We address the problem of modeling dynamic rupture on multiscale heterogeneous faults in 3D. Under the assumption of slip-weakening friction, we numerically construct effective friction laws that integrate the effects of small-scale heterogeneity during the rupture. This homogenization process is based on the description of the initial phase of the rupture by the dominant unstable spectral mode. Its dynamics is influenced by the geometry of the fault, the static friction heterogeneities and the friction law. We first define a periodic small-scale heterogeneous model, introducing heterogeneity in the distribution of the static friction coefficient. We then describe a method for constructing this effective friction law. Applying this new law homogeneously on the fault permits to reproduce the dynamic evolution of the heterogeneous fault. Furthermore, we show that the effective friction law can be used to replace small-scale heterogeneities in two-scale heterogeneous models, while preserving their effects. We study three kinds of two-scale models, with growing complexity: first periodic at both scales, then periodic only at small scale, and finally irregular at both scales. This homogenization method can be adapted to the case where the heterogeneity is introduced in the initial stress rather than in the static friction value. Finally, we show in a simple example that the effective friction law permits to reproduce the transition between subshear and supershear rupture propagation, originally produced by heterogeneities on the fault.

Conditions under which a single oscillator model coupled with Dieterich-Ruina’s rate and state dependent friction exhibits chaotic dynamics is studied. Properties of spring-block models are discussed. The parameter values of the system are explored and the corresponding numerical solutions presented. Bifurcation analysis is performed to determine the bifurcations and stability of stationary solutions and we find that the system undergoes a Hopf bifurcation to a periodic orbit. This periodic orbit then undergoes a period doubling cascade into a strange attractor, recognized as broadband noise in the power spectrum. The implications for earthquakes are discussed.

The Campus Earthquake Program (CEP) of the University of California (UC) started in March 1996, and involved a partnership among seven campuses of the UC-Berkeley, Davis, Los Angeles, Riverside, San Diego, Santa Barbara, Santa Cruz-and the Lawrence Livermore National Laboratory (LLNL). The aim of the CEP was to provide University campuses with site-specific assessments of their earthquake strong motion exposure, to complement estimates they obtain from consultants according to the state-of-the-practice (SOP), i.e. Building Codes (UBC 97, IBC 2000), and Probabilistic Seismic Hazard Analysis (PSHA). The Building Codes are highly simplified tools, while the more sophisticated PSHA is still somewhat generic in its approach because it usually draws from many earthquakes not necessarily related to the faults threatening the site under study.Between 1996 and 2001, the site-specific studies focused on three campuses: Riverside, San Diego, and Santa Barbara. Each campus selected 1-3 sites to demonstrate the methods and procedures used by the CEP: Rivera Library and Parking Lots (PL) 13 and 16 at UCR, Thornton Hospital, the Cancer Center, and PL 601 at UCSD, and Engineering I building at UCSB. The project provided an estimate of strong ground motions at each selected site, for selected earthquake scenarios. These estimates were obtained by using an integrated geological, seismological, geophysical, and geotechnical approach, that brings together the capabilities of campus and laboratory personnel. Most of the site-specific results are also applicable to risk evaluation of other sites on the respective campuses.The CEP studies have provided a critical assessment of whether existing campus seismic design bases are appropriate. Generally speaking, the current assumptions are not acknowledging the severity of the majority of expected motions. Eventually, both the results from the SOP and from the CEP should be analyzed, to arrive at decisions concerning the design-basis for buildings on UC campuses. Published by Elsevier Ltd.

To estimate the broadband strong ground motion one might expect at a given site we develop a method that includes heterogeneous slip on a finite-fault, full wave propagation with high frequencies, and site-specific material properties with nonlinear soil response. The faulting is simulated as a stochastic process with the spatial variation of the key parameters determined by probability distribution functions. The wave propagation from source to site is accounted for by using small earthquake recordings as empirical Green’s functions (EGF). This accounts for the regional effects of scattering, attenuation and structure while providing the basis for a broadband (0.5–10Hz) time history. Because we are interested in sites where the ground motion is expected to be severe, we have included nonlinear wave propagation through the soil. The material properties of the soil column have been determined from laboratory tests, borehole logs and confirmed through seismological modeling of weak motion. We have computed 240 three-component acceleration time histories to represent the range of ground motion one might expect from a M 6.8 earthquake for a site that is located 10km above the hanging wall of blind thrust (7.1km closest distance). Based on the suite of time histories we computed a S.D. of 0.45 (natural log units) for the acceleration response spectra in the passband 0.5–10Hz. The total S.D. (modeling plus parameterization) is 0.6 in natural log units. The mean acceleration response spectrum is near the median 10% in 50-year probabilistic seismic hazard analysis (PSHA) spectrum for the site; the 84% spectrum of the simulations is closer to the 5% in 50 years median spectrum.

