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a Hydrology and Water Resources, Univ. of Arizona, Tucson, AZ 85721
b Soil, Water, and Environmental Sciences, Univ. of Arizona, Tucson, AZ 85721
c Inst. Soil, Water, and Environ. Studies, ARO, Volcani Ctr., P.O. Box 6, Bet Dagan 50250, Israel
* Corresponding author (alexf{at}agri.gov.il)
Received 12 May 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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Abbreviations: EC, electrical conductivity • ERT, electrical resistivity tomography • GA, genetic algorithms
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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Monitoring of hydrological problems differs in many ways from surveying for geological purposes. The most important factor is the need to consider the measurement time in hydrologic monitoring. Other differences include the spatial scales of investigation (primarily that shallower targets are of greater interest in most hydrologic investigations) and the required spatial resolution of the image (higher resolution is needed for hydrologic characterization).
Although the time required for a single ERT measurement depends on many parameters, including the properties of the subsurface at the time of measurement, it can be treated as a constant (e.g., 
15 s). The time window available for completion of a survey is based on the physical process to be monitored. That is, all measurements comprising a survey must be made rapidly compared with the rate at which the electrical conductivity changes in the subsurface. The time window divided by the time required per measurement defines the maximum number of arrays in a survey, A. Furthermore, the rate of change of the electrical conductivity, EC, may change with time, leading to tightening or relaxation of the time frame, and a change in the number of arrays comprising a survey.
Electrical resistivity tomography (note that for the purposes of this paper only surface electrodes are considered) has long been seen as a promising noninvasive, nondestructive method (Edlefsen and Anderson, 1941). Recent advances in ERT instrumentation and inversion methods have increased the use of ERT for hydrologic investigations (Barker and Moore, 1998). These improvements include the use of multicore cable, addressed electrodes, improved control and recording, and increased measurement accuracy, allowing for the use of very small currents (i.e., tens to hundreds of milliamps).
Despite the increased use of electrical geophysical methods, including ERT, applications are largely limited to monitoring static or very slowly changing conditions. Electrical resistivity tomography has been applied to mineral exploration (e.g., Griffiths and Barker, 1993), geologic mapping (e.g., Griffiths and Barker, 1993; Storz et al., 2000), groundwater table location (e.g., Yadav et al., 1997), and groundwater contamination mapping (e.g., Buselli and Lu, 2001). Barker and Moore (1998) showed that ERT could be used to monitor transient processes in the shallow subsurface. However, their examples are limited to processes that occur slowly, on the order of hours. An alternative to time-lapse monitoring involves coupling the inversion of geophysical data with the physical description of the monitored process (i.e., Richards' equation in the case of infiltration), using the approach of Seppanen et al. (2001).
As discussed above, the successful application of ERT for hydrological monitoring needs to address the requirement for rapid measurement, accuracy, and high resolution. Additional problems include the separation of the resistivity to its components (primarily water content, chemical properties of the water, and electrochemical properties of the porous or fractured media). This can be particularly difficult under unsaturated conditions, or when high solute concentrations are involved.
Increased accuracy can be achieved through noise reduction (Ritz et al., 1999), improved inversion algorithms, assimilation of data obtained through other means (Yeh et al., 2002), and optimized selection of arrays used to compose the survey (Furman et al., 2003). Improved inversion algorithms for time-lapse data (obtained when monitoring transient process) have been developed (e.g., LaBrecque and Yang, 2000; Kemna et al., 2002).
Improved monitoring of transient hydrologic processes requires improved measurement accuracy, and measurement that is more rapid. In addition, because ERT inversion is inherently ill posed (Sun, 1994), an increase in the number of measurements contributes to the overall ERT survey quality (although ERT inversion remains a strongly ill-posed problem). Therefore, when monitoring a transient process, the number of measurements should be as large as time allows. This is true also for monitoring static problems. Maximization of the number measurements can be achieved by reduction of the single measurement time, but this often causes an increase in the measurement error. Multichannel monitoring equipment (e.g., AGI, 2002) can further increase the number of measurements for a given time frame.
