# Prediction of Combined Terrestrial Evapotranspiration Index (CTEI) over Large River Basin Based on Machine Learning Approaches

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## Abstract

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^{2}value of 0.82 and the lowest errors in terms of the Root Mean Squared Error (RMSE) (0.33) and Mean Absolute Error (MAE) (0.20), followed by the Matern 5/2 Gaussian model with an R

^{2}value of 0.75 and RMSE and MAE of 0.39 and 0.21 mm/day, respectively. Moreover, among all the five methods, the SVM and Matern 5/2 Gaussian methods were the best-performing ML algorithms in our study of CTEI predictions for the Ganga basin.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}(Figure 1).

^{2}, which is nearly 26% of the country’s total geographic area [50]. The GRB originates in the Himalayan Mountains at the Gangotri glacier’s snout, at an elevation of ~7000 m a.s.l. The Bhagirathi and Alaknanda Rivers’ confluence occurs in Devprayag, which is then officially called the Ganga River. The Ganga River’s main tributaries are the Yamuna, the Ramganga, the Gomti, the Ghaghra, the Sone, the Gandak, the Kosi, and the Mahananda. It flows for about 2510 km, generally southeastward, through a vast plain to the Bay of Bengal. The primary source of water in the Ganga River is surface runoff generated by precipitation (~66%), base flow (~14%), glacier melt (~11.5%), and snowmelt (~8.5%). The GRB receives 84% of total rainfall during the monsoon season (June to October). The monsoon season accounts for 75% of the rain in the upper basin and 85% of the rain in the lower basin [51]. The elevation range across the basin varies from sea level to the highest mountain peak (~8850 m a.s.l). Several researchers have documented drought years in the region. For example, the NRAA [52] reported that India had experienced 22 large scale droughts years—in 1891, 1896, 1899, 1905, 1911, 1915, 1918, 1920, 1941, 1951, 1965, 1966, 1972, 1974, 1979, 1982, 1986, 1987, 1988, 1999, 2000, and 2002—and also that their frequency had increased during the periods 1891–1920, 1965–1990 and 1999–2002. Rathore et al. [53] reported that India experienced three major droughts in 2002, 2004, and 2009. A drought occurred in Tharparkar, in Sindh province, starting in 2013 and reaching its most devastating point between March and August 2014 [54]. Kothawale and Rajeevan [55] documented several rainfall deficit years between 1871 and 2016. They reported four deficit years (2004, 2009, 2014, and 2015) between 2003 and 2016.

#### 2.2. Data Used

#### 2.2.1. GRACE Terrestrial Water Storage Anomaly

#### 2.2.2. Global Land Data Assimilation System (GLDAS) Observation

#### 2.2.3. Tropical Rainfall Measuring Mission

#### 2.2.4. Potential Evapotranspiration

#### 2.3. Methodology

#### 2.3.1. CTEI Description and Calculation

#### 2.3.2. Machine Learning Models

#### Support Vector Machine

_{f}represents the nonlinear transfer function, which projects the input vectors towards a very high dimension feature space; and b is the constant variable. The parameters used for the SVM algorithm were batch size = 100, C = 1, filter type = normalized training data, and kernel = poly kernel.

#### Decision Trees

- Boosted Tree

- Bagged Tree

- Random Forest

#### Matern 5/2 Gaussian Process

^{d}is assigned a random variable f(x). The Matern 5/2 kernel takes actual data densities of the stationary kernel. It creates a Fourier transform of the radial basis function (RBF) kernel. It does not have any measure problems for high-dimensional spaces. The algorithm of the Matern 5/2 GPR is as follows in Equation (7)

_{1}),...,f(x

_{N})), μ = (m(x

_{1}),...,m(x

_{N})), and K

_{ij}= κ(x

_{i},x

_{j}). m is the mean function, and it is common to use m(x) = 0 as GPs are flexible enough to model the mean arbitrarily well. κ is a positive definite kernel function or covariance function. Thus, a Gaussian process is a distribution over functions whose shape is defined by K. If points x

_{i}and x

_{j}are considered similar by the kernel, the function values at these points, f(x

_{i}) and f(x

_{j}), can be expected to be similar too. All algorithms were implemented for -CTEI modeling using MATLAB (R 2019a) software. The CTEI datasets for all algorithms were divided into training sets for those from 2003 to 2013 and testing sets for those from 2014 to 2016.

