Challenge of rainfall network design considering spatial versus spatiotemporal variations

Journal of Hydrology 574 (2019) 990–1002

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Research papers

Challenge of rainfall network design considering spatial versus spatiotemporal variations

T

Bardia Bayata, Khosrow Hosseinia, , Mohsen Nasserib, Hojat Karamia ⁎

a b

Faculty of Civil Engineering, Semnan University, Semnan, Iran School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran

ARTICLE INFO

ABSTRACT

Keywords: Genetic Algorithm Ordinary Kriging Bayesian Maximum Entropy Rain gauge monitoring network Spatiotemporal precipitation Namak Lake watershed

Precipitation data plays an important role in investigation of water-related fields of research such as water resources management, hydraulic structure design and groundwater quantity/quality parameters due to its high variability in space and time. To evaluate and investigate precipitation pattern, a set of well-designed rain gauge stations can substantially reduce the cost and increase the estimation accuracy. In the present research, a new methodology of spatiotemporal optimization is developed for a rain gauge monitoring network and the results are compared to the optimized network based on spatial variations of precipitation. The optimization process consists of two main steps of application of multi attribute decision making and heuristic approaches. The entropy is chosen as the multi attribute decision making approach to determine optimum number of stations. Then, the optimization process is implemented via coupling of Genetic Algorithm (GA) and geostatistical methods to identify the best rain gauge network configuration. To determine the spatiotemporal structure of precipitation, spatiotemporal variography and geostatistical methods known as Ordinary Kriging (OK) and Bayesian Maximum Entropy (BME) have been undertaken. Thirty years of annual precipitation data from 105 rain gauge stations within and near Namak Lake watershed in the central part of Iran are utilized in this research to optimize rain gauge stations spatially and temporally. Results showed that spatiotemporal network design considerably differs from spatial optimal rain gauge stations. Optimal locations of rain gauge stations resulted from spatiotemporal approach are almost two times better than spatial configuration to reduce rain gauge costs (installation, operation, maintenance, etc.) by avoiding data redundancy more efficiently. In addition, BMEbased network design method outperformed OK-based network.

1. Introduction Estimation of precipitation usually faces uncertainties associated with spatial and/or spatiotemporal variations of precipitation and nonuniform distribution of rain gauge networks. These uncertainties greatly affect information required for flood forecasting and design of hydraulic structures. Hence, selection of a reliable and proper network of rain gauge stations has consequently received more attention than before because of water scarcity and dominant local or even regional droughts. A well-designed network of rain gauges is the best network configuration (appropriate numbers and locations) with the least number and best location of stations which reflects the reliable and acceptable spatial/spatiotemporal variability of precipitation over a watershed. This purpose can be achieved by eliminating the redundant stations (Rodriguez-Iturbe and Mejía, 1974; Dong et al., 2005; Nunes et al., 2004) and augmentation of existing network (Cheng et al., 2008;

⁎

Chebbi et al., 2011; Delmelle and Goovaerts, 2009; Delmelle, 2014; Adhikary et al., 2015; Nazaripour and Daneshvar, 2017). Generally, various approaches can be employed to find optimal configuration of stations. These approaches can be categorized into four folds known as statistical, geostatistical, Multi Attribute Decision Making (MADM) and heuristic frameworks. In Table 1, studies related to these approaches are categorized into spatial and spatiotemporal network design studies. As this table shows, geostatistical sampling has been received more attention as a reliable and useful tool for monitoring network design which is utilized in three parts of the table known as purely geostatistical (the second row of the table), combination of MADM and geostatistical (the fourth row of the table) and combination of heuristic and geostatistical (the fifth row of the table). That is because of the fact that geostatistical approach considers variations of the studied variable both spatially and spatiotemporally; it provides Best Linear Unbiased Estimator (BLUE) and the estimated

Corresponding author at: Department of Civil Engineering, Semnan University, Iran. E-mail addresses: [email protected] (B. Bayat), [email protected] (K. Hosseini), [email protected] (M. Nasseri), [email protected] (H. Karami).

https://doi.org/10.1016/j.jhydrol.2019.04.091 Received 7 March 2019; Accepted 28 April 2019 Available online 30 April 2019 0022-1694/ © 2019 Elsevier B.V. All rights reserved.

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Table 1 Different types of spatial (rainfall network)/spatiotemporal (non-rainfall network) network design methods. Network Design Based on: Statistical

Geostatistical

MADM

Description

on the analysis of data within • Based statistical principles various statistical • Considering parameters

on statistical and variogram • Based principles values at unsampled points are • The computed as a weighted average of

•

neighboring data in which the weights are mainly based on variogram modeling Based on probabilistic principles

Spatial Studies

Correlation Approach (Jung et al., 2014; • Spatial Nazaripour and Daneshvar, 2017) Approach (Rodriguez-Iturbe • Variance-reduction and Mejía, 1974; Bras and Rodríguez-Iturbe, 1976; Papamichail and Metaxa, 1996; Dong et al., 2005)

factor and expected level of statistical • Covariance accuracy (Shih, 1982) et al., 1984; Kassim and Kottegoda, 1991; • Bastin Abtew et al., 1995; Tsintikidis et al., 2002; St-

Hilaire et al., 2003; Nour et al., 2006; Cheng et al., 2008; Shafiei et al., 2014; Adhikary et al., 2015

Theory (Krstanovic and Singh, 1992; Al• Entropy Zahrani and Husain, 1998; Yoo et al., 2008;

Vivekanandan and Jagtap, 2012; Xu et al., 2015, Wang et al., 2018)

MADM + Geostatistical

Heuristic + Geostatistical

on the combination of • Based geostatistical and MADM principles

on the combination of • Based geostatistical and Heuristic principles techniques are based on • Heuristic mathematical and random search

Factor + Clustering Technique + Kriging • Analysis (Shaghaghian and Abedini, 2013) Theory + Kriging (Chen et al., 2008; Yeh • Entropy et al, 2011; Awadallah, 2012; Mahmoudi-Meimand et al., 2016)

