Parameter Optimization of KINEROS2 Using Particle Swarm Optimization Algorithm Within R Environment for Rainfall–Runoff Simulation

5 Parameter Optimization of KINEROS2 Using Particle Swarm Optimization Algorithm Within R Environment for Rainfall
Runoff Simulation Hadi Memarian1, Mohsen Pourreza Bilondi2, Zinat Komeh3 1

FACULTY OF NATURAL RE SOURCE S A ND ENVIRONM ENT, DEPARTMENT OF WATERSHED MANAGEME NT , UNIVERSITY OF B IRJAND, BIRJAND, IRAN 2 FACULTY O F AGRICUL TURE, DEPARTME NT OF WATER E NGINEERING, UNIVERS ITY OF B IRJAND, B IRJAND, IRAN 3 F AC U LT Y OF NATURAL RE SOURCE S A ND ENVIRONM ENT, DEPARTMENT OF WATERSHED ENGINEERING, UNIVERSITY OF BIRJAND, BIRJAND, IRAN

5.1 Introduction Flooding can directly impact: the wellbeing and prosperity of natural life and domesticated animals; riverbank erosion and sedimentation; the dispersal of supplements and contamination; surface and groundwater supplies; and local landscapes and living spaces (Bonacci, 2007; Bubeck et al., 2017; Shapiro, 2016). Simulation of the rainfall
runoff (flooding) process in the watershed is particularly important in order to have a better understanding of hydrological issues, water resource management, river engineering, flood control structures, and flood storage (Neitsch, Williams, Arnold, & Kiniry, 2011). Models of different types provide a means of quantitative extrapolation or prediction that will hopefully be helpful in decisionmaking (Beven, 2011). Recently, the application of models has become an essential tool for understanding the natural processes that have occurred in the watershed (Sorooshian & Gupta, 1995). Rainfall and runoff are the important phases of the hydrological cycle, and the basis of a hydrological model is to examine the relationship between rainfall and runoff (Knapp, Ortel, & Larson, 1991). The amount of runoff, erosion, and sediment transport changes depending on various hydrological conditions, soil, and cover in the basin. For example, the amount of sediment yield due to soil erosion may be significantly influenced through water-harvesting techniques (Grum et al., 2017), slope gradient (Wu, Peng, Qiao, & Ma, 2018), and different soil and water conservation strategies (Melaku et al., 2018). Spatial Modeling in GIS and R for Earth and Environmental Sciences. DOI: https://doi.org/10.1016/B978-0-12-815226-3.00005-3 © 2019 Elsevier Inc. All rights reserved.

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Certainly, considering these factors could be useful for land-use planners who intend to implement different measures of catchment management. Therefore, simulating the above processes requires the necessary information of spatial variation of these factors (Azizian & Shokoohi, 2014). In this regard, natural process modeling based on a geographic information system (GIS) is an important tool in the study of runoff and soil erosion and consequently in the development of appropriate strategies to conserve soil and water, especially at the watershed scale (Memarian et al., 2013). Thus far, different rainfall
runoff models have been tested in watersheds with different climates, such as Identification of unit Hydrograph and Component flows from Rainfall Evaporation and Stream flow data (IHACRES), Hydrologic Engineering Center
Hydrologic Modeling System (HEC
HMS), and Hydrologiska Byrans Vattenavdelning (HBV). Different physical models have been also examined to estimate the erosion/sediment process within the watershed, including: Areal Nonpoint Source Watershed Environment Response Simulation (ANSWERS) (Beasley, Huggins, & Monke, 1980), Chemicals Runoff and Erosion from Agricultural Management systems (CREAM) (Knisel & Foster, 1981), Erosion Productivity Impact Calculator (EPIC) (Williams, 1989), Simulator for Water Resources in Rural Basin (SWRRB) (Williams, Nicks, & Arnold, 1985), and Soil and Water Assessment Tool (SWAT) (Arnold, Williams, Srinivasan, King, & Griggs, 1994). KINEROS2 (KINematic runoff and EROSion), or K2, originated at the USDA Agricultural Research Service (ARS) in the late 1960s as a model that routed runoff from hillslopes, represented by a cascade of overland-flow planes using the stream path analogy proposed by Onstad and Brakensiek (1968), and then laterally into channels (Woolhiser, Hanson, & Kuhlman, 1970). Conceptualization of the watershed in this form enables solution of the flow-routing partial differential equations in one dimension. Rovey (1977) coupled interactive infiltration to this model and released it as KINGEN. After substantial validation using experimental data, KINGEN was modified to include erosion and sediment transport as well as a number of additional enhancements, resulting in KINEROS, which was released in 1990 (Smith, Goodrich, Woolhiser, & Unkrich, 1995; Woolhiser, Smith, & Goodrich, 1990). Kalin and Hantush (2003) evaluated the efficiency of GSSHA and KINEROS2 models in simulating runoff and sediment process. Based on the results, the K2 model, due to a better formulation of the algorithm, had a better and stronger efficiency than the Gridded Surface/Subsurface Hydrologic Analysis (GSSHA) (Downer & Ogden, 2004) model in sediment routing. In another study by Smith, Goodrich, and Unkrich (1999), the ability of KINEROS2 to simulate sediment and runoff by selective rainfall events in the basin of Catsop, the Netherlands, has been investigated. According to simulation results due to the lack of data, a detailed hydrologic simulation is needed to simulate erosion successfully. In addition to the above cases that examine the simulation of sediment transport in the watershed, other studies have been done on flood simulation using this model, including the studies of Schaffner, Unkrich, and Goodrich (2010). They assessed flash flood prediction using near real-time radar-rainfall estimates of the National Weather Service. Michaud and Sorooshian (1994) employed this model to achieve a flash flood forecasting with typical Automated Local Evaluation in RealTime (ALERT) data constraints, as well. Flood risk prediction under land-use change was

