Attenuation Measurements of Split Normal Modes for the 1960 Chilean and 1964 Alaskan Earthquakes by Seth Stein and Robert

Measurements of attenuation for the Earth's longest period modes can be significantly biased by the effects of frequency splitting. Using our previously developed methods of time domain synthesis of split normal modes, we measure Q without such a bias. We also conduct numerical experiments to confirm the errors in Q measurements which result from neglecting the effects of splitting. In contrast to frequency domain this time domain technique allows us to reject data below the ambient noise level for each mode. The Q's of the longest period spheroidal (oS2-oSs) and torsional (o7"3-o7"4) modes are determined using long (500 hr) records from the Chilean and Alaskan earthquakes, INTRODUCTION Measurements of attenuation were among the earliest analyses conducted on the normal mode data for the 1960 Chilean earthquake. The primary focus of interest was on the longest period modes with periods greater than 20 min. Benioff et al. (1961) used spectral widths to measure the Q of 0S3 from the Isabella record. Alsop et al. (1961a) used a time decay method to find the Q's of the low-order spheroidal modes, and Ness et al. (1961) estimated the Q of oSo. Similar analyses were conducted on the data from the 1964 Alaskan earthquake. Smith (1972) summarized all longperiod Q results to that date, including those of Slichter (1967). Additional studies of long-period Q have recently been conducted by Sailor and Dziewonski (1978) and by Buland and Gilbert (1978). This renewed interest is motivated by the need to correct earth models for the effects of physical dispersion caused by anelasticity (Akopyan et al., 1975, 1976; Liu et al., 1976), and a desire to derive models of the Q structure of the Earth. Attenuation measurements for the longest period modes are complicated by frequency splitting (Ness et al., 1961; Benioff et al., 1961). Pekeris et al. (1961) and Backus and Gilbert (1961) showed that the splitting could be explained by the Earth's rotation. The 21 + 1 singlets in the multiplet of angular order I are split such that each one has a distinct eigenfrequency, amplitude, and phase. The individual singlets are also broadened by attenuation. Thus the splitting is observable only for low-angular order (lo,~g-period) multiplets for which the frequency separation due to splitting of the singlets is greater than the spectral line broadening due to attenuation. The combined effects of splitting and peak broadening due to attenuation pose substantial difficulty for Q measurements. For a single damped harmonic oscillator, attenuation can be easily measured in either the time or the frequency domain. In the time domain, the amplitude is given by A ( t) = e t[-w/2Q+i~] and Q can be found from the ratio of the amplitudes at two different times. Equivalently, Q can be determined from the width of the spectral peak by using