+ Kriging (Pardo-Igúzquiza, 1998; Barca et al., • SA 2008; Chebbi et al., 2011; Wadoux et al., 2017)

algorithms

MADM + Heuristic

Spatiotemporal Studies

Theory (Mogheir et al., 2009; • Entropy Owlia et al., 2011) Technique + Discriminant • Clustering Analysis (Tanos et al., 2015) Theory + Value of Information • Entropy (Maymandi et al., 2018) + Kriging + BME (Hosseini and • OWA Kerachian, 2017)

+ Kriging (Chadalavada and • GA Datta, 2008; Dhar and Patil, 2012) + Kriging (Nunes et al., 2004; • SA Heuvelink et al., 2012) sequential optimization • Amethod + Kriging (Júnez-Ferreira and Herrera, 2013)

on the combination of MADM and • Based Heuristic principles

Theory + GA (Mahjouri and • Entropy Kerachian, 2011)

Mogheir et al. (2009) assessed and redesigned spatial and temporal frequency of groundwater quality monitoring network in the northern part of Gaza Strip using entropy theory. The temporal frequency was tested using transinformation model by increasing it to once a year and once every two years. Owlia et al. (2011) recommended a methodology to redesign groundwater quality monitoring network, in an aquifer located in Iran, using information content of Transinformation-Distance (T-D) curve (extracted from entropy theory). They measured some statistical factors in each station to find stations with similar temporal trend. These included magnitude of variable in each period, direction of change, higher concentration slope trend and finally dispersion and homogeneity of data around the mean. 2. The second scenario is based on considering spatiotemporal aspect of information and using spatiotemporal simulation method (Heuvelink et al., 2012; Júnez-Ferreira and Herrera, 2013; Hosseini and Kerachian, 2017). Heuvelink et al. (2012) optimized temperature network design via SA coupled with spatiotemporal geostatistics. Their objective function was to minimize the average universal kriging variance. They examined three scenarios of considering both static (fixed location) and mobile (with different starting time) configuration of stations. Júnez-Ferreira and Herrera (2013) proposed a methodology to optimize a hydraulic head monitoring network in an aquifer located in Mexico considering spatiotemporal variations. They used combinations of coupled spatiotemporal variogram and Kalman Filter (KF) to assimilate missing wells’ data. In their methodology, optimal positions of wells were obtained yearly and separately by eliminating the redundant spatiotemporal samples. Hosseini and Kerachian (2017) optimized spatiotemporal groundwater level monitoring network using a combination of Bayesian Maximum Entropy (BME) and Ordered Weighted Averaging (OWA) as a MADM method. Firstly, they determined the priority number of each station to be removed. Then, they have

value can be calculated along with its Estimation Error Variance (EEV). Most of the previous rain gauge monitoring network designs have been aimed at prioritizing/optimizing stations according to their spatial (or temporal lumped) variations of the precipitation. All spatial studies in Table 1 investigate spatial variations of precipitation. Modeling spatial variation individually may lack some valuable information in time scales. Due to possible gaps in spatiotemporal records of precipitation data, spatiotemporal investigation of precipitation has received high attention in recent years (Bayat et al., 2013; Pulkkinen et al., 2016). It seems that no research has been done to investigate the effectiveness of rainfall network design based on spatiotemporal simulation. In other fields of environmental research (especially groundwater monitoring network), there are some studies that have investigated spatiotemporal variations of variable which are presented in spatiotemporal studies in Table 1. Different scenarios dealing with the spatiotemporal aspects of variables (spatiotemporal studies of Table 1) can be categorized into the following sections: 1. The first scenario is to deal with space and time aspects separately (Nunes et al., 2004; Mogheir et al., 2005; Chadalavada and Datta, 2008; Mogheir et al., 2009; Owlia et al., 2011; Dhar and Patil, 2012; Tanos et al., 2015; Maymandi et al., 2018). Nunes et al. (2004) used three optimization criteria (maximization of spatial accuracy, minimization of temporal redundancy and combination of these criteria) to achieve the best subset of stations in groundwater monitoring network. They used Simulated Annealing (SA) method to optimize variance estimation coupled with only spatial geostatistical approach. Chadalavada and Datta (2008) eliminated temporal redundancy in detecting groundwater pollution in a hypothetical contaminated aquifer. Their selected monitoring network design was implemented for each period of time separately. In their sequential approach, the solution of previous management period was considered as an input for the subsequent management time. 991

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tested different time lags and starting points to obtain temporal frequencies.

2.1.1. Calibration network The first fold of the stations, known as calibration rain gauge stations, consists of 105 stations; 87 stations located inside and the rest are scattered outside the watershed. In fact, the optimization process in this research is implemented on calibration network. As seen in Fig. 1, outside stations are entirely located in northern and northeastern parts of the watershed.

In most spatiotemporal researches, spatiotemporal variations have been considered as spatial configuration combined with temporal variation of their temporal snapshot separately, without any spatiotemporal correlation (scenario one). In the second scenario, it was concluded that the procedures and goals of spatiotemporal optimization were different from the methodology applied in this research. In detail, Júnez-Ferreira and Herrera (2013) reached the optimum structure for each year separately. Hosseini and Kerachian (2017) defined optimal location and frequency of stations using MADM approach. They introduced various scenarios regarding to the hexagons side lengths, time lags and starting points for temporal sampling, which can make the optimization problem more time consuming and complex. In these two researches, “time” as mean as sampling frequency was considered as another feature which was optimized throughout the research time period. In fact, in the second scenario, total number of space–time records has been optimized. This approach may not lead to reduction in existing stations. The main objective in this research was to investigate optimal positions of rain gauge stations, while a new framework of spatiotemporal variations of precipitation were considered. In fact, dimension of “time” is considered as the third aspect, which is coupled with spatial dimensions (longitude and latitude). As the most noble point of the current paper and also to achieve comprehensive spatiotemporal dynamics of precipitation, spatiotemporal geostatistical methods known as Ordinary Kriging (OK) and BME were used due for reliable and accurate estimations in the application of inherent spatiotemporal variations of precipitation. One of the important points of the current approach is its independency to missing data at the rain gauge stations. This research is organized into five sections. The first of them is the materials and methods, which consist of the study area and description of the applied methods and proposed spatiotemporal methodology. The second and third parts are related to the algorithm and performance indicators, respectively. The fourth section is the results which categorize into optimal configuration of rain gauge stations obtained from spatiotemporal modeling and also the comparison between spatial and spatiotemporal network design. The last section is dedicated to conclusions and discussion of network design based on spatiotemporal simulation.

