An algorithm SUMMOOTH for smoothing raw observations that are unequally spaced is explained. The process generalizes the method of summary values introduced into geophysics by H. Jeffreys. Smoothed data points, defined as the intersection of the local linear and parabolic least-squares fits, are computed for overlapping intervals rather than fixed sequential intervals as in previous work. A new feature is the parallel computation of (smoothed) summary gradients, essential for the Herglotz velocity integration. The interval selection is objective because the position of the smoothed values at each stage depends only on the spacing of the raw sample points. Selection of the starting interval can be made objective by addition of a principle, such as symmetry, or a rule, such as the fitted local curvature must never exceed a fixed value. In practice, selection should employ a trade-off curve between resolution and variance. An advantage of the process is that uncertainties at the summary points are independent. A comparison is given, for scarce data of Mossbauer spectra, with the smoothing method of Talmi and Gilat. Application to ragged time series for v p v s observations in earthquake prediction studies and to the construction of seismological travel-time curves illustrates the value of the method in geophysics.
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