#### 2.4. Statistical Analysis

- Root Mean Square Error.

- 2.
- Coefficient of determination (Equation (9))

- 3.
- Mean absolute error (MAE)

## 3. Results and Discussion

#### 3.1. Assessment of ML Models Performance

^{2}lower than 0.15 in all cases. The RMSEs and MAEs for all models were as high as 0.6, 80, and 65, respectively. Based on these statistical analyses, it can be observed that model 8 showed the best predictions for the CTEI when compared to the other model setups.

^{2}and the lowest values in terms of the RMSE and MAE. The SVM algorithm was characterized as the best performing method among all the five algorithms, with an R

^{2}value of 0.82 (Figure 6) and the lowest errors in terms of the RMSE (0.33) and MAE (0.20).

^{2}values of 0.75, 0.58, and 0.52, respectively. They also had very low values for the error statistics in terms of the RMSE (0.39, 0.51, and 0.58, respectively) and MAE (0.21, 0.34, and 0.38, respectively).

#### 3.2. Comparison of Actual CTEI with Predicted CTEI

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The Ganga river basin’s elevation map and its tributaries along with a scale of the varying elevation.

**Figure 2.**Mean monthly values of (

**a**) GRACE terrestrial water storage anomaly (TWSA) data (derived from three agencies), (

**b**) GLDAS data (derived from four models), and (

**c**) Tropical Rainfall Measuring Mission (TRMM) and potential evapotranspiration (PET) precipitation data averaged for the basin for the period from 2003 to 2016.

**Figure 3.**(

**a**) Example of support vector regression (SVR); errors do not matter as long as they are less than ε, while the deviations are penalized. (

**b**) Typical architecture of an SVR algorithm.

**Figure 5.**Typical architectures of bagging or random forest models. The algorithms are different depending on the way the regression trees are built.

**Figure 6.**The best model for predicting the CTEI with the implementation of the support vector machine (SVM) algorithm—model 8.

**Figure 8.**Comparison of all the models with the best machine learning (ML) algorithm (model 8, SVM) for the actual and predicted CTEI values for 2003–2016.

**Figure 9.**The correlation coefficients of the predicted CTEI values from all the 12 models with the best-fitted algorithm (model 8 with the SVM algorithm) for the training (left side) and the testing period.

Data Used | Variables | Agencies/Model (Version) | Spatiotemporal Resolution | Duration |
---|---|---|---|---|

GRACE | TWSA (averaging CSR, GFZ, JPL) | CSR (RL05) | 1° × 1°, Monthly | 2003–2016 |

GFZ (RL05) | 1° × 1°, Monthly | |||

JPL (RL05) | 1° × 1°, Monthly | |||

TWSA (averaging Mosaic, NOAH, VIC, CLM) | MOSAIC (V001) | 1° × 1°, Monthly | ||

TRMM | TWSA (averaging Mosaic, NOAH, VIC, CLM) Precipitation | NOAH (V001) | 1° × 1°, Monthly | |

VIC (V001) | 1° × 1°, Monthly | |||

CLM (V001) | 1° × 1°, Monthly | |||

3B42v7 | 0.25° × 0.25°, Daily | |||

GDAS | Potential evapotranspiration | SPEIbase v2.4 | 1° × 1°, Daily |

**Table 2.**The effectiveness of the considered models for the Combined Terrestrial Evapotranspiration Index (CTEI) estimate in the Ganga basin.

ML Model | Input Variables | ML Algorithms | ${\mathbf{R}}^{2}$ | RMSE | MAE |
---|---|---|---|---|---|

Model 1 | GRACE TWSA, ${\mathrm{T}}_{\mathrm{a}}$, P, PET | RF | 0.42 | 0.59 | 0.45 |

SVM | 0.30 | 0.65 | 0.48 | ||

Boosted trees | 0.54 | 0.53 | 0.36 | ||

Bagged trees | 0.25 | 0.67 | 0.52 | ||

Matern 5/2 GPR | 0.63 | 0.47 | 0.28 | ||

Model 2 | GRACE TWSA, GWSA, ET, R | RF | 0.41 | 0.60 | 0.45 |

SVM | 0.61 | 0.49 | 0.32 | ||

Boosted trees | 0.53 | 0.54 | 0.38 | ||

Bagged trees | 0.49 | 0.56 | 0.42 | ||

Matern 5/2 GPR | 0.56 | 0.52 | 0.35 | ||

Model 3 | GRACE TWSA$,\mathrm{GLDAS}\mathrm{TWASA},\mathrm{GWSA},\mathrm{P},{\mathrm{T}}_{\mathrm{s}}$ | RF | 0.41 | 0.60 | 0.45 |

