Evaluation of Recharge and Groundwater Dynamics of a Shallow Alluvial Aquifer in Central Ganga Basin, Kanpur (India)

Abstract

Evaluation of recharge and groundwater dynamics of an aquifer is an important step for finding a proper groundwater management scenario. This has been performed on the basis of statistical Kendall Tau test to find a relationship between groundwater levels and hydro-meteorological parameters (e.g., precipitation, temperature, evaporation). Recharge to the aquifer was estimated for identification of critical areas/locations based on the analytical Soil and Water Assessment Tool. Moreover, spatiotemporal variability of groundwater levels has been quantified using space–time variogram. The overall characterization method has been applied to the shallow alluvial aquifer of Kanpur city in India. The analysis was performed using groundwater level data from 56 monitoring piezometer locations in Kanpur from March 2006 to June 2011. Groundwater level shows relatively higher correlation with temperature. Performance of the geostatistical model was evaluated by comparing with the observed values of groundwater level from January 2011 to June 2011 for two scenarios: “with limited spatiotemporal data” and “without spatiotemporal data.” It is evident that spatiotemporal prediction of groundwater level can be performed even for the unmonitored/missing data. This analysis demonstrates the potential applicability of the method for a general aquifer system.

Appendix: Model Performance Evaluation Measures

Average Absolute Relative Error (AARE)

The average absolute relative error (AARE) is the average value of the relative errors in estimation of the total number of data points considered. The relative error in estimation RE(u, t) is given by

$$ {\text{RE}}\left( {u,t} \right) = \frac{{Z^{*} \left( {u,t} \right) - Z\left( {u,t} \right)}}{{Z\left( {u,t} \right)}} \times 100\;\%, $$

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where Z ^*(u, t) and Z(u, t) are the estimated value and observed value at a certain spatial location u and time t. The AARE is given by

$$ {\text{AARE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left| {{\text{RE}}\left( {u_{i} ,t} \right)} \right|}, $$

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where N is the total number of spatial locations where values are estimated.

Threshold Statistics (TS _x )

The threshold statistic (TS) provides the estimate of error distribution thereby framing a clear picture of the model performance. The TS is defined for a certain level (x %) of absolute relative error (ARE) and is denoted as TS_x. The TS_x for level x is given by

$$ {\text{TS}}_{x} = \frac{{n_{x} }}{N} \times 100\;\%, $$

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where n _x is the total number of values predicted in which the absolute relative error (ARE) in prediction is less than x %.

Correlation Coefficient (R)

The correlation coefficient (R) measures the strength of correlation between the observed values and the predicted values. Its value ranges between −1 and +1. A value closer to 1 indicates good correlation, while 0 indicates no correlation at all. The correlation coefficient (R) is calculated by

$$ R = \frac{{N\sum\limits_{i = 1}^{N} {Z(u_{i} ,t).Z^{*} (u_{i} ,t)} - \sum\limits_{i = 1}^{N} {Z(u_{i} ,t)} \sum\limits_{i = 1}^{N} {Z^{*} (u_{i} ,t)} }}{{\sqrt {N\left( {\sum\limits_{i = 1}^{N} {Z(u_{i} ,t)^{2} } } \right) - \left( {\sum\limits_{i = 1}^{N} {Z(u_{i} ,t)} } \right)^{2} } \sqrt {N\left( {\sum\limits_{i = 1}^{N} {Z^{*} (u_{i} ,t)^{2} } } \right) - \left( {\sum\limits_{i = 1}^{N} {Z^{*} (u_{i} ,t)} } \right)^{2} } }}. $$

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Coefficient of Efficiency (E)

The coefficient of efficiency (Nash and Sutcliffe 1970) compares the predicted and the observed values and evaluates the capability of the model to explain total variance in the data set. The coefficient of efficiency is given by

$$ E = \frac{{E_{1} - E_{2} }}{{E_{1} }}, $$

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$$ E_{1} = \sum\limits_{i = 1}^{N} {\left( {Z\left( {u_{i} ,t} \right) - \bar{Z}\left( {u_{i} ,t} \right)} \right)^{2} }, $$

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$$ E_{2} = \sum\limits_{i = 1}^{N} {\left( {Z^{*} \left( {u_{i} ,t} \right) - Z\left( {u_{i} ,t} \right)} \right)^{2} }, $$

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where $\bar{Z}\left( {u_{i} ,t} \right)$ is the mean of the observed values.

Normalized Mean Bias Error (NMBE)

The mean bias error (MBE) is calculated by

$$ {\text{MBE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {Z^{*} \left( {u_{i} ,t} \right) - Z\left( {u_{i} ,t} \right)} \right)}. $$

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The normalized mean bias error is given by

$$ {\text{NMBE}} = \frac{{\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {Z^{*} \left( {u_{i} ,t} \right) - Z\left( {u_{i} ,t} \right)} \right)} }}{{\frac{1}{N}\sum\limits_{i = 1}^{N} {Z\left( {u_{i} ,t} \right)} }} \times 100\;\%. $$

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A positive value of NMBE indicates an overall over-prediction of values, while a negative value of NMBE indicates an overall under-prediction.

Normalized Root Mean Square Error (NRMSE)

The root mean square error (RMSE) is calculated by

$$ {\text{RMSE}} = \left( {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {Z^{*} \left( {u_{i} ,t} \right) - Z\left( {u_{i} ,t} \right)} \right)} \right]^{2} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}. $$

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The normalized root mean square error is given by

$$ {\text{NRMSE}} = \frac{{\left( {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left[ {\left( {Z^{*} \left( {u_{i} ,t} \right) - Z\left( {u_{i} ,t} \right)} \right)} \right]^{2} } } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\frac{1}{N}\sum\limits_{i = 1}^{N} {Z\left( {u_{i} ,t} \right)} }}. $$

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A better model performance is indicated by a lower value of NRMSE.