A Bayesian Nonlinear Inversion of Seismic Body-Wave Attenuation Factors

k is a well-known fact that the uncertainties in measuring relative attenuation factors within a local or regional seismic network are usually high, due to noise of different kinds and unrealistic assumptions. Numerical experiments using nine synthetic seismograms, created using t* values ranging from 0.1 to 0.9 sec, reveal that the commonly used spectral ratio method is strongly affected by the selection of data processing parameters such as width of the spectral smoothing window, reference station, and so on. The numerical experiments demonstrate that a Bayesian nonlinear inversion approach that directly matches the spectra is better at finding the correct parameters used to generate the synthetic seismograms. The Bayesian inversion approach uses a priori information to simultaneously search for the t* values, the common spectrum for all the records from an event, and the near-receiver amplification factors by using all the recordings from an event. When z, the ratio of Gaussian noise to signal, _-__ 0.1, the spectral ratio and Bayesian methods yield similar results with mean t* measurement errors <0.05 sec. For 0.1 < z 0.8, the mean errors of the spectral ratio method are larger than 0.1 sec and in some cases as large as 0.6 sec, while those of the Bayesian method are less than 0.08 sec. Frequency-independent t* and near-receiver amplification factors are assumed. A multi-step procedure is proposed to reject records with a large misfit. Introduction Measurements of seismic wave attenuation factors, quantified by t* or travel time over Q, provide important information about the physical state of the earth (e.g., Knopoff, 1964; Anderson, 1967; Solomon and Toks6z, 1970; Jackson and Anderson, 1970; Der et al., 1975; Taylor et aL, 1986). However, the uncertainties of measurements are usually high, due to different kinds of noise and unrealistic assumptions. An important criterion for any practical method is the stability of the measurements with regard to data processing parameters. In this study, synthetic data are used to compare two methods for finding the t* values used in creating the data. It was found that a Bayesian nonlinear inversion method is better than the popular spectral ratio method. The amplitude spectrum of event k recorded by station i, Aik(f) can be written as (e.g., Teng, 1968) Aik( f ) = Sk(f)Gik(f )Rik(f)Ii(f), (1) where Sk(f) is the spectrum of source waveform S(t), Gikff) is that of a Green's function G(t), Rik(f) is that of the nearreceiver effects, and Ii(f) is that of the instrument response. The spectrum of the Green's function can be written as iPresent address: Department of Terrestrial Magnetism, Carnegie Institution of Washington. Gik(f) = exp[-nfl/~(D], (2) where t* is the attenuation factor that is defined as the ratio of travel time to the effective mean quality factor Q; that is, fp ds t q) = .tu Q(s, f ) V(s, f ) (3) where V(s, f ) is the instantaneous velocity of body waves. Both V and Q are in general a function of location as well as frequency. The main causes of high measurement uncertainties include the following: (1) noise caused by natural and cultural activities; (2) scattering caused by velocity fluctuations (Richards and Menke, 1983); (3)Near-receiver topographic effects (e.g., Vidale et al., 1991; Frankel and Leith, 1992); (4) focusing/defocusing of energy from sedimentary lens structures (e.g., Gao et aL, 1996); (5) frequency dependence of t~(f) when a frequency-independent approach such as the spectral ratio method is used (e.g., Anderson and Given, 1982); (6) uncertainties in the determination of source-related effects such as focal mechanisms and radiation patterns; and (7) interference from other arrivals that can be avoided by using phases that are well separated from other possible arrivals, and by careful selection of time windows for the computation. 961