A critical note on back-and-forth Data Assimilation Nudging Algorithm

TL;DR

This paper critically examines the limitations of the Back-and-Forth Nudging data assimilation algorithm, demonstrating that it cannot reliably recover initial conditions in certain dissipative systems due to indistinguishable sparse observations.

Contribution

It constructs infinitely many solutions sharing identical sparse data, revealing fundamental limitations of BFN algorithms in recovering initial conditions.

Findings

BFN cannot reliably recover initial conditions in some dissipative systems.

Infinitely many solutions can share the same sparse observational data.

Regularized BFN improves stability but still cannot recover unobserved fine scales.

Abstract

This work investigates the effectiveness of the Back-and-Forth Nudging (BFN) data assimilation algorithm, specifically its performance when employing the Azouani-Olson-Titi (AOT) continuous data assimilation downscaling nudging algorithm, for recovering initial conditions of dissipative dynamical systems. Contrary to previous reports in the literature, we show that, for several systems of interest, one can construct initial conditions that BFN cannot reliably recover. Our key finding is the construction of infinitely many distinct solutions for certain dissipative systems that share identical spatially sparse observational data. Since these observations are indistinguishable, no data assimilation method relying only on them can differentiate between these solutions or recover the correct initial condition. We illustrate these pathological initial conditions for the Lorenz 1963 model and several 1D partial differential equations: the heat equation, viscous linear transport, and viscous and inviscid Burgers equations. Our analytical results are supported by numerical simulations. To address the numerical ill-posedness of backward-in-time iterations, an essential step of BFN for dissipative models, we introduce a regularized backward step, the Voigt-regularized BFN (VBFN). We investigate its performance for the 2D Navier-Stokes equations and the viscous 1D KdV equation, comparing it with standard BFN and Diffusive BFN (dBFN). While VBFN improves numerical stability and reduces model bias relative to dBFN, it still cannot reconstruct the unobserved fine spatial scales of the reference solution. This reinforces our main conclusion: even with regularization, BFN-type algorithms are limited in recovering the full state, and in particular the initial data, from sparse spatial observations.