The land surface elevation distribution will serve as fundamental input data to any wetland flow model. As an alternative to the traditional smooth function approach to represent or interpolate elevation data, we explore the use of Levy monofractals and universal multifractals as a means for defining a statistically equivalent topography. The motivation behind this effort is that fractals, like natural topography, are irregular, they offer a way to relate elevation variations measured at different scales, and the relationships are of a statistical nature. The study site was a riparian wetland near Savannah, GA, that contained beavers, and a total of four elevation transects were examined. The elevation increments showed definite non-Gaussian behavior, with parameter values, such as the Hurst coefficient and Lévy index (α), depending on the question of presence of beaver activity. It was obvious that the data were highly irregular, especially the transects influenced by beavers. Significantly different α values were obtained depending on whether the entire data set or just the tails were examined, which demonstrated inability of the monofractal model to reflect fully the irregularity of wetland data. Further analysis confirmed definite multifractal scaling, and it is concluded that the multifractal model is superior for this data set. Universal multifractal parameters are calculated and compared to those obtained previously for more typical terrain. Although it is difficult to consider a unique universal multifractal parameter α for the entire wetland, multifractal-like scaling was evident in each transect as reflected by the nonlinear behaviors of the scaling functions. We demonstrate a good agreement between theory and measurements up to a critical order of statistical moments, q D , close to 3.5 and obtain realistic unconditioned simulations of multifractal wetland topography based on our parameter estimates. Future work should be devoted to conditioning multifractal realizations to data and to obtaining larger data sets so that the question of anisotropy may be studied.

Abstract. Over wide ranges of scale, orographic processes have no obvious scale; this has provided the justification for both deterministic and monofractal scaling models of the earth's topography. These models predict that differences in altitude (Δh) vary with horizontal separation (l) as Δh ≈ lH. The scaling exponent has been estimated theoretically and empirically to have the value H=1/2. Scale invariant nonlinear processes are now known to generally give rise to multifractals and we have recently empirically shown that topography is indeed a special kind of theoretically predicted "universal" multifractal. In this paper we provide a multifractal generalization of the l1/2 law, and propose two distinct multifractal models, each leading via dimensional arguments to the exponent 1/2. The first, for ocean bathymetry assumes that the orographic dynamics are dominated by heat fluxes from the earth's mantle, whereas the second - for continental topography - is based on tectonic movement and gravity. We test these ideas empirically on digital elevation models of Deadman's Butte, Wyoming.

Over wide ranges of scale, orographic processes have no obvious scale; this has provided the justification for both deterministic and monofractal scaling models of the earth's topography. These models predict that differences in altitude (Δh) vary with horizontal separation (l) as Δh ≈ lH. The scaling exponent has been estimated theoretically and empirically to have the value H=1/2. Scale invariant nonlinear processes are now known to generally give rise to multifractals and we have recently empirically shown that topography is indeed a special kind of theoretically predicted "universal" multifractal. In this paper we provide a multifractal generalization of the l1/2 law, and propose two distinct multifractal models, each leading via dimensional arguments to the exponent 1/2. The first, for ocean bathymetry assumes that the orographic dynamics are dominated by heat fluxes from the earth's mantle, whereas the second - for continental topography - is based on tectonic movement and gravity. We test these ideas empirically on digital elevation models of Deadman's Butte, Wyoming.

For some time, ocean wave breaking has been conceptualized as a cascade process in which the large scale wind energy flux driving the system is dissipated by wave breaking at small scales, the two seperated by the “equilibrium” scaling range. Cascades are now known to generically lead to multifractals; with special “universal” multifractals theoretically predicted. In this paper we use far red (0.95 μm) radiances at 1m resolution obtained from aircraft to test the multifractal behavior of the ocean surface and estimate the corresponding universal multifractal parameters of the radiance field.

