Keywords

1 Introduction

Groundwater is the most important source of natural resources. It is a vital source of industries, agriculture, and domestic requirements which want to be carefully managed for hard rock and drought-prone areas [1]. It has become a reliable source of water in all climatic regions of the world [2]. Groundwater is the largest available freshwater resource in the whole world. Aquifer wells provide potable water to 50% of the world's population and record 43% of overall irrigation water consumption. In addition, worldwide 2.5 billion citizens depend entirely on groundwater supplies in order to meet their everyday needs [3]. In arid and semi-arid climates, with frequent dry spells and sometimes erratic surface waters (Liamasand Martínez-Santos, 2005), groundwater is significant. Groundwater is an important medium of water supply in different regions of the world, as a result, several studies highlighted different features of groundwater such as storage potential, hydrogeology, water quality , exposure, and so on [4,5,6,7]. Furthermore, groundwater simulation has become an essential tool among scientists and engineers working on water management for optimizing and protecting the development of groundwater. Physically, during the past few years, simulations have been implemented to simulate and analyze the groundwater environment and then take remedial steps in order to allow effective use of the control of water supplies. These models act as a hydrological variableness framework and understand the physical processes within the aquifer. Hydrologists, mechanics, and environmental engineers use this frequently in computer applications but challenges range from aquifer protection yield to soil quality and clean-up. Although such models use data in highly intense, laborious, and expensive ways. As a consequence, physical models in developed countries are significantly limited because of the lack of appropriate and high-quality data.

In this paper, we have used ANN for groundwater prediction of four Blocks of the BANDA District of UP. Prediction of groundwater is very important for planning groundwater administration and water resources in any river basin. Physical-based models are widely used in groundwater simulation. Wide numbers of numerical models have already been developed for different areas with different objectives such as to express provincial groundwater behavior and to understand local hydrological processes [8,9,10]. The relevance of the ANN technique in water management ranges from event-based simulation to real-time simulation. It has been used for rainfall-runoff simulation, precipitation simulation as well as for stream flows simulation, evapotranspiration, water quality as well as groundwater [11,12,13]. In the literature, comparatively less research on the ANN-based approach in groundwater hydrology has been used in comparison to surface water hydrology. Neural networking practises are used in groundwater hydrology for the evaluation of the aquifer parameters [14,15,16,17,18,19,20], groundwater quality predictions [17, 21, 22].

2 Study Area

Banda district lies between latitude 25\(^\circ\)00′00’’ and 25\(^\circ\)59′00’’ north and longitude 80\(^\circ\)06′00’’ and 81\(^\circ\)00′00’’. The district's total area is 4460 km2. Baberu is one block of the Banda district. It consists of 570.41 km2. The area geologically comprises Precambrian Bundelkhand granites overlain by Vindhyan and quaternary alluvium. The area is roughly plain apart from some isolated granitic hillocks and the division of point bars natural levees, and flood plain. It is made up of unconsolidated deposits of Indo-Gangetic alluvium of recent age comprising silt clay, silt, Kankar, sand and their admixtures of various grades.

figure a

3 Study Period

The periods for study depend from the time of minimum to the time of maximum water table elevation as the non-monsoon period and from the time of minimum to the time of maximum water table elevation as monsoon period. For this purpose, data have been taken from 1995 to 2016 in northern India and the water year is considered from November 1 to October 31 next year. The study periods are taken as non-monsoon periods for the duration of November to May.

4 Materials and Methods

4.1 Ground Water Balance Equation

$${\varvec{R}}_{{\varvec{c}}} + {\varvec{R}}_{{\varvec{i}}} + {\varvec{R}}_{{\varvec{r}}} + {\varvec{R}}_{{\varvec{t}}} + {\varvec{S}}_{{\varvec{i}}} + {\varvec{I}}_{{\varvec{g}}} = {\varvec{E}}_{{\varvec{t}}} + {\varvec{T}}_{{\varvec{p}}} + {\varvec{S}}_{{\varvec{e}}} + {\varvec{O}}_{{\varvec{g}}} + \Delta {\varvec{S}}$$
(1)

where

R = Rainfall Recharge;

Rc = Canal seepage Recharge;

Rr = Field irrigation Recharge; Rt = Recharge from pond storage

Ig = inflow from blocks; Et = Evapo-transpiration;

Tp = Groundwater discharge from tube well;

Si, Se = influent and effluent seepage from rivers; Og = outflow to other blocks; and

ΔS = change in groundwater storage.

