## 1. Introduction

[2] Conceptual rainfall runoff models are widely used in various decision-making problems ranging from online flood forecasting to evaluation of flood-reducing strategies and hydraulic structure designs. Many parameters of such models cannot, in general, be obtained directly from measurable quantities, and hence model calibration is needed. A model is calibrated to determine values of model parameters so that the model simulates the hydrological behavior of a catchment as closely to observations as possible. Hence, model calibration is an essential step before a model can be confidently used. The confidence that can be ascribed to model simulations largely depends on how a model has been calibrated.

[3] The assessment of parameter and predictive uncertainty of hydrological models is an essential part of any hydrologic study [*Schoups and Vrugt*, 2010], and uncertainty analysis in hydrological models has attracted much attention. Uncertainty analysis in hydrological models is commonly carried out based on calibration. Model parameter uncertainties are generally estimated based on residual errors. The residual errors are summarized using a likelihood function to quantify the probability density of observed data being generated by a particular parameter set. The mapping from parameter space to likelihood space results in a range of plausible parameter sets and allows estimation of parameter and predictive uncertainty. The likelihood function can be specified through either a formal or an informal approach. An informal likelihood function is specified flexibly without strictly defined statistical assumptions on residual errors and the function is formed subjectively. A well-known example is the generalized likelihood uncertainty estimation (GLUE) methodology [*Beven and Binley*, 1992]. With a formal approach, the residual errors are assumed to obey certain statistical models using Bayesian statistics. The simpler model assumes that the errors are independent and identically distributed normal random variables with a mean of zero and a constant variance. Discussions and comparisons of informal and formal approaches are given by many researchers [e.g., *Makowski et al*., 2002; *Montovan and Todini*, 2006; *McMillan and Clark*, 2009; *Jin et al*., 2010; *Dotto et al*., 2012]. Among many possible methods for searching for possible model parameters, the Markov Chain Monte Carlo (MCMC) method has become increasingly popular, being applied in calibration for problems involving complex inference, search, and optimization [*Vrugt et al*., 2003]. The MCMC method is mostly coupled with Bayesian statistics using a formal likelihood function [e.g., *Schoups and Vrugt*, 2010], but it can also be applied using informal likelihood functions [e.g., *Blasone et al*., 2008].

[4] The above approaches calibrate models based on residual errors that typically consist of a combination of input, model structure, output, and parameter errors. Traditionally, uncertainty in parameters is mainly attributed to output errors in calibration, and the errors in inputs are not explicitly addressed. It is not realistic for real-world applications because measured forcing data (rainfall measurements in hydrological process in this case) often contain significant uncertainty [*Kavetski et al*. 2006a, 2006b; *Stransky et al*., 2007; *Vrugt et al*., 2009], and hence it is desirable to incorporate and treat different uncertainty sources appropriately. *Kavetski et al*. [2006a, 2006b] and *Vrugt et al*. [2008] made the step forward to incorporate input uncertainty in hydrological model calibration by introducing multipliers to rainfall events. *Kavetski et al*. [2006a, 2006b] developed the Bayesian total error analysis (BATEA) framework to explicitly represent each source of uncertainty in hydrological models. BATEA relies on a hierarchical Bayesian model to handle uncertainty in terms (e.g., rainfall) by using latent variables that will be identified in the calibration as other model parameters. In addition, the framework also allows modelers to introduce as much detailed information about the data accuracy and uncertainty description as is known by simply incorporating parameters in error models into the hierarchical Bayesian model. In comparison to traditional approaches that lump all uncertainties into a single error term, BATEA provides improvements on model calibration in terms of deriving consistent parameter estimation and reliable quantification of predictive uncertainty [e.g., *Kuczera et al*., 2006, 2010; *Thyer et al*., 2009; *Renard et al*., 2010, 2011; *Salamon and Feyen*, 2010; *Li et al*., 2012]. Explicit input uncertainty analysis under the Bayesian context using the BATEA framework has been applied to different extents [e.g., *Huard and Mailhot*, 2006, 2008; *Reichert and Mieleitner*, 2009; *Sikorska et al*., 2012]. *Vrugt et al*. [2008] also evaluated rainfall uncertainty simultaneously with the hydrologic model parameters by minimizing the mismatch between observed and simulated catchment responses. *Vrugt et al*. [2008] did not directly include random effects of rainfall errors as a multiplicative factor in the likelihood term, which differs from the BATEA.

[5] This paper addresses uncertainty estimation in hydrological model calibration with a focus on handling rainfall uncertainty. Three approaches based on the Bayesian concept, considering input uncertainty in hydrological model calibration in various ways, are presented and compared. This paper focuses on the calibration of small urban catchments (<200 ha with high imperviousness of surfaces) with a small temporal scale (1 -min time step), while most literature studies model larger catchments at hourly or, more frequently, daily time steps. The paper investigates the ability of different approaches for providing parameter estimation and uncertainty quantification. The merits of explicit separation of input and output uncertainties are discussed. In addition, when explicitly incorporating rainfall errors in calibration, rainfall measurements at 1 min time step are grouped to share parameters to be calibrated to keep down the dimension of the calibration problem. The proper grouping strategies are studied. The remainder of the paper is organized as follows: section 2 presents the general hydrological calibration problem and the three approaches that account for input uncertainty in different ways; in section 3, application cases are introduced; section 4 then gives the results and discussion, focusing on comparison between different approaches and explanations of observed results; lastly conclusions are drawn.