent noise deletion for random background noise levels of 0, 10, and 20% and data deletion densities of 10% (Fig. 16a) and 20% (Fig. 16b). These are shown as white squares. For the same set of parameters a total of 10 realizations of random data deletion for each data deletion density are analyzed and relevant statistics are shown as black squares (mean) and whiskers (three standard deviations). A statistically significant, least-squares based, optimal linear fit for the coherent noise data is also shown for completeness.
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Fig. 16. Mean inversion RMSE vs. percentage of random background noise with (a) 10% and (b) 20% of data randomly and coherently deleted and restored through kriging. The mean RMS and three standard deviation error bars have been calculated over 10 realizations of random point deletion.
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On the basis of these results we conclude that, consistent with the qualitative observations made in the previous section, there exists little difference between the coherent and random deletion cases in regard to the efficacy of kriging for resistivity reconstruction. However, it should be noted that this difference is consistently manifested as a relative deterioration of interpretability of the geological structure for the coherent noise deletion.
Finally, Fig. 17
provides yet another way of comparing the effects of random background noise addition and kriging. The original synthetic data is bimodal with a sharply defined range. Addition of random background noise increases the range of the data, and changes the bimodal distribution to one that is approximately unimodal with a positive skew. Use of the kriging procedure to restore the data increases the range, and restores the bimodal distribution, although the peak frequencies are shifted slightly from the original distribution.
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Fig. 17. Smoothed histograms showing the effects of random background noise addition and kriging. Compared are the original fault model synthetic data, data containing 30% random background noise, and data containing 30% noise after 40% of data have been randomly deleted and restored by kriging.
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SUMMARY AND CONCLUSIONS
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Recent advances in the theoretical and practical development of two-dimensional resistivity imaging approaches have led to an increased use of resistivity methods for near-surface earth imaging applications. The need to reduce the effects of various types of noise observed in the field data motivated this study to determine if the geostatistical method of kriging could be used to restore noisy field data eliminated from source files before inversion.
We used resistivity forward and inversion modeling (for a simple fault earth model) to produce and invert synthetic data sets that were generated using a MATLAB program developed for this study. The effects of random background noise, data deletion density levels, and noise/gap structure were examined to determine how these factors affect the inversion of resistivity data and the subsequent interpretability of the geologic structure.
The addition of random background noise increases inversion RMSE. The relationship between random background noise and RMSE appears to be linear for the tested model. The resistivity inversion procedure tolerates the addition of more than 20% random background noise before the essential geoelectric structure is obscured. As random background noise levels increase, spurious resistivity anomalies of increasing severity develop, and the resolution of the basement structure is degraded. For the range of noise levels tested, the upper noise limit where the basement structure could be reliably interpreted was 30%
The effect of data removal on the resistivity inversions appears to be correlated with the level of random background noise present in the data. For the fault model tested, inversion of files containing moderate levels of noise (<20%), produced geoelectric sections where basement, overburden structure and fault geometry can be observed even when the majority of the field data is eliminated. For the case of 10% random background noise, interpretable sections were produced until 90% of the data were deleted, although resolution of the vertical fault structure and location degrades after the 40% data deletion level is reached.
As the random background noise level increases, the level of data deletion that can be tolerated before serious interpretation degradation is reduced. For the synthetic data files containing 20% random background noise, inversion interpretability is lost when the 60% data deletion level is reached. For the synthetic data files containing 30% random background noise, inversion interpretability is lost when the 40% data deletion level is reached.
Our numerical code enabled the testing of several different types of variogram models on the synthetic resistivity data, including exponential, Gaussian, linear, and spherical. The variography tests conducted before kriging indicate that the Gaussian model can be used effectively to describe the spatial variability of the fault model synthetic resistivity data.
The kriging method seems to be useful for the task of data restoration. However, the efficacy of kriging depends on the level of noise and the amount of data deleted. For files containing 10% random background noise, kriging results in a substantial restoration of the geologic structure observed in the original data files compared with the inversions performed on files where the data had been deleted. Even where 80% of the original data was eliminated, the kriging procedure resulted in significant improvement in the ability to resolve the basement and overburden structure, correctly place the orientation and location of the fault, and identify the downthrown block.
At random background noise levels of 20 and 30% kriging remains effective, but to a lesser degree. The efficacy of the kriging procedure performed on the noisiest data appears to be a function of the location and magnitude of data gap induced by editing or missing strings. Large gaps of data that occur where the gradient of the field data is significant are not restored effectively. However, we found little difference in both RMSE and interpretability of the geologic structure between the coherent and random deletion schemes in regards to the efficacy of kriging in the resistivity inversion and reconstruction process.
The inversion RMSE of the kriged files is less than that of the original random background noise files containing all data; the magnitude of the improvement increases as random background noise increases. The addition of random background noise and the kriging procedure have significant effects on the data. The original noise-free synthetic data generated for this study are bimodal with a sharply defined range. Addition of random background noise dispersed the data and altered the data distribution from a bimodal to unimodal distribution. Use of the kriging procedure to restore the data increased the dispersion further while appearing to restore at least a semblance of the original bimodal distribution.
Our results suggest that geostatistical methods might provide the means for restoring data deleted from noisy resistivity field data in cases where moderate noise levels and data deletion warrant such reconstruction.
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ACKNOWLEDGMENTS
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The authors wish to thank John W. Lane, Jr. of the USGS Office of Ground Water, Branch of Geophysics for numerous contributions during this research effort; without his help this work could not have been completed. Thanks also go to Koray Ergun who contributed to programming efforts in the early stages of this study. Constructive reviews provided by L.R. Bentley (University of Calgary) and L. Liu (University of Connecticut) on a preliminary version of this manuscript helped the authors improve the quality of the presentation of their work. Finally, the authors acknowledge the thorough reviews, comments, and suggestions received by two anonymous referees and the associate editor of the journal.
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