We are often interested in phases of complex quantities; e.g., in non-destructive testing of aerospace structures, important information comes from phases of Eddy current and magnetic resonance.
For each measurement, we have an upper bound D on the measurement error dx=X-x, so when the measurement result is X, we know that the actual value x is in [X-D,X+D]. Often, we have no information about probabilities of different values, so this interval is our only information about x. When the accuracy is not sufficient, we perform several repeated measurements, and conclude that x belongs to the intersection of the corresponding intervals.
For real-valued measurements, the intersection of intervals is always an interval. For phase measurements, we prove that an arbitrary closed subset of a circle can be represented as an intersection of intervals.
Handling such complex sets is difficult. It turns out that if we have some statistical information, then the problem often becomes tractable. As a case study, we describe an algorithm that uses both real-valued and phase measurement results to determine the shape of a fault. This is important: e.g., smooth-shaped faults gather less stress and are, thus, less dangerous than irregularly shaped ones.