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Abstract Satellite precipitation products, as all quantitative estimates, come with some inherent degree of uncertainty. To associate a quantitative value of the uncertainty to each individual estimate, error modeling is necessary. Most of the error models proposed so far compute the uncertainty as a function of precipitation intensity only, and only at one specific spatiotemporal scale. We propose a spectral error model that accounts for the neighboring space–time dynamics of precipitation into the uncertainty quantification. Systematic distortions of the precipitation signal and random errors are characterized distinctively in every frequency–wavenumber band in the Fourier domain, to accurately characterize error across scales. The systematic distortions are represented as a deterministic space–time linear filtering term. The random errors are represented as a nonstationary additive noise. The spectral error model is applied to the IMERG multisatellite precipitation product, and its parameters are estimated empirically through a system identification approach using the GV-MRMS gauge–radar measurements as reference (“truth”) over the eastern United States. The filtering term is found to be essentially low-pass (attenuating the fine-scale variability). While traditional error models attribute most of the error variance to random errors, it is found here that the systematic filtering term explains 48% of the error variance at the native resolution of IMERG. This fact confirms that, at high resolution, filtering effects in satellite precipitation products cannot be ignored, and that the error cannot be represented as a purely random additive or multiplicative term. An important consequence is that precipitation estimates derived from different sources shall not be expected to automatically have statistically independent errors. Significance Statement Satellite precipitation products are nowadays widely used for climate and environmental research, water management, risk analysis, and decision support at the local, regional, and global scales. For all these applications, knowledge about the accuracy of the products is critical for their usability. However, products are not systematically provided with a quantitative measure of the uncertainty associated with each individual estimate. Various parametric error models have been proposed for uncertainty quantification, mostly assuming that the uncertainty is only a function of the precipitation intensity at the pixel and time of interest. By projecting satellite precipitation fields and their retrieval errors into the Fourier frequency–wavenumber domain, we show that we can explicitly take into account the neighboring space–time multiscale dynamics of precipitation and compute a scale-dependent uncertainty. 

Abstract. This paper presents the results of the ensemble Riemannian data assimilation for relatively high-dimensional nonlinear dynamical systems, focusing on the chaotic Lorenz-96 model and a two-layer quasi-geostrophic (QG) model of atmospheric circulation. The analysis state in this approach is inferred from a joint distribution that optimally couples the background probability distribution and the likelihood function, enabling formal treatment of systematic biases without any Gaussian assumptions. Despite the risk of the curse of dimensionality in the computation of the coupling distribution, comparisons with the classic implementation of the particle filter and the stochastic ensemble Kalman filter demonstrate that, with the same ensemble size, the presented methodology could improve the predictability of dynamical systems. In particular, under systematic errors, the root mean squared error of the analysis state can be reduced by 20 % (30 %) in the Lorenz-96 (QG) model. 

Abstract As more global satellite-derived precipitation products become available, it is imperative to evaluate them more carefully for providing guidance as to how well precipitation space-time features are captured for use in hydrologic modeling, climate studies and other applications. Here we propose a space-time Fourier spectral analysis and define a suite of metrics which evaluate the spatial organization of storm systems, the propagation speed and direction of precipitation features, and the space-time scales at which a satellite product reproduces the variability of a reference “ground-truth” product (“effective resolution”). We demonstrate how the methodology relates to our physical intuition using the case study of a storm system with rich space-time structure. We then evaluate five high-resolution multi-satellite products (CMORPH, GSMaP, IMERG-early, IMERG-final and PERSIANN-CCS) over a period of two years over the southeastern US. All five satellite products show generally consistent space-time power spectral density when compared to a reference ground gauge-radar dataset (GV-MRMS), revealing agreement in terms of average morphology and dynamics of precipitation systems. However, a deficit of spectral power at wavelengths shorter than 200 km and periods shorter than 4 h reveals that all satellite products are excessively “smooth”. The products also show low levels of spectral coherence with the gauge-radar reference at these fine scales, revealing discrepancies in capturing the location and timing of precipitation features. From the space-time spectral coherence, the IMERG-final product shows superior ability in resolving the space-time dynamics of precipitation down to 200 km and 4 h scales compared to the other products. 

Optimal transport began as the problem of how to efficiently redistribute goods between production and consumers and evolved into a far-reaching geometric variational framework for studying flows of distributions on metric spaces. This theory enables a class of stochastic control problems to regulate dynamical systems so as to limit uncertainty to within specified limits. Representative control examples include the landing of a spacecraft aimed probabilistically toward a target and the suppression of undesirable effects of thermal noise on resonators; in both of these examples, the goal is to regulate the flow of the distribution of the random state. A most unlikely link turned up between transport of probability distributions and a maximum entropy inference problem posed by Erwin Schrödinger, where the latter is seen as an entropy-regularized version of the former. These intertwined topics of optimal transport, stochastic control, and inference are the subject of this review, which aims to highlight connections, insights, and computational tools while touching on quadratic regulator theory and probabilistic flows in discrete spaces and networks. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates. 

Bedload particle hops are defined as successive motions of a particle from start to stop, characterizing one of the most fundamental processes of bedload sediment transport in rivers. Although two transport regimes have been recently identified for short and long hops, respectively, there is still the lack of a theory explaining the mean hop distance–travel time scaling for particles performing short hops, which dominate the transport and may cover over 80 % of the total hop events. In this paper, we propose a velocity-variation-based formulation, the governing equation of which is intrinsically identical to that of Taylor dispersion for solute transport within shear flows. The key parameter, namely the diffusion coefficient, can be determined by hop distances and travel times, which are easier to measure and more accurate than particle accelerations. For the first time, we obtain an analytical solution for the mean hop distance–travel time relation valid for the entire range of travel times, which agrees well with the measured data. Regarding travel times, we identify three distinct regimes in terms of different scaling exponents: respectively, $$\sim$$ 1.5 for the initial regime and $$\sim$$ 5/3 for the transition regime, which define the short hops, and 1 for the Taylor dispersion regime defining long hops. The corresponding distribution of the hop distance is analytically obtained and experimentally verified. We also show that the conventionally used exponential distribution, as proposed by Einstein, is solely for long hops. Further validation of the present formulation is provided by comparing the simulated accelerations with measurements.