SVM | 0.39 | 0.61 | 0.46 | ||

Boosted trees | 0.50 | 0.56 | 0.41 | ||

Bagged trees | 0.42 | 0.60 | 0.45 | ||

Matern 5/2 GPR | 0.60 | 0.50 | 0.32 | ||

Model 4 | $\mathrm{u},{\mathrm{R}}_{\mathrm{N}},$LWN, PET | RF | 0.02 | 0.79 | 0.62 |

SVM | 0.15 | 0.72 | 0.56 | ||

Boosted trees | 0.05 | 0.76 | 0.60 | ||

Bagged trees | 0.01 | 0.78 | 0.61 | ||

Matern 5/2 GPR | 0.03 | 0.77 | 0.60 | ||

Model 5 | $\mathrm{GWSA},\mathrm{E},{\mathrm{R}}_{\mathrm{N}}$, SWN | RF | 0.31 | 0.65 | 0.46 |

SVM | 0.60 | 0.49 | 0.36 | ||

Boosted trees | 0.51 | 0.55 | 0.40 | ||

Bagged trees | 0.23 | 0.69 | 0.53 | ||

Matern 5/2 GPR | 0.53 | 0.54 | 0.37 | ||

Model 6 | $-\mathrm{GRACE}\mathrm{TWSA},\mathrm{GLDAS}\mathrm{TWASA},$ SWN, LWN, P, PET | RF | 0.19 | 0.70 | 0.52 |

SVM | 0.71 | 0.42 | 0.29 | ||

Boosted trees | 0.51 | 0.55 | 0.39 | ||

Bagged trees | 0.22 | 0.69 | 0.52 | ||

Matern 5/2 GPR | 0.70 | 0.42 | 0.25 | ||

Model 7 | $\mathrm{ET},\mathrm{R},{\mathrm{T}}_{\mathrm{a}}$ | RF | 0.05 | 0.76 | 0.61 |

SVM | 0.04 | 0.77 | 0.61 | ||

Boosted trees | 0.00 | 0.79 | 0.65 | ||

Bagged trees | 0.03 | 0.77 | 0.62 | ||

Matern 5/2 GPR | 0.02 | 0.77 | 0.62 | ||

Model 8 | $-\mathrm{GRACE}\mathrm{TWSA},\mathrm{GLDAS}\mathrm{TWASA},$$\mathrm{GWSA},\mathrm{ET},\mathrm{R},{\mathrm{T}}_{\mathrm{a}}$$,{\mathrm{R}}_{\mathrm{N}}$,SWN, LWN, P, PET | RF | 0.33 | 0.63 | 0.45 |

SVM | 0.82 | 0.33 | 0.20 | ||

Boosted trees | 0.58 | 0.51 | 0.34 | ||

Bagged trees | 0.52 | 0.54 | 0.38 | ||

Matern 5/2 GPR | 0.75 | 0.39 | 0.21 | ||

Model 9 | $-\mathrm{GRACE}\mathrm{TWSA},\mathrm{GLDAS}\mathrm{TWASA},$ GWSA, P, ET, R, ${\mathrm{T}}_{\mathrm{a}}$$,{\mathrm{T}}_{\mathrm{S}}$$,{\mathrm{R}}_{\mathrm{N}}$$,\mathrm{SWN},\mathrm{LWN},\mathrm{PET},\mathrm{u}$ | RF | 0.35 | 0.63 | 0.44 |

SVM | 0.60 | 0.49 | 0.36 | ||

Boosted trees | 0.46 | 0.57 | 0.38 | ||

Bagged trees | 0.48 | 0.57 | 0.40 | ||

Matern 5/2 GPR | 0.69 | 0.43 | 0.24 | ||

Model 10 | $\mathrm{GRACE}\mathrm{TWSA}$$,\mathrm{GWSA},{\mathrm{T}}_{\mathrm{S}}$$,\mathrm{P},\mathrm{u}$ | RF | 0.24 | 0.69 | 0.50 |

