COKRIGING OPTIMIZATION OF MONITORING NETWORK CONFIGURATION BASED ON FUZZY AND NON-FUZZY VARIOGRAM EVALUATION G. PASSARELLA1∗ , M. VURRO1 , V. D’AGOSTINO2 and M. J. BARCELONA3 1 Water Research Institute, CNR, Bari, Italy; 2 Tecnopolis, NOVUS ORTUS, s.s. Casamassima, Valenzano, Bari, Italy; 3 Western Michigan University, Department of Chemistry, Kalamazoo, Michigan, U.S.A. (∗ author for correspondence, e-mail:
[email protected])
(Received 23 July 2001; accepted 14 March 2002)
Abstract. A number of optimization approaches regarding monitoring network design and sampling optimization procedures have been reported in the literature. Cokriging Estimation Variance (CEV) is a useful optimization tool to determine the influence of the spatial configuration of monitoring networks on parameter estimations. It was used in order to derive a reduced configuration of a nitrate concentration monitoring well network. The reliability of the reduced monitoring configuration suffers from the uncertainties caused by the variographer’s choices and several inherent assumptions. These uncertainties can be described considering the variogram parameters as fuzzy numbers and the uncertainties by means of membership functions. Fuzzy and non-fuzzy approaches were used to evaluate differences among well network configurations. Both approaches permitted estimates of acceptable levels of information loss for nitrate concentrations in the monitoring network of the aquifer of the Plain of Modena, Northern Italy. The fuzzy approach was found to require considerably more computational time and numbers of wells at comparable level of information loss. Keywords: geostatistics, groundwater, monitoring networks
1. Introduction Groundwater data provide a basis for cost-effective water resource management. The design of reliable monitoring well networks and use of effective sampling procedures insures documented data quality. Usually, at the beginning of a monitoring program design, a census of existing wells in the area of concern must be achieved. This is the least expensive way to create a new monitoring network. The most favorable case is when the number of available wells is large and the water authority may use a reduced set of them. If existing wells are insufficient, new ones must be drilled to fill data gaps. The optimization problem may then be posed as two questions. Which existing wells are to be excluded from a large network? and; where should new wells in a sparse network be drilled? Many real cases are characterized by a number of existing wells, irregularly spread over the considered area. This is a typical case of optimization and it may Environmental Monitoring and Assessment 82: 1–21, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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be solved seeking among several a priori alternatives by using search algorithms (Loaiciga, 1989). Insight on this optimization problem can be gained from the use of geostatistics, particularly from using Cokriging Estimation Variance (CEV). Cokriging is often preferable to ordinary kriging. This is because if a considered parameter shows appreciable temporal persistence, it allows one to reduce the uncertainty in the evaluation of the spatial behavior of an under-sampled parameter using the values of the parameter sampled at another time (D’Agostino et al., 1998), or of another parameter sampled at the same time (Istok et al., 1993). Consequently, sampling of a water quality parameter can be planned and scheduled in a reduced number of wells without loss of accuracy. Furthermore, the cokriging equations allow the estimation of the CEV, evaluated in a generic location x0 , to be dependent only on the relative positions of x0 with respect to data sample locations through the direct- and cross-variograms. This property of the CEV might be used to evaluate the benefit for the estimation in x0 of an additional sampling point or respectively the loss due to dropping a point once the geographical location of the data points and the direct- and cross-variograms are known. A methodology based on a sequential search algorithm has been applied to determine a reduced arrangement of an existing network of monitoring wells in an aquifer. Nitrate concentrations have been considered as unique variables varying in space and correlated in time, and appropriate co-variogram models have been used to describe spatial and temporal behavior. The approach reduces the number of the monitoring wells by eliminating the ‘least useful’ for the estimation, by cokriging, at several critical sites (Pan et al., 1992). This methodology is based on the assumption that the spatial behavior of the considered variable in the previous year is known and may be considered persistent in time. This consideration allows one to use the variogram model of the same season of the previous year as representative of the considered variable in the current season (D’Agostino et al., 1997). Practically, if temporal persistence of the parameter concentration is assumed in the groundwater then each seasonal data set can be interpreted as an individual realization of different random correlated functions enabling the geostatistical approach (Myers, 1982, 1988; Wackernagel, 1989). Goovaerts and Sonnet (1992) have applied the model of coregionalization to hydrogeochemical data collected in the wells of a monitoring network, at six different time periods. The assumption of temporal persistence of nitrate concentrations can be considered strong compared to the natural phenomenon. Since the selection of the variogram characteristics plays a critical role in the final reliability of the estimation, the uncertainty implied by this assumption should not be neglected in the approach. This uncertainty has to be added to that related to the interpolation practices of variography for the selection of a robust theoretical variogram model, which correctly describes the spatial behavior of the regionalised variable.
