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We propose an approximation to the forward filter backward sampler (FFBS) algorithm for large‐scale spatio‐temporal smoothing. FFBS is commonly used in Bayesian statistics when working with linear Gaussian state‐space models, but it requires inverting covariance matrices which have the size of the latent state vector. The computational burden associated with this operation effectively prohibits its applications in high‐dimensional settings. We propose a scalable spatio‐temporal FFBS approach based on the hierarchical Vecchia approximation of Gaussian processes, which has been previously successfully used in spatial statistics. On simulated and real data, our approach outperformed a low‐rank FFBS approximation. [ABSTRACT FROM AUTHOR]

Point forecasts can be interpreted as functionals (i.e., point summaries) of predictive distributions. We extend methodology for the identification of the functional based on time series of point forecasts and associated realizations. Focusing on state‐dependent quantiles and expectiles, we provide a generalized method of moments estimator for the functional, along with tests of optimality under general joint hypotheses of functional relationships and information bases. Our tests are more flexible, and in simulations better calibrated and more powerful than existing solutions. In empirical examples, economic growth forecasts and model output for precipitation are indicative of overstatement in anticipation of extreme events. [ABSTRACT FROM AUTHOR]

Surface soil moisture (SSM) has been identified as a key climate variable governing hydrologic and atmospheric processes across multiple spatial scales at local, regional, and global levels. The global burgeoning of SSM datasets in the past decade holds a significant potential in improving our understanding of multiscale SSM dynamics. The primary issues that hinder the fusion of SSM data from disparate instruments are (1) different spatial resolutions of the data instruments, (2) inherent spatial variability in SSM caused due to atmospheric and land surface controls, and (3) measurement errors caused due to imperfect retrievals of instruments. We present a data fusion scheme which takes all the above three factors into account using a Bayesian spatial hierarchical model (SHM), combining a geostatistical approach with a hierarchical model. The applicability of the fusion scheme is demonstrated by fusing point, airborne, and satellite data for a watershed exhibiting high spatial variability in Manitoba, Canada. We demonstrate that the proposed data fusion scheme is adept at assimilating and predicting SSM distribution across all three scales while accounting for potential measurement errors caused due to imperfect retrievals. Further validation of the algorithm is required in different hydroclimates and surface heterogeneity as well as for other data platforms for wider applicability. Plain Language Summary: Surface soil moisture (SSM) is an essential climate‐variable governing land‐atmosphere interactions. SSM is spatially variable in the presence of changing atmospheric factors such as rainfall and land‐surface characteristics such as soil, vegetation, and topography. SSM is measured using various instruments from point to satellite resolutions (25–40 km) and each instrument is accompanied by its own set of errors. Due to the importance of SSM, it would be beneficial to combine the SSM measurements from all available instruments in a region while accounting for the spatially varying nature of SSM and the measurement errors caused due to instruments. We present a novel framework to achieve the abovementioned objective and successfully apply it to a watershed in Manitoba, Canada to combine data from point, airborne, and satellite instruments. We demonstrate that the proposed framework can be used to optimally combine and predict SSM across different spatial resolutions in the presence of uncertainty. Key Points: Proposed a multi‐scale data fusion framework accounting for spatial variance/correlation of soil moistureThe proposed framework optimally separates the inherent soil moisture dynamics and measurement errors in instrumentsThe framework is applied to combine point, airborne and satellite data in a heterogeneous watershed [ABSTRACT FROM AUTHOR]

Soil moisture is spatially variable due to complex interactions between geologic, topographic, vegetation, and atmospheric variables. Correct representation of subgrid soil moisture variability is crucial in improving land surface modeling schemes and remote sensing retrievals. In addition to the mean structure, the variance and correlation of soil moisture are affected by the underlying land surface heterogeneity. This often violates the underlying assumption of stationarity/isotropy made by classical geostatistical models. The present study proposes a geostatistical framework to predict and upscale soil moisture in a nonstationary setting using a flexible spatial model whose variance/correlation structure varies with changing land surface characteristics. The proposed framework is applied to model soil moisture distribution using in situ data in the Red River watershed in Southern Manitoba, Canada. It is seen that both the variance and correlation structure exhibits spatial nonstationarity for the given surface heterogeneity driven primarily by vegetation and soil texture. At the beginning of the crop season, soil texture plays a critical role in the drying cycle by decreasing variance and increasing correlation as the soil becomes drier. Once the crops begin to mature, vegetation becomes the dominant driver, promoting spatial correlation and reducing SM variance. We upscale our point scale soil moisture predictions to the airborne extent (∼1.5 km) and find that the upscaled soil moisture agrees well with the observed airborne data with root‐mean‐square error values ranging from 0.04 to 0.08 (v/v). The proposed framework can be used to predict and upscale soil moisture in heterogeneous environments. Plain Language Summary: Soil moisture (SM) is a critical variable governing the global water and energy cycles. Understanding how SM varies in space is therefore critical. This spatial variation of SM can be typically defined by three statistical quantities: mean (average value), variance (how far the individual SM values are from the average value), and correlation (how individual SM values are related to each other). Variance/correlation of SM are typically assumed to be constant in traditional geostatistics methods. This is a major shortcoming because it has been well established that land surface characteristics such as soil, vegetation, and topography affects the spatial variability of SM. In this study, we propose a framework that accounts for the effect of these characteristics on the variance/correlation of SM. We apply our framework to a watershed in Manitoba, Canada, and find that our framework performs significantly better than the traditional method. We find that soil texture and vegetation affect SM distribution at different stages of crop growth. We aggregate our point scale SM predictions to 1.5‐km (airborne) scale and find that our predictions mimic observed SM data at this scale. We conclude that our framework can be used to predict and aggregate SM using surface data. Key Points: Proposed a framework to assess spatial nonstationarity of soil moistureOptimal prediction and upscaling of soil moisture under nonstationarityQuantified the effects of soil texture and vegetation on the spatial variance/correlation of soil moisture [ABSTRACT FROM AUTHOR]