2.1.2. Validation network The second fold of stations accounts for 30% of the calibration network, with 31 rain gauges, which are chosen randomly so that at least one station is considered in each region of the watershed. Introducing of this network is for evaluation purposes of the optimized networks. Apart from calibration network, validation stations are more uniformly distributed throughout the watershed. 2.2. Methodology In this research, Genetic Algorithm (GA), as a heuristic method, and entropy theory, as a MADM, have been used as useful tools to reach optimal network configuration. In the current section, spatiotemporal geostatistical framework, entropy theory, GA and the proposed spatiotemporal methodology are briefly described. 2.2.1. Geostatistical framework Combining spatial/spatiotemporal structure of random variables via statistical approaches have been received more attention (Snepvangers et al., 2003; Heuvelink and Griffith, 2010) and different models have been developed to analyze spatial/spatiotemporal random structures (e.g., Goovaerts, 2010; Cameletti, 2012; Pebesma, 2012). Generally, three objectives can be applied regarding to optimal spatial/temporal sampling; variogram estimation, geostatistical formulation and sampling in the multivariate field. Reliable variogram modeling plays an important role in spatial estimation accuracy. In the second case, the goal is to minimize kriging variance. Sampling from the multivariate field (third case), the secondary information can be applied in case of difficult and expensive measurement of the primary variable (Delmelle, 2014). A realization of spatiotemporal random function (RF), Z(s,t), represents continuous spatiotemporal domain where Z demonstrates attribute value at location s (for example any station) at time t. The experimental space–time variogram ( ST ) is given by:

2. Materials and methods

ST (hs ,

2.1. Study area and data description

ht ) = 0.5 Var [Z (s + hs , t + ht )

Z (s, t )]

(1)

where hs and ht are spatial and temporal increments and Var represents variogram. Finding suitable and accurate mathematical variogram structure to represent spatiotemporal domain accurately plays crucial role in geostatistical estimation discourse. To achieve this goal, two types of mathematical spatiotemporal variogram models have been proposed known as separable and non-separable spatiotemporal variograms. Earlier attempts used separable models to construct spatiotemporal covariance functions with some simplified assumptions. These models separate the dependence of space and time. Development of non-separable covariance functions first began by Dimitrakopoulos and Luo (1994). Different researchers have utilized various types of covariance functions, which are known as sum, product, and product-sum mathematical forms (De Iaco et al., 2001; De Cesare et al., 2001a, 2001b). The next step after determination of spatiotemporal variogram/covariance is to implement geostatistical methods; OK and BME, as classic and modern geostatistics, which are utilized in the current research. The estimated variable in geostatistical framework is represented as a weighted linear combination of observed values as follows:

Namak Lake watershed is located between 51° and 52° east longitudes and 39° and 40° north latitudes. The altitude values range from 752 m in eastern areas up to 4330 m in the northeastern regions. Its area is approximately 90,000 km2. This watershed is regarded as one of the most important watersheds in Iran, because Tehran (capital of Iran) is located in it. This altitude variation leads to high variability of precipitation amounts over the watershed; for example, large precipitation amounts (more than 1000 mm) are observed in the north-eastern part of the watershed. Western and south-western areas receive less-intense precipitation (lower than 100 mm) and the remaining regions have median range of precipitation. In the current paper, the rain gauge network has been divided into two sections for calibration and validation purposes, which are illustrated in Fig. 1. Digital Elevation Model (DEM), plotted in this figure, shows that the majority of the watershed has elevation ranging from 1000 m to 2000 m. There are very few areas, located in north-eastern parts, with elevation higher than 3000 m. Eastern regions of the watershed are lowlands with elevation lower than 1000 m. Annual precipitation of 30-year period (1976–2005) are considered in this research.

n

Zk ( s , t ) =

i(s i=1

992

, t ) Zi ( s , t )

(2)

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Fig. 1. Locations of calibration and validation rain gauge stations plotted on DEM map.

and S. At this stage, the Bayesian conditionalization rule is applied. From the posterior PDF, BMEmode (the most likely value at the estimation value) and BMEmean (minimizes the mean square estimation error) can be calculated. In this paper, BMElib package is implemented to analyze space and space-time geostatistical section of monitoring network design. For more details about the BME method, readers are referred to Christakos et al. (2002).

where Zk ( s , t ) is estimated value at unknown point k; Zi ( s , t ) represents the observed value at sampled points i; n is total number of observed points; and i ( s , t ) indicates the weight associated with sample point i. Ordinary Kriging (OK): The OK estimation method, known as classical geostatistics, is derived by solving the following system equations: n j=1

j

( si , sj )

µ = ( si , sk ) n i=1

i

=1

2.2.2. Entropy theory Entropy theory has been widely used for achieving optimal rainfall gauge network (related papers are presented in MADM type of network design in Table 1). Entropy theory is characterized based on information content of station’s distributions which can also describe uncertainty. The main objective in entropy theory is to minimize the redundant information (or to maximize information transmitted by minimum number of stations) which is described by transinformation (T) function. The discretized format of transinformation between two rain gauge stations (x and y) is expressed as:

i = 1, 2, .....,n (3)

where ( si , sj ) is the variogram between two observed points i and j; ( si , sk ) represents the variogram between observed and unknown points i and k; µ demonstrates the Lagrange multiplier. The EEV is expressed as: n 2 R

