Optimal estuarine sediment monitoring network design with simulated annealing

Abstract

An objective function based on geostatistical variance reduction, constrained to the reproduction of the probability distribution functions of selected physical and chemical sediment variables, is applied to the selection of the best set of compliance monitoring stations in the Sado river estuary in Portugal. These stations were to be selected from a large set of sampling stations from a prior field campaign. Simulated annealing was chosen to solve the optimisation function model. Both the combinatorial problem structure and the resulting candidate sediment monitoring networks are discussed, and the optimal dimension and spatial distribution are proposed. An optimal network of sixty stations was obtained from an original 153-station sampling campaign.

Introduction

A well designed, ongoing monitoring program is fundamental for the evaluation of environmental management of natural systems (Kay and Alder, 2000). The design of an effective monitoring program depends on the management objectives, resources (funding and staff) and available technology. Monitoring programmes should be designed to contribute to a synthesis of information or to evaluate impacts, or analyse the complex cross-linkages between environmental quality aspects, impacts and socio-economic driving forces (RIVM, 1994).

The technical design of monitoring networks is related to the determination of: (i) monitoring sites; (ii) monitoring frequencies; (iii) variables to be sampled; (iv) duration of sampling (the last two variables are not discussed here because they are case-specific). Most of the research results in this area have been obtained in the context of statistical procedures (Sanders et al., 1983; Moss, 1986, IAHS, 1986; Cochran, 1977). These rely in the principle that there are several sources of uncertainty, due to measuring errors, inherent heterogeneities of the involved variables, and in the cases where modelling is involved, also simplifications and errors in both the modelling and numerical/analysis solution phase. McBratney et al., 1981), as well as many other authors after them, indicated that uncertainties are the result of lack, in quality and quantity, of information concerning the systems under study, or as a result of spatial and temporal variations of parameters.

In many monitoring programs a first sampling stage with a large number of locations is undertaken, either because there is no prior information or it is considered necessary to collect more data. This stage is usually planned to give statistical information about the variables under study and to calculate their spatial covariance. A second stage is needed to transform the original set of sampling stations, with high cardinality, into a lower cardinality set of monitoring stations. Probably the methods used most to reduce cardinality are those based on the maximisation of spatial accuracy, or in other words, on the minimisation of the variance of the estimation error, also known as variance reduction methods. This is usually carried out in the context of geostatistical theory (Matheron, 1963, Matheron, 1965) and most frequently by interpolation with an unknown mean, i.e. by ordinary kriging. Other promising methods have been proposed for optimising the monitoring network design, in particular those based on information theory, as in articles such as those by Amorocho and Espildora, 1973, Caselton and Husain, 1980, Caselton and Zidek, 1984, Harmancioglu and Yevjevich, 1987, Husain, 1989, Harmancioglu and Alspaslan, 1992. Despite the elegance of these methods, they are limited by the need to assume a probability distribution for the variables, which may be unknown or difficult to determine. Moreover the method is particularly well adapted to variables with equal probability distributions (usually normal or lognormal). When soft and other sources of information are available then the Bayesian Maximum Entropy geostatistical method, first developed by George Christakos (Christakos, 1990, Christakos, 1992), have proven to outperform ordinary kriging (D'Or et al., 2001), and also have the advantage over the latter that they do not require the specification of particular probability distributions.

Kriging variance has been extensively used for monitoring network design. Examples can be found in the work of Bras and Rodríguez-Iturbe, 1976, Rouhani, 1985, Loaiciga, 1989, Rouhani and Hall, 1988, Pardo-Igúzquiza, 1998, van Groenigen et al., 1999, van Groenigen and Stein, 1998, and Nunes et al., 2004a, Nunes et al., 2004b.

Two categories for monitoring optimisation with variance reduction have been proposed: (i) the local approach (e.g. Amorocho and Espildora, 1973); and (ii) the global approach (e.g. Ahmed et al., 1988). In the first the influence of each additional point is analysed separately. Total variance reduction after adding one point is easily computed by considering the individual values at each initial location or at the points in the vicinity of the point being estimated. In the global approach average estimation variances are used. Therefore, global approaches provide only average answers to monitoring designs. It is useful to analyse designs still on the drawing board or to perform extensive redesigns aimed at maintaining the efficiency of a monitoring network, which may require removal of poorly located sites. The local approach, on the other hand, is better suited to optimally expanding an existing network. The optimality in this case only relates to the additional points, which may not be acceptable if the original points are not optimal (Markus et al., 1999).

Minimisation of the average kriging variance approach was applied here to select the number and positions of sediment monitoring stations in the Sado river estuary located in the southwest coast oft Portugal (Fig. 1), such that different physically and chemically homogeneous areas identified in a prior sampling campaign were considered. This monitoring network will be further integrated into an environmental data management system for the Sado Estuary as a decision support tool for local authorities. The Sado Estuary in Portugal is an example where environmental problems are not well managed owing to the high natural values and diverse pressures for development and where the right tools to help evaluating the environmental quality status need to be developed. The objective here was on the development of a monitoring network that constitutes one the information sources of the Sado Estuary management system (physic-chemical data of sediment quality).

For practical and budgetary reasons the number of monitoring stations should be reduced to a minimum. The optimisation problem can be stated in a very simple way: maximising the spatial accuracy, constrained to a maximum number of stations, given the information collected in a prior sampling program (153 sampling sites). Maximisation of spatial accuracy is easily attained by minimising the variance of estimation error, though incorporating the patchiness of homogeneous areas is a more difficult problem. One alternative would be to fix several locations inside the different homogeneous areas, but then the choice of stations would be arbitrary. Another way is to use stratification, considering that a defined number of stations must be placed inside homogeneous areas. Stratification is a well-known statistical technique used for designing monitoring (or sampling) programs with denser networks in some areas than in others. The difference in probability density may be based, for example, on spatial autocovariances, statistical risk of contamination, plume detection probabilities or empirical judgement, among many others. Here we propose a statistically based stratification: homogeneous areas are monitored according to the frequency with which they appear in the prior sampling program. The inclusion of homogeneous areas was considered important by the manager because sediment granulometry and physical and chemical characteristics have strong correlations with the amount of xenobiotics the sediment can retain and because these areas were planned to be geographic spatial units in an environmental management system. Hence, four types of sediments were established on the basis of three physical and chemical variables and the manager demanded that the proportion of stations in the four types of sediments in the monitoring network be similar to that of the sampling campaign (thus the constraint on the proportions).

Optimisation consists, then, of finding an optimal subset with a combination of stations taken from a larger set. Even for relatively small set cardinalities the number of combinations is too high to allow them all to be exhaustively evaluated in a reasonable amount of time. One of the most well known algorithms for solving combinatorial problems is simulated annealing, in particular in sampling/monitoring network optimisation (e.g. Meyer et al., 1994, Pardo-Igúzquiza, 1998, van Groenigen et al., 1999, Brus et al., 2000, Brus et al., 2002, Nunes et al., 2004a, Nunes et al., 2004b).

The article is divided in five sections. This Introduction is followed by a second section where the theoretical geostatistical and optimisation framework is presented. In this section the geostatistical parameter most frequently used to measure accuracy, the kriging estimation error variance, is explained and compared with another geostatistical measure of accuracy, the fictitious point estimation error variance. Also the simulated annealing heuristic used to solve the optimisation problem is introduced. In the third section a case-study is presented and data transformations are explained, while, in the fourth section, optimisation results are discussed. Finally, in the last section, the most important conclusions are drawn.