## Abstract

The climate change effects on the variation and distribution of precipitation at 15 vital dam sites (large dams) located over the Godavari River Basin (GRB) are investigated in this study. The Time-Varying Downscaling Model (TVDM) technique is employed for this purpose. The data for daily rainfall at 0.25 × 0.25 degrees grid size are obtained from IMD, Pune. The CanESM2 outputs were considered as predictor variables. The calibration and validation of TVDM was carried out using the historical rainfall data (1951–2010). Next, the future variation (2011–2100) of rainfall over the GRB is analyzed using three Representative Concentration Pathway (RCP) climate change scenarios viz. RCP2.6, 4.5 and 8.5. Further, the extreme rainfall events which are generating the floods in the selected locations are analyzed using the extreme percentile values (90th and 95th). The variations of these extreme values in the future (2011–2100) are judge against the baseline (1971–2000) data for analyzing the possible impacts of climate change. The results reveal that the magnitude and the count of extreme events are expected to intensify in the future period. In comparison with the baseline data, all the study locations are expected to experience an increase of 3.61–38.93% in the 95th percentile values according to the RCP8.5 scenario. Furthermore, rainfall frequency analysis is also conducted to evaluate the potential hazards associated with flooding. The results show that the extreme events are more frequent with high magnitude during the future period compared to the baseline data. It implies a vulnerable situation to most of the dams in the GRB during the future period and effect will be worst as per RCP8.5 scenario. The outcomes of this research are expected to aid stakeholders to develop good strategies for safeguarding through the better maintenance, management of the dams and reservoirs in the wake of Climate Change.

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## Data availability

All the data sets used in this study are freely available. However, if readers/users face any difficulty to access the data sets, the authors are happy to provide the same upon gentle request for the same. This study uses the observed rainfall data from IMD, Pune, it can be accessible at https://www.imdpune.gov.in/cmpg/Griddata/Rainfall_25_Bin.html. The large-scale GCM data sets are downloaded from the CMIP-5 web portal which are available upon successful registration to the CMIP web portal at https://esgf-node.llnl.gov/projects/cmip5/.

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## Acknowledgements

The Department of Science and Technology (DST), Ministry of Science and Technology, Government of India, provided financial support (SRG/2020/000823) for conducting this research, for which the authors express their gratitude. The authors also extend their thanks to the India Meteorological Department (IMD) for providing the required data for this study.

## Funding

DST SERB, SRG/000823/2020, Subbarao Pichuka.

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## Appendices

### Annexure – I

Table A1 (see Table 6)

Table A2 (see Table 7)

### Outline of the time-varying downscaling model (TVDM)

TVDM is developed by considering the time-varying association between large-scale (causal) variables and the target variable. The methodology is developed based on the parameter updating ability of the Bayesian approach (West and Harrison 1997). Many recent studies related to climate change have adopted the Bayesian approach. For instance, the prediction of hydroclimatic variables, the quantification of water quality, uncertain runoff calculations are some of them (Maity and Nagesh Kumar 2006; Nagesh Kumar and Maity 2008; Sarhadi et al. 2016; Tyralis and Koutsoyiannis 2014; Vrugt et al. 2009; Yang et al. 2007).

The monthly meteorological mean of the target time series is hypothesized to have a deterministic component and a stochastic component. The information about causal variables has been used to estimate the stochastic component, which is added to the deterministic part. Then the causal and target variables are standardized. For standardizing a variable, the long-term mean (*µ*) is subtracted from the variable value at any time step and then it is divided by the long-term standard deviation (*σ*). Hereafter the causal and target variables are referred as standardized variables. The methodology of TVDM can be found from Pichuka and Maity (2018). For brevity, the key equations of TVDM are presented in this paper as follows

The downscaled target variable at time step *t* is expressed in the form of

where \(Y_{t}\) is the target variable to be downscaled at time step *t*,

\(D_{t - 1}\) is the initial information provided by the modeler to initialize the TVDM,

*n* is degree of freedom.

*n* for *t*th time step is given as

The expressions for \(F_{t}\) and \(Q_{t}\) are expressed as –

where \(\overline{{Y_{t} }}\) is the climatological mean value of the target variable at *t*th time step,

\(x_{t}^{1} ,\,x_{t}^{2} ,\,x_{t}^{3} ,....x_{t}^{z}\) are the standardized causal variables at the *t*th time step,

\(m_{t - 1}^{1} ,\;m_{t - 1}^{2} ,....m_{t - 1}^{z}\) are the model parameters supplied by the modeler at initial time step.

\(R_{t}\) is expressed as

where \(\delta\) is termed as discount factor and ranges between 0 and 1. The value of \(\delta\) denotes the fact that the system evolution variance increases from time to time (i.e., *t* to *t* + 1), i.e., more uncertainty present in the future information. The optimum value of \(\delta\) is obtained based on model performance. The higher the value, the lower the rate of decay of past information and vice versa (West and Harrison 1997).