=

i i=1

( si , sk ) + µ

(4)

n

For more details about the OK method, readers are referred to Kitanidis (1997) and Goovaerts (1997). Bayesian Maximum Entropy (BME): The BME method, known as modern geostatistics, provides a mathematical/ probabilistic framework to accurately estimate the stochastic variable in space and time along with uncertainty information. It takes two knowledge bases; general knowledge base (G) and site-specific knowledge (S). The former demonstrates general characterization of space-time random function such as mean and covariance functions. The latter considers both hard (monitoring data) and soft (measurement errors, model prediction, uncertain information, etc.) data, which can be considered as the difference from OK. Briefly speaking, BME consists of three main stages. At the prior stage, the general knowledge base is characterized to form prior Probability Density Function (PDF). At this stage, the concept of maximum entropy is applied. At the meta-prior stage, the site-specific knowledge is organized into hard and soft data. Finally, at the integration (posterior) stage, posterior PDF is constructed by blending G

n

T (x , y ) =

p (xi , yj ) ln i=1 j =1

p (x i , yj ) p (x i ) p (yj )

(5)

where n is the number of stations; p(xi) and p(xj) are the discrete probabilities of occurrence of two rain gauge stations xi and yj, and P (xi,yj) is the joint (or conditional) probability. The joint probability is measured from a contingency table which is characterized from marginal frequencies of rainfall time series. The procedure to construct T-D relationship is fully described by Mogheir et al. (2006). 2.2.3. Genetic Algorithm The concept and procedure of GA is firstly introduced by Goldberg (1989). It is a random-based Evolutionary Algorithm (EA) searching through possible solutions to find the optimal one. One of the major advantages in GA is to deal with discrete variables with infinite number of possible values which is the purpose of this study. GA consists of three main operators known as initial population, mutation and 993

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Fig. 2. Schematic diagram of proposed spatiotemporal structure.

Fig. 3. The procedure for the assessment of the rain gauge network.

994

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crossover. A binary code to represent the combination of rain gauge stations has been used. Different values of GA parameters including crossover probability (Pc) and mutation probability (Pm) have been evaluated. These parameters range from 0.7 to 0.9 and 0.01–0.1, respectively. The algorithm is stopped when reaches a constant iteration or when no improvement has been observed between a set of iterations.

Bayat et al., 2018). The output of this step is the final number of stations, which are recommended to be removed (N). The detailed process of step 1 approach is fully presented in Bayat et al. (2018). Step 2- Selection of removed stations at GA population stage: As the objective in this paper is to find redundant stations with little information, GA initial population stage is applied on removed stations (N) from the pool of calibration stations (n), which are fully described in the “study area and data description” section. Then, for each chromosome, a set of random N stations were chosen. As can be seen from the example below, step 2 in Fig. 3, stations 5 to n (from N stations) are assumed to be removed for the sample chromosome number 1 (j = 1). Step 3- Removing of entire N stations and returning of them individually: After determination of the selected removed N stations, in this step firstly the whole selected stations are omitted. Then, these removed stations are returned into remaining stations individually. For this configuration of stations, Estimation Error Variance (EEV) and also annual precipitation of the individual removed station (Est) are calculated for each time step (t) via OK and BME estimators. After calculation of these required parameters (EEV and Est) for the selected removed station, this station is being removed from remaining stations and the next station is added. This process is repeated for all the stations. The proposed procedure is also illustrated under this step for the three removed stations (i) as an example. Obs in this example is represented as observed value, which is known. As depicted in Fig. 3, geostatistical framework is implemented to simulate the attribute value in space and time. The precipitation information is classified into two classes known as hard and soft datasets. Among the entire stations (105) and simulation periods (30 years) (105 × 30 = 3150 records), 2903 records of annual precipitation are available to consider as hard data and the remaining part is soft data. Spatiotemporal variogram is calculated based on available annual precipitation. For experimental spatiotemporal covariance, hard data of 30-year time period are collected consecutively. To fit the best covariance function over experimental covariance, various separable product theoretical covariance models for spatial and temporal structures (spherical, Gaussian, etc.) are tested, and finally, the best mathematical spatiotemporal variogram is chosen based on the highest goodness of fit. The best combination of exponential and spherical for space and time are selected. The following equation is assumed for the covariance function:

2.2.4. The proposed spatiotemporal framework As stated in the “introduction” section, the new methodology of considering “time” dimension is utilized in this paper which differs from other approaches in concept and structure. In this approach, the thirty-year annual precipitation data are considered jointly and simultaneously with their missing values. The spatiotemporal covariance is fitted to the existing datasets and then spatiotemporal network design is implemented based on the optimization procedures applied in this paper. After identification of the optimum positions of rain gauge stations, precipitation on grid points are estimated yearly based on the selected geostatistical estimators (OK and BME). Then, grid-wise mean annual precipitation (over the time period) has been calculated. Fig. 2 illustrates schematic spatiotemporal structure of the proposed methodology in 2D + 1 (spatial + temporal) dimensions. The main difference between the proposed framework and the previous studies is that the selected removed stations are considered to be constant for each time period. Time series of annual precipitation for two stations are also drawn in Fig. 2 which shows that stations may consist of missing data during studied time period. 3. Modeling procedure Achieving the best configuration of network via network design methods is based on two different strategies. One of them is founded on adding new station(s) to fill its gap of information and the second strategy is based on removing stations(s) to reduce cost or redundant information. In the current paper, the second area has been chosen to evaluate presented spatiotemporal network design procedure. Fig. 3 illustrates the process of spatiotemporal optimization to find optimal configuration of rain gauge network. As shown in this flowchart, it consists of six main steps which are described in the following steps. Step one relates to entropy theory and other steps demonstrate GA process. For more comprehensive representation of the flowchart, an example of n total stations with N removed stations are also presented under each step. In this example, i and j represent removed stations and chosen chromosomes in GA, respectively. The presented example is supposed to be one of the chosen chromosomes (for instance j = 1). In this example, circular and triangle symbols represent remaining and removed stations in order. Step 1- Achieving the optimum number of rain gauge stations: The main goal of the heuristic optimization process in network design is to identify the best network configuration (number of stations and their locations). In this paper, firstly the removed number of stations is determined based on the application of entropy theory, a widely used method, especially in rainfall monitoring network design. The procedure of entropy-based optimization applied in this paper is in line with that of Mogheir et al. (2006) who used T-D (transinformation-distance) curve. T-D curve and its parameter play important role in selection of optimal stations. After calculation of pair-wise transinformations and distances in rain gauge stations, T-D points are drawn and then the best function is fitted to experimental values. Exponential decay curve is selected as the best fitting function. From this curve, the optimum distance (L) and then optimum grid size (a) can be identified. After fitting the best grid network to the watershed and calculation of one station per grid size based on geostatistical framework, the optimum positions of rain gauge stations can be identified (Mogheir et al., 2009;

