## 1. Introduction

[2] Precipitation is one of the most inhomogeneous and fast evolving hydrometeorological processes in space and time. The multiscale variability observed in precipitation is due to the nesting of small, transient storm elements within larger long-lived elements. In order to increase the accuracy of atmospheric and hydrologic predictions, accurate precipitation estimates are required for model initialization, data assimilation, and also model verification. A variety of sensors, for example, rain gauges, radars, and satellites are used to obtain precipitation related measurements. Each measurement technique has some advantages and limitations. Rain gauges and radars, for example, provide relatively the most accurate precipitation measurements but with limited coverage. On the other hand, infrared sensors on geostationary satellites provide a broad and continuous coverage but with limited accuracy, and microwave sensors on polar orbiting satellites stand somewhere in between. In order to produce accurate precipitation estimates, an obvious solution is to merge these disparate sources of measurements and exploit the advantages that each measurement technique has to offer.

[3] Scale-recursive estimation (SRE) (see *Chou et al.* [1994a, 1994b] for original references) has recently been proposed as a methodology for merging multisensor, multiscale precipitation measurements in order to obtain estimates of precipitation and their error statistics at desired spatial scales, for the purpose of model verification [*Tustison et al.*, 2002] or data assimilation [*Kumar*, 1999; *Primus et al.*, 2001]. The SRE methodology, which has its roots in Kalman filtering, explicitly takes into account the disparate (in scale) measurement sources and their sensor-dependent uncertainty. This methodology requires a multiscale stochastic model to describe the scale-to-scale variability of spatial precipitation. Several such models have been explored in the past (e.g., see *Gupta and Waymire* [1993], *Lovejoy and Schertzer* [1991], *Kumar and Foufoula-Georgiou* [1993a, 1993b], and *Harris et al.* [1997], among others) but a class of models that naturally fits into the SRE framework (because they can be brought into the recursive additive form required by SRE) is that of multiplicative cascade models [e.g., *Gupta and Waymire*, 1993; *Over and Gupta*, 1994]. These models have been used for rainfall applications within the SRE framework by *Primus et al.* [2001] and *Tustison et al.* [2002].

[4] There are two (interrelated) problems that arise in using multiplicative cascade models for spatial rainfall within the SRE methodology. First, in order to bring these models into the additive state scale-recursive equation required by SRE, one has to work in the logarithmic space. Since spatial rainfall fields contain zero values (because of intermittency), a small threshold is usually used to replace the zeros with nonzero values during the SRE procedure. Sensitivity of the fitted model parameters and SRE estimates to the chosen threshold value was reported by *Tustison et al.* [2002] although this issue was not pursued further. Second, by construction, multiplicative cascade models produce fields, which are nonzero everywhere within the modeling domain. If an imposed small threshold value were to be used to define “zeros” (as values below the chosen threshold) the statistics of these zeros would be completely predetermined by the cascade model parameters and would follow a power law distribution (i.e., zero areas of all sizes would be expected to be present). This might be a restriction, if the statistics of the zero areas do not follow power law distributions. Thus both of the above issues pose limitations in considering multiplicative cascade models for rainfall within the SRE framework.

[5] Motivated by these limitations, this paper proposes an alternative approach to scale-recursive estimation based on a data-driven system identification methodology, which operates directly on the data (and not their logs) and does not require a prespecified multiscale model structure (in that sense, the proposed approach is referred to as “nonparametric”). This is accomplished by a likelihood-based expectation-maximization (EM) on scale-recursive dynamics on trees [e.g., *Kannan et al.*, 2000], which identifies and estimates the model recursively (and dynamically) from the available multiscale/multisensor observations with no fixed structure of the process dynamics. As such, it provides a valuable alternative in many practical situations. The merits of the proposed nonparametric EM-SRE approach compared to parametric approaches are documented on the basis of a suite of numerical experiments.

[6] In practical applications, merging of multiscale observations has to be performed continuously over time giving rise to the need to have the temporal structure of precipitation also taken into account. Extending the SRE methodology to dynamic (spatiotemporally varying) fields is not a simple task. The challenge of multiscale estimation of dynamic systems lies in the prediction step, which requires untangling the spatial mixing due to temporal dynamics. This step can be involved even in simple dynamics such as diffusion processes. Research on SRE of dynamic fields includes that of *Ho et al.* [1996]. The idea behind their approach is that the multiscale models for the updated and predicted estimation errors are propagated through time in the same way that Kalman filter propagates the error covariances, but in a more computationally efficient manner, i.e., without computing or storing the full error covariance matrix. They introduced a reduced-order spatially interpolated multiscale model and its efficiency was demonstrated in several applications. Before these methodologies are explored toward the problem of merging multiscale spatiotemporal precipitation observations, it is worth considering simpler methodologies, which can provide insight into the problem. Such a simple methodology which relies on combining the EM-SRE methodology with a downscaling (spatial or spatiotemporal) scheme to produce space-time merged precipitation products is explored in this paper via an example motivated by the sampling specifications of the Global Precipitation Measuring (GPM) mission.

[7] This paper is structured as follows. In the next section, a brief overview of the SRE framework is presented while leaving the mathematical details for the appendix. Section 3 focuses on lognormal and bounded lognormal multiplicative cascades and numerical experiments are carried out to determine their merits and limitations for precipitation representation within SRE. In section 4, the nonparametric EM-SRE methodology is presented. Section 5 demonstrates, through numerical experiments, some advantages of the nonparametric over the multiplicative cascade parametric models. Section 6 presents a case study that is of potential relevance to the GPM mission. Namely, the EM-SRE framework is combined with spatial downscaling to accommodate the merging of observations available at different scales and different times. Finally, conclusions and open problems for future research are presented in section 7.