Fractal Dimension of the 1999 Chamoli Earthquake from Aftershock Studies in Garhwal Himalaya

The Aftershock sequence of Chamoli earthquake (M _w 6.4) of 29 March 1999 is analyzed to study the fractal structure in space, time and magnitude distribution. The b value is found to be 0.63 less than which is usually observed worldwide and in the Himalayas. This indicates that the numbers of smaller earthquakes are relatively less than the larger ones. The spatial correlation is 1.64, indicating that events are approaching a two-dimensional region meaning that the aftershocks are uniformly distributed along the trend of the aftershock zone. Temporal correlation is 0.86 for aftershocks of M 1, indicating a nearly continuous aftershock activity. However, it is 0.5 for aftershocks of M 1.75, indicating a non continuous aftershock activity. From the assessment of slip on different faults it is inferred that 70% displacement is accommodated on the primary fault and the remainder on secondary faults.     

Curie-temperature depth estimation using a self-similar magnetization model

The Earth's crust is magnetized down to the Curie-temperature depth at about 10 to 50 km. This limited depth extent of the crustal magnetization is discernible in the power spectra of magnetic maps of South Africa and Central Asia. At short wavelengths, the power increases as rapidly towards longer wavelengths as expected for a self-similar magnetized crust with unlimited depth extent. Above wavelengths of about 100 km the power starts increasing less rapidly, indicating the absence of deep-seated sources. To quantify this effect we derive the theoretical power spectrum due to a slab carved out of a self-similar magnetization distribution. This model power spectrum matches the power spectra of South Africa and Central Asia for a self-similarity parameter of β = 4 and Curie temperature depths of 15 to 20 km.     

Broad-band power-law spectra of well-log data in Japan

For the purpose of revealing the statistical characteristics of P -wave velocity, S -wave velocity and density in the uppermost part of the crust, we analysed well-log data obtained from five deep wells in different tectonic regions in Japan: three wells through the mainly sedimentary rocks in the Kanto plain and two wells in the Kuju volcano group in Kyushu Island. In the Kanto plain, the power spectral density of fractional fluctuation of P -wave velocity and that of density are proportional to a power of the spatial wavelength from a few metres to 100 m. where the power index (slope of the power spectral density at double logarithmic scale) is 1.1-1.3. At the Kuju volcano group, that of P - and S -wave velocity and density also obey a power law, with a power index of 1.3-1.6 for wavelengths from a few metres to few hundred metres. Correcting the effect of the moving box-car observation window which corresponds to the separation of two receivers of the logging tool, we find that the power-law characteristics hold for wavelengths down to a few tens of centimetres. The 1-D sections of the elastic inhomogeneities follow a kind of band-limited self-affine random process. Comparing the power spectral densities, we find smaller values of the power index in stable areas and larger values in tectonically active areas. The difference in the power index arises from long-wavelength components.     

Simulated envelopes of non-isotropically scattered body waves as compared to observed ones: another manifestation of fractal heterogeneity

Envelopes of scalar waves are simulated at various distances from an instant point source embedded in a random uniformly scattering medium by means of direct Monte-Carlo modelling of wave-energy transport. Three types of scattering radiation pattern ('indicatrix') are studied, for media specified by (1) a Gaussian autocorrelation function of inhomogeneities, (2) a power-law ('fractal', k -^α) inhomogeneity spectrum and (3) the mix of case (1) and the isotropic indicatrix (very small + large inhomogeneities). We look for a model that can qualitatively reproduce the two most characteristic features of real S-wave envelopes of near earthquakes, namely (1) the broadening of the 'direct' wave group with distance and (2) the monotonously decaying shape of the coda envelope that does not deviate strongly from that expected in the isotropic scattering case. Both properties are observed for any band over a wide frequency range (1-40 Hz). The well-studied isotropic scattering model realistically predicts the appearance of codas but fails to predict pulse broadening. The model of large-scale inhomogeneity realistically predicts the mode of pulse broadening but fails to predict codas. We have found that, for a particular frequency band, within each class of inhomogeneity studied, both requirements can be qualitatively satisfied by a certain choice of parameters. In the Gaussian-ACF case, however, this match can be obtained only for a narrow frequency range. In contrast, the fractal case (with a value of exponent a of about 3.5-4) reproduces qualitatively the observed wide-band behaviour, and we consider it a reasonable representation of the gross properties of the earth medium.