COV (r , t ) = nugget + sill [exp(

3 r / sr ) 2] × [1

1.5(t / tr )

0.5(t / tr )3] (6)

where r and t are spatial and temporal lags, nugget is spatiotemporal covariance nugget effect, sill is spatiotemporal covariance sill (variance of spatiotemporal), sr is spatial covariance range and tr is temporal covariance range. The 3D covariance function is illustrated in Fig. 4. As shown in this figure, theoretical covariance has good relationship with experimental covariance values. The values of fitted parameters are nugget = 3078, sill = 2.37 × 10 4 , sr = 1.18 × 105 m and tr = 200 years. Step 4- Objective functions: After determining the removed stations, three Objective Functions (OFs) are calculated over the time of simulation (t) as follows: 1. Estimation Error Variance (EEV) (must be minimized) 2. Mean Square Error (MSE) (must be minimized) 3. Coefficient of Determination (R2) (must be maximized) (7)

EEV (j) = Max {Max [EEVit ]Tt =1 1}iN= 1

MSE (j) =

995

1 N

N i=1

1 T2

T2 t=1

[Estit

Obsit ]2

(8)

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Fig. 4. Spatiotemporal experimental and theoretical covariance functions.

R (j) =

1 N

N i=1

T2 × [

{T2 × Tt =21 [Estit Obsit ]} ( Tt =2 1 Estit )( Tt =2 1 Obsit ) T2 T2 t 2 t 2 T × [ T2 (Obs t )2] ( T2 Obs t )2 2 i i t=1 t =1 t = 1 (Esti ) ] ( t = 1 Esti )

NMSE = (9)

CC (X , Y ) =

where

(12)

Based on the proposed procedure of spatiotemporal network design, the best set of removed stations considering two geostatistical estimators (OK and BME), and three OFs (EEV, MSE and R2) have been presented in Table 2. In this table, optimum GA parameters, optimum fitness function and run time are also viewed, which have been obtained by evaluating various GA parameters. As the fitness function in all OFs values shows, BME results outperformed OK. Generally, run time of OK vs. BME and Spatial (S) vs. Spatiotemporal (ST) has shown considerable lower run time of OK against BME and S against ST. The statistical results of the three OFs (EEV, MSE and R2) were compared separately according to the validation network indicators known as MAE, NMSE and CC. Then, the final conclusion was drawn

V

Est ( si )}

cov (X , Y ) X . Y

(11)

5.1. Results of spatiotemporal network design

To evaluate the performance of proposed network design approach considering spatiotemporal variations of annual precipitation and also the three OFs, validation network (which are fully described in “study area and data description” section) criteria are applied. After determination of estimated (Est ( si ) ) and observed (Obs ( si ) ) precipitation at validation stations ( s = (s1, s2) ), Mean Absolute Error (MAE), Normalized Mean Square Error (NMSE) and Coefficient of Correlation (CC) are calculated as follows:

abs {Obs ( si )

Est ( si )} 2

As stated in the modeling section, the required number of removed rain gauge stations was obtained via entropy theory approach. Bayat et al. (2018) comprehensively described the procedure and pre-results of number of removed stations. According to Bayat et al. (2018), optimum number of removed stations was equal to 15, which will be used in network design based on spatiotemporal simulation coupled with GA with different OFs (EEV, MSE and R2). To achieve the comprehensive perspective, two spatiotemporal geostatistical methods (OK and BME) were also employed to validate the proposed spatiotemporal network design as well. As stated earlier, the following results of spatiotemporal estimation of precipitation over grid points of the watershed have been averaged throughout time. For the sake of simplification while comparing the results, space and time-averaged spatiotemporal variations of annual precipitation were labeled as S and ST. In the current section, firstly, the outcomes of spatiotemporal network design (Section 5.1) are presented and secondly similarity/dissimilarity of the spatial vs. spatiotemporal (Section 5.2) approaches is discussed.

4. Evaluation of the proposed methodology

i=1

{Obs ( si ) i=1

5. Results and discussion

The first OF (EEV) relates to spatial estimation variances obtained from existing stations which depends on the structure and the location of the stations. The second and third OFs are in relation with the accuracy of the candidate removed stations. Step 5- Calculation of OFs in GA crossover and mutation stages: After construction of initial population in GA and related OFs, crossover and mutation stages should be implemented. The procedure for stations configuration and OF calculations is the same as steps 3 and 4. Step 6- Selection of the best OF: In this step, the calculated population, crossover and mutation OFs are put together and the best individuals are obtained based on elitism selection strategy.

1 V

V

where V is the total number of validation dataset, abs operator returns absolute value, S 2 represents the variance of precipitation values of the validation network, cov demonstrates covariance and σX is standard deviation of variable X.

EEVit = Estimation Error Variance of the removed station (i) in time step (t) based on the remaining stations (N-n) Estit = Estimation of precipitation in removed station (i) in time step of (t) based on the remaining stations (N-n) Obsit = Observed precipitation value in the removed station (i) in time step of (t) n = total number of stations N = total number of removed stations T1 = total interested periods of time T2 = total time of available observed precipitation values

MAE =

1 S 2V

(10)

996

997 Pc = 0.8 Pm = 0.1 4 min 103.4

5,9,10,12,24,30,32,51,59,62,69,89,100,102,105 Pc = 0.7 Pm = 0.1 4 min 9719.1

Removed Stations

Optimum GA parameters Average run time for each generation Optimum Fitness Function

Pc = crossover probability. Pm = mutation probability.

