Limits of the L-Curve Criterion for Adaptive Regularization-Parameter Selection in Sparse-Sensor Source Term Estimation

Source term estimation (STE) from sparse atmospheric sensor networks is an ill-posed inverse problem that is typically stabilized by Tikhonov-type regularization with a hand-picked regularization parameter α, whose value is rarely justified or tested for robustness — a gap in an otherwise well-studied area of atmospheric transport modelling to which the authors' own research group has contributed [1]. This note reports a preliminary numerical test of the most classical automatic alternative to a fixed α — the L-curve criterion [2] — against a naive fixed-α = 1 baseline, on a closed-form Gaussian-plume advection–diffusion test problem across 4 sensor counts (N = 5–20), 4 noise levels (1–20%) and 15 repetitions each (480 trials in total). The L-curve criterion collapsed onto a single α value in 62% of trials, matched the fixed baseline's mean accuracy but with substantially higher variance, and specifically underperformed it under the combination of high noise and small N that characterizes the target application. A regularization-parameter strategy tailored to this sparsity/noise regime remains an open problem for the ongoing dissertation.