P. R. Mora

Nonlinear two-dimensional elastic inversion of multioffset seismic data

The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena, such as amplitude‐offset variations and shearwave events, which can only be explained by using the more correct elastic wave equation. Not only are such phenomena ignored by acoustic theory, but they are also treated as undesirable noise when they should be used to provide extra information, such as S‐wave velocity, about the subsurface. The problems of using the conventional acoustic wave equation approach can be eliminated via an elastic approach. In this paper, equations have been derived to perform an inversion for P‐wave velocity, S‐wave velocity, and density as well as the P‐wave impedance, S‐wave impedance, and density. These are better resolved than the Lame parameters. The inversion is based on nonlinear least squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between the observed data and t...

Anisotropic wave propagation through finite-difference grids

An algorithm is presented to solve the elastic-wave equation by replacing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models. The space derivatives are calculated using discrete convolution sums, while the time derivatives are replaced by a truncated Taylor expansion. A centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to the partial derivatives results in a frequency-dependent error in the group and phase velocities of waves. For anisotropic media, the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the Hook sum. This interpolation induces an additional error in the wave properties. The overall error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the degree of anisotropy. The dispersion relation for the homogeneous case was derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and group velocities of the three wave types can be determined in any direction. We demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed.

Inversion = migration + tomography

Seismic inversion, broadly enough defined, is equivalent to doing migration and reflection tomography simultaneously. Diffraction tomography and inversion work best when sources and receivers surround the region of interest, as in medical imaging applications. Theoretical studies have shown that high vertical wavenumber velocity perturbations are resolved by inverting surface seismic reflection data, but the low vertical wavenumbers must be obtained using a separate step, such as velocity analysis or reflection tomography. I propose that a nonlinear iterative inversion that updates a varying background velocity obtains all wavenumbers that are resolvable separately by migration and tomography. The background velocity must contain reflectors to provide data on both upward and downward transmission paths through the earth and hence the low wavenumbers. By considering the downward transmission paths to be between surface sources and buried image geophones and the upward transmission paths to be between surfa...

Simulation of the frictional stick-slip instability

A lattice solid model capable of simulating rock friction, fracture and the associated seismic wave radiation is developed in order to study the origin of the stick-slip instability that is responsible for earthquakes. The model consists of a lattice of interacting particles. In order to study the effect of surface roughness on the frictional behavior of elastic blocks being rubbed past one another, the simplest possible particle interactions were specified corresponding to radially dependent elastic-brittle bonds. The model material can therefore be considered as round elastic grains with negligible friction between their surfaces. Although breaking of the bonds can occur, fracturing energy is not considered. Stick-slip behavior is observed in a numerical experiment involving 2D blocks with rough surfaces being rubbed past one another at a constant rate. Slip is initiated when two interlocking asperities push past one another exciting a slip pulse. The pulse fronts propagate with speeds ranging from the Rayleigh wave speed up to a value between the shear and compressional wave speeds in agreement with field observations and theoretical analyses of mode-II rupture. Slip rates are comparable to seismic rates in the initial part of one slip pulse whose front propagates at the Rayleigh wave speed. However, the slip rate is an order of magnitude higher in the main part of pulses, possibly because of the simplified model description that neglected intrinsic friction and the high rates at which the blocks were driven, or alternatively, uncertainty in slip rates obtained through the inversion of seismograms. Particle trajectories during slip have motions normal to the fault, indicating that the fault surfaces jump apart during the passage of the slip pulse. Normal motion is expected as the asperities on the two surfaces ride over one another. The form of the particle trajectories is similar to those observed in stick-slip experiments involving foam rubber blocks (Bruneet al., 1993). Additional work is required to determine whether the slip pulses relate to the interface waves proposed by Brune and co-workers to explain the heat-flow paradox and whether they are capable of inducing a significant local reduction in the normal stress. It is hoped that the progressive development of the lattice solid model will lead to realistic simulations of earthquake dynamics and ultimately, provide clues as to whether or not earthquakes are predictable.