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Evaluating these uncertainties strongly depends on the quality and quantity of the available information. Bardossy et al. (1990a) pointed out that the scarcity of measures may be integrated by a priori knowledge and experience to determine the variogram model parameters. However, Englund (1990) proposed the same data sets to 12 independent geostatisticians who obtained substantially different variogram models. Possible causes of this result can be the difficulty in representation of a probabilistic framework of the uncertainties due to variogram choice (Bardossy et al., 1990a) or the strong assumptions on the underlying distributions made by the modelers (Kitanidis, 1986). Many methods have been used to evaluate the uncertainty due to the choice of a variogram model but most of them are often not able to furnish a totally satisfactory outcome. A simple method adopted to evaluate the uncertainty is to perform an interval analysis (Bogardi and Bardossy, 1990). The propagation of the uncertainty characteristics may be estimated as intervals in a traditional sensitivity analysis (Varljen et al., 1999). Geostatistical cross-validation allows one to compare the impact of different models on interpolated result but it does not provide any useful information about the comparability of the modeled covariance (variogram) function to the true one (Solow, 1990). Consequently it is not possible to eliminate the uncertainty in the choice of the variogram model by this method. Gutjahr (1994) highlights that the effects of correlation and bias can be minimized, however, by fitting the standard models to the experimental points by sight. Several methods for evaluating the robustness of the results of the structural analysis exist (Dowd, 1984; Srivastava and Parker, 1989), but they may lead to substantially different fits. Woldt et al. (1990) used fuzzy set theory to describe the degree of imprecision resulting from the fitting of a theoretical variogram to an experimental one. This methodology introduced the concept of an experimental variogram mist as a means to locate possible theoretical variogram limits. Bardossy et al. (1990b) used a kriging with nugget effect, sill, and range interpreted as fuzzy sets. Both kriging estimates and estimation variances are calculated as fuzzy numbers from the fuzzy variogram and data points. The main purpose of this study was the comparison between a fuzzy and a nonfuzzy approach used for assessing a reduced arrangement of a monitoring wells network by means of a methodology based on cokriging, in order to show positive and negative aspects. The non-fuzzy approach has been already applied (D’Agostino et al., 1997) and it has been proved to be quick, easy and rather affordable. Nevertheless, the assumption must be made that the physical behavior of the monitored variable, described by the variograms, is persistent during the year and almost the same each season, year after year. This assumption is rather rigid since it implies that the variogram models are invariant during the years. The fuzzy approach has been explored in order to take into account the uncertainty related to the choice of the variograms. This approach assumes time
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invariance of the variogram models but not of their parameters, which are treated as fuzzy numbers. The choice of the method to be used remains an option for people responsible of decisions related to groundwater quality control.
2. Materials and Methods 2.1. C OREGIONALIZATION ANALYSIS The proposed approach for monitoring network optimization is based on the analysis of the CEV at several ‘critical sites’, taking into account its monotonic property. Increasing (or decreasing) a network by one well implies a decreasing (increasing) CEV at a critical site. First one identifies two sets of measured values of the same variable collected at different times T1 and T2 , whose dimensions are, respectively, n and m. Then assuming that the evolution of the underlying phenomenon allows one to consider the measured values spatially correlated at different times, then the whole sample may be interpreted as a unique realization of two inter-correlated random functions {Z1 ,Z2 }. This is an example of simultaneous regionalization or coregionalization. The probabilistic approach to a coregionalization is similar to that of the regionalization of a single variable. This approach is based on the determination of two variograms and one cross-variogram (Journel and Huijbregts, 1978). In fact, the spatial behavior of each data set is characterized by its own sample variogram and the pair of data set by its own sample cross-variogram. The model for each of these sample variograms may consist of one or more basic models (Isaaks and Srivastava, 1989). Nevertheless, just as any function cannot be considered as the variogram of a second order stationary random function, any set of functions cannot be