Global Independence, Possible Local Dependence: Towards More Realistic Error Estimates for Indirect Measurements

In many practical situations, it is not realistically possible to directly measure the desired physical quantity. In such situations, we have to measure this quantity indirectly, i.e., measure related quantities and use the known relation to estimate the value of the desired quantity. How accurate it the resulting estimate? The traditional approach assumes that the measurement errors of all direct measurements are independent. In many practical situations, this assumption works well, but in many other practical situations, it leads to a drastic underestimation of the resulting estimation error: e.g., when we base our estimate on measurements performed at nearby moments of time, since there is usually a strong correlation between the corresponding measurement errors. An alternative approach is when we make no assumptions about dependence. This alternative approach, vice versa, often leads to a drastic overestimation of the resulting estimation error. To get a more realistic estimate, it is desirable to take into account that while on the local level, we may have correlations, globally, measurement errors are usually indeed independent – e.g., measurements sufficiently separated in time and/or space. In this paper, we show how to analyze such situations by combining Monte-Carlo techniques corresponding to both known approaches. On the geophysical example, we show that this combination indeed leads to realistic estimates.