Typically, ERT surveys include only one type of array (e.g., Wenner arrays). Zhou et al. (2002) and Furman et al. (2003) suggested that a combination of different array types may increase ERT survey quality. Once the concept of a survey composed of a single array type is abundant, it is clear that survey optimization requires the choice of the optimal set of arrays to form a survey. We propose that improvements in instrumentation alone are not enough to allow for monitoring of more rapid processes with ERT. Rather, these improvements must be coupled with improved survey design to optimize information content.
Furman et al. (2003) (see also Barker, 1979; McGillivray and Oldenburg, 1990; Park and Van, 1991; Spitzer, 1998; Kemna, 2000) showed that each ERT array is characterized by a unique sensitivity distribution, and that this sensitivity distribution can be used to compare arrays directly. The sensitivity function may also be used directly in inversion process (e.g., Geselowitz, 1971; Murai and Kagawa, 1985; Kotre, 1994; Wang, 2002). We propose that these sensitivity distributions can be used to design an ERT survey with a user-defined optimal sensitivity. This could then allow for the design of the optimal survey for specific monitoring needs. Specifically, we present a method to identify the optimal set of arrays to form an ERT survey composed of a fixed number of measurements. The method makes use of genetic algorithms (GA) to identify the array set with the optimal cumulative survey spatial sensitivity based on user-defined criteria.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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ERT as an Optimization Problem
A typical ERT survey is comprised of a set of four-electrode arrays. For simplicity, surveys using a reduced number of electrodes are not considered here. Given that each array has a unique sensitivity and a unique sensitivity distribution, sets of arrays can be chosen that give optimal cumulative sensitivity based on a user-defined total number of arrays and a user-defined preferred sensitivity distribution. The purpose of this investigation is to develop a method to identify these.
The sensitivity, Sja (V), of an array, a, to a finite circular perturbation at location, j, which has an electrical conductivity contrast with the background, 
K (
–1 m–1), is defined here as the absolute change in apparent resistivity due to the perturbation, divided by the geometric factor and multiplied by the applied current (Telford et al., 1990). Sja can be calculated as
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The weighted cumulative sensitivity of an array can be defined as the weighted sum of the Sja values for a large number (J) of perturbations distributed throughout the subsurface:
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is a weighting factor. Through the use of these weighting factors, a common set of perturbation locations, distributed equally throughout the subsurface, can be used for all optimizations. That is, specific regions of the subsurface can be targeted by simply manipulating the weighting factors in these regions. For a "no preferences" optimization, = 1 for all perturbations.
The cumulative sensitivity of a survey, Sc, comprised of A arrays can then be defined as
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It is convenient to normalize this survey cumulative sensitivity by the number of arrays to allow for more direct comparison of surveys with different numbers of arrays. The mean survey sensitivity is then defined as
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If all perturbations are weighted equally, the mean survey sensitivity is
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Alternatively, one can sum the sensitivity at a point, j, across all arrays comprising a survey. In this way, a sensitivity map of a survey is achieved (assuming all arrays contribute equally). The survey sensitivity at a point j is then defined as
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Now, the standard deviation of the cumulative sensitivity, 
s, can be computed as
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As an example, consider a survey that is designed to find the most representative electrical conductivity (EC) of the subsurface using only 15 arrays. To achieve this, perturbations should be located uniformly throughout the subsurface and weighted equally when analyzing the survey sensitivity. Arrays with small electrode separations have sensitivities that are highly focused within the shallow subsurface. In contrast, wider arrays have moderate sensitivity over a much larger sample volume. As a result, it can be shown that wider arrays will have the highest cumulative sensitivities (Furman et al., 2003). Therefore, as may be expected, a survey comprised of 15 wide arrays will give the best measure of the average electrical conductivity of the subsurface.
The optimization approach described above does not take into account the spatial distribution of sensitivity of the optimal survey, but only its average value. Naturally, shallow regions will show higher sensitivity. A similar approach can be designed to identify the survey that gives the best representation of the electrical conductivity of the subsurface while maintaining the most evenly distributed measurement sensitivity. This becomes increasingly important with increasing heterogeneity of the subsurface electrical conductivity. To achieve greater uniformity of the cumulative sensitivity of all of the arrays comprising a survey, a component is added to the objective function that states that the standard deviation of the sensitivity values, S, must be minimized as well as maximizing the mean survey sensitivity. The measure of performance of a survey, Z, is therefore based on two components: the survey average sensitivity and the survey standard deviation. The general form of the objective function, formatted to a maximization problem, is then
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. Then, the 15 most sensitive can be chosen to form the survey. However, if ß is not one, the optimization requires a more sophisticated approach.