**

*

2511.9

12332.9

1,6,10,11,12,26,27,47,49,56,66,67,78102,104

Pc = 0.8 Pm = 0.05 2 h 30 min

Pc* = 0.8 Pm** = 0.01 2h

Optimum GA parameters Average run time for each generation Optimum Fitness Function

Spatial

4,6,15,26,31,47,49,53,56,67,72,77,78,92,102

5,10,17,43,44,51,52,64,65,69,75,80,90,96,104

Removed Stations

MSE

Spatiotemporal

EEV

OK

Optimal Items

Network Design Approach

0.9994

Pc = 0.9 Pm = 0.01 4 min

1,6,9,15,26,35,53,62,64,67,72,83,86,92,102

0.7954

Pc = 0.8 Pm = 0.01 2 h 30 min

5,19,31,48,51,57,62,65,72,74,80,82,84,95,104

R2

Table 2 Optimum removed rain gauge stations and GA parameters obtained from spatiotemporal and spatial variations of precipitation.

10134.9

Pc = 0.8 Pm = 0.1 31 min

5,10,12,17,24,30,43,49,57,64,77,80,86,89,100

11100.3

Pc = 0.9 Pm = 0.1 13 h

5,9,10,17,36,51,52,59,65,69,75,80,100,102,105

EEV

BME

53.7

Pc = 0.7 Pm = 0.1 25 min

1,6,10,11,27,47,49,53,56,66,73,90,98,102,104

2347.4

Pc = 0.8 Pm = 0.08 13 h

1,4,35,47,51,53,55,56,67,74,77,78,82,92,102

MSE

0.9991

1,6,15,26,35,53,56,62,64,65,74,83,86,92,105 Pc = 0.8 Pm = 0.1 28 min

0.8014

1,5,19,36,51,57,62,64,65,74,77,84,90,95,100 Pc = 0.9 Pm = 0.05 12 h

R2

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– Comparing the results of OK and BME estimators have shown that BME experienced better results, which were on average 7.7%, 21.8% and 3.3% for MAE, NMSE and CC, respectively. As an example in MAE indicator, MAE value for OK-EEV-ST and BME-EEVST were 37.6 and 34.3; a superiority of 8.58%. NMSE values for OKR2-ST and BME-R2-ST were 0.241 and 0.187; a progress of 22.4%. Finally, CC values for OK-R2-ST and BME- R2-ST were 0.87 and 0.90 which have shown 3.94% superiority.

Table 3 Statistical indicators of the optimized network based on spatial (Bayat et al. 2018) and averaged spatiotemporal variations. Operators

OK

Objective Functions

Spatial/ Spatiotemporal

Validation Network MAE

NMSE

CC

EEV

Spatial (S) Spatiotemporal (ST) Spatial (S) Spatiotemporal (ST) Spatial (S) Spatiotemporal (ST)

65.3 37.6 59.9 35.3 64.3 38.6

0.442 0.238 0.347 0.209 0.404 0.241

0.77 0.88 0.81 0.89 0.78 0.87

Spatial (S) Spatiotemporal (ST) Spatial (S) Spatiotemporal (ST) Spatial (S) Spatiotemporal (ST)

61.1 34.3 59.6 32.9 64.5 35.6

0.333 0.182 0.322 0.168 0.376 0.187

0.81 0.91 0.82 0.92 0.79 0.90

MSE R2 BME

EEV MSE R2

As can be seen from the described evaluation remarks, it can be totally concluded that MSE outperformed others and R2 received the lowest accuracy. 5.2. Spatial and spatiotemporal: similarities and dissimilarities In the current section, network design based on spatial (S) and timeaveraged spatiotemporal (ST) approaches have been compared. The records of validation network indicators in Table 3 show that the differences between the two approaches ware highly significant and ST outcomes are almost two times better than S records.

after analyzing the results. The results are shown in Table 3. Discussions of the results only for ST network design part are as follows:

– The superiority of ST than S in MAE and NMSE indicators (averaged on the whole OFs) was observed 70% and 72% by means of OK estimator, respectively. These percentages were also confirmed by BME estimator which was reported as 80% and 91% superiority of ST vs. S in MAE and NMSE indicators, respectively. As an example in OK estimator, MAE values in OK-MSE-S and OK-MSE-ST were reported 59.9 and 35.3 which have been shown 69.7% more accuracy of the ST relative to S. As an another case in BME method, NMSE values in BME-R2-S and BME-R2-ST were equal to 0.376 and 0.187, which implies 103.2% higher performance of ST against S.

– Comparison of MAE, NMSE and CC values implied the relatively low superiority of MSE against EEV and R2. The MAE, NMSE and CC for OK-MSE-ST were 35.3, 0.209 and 0.89, respectively. These values for the worst OF, OK-R2-ST, were 38.6, 0.241 and 0.87, respectively, which have shown 9%, 13% and 3% greater accuracy of OK-MSE-ST than those obtained from OK-R2-ST. These conclusions were also true for BME estimator, which has shown around 8%, 10% and 2% superiority of BME-MSE-ST (the best OF) compared with BME-R2-ST (the worst OF).

Fig. 5. Distribution of removed rain gauge stations based on spatial (S) network design approach for whole OFs based on OK (a) and BME (b) estimators. 998

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Fig. 6. Distribution of removed rain gauge stations based on spatiotemporal (ST) network design approach for whole OFs based on OK (a) and BME (b) estimators.

– Considering CC values, they were higher in ST approach compared to S. The greatest increase in OK estimator was stated by OK-EEV-S vs. OK-EEV-ST, which were reported as 0.77 and 0.88; a surge of 0.11 unit. This approach based on BME was between BME-R2-S and BME-R2-ST, which was the same enhancement of 0.11 unit. Comparison of OK and BME estimators by means of CC values also confirmed the superiority of BME against OK method. The most recognized case was related to OK-EEV-S vs. BME-EEV-S, which was recorded 0.04 unit more rigorous results of BME than OK.

