Power spectra of random heterogeneities in the solid earth

Abstract

Abstract. Recent seismological observations focusing on the collapse of an impulsive wavelet revealed the existence of small-scale random heterogeneities in the earth medium. The radiative transfer theory (RTT) is often used for the study of the propagation and scattering of wavelet intensities, the mean square amplitude envelopes through random media. For the statistical characterization of the power spectral density function (PSDF) of the random fractional fluctuation of velocity inhomogeneities in a 3-D space, we use an isotropic von Kármán-type function characterized by three parameters: the root mean square (RMS) fractional velocity fluctuation, the characteristic length, and the order of the modified Bessel function of the second kind, which leads to the power-law decay of the PSDF at wavenumbers higher than the corner. We compile reported statistical parameters of the lithosphere and the mantle based on various types of measurements for a wide range of wavenumbers: photo-scan data of rock samples; acoustic well-log data; and envelope analyses of cross-hole experiment seismograms, regional seismograms, and teleseismic waves based on the RTT. Reported exponents of wavenumber are distributed between −3 and −4, where many of them are close to −3. Reported RMS fractional fluctuations are on the order of 0.01–0.1 in the crust and the upper mantle. Reported characteristic lengths distribute very widely; however, each one seems to be restricted by the dimension of the measurement system or the sample length. In order to grasp the spectral characteristics, eliminating strong heterogeneity data and the lower mantle data, we have plotted all the reported PSDFs of the crust and the upper mantle against wavenumber for a wide range (10−3–108 km−1). We find that the spectral envelope of those PSDFs is well approximated by the inverse cube of wavenumber. It suggests that the earth-medium randomness has a broad spectrum. In theory, we need to re-examine the applicable range of the Born approximation in the RTT when the wavenumber of a wavelet is much higher than the corner. In observation, we will have to carefully measure the PSDF on both sides of the corner. We may consider the obtained power-law decay spectral envelope as a reference for studying the regional differences. It is interesting to study what kinds of geophysical processes created the observed power-law spectral envelope at different scales and in different geological environments in the solid earth medium.