In general, weights (i.e., , ß) may be used to tailor a survey to achieve a specific desired distribution of the survey sensitivity. However, if a lesser degree of freedom is required in the design of a survey (i.e., no specific target in the subsurface is considered) it is convenient to use a normalized offset instead of the sensitivity, as described below. Weighting (i.e., ) still can be used with the offset to target specific zones within the subsurface. Considering all arrays that can be formed from a given set of electrodes, one array will have the maximum sensitivity to a perturbation at location, j, as described by Furman et al. (2002). The sensitivity of this array is referred to as SMAXj. We define the offset, eaj, of array a to a perturbation centered at point, j, as the difference between the sensitivity of array a and SMAXj, normalized by SMAXj:
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The use of the offset in the optimization objective function is identical to the use of the sensitivity in Eq. [2] through [5]. We use the notations Ea and Ec for array and survey cumulative offset as the equivalents of Sa, and Sc, respectively, and the mean survey offset, 
, in equivalence to . The objective function, Z, is defined in this case as
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As the offset is a normalized quantity, the standard deviation of the offset may be omitted, resulting in a simplified objective function:
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The solution to the optimization problem in this case is simply the selection of the A arrays showing the lowest array offset, Ea.
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METHODS |
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TOPABSTRACTINTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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Genetic algorithms have been applied previously in the field of water resources to find the optimal design of water distribution systems (e.g., Savic and Walters, 1997), optimal reservoir operation (e.g., Wardlaw and Sharif, 1999), and optimal groundwater management (e.g., Wang and Zheng, 1998). Genetic algorithms can be described as numerical optimization methods that simulate the evolutionary process (Holland, 1975). Specifically, a population is defined as a number of different chromosomes. In the case of ERT, several surveys are defined using different combinations of arrays; each survey is a chromosome. Each chromosome is comprised of a number of genes. (Most genetic algorithms use binary or scalar variables to represent genes). We introduce the use of a vectored description of electrode locations, with four numbers representing the electrode numbers used as the two current and two potential electrodes, as the gene. The optimization marches forward through successive generations until the objective function is satisfied within some convergence criterion. The population begins with N initial guesses. For ERT, the initial population may be chosen at random or based on some combination of Wenner, Schlumberger, double-dipole, or partially overlapping (CPCP configuration, Furman et al., 2002) arrays. Alternatively, the approach of Furman et al. (2002) can be used to define the locally optimized survey as an initial guess. Following the optimization procedure outline above, the surveys comprising the initial guess are ranked in order of their performance, Z. Mimicking natural selection, only the strongest (highest performance) chromosomes are carried forward into the next generation as an elite group (Coley, 1999). The remaining chromosomes are subject to a range of regeneration processes, or mutations, based on their relative fitness.
We distinguish between several mechanisms that may create (mutate) new chromosomes to enter the active "chromosome pool." Figure 1 illustrates the different mechanisms by which new chromosomes are created. The gray regions represent parts that are altered during the evolutionary process. A gray column indicates complete gene replacement. A gray rectangle indicates a single electrode that was replaced. The rate at which the GA will converge toward the global optimum is primarily determined by the rate that new chromosomes are created through these mechanisms.
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To demonstrate the application of a genetic algorithm optimization to the selection of an optimal ERT survey, we will search for the 15-array survey that gives the best representation of the electrical conductivity of the subsurface. One initial guess will be the locally optimized set determined using the approach of Furman et al. (2002)(.) A locally optimal survey is defined as a survey comprised of individual arrays, each of which is chosen based on having the highest array sensitivity to each of a number of individual perturbations. The contribution of this initial guess to the convergence rate will be examined. All other initial guesses will be comprised of random selections of electrodes.
We use a chromosome pool of fixed size, N. Four adaptive mechanisms and one "elite" group, described below, generate N1, N2, N3, N4, and N5 chromosomes at each generation, where N1 + N2 + N3 + N4 + N5 = N. In general, the size of the chromosome pool as well as the size of each of the subgroups may be changed at each generation. However, for simplicity, we use fixed values of N1 = N2 = N3 = N4 = N5 = 20.
