## 1. Introduction

[2] Precipitation simulation is one of the key features of hydrological models, agricultural models, and climate impact studies [*Kleiber et al.*, 2011]. Sufficiently long series of precipitation records are needed for catchment water management, drought characterization and prediction, and crop growth simulation. However, historical records of precipitation with desired spatial and temporal resolution are almost always insufficient. Moreover, it is difficult to quantify the uncertainty of model results from only a single sequence of realizations. On the other hand, there is considerable discussion these days that climate change is contributing to the increase in frequencies and magnitudes of precipitation extremes, leading to floods or droughts, and hence evaluating changes in precipitation extremes is receiving significant attention [*Solomon et al.*, 2007; *Lenderink and Meijgaard*, 2008; *Hardwick Jones et al.*, 2010]. Therefore, realistically modeling the full spectrum of precipitation is desired.

[3] Precipitation simulation dates back to the 1950s. Over the past decades, many simulation techniques have been developed [e.g., *Gabriel and Neumann*, 1962; *Katz*, 1974, 1977; *Todorovic and Woolhiser*, 1975; *Richardson*, 1981; *Stern and Coe*, 1984; *Lall and Sharma*, 1996; *Wilks*, 1998; *Rajagopalan and Lall*, 1999; *Parlange and Katz*, 2000; *Yan et al.*, 2002; *Harrold et al.*, 2003a, 2003b; *Chandler*, 2005; *Mehrotra and Sharma*, 2007a, 2007b; *Furrer and Katz*, 2007; *Zheng and Katz*, 2008a, 2008b; *Brissette et al.*, 2007]. Typically, daily precipitation is represented as a mixture of two distributions in a parametric, nonparametric, or semiparametric framework. One is discrete binary distribution modeling the wet or dry state of a given day, and the other is continuous distribution modeling nonzero precipitation amounts on wet days. A most recent review on precipitation simulation can be found by *Sharma and Mehrotra* [2010]. Overall, there are two acknowledged challenges in daily precipitation simulation. One is referred to as overdispersion [*Katz and Zheng*, 1998]. The other one is the loss of extreme precipitation events. The first problem concerns both the occurrence and amount processes of precipitation, whereas the second one mainly concerns the amount process. This paper focuses on the second problem.

[4] Since daily precipitation amount always shows a skewed distribution with a bias toward low values, it is usually modeled by distribution families which have right-skewed property [*Hundecha et al.*, 2009]. Different distributions, such as Kappa [*Mielke*, 1973], exponential [*Todorovic and Woolhiser*, 1975; *Roldan and Woolhiser*, 1982], gamma [*Ison et al.*, 1971; *Katz*, 1977; *Schoof et al.*, 2010], mixed exponential [*Roldan and Woolhiser*, 1982; *Wilks*, 1998, 1999], and truncated and power transformed normal distributions [*Bárdossy and Plate*, 1992; *Hutchinson*, 1995] have been used to model daily precipitation amount. The aforementioned families perform reasonably well in terms of reproducing averaging characteristics of precipitation. Nevertheless, none of them necessarily performs well in terms of simulating extremes [*Wilks*, 1999; *Furrer and Katz*, 2008]. Besides parametric approaches, nonparametric approaches also have been used for daily precipitation simulation. Synthetic precipitation is sequentially sampled from historical observations with replacement. Several limitations, especially with respect to extremes, inherent to the sampling scheme [*Furrer* and *Katz*, 2008], have been recognized, and corrected via nonparametric kernel density estimator (KDE) [*Lall and Sharma*, 1996; *Rajagopalan and Lall*, 1999]. Nevertheless, the likelihood for extremes to be generated is low [*Markovich*, 2007], leading to underestimated extreme rainfall. Reproducing the entire range of precipitation in synthetic series has been identified as a critical research need in both simulation and downscaling, and has inspired a recent flurry of research, like *Vrac and Naveau* [2007], *Furrer and Katz* [2008], and *Hundecha et al.* [2009], in which compound distributions are used for modeling precipitation amount. Problems involved in fitting these distributions include numerical instability, data sensitivity, supervised learning, and computational demand.

[5] The objective of this study therefore is to develop an efficient, reliable and relatively easy-to-implement method for simulating both the low to moderate and extreme rainfall. To that end, the specific objectives are to: (1) examine if the existing distributions are reliable to model precipitation amount in different climate divisions, (2) present a hybrid distribution to model the full spectrum of daily precipitation, (3) validate the hybrid distribution model, and (4) describe approaches for estimating parameters of the hybrid distribution. It is noted that the proposed distribution is not intended, however, to replace existing distributions, like those developed by *Vrac and Naveau* [2007] and *Furrer and Katz* [2008], but rather to complement them and to provide a more efficient, reliable, and less complicated approach to model the heavy tail distribution of precipitation amount without losing any goodness-of-fit.

[6] The paper is organized as follows. Formulating the objectives of the study in section 1, a short discussion of data to be used is given in section 2. Fundamental to the simulation of daily rainfall is the choice of a probability distribution which is described in section 3. A hybrid probability distribution is presented in section 4. Its evaluation is presented in section 5. Based on problems raised from real cases, three estimation approaches are presented in section 6. The paper is concluded in section 7.