Inverse Models for Estimating the Initial Condition of Spatio-Temporal Advection-Diffusion Processes

Inverse problems involve making inference about unknown parameters of a physical process using observational data, and are widely found in scientific and engineering applications. For example, in urban air quality and environmental monitoring, inverse problems aim at quickly pinpointing the sources of instantaneous emissions of gaseous pollutants that cause public health concerns (Eckhardt et al., 2008; Martinez-Camara et al., 2014; Hwang et al., 2019), or detecting fugitive emissions due to accidental releases from industrial operations (Hosseini and Stockie, 2016; Klein et al., 2016). In healthcare applications, inverse models have been employed to obtain heart-surface potentials from body-surface measurements, known as the inverse ECG problem (Yao and Yang, 2021). In Seismology, inverse problems aim at getting information about the structure of the forces acting in the earthquake's focus from seismic waves at Earth's surface (Apostol, 2019). Inverse modeling has also found its applications in detecting the impact location of the missing Malaysian Airlines MH370, using the drift of marine debris (Miron et al., 2019) or acoustic-gravity waves (Kadri, 2019). This paper investigates an important class of statistical inverse problems--the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Consider the detection of accidental releases of fugitive emissions from industrial operations (Hosseini and Stockie, 2016).