The first group of N1 chromosomes is the group that shows the highest performance at any iteration (generation), which is termed the elite group. This group plays a crucial part in the GA, as members of this group are used to create some of the mutants that will enter the chromosome pool in the next generation. In evolutionary terms, these strong chromosomes are the only members to reproduce. The inclusion of such a group in the GA is called elitism. In this paper, chromosomes are chosen from the N1 group on a random basis. Random selection allows the variation in the size of the different groups, as well as preventing oscillations near a local optimum.
The process is repeated until a certain number of iterations (generations) have passed, or until no change is observed in the top-ranked survey. Then, the top-ranked survey is selected as the optimal survey set. In cases where the theoretical optimal value of the performance is known (but not the structure of the optimal solution), a numerical convergence criteria may be defined.
The First Adaptive Mechanism
is the random generation of new chromosomes. This generation of new initial guesses is one of the unique characteristics of GA that distinguishes it from many other optimization techniques. In this paper, we generate N2 new chromosomes at each generation (i.e., iteration), replacing N2 chromosomes from the previous generation.
All other mechanisms are essentially perturbations of existing chromosomes. This alternation of existing chromosomes is another aspect of GA that differs from other simple search methods. To speed convergence, chromosomes of group N1 are used as the bases for these alternations.
The Second Adaptive Mechanism
is complete gene perturbation. In each of the chromosomes in the N3 group some N3P (N3P = 2 in our example) genes (ERT arrays) are replaced by other randomly generated arrays. This keeps the majority of each of the best performing survey sets and changes only a few of the arrays composing it.
The Third Adaptive Mechanism
is the creation of new chromosomes through marriage. Two different parent chromosomes, typically (in this study strictly) from the highest ranking group, N1, are both broken at the same arbitrary point, and the four pieces are merged into two new chromosomes by switching the two "tails" of the original chromosomes. In this way, N4 new chromosomes are generated at every iteration. In general any part of the chromosomes may be switched, but in this study we limited the mechanism to tail switching.
The Fourth Adaptive Mechanism
is alteration of internal parts of the gene. Unlike previous mechanisms, where a complete gene was replaced, here only a few electrodes in some of the genes composing a chromosome are changed. In each of the chromosomes in the N5 group some N5P (N5P = 2 in our example) electrodes are replaced by other randomly generated electrodes. Through this mechanism, N5 chromosomes are generated at each iteration.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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Consider a two-dimensional ERT survey that consists of 21 electrodes, designed to determine the subsurface EC with no consideration, at this point, of the distribution of sensitivity (ß = 1). Electrodes are placed along a line with 1-m separations between the electrodes. Assuming that we have no information regarding the EC distribution in the subsurface, we evenly distribute 28 perturbations in a triangular region of the subsurface beneath the electrodes, approximately filling the region where most current flows when the outermost current electrodes are used (see Fig. 3). Note that all results may be scaled by the electrode separation.
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Twenty-one electrodes generate a gene space of 35910 arrays. The total number of 15-array surveys that can be composed with these electrodes is enormous. It is clear, then, that a systematic search for the optimal solution is not possible. Our algorithm only searches a maximum of 2.5 x 10 6 sets, which is a very small percentage of all possible sets. For the case of ß = 1 and uniform (with Eq. [8] as objective function), the optimal survey can be identified directly by choosing the 15 arrays with the maximal average sensitivity. That is, the sensitivity of each array to all perturbations is calculated and averaged over all perturbations. Next, arrays are sorted by their average sensitivities, and the highest ranked arrays are selected.