With the proliferation of modern high-resolution measuring instruments mounted on satellites, planes, ground-based vehicles, and monitoring stations, a need has arisen for statistical methods suitable for the analysis of large spatial datasets observed on large spatial domains. Statistical analyses of such datasets provide two main challenges: first, traditional spatial-statistical techniques are often unable to handle large numbers of observations in a computationally feasible way; second, for large and heterogeneous spatial domains, it is often not appropriate to assume that a process of interest is stationary over the entire domain. We address the first challenge by using a model combining a low-rank component, which allows for flexible modeling of medium-to-long-range dependence via a set of spatial basis functions, with a tapered remainder component, which allows for modeling of local dependence using a compactly supported covariance function. Addressing the second challenge, we propose two extensions to this model that result in increased flexibility: first, the model is parameterized on the basis of a nonstationary Matérn covariance, where the parameters vary smoothly across space; second, in our fully Bayesian model, all components and parameters are considered random, including the number, locations, and shapes of the basis functions used in the low-rank component. Using simulated data and a real-world dataset of high-resolution soil measurements, we show that both extensions can result in substantial improvements over the current state-of-the-art. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Spatio-temporal statistics is prone to the curse of dimensionality: one manifestation of this is inversion of the data-covariance matrix, which is not in general feasible for very-large-to-massive datasets, such as those observed by satellite instruments. This becomes even more of a problem in fully Bayesian statistical models, where the inversion typically has to be carried out many times in Markov chain Monte Carlo samplers. Here, we propose a Bayesian hierarchical spatio-temporal random effects (STRE) model that offers fast computation: Dimension reduction is achieved by projecting the process onto a basis-function space of low, fixed dimension, and the temporal evolution is modeled using a dynamical autoregressive model in time. We develop a multiresolutional prior for the propagator matrix that allows for unknown (random) sparsity and shrinkage, and we describe how sampling from the posterior distribution can be achieved in a feasible way, even if this matrix is very large. Finally, we compare inference based on our fully Bayesian STRE model with that based on an empirical-Bayesian STRE-model approach, where parameters are estimated via an expectation-maximization algorithm. The comparison is carried out in a simulation study and on a real-world dataset of global satellite CO2 measurements. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The use of satellite measurements in climate studies promises many new scientific insights if those data can be efficiently exploited. Due to sparseness of daily data sets, there is a need to fill spatial gaps and to borrow strength from adjacent days. Nonetheless, these satellites are typically capable of conducting on the order of 100,000 retrievals per day, which makes it impossible to apply traditional spatio-temporal statistical methods, even in supercomputing environments. To overcome these challenges, we make use of a spatio-temporal mixed-effects model. For each massive daily data set, dimension reduction is achieved by essentially modelling the underlying process as a linear combination of spatial basis functions on the globe. The application of a dynamical autoregressive model in time, over the reduced space, allows rapid sequential computation of optimal smoothing predictions via the Kalman smoother; this is known as Fixed Rank Smoothing (FRS). The dimension-reduced mixed-effects model contains a number of unknown parameters, including covariance and propagator matrices, which describe the spatial and temporal dependence structure in the reduced-dimensional process. We take an empirical-Bayes approach to inference, which involves estimating the parameters and substituting them into the optimal predictors. Method-of-moments (MM) parameter estimation (currently used in FRS) is typically inefficient compared to maximum likelihood (ML) estimation and can result in large sampling variability. Here, we develop ML estimation via an expectation-maximization (EM) algorithm, which offers stable computation of valid estimators and makes efficient use of spatial and temporal dependence in the data. The two parameter-estimation approaches, MM and ML, are compared in a simulation study. We also apply our methodology to global satellite CO measurements: We optimally smooth the sparse daily CO maps obtained by the Atmospheric InfraRed Sounder (AIRS) instrument on the Aqua satellite; then, using FRS with EM-estimated parameters, a complete sequence of the daily global CO fields can be obtained, together with their associated prediction uncertainties. [ABSTRACT FROM AUTHOR]

Extreme weather events such as thunderstorms and tornadoes are of great concern as these events pose a significant threat to life, property, and economic stability. Because of the difficulty of gathering data on extreme events, this paper proposes modeling the conditions for extreme weather through large-scale indicators. The advantage of using large-scale indicators is that climate models can be used to generate data whereas climate models cannot generate data on extreme events themselves. This paper focuses on comparing spatio-temporal models for reanalysis data of large-scale indicators for extreme weather observed across the continental United States and Mexico. Results indicate that rigorous treatment of spatial and temporal dynamics is necessary. The models find that the intensity of conditions for extreme weather is particularly high for the central United States and the intensity of these conditions is increasing over time but the amount of increase may not be practically significant. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]