## 1. Introduction

[2] In hydrological modeling, prior knowledge about the physical system is often represented by defining prior ranges of feasible parameter values [*McIntyre et al*., 2005; *Bárdossy*, 2007; *Kollat et al*., 2012; *Samuel et al*., 2012]. Estimation often proceeds using Bayes' equation (while there are less formal estimation methods, this note focuses primarily on a formal use of Bayes'). For gauged catchments, measured hydrological states and fluxes, usually flows, together with assumptions about error properties are used to calculate the likelihoods. For ungauged catchments, regionalized data can instead be used to feed the likelihood function. Signatures of catchment responses, which reflect hydrological response characteristics of a particular catchment (e.g., mean annual discharge, base flow index and runoff ratio), are commonly used forms of regionalized data. Estimates of such signatures are calculated for numerous gauged catchments and regressed against a set of catchment descriptors; thus, the signature distributions for an ungauged location, for which the same descriptors are available, can be estimated and used to calculate the likelihood [*Bulygina et al*., 2009, 2012]. Developing the use of regionalized response signatures for hydrological model estimation is recognized as an important scientific challenge because the vast majority of catchments are ungauged, and the method also has potential for prediction under environmental change [*Buytaert and Beven*, 2009; *Wagener and Montanari*, 2011].

[3] When using a Bayesian procedure, assumptions about the prior distribution can have a considerable influence on the estimation of model parameters, especially in cases where data to calculate the likelihood are few and/or have high variance. It is therefore important not to introduce unintended or unjustified information into the prior, and in this respect it is common to exercise caution and aim for a noninformative prior. Introducing unjustified information can result in a disinformative prior that can counteract the information gained from the regionalization. *Beven and Westerberg* [2011] highlight that disinformation (in the context of disinformative data that ideally should be isolated and rejected prior to model calibration) should be avoided by all means. In a Bayesian context, a parametric prior is often expressed as a uniform distribution of parameters [*Winsemius et al*., 2009; *Bulygina et al*., 2011], where all parameter values fall within plausible bounds and different realizations of model parameter sets are considered equally likely a priori. However, as will be shown in this technical note, there are classes of problems where the prior is more usefully defined in terms of a uniform distribution of system behavior and where uniform distributions of model parameters should therefore not be used as the universal definition of prior lack of knowledge [*Carlin and Louis*, 2009].

[4] The problem of the prior distribution choice when little or no prior information is available has received much attention in the Bayesian statistical literature [*Box and Tiao*, 1992; *Robert*, 2007]. In many applications, uniform prior on parameters has been applied in an attempt to reflect equiprobability. However, a uniform prior does not always reflect prior ignorance, as it is not invariant under reparameterization. In the Bayesian literature, several methods have been suggested to formulate invariant noninformative priors, such as the Jeffreys prior [*Jeffreys*, 1946, 1961] and the reference priors [*Bernardo*, 1979; *Berger and Bernardo*, 1989] (for an overview of methods for constructing noninformative priors, see *Kass and Wasserman* [1996]). However, these methods may require determination of complex derivatives, and therefore are often not straightforward to apply in a hydrological modeling context, where we are dealing with multidimensional and not necessarily continuous problems. Alternatively, *Box and Tiao* [1992] introduced what they call data-translated likelihood, which gives a very intuitive idea of what makes a uniform prior noninformative. For the problem tackled in this technical note, model parameters are conditioned on regionalized data and likelihood is expressed in terms of modeled response signatures. We argue here that we should express our initial lack of knowledge in the same space that we are getting information on, i.e., the response signature space, thus avoiding introducing unjustified information from other sources (e.g., the subjective choice of model structure or prior parameter distribution). Due to the difficulties associated with approaches such as Jeffreys and reference priors in this particular context, we suggest here to use a uniform prior on response signature space to avoid disinformation. This contrasts with the usual practice in hydrological modeling to assume a uniform prior on parameters, especially when using conceptual-type models where the parameters have limited physical interpretation and thus limited prior knowledge. It is generally not possible to sample directly from a uniform signature distribution. In this technical note, we therefore propose a method for transforming an initial uniform prior on parameters to a distribution that maps to a uniform prior in terms of response signatures. A set of 84 catchments in the eastern USA, taken from the Model Parameter Estimation Experiment (MOPEX) database [*Duan et al*., 2006], for which a variety of regional response signature models and likelihood functions have previously been determined [*Almeida et al*., 2012], are used to test the method. The potential value of the method for a wider range of hydrological applications is subsequently discussed.