An integrated stochastic approach for extreme rainfall analysis in the National Capital Region of India

Abstract

The National Capital Region of India (NCR Delhi) receives around 26 rainy days with majority of short duration high intensity rainfall events. Yet, the city faces severe waterlogging during south-west monsoon, and shortage of water in other seasons due to rapid urbanization and changing hydrological flow patterns. In an uncertain scenario, where spatiotemporal variability of rainfall at local scale is not very well understood, a location-specific robust model applicable for the megacity through interpretation of parameter estimates will improve understanding of extreme rainfall pattern with duration. Identification of the best-fit statistical model for prediction of short duration extreme events is done and parameters of the model are evaluated for different durations. The study finds that the 2-parameter gamma distribution and 3-parameter generalized extreme value (GEV) predict similar return levels of extreme intensity for short durations and short return periods. The shape parameter in GEV and shape and scale parameters in gamma explain the extreme quantile in the distribution responsible for prediction of high magnitude events. The more generic gamma model is robust and applicable at local scale, with pronounced shape parameter variations across durations (1–11.534, 2–8.264, 3–6.609 h). It is concluded that the knowledge of hourly variation in extreme rainfall events will help in informed decision making in this acutely water-stressed region of the world.

Highlights

• Short duration extreme rainfall events are frequent in semi-arid urban regions and characterization of these storms is important. The extreme rainfall pattern characteristics is defined through the parameter estimates of shape and scale of gamma distribution.

• Within 1 h duration rainfall events pattern, the higher shape factor identifies storms in a more wet regime (α-14.05) than the mean (α-11.534), while the lower shape factor identifies dry regime (α-9.018).

• The 4 h and 6 h events have the lowest shape factor and high scale factor, implying variation, and unpredictability.

• Short duration storms have the potential to cause flash floods, the hydrological insights provided by this study will be useful in similar geographical regions.

Appendices

Appendix 1

The pdf of GEV, gamma and lognormal distributions are given below.

The probability density function (pdf) of GEV is given by:

$$ f\left( {x;\mu ,\sigma ,\xi } \right) = \left( {1/\sigma } \right)\left[ {1 + \xi \left( {x - \mu } \right)/\sigma } \right]^{{\frac{1}{\xi } - 1}} \times{ \exp }\left\{ { - \left[ {1 + \xi \left( {x - \mu } \right)/\sigma } \right]^{ - 1/\xi } } \right\},\;{\text{when}}\; \xi \ne 0 $$

(A1.1)

where the three parameters of the GEV are defined as $\mu$ = location parameter, $\sigma$ = scale parameter, ${{\xi }}$ = shape parameter and x is the independent rainfall intensity variable. When the shape factor (${{\xi }}$) is zero, the distribution reduces to 2-parameter distribution Gumbel which is the most common EV1 distribution.

The gamma probability distribution function (pdf) is shown in equation (A1.2) as follows (Wilks 2002):

$$ f\left( {x; \alpha ,\beta } \right) = \frac{{\left( {\frac{x}{\beta }} \right)^{\alpha - 1} e^{{ - \frac{x}{\beta }}} }}{{\beta {{\varGamma }}\left( \alpha \right)}};\quad x > 0. $$

(A1.2)

The parameters of the gamma distribution are α: shape factor, β: scale factor and Γ(α): the gamma function.

The pdf of lognormal distribution is shown in equation (A1.3) as follows (Kottegoda and Rosso 2008):

$$ f\left( {x;\mu_{{{ \ln }\left( {{x}} \right)}} ,{\sigma}_{{{ \ln }\left( {{x}} \right)}} } \right) = \frac{1}{{x\sigma_{{{ \ln }\left( {{x}} \right)}} \sqrt {2\varPi } }}\times e\left\{ { - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\left( {\frac{{\ln \left( x \right) - \mu_{{{ \ln }\left( {{x}} \right)}} }}{{\sigma_{{{ \ln }\left( {{x}} \right)}} }}} \right)^{2} } \right\} $$

(A1.3)

where the location factor is μ_ln(x) and scale factor is σ_ln(x), respectively.

Appendix 2: Akaike Information Criterion (AIC)

The Akaike Information Criterion (Cahill 2003) uses the discrepancy measure between the true model (which is unknown) and the approximate model, in terms of the negative log likelihood (nllh) value. It is an alternative procedure for statistical tests for flood frequency analysis (Laio et al. 2009). The AIC is considered robust as the method measures the amount of information lost, so the model which loses the least information has the best model quality. Thus, the model with least AIC value is chosen as the best model. AIC(p) is sometimes used instead of AIC, which adds the sum of number of parameter (as p) to the nllh value of the distribution, it has been used to choose the best-fit distribution for estimating intensity duration frequency (IDF) of extreme rainfall in India at regional scale (Mondal and Mujumdar 2015). It is a method of selecting a model from a set of models by calculating the Kullback–Leibler distance between the true model and the observed (Laio et al. 2009) as follows:

$$ {\text{AIC}}\left( {{p}} \right) = 2{\text{nllh}}\left( {{p}} \right) + 2{{p}}. $$

(A2.1)

2.1 Index of agreement

The prediction ability of the gamma and GEV models with the observed AMS may be evaluated using the index of agreement method-d (Willmott 1982) which is sensitive to extreme values. It has been used to evaluate model performance in the field of environment, hydrology and agrometeorology (Sharma et al. 1999).

$$ {{d}} = 1 - \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {P_{i} - O_{i}} \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{n} \left( {\left| {P_{i} - O} \right| + \left| {O_{i} - O} \right|} \right)^{2} }} $$

(A2.2)

d varies between 0 and 1, observed value in AMS is O_i, O is mean of observed values, P_i is the predicted model value. The higher the value of index d, better is the model performance, with high values of d as the best fit, indicating better predicted model fit to the observed values.

Author statement

The research methodology was designed by Prof Prateek Sharma, Ms Ranjana Ray Chaudhuri carried out the research and data analysis and wrote the manuscript. Prof Prateek Sharma supervised the research and data analysis and refined the ideas and contributed in finalizing the paper.