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also achieved as an application of KINEROS2 (Nikolova et al., 2009). The application of the event-based physical model, KINEROS2, on a developed tropical watershed in Malaysia was evaluated (Memarian et al., 2012). Three storm events of different intensities and durations were applied for K2 calibration. K2 validation was done using two other rainfall events before and after the calibration year. The results established that K2 could simulate runoff well, but its capability in sediment load estimation was mostly limited to the accuracy of input data, mainly land-use maps. Namavar (2011) evaluated the efficiency of KINEROS2 in predicting runoff in the Kameh watershed, Iran, through the AGWA (Automated Geospatial Watershed Assessment) toolbox (Miller et al., 2007) in the ArcGIS environment. According to the results, KINEROS2 satisfactorily simulated hydrograph shape and peak magnitude. In addition, the ability of sediment estimation was used in the watershed modeling, but was not calibrated due to the lack of sediment data. Molaeifar (2013) evaluated the efficiency of KINEROS2 in hydrograph simulation in the Ziarat watershed, Iran, and concluded that the model can estimate the hydrological components with acceptable accuracy. KINEROS2 was also integrated with a flow/sediment transport solver to analyze the relationship between hydrologic response, management, and geomorphometrics (Norman et al., 2017). Gabriel et al. (2016) employed the KINEROS2/AGWA interface to compare several spatial and temporal rainfall representations of postfire rainfall
runoff events. They determined the effect of differing representations on modeled peak flow and obtained at-risk locations within a watershed. Small-scale researches have also been conducted using K2. Kim et al. (2014) conducted their experiments during four seasons in mixed tall fescue
Bermuda grass coarse sandy loam soils with 3%
10% slopes and variable rainfall rates. Parameter ESTimation (PEST) algorithm (Doherty, Brebber, & Whyte, 1994) was also used to assess and calibrate the nine KINEROS2/STWIR parameters on 36 plots. Mirzaei et al. (2015) also studied the uncertainty in rainfall and the input parameters of the KINEROS model which affect the model output (runoff). They aimed at better quantifying the magnitude and uncertainty of extreme precipitation
runoff events. Manual calibration of hydrological models has been used since the early 1960s, but due to its complexity and being time consuming, automatic calibration has been available since the end of the 1960s. Autocalibration needs an appropriate objective function, search algorithm, and a criterion to complete the algorithm. However, in the early years, using this method has not been very successful (Gupta, Sorooshian, & Yapo, 1999). From one side, most of the obtained parameters have not been real conceptually; on the other hand, the efficiency of the model on various data was different and calibration results were affected by the selected data, the initial guess for the parameters, the objective function, and the search process (Sorooshian & Gupta, 1983). Currently, there are a few known issues that have made some serious problems for studies related to the optimum parameters set. These problems include several local optimum set, numerical granularity, nonconvex response surface, nonlinear dependence of parameters, interaction of parameters on each other, creating a saddle point where the first derivative is toward zero, outlier data and deviation, autocorrelation, anisotropy, and variance in the residual error (Beven & Binley, 1992; Zambrano-Bigiarini & Rojas, 2013). In order to solve the problems mentioned above, advanced calibration and optimization algorithms and techniques have been proposed. These techniques include

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simulated annealing (SA), genetic and evolutionary programming (GP and EP) (Goldberg, 1989), particle swarm optimization (PSO) (Kennedy & Eberhart, 1995), ant colony optimization (Dorigo & Stutzle, 2004), differential evolution (DE) (Storn & Price, 1997), and adaptive multimethod searching or AMALGAM (Vrugt & Robinson, 2007). Recently, hybrid algorithms have attracted much consideration and led to the identification of some new algorithms, such as Bees Algorithm Hybrid with Particle Swarm Optimization (BAHPSO) (Zarea, Kashkooli, Soltani, & Rezaeian, 2018) and Shuffled Complex-Self Adaptive Hybrid EvoLution (SC-SAHEL) (Naeini et al., 2018). Among the methods mentioned above, the PSO algorithm, due to its flexibility, easy implementation, and high performance, has been favored by many researchers in recent years. This method has a high rate of convergence and suitable computational cost (Parsopoulos & Vrahatis, 2002). The R package is the most important software which uses the PSO algorithm to optimize hydrological models and to implement sensitivity analysis, model calibration, and results analysis using the hydroPSO tool as an independent package. This package is able to be connected with various hydrological models. Thus far, the connection of hydroPSO has been conducted by SWAT (Abdelaziz & Zambrano-Bigiarini, 2014) and MODFLOW (Zambrano-Bigiarini & Rojas, 2013) models. In recent years, the PSO algorithm has been increasingly applied in the estimation of parameters of hydrological models (Baltar & Fontane, 2004; Gill, Kaheil, Khalil, McKee, & Bastidas, 2006; Jiang, Liu, Huang, & Wu, 2010). Kamali, Mousavi, and Abbaspour (2013), in a study, conducted automatic calibration on HEC
HMS using single- and multiobjective PSO algorithms to model the rainfall
runoff process in Tamar basin. For this purpose, they used three events to calibrate the model and one event to validate the model. According to the results, multiobjective calibration could outperform the single-objective calibration technique. Gill et al. (2006) employed multiobjective particle swarm optimization (MOPSO) to optimize 13 parameters of a rainfall
runoff model in Sacramento. They also used MOPSO to calibrate a vector model with three parameters to predict soil moisture. Jiang, Li, and Huang (2013) applied PSO for calibration of the rainfall
runoff model HIMS. They compared classical PSO algorithm with distributed PSO versions, which are using the complexes and shuffling mechanism. Results indicated that the distributed PSO variants were significantly better than the original one. Abdelaziz and Zambrano-Bigiarini (2014) studied adaptability and capability of hydroPSO to optimize hydrological models within R software in the Geneiss watershed, Germany. According to the results, PSO was useful to optimize MODFLOW. Furthermore, due to the use of parallel processing systems, the application of hydroPSO could reduce the number of iterations required to achieve optimum value, and consequently could reduce the time of the modeling computation to one-eighth of the total time. Zambrano-Bigiarini and Rojas (2013) used the hydroPSO package as an independent package in R software to calibrate SWAT and compared hydroPSO with standard algorithms using a series of specific functions in two different watersheds. According to the results, hydroPSO is an efficient and suitable method compared to other common optimization algorithms. Moreover, this is a scalable software package, that is, the efficiency of the model is preserved by increasing the dimension of the problem and is adaptable to different problems.