The optimum value of \(\delta\) is estimated on the basis of model performance. It is also noteworthy that higher values of \(\delta\) indicate slower rate of decay of previous information and vice versa (West and Harrison 1997).

\(S_{t - 1}\) is expressed as

The parameters \(m_{0}\),\(C_{0}\), \(n_{0}\), and \(d_{0}\) are to be supplied to the TVDM as the initial information for all the causal variables. The parameter values (henceforth *m* values) are updated through a system of equations till the development period. The target variable is obtained at every time step based on Eq. 4. Further, the *m *values are projected for the future period to obtain the target variable. The projection is carried out by considering the deterministic part (trend and periodicity) and stochastic part of the *m* values from development period. Once the future *m* values are projected, the target variable is downscaled from Eq. 4. The required causal variables are obtained from various scenarios (RCPs) of selected GCM. The procedure of updating *m* values, system of equations to update these *m* values, and their projection during the future period are presented in the following section.

### Updating and projection of the parameter values (*m* values)

To update the parameters from the time step \(\left( {t - 1} \right)\) to \(t\), where \(t = 2,\,3,\,4,\,...\), *N*, the value of \(R_{t}\) (given in Eq. 6) is updated for next time step \(t\); to do so, the value of \(C_{t}\) should be known and is given as

Finally, the downscaled target variable at time step *t* is expressed in the form of Eq. (2) and the downscaled values are obtained using Eqs. (3–5).

The *m values* are updated through system of equations presented below.

where \(A_{t}\) is given as

For the next step, i.e., \(t + 1\), \(n_{t}\), and \(d_{t}\) are required to calculate \(Q_{t}\) using \(S_{t}\) (Eq. 7). These are expressed as:

The *m values*, \(m_{t}^{p} \left( {t = 1,\,2, \cdots \,,n\,\;and\,\,p = 1, \cdots ,z} \right)\) develop over time for all the input variables.

### Projection of m values for future period

The projection of *m values* is carried out by considering the deterministic part (trend and periodicity) and stochastic part of the *m values* from development period. The stochastic part is captured by means of auto-regressive (AR) model. Thus, the time series of historical *m values* is modeled as:

where \(m_{t}^{{{\text{tr}}}}\), \(m_{t}^{{{\text{pr}}}}\), and \(m_{t}^{{{\text{st}}}}\) represent the linear trend component, periodic component, and the stochastic component, respectively, of the *m* values at time *t*. The \(m_{t}^{{{\text{tr}}}}\) is given by

where \(p_{1}\) and \(p_{2}\) are the regression coefficients those are obtained by least square method taking time as the independent variable. The periodic component, i.e., \(m_{t}^{{{\text{pr}}}}\) is modeled after separation of trend from the *m values* time series.

The equation for \(m_{t}^{{{\text{pr}}}}\) is expressed as

where \(m_{t}^{{{\text{pr}}}} (T)\) is harmonically fitted means at period \(T\) (\(T\) = 1, 2,…. \(P\)) and \(P\) is base period, which is calculated from the periodogram of the detrended time series of *m* values which is given as

where \(w\) is the index value corresponding to the peak of the periodogram; \(A_{0}\) = mean of historical *m values* and is given as

where \(X(T)\) is the detrended *m values*; *h* is number of harmonics, which is expressed as

\(A_{k}\) and \(B_{k}\) are known as sine and cosine Fourier coefficients, respectively, and written as

where *k* = 1, 2,…, *h*.

The residual left (after take out of trend and periodicity) is modeled by an AR model. The AR(1) model is obtained to be sufficient from the autocorrelogram and partial autocorrelogram analyses. Thus, an AR(1) model is given by

where \(m_{t}^{{{\text{st}}}}\) is the modeled stochastic component, \(b\) is the coefficient of the AR(1) model fitted to the residuals, and \(m_{t}^{{\text{e}}}\) is the error at the *t*th time step. Since the error \(m_{t}^{{\text{e}}}\) is assumed to be normally distributed with a mean 0 and standard deviation \(\sigma_{{{\text{e}}_{t} }}\) the expected value of \(m_{t}^{{{\text{st}}}} \left( { = b \times m_{t - 1}^{{{\text{st}}}} } \right)\) is used in Eq. (14).

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Pichuka, S., Roulo, D. Dams and their safety under the extreme climate conditions: study of dams on Godavari River.
*Sustain. Water Resour. Manag.* **10**, 32 (2024). https://doi.org/10.1007/s40899-023-01018-7

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DOI: https://doi.org/10.1007/s40899-023-01018-7