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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. 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. 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

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. 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. 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

Regression‐based detection and attribution methods continue to take a central role in the study of climate change and its causes. Here we propose a novel Bayesian hierarchical approach to this problem, which allows us to address several open methodological questions. Specifically, we take into account the uncertainties in the true temperature change due to imperfect measurements, the uncertainty in the true climate signal under different forcing scenarios due to the availability of only a small number of climate model simulations, and the uncertainty associated with estimating the climate variability covariance matrix, including the truncation of the number of empirical orthogonal functions (EOFs) in this covariance matrix. We apply Bayesian model averaging to assign optimal probabilistic weights to different possible truncations and incorporate all uncertainties into the inference on the regression coefficients. We provide an efficient implementation of our method in a software package and illustrate its use with a realistic application. Novel computationally efficient Bayesian hierarchical approach for detection and attributionUncertainty from covariance estimation incorporated in regression coefficient estimatesA range of EOF truncations is considered and optimally weighted using Bayesian model averaging

A probabilistic forecast takes the form of a predictive probability distribution over future quantities or events of interest. Probabilistic forecasting aims to maximize the sharpness of the predictive distributions, subject to calibration, on the basis of the available information set. We formalize and study notions of calibration in a prediction space setting. In practice, probabilistic calibration can be checked by examining probability integral transform (PIT) histograms. Proper scoring rules such as the logarithmic score and the continuous ranked probability score serve to assess calibration and sharpness simultaneously. As a special case, consistent scoring functions provide decision-theoretically coherent tools for evaluating point forecasts. We emphasize methodological links to parametric and nonparametric distributional regression techniques, which attempt to model and to estimate conditional distribution functions; we use the context of statistically postprocessed ensemble forecasts in numerical weather prediction as an example. Throughout, we illustrate concepts and methodologies in data examples.

Generalized Gaussian processes (GGPs) are highly flexible models that combine latent GPs with potentially non-Gaussian likelihoods from the exponential family. GGPs can be used in a variety of settings, including GP classification, nonparametric count regression, modeling non-Gaussian spatial data, and analyzing point patterns. However, inference for GGPs can be analytically intractable, and large datasets pose computational challenges due to the inversion of the GP covariance matrix. A Vecchia–Laplace approximation for GGPs is proposed, which combines a Laplace approximation to the non-Gaussian likelihood with a computationally efficient Vecchia approximation to the GP, resulting in simple, general, scalable, and accurate methodology. Numerical studies and comparisons on simulated and real spatial data are provided. The methods are implemented in a freely available R package. [ABSTRACT FROM AUTHOR]