{'fraction_non_alphanumeric': 0.03505535055350553, 'fraction_numerical': 0.007380073800738007, 'mean_word_length': 4.14873417721519, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stable determination of (Brune) stress drops

Abstract The Brune stress drop, σ B , is generally calculated from two spectral observables: Ω 0 , the zero-frequency level of the far-field displacement amplitude spectrum, and v c , the corner frequency of that spectrum. σ B takes the form σ B = h Ω 0 v c 3 , (a) where h contains kinematic factors. In some recent studies, other estimates of the stress drop, such as the dynamic stress drop and the apparent stress, are found to be more stable and/or reliable than σ B . We find that these apparent shortcomings of σ B can be remedied if we replace the dependence of σ B on v c as a directly determined spectral observable with a dependence on a parameter which can be determined more reliably and which appears linearly (instead of as the third power) in the expression for σ B . This observable is J , the integral of the square of the ground velocity, which, by Parseval9s theorem, is the second moment of the power spectrum. For spectra from broadband seismographs, the determination of J is totally objective; it is an integral for which the integrand approaches zero at both limits. For Brune9s model, v c 3 = qJ /Ω 0 2 with q = 1/(2π 3 ), and equation (a) becomes σ b = h q J Ω 0 ( b ) If the P -wave contribution to the radiated energy as well as the angular dependence of J are neglected, the right-hand side of equation (b) is a constant multiple of the apparent stress. Hence, studies which show that apparent stress determinations provide more robust estimates of the stress drop than σ B determined through equation (a) support evaluating σ B through equation (b).