MSE-based removed stations were more scattered compared to EEVbased and R2 –based removed stations. For more comprehensive analyses regarding the removed stations, Table 4 demonstrates the percentage of similarity in removed rain gauge stations. Discussion about optimization approaches (S/ST) and OFs (EEV, MSE and R2) in this table can be demonstrated from two points of view. – In the first view of Table 4, the main focus is to compare the three OFs in each estimator/optimization approach separately (OK-S, BME-S, OK-ST and BME-ST). In OK-S, the highest percentage was among MSE and R2 (OK-MSE-S and OK-R2-S) with 33.3% similarity. This conclusion was also proven in spatial removed stations distribution presented in Fig. 5a. In BME-S, whole of the OFs experienced the same similarity of 13.3%. The records from OK-ST and BME-ST indicated that EEV and R2 had the highest relationship with 40% and 33.3% similarities. These findings were also identified from Fig. 6a and b for OK-ST and BME-ST methods sequentially. – In the second view of Table 4, the percentage of similarity in the same OF (OK-S vs. BME-S, OK-ST vs. BME-ST, OK-S vs. OK-ST and BME-S vs. BME-ST) was presented. As can be seen from this table, OK-S and BME-S had the highest similarities, which were 46.7%, 73.3% and 73.3% for EEV, MSE and R2 in order. OK-ST and BME-ST received the second highest similarities, which were 53.3%, 60% and 66.7% for EEV, MSE and R2, respectively. The comparison of OK (OK-S vs. OK-ST) and BME (BME-S vs. BME-ST) results considering whole OFs have shown higher similarity of BME estimator. As an example, BME-EEV-S vs. BME-EEV-ST percentage of similarity was 33.3%, while this percentage for OK-EEV-S vs. OK-EEV-ST was stated 26.7%. As the previous results presented, MSE was considered as the most accurate in both S and ST approaches and also both OK

Distribution of the removed rain gauge stations for S and ST optimization approaches are illustrated in Figs. 5 and 6, in which, square, cross and circle symbols are represented as EEV, MSE and R2, respectively. In these figures, left and right figures relate to OK and BME estimators, respectively. As shown in both S and ST distributions and whole OFs, the spread of removed rain gauge stations estimated by both OK and BME methods were almost the same. More specifically, removed stations in EEV-S, EEV-ST and R2-ST (both OK and BME) were more located in northern regions of the watershed, which were mountainous areas. Others, MSE-S, MSE-ST and R2-S (both OK and BME), were more scattered. The comparison between three OFs of the removed stations in S approach (Fig. 5) demonstrates that MSE and R2 have almost the same spread of stations and extended into the whole areas. On the other hand, removed stations based on EEV OF were more concentrated in northern parts of the watershed than the other OFs. Locations of MSE-based removed stations in ST approach (Fig. 6) have shown that they were mostly distributed in central and northern regions of the watershed. Distribution of both EEV-based and R2 –based removed stations have demonstrated that they were more localized in northern parts of the study area. It can be generally concluded that 999

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Table 4 Percentage of similarity between optimal configurations of removed stations based on applied optimization approaches. OK-S EEV OK-S

EEV MSE R2

BME-S

EEV MSE R2

OK-ST

EEV MSE R2

BME-ST

BME-S MSE

R

2

EEV

20.0

20.0 33.3

46.7

OK-ST MSE

R

2

EEV 26.7

73.3

73.3

13.3

BME-ST MSE 53.3

R

2

33.3

0.0

MSE

40.0 13.3

53.3

73.3

60.0 13.3

and BME estimators. Table 4 results also confirm the same conclusions so that MSE in whole cases (OK-S vs. BME-S, OK-ST vs. BMEST, OK-S vs. OK-ST and BME-S vs. BME-ST) received the highest percentage of similarity. The percentages are reported as 73.3%, 60%, 53.3% and 73.3%, respectively.

R2

20.0

13.3 13.3

EEV MSE R2

EEV

26.7

66.7 33.3 20.0

(OK-MSE-S vs. OK-MSE-ST) and BME (BME-MSE-S vs. BME-MSE-ST) estimators. The inferred points of OK q-q plot (Fig. 8a) have shown that S versus time-averaged ST had almost the same statistical probability of exceedance up to 450 mm of annual precipitation. After this critical point, the values of time-averaged ST in OK estimator have been shifted upward about 50 mm and have got away from the bisector line. But in BME estimator, precipitation values of more than 450 mm are fluctuating around the bisector line. This shows more compatibility of S and ST precipitation values in BME method compared to OK. Grid points of precipitation values more than 450 mm were all located in northeastern parts of the watershed with the highest elevation data.

5.2.1. Annual precipitation pattern In this part, spatial pattern of annual precipitation for the best OFs is depicted to investigate the effect of removed rain gauge stations on contour lines of precipitation. As described earlier, MSE has been considered as the best OF in both S and ST network design approaches and also both OK and BME estimators, which were extracted from Table 3. As an example, spatial precipitation patterns of BME-MSE-S and BME-MSE-ST are illustrated in Fig. 7, which have demonstrated the highest differences. In these figures, white circles and black squares are the optimized combination of removed and remaining stations, respectively. The three most recognizable differences in contour lines are spotted by square. In these locations, the effect of rain gauge removed stations on contour lines can be identified clearly. These observations have demonstrated that how loss of one or two stations can change precipitation information and pattern in those areas.

