## 1. Introduction

[2] There are several factors that could affect the mean and peak of flow in a basin. Changes in precipitation and temperature as a result of climate change can directly influence the streamflow trend [*Arnell et al*., 2001; *Moradkhani et al*., 2010; *Gao et al*., 2011; *Kundzewicz et al*., 2013]. Due to the long-term increase in climate temperature, the snowmelt, which is dependent on seasonal temperature, may shift back from spring to winter and that may affect the peak flows [*Jung et al*., 2012]. Although analyzing the contributing factors in streamflow generation provides a general view of its behavior, it does not suffice as an accurate image of the characteristics of a flow regime in a basin. Studying the historical records of streamflow is a meaningful way to detect long-term trends in the face of natural variations as a result of climate and land use change [*Fu et al*., 2010; *Karl et al*., 2009; *Piao et al*., 2011a].

[3] The U.S. Geological Survey (USGS) maintains streamflow records at gage sites across the United States as a valuable resource for flood estimation and water resource management. However, gage stations are not uniformly distributed across the river basins with fewer stations in mountainous regions [*Durrans and Tomic*, 1996]. Also, the recently installed gages provide short records of data that makes it almost impossible to perform a robust flood assessment.

[4] The analysis of runoff extremes would be possible through the extreme value theory [*Beirlant et al.*, 2004; *Coles*, 2001] comprising of various extreme value distributions. Commonly, the block maximum [*Huerta and Sansó*, 1997; *Kharin and Zwiers*, 2012; *Sang and Gelfand*, 2010] and the extreme over a specific threshold [*Durman et al*., 2001] are adopted in hydrologic applications. Several studies have performed at-site analysis of extreme events [*Frei et al*., 2006; *Guttorp and Xu*, 2007; *Halmstad et al*., 2011; *Kharin et al*., 2009; *Towler et al*., 2009; *Villarini et al*., 2010]. However, one may expect similar distributions for extreme runoff records at gages that are close to each other. Furthermore, limited records of hydroclimatic extremes in space and time [*Fuentes et al*., 2013] require the collection of information from different locations in order to reduce the uncertainty of the simulations and provide more reliable results. Methods have been developed for regional frequency analysis (RFA) that are shown to be superior to the at-site flood estimations [*Burn*, 1990; *Chokmani and Ouarda*, 2004; *Dalrymple and Survey*, 1960; *GREHY*, 2011; *Gupta et al*., 1996; *Ouarda and El-Adlouni*, 2011; *Stedinger and Tasker*, 2011]. Index flood method [*Dalrymple and Survey*, 1960] is an approach to combine extreme data from different locations in order to improve the accuracy of the estimates and to predict flood at ungagged sites. This method was further improved by *Hosking and Wallis* [1999] who used the L-moments approach to estimate the parameters of the extreme value distribution. They divided the method into three steps: outline a homogeneous region, divide the extreme data at each gage by the index flood, and then fit a distribution to the combined data from all gages. RFA, however, does not consider the spatial components of the point data (i.e., geographic coordinate, elevation) and cannot incorporate additional variables (i.e., covariates) into the analysis. Besides, it is not possible to explicitly estimate the uncertainties based on the L-moments approach.

[5] Recently, with the accessible records of spatially scattered or gridded data and high-performance computing machines, there is growing interest in the analysis of spatially distributed extremes. Applications are found in the studies of wind [*Fawcett and Walshaw*, 2006], precipitation, and temperature [*Aryal et al*., 2009; *Cooley et al*., 2007; *Sang and Gelfand*, 2010; *Schliep et al*., 1975] among others. For this purpose, it is possible to consider a univariate extreme value distribution at each point (or grid) generating a spatial model on its parameters [*Cooley and Sain*, 2010; *Cooley et al*., 2007; *Lima and Lall*, 2008; *Renard*, 2011; *Sang and Gelfand*, 2004; *Schliep et al*., 1975]. The spatial dependence can also be modeled using the theory of max-stable process [*Coles*, 1993; *De Haan and Pereira*, 2006; *Padoan et al*., 2010]. In addition, Bayesian approach is a formal way to quantify the uncertainties [*Majda and Gershgorin*, 1999; *Najafi et al*., 2010; *Tebaldi et al*., 1957; *Moradkhani et al.*, 2012] and is flexible in combining different sources of uncertainties [*Tebaldi et al*., 1957]. By considering the parameters of the extreme distributions as random variables, one would utilize the Bayesian method to find their corresponding distributions.

[6] Attempts to develop models for characterizing the spatial dependencies in hydroclimate data and the underlying processes root back to 1950s [*Cressie*, 1992; *Robinson and Bryson*, 2010]. *Besag* [1974] introduced the concept of Markov random fields that consist of random variables with Markov properties. Hierarchical spatial models became popular after the introduction of Markov Chain Monte Carlo methods [*Berliner*, 1996; *Wikle et al*., 2013]. *Arab et al*. [2007] and *Banerjee et al*. [2004] provide detailed explanation of the origins of the hierarchical spatial models. *Casson and Coles* [1999] performed one of the earliest studies on hierarchical spatial modeling over the hurricane wind speed. Recently, there has been growing interest in the development and application of hierarchical spatial models for climate variables [*Fawcett and Walshaw*, 2006; *Sang and Gelfand*, 2004, 2010; *Schliep et al*., 1975]. *Cooley et al*. [2007] analyzed the precipitation extremes over Colorado using 56 gage records. *Cooley and Sain* [2010] studied the changes in precipitation extremes using regional climate model data for historical and future periods. *Sang and Gelfand* [2004] and *Schliep et al*. [1975] utilized a spatial autoregressive model for the annual maximum rainfall. *Renard* [2011] developed a procedure to account for spatial dependency using copulas in the analysis of rainfall extremes.

[7] Although much attention has been directed toward climate variable extremes such as precipitation and temperature and their spatial variations, fewer attempts have been made to provide more reliable models for hydrologic extremes such as streamflow. *Lima and Lall* [2008] employed a hierarchical Bayesian model based on a block maxima distribution and incorporated the drainage area as an additional variable to illustrate the parameters of the distributions at each gage station. However, no spatial analysis was performed for the hierarchical model. Furthermore, this approach disregards other extreme events that are lower than the block maximum. Considering the fact that extreme events are rare, the peaks over threshold approach would provide additional data for the analysis [*Coles*, 2001].

[8] In this study, a procedure is developed to model the runoff extremes recorded at USGS gage stations given their spatial variations (e.g., latitudes and longitudes), drainage areas, and elevations. This is done by spatially modeling the parameters of an extreme distribution through a hierarchical Bayesian process with latent parameters considered as random variables and simulated using Markov Chain Monte Carlo techniques. Estimates of the return levels for gages not being used in model fitting process are compared with point fit model results in order to validate the procedure. Furthermore, the trend in runoff extreme is assessed using the estimated parameters for time windows of 15 years, starting from 1906 until 2011.

[9] Section 2 will start by the extreme value theory and the generalized Pareto distribution that builds the basis for the hierarchical Bayesian model. The spatial hierarchical Bayesian modeling is then illustrated along with the Markov Chain Monte Carlo (MCMC) parameter estimation. In section 3, the case study is presented along with the model test and trend analysis of the runoff extremes followed by concluding remarks, a brief summary, and future research in section 4.