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This research was planned to achieve the following main goals: 1. To model rainfall
runoff processes in the GIS environment and to extract flood peakrelated maps at the two levels of plan and channel. 2. To connect K2, as a distributed hydrological model based on separate storm events to hydroPSO optimization package (within R environment) in the Tamar watershed, Iran, to overcome the problems resulting from the model calibration by common algorithms.

5.2 Materials and Methods 5.2.1 Study Area Tamar watershed as a subwatershed of the Gorganrood basin is located in Golestan province, Iran. The watershed area is 1525.3 km2 and geographically is located in the range of 37 240 to 37 490 northern latitude and 55 290 to 56 040 eastern longitude (Fig. 5-1). The highest point of the Tamar watershed is located in the Khoshyeylagh region, with an altitude of 2168 m above the sea level and the lowest point, with an altitude of 107 m, is located at the Golestan dam. The average altitude of the Tamar watershed is 754.35 m. There are a limited number of evaporation and hydrometric stations in this basin. Most of these have a shortterm inventory, except Tamar station, which has a 40-year inventory including daily rainfall and temperature data. The watershed climate type is categorized within the subhumid class, with a precipitation range of 400
800 mm (Gholami & Mohseni Saravi, 2010).

FIGURE 5-1 Geographic location of the study area.

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Table 5-1

Properties of Selected Storm Events

Event # Date 1 2 3 4

Duration (h) Rainfall Depth (mm)

September 19, 2004 17 May 6, 2005 34 October 8, 2005 14 August 9, 2005 20

50.28 57.43 41.17 59.6

Rainfall Volume (MCM) I60_max (mm/h) 76.7 87.6 62.8 90.9

13.13 8.14 7.67 9.93

I60_max: maximum 60 min intensity.

5.2.2 Data Set A set of hydrometeorological data of water flow and rainfall records in four different storms, that is, September 2004, May 2005, October 2005, and August 2005, has been collected from the Tamar hydrometric station (Table 5-1). The land-use map was prepared based on field observation and visual interpretation of SPOT images applied in Google Earth (Fig. 5-2). The available data and FAO digital maps in the form of the Harmonized World Soil Database (HWSD) (Nachtergaele et al., 2008) have been used to map soil series. The Digital Elevation Model was extracted based on the Aster satellite data set with a resolution of 30 m (available online at: http://gdex.cr.usgs.gov/gdex).

5.2.3 Methodology 5.2.3.1 KINEROS KINEROS, as a physical model, examines the amount of runoff and erosion and simulates routing of surface runoff at the watershed scale. In this model, the movement of water is evaluated using a kinematic wave approximation of Saint-Venant equations and the resulting runoff is estimated based on the Horton equation. According to this equation, runoff occurs when the rainfall intensity is higher than the infiltration speed. Infiltration equations employed in KINEROS are based on the Smith and Parlange (1978) infiltration model (Memarian et al., 2013). In the KINEROS model, the watershed is divided into several subwatersheds and each one of these is simulated based on similar surface flow planes and channels. In each subwatershed, surface flow planes are in the form of a rectangle and regular surfaces with similar input parameters. The parameters of the model may be changed from one plane/channel to another, but the specifications in each element are assumed to be similar. These specifications mainly include hydraulic attributes of soil, rainfall properties, topography, geometric shape of the earth, and land-use and land-cover characteristics. In this model, the surface flow plane is created based on the general slope of the earth by selecting the maximum and minimum altitude of the area. The channels with specific slope and assumed trapezoidal shape are directed toward the outlet of basin (Memarian et al., 2013). In the conceptual model of overland-flow, small-scale changes of infiltration and microtopography are parameterized and considered in the simulation.

FIGURE 5-2 Land-use map of the Tamar watershed.