6. Conclusions In this research, a new framework of rain gauge monitoring network optimization considering spatiotemporal variations of annual precipitation is presented. The methodology was applied to Namak Lake watershed located approximately in central part of Iran. The proposed methodology in the current paper is systematically and methodologically different from previous approaches in the context of rainfall network design methods. In the previous researches of spatiotemporal network design, temporal frequency and spatial position of stations were separately identified for each time lag. However, in the current paper, “spatial” and “temporal” dimensions were considered as a unified information structure. The proposed optimization process is the hybrid of GA and

5.2.2. Distribution assessment of spatial versus spatiotemporal results To assess distribution of the results, q-q plots of S versus timeaveraged ST values of annual precipitation are illustrated in Fig. 8. These figures were related to the best configurations resulted from OK

Fig. 7. Annual precipitation pattern for the best objective function obtained from BME-MSE-S (a) and BME-MSE-ST (b) (white circles, black squares and black crosses are removed, remaining and auxiliary stations, respectively). 1000

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during time period. In the second challenge, decision makers should keep in mind the accuracy, computational time and complexity of each optimization approaches and select the best approach based on the their project goals. Among spatial and spatiotemporal objective functions, MSE performed the best in both OK and BME simulations. This conclusion can also be confirmed by similarity percentages in Table 4 which were reported as 73.3%, 53.3%, 73.3% and 60% in OK-S vs. BME-S, OK-S vs. OK-ST, BME-S vs. BME-ST and OK-ST vs. BME-ST, respectively. As a further investigation of this research, other heuristic methods and objective functions can be dedicated to investigate the effects of searching algorithms presented in other optimization methods. Another area for future investigation is evaluation of the proposed spatial and spatiotemporal network design considering collocated geostatistical methods (such as co-kriging) and also space/spatiotemporal clustering to assess their effects on network design. In addition, to quantify uncertainties related to the observed and estimated precipitation, fuzzy mathematics can be recommended. Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement Authors would like to thank the anonymous reviewers for their invaluable comments and recommendations that improved the quality and clarity of the manuscript. References Abtew, W., Obeysekera, J., Shih, G., 1995. Spatial variation of daily rainfall and network design. Trans. ASAE 38 (3), 843–845. Adhikary, S.K., Yilmaz, A.G., Muttil, N., 2015. Optimal design of rain gauge network in the Middle Yarra River catchment, Australia. Hydrol. Process. 29 (11), 2582–2599. Al-Zahrani, M., Husain, T., 1998. An algorithm for designing a precipitation network in the south-western region of Saudi Arabia. J. Hydrol. 205 (3–4), 205–216. Awadallah, A.G., 2012. Selecting optimum locations of rainfall stations using kriging and entropy. Int. J. Civil Environ. Eng. 12 (1), 36–41. Barca, E., Passarella, G., Uricchio, V., 2008. Optimal extension of the rain gauge monitoring network of the Apulian Regional Consortium for crop protection. Environ. Monit. Assess. 145 (1–3), 375–386. Bastin, G., Lorent, B., Duque, C., Gevers, M., 1984. Optimal estimation of the average areal rainfall and optimal selection of rain gauge locations. Water Resour. Res. 20 (4), 463–470. Bayat, B., Zahraie, B., Taghavi, F., Nasseri, M., 2013. Evaluation of spatial and spatiotemporal estimation methods in simulation of precipitation variability patterns. Theor. Appl. Climatol. 113 (3–4), 429–444. Bayat, B., Nasseri, M., Hosseini, Kh., Karami, H., 2018. Revisited rainfall network design: Evaluation of heuristic versus entropy theory methods. Arabian J. Geosci. 11, 561. Bras, R.L., Rodríguez-Iturbe, I., 1976. Network design for the estimation of areal mean of rainfall events. Water Resour. Res. 12 (6), 1185–1195. Cameletti, M., 2012. Spatio-Temporal Models in R Package Vignettes. CRAN, Universit‘a di Bergamo. Chadalavada, S., Datta, B., 2008. Dynamic optimal monitoring network design for transient transport of pollutants in groundwater aquifers. Water Resour. Manage. 22 (6), 651–670. Chebbi, A., Bargaoui, Z.K., Cunha, M.D.C., 2011. Optimal extension of rain gauge monitoring network for rainfall intensity and erosivity index interpolation. J. Hydrol. Eng. 16 (8), 665–676. Chen, Y.C., Wei, C., Yeh, H.C., 2008. Rainfall network design using kriging and entropy. Hydrol. Process. 22 (3), 340–346. Cheng, K.S., Lin, Y.C., Liou, J.J., 2008. Rain-gauge network evaluation and augmentation using geostatistics. Hydrol. Process. 22 (14), 2554–2564. Christakos, G., Bogaert, P., Serre, M.L., 2002. Temporal GIS. Springer-Verlag Press, New York. De Cesare, L., Myers, D.E., Posa, D., 2001a. Estimating and modeling space–time correlation structures. Stat. Prob. Lett. 51 (1), 9–14. De Cesare, L., Myers, D.E., Posa, D., 2001b. Product-sum covariance for space-time modeling: an environmental application. Environmetrics 12 (1), 11–23. De Iaco, S., Myers, D.E., Posa, D., 2001. Space–time analysis using a general product–sum model. Stat. Prob. Lett. 52 (1), 21–28. Delmelle, E.M., 2014. Spatial sampling. In: Handbook of Regional Science. Springer,

Fig. 8. q-q plot of the best network for OK (a) and BME (b) estimators.

spatiotemporal geostatistical method to identify optimal position of rain gauge stations in terms of three objective functions known as minimizing the EEV, MSE and maximizing the R2. These OFs mainly correspond to the variances and accuracy of the model. The most crucial issue in developed methodology is that no gap filling was implemented throughout the optimization process, not to include another source of error related to estimation process. In this research, different configurations of rain gauge stations obtained from spatial and spatiotemporal precipitation variations can be considered as a controversial issue among decision makers. In the developed approach, spatiotemporal annual precipitation was modeled considering spatiotemporal covariance function and then estimation process was implemented via classical and modern geostatistical methods known as OK and BME. It was concluded that spatiotemporal framework of rain gauge network design can substantially improve the accuracy of estimation compared with the network design with spatial variations of precipitation. Records of validation network indicators have indicated that optimized network obtained from spatiotemporal precipitation variations were almost two times better than spatial variations. This issue can be challenged into two folds. The first is the fact that spatiotemporal variations of annual precipitation can be modeled more reliably and truly by considering time series of precipitation with missing data. While, spatial variations of annual precipitation are averaged 1001

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