Figure 2 shows the evolution of the optimal survey performance through successive generations for several runs. The convergence rate of the algorithm depends on some arbitrary factors, such as the rate at which new arrays and new surveys enter the set, but also on the wisdom in the selection of these arrays (i.e., how well the adaptive mechanisms perform with regard to the specific problem). A statistical description of the distribution of sensitivity within each survey set may allow for a more informed selection of arrays, especially regarding which existing chromosomes (survey sets) are used to create new chromosomes. However, analysis of the performance of the GA is beyond the scope of this paper, and we limit ourselves here to demonstration of the applicability of GA to the identification of optimal ERT surveys. As the algorithm relies on random number generation (e.g., for gene alternation), each run is unique and cannot be repeated. Therefore, each value or graph presented here is a single realization of the optimization process. The performance of the true optimal survey is Z = 0.004568 (shown as dashed line in Fig. 2). In some realizations, the convergence toward the optimal solution may be immediate. It is important to note that the GA cannot a priori guarantee reaching the global optimum within a certain number of generations, but, as shown in Fig. 2, the routine reaches the vicinity of the optimal solution within a relatively small number of sets searched compared with the solution space size.
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In Fig. 2 three more realizations that include a locally optimal initial guess are also presented. Note that the initial advantage of those surveys diminishes during optimization.
Initially, the GAs rapidly approaches the optimum (note that generations are plotted in log scale). Further iterations introduce relatively limited improvement. The incorporation of a wise initial starting point leads to slightly more rapid initial converge, but the final solution (i.e., after 25000 iterations) is not improved. This suggests that the GA optimization is not sensitive to the initial guess. However, since the incorporation of the initial guess requires minimal computational effort, there is no reason not to include it in the algorithm.
An important point for practical applications is the computation of time required for optimization. Although the exact number of processor operations ("flops") was not measured, the following computational times were recorded for the runs presented in Fig. 2:
All calculations were all made on an Intel Pentium 4 machine with a processor speed of 2.4 GHz, and with 512 MB of RAM. Note that the computational effort for the first stage is more or less of order Ne4, where Ne is the number of electrodes (computational effort is a linear function of the number of arrays that can be made of Ne electrodes, which is of order Ne4). Computational effort increases approximately linearly with the number of perturbations.
Figure 3 (top) presents the sensitivity map corresponding to a pseudo-optimal solution (i.e., that was obtained after 25000 generations) for the 15-array survey with ß = 1. The sensitivity is distributed relatively homogeneously in the region where the perturbations are located, but very little sensitivity is associated with the deeper perturbations (the six deepest perturbations are within the 90% cumulative sensitivity region). Note that the contours in Fig. 3 through 5 represent percentages of the cumulative sensitivity, following Furman et al. (2003).
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s, we make the distribution of the sensitivity more homogeneous. Figure 3 (bottom) presents the sensitivity distributions of the 15-array optimal surveys using ß = 0.05. Both cases (i.e., ß = 1 and ß = 0.05) are presented "per measurement" (i.e., values are normalized by the number of arrays). Both solutions are presented after 25000 GA iterations. Note that as only 25000 GA iterations were used, and only 15 arrays are considered, the solutions are not as smooth as a global solution of a full-scale problem (e.g., Fig. 4) looks. Increasing the number of generations used would reduce this effect.
The survey sensitivity (Sj) distribution for the case of ß = 0.05 is clearly more homogeneous, as expressed by the standard deviation (3 x 10–6 for ß = 0.05 compared with 4.5 x 10–3 for ß = 1), and as can be seen clearly by comparing the 0.9 cumulative sensitivity line in both panes of Fig. 3. The fractions of the domain that contain the most sensitive 25, 50, 75, and 90% of the cumulative sensitivity are 1.0, 2.8, 7.6, and 17.9%, respectively, for the ß = 1 case, and 1.1, 3.2, 8.7, and 20.4% for the ß = 0.05 case. The increase in sample areas due to the use of standard deviation weight is not huge; however, it provides increased sensitivity mostly in the deep regions of the subsurface, leading to deeper investigation ability (in higher accuracy) of ERT.
The immediate conclusion is that the use of ß = 1 (i.e., optimization for average sensitivity only) may result in a survey that is relatively shallow in penetration. This depends also on the spatial distribution of the perturbations in the subsurface.
While the use of nonzero weight increases the uniformity of the sensitivity, it also reduces the absolute value of the average sensitivity. Maximal values of sensitivity for the ß = 1 case are about S = 0.0035, while for the ß = 0.05 case the maximal values are around S = 0.0025 (about 28% reduction).
