## 1. Introduction

[2] Hydrometric information, which is mainly collected by monitoring networks, constitutes the fundamental input for planning, design, operation, and management of water resources systems. Optimally siting of monitoring gauges such that they are effective and efficient in gathering data (information) has received considerable attention. Although there are myriad concerns in hydrometric network design, this study focuses on the fundamental theme, i.e., selecting an optimum number of stations and their optimum locations. Many approaches have been developed for that purpose. A comprehensive review can be found in the work of *Mishra and Coulibaly* [2009]. Among others, one type of approach is based on entropy theory. The merit of entropy theory is that it directly defines information and quantifies uncertainty [*Harmancioglu and Singh*, 1998; *Mogheir et al.*, 2006]. A significant body of literature employing entropy theory for the design of monitoring networks has been reported, as briefly reviewed in what follows.

[3] *Caselton and Husain* [1980] and *Husain* [1987] used an information maximization principle for identifying optimum locations of rainfall gauges to be retained in a dense network. When an existing network is sparse, *Husain* [1989] proposed a methodology for expanding it by means of information interpolation. *Krstanovic and Singh* [1992a]developed an entropy-based approach for hydrologic network evaluation. This approach was then used to spatiotemporally evaluate the rainfall network in Louisiana [*Krstanovic and Singh*, 1992b]. *Yang and Burn* [1994] presented a method for data collection design in which a concept of directional information flow was employed. Also in terms of entropy theory, *Mogheir et al.* [2006] evaluated the optimality of the groundwater quality monitoring network in Gaza Strip, Palestine. *Mishra and Coulibaly* [2010] assessed streamflow network in different Canadian watersheds using entropy, joint entropy and transinformation.

[4] Common with the aforementioned studies is that entropy terms were computed using univariate or bivariate formulations. However, joint information retained by multiple stations and their dependence are always required for a more objective evaluation. A few studies did use multivariate distributions but assumed that the data were normally or lognormally distributed as, for example, did *Husain* [1987, 1989], and *Krstanovic and Singh* [1992a, 1992b]. This assumption is debatable, since many natural phenomena are heavy tail distributed, like streamflow and precipitation [*Bernadara et al.*, 2008; *Carreau et al.*, 2009; *Li et al.*, 2012].

[5] Similar to *Krstanovic and Singh* [1992a, 1992b], *Alfonso et al.* [2010a]introduced several adaptations to make the entropy-based method applicable to the design of water level monitors for highly controlled polder system in the Netherlands. They first used multivariate total correlation to assess the performance of three pairwise dependence criteria. The selection of optimal monitors was restricted to low dimensional analysis (less than 2).

[6] Later, *Alfonso et al.* [2010b] proposed another criterion by maximizing multivariate joint entropy and minimizing total correlation for optimally siting water level monitors. Yet, this approach failed to account for the information transition ability (transinformation) of a network. It is acknowledged that transferring hydrologic information from points where it is available to those where it is required is one of the purposes of collecting hydrometric information [*Harmancioglu and Yevjevich*, 1987].

[7] Moreover, optimally siting hydrometric monitors is a multiobjective problem. Solving this problem is tricky in practice. *Alfonso et al.* [2010b]exploited a genetic algorithm to approach the multiobjective optimization. The advantage of multiobjective optimization is that it provides different feasible solutions under different scenarios. Nevertheless, selection of the final network is not straightforward. To assist the decision making processes, it is quite admirable to find an easy-to-implement way to solve the multiobjective problem and to provide the end user a unique solution with decent performance.

[8] Our objective therefore is to develop an easy-to-implement approach for the design (or evaluation) of hydrometric networks. To that end, we first propose an entropy theory-based criterion, named as maximum information minimum redundancy (MIMR), satisfying three norms: (1) maximum overall information (joint entropy), (2) maximum information transition ability (transinformation), and (3) minimum redundant information (total correlation). These entropy terms are calculated at multivariate level without any distributional assumption. Thereby interactions among stations can be properly accounted for. Then we present a straightforward greedy selection algorithm to rank the candidate gauges based on MIMR. During selection, the three commensurable norms are additively unified, which circumvents the complexity of multiobjective optimization, while preserving its advantage in achieving different feasible solutions through information-redundancy tradeoff weights.

[9] The paper is organized as follows. Formulating the objectives of the study in this section, basic entropy theory is briefly presented in section 2 for the ease of understanding the MIMR criterion, which, together with a selection algorithm, is discussed in section 3. Through two case studies, section 4 illustrates the applicability of MIMR to the evaluation and design of hydrometric networks. Merits and demerits of MIMR are discussed in section 5. Conclusions are generalized in section 6.