Study Area

Kalahandi district lies between 19.175489-20.454517°N latitudes and 82.617767— 83.794874°E longitudes occupying southwestern part of Odisha (Figure 10.1). It is bordered by Balangir and Nuapada districts in North, by Nabarangpur, Koraput, and Rayagada districts in South, and by Rayagada, Kandhamal, and Boudh districts in East. Climate in Kalahandi district is of extreme kind. It is very dry excluding the monsoon period. Maximum temperature here is about more than 45°C, while minimum temperature is 4°C. Kalahandi experiences average annual precipitation of 1378.20 mm. Monsoon starts late in June and usually continues till September. It is principally an agriculture-based economy. Dharamgarh and Narla subdivisions are considered in present study for GWL prediction.

Proposed watershed

FIGURE 10.1 Proposed watershed.

Methodology

RBFN

Moody and Darken proposed RBFN in late 1980s. Since then, it is being extensively utilized to classify and approximate different linear and non-linear functions (Chu et al., 2013). RBFN possesses a resilient biological training and has the capability for approximating any random non-linear function (Schilling et al., 2001). Another advantage of this network is that it comprises optimum estimate point. Fundamentally, RBFN consists of many modest and vastly interrelated simulated neurons which are systematized to several layers, i.e. input, hidden, and output layers as presented in Figure 10.2 (Samantaray and Sahoo, 2020c; Sahoo et al.. 2020; Samantaray and Ghose, 2019, 2020a,b).

A single output RBFN with К hidden layer neurons is articulated as

Architecture of RBFN

FIGURE 10.2 Architecture of RBFN.

where yk is the kih output node on output layer; cort is the weight connection amid fth hidden node and &th output node; and 9* is the threshold value of kth output node.

Gaussian function is the most common utilized function in hidden layer, as shown in Equation 10.1:

where x is the input vector with n dimension; c, is the fth RBF centre; 0, is the RBF spread in i th hidden node, indicating radiated distance from RBF centre; m is the number of hidden nodes; and ||x-c, || is the radiated distance from X to RBF centre. Quantity of hidden neurons have a big influence on the output of the RBF.

SVM

Among many soft computing techniques, SVM is one of the most recently used techniques applied to many research areas, for example computational field, hydrological field, and in the field of environment studies (Asefa et al., 2006; Sun, 2013). Essentially SVM is used in recognizing various patterns, prediction, forecasts, regression analysis, and different classification. Evolution of SVM theory was proposed by (Vapnik, 2000). While customary approaches aim at minimizing error in local training nodes, SVM focuses to minimize upper bound for simplification error (Vapnik and Vapnik, 1998). Concept of SVM is based on the principle of structural risk minimization, which is a major advantage over traditional soft computing techniques. Moreover, it provides a unique solution because of its convex nature. SVM consists of a high-dimensional kernel function space discreetly containing non-linear transformation where the input data is mapped (Samantaray and Sahoo, 2020b; Samantaray et al., 2020). Different equations related to SVM in accordance to Vapnik’s theory is presented in Equations 10.3-10.5. Considering n points in a dataset specified by {xj,di}", SVM approximate functions specified in Equations 10.3 and 10.4:

where x, is the input space vector, d, is the target value, x, w is the normal vector, b is the scalar vector,

C—^'L(xj,di) signifies empirical error, and tv and b are linear function f(x) n /=1 constant.

Constraints tv and b are calculated utilizing Equation 10.4. %i* and qi are positive slack variables signifying lower and upper additional deviance.

1 2

where — ||w||‘ is the regularizing term, £ is the loss function equating to approximate accurateness of training data point, C is the error penalty factor, and / is the training dataset size. A typical architecture of proposed SVM is shown in Figure 10.3.

Determining data correlation using non-linear mapping method is the main aim of SVMs. Kernel techniques permit functions in a higher-dimensional and inherent feature space without computing coordinates of data in corresponding space. Results attained from high-dimensional feature space associates with original lowdimensional data from input space.

Architecture of SVM

FIGURE 10.3 Architecture of SVM.

SVM-FFA

Nature has inspired many researchers in developing several optimization algorithms to solve traditional problems (Yang and Deb, 2014; Assareh et al., 2010). Among these developed algorithms, ant colony optimization, PSO, cuckoo search (CS), and GA are a few examples. Fundamental concept of such algorithms are based on existence and assortment of fittest species in nature. Most recent biological motivated optimization algorithm was developed by Yang (2009) known as FFA. It was founded based on social movement of fireflies in their natural setting. FFA shows some interactive pattern followed by the fireflies such as their flashing characteristics. A firefly attracts other fireflies and victims utilizing its bio-luminous quality. Luminance yielded by the firefly with high intensity is trailed by the other fireflies following certain path. Research studies recommend that FFA is robust and effective compared to other biologically motivated algorithms (Yang, 2013; Pal et al., 2012). Formulation of attraction and light intensity difference are major shortcomings of FFA. For maximizing the objective function, there is a need to refine the design specifications (Samantaray and Sahoo. 2020a; Samantaray et al., 2020b). The objective function has to be proportional to light intensity generated by a firefly. Light intensity with changing distance in Gaussian form is described as follows:

where e is the exponential function. I is the light intensity at a distance of r in reference to firefly, /0 is the intensity of light at distance r = 0 from reference firefly, and Y is the light absorption coefficient.

Attractiveness of a firefly trails a light intensity proportion when detected by other fireflies. Attraction x at a distance equal to r from reference firefly is defined as

where co0 is the attractiveness at r = 0 and со is the light absorption coefficient.

The distance amid any two fireflies i and j at x, and Xj, correspondingly, is well- defined as the Cartesian distance:

where // denotes dimensionality of problem, x,.* is the Ath constituent of spatial coordinate x, of /th constituent.

Data Collection and Model Performance

Altogether, water table depth data of 20 years (2000-2019) were collected from India Meteorological Department (IMD), Groundwater Survey and Investigation (GWSI), Kalahandi district, Orissa. Collected data are utilized to develop models for studying effects of GW fluctuation. The data from 2000 to 2015 (80% of dataset) are utilized to train and data from 2016 to 2019 (20%) are utilized to test the network. Monthly data are transformed from daily data that helps in training and testing proposed model. Input and output data are arranged in such a manner that every data falls inside a quantified series before training. This procedure is known as normalization, with normalized values confined to 0-1 range. Normalization equation which is utilized to scale data is

where M, = transformed data series, M = actual data set, Mmm = minimum of actual dataset, Mmax = maximum of actual dataset. Subsequent GWL arrangements are employed as input:

Input

RBFN

SVM

SVM-FFA

RBFN: 1

SVM: 1

SVM-FFA: 1

RBFN:2

SVM:2

SVM-FFA:2

RBFN:3

SVM:3

SVM-FFA:3

RBFN:4

SVM:4

SVM-FFA:4

RBFN:5

SVM:5

SVM-FFA:5

where

//,_1: one month lag GWL Ht-2: two month lag GWL H, ?: three month lag GWL

Hi__j: four month lag GWL

H,_5: five month lag GWL

Evaluating Criteria

RMSE, MSE, and R1 are the evaluating standards to determine the best model. To select the perfect model for desired area of study, the criteria followed are MSE and RMSE must be minimum and R2 must be maximum.

where

Vcomp = predicted data Vobs = observed data Vcomp = mean predicted data Vohs = mean observed data

 
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