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Prieto, GA, Parker RL, Vernon FL, Shearer PM, Thomson DJ.  2006.  Uncertainties in earthquake source spectrum estimation using empirical Green functions. Earthquakes; radiated energy and the physics of faulting. 170( Abercrombie RE, McGarr A, Kanamori H, Di Toro G, Eds.).:69-74., Washington: American Geophysical Union   10.1029/170gm08   Abstract

We analyze the problem of reliably estimating uncertainties of the earthquake source spectrum and related source parameters using Empirical Green Functions (EGF). We take advantage of the large dataset available from 10 seismic stations at hypocentral distances (10 km < d <50 km) to average spectral ratios of the 2001 M5.1 Anza earthquake and 160 nearby aftershocks. We estimate the uncertainty of the average source spectrum of the M5.1 target earthquake by performing propagation of errors, which, due to the large number of EGFs used, is significantly smaller than that obtained using a single EGF. Our approach provides estimates of both the earthquake source spectrum and its uncertainties, plus confidence intervals on related source parameters such as radiated seismic energy or apparent stress, allowing the assessment of statistical significance. This is of paramount importance when comparing different sized earthquakes and analyzing source scaling of the earthquake rupture process. Our best estimate of radiated energy for the target earthquake is 1.24×1011 Joules with 95% confidence intervals (0.73×1011, 2.28×1011). The estimated apparent stress of 0.33 (0.19, 0.59) MPa is relatively low compared to previous estimates from smaller earthquakes (1MPa) in the same region.

Parker, RL.  1977.  Understanding inverse theory. Annual Review of Earth and Planetary Sciences. 5:35-64.   10.1146/annurev.ea.05.050177.000343   AbstractWebsite
Huestis, SP, Parker RL.  1979.  Upward and downward continuation as inverse problems. Geophysical Journal of the Royal Astronomical Society. 57:171-188.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1979.tb03779.x   AbstractWebsite

Summary. The formalism of Backus & Gilbert is applied to the problems of upward and downward continuation of harmonic functions. We first treat downward continuation of a two-dimensional field to a level surface everywhere below the observation locations; the calculation of resolving widths and solution estimates is a straightforward application of Backus—Gilbert theory. The extension to the downward continuation of a three-dimensional field uses a delta criterion giving resolving areas rather than widths. A feature not encountered in conventional Backus—Gilbert problems is the requirement of an additional constraint to guarantee the existence of the resolution integrals. Finally, we consider upward continuation of a two-dimensional field to a level above all observations. We find that solution estimates must be weighted averages of the field not only on this level, but also on a line passing between the observations and sources. Weighting on the lower line may be traded off against resolution on the upper level.