considered as a matrix of variograms of a set of coregionalized random functions. The matrix of variograms must be positive definite so as to ensure that the variances of all finite linear combinations of the basic random functions are positive. Anyway, if the three variogram models satisfy the Cauchy-Schwarz (C-S) inequality, |γij (h)|≤ γii (h)γjj (h) , this can be considered as a necessary and sufficient condition for the positiveness of the matrix of a linear model of coregionalization (Journel and Huijbregts, 1978; Wackernagel, 1995). Assuming the intrinsic hypothesis (Matheron, 1978), the two direct variograms and the cross variogram, depend only on the separation distance between two points. All the data samples available at time T1 and T2 , and space and time information contained in the variograms may then be cokriged to give the best linear unbiased estimate of the variable at the under-sampled time, at a given critical site x0 . This
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is the primary advantage of cokriging over kriging. The best estimator of x0 means that cokriging provides an estimate having the minimum variance among all the linear unbiased estimators. Furthermore, because of the conditions imposed to the estimator, the CEV then represents the confidence in the estimation in x0 . The cokriging equations allow the CEV of x0 to be dependent only on the relative positions of x0 and the data sample (x i ) locations through the direct and cross variograms. This property of the CEV might be used to evaluate the benefit in x0 of an additional data sample point (or respectively the loss when a point is dropped), when just the position of the data point to add (respectively to drop) is known. In other words, once the direct and the cross variograms have been selected, the improvement of the CEV in x0 caused by an additional point (xn+1 ) can be evaluated without the chemical parameter being sampled at this point. It follows that, the evaluation of the CEV at the critical sites (x0i ) requires that the two direct variograms for two seasons and the cross variogram are known. This means that the chemical parameter has to be sampled during both the seasons in the entire monitoring network. Unfortunately, the actual values of the future season are unknown since this is the season when the sampling must be optimized. The assumption must be made that the physical behavior described by the variograms are persistent during the year and almost the same each season, year after year. In practice, the variogram of the same season of the previous year can be used instead of that, unknown, in the second season. This assumption is rather rigid since it implies that the variogram models are invariant during the years (D’Agostino et al., 1997). 2.2. F UZZY NUMBERS A less stringent assumption treats the time invariance of the variogram models and not their parameters, which are then treated as fuzzy numbers. In fact, the variograms depend on their three parameters and on the separation vector h: γ (h) = γ (c0 , c, a, h) The set of possible variogram parameters can be supposed to be a fuzzy number in the space of real numbers 3 . In order to introduce the role of a fuzzy number a brief introduction to the fuzzy sets is necessary (Pedrycz, 1989). A fuzzy set is represented mathematically by a function that measures the grade of membership of an element in the set. Formally, a fuzzy set A defined in a universe of discourse X is expressed by its membership function A: X→[0, 1], where A(x) expresses the extent to which x fulfills the category specified by A. Fuzzy numbers are a special form of fuzzy sets. Properly, they are fuzzy sets defined in the space of real numbers. They are characterized by some additional
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properties related to the shape of their membership functions. Every membership level α∈[0, 1] cuts an interval of values in X, called the α-interval. The combination of fuzzy numbers, through functional dependencies, still produces fuzzy numbers, consequently, for each α∈[0, 1], the combination of α-intervals in input must produce α-intervals in output. The basic tool for development of fuzzy arithmetic is the extension principle (Zadeh, 1965). If X and Y are two sets, and f is a mapping from X to Y then f can be extended to operate on fuzzy subsets of X in the following way. Let A be a fuzzy subset of X with membership function µA (x). The image of A in Y is the fuzzy subset B with the membership function µB (y) equal to 0, if no x∈ X such that f (x) = y, or µB (y) = {sup[µA (x); x∈X, y = f (x)]} . Sometimes it can be useful to convert the output fuzzy number to a crisp number. In these cases several methods of defuzzification can be applied. One of these methods ˆ of a fuzzy number A: consists in computing the average M(A) xµ (x)dx ˆ = R A . M(A) R µA (x)dx ˆ of a fuzzy number A: Another method consists in computing the centroid C(A)
ˆ C(A) −∞
µA (x)dx =
+∞ ˆ C(A)
µA (x)dx .