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KINEROS2 (K2) is an updated version of the KINEROS model (Woolhiser et al., 1990) implemented under a graphical user interface (AGWA) in the ArcGIS environment. Modeling in the urban region is based on runoff estimation of pervious and impervious sections. In the K2 model, infiltration is dynamic and is associated with rainfall and runoff. The conceptual model is able to incorporate two layers in the soil profile and redistributes soil moisture during storm hiatus (Semmens et al., 2008). Based on the topography map, AGWA discretizes the watershed into subwatersheds (or planes) according to the contributing source area (CSA), as defined by the user. The CSA is the minimum area that is required for initiation of channel flow (Gal, Grippa, Hiernaux, Pons, & Kergoat, 2017). In this model, the surface flow is considered as a one-dimensional flow, as follows: @h @Q 1 5 qðx; tÞ @t @x

(5-1)

where Q is discharge per unit width, h is the depth of surface runoff, and q is the difference between rainfall and infiltration intensity (Memarian et al., 2012; Semmens et al., 2008; Smith et al., 1999). Using the kinematic wave approximation, Q (in Eq. 5-1) is replaced with Eq. (5-2) and the resulting differential equation (Eq. 5-3) is solved by the finite difference method. In Eq. (5-2), the coefficients m and α depend on the amount of slope (s), roughness (n), and surface flow regime on the planes (Memarian et al., 2012; Semmens et al., 2008). Q 5 αhm

(5-2)

@A @h 1 αmhm21 5 qðx; tÞ @t @x

(5-3)

Given the boundary conditions upstream and downstream of the planes, Eq. (5-3) will be solved. In K2, the flow equation (Eq. 5-4) in channels is estimated through the equation of Saint-Venant: @A @Q 1 5 qc ðx; t Þ @x @x

(5-4)

where Q is water discharge in the channel, A is the cross-sectional area of the channel, and qc is lateral flow. Using the kinematic wave approximation of Eq. (5-5) and substituting in Eq. (5-4), differential Eq. (5-6) can be obtained and will be resolved through a finite difference method given the boundary conditions upstream and downstream of the channel (Memarian et al., 2012; Semmens et al., 2008; Smith et al., 1999). Q 5 αRm21

(5-5)

@A @A 1 αmRm21 5 qc ðx; tÞ @t @x

(5-6)

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In the above equation, the values of m and α can be calculated using the Manning and Chezy equations in the channel: 1

α 5 1:49

S2 5 3 m5 m5 3 2 n 1

CS2 5 α

(5-7) (5-8)

where S is the channel slope, n is the Manning roughness coefficient, and c is the Chezy roughness coefficient. In this work, the local minimum method was applied on flow data to separate the base flow (McCuen, 1989). The 58 planes with an average area of 27.65 km2 and 22 channels with an average length of 10 km were discretized using the AGWA interface.

5.2.3.2 Optimization Algorithm In this work, the PSO algorithm was utilized to determine the optimum values of K2 model parameters. Initially, this algorithm is started by a swarm of random solutions. Each member of this swarm is identified as a particle. Particle conducting is done in a way that all particles store the best position during the searching process in the memory. On the other hand, the best position obtained in each stage by all particles is stored. In this algorithm, all particles move towards better solutions based on a weighted average with random components to eventually converge to a single point (Kennedy & Eberhart, 1995; Poli, Kennedy, & Blackwell, 2007). In employed algorithm, a random point for each particle is defined in the hypersphere
t  t ~ ; :G ~t 2 X ~ H G : . The particle velocity is computed using the following equation: i i i
t  ~ ; :G ~t 2 X ~ ti 1 H G ~ ti : 2 X ~ ti ~ t11 5 ωV V i i i

(5-9)

The particle’s position is updated by Eq. (5-10), as follows: ~ ti 1 V ~ t11 ~ t11 5X X i i

(5-10)

~ ti and V ~ t11 ~ ti and X ~ t11 where V are the previous and new particle velocities, respectively; X i i t ~ is the previous gravity denote the previous and new position of each particle, respectively; G i center of each particle; i 5 1, 2, . . ., N, specify the swarm size, and t 5 1, 2, . . ., T, denotes the number of iterations. ω is the inertia weight which controls the impact of earlier particle velocity on its present one (Abdelaziz & 2014).  Zambrano-Bigiarini,  At each iteration the swarm radius δt is computed using Eq. (5-11), as follows: ~: ~i 2 G δt 5 median:X t

t

(5-11)

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where :∙: denotes the Euclidean norm (Abdelaziz & Zambrano-Bigiarini, 2014; Evers & Ghalia, 2009). The diameter of the search space is determined using the following equation: diamðΨÞ 5 :ranged ðΨÞ:

(5-12)

  Then the normalized swarm radius δtnorm is used as a degree of convergence: δtnorm 5

δt diamðΨÞ

(5-13)

The hydroPSO package in R software environment was employed to implement the PSO optimization algorithm. The possibility to develop R capabilities by adding the produced packages by the users is one of the most important specifications of this software (Bloomfield, 2014). The hydroPSO package includes the following key functions (ZambranoBigiarini & Rojas, 2013): 1. The lhoat function implements sensitivity analysis based on Latin Hypercube One factor At a Time (LH-OAT) technique (Van Griensven et al., 2006). In this technique, the most effective parameter on model output receiving a rank of 1 and the parameter with the lowest efficiency receiving a rank equal to the number of parameters (D). 2. The hydromod function controls model implementation. Initially, this function reads a set of the parameter’s value written by the user in a file named Paramfiles.txt. Then, the hydromod function recalls the model executable file to produce some outputs. These outputs are read through the out.FUN function. Finally, simulated outputs are compared to observed outputs through the gof.FUN function (fitness function). In this study, the objective function of Nash
Sutcliffe efficiency (NSE) has been employed. 3. The hydroPSO function is the main driver of calibrating the hydrologic model. In the first iteration of the algorithm, the parameters are sampled in a defined range by the user in ParamRanges.txt file. Then, hydromod is recalled to estimate the fitness for each particle and the location/speed of each particle is improved and evolved based on the setting defined by the user to estimate the final standard of fitness and optimization. Finally, hydroPSO collects and saves optimum parameters, sampled parameters, goodness of fitness of the parameters, speed of the particles, and convergence measures. 4. The Plot-results function implements postprocessing of results and gives the plots with high quality to the user to evaluate the results of the calibration. 5. The Verification function validates a set of parameters defined by the user using the goodness-of-fit estimation.

