Monte-Carlo-Type Techniques for Processing Interval Uncertainty, and Their Geophysical and Engineering Applications

To determine the geophysical structure of a region, we measure seismic travel times and reconstruct velocities at different depths from this data. There are several algorithms for solving this inverse problem, but these algorithms do not tell us how accurate these reconstructions are. Traditional approach to accuracy estimation assumes that the measurement errors are independently normally distributed. Problem: the resulting accuracies are not in line with geophysical intuition. Reason: a typical error is when we miss the first arrival of the seismic wave; it is not normal (bounded by the wave period ∆) and not independent. Typically, all we know is the upper bound ∆ on the measurement error, so when the measured value is x̃, we conclude that x ∈ [x̃−∆, x̃+∆]. For this interval uncertainty, the resulting velocity accuracy is, qualitatively, in much better accordance with geophysics. Interval uncertainty naturally appears in other applications as well. In this paper, we describe Monte-Carlo-Type techniques for processing interval uncertainty, and their geophysical and engineering applications.