Double-scale diffusive wave model dedicated to spatial river observation and associated covariance kernel for variational data assimilation

<p>The spatial altimetry provides an important amount of water surface height data from multi-missions satellites (especially Jason-3, Sentinel-3A/B and the forthcoming NASA-CNES SWOT mission). To exploit at best the potential of spatial altimetry, the present study proposes on the derivation of a model adapted to spatial observations scale; a diffusive-wave type model but adapted to a double scale [1].</p><p>Moreover, Green-like kernel can be employed to derived covariance operators, therefore they may provide an approximation of the covariance kernel of the background error in Variational Data Assimilation processes. Following the derivation of the aforementioned original flow model, we present the derivation of a Green kernel which provides an approximation of the covariance kernel of the background error for the bathymetry (i.e. the control variable) [2].</p><p>This approximation of the covariance kernel is used to infer the bathymetry in the classical Saint-Venant’s (Shallow-Water) equations with better accuracy and faster convergence than if not introducing an adequate covariance operator [3].</p><p>Moreover, this Green kernel helps to analyze the sensitivity of the double-scale diffusive waves (or even the Saint-Venant’s equations) with respect to the bathymetry.</p><p>Numerical results are analyzed on real like datasets (derived from measurements of the Rio Negro, Amazonia basin).</p><p>The double-scale diffusive wave provide more accurate results than the classical version. Next, in terms of inversions, the derived physically-based covariance operators enable to improve the inferences, compared to the usual exponential one.</p><p>[1] T. Malou, J. Monnier "Double-scale diffusive wave equations dedicated to spatial river observations". In prep.</p><p>[2] T. Malou, J. Monnier "Physically-based covariance kernel for variational data assimilation in spatial hydrology". In prep.</p><p>[3] K. Larnier, J. Monnier, P.-A. Garambois, J. Verley. "River discharge and bathymetry estimations from SWOT altimetry measurements". Inv. Pb. Sc. Eng (2020).</p>

Efficient multi-scale Gaussian process regression for massive remote sensing data with satGP v0.1.2

Satellite remote sensing provides a global view to processes on Earth that has unique benefits compared to making measurements on the ground, such as global coverage and enormous data volume. The typical downsides are spatial and temporal gaps and potentially low data quality. Meaningful statistical inference from such data requires overcoming these problems and developing efficient and robust computational tools. We design and implement a computationally efficient multi-scale Gaussian process (GP) software package, satGP, geared towards remote sensing applications. The software is able to handle problems of enormous sizes and to compute marginals and sample from the random field conditioning on at least hundreds of millions of observations. This is achieved by optimizing the computation by, e.g., randomization and splitting the problem into parallel local subproblems which aggressively discard uninformative data. We describe the mean function of the Gaussian process by approximating marginals of a Markov random field (MRF). Variability around the mean is modeled with a multi-scale covariance kernel, which consists of Matérn, exponential, and periodic components. We also demonstrate how winds can be used to inform covariances locally. The covariance kernel parameters are learned by calculating an approximate marginal maximum likelihood estimate, and the validity of both the multi-scale approach and the method used to learn the kernel parameters is verified in synthetic experiments. We apply these techniques to a moderate size ozone data set produced by an atmospheric chemistry model and to the very large number of observations retrieved from the Orbiting Carbon Observatory 2 (OCO-2) satellite. The satGP software is released under an open-source license.

Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach

In this paper we investigate scaling limits of the odometer in divisible sandpiles on d-dimensional tori following up the works of Chiarini et al. (Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits, 2018), Cipriani et al. (Probab Theory Relat Fields 172:829–868, 2017; Stoch Process Appl 128(9):3054–3081, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalized Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $$(-\varDelta )^{-s/2} W$$ ( - Δ ) - s / 2 W for $$s>2$$ s > 2 and W a spatial white noise on the d-dimensional unit torus.

On the positivity and magnitudes of Bayesian quadrature weights

This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g. Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g. Matérn kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.

Efficient multi-scale Gaussian process regression for massive remote sensing data with satGP v0.1

Satellite remote sensing provides a global view to processes on Earth that has unique benefits compared to measurements made on the ground. The global coverage and the enormous amounts of data produced come, however, with the price of spatial and temporal gaps and less than perfect data quality. Meaningful statistical inference from such data requires overcoming these problems and that calls for developing efficient computational tools. We design and implement a computationally efficient multi-scale Gaussian process (GP) software package, satGP, geared towards remote sensing applications. The software is designed to be able to handle problems of enormous sizes and is able to compute marginals and sample from a random process with at least over hundred million observations. The mean function of the Gaussian process is described by approximating marginals of a Markov random field (MRF). For covariance functions, Matern, exponential, and periodic kernels are utilized in a multi-scale kernel setting to describe the spatial heterogeneity present in data. We further demonstrate how winds can be used to inform the covariance kernel formulation. The covariance kernel parameters are learned by calculating an approximate marginal maximum likelihood estimate and this is utilized to verify the validity of the multi-scale approach in synthetic experiments. For demonstrating the techniques above, data from the Orbiting Carbon Observatory 2 (OCO-2) satellite is used. The satGP program is released as open source software.

Gaussian Process Based Bayesian Inference System for Intelligent Surface Measurement

This paper presents a Gaussian process based Bayesian inference system for the realization of intelligent surface measurement on multi-sensor instruments. The system considers the surface measurement as a time series data collection process, and the Gaussian process is used as mathematical foundation to establish an inferring plausible model to aid the measurement process via multi-feature classification and multi-dataset regression. Multi-feature classification extracts and classifies the geometric features of the measured surfaces at different scales to design an appropriate composite covariance kernel and corresponding initial sampling strategy. Multi-dataset regression takes the designed covariance kernel as input to fuse the multi-sensor measured datasets with Gaussian process model, which is further used to adaptively refine the initial sampling strategy by taking the credibility of the fused model as the critical sampling criteria. Hence, intelligent sampling can be realized with consecutive learning process with full Bayesian treatment. The statistical nature of the Gaussian process model combined with various powerful covariance kernel functions offer the system great flexibility for different kinds of complex surfaces.