Why Differential Data Work

This article explains the features of differential data that make them attractive, their shortcomings, and the situations for which they are best suited. The use of differential data is ubiquitous in the seismological community, in which they are used to determine earthquake locations via the double-difference method and the Earth's velocity structure via geotomography; furthermore, they have important applications in other areas of geophysics, as well. A common assumption is that differential data are uncorrelated and have uniform variance. We show that this assumption is well justified when the original, undifferenced data covary with each other according to a two-sided exponential function. It is not well justified when they covary according to a Gaussian function. Differences of exponentially correlated data are approximately uncorrelated with uniform variance when they are regularly spaced in distance. However, when they are irregularly spaced, they are uncorrelated with a nonuniform variance that scales with the spacing of the data. When differential data are computed by taking differences of the original, undifferenced data, model parameters estimated using ordinary least squares applied to the differential data are almost exactly equal to those estimated using weighed least squares applied to the original, undifferenced data (with the weights given by the inverse covariance matrix). A better solution only results when the differential data are directly estimated and their variance is smaller than is implied by differencing the original data. Differential data may be appropriate for global seismic travel-time data because the covariance of errors in predicted travel times may have a covariance close to a two-sided exponential, on account of the upper mantle being close to a Von Karman medium with exponent kappa MUCH LESS-THAN12.