All these parameters are calculated by Central Groundwater norms [Ref].

figure b

ANN Architecture

For the prediction of groundwater resources, ANN model is proposed the proposed models have been built using MATLAB The proposed ANN model consists of only a hidden layer in between input and output layers. Transfer function used on behalf of the hidden layer is sigmoid whereas used for output layer it is linear. Four different algorithms Levenberg Marquardt, Gradient Descent, Scaled Conjugate Gradient, and Bayesian Regularization backpropagation algorithm are used for training. The proposed model has been trained, tested, and validated with recharge and discharge and groundwater level data. The block diagram of the proposed two inputs and one output ANN model is shown in Fig. 1. The structure of an ANN is usually prejudiced by the nervous structure of humans.

Fig. 1
figure 1

Actual and predicted groundwater level through Levenberg–Marquardt for Non-Monsoon season

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4.2 Levenberg–Marquardt (LM)

The Levenberg–Marquardt technique is a modification of the typical Newton algorithm for ruling an optimum answer to minimize complexity. It employs approximation to the Hessian matrix in the subsequent Newton-like weight update

$$x_{k + 1} = x_{k } {-} \left[ { J^{T} J + \mu I } \right]^{ - 1} J^{T} e$$
(2)

when neural network x is the weights, J of Jacobian matrix minimizes the presentation criterion, μ of a scalar emphasizes the phase of learning, and e is the vector of the residual error. When μ is bigger, Eq. 1 is decent in the gradient for a limited stage scale. The Newton method is faster and more reliable, near to minimum error, because the objective is to change size. The scalar μ is zeros equation 1 automatically is the Newton method. Newton's method is quick and more accurate because of the shifting toward the Newton method quickly. Levenberg–Marquardt has computational requirements so it can be used for small networks [23].

4.3 Bayesian Regularization (BR)

The Bayesian regularization is an algorithm that mechanically sets optimum standards in support of the parameter of the point function. The weight and bias of the network be understood to be a random variable with specified circulation. The benefit of Bayesian control is that the feature should not surpass the scale of the network. The effective usage of Bayesian regularization in literature [24].

4.4 Gradient Descent by Means of Momentum and Adaptive Learning Rate Back Propagation (GDX)

In order to measure the derivative of the output cost function according to the arbitrary weights and bias of the network, this technique utilizes a standard back propagation algorithm. This strategy utilizes gradient descent with momentum to control each variable. With each level of shift, the learning rate is increased if efficiency declines, one of the simplest and most popular ways to train a network [25].

4.5 Scaled Conjugate Gradient (SCG)

The scaled conjugate gradient (SCG) algorithm [26] determines the quadratic error calculation in the neighborhood. Moller [26] proved this hypothetical base work to be the primary order approach for the primary derivative, such as regular back propagation, and found an important way to obtain a local minimum of second-order technique in the second derivatives. SCG is a second-order combination of gradient algorithms that has helped to reduce a multidimensional target function. SCG is a simple algorithm and employs a scaling method that holds the search through information iteration away from the time-consuming line [26, 27] has shown that the SCG approach presents super linear convergence for major problems.

4.6 Criteria for Evaluation

The following statistical indices such as R2 efficiency criteria, root mean square error (RMSE), Mean Absolute Error (MAE), Mean Square Error (MSE), and coefficient of correlation (r) were used to evaluate the performance.

