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Abstract
Determining the location, release strength and temporal profile of an unknown atmospheric pollutant source from concentration measurements recorded by a small number of ground-based sensors is a core problem of environmental monitoring, industrial safety and emergency response. Because the number of independent observations is typically far smaller than the number of unknown source parameters, source term estimation (STE) from sparse networks is an ill-posed inverse problem in the sense of Hadamard, and its solution has developed along three largely independent methodological lines: deterministic approaches built on adjoint transport equations and Tikhonov-type regularization; Bayesian approaches relying on Markov chain Monte Carlo (MCMC) sampling; and, over the last five years, machine-learning surrogate and physics-informed neural network (PINN) approaches that accelerate the repeated forward-model evaluations required inside an iterative inversion loop. This paper reviews the primary literature across all three streams, classifies it by the degree of information deficiency a method was designed to tolerate — the ratio of independent sensor readings to unknown source parameters — and identifies a specific, still largely open regime: reconstruction from 5–15 non-uniformly placed sensors under weak-wind, near-neutral atmospheric stability, conditions characteristic of arid inland regions such as Central Asia. Unlike a literature-only survey, the comparative claim is tested directly: four representative methods (unregularized least squares, adjoint-equivalent Tikhonov regularization, Bayesian MCMC, and a quadratic-surrogate-accelerated inversion) are implemented and run on a common closed-form advection–diffusion test problem across 5–20 sensors and 1–20% observation noise (192 independent trials). The experiment confirms the expected degradation of all methods as the network sparsifies, shows that a fixed regularization parameter introduces an avoidable bias relative to the unregularized estimate under low noise while stabilizing it under high noise, and shows that a naively chosen low-order polynomial surrogate fails outright on this nonlinear, compactly-supported forward map — a cautionary result that supports, rather than merely asserts, the recent shift toward neural-network and physics-informed surrogates in the reviewed literature. Two open problems follow directly from the combined review and experiment: automatic, data-driven regularization-parameter selection under sparse, noisy data, and a principled criterion for detecting when a machine-learning surrogate is operating outside its training domain.
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