IDF-package {IDF}

Introduction

Description

This package provides functions to estimate IDF relations for given precipitation time series on the basis of a duration-dependent generalized extreme value distribution (d-GEV). The central function is gev.d.fit, which uses the method of maximum-likelihood estimation for the d-GEV parameters, whereby it is possible to include generalized linear modeling for each parameter. This function was implemented on the basis of gev.fit. For more detailed information on the methods and the application of the package for estimating IDF curves with spatial covariates, see Ulrich et. al (2020).

Details

• The d-GEV is defined following Koutsoyiannis et al. (1998):

  G(x)= \exp[-( 1+\xi(x/\sigma(d)- \tilde{\mu}) ) ^{-1/\xi}]

  defined on \{ x: 1+\xi(x/\sigma(d)- \tilde{\mu} > 0) \} , with the duration dependent scale parameter \sigma(d)=\sigma_0/(d+\theta)^\eta > 0, modified location parameter \tilde{\mu}=\mu/\sigma(d)\in R and shape parameter \xi\in R, \xi\neq 0. The parameters \theta \leq 0 and 0<\eta<1 are duration offset and duration exponent and describe the slope and curvature in the resulting IDF curves, respectively.

• The dependence of scale and location parameter on duration, \sigma(d) and \mu(d), can be extended by multiscaling and flattening, if requested. Multiscaling introduces a second duration exponent \eta_2, enabling the model to change slope linearly with return period. Flattening adds a parameter \tau, that flattens the IDF curve for long durations:

  \sigma(x)=\sigma_0(d+\theta)^{-(\eta+\eta_2)}+\tau

  \mu(x)=\tilde{\mu}(\sigma_0(d+\theta)^{-\eta_1}+\tau)

• A useful introduction to Maximum Likelihood Estimation for fitting for the generalized extreme value distribution (GEV) is provided by Coles (2001). It should be noted, however, that this method uses the assumption that block maxima (of different durations or stations) are independent of each other.

References

• Ulrich, J.; Jurado, O.E.; Peter, M.; Scheibel, M.; Rust, H.W. Estimating IDF Curves Consistently over Durations with Spatial Covariates. Water 2020, 12, 3119, https://doi.org/10.3390/w12113119

• Demetris Koutsoyiannis, Demosthenes Kozonis, Alexandros Manetas, A mathematical framework for studying rainfall intensity-duration-frequency relationships, Journal of Hydrology, Volume 206, Issues 1–2,1998,Pages 118-135,ISSN 0022-1694, https://doi.org/10.1016/S0022-1694(98)00097-3

• Coles, S.An Introduction to Statistical Modeling of Extreme Values; Springer: New York, NY, USA, 2001, https://doi.org/10.1198/tech.2002.s73

Examples

## Here are a few examples to illustrate the order in which the functions are intended to be used.

## Step 0: sample 20 years of example hourly 'precipitation' data