5 Results and Discussion

In Babeu Block of BANDA, part of the Yamuna river basin, the purpose of ANN is to measure the capacity to predict a fluctuation of the groundwater level. The network has the following input parameters, Recharge and Discharge. In recharge all the parameters are included like recharge from rainfall, recharge from canal seepage, recharge from field irrigation, recharge from pond storage and in discharge all the parameters are included like groundwater discharge from tube well, influent and effluent seepage from rivers, and for the output parameters, groundwater levels were taken. The four wells' groundwater levels were estimated by using the feed-forward network with a back propagation algorithm. Minimum errors were saved in the trained networks. The neural networks of each wells producing maximum value for R2. was selected as the best network.

For ALIHA well LAT = 25.495 LONG = 80.525

Year

Recharge in Ham

Discharge in Ham

Groundwater level in MBGL

1995

2776.139

74.557

6.53

1996

2594.47

74.63

5.09

1997

2488.79

71.615

5.1

1998

2903.234

71.610

5.28

1999

2352.035

80.709

5.33

2000

3168.478

80.704

7.43

2001

3436.0.904

80.700

4.08

2002

3435.626

80.695

5.73

2003

3137.422

80.535

1.83

2004

4802.41

81.270

5.23

2005

1735.716

81.717

5.3

2006

3301.524

82.368

5.91

2007

2686.633

82.156

5.5

2008

3983.97

82.704

6.09

2009

3155.92

83.233

8.03

2010

3077.607

83.802

7.02

2011

3556.657

109.231

6.05

2012

3294.387

109.784

8.02

2013

3152.837

111.968

6.11

2014

2603.938

113.001

6.5

2015

3019.837

114.034

6.8

2016

3593.046

114.146

8.3

HAM = Hectare Metre, MBGL = Metre Below Groundlevel

For Mural well LAT = 25.51, LONG = 80.562

Year

Recharge in HAM

Discharge in HAM

Groundwater level in MBGL

1995

7082.457428

190.2115222

4.3

1996

6618.994941

190.410659

3.9

1997

6349.392655

182.7041859

2.1

1998

7406.700558

182.6928576

4.7

1999

6000.488148

205.9046163

3.1

2000

8083.388163

205.8932881

2.42

2001

8768.194147

205.8819598

2.6

2002

8764.93384

205.8706316

9.65

2003

8004.157891

205.4621211

0

2004

12,251.87134

207.3352643

1.33

2005

4428.140221

208.4777991

2.87

2006

8422.813025

210.1386775

8.52

2007

6854.111735

209.5955874

5.97

2008

10,163.88281

210.9957958

5.95

2009

8051.360821

212.3960043

5.36

2010

7851.559325

213.7962128

6.3

2011

9073.706649

278.6707427

6.13

2012

8404.605956

280.0800508

3.6

2013

8043.484473

285.6519149

2.34

2014

6643.138717

288.287559

3.31

2015

7704.17571

290.923203

5.33

2016

9166.541755

291.2091086

2.26

For Patwan well LAT = 25.59 LONG = 80.56

Year

Recharge in HAM

Discharge in HAM

Groundwater level in MBGL

1995

13,691.21989

367.7011551

4.3

1996

12,795.29261

368.0861097

7.2

1997

12,274.11981

353.1885944

6.5

1998

14,318.01985

353.1666955

7.9

1999

11,599.64652

398.0377444

6.3

2000

15,626.13626

398.0158455

6.52

2001

16,949.94645

397.9939467

7.87

2002

16,943.64389

397.9720479

8.74

2003

15,472.97486

397.1823492

0

2004

23,684.30256

400.8033545

7.93

2005

8560.113787

403.012008

11.66

2006

16,282.28428

406.2226805

11.05

2007

13,249.80092

405.1728238

16.6

2008

19,647.97613

407.8795908

17.35

2009

15,564.22365

410.5863577

17.52

2010

15,177.98396

413.2931247

17.5

2011

17,540.5379

538.7031909

14.8

2012

16,247.08788

541.4275485

11.3

2013

15,548.99775

552.1986145

5.67

2014

12,841.96536

557.2936233

10.25

2015

14,893.07416

562.3886321

15.55

2016

17,719.99904

562.9413209

13.65

For Baberu well LAT = 25.54 LONG = 80.71