5.2.3.3 Model Evaluation The statistical measures used in this work are model bias (MB), modified correlation coefficient (rmod), and NSE. These metrics are the most common evaluation criteria in the literature. Capability of the model in water discharge estimation can be assessed by MB, while rmod signifies the differences both in hydrograph size and shape (McCuen & Snyder, 1975;

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Safari, De Smedt, & Moreda, 2012; Memarian et al., 2013). Moreover, the skill of the model for imitating the hydrograph can be examined utilizing the NSE (Memarian et al., 2013; Nash & Sutcliffe, 1970; Safari et al., 2012). The following equations define these measures: 3 ðQsi 2 Qoi Þ 7 6i51 7 MB 5 6 n 5 4 P Qoi 2P n

(5-14)

i51

 rmod 5

minfσo ; σs g r max fσo ; σs g 2P n

6 i51 NSE 5 1 2 6 n 4P i51

 (5-15)

ðQsi 2Qoi Þ2 ðQOi 2Q O Þ

2

3 7 7 5

(5-16)

where Qsi and Qoi are the simulated and observed water discharges at the time step i, Q O is the mean of measured flow in the simulation period, σo andσs describe the standard deviations of observed and simulated discharges, respectively, r is the correlation coefficient between observed and simulated data, and n is the number of observations during the simulation period. The perfect rate for MB is 0 and for other assessors is 1. NSE is a normalized statistic, extending between 2 N and 1, which defines the relative amount of the residual variance compared to the observed data variance. NSE values between 0.75 and 0.36 reflect satisfactory simulation, while values $ 0.75 are considered excellent (Geza, Poeter, & McCray, 2009; Musau, Sang, Gathenya, Luedeling, & Home, 2015). For evaluating the size, shape, and volume of simulated hydrographs, an aggregated measure (AM) can be computed as follows: AM 5

rmod 1 NSE 1 ð1 2 jMBjÞ 3

(5-17)

An AM value of 1 reveals a perfect fit. Table 5-2 shows classes of goodness of fit based on AM value.

5.2.3.4 Parameters of Model in Optimization Process In this work, 16 parameters listed in Table 5-3 have been introduced as the effective parameters on flood simulation by K2. These parameters were calibrated using the hydroPSO optimization package within R environment, which benefits from a parallel processing capability and a higher speed of computations, as compared with other software environments like MATLAB. The common parameters in the calibration process involved in the main code of K2 program include Ks, n, CV, G, and In. In this study, by changing some codes in K2 through the FORTRAN programming language, calibration parameters were increased by 16 parameters (Table 5-3). Therefore, the response of a watershed to the variations of these

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Table 5-2

Model Performance Categories

Goodness of Fit

Aggregated Measure (AM)

Excellent Very good Good Poor Very poor

. 0.85 0.70
0.85 0.55
0.70 0.40
0.55 , 0.4

Source: Adapted from Safari, A., De Smedt, F., & Moreda, F. (2012). WetSpa model application in the distributed model intercomparison project (DMIP2). Journal of Hydrology, 418, 78
89.

parameters, separated for channel and plane, can be well evaluated. As expressed in Table 5-3, due to semidistributed simulation of K2, changing the amount of each parameter was done through “relative changes” in the initial value using a multiplier approach.

5.3 Results and Discussion One of the most common ways to measure the performance of any model is by plotting the simulated values against its corresponding observations. Fig. 5-3 shows water discharge (Qw) curves, hyetographs (PCP), and scatterplots for the best model output obtained by hydroPSO. In this study, besides the NSE as the objective function, the coefficient of determination (R2) was utilized to compare observed and simulated flow. The R2 measures the degree of colinearity between simulated and measured values and ranges from 0 to 1, whereby values greater than 0.5 are generally considered acceptable (Moriasi et al., 2007; Musau et al., 2015). According to R2, the results indicated better fitness of simulated water flow to observed flow for event #3 (Fig. 5-3). The R2 resulting by comparison of the simulated flow with measured flow was equal to 0.9114. This indicates that a large part of the variance of the response variable, that is, water flow, is explained and justified by the model. After this event, the best coefficient of determination (R2 5 0.9084) was obtained for event #2. In terms of goodness of fit, the event #4 with an R2 of 0.8946 ranks after events #2 and #3. However, the weakest result of optimization by hydroPSO was observed for event #1, with a coefficient of determination of 0.6368. In all simulated events, due to higher R2 than a threshold of 0.5, the result of hydroPSO simulation was acceptable in terms of collinearity (Moriasi et al., 2007; Musau et al., 2015). As shown in Fig. 5-3, the estimated peak flow compared to simulated peak flow was different for various events. This difference for the first event was 9%, which indicated that the estimated peak flow was higher than the actual peak flow for 9%. In the second, third, and fourth events, simulated peak flows were lower than observed hydrograph peaks, at 17%, 16%, and 30%. Therefore, the highest difference was observed in the second and fourth events and the lowest difference was observed in the first event. Generally, the model

