Full field reconstruction of turbulent systems from sparse observations

We investigate the ability of two data assimilation (DA) methods – ensemble Kalman filter (EnKF) and variational smoother (4D-Var) – to reconstruct the Kuramoto–Sivashinsky and complex Ginzburg–Landau systems from sparse observations. While turbulent systems can reconstruct small scales below a critical resolution by substituting larger scales into governing equations, we examine whether DA methods can accurately recover the full field with observations sparser than this threshold, and consider which method performs better. Our findings show that both methods can accurately reconstruct the full field even with observations much sparser than the substitution-based critical resolution. However, likelihood of successful reconstruction within a fixed assimilation time decreases as sparsity increases. The EnKF method needs smaller assimilation time and lower temporal sampling rate than 4D-Var, but needs ad hoc stabilisation (e. g. inflation) and higher memory for ensemble storage. We validate these results by applying EnKF to turbulent two-dimensional Kolmogorov flows at Reynolds numbers from 200 to 2000, and forcing at wavenumbers 4 and 5. For these flows, we achieve full field reconstruction from observations as sparse as 4 4, outperforming existing 4D-Var and machine learning results where denser observations are required for reconstruction. These findings highlight the strengths and trade-offs of DA methods, offer guidance for reconstructing turbulent flows, and establish benchmarks for evaluating alternative methods.