Numerical simulation of earthquake faults with gouge: Toward a comprehensive explanation for the heat flow paradox

The particle-based lattice solid model is used to simulate transform faults with and without fault gouge. Stick-slip frictional behavior is observed in two-dimensional numerical experiments of model faults both with and without gouge. When no gouge is present, the fault is strong, and the heat generation and stress drops are correspondingly high, in disaccord with observations surrounding the heat flow paradox. In contrast, when a gouge is specified, the fault is weak, and the heat generation as well as stress drops are low, in quantitative agreement with observational constraints. The heat flow is low on average and during short periods of aseismic creep. Seismic efficiencies are compatible with observationally based bounds. Counter intuitively, the fault strength decreases as the intrinsic friction between particles is increased beyond a given threshold. The mechanism for low fault strength and heat is rolling and jostling of fault gouge grains during slip. This allows macroscopic movement of the fault with only minimal slip between surfaces of the gouge grains. As this dynamical mechanism operates during seismic and aseismic slip, it provides an explanation for the lack of a heat flow anomaly in both the seismic and creeping parts of the San Andreas fault. The simulation results provide the first comprehensive and quantitative possible explanation of the heat flow paradox and suggest that fault gouge plays a fundamental role on the dynamics of earthquake faults. Whether rolling and jostling of fault gouge particles provides the explanation for the heat flow paradox in nature remains to be validated by observation evidence.

Effect of rolling on dissipation in fault gouges.

Sliding and rolling are two outstanding deformation modes in granular media. The first one induces frictional dissipation whereas the latter one involves deformation with negligible resistance. Using numerical simulations on two-dimensional shear cells, we investigate the effect of the grain rotation on the energy dissipation and the strength of granular materials under quasistatic shear deformation. Rolling and sliding are quantified in terms of the so-called Cosserat rotations. The observed spontaneous formation of vorticity cells and clusters of rotating bearings may provide an explanation for the long standing heat flow paradox of earthquake dynamics.

A LATTICE SOLID MODEL FOR THE NONLINEAR DYNAMICS OF EARTHQUAKES

A lattice solid model is presented that is capable of simulating the nonlinear dynamical processes (friction and fracture) associated with earthquakes. It is based on molecular dynamics principles to model interacting particles by numerically solving their equations of motion. Particles represent indivisible units of the system such as grains and interactions are described through effective potential functions. In this initial work, particles interact through radial pairwise potentials and the solid is made of particles arranged in a two–dimensional triangular lattice which corresponds to an isotropic elastic medium in the macroscopic limit. Simple and tractable potentials are specified to model brittle and ductile material. Numerical experiments of flawed brittle and ductile blocks subjected to uni–axial compression yield mode II fracturing behavior and characteristic stress–strain curves. In another experiment involving brittle blocks with rough surfaces being dragged past one another, stick–slip frictional behavior is observed. These results suggest that earthquakes can be simulated using the particle based modeling approaches even when the particles and their interactions are highly simplified.

The weakness of earthquake faults

Numerical experiments using: the particle based lattice solid model produce simulated earthquakes. Model faults with a thin gouge layer are sufficiently weak relative to those without gouge to explain the heat flow paradox (HFP). Stress drop statistics are in agreement with field estimates. Models with a thick granular fault zone exhibit a strong evolution effect. Results are initially similar to those of laboratory experiments but after a sufficient time, the system self-organizes into a weak state. The long time :required for self-organization could explain why weak gouge has not been observed in the laboratory. The new results suggest an HFP explanation without the so called fatal flaws of previously proposed solutions. They demonstrate that fault friction potentially undergoes a strong evolution effect and could be dependent on gouge microstructure. This raises questions about the extent to which laboratory derived friction laws can be used in macroscopic domain earthquake simulation studies.

Finite differences on minimal grids

Conventional approximations to space derivatives by finite differences use orthogonal grids. To compute second-order space derivatives in a given direction, two points are used. Thus, 2N points are required in a space of dimension N; however, a centered finite-difference approximation to a second-order derivative may be obtained using only three points in 2-D (the vertices of a triangle), four points in 3-D (the vertices of a tetrahedron), and in general, N + 1 points in a space of dimension N. A grid using N + 1 points to compute derivatives is called minimal. The use of minimal grids does not introduce any complication in programming and suppresses some artifacts of the nonminimal grids. For instance, the well-known decoupling between different subgrids for isotropic elastic media does not happen when using minimal grids because all the components of a given tensor (e.g., displacement or stress) are known at the same points. Some numerical tests in 2-D show that the propagation of waves is as accurate as when performed with conventional grids. Although this method may have less intrinsic anisotropies than the conventional method, no attempt has yet been made to obtain a quantitative estimation.

Evolution of Stress Deficit and Changing Rates of Seismicity in Cellular Automaton Models of Earthquake Faults

Abstract—We investigate the internal dynamics of two cellular automaton models with heterogeneous strength fields and differing nearest neighbour laws. One model is a crack-like automaton, transferring all stress from a rupture zone to the surroundings. The other automaton is a partial stress drop automaton, transferring only a fraction of the stress within a rupture zone to the surroundings. To study evolution of stress, the mean spectral density