Linear Algebra Analysis for Contaminant Source Identification in Water Distribution Systems

The problem of identifying the locations (or network areas) that may be the source of a detected contamination event in drinking water distribution systems, is typically ill-posed and an infinite set of solutions is possible. For a conservative (or first order reactive) contaminant, the observed concentration behavior at any network node can be written as linear combination of contaminant injections at previous times. Linear algebra analysis is thus employed to account for prior information and pipe grouping to improve problem invertibility. Given the possible multiple solutions and presence of uncertainties, the minimum relative entropy method is suggested for solving the inverse problem. Thus, a probability density function rather than a specific contaminant injection value is found for each potential contamination source node. Several analytical results may be derived when this method is applied to linear systems. Moreover, this entropic formulation provides significant advantages when dealing with a prior bias in the estimated probability function, it may allow for the less committed prior distribution with respect to unknown information, and it does not introduce spurious structures that do not reflect the real physical system.