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

Emine Celik,Eric Olson,Edriss S Titi

doi:10.1137/18m1218480

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.

Highlights

The goal of data assimilation is to optimally combine known information about the dynamics of a solution with low-resolution observational measurements of that solution over time to create better and better approximations of the current state
Our results extend the work of Hayden, Olson and Titi [17] on discrete-in-time data assimilation from the case where the low-resolution observations are given by projection onto the low Fourier modes to both the first and second type of general interpolant observables that appear in Azouani, Olson and Titi [3], see Bessaih, Olson and Titi [4]
We introduce the spectrally-filtered discrete-in-time data assimilation algorithm which forms the focus of our study

Summary

Introduction

The goal of data assimilation is to optimally combine known information about the dynamics of a solution with low-resolution observational measurements of that solution over time to create better and better approximations of the current state. Our results extend the work of Hayden, Olson and Titi [17] on discrete-in-time data assimilation from the case where the low-resolution observations are given by projection onto the low Fourier modes to both the first and second type of general interpolant observables that appear in Azouani, Olson and Titi [3], see Bessaih, Olson and Titi [4]. Not the focus of the present research, the algorithm stated above may be used to stabilize the growth of numerical error Putting such numerical considerations aside, we view the data assimilation algorithm given in Definition 1.2 as a way of improving estimates of the unknown state of U at time t by means of known dynamics and a time-series of low-resolution observations. We finish with some concluding remarks concerning the dependency of h and λ on δ and the other physical parameters in the system

Preliminaries

Type-I Interpolant Observables

Type-II Interpolant Observables

Conclusions

A Estimates for Time Dependent Forcing