Year

Recharge in HAM

Discharge in HAM

Groundwater level in MBGL

1995

4581.922195

123.0553666

3.15

1996

4282.089958

123.1841961

1.95

1997

4107.673562

118.1985734

2.5

1998

4791.687917

118.1912447

2.05

1999

3881.953419

133.2078507

2.15

2000

5229.463927

133.200522

2.89

2001

5672.492038

133.1931933

2.65

2002

5670.382817

133.1858646

1.85

2003

5178.206727

132.921583

1.45

2004

7926.220778

134.1333936

1.95

2005

2864.739275

134.8725445

3.46

2006

5449.051313

135.9470325

5.62

2007

4434.196323

135.595686

5.2

2008

6575.418306

136.5015363

6.37

2009

5208.744171

137.4073867

5.5

2010

5079.484671

138.313237

5.25

2011

5870.140175

180.2831396

2.75

2012

5437.272439

181.1948768

3.65

2013

5203.64865

184.7995363

2.84

2014

4297.709523

186.5046389

3.93

2015

4984.136373

188.2097414

4.45

2016

5930.198882

188.394705

4.32

For ALIHA Well, all recharge and discharge data were calculated according to the groundwater estimation committee norms. In the year 2002, recharges were the most, i.e., 3435.626 and the discharges were the most in the year 114.146. For Murwal well, maximum recharge was found in the year 2008, that is, 10,163.88281 HAM and maximum discharge was found in the year 2016 that is 291.2091086 HAM. For Patwan well, maximum recharge was found in the year 2004, that is, 23,684.30256 HAM and maximum discharge was found in the year 2016, that is, 562.9413209 HAM. For Baberu well, maximum discharge was found in the year 2016, that is, 188.394705. HAM and maximum recharge were found in the year 2004 that is 7926.220778 HAM.

For ALIHA Well

See Figs. 1, 2, 3 and 4.

Fig. 2
figure 2

Scatter diagram for actual and predicted groundwater level for R2 = 0.88 for testing

Full size image
Fig. 3
figure 3

Actual and predicted groundwater level through Bayesian Regularization for Non-Monsoon season

Full size image
Fig. 4
figure 4

Scatter diagram for actual and predicted groundwater level for R2 = 0.85 for testing

Full size image

For Baberu well

See Figs. 5 and 6.

Fig. 5
figure 5

Actual and Predicted groundwater level through Bayesian Regularization for Non-Monsoon season

Full size image
Fig. 6
figure 6

Scatter diagram for actual and predicted groundwater level for R2 = 0.77 for testing

Full size image

For Murwal Well

See Figs. 7 and 8.

Fig. 7
figure 7

Actual and predicted groundwater level through Levenberg- Marquardt for Non-Monsoon season

Full size image
Fig. 8
figure 8

Scatter diagram for actual and predicted groundwater level for R2 = 0.94 for testing

Full size image

For Patwan Well

See Figs. 9 and 10.

Fig. 9
figure 9

Actual and predicted groundwater level through Levenberg-–Marquardt for Non-Monsoon season

Full size image
Fig. 10
figure 10

Scatter diagram for actual and predicted groundwater level for R2 = 0.96 for testing

Full size image

6 Conclusion

The function of the artificial neural network of feed-forward back propagation into groundwater prediction has been investigated in this research paper. Input and output data are grouped into hydro-geological well classes and the LM, SCG, BR and GD have been trained for each well sheet. The findings demonstrate explicitly that the LM algorithm works well for all four wells. Results demonstrate that the ANN model is capable of predicting the virtual physical structure's complex response. A major advantage of this ANN technique is that it can provide good predictions by means of limitations of groundwater data (Table 1).

Table 1 Comparison of performance of models developed for all wells, training, testing and validation
Full size table