Discrete Gaussian Vector Fields on Meshes

Introduction

Statistical downscaling of climate models focuses on the statistical relationship between spatial locations and dynamical models. This is useful when the outputs of climate models are fed into other dynamical models.

A GP can loosely be considered to be a distribution over functions, and more concretely be considered a multivariate distribution where the joint distribution of every set of variables is a Gaussian distribution .

There are many examples of the application of this methodology to the problem of statistical downscaling in the literature; see Xiong et al. and Fuentes and Raftery .

Methodology

Formulated in this way it is a regression problem, and a common methodology is to use Gaussian processes (GPs) .

Study Design

Results & Findings

High-quality inference procedures and the availability of historic weather data allow us to make inferences about what the model outputs would be at unobserved spatial locations. The goal of downscaling is to take observations of some variable(s) of interest at some spatial locations s 1 , . . . , s n and infer the value of the variable(s) at some unobserved locations s n+1 , . . . .

• High-quality inference procedures and the availability of historic weather data allow us to make inferences about what the model outputs would be at unobserved spatial locations.
• The goal of downscaling is to take observations of some variable(s) of interest at some spatial locations s 1 , . . . , s n.
• When considering random fields, a GP becomes a prior over functions of space, where each unobserved spatial index has a mean and variance derived from the.
• An obvious choice here would be to model the u and v components of the vector with independent scalar GPs, but this approach has limitations.
• Berlinghieri et al. establish a univariate spatial GP prior on the decomposed components, which allows them to create a closed-form prior on the reconstructed field.

Practical Applications

This allows parameterisation based on curling and diverging velocities, and accounts for instances where the data may be a vector field with only one of these parts. However, there are some possible extensions to this work in its current form.

Another possible extension concerns precision formulations of the kernels.

Background

This section covers intrinsic Gaussian processes and their mathematical foundations, including the Mat\u00e9rn GP and its application to smooth manifolds. It discusses the spectral decomposition of the Laplace-Beltrami operator and its relevance to defining covariance kernels on Riemannian manifolds.

Discrete Exterior Calculus

The section introduces Discrete Exterior Calculus (DEC), which generalizes calculus to higher-degree differential forms on discretized manifolds. It explains how DEC provides a discrete analogue to exterior calculus and defines differential operators on meshes.

Cotangent Laplacian

This section discusses the cotangent Laplacian as a discrete Laplace-Beltrami operator, emphasizing its convergence properties. It describes the construction of the cotangent Laplacian in the DEC setting and its spectral decomposition.

Figures Explained

The paper’s visual material highlights the workflow and the main system components.

• Fig. 1 :: Fig. 1: Sampled vector fields on a variety of meshes. Colour and length of cones indicates magnitude. Left: an icosphere subdivided 4 times, giving 2562 vertices and 5210 faces. Middle: A torus with 96 major sections and 24 minor sections, giving 2304 vertices and 4608 faces. Right: Stanford Bunny, reduced to have 5048 vertices and 10000 faces.
• Fig. 2 :: Fig. 2: 0-eigenspace of Hodge-Laplacian on a subset of R 2 . Colour indicates magnitude of the vector. There are obvious boundary effects due to the formulation of the Hodge-Laplacian as a DEC operator.
• Fig. 4 :: Fig. 4: Downscaled monthly average wind data from ERA-5 on the plate carr\u00e9e projection. Red arrows indicate observed data for the downscaling procedure, with size indicating magnitude. Other arrows are the posterior mean of the GP, with colour and size indicating magnitude. MSE = 0.027, NLL = -652.992.
• Fig. 5 :: Fig. 5: Downscaled monthly average wind data from ERA-5 on an orthographic projection. Arrow colour and size indicates magnitude.
• Fig. 6 :: Fig. 6: Squared error of downscaled monthly average wind data. Arrow length and direction are the posterior mean values, with colour giving the error magnitude.