Optimal Development of Regional Rain Network Using Entropy and Geostatistics Hadi Mahmoudi-Meimand, Sara Nazif and Hasan-Ali Faraji-Sabokbar
1 Introduction Rainfall is hydro-meteorological variable with drastic spatial and temporal variations. In recent years, more attention is paid to application of entropy theory and geostatistical method in rain gauge network design. Karamouz et al. [1], utilized the measure of transinformation in entropy theory for selecting the best water quality monitoring stations from a set of potential monitoring sites along a river. Yeh et al. [2] introduced a model composed of kriging and entropy with probability distribution function to relocate the rainfall network and to obtain the optimal design with the minimum number of rain gauges. Chen et al. [3] proposed a method composed of kriging and entropy that can determine the optimum number and spatial distribution of rain gauge stations in catchments. In this study the entropy and geostatical methods are used in combination to improve the rain gauge network density.
2 Materials and Methods In this paper, the optimal locations of rain gauges development in a monitoring network are determined using the entropy theory by considering the maximum uncertainty (minimum redundant information in the system) and the maximum rainfall estimation error based on Kriging method. H. Mahmoudi-Meimand (B) · H.-A. Faraji-Sabokbar Department of Cartography, University of Tehran, Tehran, Iran e-mail:
[email protected] H.-A. Faraji-Sabokbar e-mail:
[email protected] S. Nazif School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran e-mail:
[email protected] E. Pardo-Igúzquiza et al. (eds.), Mathematics of Planet Earth, Lecture Notes in Earth System Sciences, DOI: 10.1007/978-3-642-32408-6_108, © Springer-Verlag Berlin Heidelberg 2014
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Fig. 1 Location and rain gauge of the study area
2.1 Calculating the Rainfall Estimation Variance Using Kriging and the Transformation Entropy In this study, the rainfall is spatially interpolated using ordinary kriging. In kriging the error associated with the estimates in points with no measurements are calculated. Rainfall Kriging Variance is calculated by the following formula: 2
σ=μ+ z˜
N i=1
λi γi0 .
(1)
where σ2z˜ is the kriging variance, which provides a measure of the error associated with the kriging estimator, μ is the Lagrange parameter and γi0 is Variogram values i between the i sample and estimated spot and λi is the weight of rain gauge i. Due to high variability of rainfall in different months, the weighted average of monthly rainfall is estimated. The final layer of rainfall estimation error at the basin for the maximum kriging error as a measure of choice in network structure optimization model for locating new stations rain guage used. The transferable information T(X,Y) of two rain gauge stations X and Y is the mutual information or overlapped information of X and Y, where f(x, y) is the joint
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Fig. 2 The final layer of the variance estimates at the basin
entropy of x and y, f(x) and f(y) are the probability density function of the variables x and y, respectively. Transinformation (T(x, y)) in the discrete form can be expressed as follows: T(x, y) = T(y, x) = −
∞ −∞
f(x, y) . f(x, y)Ln f(x)f(y) −∞ ∞
(2)
Equation (2) for variables with normal distribution can be simplified as follows [4]: 1 2 T(x, y) = − Ln 1 − rxy 2
(3)
where rxy is the correlation coefficient between x and y. The final layer of entropy transfer at the basin for the minimum transformation entropy as a measure of choice in network structure optimization model for locating new stations rain guage used.
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Fig. 3 The final layer of entropy transfers at the basin level
3 Rain Gauge Network Optimization Model The objective of this model is combination of maximizing the minimum transinformation entropy and minimizing the maximum Error variance which is quantified as follows: n m
2
2 . aij α1 Errsij − Errsmax + α2 Entsij − Entsmin Min Z = j=1
i=1
(4) The main constraints are as follows: r −Ents Errsij −r2 r1 , cij > , cij < aij = bij × cij , bij > 1 r1 ij , bij < Ents r2 ij aij , bij , cij ∈ [0, 1] .
Errsij 2 i=1 αi =1, r2 ,
(5) To standardize these criteria, the following relations are used: Entsij =
Ent ij − Ent min Ent max − Ent min
Errsij =
Err ij − Err min Err max − Err min
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Fig. 4 Location of 17 selected rain gauge stations of the study area
where Entij and Errij are the real value of entropy and error kriging, respectively. Errmax = max(Errij ) and Entmin = min(Entij ). Errsmax = max(Errsij ), Entsmin = min(Entsij ). r1 is the threshold value for the selected range of data entropy and r2 is the threshold value for the selected range of data error kriging. aij , bij and cij are auxiliary binary variables. α1 and α2 coefficients are between 0 and 1. Their values are determined by the decision maker.
4 Results and Discussion Rainfall data of the 49 stations in the Karkhe located at the south western part of Iran (Fig. 1) in months October to April calculated for each month. The developed layers of rainfall estimation variance are weighted based on monthly rainfall and overlaid. The final layer of the overlapping layers are obtained and as a measure of the objective function be considered (Fig. 2). The transformation entropy layer is developed in the similar way (Fig. 3). The estimation variance and entropy layers are combined based on the objective function and the optimal points of rain gauges
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development are determined based on the optimization model. In the considered case study, the application of the proposed method to an existing rain network over the Karkhe catchment region under a minimum transformation entropy of 30 % and maximum Kriging error of 60 % resulted in 17 new rain stations to be added to the original network (Fig. 4).
References 1. Karamouz, M., Nokhandan, A. K., Kerachian, R., & Maksimovic, C. (2009). Design of online river water quality monitoring systems using the entropy theory: A case study. Journal of Environmental Monitoring and Assessment, 155, 63–81. 2. Yeh, H. C., Chen, Y. C., Wei, C., & Chen, R. H. (2011). Entropy and kriging approach to rainfall network design. Journal of Paddy and Water Environment, 9, 343–355. 3. Chen, Y.-C., Wei, C., & Yeh, H.-C. (2008). Rainfall network design using kriging and entropy. Journal of Hydrological processes. 22, 340–346. 4. Mogheir, Y. (2003). Assessment and redesign of groundwater quality monitoring networks using the entropy theory-Gaza strip case study. PhD thesis, University of Coimbra, Coimbra, Portugal.
