Title
Interval Approach to Phase Measurements Can Lead to Arbitrarily Complex Sets A Theorem and Ways Around It
Publication Date
12-2004
Document Type
Article
Abstract
We are often interested in phases of complex quantities; e.g., in nondestructive testing of aerospace structures, important information comes from phases of Pulse Echo and magnetic resonance. For each measurement, we have an upper bound Δ on the measurement error Δx=x~" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">x̃ x~−x, so when the measurement result is x~" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">x̃ x~, we know that the actual value x is in [x~" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">x̃ x~−Δ,x~" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">x̃ x~+Δ]. 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.
Comments
Mulupuru, B.C., Kreinovich, V. & Osegueda, R. Numerical Algorithms (2004) 37: 285. https://doi.org/10.1023/B:NUMA.0000049475.44047.c8