Use of Offset in Objective Function
As discussed above, the objective function of the optimization problem may be simplified by using the normalized offset instead of the sensitivity. In such cases, the optimal solution may be obtained directly by calculating the average normalized offset of each individual array. (Although this requires large computational effort, it is still manageable even for large electrode sets, as shown by the above estimations of computational time.) Results presented in Fig. 4 and Table 1 make use of the offset in the objective function (Eq. [11]).
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For more direct comparison, we compare optimal surveys comprised of the same number of arrays as each of the other survey types (i.e., 63 for Wenner, 217 for Schlumberger). All results are obtained using Eq. [11] as the objective function.
None of the typically used survey types has a performance close to that of the optimal survey (Table 1). The partially overlapping survey is an improvement compared with the classic surveys, but is still far from optimal.
While performance is a quantitative measure, it is difficult to relate it directly to survey sensitivity, or even to survey resolution. Furthermore, although the performance is an easy to use comparative parameter, its numerical value has only comparative meaning given surveys are compared on equal basis (e.g., same targets, same resistivity model, etc.). Therefore, we compare directly the sensitivity distributions and sample areas for optimal and typical surveys. Figure 4 compares the spatial sensitivity distribution of the optimal survey with that of a Wenner survey. Shown are integrated sensitivity (i.e., summed over all arrays) and its distribution within the domain. For comparison with other surveys see Furman et al. (2003)(note that figures of Furman et al. are normalized by the number of arrays in each survey).
The sensitivity values are higher for the optimal solution than for the Wenner survey. The 25, 50, 75, and 90% sample areas of the optimal survey are much larger than those of the Wenner survey (2.2, 6.8, 15.8, and 30.9%, respectively, for the optimal case, compared with 1.1, 3.2, 8.6, and 20.4%, respectively, for the Wenner survey). This indicates that the sensitivity distribution of the optimal survey is more uniform than that of the Wenner survey (Furman et al., 2003), and that the sensitivity is higher throughout the domain.
Looking into the types of arrays composing the optimal survey (Fig. 4, Table 1) emphasizes the idea that an optimal survey is not to be composed of a single array type as are classic surveys. For the case of 63 arrays in a survey, and with ß = 1, the optimal set is composed of 18 Wenner–Schlumberger-like arrays (i.e., CPPC configuration, where C indicates a current electrode, and P indicates a potential electrode), 15 inverse Schlumberger-like arrays (i.e., PCCP configuration, which are essentially reciprocals of the Schlumberger-like arrays in terms of sensitivity distribution), and 30 partially overlapping arrays (i.e., CPCP or PCPC configuration). About the same ratio of array types is kept for wide range of survey sizes. None of the arrays composing the optimal survey is a double-dipole array (i.e., CCPP or PPCC configuration). The first double-dipole array comes to the optimal set only if 14559 arrays or more were used to compose a survey. Although it contradicts common practice, the absence of double-dipole arrays is reasonable because the current density near the potential electrodes is very small for these arrays, leading to low sensitivities for these arrays.
For all solutions presented in Fig. 4 and in Table 1, almost all electrodes are used, but outer electrodes are used more often, leading to clear preference of the large arrays. Small arrays show an offset close to one for all deep perturbations. As a result, the optimal survey uses only the outer electrodes. This may be an advantage in terms of cabling and electrode mobilization. However, we speculate that it is likely that measurements that do not use wide variation of electrode spacings will not lead to a unique, or stable, inversion of EC distribution. This is partially due to background noise, which may diffuse the independence of measurements, leading to linearly dependent equations (in the inversion) making the inverse solution highly underspecified.
Spatial Distribution of Perturbations
One way to control the distribution of sensitivity is the use of the standard deviation of the sensitivity in the objective function. This is, however, a very limited method, which allows mostly the enforcement of a homogeneous sensitivity distribution. In some cases, an ERT survey may be conducted to obtained information from a specific region of the subsurface. Examples vary, from archeological surveys, with prior knowledge of target depth, to the tracking of an infiltration wetting front, where the approximate depth of the wetting front with time may be easily estimated. By specifying perturbation locations in the optimization process according to the desired regions in the subsurface where data are to be gathered, the ERT survey may be specifically targeted.