Table 5-3

Optimization Parameters Used in HydroPSO

No. Symbol Parameter

Values Suggested or Used in the Reference

1

Ks_p

Saturated hydraulic conductivity (mm/h)_planes

0.6
210 0.22
266.3 0.3
73.3

2

Ks_c

Saturated hydraulic conductivity (mm/h)_channels

17.2
48.3 0
10

3

n_p

4

n_c

5

CV_p

Manning’s roughness coefficient_planes Manning’s roughness coefficient_channels Coefficient of variations of Ks_planes

6

G_p

7

G_c

8

In

9

Cov

1.46
63.27 0.1
0.63 0.053
0.8 0.01
0.1 0.09
0.64 0.1
2.0 0.02
27.3 1.6
7.6 0.57
0.95 Mean capillary drive (mm)_ planes 50.0
410 46.0
407 Mean capillary drive (mm)_ channels 1.0
263 100
306 1.0
10.0 Interception depth (mm) 0.5
4.1 4.77
101.3 Percent of surface covered by 1.0 intercepting cover 34.5
46.5 5.0
90

Multiplier Range Used in This Work References

Initial Values

Lower Upper

Woolhiser et al. (1990) Meyer, Rockhold, and Gee (1997) Guber, Yakirevich, Sadeghi, Pachepsky, and Shelton (2009) Guber et al. (2011) Al-Qurashi, McIntyre, Wheater, and Unkrich (2008) Memarian et al. (2012) Woolhiser et al. (1990) Al-Qurashi et al. (2008) Memarian et al. (2012)

5
24.21

0.2

2

210

0.2

2

0.102
0.149

0.3

4

0.035

0.5

5

http://www.tucson.ars.ag.gov/kineros/ Guber et al. (2011) Memarian et al. (2012) Wagener and Franks (2005) http://www.tucson.ars.ag.gov/kineros/ Woolhiser et al. (1990) Guber et al. (2009) Guber et al. (2011) Memarian et al. (2012) Woolhiser et al. (1990) Wagener and Franks (2005) Kasmaei, Van Der Sant, Lane, and Sheriadan (2015) Vatseva, Nedkov, Nikolova, and Kotsev (2008) Koster (2013)

0.75
1.4

0

2

120.67
240.87 0.3

3

101

0.3

3

0.5
1.27

0.1

2

0.229
0.66

0.5

2

(Continued)

Table 5-3

(Continued)

No. Symbol Parameter 10

Rock

11

Por_p

12 13

Por_c Dist_p

14

Dist_c

15

Smax

16

Sat

Volumetric rock fraction

Values Suggested or Used in the Reference

0.57
0.62 0.1 0.011
0.193 Porosity_planes 0.44
0.46 0.25
0.35 Porosity_channels 0.42
0.56 Pore size distribution index_planes 0.15
0.694 0.14
1.43 Pore size distribution index_channels 0.25
0.54 0.16
0.40 Maximum soil saturation 0
10 0.85 0.4
0.58 Initial soil saturation 0
0.5 0.4 0.19
0.32

Multiplier Range Used in This Work References

Initial Values

Lower Upper

Wagener and Franks (2005) Kennedy, Goodrich, and Unkrich (2012) Koster (2013) Wagener and Franks (2005) Kasmaei et al. (2015) Koster (2013) Meyer et al. (1997) Wagener and Franks (2005) Koster (2013)

0
0.32

0.5

2

0.456
0.468

0.5

2

0.44 0.26
0.34

0.5 0.5

2 2

0.545

0.5

2

Al-Qurashi et al. (2008) Memarian et al. (2012) Koster (2013) Al-Qurashi et al. (2008) Wagener and Franks (2005) Koster (2013)

0.88
0.92

0.1

1

0.2

0.5

5

Chapter 5 • Parameter Optimization of KINEROS2

FIGURE 5-3 Observed versus simulated water discharge of selected storm events.

131

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SPATIAL MODELING IN GIS AND R FOR EARTH AND ENVIRONMENTAL SCIENCES

Table 5-4

Fitting Metrics of Selected Storm Events for Runoff Modeling

Fitting Metrics

Event #1

Event #2

Event #3

Event #4

MB rmod NSE AM Goodness of fit

2 0.44 0.74 0.39 0.56 Good

0.04 0.90 0.91 0.92 Excellent

0.20 0.81 0.89 0.83 Very good

0.05 0.73 0.86 0.85 Excellent

showed more intention toward underestimation of peak flow to fit simulated hydrograph with observed hydrograph. As shown in Table 5-4, events #2 and #4, with NSE equal to 0.92 and 0.85, respectively, had the best fitness of simulated flow compared to observed flow. Event #3, with an NSE of 0.83, was placed next in order. However, event #1, with an NSE of 0.39, had the lowest fitness among the simulated events. According to the AM measure, the best fitness was observed in the simulation of event #2 (AM 5 0.92). The events #4, #3, and #1 ranked next in order, respectively (with AM 5 0.85, 0.83, and 0.56). According to the MB measure, hydroPSO overestimated flood magnitude in simulation of events #2
4. However, underestimation of flood magnitude was observed in simulation of event #1. Some diversions are observed in rising and recession limbs of the simulated hydrographs than the real data which are higher for event #1 than those for other events. These diversions or overestimation/underestimation of water discharge could be caused by the fact that only one rain gauge station was used, and only one isolated storm event on the watershed surface was considered (Hernandez et al., 2000; Memarian et al., 2012; Memarian et al., 2013). Using the optimized parameters and according to event #2, the discretized map in both planes and channels was classified based on the simulated peak flow (Fig. 5-4). The sensitivity analysis of K2 parameters was accomplished through following the evolution and convergence of parameter values, global optimum, and the normalized swarm radius. Fig. 5-5 shows the evolution of the 16 parameters employed in K2 calibration. This shows that the parameters Ks_p, Ks_c, n_p, n_c, CV_p, and Sat were the most effective parameters in K2 calibration, respectively. This was also reported and confirmed in previous studies by Nearing et al. (2005), Canfield and Goodrich (2006), Martínez-Carreras, Soler, Hernández, and Gallart (2007), Al-Qurashi et al. (2008), and Memarian et al. (2012). Fig. 5-6A illustrates the frequency histograms of the 16 parameter values employed in K2 calibration. Sporadic and level states of the histograms suggest the uncertainty about the likely optimal values of the parameters (Musau et al., 2015). In this work, the parameters are moderately well defined due to the sharp distribution of the peak around the best value in all parameters expect In, COV, Por_p, and Dist_p. The empirical cumulative distribution functions (ECDFs) were utilized to estimate the factual primary cumulative distribution function of the sampled points (Fig. 5-6B). Fig. 5-6A, B endorses that the parameters n_p, n_c, CV_p, G_c, Rock, Por_c, Dist_c, Smax, and Sat track normal and near-normal distributions,