In our approach, the functional relationship among the fuzzy numbers chosen to represent the variograms parameters, is the CEV expression: 2 (x0 ) = µ1 + CEV = σck
m
λ1i γ11 (x1i − x0 ) +
i=1
n
λ2j γ21 (x2j − x0 )
j =1
where x1i for i = 1, . . ., m, and x2j for j = 1, . . ., n are, respectively, the locations of the samples collected during the principal and the auxiliary seasons, x0 is the location of the target point, µi is the Lagrange’s multiplier, λ1i and λ2j are the cokriging weights, γ11 is the direct variogram of the principal season, and γ21 is the cross variogram. 2.3. T HE PROPOSED APPROACH The first sampling campaign of the year (auxiliary variable) has to be performed over the entire network to use the full set of samples as auxiliary variable in the cokriging analysis. The assumption of annual persistent behavior of the considered quality parameter has been made so that both the direct variogram of the principal variable
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and cross variogram may be determined using the data collected in the previous year. In order to take into account possible changes in the values of the sills of these variograms from the previous year, or, better, to improve the validity of the assumption reported above, these parameters were considered as a fuzzy numbers, and a membership function was chosen to describe the uncertainty. In particular the value from the previous year was considered as that corresponding to the maximum membership level (α = 1) of an isosceles triangular membership function. The base of this function (α = 0) has been set using percent changes of the central value. Once the direct and the cross variograms are chosen and a configuration of the monitoring network has been set, the next step of the methodology consists in evaluating the CEVs in the target points, in order to eliminate the ‘least useful’ well from the set. As stated above, the combination of fuzzy numbers (variograms), through functional dependencies, still produces fuzzy numbers (CEVs). Consequently, after the membership function of the CEVs is evaluated, its average may be computed and sampling point elimination is conducted on the basis of this crisp parameter. In summary detail, the proposed fuzzy approach goes throughout the following steps: (1) membership functions of the CEVs are evaluated at each critical site using the full sets of coordinates of the principal season (n wells) and the auxiliary season (m wells, m = n). These membership functions are converted to a crisp number, throughout a defuzzification method. Since they use the full data set, these crisp numbers represent the least spatial uncertainty at each of the critical sites. Then the array of the defuzzified CEVs are used as reference values in the following steps; (2) membership functions are evaluated k times at each of the critical sites, using the subset of coordinates of the principal season obtained by eliminating one well each time (k = n − 1) and the full set of the auxiliary season; (3) difference (in percent) between the defuzzified CEVs estimated at the second step and the reference values are then calculated. The greater the change in the defuzzified CEV due to the elimination of a well, a greater value is assigned to results from that well for the monitoring network; (4) it follows that, the well whose change in defuzzified CEV is the least is a good candidate for elimination. If the percent value of this change is lower than a given threshold, it may then be eliminated with minimal information loss; (5) repeat step 2 with k = k − 1 and steps 3 and 4 until the lowest change of defuzzified CEV becomes greater than the threshold. In order to test the methodology a computer code (WellOpt) was written and interfaced as a Windows© application. This software permits easy introduction of the data in form of data files and exploration of several options. The fundamental
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Figure 1. Map of the Modena study area.
option was the choice of either a computational non-fuzzy or fuzzy approach. The fuzzy approach was chosen to consider the error resultant from considering the variogram of the fall of the previous year instead of the current year (unknown) as well as the uncertainty in variogram parameter evaluation. Anyway, the computational requirements of this approach were greater than non-fuzzy approach.
3. Description of the study area The study area (Figure 1) comprises a major portion of the Plain of Modena situated in Northern part of Italy. It covers a surface of about 24 000 ha. The Western and the Eastern boundaries are marked by the rivers Secchia and Panaro. On the Southern side, it reaches to the Apennines. While the Northern boundary may be represented
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by an imaginary E–W line crossing the city of Modena. The topography ranges from 30 m above sea level near Modena to 150 m above sea level at the Southern boundary. From a geologic standpoint, the plain between the Apennines on the south and the sedimentary basin of river Po on the north is characterized by a system of alluvial fans. These fans have been produced by sediments from the main rivers (Secchia and Panaro) and from deposition in the secondary tributary network (Barelli et al., 1990). The Apennines are composed mainly of clay sequences that date back to the Pliocenic-Calabrian marine cycle. The sediments of the low alluvial plain, which extends to the river Po, consist of sand with a significant fraction of mud and clay. In the apical part, gravel and sand alternate with pelitic bodies whose thickness gradually increases approaching the distal section of the fans. The total aquifer thickness of the deposits of the Modena plain is variable and generally increases from 100 m near the Apennine to the 400 m towards the north. This pattern is due to deep tectonic structures, which caused differential subsidence across the plain (Paltrinieri and Pellegrini, 1990). The water table decreases from the southwestern part of the valley to the northeast one, near Modena, and it is normally higher in spring than in autumn (Figure 2). The flow pattern is strongly affected by recharge from rivers Secchia and Panaro and by groundwater extraction in the Modena area. A change in hydraulic conditions from unconfined to confined or semi-confined aquifer, in the central part of the valley, was pointed out in a former study (Paltrinieri and Pellegrini, 1990). Precipitation in the Modena area is about 700 mm yr−1 , the average yearly temperature varies between 10 and 14 ◦ C, the potential evapotranspiration is about 720 mm yr−1 and the actual evapotranspiration is about 540 mm yr−1 . A land use map derived from aerial photographs was provided by the Servizio Cartografico della Provincia di Modena (Regione Emilia Romagna, 1993). Agricultural land use extends over the 80% of the area. Principal crops are wheat, other cereals (maize), fodder, vegetables, vineyard and orchard (ISTAT, 1992). The yearly amount of nitrogen applied to the different crops was deduced from literature. With regard to water quality, ammonia concentrations are low (i.e.