The unified scaling model of the atmosphere links the large ans small scale dynamics by a single scaling but anisotropic regime, rather than distinct isotropic two- and three-dimensional turbulent regimes as posited in the standard model. We argue that the study of mesoscale clouds is a particularly stringent test for the standard model and we present (perhaps the first) systematic analysis of the scaling of energy spectra of satellite radiances over five wavelength channels and spanning the range of scales from 160 m to 4000 km (the entire mesoscale). The study mostly involved 15 consecutive scenes of AVHRR data (1.1 km resolution, 512×512 pixels) taken over the same location at the same local time in February 1986

The unified scaling model of the atmosphere links the large and small scale dynamics by a single scaling but anisotropic regime, rather than distinct isotropic two- and three-dimensional turbulent regimes as posited in the standard model. We argue that the study of mesoscale clouds is a particularly stringent test for the standard model and we present (perhaps the first) systematic analysis of the scaling of energy spectra of satellite radiances over five wavelength channels and spanning the range of scales from 160 m to 4000 km (the entire mesoscale). The study mostly involved 15 consecutive scenes of AVHRR data (1.1 km resolution, 512 x 512 pixels) taken over the same location at the same local time in February 1986. This data was chosen because it was expected to provide a very sensitive indicator of the mesoscale break in the scaling predicted by the standard model of atmospheric dynamics (the "mesoscale gap"). Over the entire range, with surprisingly little scene-to-scene variation, the (isotropic) energy spectrum (E (k)) was found to follow the scaling form E (k) almost-equal-to k(-beta) where k is a wavenumber, and beta is the spectral exponent. This type of behaviour is exactly as predicted by the unified scaling model of the atmosphere as the outcome of anisotropic nonlinear cascade dynamics. It is hard to see how these results can be reconciled with the standard model.

We report the first direct empirical estimates of the Levy indices (alpha) characterizing the intermittency of turbulent velocity and temperature fields in the framework of universal multifractals. Using the double trace moment analysis technique, we find alpha(upsilon) almost-equal-to 1.3 +/- 0.1, alpha(GAMMA) almost-equal-to 1.2 +/- 0.1, which shows that turbulence has behavior in between that of lognormal (alpha = 2) multifractal and beta model (alpha = 0) monofractal. It is an unconditionally hard multifractal process (since alpha > 1). These results permit a new estimation for wind turbulence of the intermittency parameter mu(almost-equal-to 0.35 +/- 0.1).

It is now apparent that the two principal models of turbulence (the "beta" and "lognormal" models) are the extremes of a continuous family of (stable, attractive, hence "universal") multifractals characterized by Levy indices alpha = 0 and 2, respectively. Using a technique called double trace moment analysis, and turbulent velocity data, we empirically obtain alpha almost-equal-to 1.3 +/- 0.1: As has long been suspected, turbulence really is "in between" the beta and lognormal models. This describes the entire hierarchy of singularities of the Navier-Stokes equations.

In the 1970's it was found that; for low frequencies (<10 Hz), speech is scaling: it has no characteristic time scale. Now such scale invariance is associated with multiscaling statistics, and multifractal structures. Just as Gaussian noises frequently arise because they are generically produced by sums of many independent noise processes, scaling noises have an analogous universal behavior arising from nonlinear mixing of processes. We show that low frequency speech is consistent with these ideas, and use the measured parameters to produce stochastic speech simulations which are strikingly similar to real speech.

Our group has been very active over the last year, analyzing a number of data sets to characterize multifractal cloud properties and assess the effects of clouds on surface radiation properties (spectral and broadband). The data sets analyzed include: AVHRR observations of clouds over the ocean, SPOT observations of clouds over the ocean, SSM/I observations of clouds over the ocean, pyranometer data with all-sky photographs, pyrgeometer data all-sky photographs, and spectral surface irradiance all-sky photographs. A number of radiative transfer computations have been performed to help in the interpretation of these observations or provide theoretical guidance for their analysis. Finally 4 number of radiative transfer models have been acquired and tested to prepare for the interpretation of ARM/CART data.