A Model for the Shape of the Fourier Amplitude Spectrum of Acceleration at High Frequencies By

At high frequencies f the spectrum of S-wave accelerations is characterized by a trend of exponential decay, e -~". In our study, the spectral decay parameter shows little variation at a single station for multiple earthquakes at the same distances, but it increases gradually as the epicentral distance increases. For multiple recordings of the San Femando earthquake, x increases slowly with distance, and x is systematically smaller for sites on rock than for sites on alluvium. Under the assumption that the Fourier spectrum of acceleration at the source is constant above the comer frequency (an , 2 source model), the exponential decay is consistent with an attenuation model in which Q increases rapidly with depth in the shallow crustal layers. INTRODUCTION The shape and amplitude of the Fourier amplitude spectrum of strong ground acceleration is recognized as useful for various applications to earthquake engineering (McGuire, 1978). This acceleration spectrum also contains fundamental information about physical processes at the earthquake source and wave propagation in the crust of the earth. Yet at high frequencies, we still do not have a satisfactory model for the shape of the acceleration spectrum. By the shape of a spectrum we refer to a smooth trend through the spectrum; the fine structure which is superimposed on this trend is not meant to be included. At low frequencies and sufficiently far from the fault, the inevitable result of an elastic rebound source model is that the acceleration spectrum increases as w2, where ~ = 2~[ and [ is the frequency of ground motion. For example, a widely employed model by Brune (1970) relates the coefficient of this w2 trend to the seismic moment, Mo, and relates the corner frequency ([0) where this w2 trend terminates to a stress drop parameter at the source. Above the corner frequency, Trifunac (1976) and McGuire (1978) have carried out empirical regressions for the shape of the acceleration spectrum but these regressions do not yield much insight into the physical processes which are involved. Hanks (1979, 1982) suggests that, in general, the acceleration spectrum is fiat above the corner frequency to a second corner frequency (/max) above which the spectrum decays rapidly. In the next section, we propose a parametric shape for the acceleration spectrum at high frequency. Our model is characterized by one parameter, which we designate as the spectral decay parameter K. Recognition and study of this parameter were motivated in part by the observations that most spectra observed in the 1981 Santa Barbara Island earthquake appear to fall off exponentially (Anderson, 1984). Subsequent sections explore the systematic behavior of K for the S-wave portion of the accelerogram. We also recognize a plausible attenuation model to explain the observations but intentionally avoid introducing the terminology and notation of that model into the observation sections of this paper. SPECTRAL SHAPE AT HIGH FREQUENCIES Figure 1 shows the Fourier amplitude spectrum of acceleration for the $16°E component of the Pacoima Dam accelerogram from the 1971 San Fernando, Cali1969 1970 JOHN G. ANDERSON AND SUSAN E. HOUGH fornia, earthquake. Figure 1A shows the spectrum plotted on log-log axes. Based on a figure of this type, Hanks (1982, Figure 2) selects/max for this record to be near 10 Hz. In Figure 1B, the frequency axis is linear. On these axes, the dominant trend is a linear decrease of the log of spectral amplitude with frequency, and there is no apparent additional slope break in the vicinity of 10 Hz. In some cases, the dominant trend of exponential decay is initiated near/o, but on other spectra it begins at some higher frequency. It is, therefore, useful to label the frequency above which the spectral shape is indistinguishable from exponential decay. Here we call this frequency rE. We do not ascribe any fundamental importance to rE, and pay little attention to it in the rest of this paper. Considering the amplitude of the fine n~ I-0 m n (,0 h 0 (b 0