## 1. Introduction

### 1.1. A Critique of Parametric Rainfall-Runoff Modeling

[2] Catchments are nonlinear dynamic hydrological systems. A natural and long-established framework for modeling their rainfall-runoff relationships is the input-state-output parametric approach, i.e., the internal fluxes and the output are modeled through parametric relationships involving a set of state variables. The recourse to input and output data for parameter estimation appears as unavoidable for presently available models [*Beven*, 2000, 2002]. The data and model errors play a central role in calibration, as, explicitly or implicitly, inference is based on the nature of the distribution of modeling errors. There are good reasons to believe that hydrological modeling errors are both heteroscedastic and serially correlated [*Sorooshian and Dracup*, 1980]. Also, the presence of outliers (large errors in data) is likely. Despite this, the most widely used performance measures, such as the Nash-Sutcliffe criterion, implicitly assume a Gaussian independent and identically distributed (i.i.d.) additive error model.

[3] A critical issue, often raised in the hydrological modeling literature, is that of over-parameterization or over-fit [e.g., *Jakeman and Hornberger*, 1993; *Hornberger et al.*, 1985; *Young*, 2001]. This essentially means that the flexibility or the capacity of the model structure is too high with respect to the information content of the available data. A related issue is that of interference. When global parametric relationships are used to model the mappings from the space of predictor variables to that of response variables, introducing the information content for a new evaluation data point may cause a change in the estimated model parameters; that is, it has a global effect [e.g., *Schaal*, 1994]. Estimating parameters using data from one region of the data space thus has a potential effect on how well the model will fit the observations in other regions. This leads to potential trade-offs between over-fitting parts of the mapping, especially those that contribute most to the performance measure, and under-fitting (introducing bias) in other parts. Interference has received less attention in the hydrological literature, although recent research [e.g., *Gupta et al.*, 1998; *Wagener et al.*, 2003] has questioned the use of a “global” performance measure on the grounds that it entails a loss of information through the aggregation of model residuals, i.e., evaluating the model globally on all the regions of the input space. Generally, global parametric methods are successful when the investigated structures of the model and the errors are sufficiently close to the “true” ones, but there is no guarantee that this will be the case in applications of rainfall-runoff models to real data sets (see discussion by *Beven and Young* [2003]).

[4] Accumulated experience in the use of parametric models suggests that only a model structure with less than 10 parameters can be supported by rainfall-runoff data identified using a single objective function for the prediction of runoff, with a suggestion that most of the information is present in 1–3 years of data (“wet,” “dry,” “average”) [e.g., *Kirkby*, 1975; *Jakeman and Hornberger*, 1993; *Young*, 2001], although further research [e.g., *Yapo et al.*, 1996] has suggested that longer records are needed for conceptual rainfall-runoff models having larger parameterizations. The use of simpler model structures, which attempt to retrieve the “dominant modes” of behavior at the catchment scale, contrasts sharply with the perceived complexity of the physical characteristics and response of even “homogeneous” small basins or hill slopes when viewed from a bottom-up or process-based modeling viewpoint. However, it has been proven very difficult to take advantage of the prior information on processes and physical characteristics that should be expected from a process-based approach, albeit for good reasons [*Beven*, 2000, 2002].

[5] It has also proven difficult to take advantage of good quality and extensive data sets from experimental catchments in refining model description. The availability of such data sets has not resulted, as might have been expected, in better predictive ability, in particular in allowing increased complexity in modeling with respect to shorter/poorer quality data sets. This is certainly due to continuing deficiencies with the current generation of model structures.

[6] The question is still open whether (1) the hydrological response at the catchment scale is intrinsically “simple” or (2) there is a fundamental limitation in the available data (quantity and quality) that allows only for the specification of simple (“dominant mode”) models, or (3) a simple parametric approach has significant limitations in consistently using the available information across the full range of catchment responses due to different event characteristics and spatial patterns of antecedent conditions. The answer to this question has far-reaching theoretical and practical implications. In our view, the hypothesis that our actual knowledge of system characteristics, especially model structure and errors, is too weak in order to fully exploit the available information in the data within this paradigm deserves further consideration.

### 1.2. An Alternative Nonparametric Input-Output Approach

[7] In this contribution we explore a new data-driven approach to modeling the rainfall-runoff relationship that differs radically from that outlined above by two characteristics. First, we hypothesize (reasonably) that the response of a catchment should be determined by its trajectory in the space of historical input forcing variables. As a first approximation, we use as predictor variables only linear combinations of inputs at previous time steps. That is, we do not use state variables and thus are able to circumvent the issues that can arise from their recurrent calculation in a nonlinear model [*Sjöberg et al.*, 1995; *Kavetski et al.*, 2003]. Second, we use as the identification algorithm regression trees [*Breiman et al.*, 1984], a nonparametric identification method. Local and nonparametric methods avoid the need to find a potentially complicated parametric function capable of representing all the data by dividing the input space into many fixed or adaptive partitions and by modeling them locally with much simpler functions, based directly on the data [*Schaal*, 1994]. The partitions will reflect the observed responses under different sets of hydrologically similar conditions. These methods also allow the quality of a model to be evaluated locally, which is a significant advantage when the global error structure is poorly known and potentially far from the Gaussian i.i.d. assumption. The regression tree needs to be trained on the observed data, like a neural network, but avoids the introduction of weighting coefficients or other parameter values that require explicit calibration. However, these advantages come at a price, in particular the need for large data sets in order to identify complex relationships, especially in high-dimensional input spaces.

[8] Local and nonparametric inductive methods seem to have gotten little attention in the hydrological literature. Nevertheless, they have the potential to better exploit the long, good quality data sets that are available. In particular they could give better insight in the measurement error structure in that they avoid aggregating all residuals and reduce the potential for a model structural component to the prediction error due to an a priori choice of conceptual model components. There is no requirement to specify a priori definitions of state variables and parametric relationships in making predictions and therefore model structure to be wrong. However, there will be new sets of conditions, not well represented in the calibration data set, that will require extrapolation within the space defined by the identified regression tree.