## 1. Introduction

[2] Deterministic models have been widely used in hydrology for both forecasting and simulation purpose. The assessment of their uncertainty is a major research issue. For managers and decision makers, quantification of uncertainty associated to model estimates is a valuable information for both alarm and operational purposes. From the modeler point of view, it provides indications for model diagnosis and improvement [*Gupta et al.*, 2008; *Reichert and Mieleitner*, 2009], and targeted data collection. Theoretically, since the forecast value by a deterministic model will never be exact, associating it with some kind of characterization of its error is the only way to assess its quality [*Weijs et al.*, 2010]. The lower the error the better the model: this is what we implicitly do in practice every time we use a deterministic model, since we regard the model prediction as the expected, or most probable value of the estimated variable. Quantification of model uncertainty makes this assumption explicit and provides the model user with a more formal and accurate evaluation of the residual error.

[3] Sources of uncertainty in deterministic models are often classified in measurement errors, both in the input and output, uncertainty in the parameters and uncertainty in the model structure, including selection of the input and the mathematical relation between input, state, and output variables. Many approaches to uncertainty assessment rely on such decomposition. One or more sources of uncertainty are given a statistical description and uncertainty is propagated in the model via random sampling and simulation, to obtain a sample or distribution of model predictions in place of a single value [see, e.g., *Thyer et al.*, 2009; *Kuczera and Parent*, 1998; *Kavetski et al.*, 2006]. The application of these methods may require collecting several information for the statistical characterization of the different uncertainty sources (e.g., the accuracy of the measurement devices, the analysis of the error induced by data preprocessing), and can be limited by the computational cost of model simulation.

[4] On the other hand, “model residual” approaches skip any distinction of uncertainty sources and directly analyze the time series of model residuals to build a model of the global predictive uncertainty, which is often sufficient for practical purposes. The drawback of model residual approaches is that, while they do not require assumptions about the different sources of uncertainty, they usually do for the characterization of the model residual. Historically, the most common approach is to assume that the residual be an independent identically distributed process, usually zero mean and Gaussian. The approach has been widely criticized because most of these assumptions are violated in hydrological applications, especially autocorrelation and homoscedasticity of the model residuals [*Sorooshian and Dracup*, 1980]. Many methods have been proposed either to manipulate model residuals so that they satisfy such assumptions (e.g., using Box-Cox transformations [*Box and Tiao*, 1973; see *Kuczera and Parent*, 1998; *Bates and Campbell*, 2001] or the normal quantile transform [e.g., *Montanari and Brath*, 2004]) or to relax some of these assumptions [e.g., *Romanowicz et al.*, 2006; *Schaefli et al.*, 2007; *Schoups and Vrugt*, 2010].

[5] In this paper we will present a novel “model residual” approach for estimating the predictive uncertainty of deterministic hydrological models. In our approach, we will assume that the residual error of the model be uncorrelated in time or that it can be described by an autoregressive model with uncorrelated residual. The assumption is very useful because the statistical description of the residual process reduces to providing a sequence of marginal probability distribution functions (pdfs). We use the same distribution type at all time steps while allowing for the residual variance to change in time and reproduce the heteroscedastic behavior that is often observed in hydrological time series. For several practical reasons that will be clarified throughout the paper, we will use Gaussian distributions, although the approach can be extended to other distributions if Gaussian proved to be unsatisfactory, provided that they are symmetric. Under these hypotheses, the identification of the residual error pdf is reduced to the estimation of the error variance (or standard deviation). The latter can be a function of time (the season) or other hydrometeorological inputs, depending on the case study under exam. The novelty of our approach is that we will not assume a priori the input variables of the variance model nor the type of relation between these inputs and the variance, but rather we will infer such information from data analysis and consideration of the features of the hydrological system under exam. This is possible if one regards the variance model identification as a regression analysis over the time series of the model residual errors. Further, we will show that under the Gaussian assumption it is possible (and numerically more efficient) to identify a model of the error standard deviation from the time series of absolute errors, rather than a model of the variance from the time series of squared errors. Another contribution of our paper is that we will show the effectiveness of introducing past absolute errors among the input of the standard deviation model, which means that the error standard deviation is modeled as a dynamic process. For all these reasons, we named the proposed approach dynamic uncertainty model by regression on absolute error (DUMBRAE).

[6] The paper is organized as follows. In section 2, the DUMBRAE approach is fully described from the methodological standpoint. Then the issue of how to evaluate the quality of an uncertainty model is discussed, from visual inspection of the inferred confidence bounds to formal methods. Relying on these results, we can demonstrate the effectiveness of the proposed approach through the application to two case studies.