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Geostatistical Reconstruction of Gaps in Near-Surface Electrical Resistivity Data

^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

View larger version (20K): [in a new window]   Fig. 1. Diagram showing three-layer earth model with fault used for this study.

View larger version (36K): [in a new window]   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.

View larger version (30K): [in a new window]   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.

View larger version (62K): [in a new window]   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.

View larger version (32K): [in a new window]   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.

View larger version (51K): [in a new window]   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.

View larger version (12K): [in a new window]   Fig. 7. Measured apparent resistivity with 10% random background noise showing 20% of randomly deleted data points.

View larger version (51K): [in a new window]   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.

View larger version (56K): [in a new window]   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.

View larger version (57K): [in a new window]   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.

View larger version (58K): [in a new window]   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.

View larger version (53K): [in a new window]   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.

View larger version (15K): [in a new window]   Fig. 13. Measured apparent resistivity results with coherent noise from Electrodes 3, 7, 11, and 15 resulting in a 10% data deletion density.

View larger version (27K): [in a new window]   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.

View larger version (37K): [in a new window]   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.

View larger version (30K): [in a new window]   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.

View larger version (21K): [in a new window]   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.