Terminal-Guided Back-and-Forth Nudging: Time-Reversed Smoke-Plume Reconstruction with Fourier Neural-Operator Acceleration

Abstract. Reconstructing an earlier smoke plume from a later observation is an ill-posed inverse problem. Diffusion is not reversible, so backward integration amplifies small-scale perturbations rather than recovering structures already erased from the terminal field. Quasi-reversibility (QRV) provides a stabilized backward baseline for the source-forced advection–diffusion–removal equation, but it does not ensure that a reconstructed earlier plume remains dynamically consistent with the observed terminal plume when replayed forward. We introduce terminal-guided back-and-forth nudging (TBFN) to reconstruct the plume state when the source history is known or estimated independently. In each short backward window, a QRV-reconstructed candidate is replayed forward to the terminal time, and the resulting mismatch is transported back through the homogeneous QRV dynamics to correct the window’s starting condition, iterated until the correction stabilizes. The selected reconstruction is then passed backward to the next window, and the procedure continues across the full interval. Across noise-free identical-twin experiments comprising a reference plume, a transport-dominated high-Péclet stress test, and a close multi-source plume, TBFN reduces QRV hidden-state (withheld forward truth) errors by approximately one to two orders of magnitude in these controlled configurations. The improvement is concentrated in coherent low-to-intermediate spectral content and does not circumvent the diffusive loss of unsupported short wavelengths. A source-free ablation shows that terminal consistency under an incorrect forcing model can coexist with a severely inaccurate reconstructed history. When the emission history is not independently known, its parameters must therefore be estimated rather than omitted or prescribed without justification. In a separate experiment with fixed transport physics, we train a conditioned Fourier neural operator (Source-FNO) to infer location, ignition time, and strength from a densely sampled plume-evolution history. Across 1,200 held-out test events, Source-FNO recovers these parameters very accurately. However, when the estimated parameters are used to drive the TBFN reconstruction instead of the reference ones, the resulting hidden-state errors are larger near ignition. We also train a chain of window-local FNOs to approximate the physics-based TBFN reconstruction maps and compose them backward in time. Across 150 held-out events under the same fixed transport physics, the learned chain attains mean hidden-state endpoint errors of 8.83 % with the configured event parameters and 9.95 % with Source-FNO estimates. The TBFN teacher generated with the configured parameters has a corresponding error of 9.54 %. Once trained, the chain provides a roughly 10⁵-fold inference speedup. This combination points toward reconstructing the earliest post-ignition plume state, the hardest part of after-the-fact fire-behavior analysis, fast enough for routine use once the neural operators are trained.