Estimation of the quantities of harmful substances emitted into the atmosphere is one of the main challenges in modern environmen- tal sciences. In most of the cases, this estimation requires solving a linear inverse problem. A key difficulty in evaluating the performance of any algorithm to solve this linear inverse problem is that the ground truth is typically unknown. In this paper we show that the noise encountered in this linear inverse problem is non-Gaussian. Next, we develop an algorithm to deal with the strong outliers present in the measurements. Finally, we test our approach on three different experiments: a simple synthetic experiment, a controlled real-world experiment, and real data from the Fukushima nuclear accident.