arXiv subjects

Anthony Weaver

Publications and source records attributed to Anthony Weaver.

6 recordsLinked to original sources

arXiv2025

On the impact of observation error correlations in data assimilation, with application to along-track altimeter data

Data assimilation involves estimating the state of a system by combining observations from various sources with a background estimate of the state. The weights given to the observations and background state depend on their specified error covariance matrices. Observation errors are often assumed to be uncorrelated even though this assumption is inaccurate for many modern data-sets such as those from satellite observing systems. As methods allowing for a more realistic representation of observation-error correlations are emerging, our aim in this article is to provide insight on their expected impact in data assimilation. First, we use a simple idealised system to analyse the effect of observation-error correlations on the spectral characteristics of the solution. Next, we assess the relevance of these results in a more realistic setting in which simulated alongtrack (nadir) altimeter observations with correlated errors are assimilated in a global ocean model using a three-dimensional variational assimilation (3D-Var) method. Correlated observation errors are modelled in the 3D-Var system using a diffusion operator. When the correlation length scale of observation error is small compared to that of background error, inflating the observation-error variances can mitigate most of the negative effects from neglecting the observation-error correlations. Accounting for observation-error correlations in this situation still outperforms variance inflation since it allows small-scale information in the observations to be more effectively extracted and does not affect the convergence of the minimization. Conversely, when the correlation length scale of observation error is large compared to that of background error, the effect of observation-error correlations cannot be properly approximated with variance inflation. However, the correlation model needs to be constructed carefully to ensure the minimization problem is adequately conditioned so that a robust solution can be obtained. Practical ways to achieve this are discussed.

math.NA↗

arXiv2023

Application of deep learning to the estimation of normalization coefficients in diffusion-based covariance models

Variational data assimilation in ocean models depends on the ability to model general correlation operators in the presence of coastlines. Grid-point filters based on diffusion operators are widely used for this purpose, but come with a computational bottleneck - the costly estimation of normalization factors for every model grid point. In this paper, we show that a simple convolutional neural network can effectively learn these normalization factors with better accuracy than the current operational methods. Our network is tested with a two-dimensional diffusion operator from the NEMOVAR ocean data assimilation system, applied to a global ocean grid with approximately one degree horizontal resolution. The network is trained on exact normalization factors estimated by a brute-force method. Knowing that convolutional networks can only model translation-equivariant functions, we ensure that the normalization estimation problem is indeed translation-equivariant. Specifically, we show how the number of inputs of this problem can be reduced while preserving translation equivariance. Adding the distance to the coastline as an input channel is found to improve the performance of the network around coastlines. Extensions to three-dimensional diffusion and to higher horizontal resolutions are discussed. Removing the computational bottleneck associated with normalization opens the way to using adaptive correlation models for operational ocean data assimilation. The code for this work is publicly available at https://github.com/FolkeKS/DL-normalization/tree/core-features

physics.data-an↗

arXiv2017

Zero-sum multisets mod p with an application to surface automorphisms

We solve a problem in enumerative combinatorics which is equivalent to counting topological types of certain group actions on compact Riemann surfaces. Let $V_2(F_p)$ be the two-dimensional vector space over $F_p$, the field with $p$ elements, $p$ an odd prime. We count orbits of the general linear group $GL_2(F_p)$ on certain multisets consisting of $R \geq 3$ non-zero columns from $V_2(F_p)$. The $R$-multisets are `zero-sum,' that is, the sum (mod $p$) over the columns in the multiset is $[\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}]$. The orbit count yields the number of topological types of fully ramified actions of the elementary abelian $p$-group of rank $2$ on compact Riemann surfaces of genus $1+ Rp(p-1)/2-p^2.$

math.CO↗

arXiv2012

Classical curves via one-vertex maps

One-vertex maps (a type of dessin d'enfant) give a uniform characterization of certain well-known algebraic curves, including those of Klein, Wiman, Accola-Maclachlan and Kulkarni. The characterization depends on a new classification of one-vertex (dually, one-face or unicellular) maps according to the size of the group of map automorphisms. We use an equivalence relation appropriate for studying the faithful action of the absolute Galois group on dessins, although we do not pursue that line of inquiry here.

math.AG↗

arXiv2009

A Diophantine Frobenius problem related to Riemann surfaces

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular $pq$-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order $pq$. The general $t$-dimensional Frobenius problem ($t \geq 3$) is $NP$-hard, and it may be that our restricted problem retains this property.

math.NT↗

arXiv2007

Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.

math.AG↗