Consider as an example the design of a survey tailored to gather information from a smaller region than the original triangle presented in Fig. 3. Figure 5 (top) presents the sensitivity map created using the optimal 15-electrode survey (using Eq. [11] as the objective function). Also marked are the locations of the 11 perturbations that we use to designate the target region for this case. To demonstrate the effect of targeting survey sensitivity, the spatial sensitivity of the targeted survey (Fig. 5, top) can be compared with the optimized spatial sensitivity based on a homogeneous distribution of perturbations (Fig. 4, top). The difference in sensitivities is shown in Fig. 5 (bottom). Blue regions have increased sensitivity for the targeted case, and red regions have reduced sensitivity.
Figure 5 is somewhat similar to Fig. 4 (top), except for the scale. The main difference is that the targets span a shorter portion of the electrodes, forcing smaller arrays to enter the optimal set and therefore creating a more homogeneous distribution of the sensitivity near the surface (above the target). It is clear, however, that the use of target perturbations created focusing of the sensitivity to that region. Note the sensitivity values that range between approximately 0.003 to 0.005 in the targeted case, compared with approximately 0.0015 to 0.0025 for the same region in Fig. 4 (top). As seen in Fig. 5 (bottom), the increase of sensitivity in the vicinity of the targeted perturbations comes mostly at the expense of sensitivity in outer regions.
The optimization of an ERT survey is performed by the perturbation approach, using an initial guess of the subsurface resistivity structure. If no other information is available, the initial guess is a homogeneous distribution. However, the actual resistivity structure of the subsurface may be very different than this initial guess; hence, it is possible that the selected optimal survey actually lies far from the true optimal one. In general, the optimized survey should be preferable to a nonoptimized survey, but for a specific case it is possible that the inverted resistivity structure of the subsurface is not more accurate than the structure that would have been obtained using a simple survey (e.g., one comprised only of Wenner arrays).
To overcome this problem, and to increase the accuracy of the resistivity survey, an iterative approach is suggested. An updated survey optimization should be determined based on the measured and inverted resistivities. This process can continue, in real time, for several measurement cycles.
For monitoring stationary conditions, it is possible to include all measurements as the input for the secondary inversion (i.e., from first and second optimizations). This way the image obtained will be based on many measurements, increasing its reliability and accuracy.
When monitoring transient conditions, there typically is less time to conduct a secondary survey. However, the conditions at each measurement can be approximated using the conditions measured (and inverted) at the previous measurement. Further improvement can be achieved using hydrological models, simulating the movement of water (and solutes) in the subsurface between the two measurements.
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CONCLUSIONS AND SUMMARY |
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TOPABSTRACTINTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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The optimal survey is composed of a mixture of arrays of different types. One clear result is that double-dipole arrays are seldom part of the optimal solution. In cases where the objective function looks only into the average sensitivity (i.e., without considering its distribution), clear preference for wide arrays was observed.
Two alternatives are suggested for the control of the distribution of the sensitivity. The first makes use of a normalized form of the offset of an array from the perturbation's maximal sensitivity, resulting in a simplified objective function. The second alternative (which can also be combined with the first) makes use of a targeted allocation of perturbations to achieve a desired sensitivity distribution. The second approach opens the way to targeted ERT survey designs. This use of smart allocation of perturbations allows for the specific tailoring of an ERT survey according to survey goals, which may change with time when monitoring temporally changing field conditions.
The objective function and GA presented above consider the cumulative sensitivity of a survey and the desire for equal confidence in results for every part of the subsurface, expressed as homogeneity of the sensitivity distribution or homogeneity of the offset function. It is clear, however, that other considerations should be included to account for the effects of the inversion process. It is highly likely that a stable and accurate inversion will require some additional measure of the independence of the arrays included in a survey (Wang 2002) to ensure that each array adds constructively to the overall information content of the survey.
This diversity of the arrays comprising a survey is not taken into account in this paper. Methods for expressing this array diversity will be addressed in future studies following the example of Wang (2002) or using the distinguishability (e.g., Isaacson and Cheney, 1990). The question of how the optimality criteria can be modified to account for the effects of nonlinear inversion will also be addressed in future work.
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONTHEORYMETHODSRESULTS AND DISCUSSIONCONCLUSIONS AND SUMMARYREFERENCES |
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