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Information-theory provides, among others, conceptual methods to quantify the amount of information contained in single random variables and methods to quantify the amount of information contained and shared among two or more variables. Although these concepts have been successfully applied in hydrology and other fields, the evaluation of these quantities is sensitive to different assumptions in the estimation of probabilities. An example is the histogram bin size used to estimate probabilities to calculate Information Theory quantities via frequency methods. The present research aims at introducing a method to take into consideration the uncertainty coming from these parameters in the evaluation of the North Sea’s water level network. The main idea is that the entropy of a random variable can be represented as a probability distribution of possible values, instead of entropy being a deterministic value. The method consists of solving multiple scenarios of Multi-Objective Optimization Problem in which information content is maximized and redundancy is minimized. Results include probabilistic analysis of the chosen parameters on the resulting family of Pareto fronts, providing additional criteria on the selection of the final set of monitoring points.

Data collection is crucial in hydrology and water resources because it is an activity that generates information about past and current states of water systems to ultimately assist informed decisions. For this purpose, monitoring sensors are positioned in strategic places in such a way that the highest information content about the state of an area is obtained, observing the limitations in the number of available sensors to do so.

Literature on design of hydrometric monitoring network started to be popular in the 1960s after the International Hydrological Decade (1965–1974) brought global attention to the need for hydrometric data [

Information-theory provides conceptual methods to quantify the amount of information contained in single random variables and contained and shared among two or more random variables. Information content of single variables can be estimated using the concept of Marginal Entropy (H), as described by [

Entropy-based methods for hydrology studies started to be popular after the seminal paper [

In recent years different authors have exploited the concept of Total Correlation to quantify the information content that is shared within a set of two or more stations, as a generalization of the Mutual Information concept used studies applying pair-wise station analysis [

Although Information Theory concepts have been successfully applied in hydrology and other fields [

This paper introduces a method to take into consideration the uncertainty coming from these assumptions in the evaluation of the North Sea’s water level network. The network is evaluated in a multi-objective optimisation framework to ensure that the set of resulting sensors are simultaneously informative and non-redundant. Entropy parameters are then sampled for the optimised network in order to estimate the robustness of the solutions given the changes in the baseline assumptions.

From the Information Theory perspective, an accepted approach to set of stations that form a monitoring network is to consider that the information content of the set is maximum, whereas, at the same time, the redundancy among each station of the set is minimum [

where _{i}_{1}..._{iM}

where _{i}

where _{x}_{i}_{i}

MO consists of searching for a set of decision variables such that simultaneously optimize independent objective functions. Usually objective functions conflict with each other, so the word optimisation suggests having a compromise among the objectives. According to the discussion in previous section, the MO problem can be posed as shown in

where _{i}_{i}_{i}

Although the approach described so far has been applied in different studies (see e.g., [_{q}

where

This paper aims at introducing a method to take into consideration the uncertainty coming from assumptions of parameters γ and

The main idea of the method is that the entropy of a random variable can be represented as a probability distribution of possible values, instead of entropy being a deterministic value. The method is called ensemble entropy, and it consists of the following steps:

Assume a value for parameters γ and

Obtain the transformed (_{i}

Solve the optimisation problem formulated in _{i}_{i}

Take

For each sample combination

Obtain the transformed (_{ij}_{i}

Evaluate _{ij}

Evaluate the two-dimensional distribution of

The method can consider any sampling strategy for parameters γ and

The city of Rotterdam is one of the most important ship transit cities in the world, an important port that is significant for the economy of The Netherlands,

This section presents the results of the methodology presented in Section 2.3, for the optimisation of the locations of three monitoring sensors. In the first place, both parameters γ and _{1}, _{2}, _{3} using the multi-objective optimisation algorithm NSGA-II, obtaining the Pareto quasi-optimal shown in

The following step in the proposed methodology is to take

Because

In this work a method to take into consideration the uncertainty coming from water monitoring sensors optimization through Information Theory is presented. The main idea is that the entropy of a random variable can be represented as a probability distribution of possible values, instead of entropy being a deterministic value. Entropy determination implies the estimation of marginal and joint probability of each data series. The problem is faced transforming each data series value in a quantized one through two parameters γ and

Part of this research was carried out with funds from the FP7 WeSenseIt project. The data used is made freely available by the Dutch Ministry of Infrastructure and Environment (

The authors declare no conflict of interest.

All the authors contributed to the manuscript. Alfonso, Ridolfi, Napolitano and Russo have contributed to the research methods and the results have been discussed among all authors. The contributions by sections are: Alfonso: Abstract, Introduction, Methods, Results, Discussion and Conclusions; Ridolfi: Methods, Results and Discussion; Gaytan-Aguilar: Case study; Napolitano: Introduction; Russo: Conclusions.

The Netherlands map, the delta of The Netherlands is highlighted by the red box.

Water level monitoring network in the North Sea and the delta of The Netherlands.

Pareto front obtained from step 2 of proposed methodology. Each point corresponds to a potential set of 3 monitors. Ideal point is such with the maximum (negative) Joint Entropy and zero Total Correlation, represented at the origin of the figure.

2D distribution of six selected solutions out of the 100 obtained. As in

Summarized 2D distribution of six selected solutions out of the 100 obtained.

Resulting monitoring network. Green triangles solutions 1, 2, 16; red stars solutions 1, 3, 16 and blue circles indicate monitors 1, 9, 11.