We used broadband waveforms collected at short hypocentral distances (r ≤ 40 km) during the Umbria-Marche (Italy) seismic sequence of September–November, 1997, in order to calculate the scaling relationships for the ground motion within the meizoseismal area, in the 0.5–16.0 Hz frequency band.

Data were collected by a 10-station portable seismic network deployed by the Istituto Nazionale di Geofisica (Rome, Italy) shortly after the occurrence of the first mainshock of the sequence, on 26 September 1997. Among the thousands of events recorded, we selected 142 earthquakes characterized by good signal-to-noise ratios at all frequencies, and by the absence of multiple shocks within the time window spanned by each recording. The data set of the selected waveforms was made of 2030 horizontal-component seismograms.

The logarithm of the peak values of narrow bandpass-filtered versions of the velocity time histories are modeled at each frequency as  
EXC(f,rref) is the excitation term at an arbitrary reference hypocentral distance, rref; SITE(f) is a site term. The empirical attenuation functional, D(r,rref, f), represents an estimate of the average crustal response for the region, at the hypocentral distance r, at the frequency f. It is modeled by using the following functional form:  
\[D(r,r_{\mathrm{ref}},f)=\mathrm{log}{\ }g(r)-\mathrm{log}{\ }g(r_{\mathrm{ref}})-\frac{{\pi}f(r-r_{\mathrm{ref}})}{{\beta}Q_{0}(f{/}f_{\mathrm{ref}})^{{\eta}}};{\ }(f_{\mathrm{ref}}=1.0{\ }\mathrm{Hz},{\ }r_{\mathrm{ref}}=10{\ }\mathrm{km}).\]
g(r) = r-1 is the body-wave geometrical spreading function; β = 3.5 km/sec is the shear-wave velocity in the crust. Due to the constraints applied to the system prior to the regressions, the excitation term represents the expected peak ground motion at the reference distance, as it would be observed at a site representative of the average site response of the network.
The random vibration theory (RVT) is used to obtain a theoretical prediction of the attenuation functional. For reproducing D(r,rref, f) we use the crustal attenuation parameter  
obtained by Malagnini et al. (2000) from the analysis of a regional data set representative of the entire Apennines, in the (0.24–5.0 Hz) band.
Two parameters are used to predict shapes and levels of the seismic spectra, the stress drop Δσ, and a high-frequency attenuation parameter κ0. The values used to reproduce the observed velocity spectra are  
\[{\Delta}{\sigma}=200{\ }\mathrm{bars};{\ }{\kappa}_{0}=0.04{\ }\mathrm{sec}.\]
The indicated stress drop was estimated in this region by Castro et al. (2000), on recordings of the largest shock of the Umbria-Marche sequence.
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