Chapter 5 • Parameter Optimization of KINEROS2

133

FIGURE 5-4 The discretized map, classified based on the peak flow in both planes and streams (based on event #2).

while Ks_p and Ks_c depict an experimented distribution skewed toward the lower frontier utilized for K2 calibration. Furthermore, the parameters In, COV, and Dist_p display a uniform distribution of sampled values. The posterior distributions of some parameters, for example, Ks_p and n_c, seem to be more sharply peaked than the other parameters, which designates less uncertainty in hydrological modeling. However, some other parameters such as In, Cov, Por_p, and Dist_p did not extensively transform from their earlier uniform distributions. This conduct may mean two sorts of errors, which are whether the efficient blunders of input data or making up for auxiliary needs in the model (Shafiei et al., 2014; Vrugt, Ter Braak, Clark, Hyman, Robinson, 2008). Boxplots in Fig. 5-7A illustrate the statistical distribution of sampled values. In each box, the top and bottom lines are the first and third quartiles, respectively. The horizontal line within the box embodies the second quartile as the median. The notch extent is computed to the range equal to 6 1.58UIQR/sqrt(n), where IQR is the interquartile range and n is the number of points. Dot plots in Fig. 5-7B show the parameter value against its corresponding goodness-of-fit value (NSE) achieved during the K2 optimization. They are important for perceiving parameter runs that create the best model execution (Abdelaziz & Zambrano-Bigiarini, 2014; Beven & Binley, 1992). A visual check of Fig. 5-7B demonstrates that for 6 out of 16 parameters (Ks_c, n_c, G_c, Rock, Dist_c, and Smax) the optimal value found amid the optimization

FIGURE 5-5 Parameters values for each run during the model calibration process based on event #2.

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FIGURE 5-6 Graphical representation of parameters’ values sampled amid the optimization based on event #2. (A) Histograms displaying the incidences of the parameter values. Vertical red line shows the finest value established for each parameter. (B) ECDFs of parameter values. Horizontal gray dotted lines denote a cumulative probability of 0.5 as the median of distribution. Vertical gray dotted lines indicate a cumulative probability of 0.5, shown in the upper part of each figure (Abdelaziz & Zambrano-Bigiarini, 2014).

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FIGURE 5-7 Graphical representation of parameters’ values sampled amid the optimization based on event #2. (A) Box-and-whisker plots (or boxplots). (B) Parameter values versus their equivalent NSE. Horizontal and vertical red lines specify the optimal value explored for each parameter (Abdelaziz & Zambrano-Bigiarini, 2014).

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matches with the median of all the sampled values. This condition establishes that a large portion of the particles met in a small locale of the solution space. For In and Sat, examined values were located inside the second quartile. Fig. 5-7B depicts that the optimum values discovered for the parameters Ks_p, Ks_c, n_c, CV_p, and Sat are relatively well defined, though other parameters demonstrate a more extensive district around their ideal. The development of the global optimum (best model performance in each iteration, i.e., largest NSE) and the normalized swarm radius (a quantity of swarm ranged on the search region) versus the number of iterations is illustrated in Fig. 5-8. Obviously, both the global optimum and the normalized swarm radius decrease with an increasing number of repetitions. This condition demonstrates that most of the particles met in a small state of the answer region (Zambrano-Bigiarini & Rojas, 2013). Moreover, Fig. 5-8 shows that only eight iterations (i.e., 8 3 200 5 1600 model executions) were needed to find the region of the global optimum, and the remaining iterations were only utilized for search refining. Fig. 5-9 shows three-dimensional dot plots, which show the interactions among parameters by plotting the NSE reaction surface onto the parameter space (for various pairs of parameters). In general, it can be seen that particles are spread everywhere throughout the parameter space, showing a decent exploratory capacity of PSO. The areas with the powerless model execution have a low mass of focus, while locales with higher model execution are more densely sampled, affirming the great capacity of the PSO exploitation (ZambranoBigiarini & Rojas, 2013). This figure demonstrates that the optimal values explored for Ks_p, Ks_c, and n_c characterize a limited scope of the parameter space with high model performance. Also, it can be revealed that the model performance is more impacted by the interaction of Ks and n parameters. The parameters CV_p and n_p represent a more extensive

FIGURE 5-8 Evolution of the global optimum and the normalized swarm radius (δ norm) over the 50 model iterations based on event #2.

FIGURE 5-9 Model performance (NSE) anticipated onto the parameter space for various pairs of parameters, based on event #2.

