## 1. Introduction

[2] Studies on verification of Quantitative Precipitation Forecasts (QPF) are commonplace in the literature and serve several purposes. Among these are: (1) comparison of the performance of different models, (2) assessment of model improvements when new parameterizations are introduced, (3) assessment of model performance when resolution of the model is changed, and (4) estimation of errors of re-analysis products when model outputs are merged with observations to produce gridded fields for model initialization and other uses. Typically, the observations available for QPF verification are at scales different than the scale (grid size) of the model and comparison is not straightforward since the variability of precipitation fields strongly depends on the scale at which the fields are considered. For example, for a typical summer convective storm, the standard deviation of instantaneous precipitation more than halves if one goes from 2 km to 10 km pixel scale [e.g., see *Tustison et al.*, 2001, Figure 1]. How this variability changes with scale is a function of the inherent characteristics of the storm (storm type) and the temporal integration scale (e.g., 5-min versus 1-hour accumulations etc.).

[3] In a recent study, *Tustison et al.* [2001] demonstrated the importance of accounting for the multiscale variability of precipitation when observations at one scale are compared with model output at another scale. They showed that using typical methods of QPF verification to change the scale of observations to the scale of model output (e.g., point-to-area conversion) or vice versa (area-to-point conversion) imposes a “representativeness error” which is non-zero even in the case of a “perfect” model. (“Perfect” model outputs were created in the numerical experiment by simply averaging the underlying field at several scales, and “perfect” observations were created by randomly subsampling the underlying field at various sampling densities.) As it was shown by *Tustison et al.* [2001], the representativeness error had significant magnitude (up to 50% of the spatial average of the precipitation field) and considerable scale dependency within the typical mesoscale ranges of 5–50 km. Also, the magnitude of the error was found to depend on the variability of the underlying field: the smoother the field (indicated by larger values of spectral slope), the smaller the representativeness error. It is stressed that the representativeness error results from the treatment of scale effects in the comparison method and is non-zero even in the case of a “perfect” forecast.

[4] We believe that a rigorous methodology which can explicitly account for the multiscale variability of precipitation is needed such that observations and model outputs at different scales can be compared or optimally merged while explicitly accounting for their scale-dependent variability and uncertainty. Such a methodology, based on scale-recursive estimation (SRE), is explored in this paper for the purpose of QPF verification from multisensor observations, typically available at different scales and with different uncertainties. This problem is illustrated in Figure 1.

[5] The SRE technique was introduced in the signal processing literature [*Chou*, 1991; *Chou et al.*, 1994a, 1994b] as a technique which can produce the best estimate (in terms of minimum variance of the estimation error) of the field at any desired scale, and also the uncertainty of the estimate, given sparse observations of the process and their uncertainties at different scales. SRE has its philosophical roots in the optimal recursive estimation technique known as Kalman filtering [e.g., see *Anderson and Moore*, 1979; *Bras and Rodriguez-Iturbe*, 1993] but instead of applying the estimation and recursion in time, it applies it in scale. The reader is referred to the original publications of *Chou* [1991] and *Chou et al.* [1994a, 1994b] for mathematical details and for demonstration of the success and computational efficiency of this technique. Since its introduction, SRE has found application in soil moisture estimation [*Kumar*, 1999], precipitation data assimilation [*Primus*, 1996], estimation of solute travel time distributions [*Daniel et al.*, 2000], imaging and remote sensing problems [*Fieguth*, 1995], assimilation of remote sensing data [*Daniel and Willsky*, 1997], and estimation of satellite altimetry [*Fieguth et al.*, 1995].

[6] In this work, SRE is explored for the problem of QPF verification using precipitation observations from multiple sensors. Special emphasis is placed on the selection of the model describing the multiscale variability of precipitation and on its parameter estimation from sparse or noisy data. Since the accuracy of SRE greatly depends on the above two factors, a simulation study is presented which quantifies the sensitivity of the SRE estimates on misspecification of model structure and model parameters. The results point out to the great potential of SRE for multisensor QPF verification, but also to the importance of performing controlled background sensitivity studies before this technique can be used with confidence in an operational setting.

[7] This paper is structured as follows. In the next section, a brief background on the SRE framework is presented while leaving some of the details for an appendix. Since SRE requires the specification of a model describing the multiscale structure of precipitation, section 3 provides information on a popular class of precipitation models (multiplicative cascades) and how these can be adapted for incorporation in the SRE framework. Section 4 presents the results of fitting two types of lognormal cascades to precipitation fields available from radars at 2 km resolution. In section 5, the sensitivity of SRE to multiscale model selection and uncertainty in the fitted parameters is investigated. In section 6, the effect of available observations (sparsity, measurement error, scales available) on QPF verification via SRE is studied and trade-offs between dense observations at a fine scale versus sparse observations at that scale and observations at another larger scale are quantified. Finally, conclusions and open problems are discussed in section 7.