SVM | 0.44 | 0.59 | 0.44 | ||

Boosted trees | 0.39 | 0.61 | 0.44 | ||

Bagged trees | 0.15 | 0.73 | 0.54 | ||

Matern 5/2 GPR | 0.56 | 0.52 | 0.35 | ||

Model 11 | $\mathrm{GRACE}\mathrm{TWSA},{\mathrm{T}}_{\mathrm{a}},{\mathrm{T}}_{\mathrm{S}},$ PET, SWN | RF | 0.13 | 0.73 | 0.52 |

SVM | 0.53 | 0.53 | 0.37 | ||

Boosted trees | 0.39 | 0.61 | 0.41 | ||

Bagged trees | 0.25 | 0.67 | 0.51 | ||

Matern 5/2 GPR | 0.58 | 0.51 | 0.31 | ||

Model 12 | $\mathrm{GWSA},{\mathrm{T}}_{\mathrm{S}}$, ET | RF | 0.43 | 0.59 | 0.43 |

SVM | 0.50 | 0.55 | 0.41 | ||

Boosted trees | 0.52 | 0.54 | 0.39 | ||

Bagged trees | 0.28 | 0.66 | 0.51 | ||

Matern 5/2 GPR | 0.59 | 0.50 | 0.36 |

**Table 3.**Average actual and predicted CTEIs and differences and deviations for the best machine learning models in the training and testing periods in the Ganga basin.

Year | Actual CTEI | Model 8 | ||
---|---|---|---|---|

Predicted CTEI | Difference | Deviation | ||

2003 | 1.03 | 1.06 | 0.03 | 0.03 |

2004 | 0.82 | 0.78 | −0.04 | −0.05 |

2005 | 0.49 | 0.44 | −0.05 | −0.10 |

2006 | 0.18 | 0.14 | −0.05 | −0.25 |

2007 | 0.43 | 0.28 | −0.14 | −0.33 |

2008 | 0.44 | 0.23 | −0.21 | −0.48 |

2009 | −0.44 | −0.29 | 0.15 | −0.34 |

2010 | −0.51 | −0.39 | 0.12 | −0.24 |

2011 | 0.26 | 0.14 | −0.12 | −0.47 |

2012 | −0.35 | −0.19 | 0.16 | −0.45 |

2013 | −0.02 | −0.20 | −0.18 | 10.31 |

2014 | −0.42 | −0.28 | 0.14 | −0.34 |

2015 | −0.71 | −0.70 | 0.01 | −0.02 |

2016 | −1.21 | −1.27 | −0.06 | 0.05 |

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**MDPI and ACS Style**

Elbeltagi, A.; Kumari, N.; Dharpure, J.K.; Mokhtar, A.; Alsafadi, K.; Kumar, M.; Mehdinejadiani, B.; Ramezani Etedali, H.; Brouziyne, Y.; Towfiqul Islam, A.R.M.;
et al. Prediction of Combined Terrestrial Evapotranspiration Index (CTEI) over Large River Basin Based on Machine Learning Approaches. *Water* **2021**, *13*, 547.
https://doi.org/10.3390/w13040547

**AMA Style**

Elbeltagi A, Kumari N, Dharpure JK, Mokhtar A, Alsafadi K, Kumar M, Mehdinejadiani B, Ramezani Etedali H, Brouziyne Y, Towfiqul Islam ARM,
et al. Prediction of Combined Terrestrial Evapotranspiration Index (CTEI) over Large River Basin Based on Machine Learning Approaches. *Water*. 2021; 13(4):547.
https://doi.org/10.3390/w13040547

**Chicago/Turabian Style**

Elbeltagi, Ahmed, Nikul Kumari, Jaydeo K. Dharpure, Ali Mokhtar, Karam Alsafadi, Manish Kumar, Behrouz Mehdinejadiani, Hadi Ramezani Etedali, Youssef Brouziyne, Abu Reza Md. Towfiqul Islam,
and et al. 2021. "Prediction of Combined Terrestrial Evapotranspiration Index (CTEI) over Large River Basin Based on Machine Learning Approaches" *Water* 13, no. 4: 547.
https://doi.org/10.3390/w13040547