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scope of the optimized levels. Good model performance for an extensive variety of estimations of other parameters affirms that these parameters are not well distinguished and enforce a few uncertainties on simulation outcomes (Shen, Chen, & Chen, 2012). Finally, a correlation matrix among parameter values and model performance (NSE) is shown in Fig. 5-10. The slice over the diagonal illustrates the Pearson’s correlation coefficient between paired samples in addition to their statistical significance as stars. The bottom slice depicts bivariate scatterplots between each column and row of the matrix, with a fitted line assimilated using locally weighted polynomial regression (Abdelaziz & Zambrano-Bigiarini, 2014; Cleveland, 1979). A histogram of each factor sampled throughout the optimization is illustrated on the diagonal. This figure shows that the highest linear correlation between the NSE and K2 parameters is obtained for the Ks_p, Ks_c, and n_p, followed by CV_p, G_c, Por_p, Dist_p, and Smax. Moreover, significant linear correlations are recognized between the parameters In and Cov; Ks_p and Ks_c; Ks_p and n_p; and Ks_c and n_p. Based on Fig. 5-10, it can be seen that nonlinear relationships are evident between some calibrated parameters, for example, G_p versus Cov, Por_c versus Dist_c, and G_c versus n_c. Looking at the outcomes in this work and in the previous study by Kamali et al. (2013), where PSO and multiobjective PSO were executed for a similar case utilizing HEC
HMS, we found that the utilization of hydroPSO integrated with KINEROS2 gave a noteworthy change to the simulated water discharge based on events #2, #3, and #4. However, HMS-PSO outperformed K2-PSO for hydrological modeling based on event #1. Finally, it is clear that there are some constraints concerning the absence of adequate data, particularly soil characteristics and flood events, which have significant impacts on the results of this study. In this regard, the outcomes and their generality are evidently limited. However, any applications should use accessible data as much as possible, and the shortage of sufficient records for a basin does not imply that something partially useful cannot be done. Nevertheless, the gained results should be updated when new information becomes accessible. Additionally, combining hydroPSO and uncertainty analysis would be an imperative subject to which future works have to allude.

5.4 Conclusion The proficient estimation of optimum parameter values is unavoidable in modeling of hydrological phenomena. In this chapter, the hydroPSO package was applied to the KINEROS2 model in R software to assess parameter identification and calibration in the Tamar watershed, Iran. Sixteen parameters, representing the overland flow and channel flow for simulation based on four storm events, were selected for model optimization. The following conclusions can be drawn from the results of this study: • The K2 model effectively simulated water discharge in the study area, considering the three main performance evaluation metrics used. The results indicated better efficiency of K2 based on event #2 with the coefficient of determination and NSE of 0.9084 and 0.92, respectively. Events #3 and #4, with NSE of 0.89 and 0.86, showed excellent and very

FIGURE 5-10 Correlation matrix between parameters and model performance (NSE), based on event #2.

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good fitness of the simulated flow compared to observed flow, respectively. However, the model had more intension toward underestimation of peak flow to fit the simulated hydrograph with the observed hydrograph. According to the MB measure, hydroPSO overestimated flood magnitude in the simulation of events #2
4. However, underestimation of flood magnitude was observed in the simulation of event #1. The diversions or overestimation/underestimation of water discharge could be caused by the fact that only one rain gauge station was used, and only one isolated storm event on the watershed extent was considered. • Sensitivity analysis established that the parameters Ks_p, Ks_c, n_p, n_c, CV_p, and Sat were the most effective parameters in K2 calibration, respectively. The posterior distributions of some parameters, such as Ks_p and n_c, appeared to be more sharply peaked than other parameters, which established less uncertainty in hydrological modeling. However, some other parameters, such as In, Cov, Por_p, and Dist_p, did not significantly change from their prior uniform distributions. This behavior represented two types of error which were whether the systematic errors of input (forcing) data or compensating for structural deficiencies in the model. Visual inspection of boxplots showed that for 6 out of 16 parameters (Ks_c, n_c, G_c, Rock, Dist_c, and Smax) the optimum value found during the optimization coincided with the median of all the sampled values, confirming that most of the particles converged into a small region of the solution space. Dot plots showed that the optimum values found for Ks_p, Ks_c, and n_c define a narrow range of the parameter space with high model performance. On the other hand, the model performance was more impacted by the interaction of Ks and n parameters. The parameters CV_p and n_p showed a wider range of optimized levels. Furthermore, during optimization, both global optimum and normalized swarm radius decreased with an increasing number of iterations, indicating that most of the particles converged into a small region of the solution space. Correlation analysis revealed that the highest linear correlation between the NSE and K2 parameters was obtained for Ks_p, Ks_c, and n_p, followed by CV_p, G_c, Por_p, Dist_p, and Smax. • The HydroPSO R package can be successfully integrated with the K2 model in R software to harness the combined benefits of a distributed hydrological model and flexible computing capability of the open source R software.

Acknowledgments The authors would like to thank the University of Birjand for financial support of this study. The corresponding contract number is 1395 1209.

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Further Reading Kirckpatrick, S., Gelatt, C., & Vecchi, M. (1983). Optimization by simulated annealing. Science, 220, 671
680. Sidman, G., Guertin, D. P., Goodrich, D. C., Unkrich, C. L., & Burns, I. S. (2016). Risk assessment of postwildfire hydrological response in semiarid basins: The effects of varying rainfall representations in the KINEROS2/AGWA model. International Journal of Wildland Fire, 25(3), 268
278.