Spectral Filtering of Interpolant Observables for a Discrete-in-time Downscaling Data Assimilation Algorithm

TL;DR

This paper introduces a spectral filtering data assimilation algorithm for the 2D Navier--Stokes equations that effectively incorporates various types of observational data without relying on orthogonal projections.

Contribution

It extends previous methods by allowing general interpolants satisfying a natural approximation property, improving the flexibility of downscaling data assimilation techniques.

Findings

Proves the algorithm's effectiveness for a broad class of interpolants.

Demonstrates the method's applicability to local spatial averages and point measurements.

Extends theoretical results beyond orthogonal projection-based observations.

Abstract

We describe a spectrally-filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier--Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.