## Abstract

The delta region of the Mahanadi river basin is faced with high floods having arisen many questions like (1) due to the release of Hirakud dam, (2) due to the middle region catchment contribution (Tel tributary at Kantamal station and Ong tributary at Salebhata station) and (3) due to an increase in heavy rainfalls over the basin. To answer this, we choose the partial duration series (PDS) having an average number of flood peaks per year (*λ* = 3) at Naraj based on the capacity of the existing embankment which mitigates floods up to 28,400 m^{3}/s and analyze the trends of peak discharge of PDS having *λ* = 3 at the Naraj, Hirakud dam and middle region to know the flow behavior which is responsible for flooding problem in the delta of Mahanadi basin. Subsequently, the trend of extreme rainfall over the entire basin and at the middle region of the basin is analyzed. Analysis of trends of extreme floods in relation to the trends of extreme rainfall to have a proper understanding of the causes of changing floods over the basin is important. Then, runoff at Naraj is estimated based on two techniques: artificial neural networks with different algorithms and an adaptive neuro-fuzzy inference system (ANFIS) using five different input combinations of peak runoff data and rainfall data of upstream locations and compare their performance based on different indices, namely correlation coefficient (*R*), index of agreement (*d*), root-mean-square error and Nash–Sutcliffe efficiency (*E*) to obtain the best input–output model. From the analysis, it is shown that ANFIS techniques as well as the neural network with Levenberg–Marquardt algorithm performs better for estimating runoff at Naraj and that gives the clear idea about the delta flooding. Also, the analysis revealed the answer arisen by many questions regarding the delta flood problem and which are due to the middle region catchment contribution and an increase in extreme rainfall.

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## Acknowledgements

The authors would like express their thanks to the National Institute of Technology, Rourkela, for providing the research environment.

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## Appendix

### Appendix

### Mann–Kendall Test

The Mann–Kendall test is a rank-based test first used by Mann (1945), and then the test statistic was derived by Kendall (1975). The Mann–Kendall S Statistic is computed as follows:

where

where *T*_{j} and *T*_{i} are the annual values in years *j* and *i*, *j* > *i*, respectively.

If *n* < 10, the value of |*S*| is compared directly to the theoretical distribution of S derived by Mann and Kendall. The two-tailed test is used. At certain probability level, *H*_{0} is rejected in favor of *H*_{1} if the absolute value of *S* equals or exceeds a specified value *S*_{α/2}, where *S*_{α/2} is the smallest *S* that has probability less than *α*/2 to appear in case of no trend. A positive (negative) value of *S* indicates an upward (downward) trend.

For *n* ≥ 10, the statistic *S* is approximately normally distributed with the mean and variance as follows:

The variance (*σ*^{2}) for the *S* statistic is defined by:

in which *t*_{i} denotes the number of ties to extent *i*. The summation term in the numerator is used only if the data series contains related values. The standard test statistic *Z*_{s} is calculated as follows:

The test statistic *Z*_{s} is used a measure of the significance of the trend. In fact, this test statistic is used to test the null hypothesis, *H*_{0}. If | *Z*_{s}| is greater than *Z*_{α/2}, where α represents the chosen significance level (e.g., 5% with *Z* 0.025 = 1.96); then, the null hypothesis is invalid, implying that the trend is significant.

### Liner Regression

Liner regression is a simple technique which stated the positive or negative slope values represent the direction, and the magnitude of the trend was also analyzed. A regression line of the order *y *= *a *+ *bx* is fitted to the sample data. If *b* is found significantly different from zero, then we assume trend to be present in the data. The linear regression-based trends are analyzed using the *R*-square values.

### Levenberg–Marquardt (LM)

The algorithm uses the Hessian matrix which is a second-order training (Demuth and Beale 1998). The LM method is the standard method for minimization of the MSE criterion, due to its rapid convergence properties and robustness.

### Gradient Descent with Momentum and Adaptive Learning Rate Algorithm (GDX)

The algorithm calculates derivatives of performance (*p)* with respect to the weight and bias variables (*X*) using various epochs. Hagan et al. (1996) provided the detailed description of this training method.

### Conjugate Gradient Backpropagation Algorithm (CGF)

The algorithm can use the expression *X *= *X* + *adX*, where *dX* is the search direction. The parameter *a* is a constant that minimizes the error functions along the search direction. Fletcher (1987) provided the details of this algorithm.

*Root-Mean-Square Error (RMSE)* Root-mean-squared error is a measure of the differences between values predicted by model or an estimator and the actually observed values.

where *O*_{i} is the observed value, *P*_{i} is the model-simulated value and *n* is the total number of observations of the dataset.*The Nash–Sutcliffe Efficiency (E) Defined as*

where *O*_{i} is the observed value, *P*_{i} is the model-simulated value and \(\overline{O}\) is the mean observed value. Essentially, the closer the model efficiency is to 1, the more accurate the model is.

*Index of agreement (d)* The index of agreement represents the ratio of the mean square error and the potential error and is defined as:

The range of *d* is similar to that of *r*^{2} and lies between 0 (no correlation) and 1 (perfect fit).

*Correlation coefficient (R)* The correlation coefficient (*R*) is given below:

where *Q*_{0} and *Q*_{P} are the observed and estimated value, *M*_{0} and *M**P* are the means of the observed and estimated value, respectively, and *n* is the total number of observations of the dataset.

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Guru, N., Jha, R. Application of Soft Computing Techniques for River Flow Prediction in the Downstream Catchment of Mahanadi River Basin Using Partial Duration Series, India.
*Iran J Sci Technol Trans Civ Eng* **44, **279–297 (2020). https://doi.org/10.1007/s40996-019-00272-0

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### Keywords

- Trend analysis
- Flow prediction
- ANN
- ANFIS
- Mahanadi river