Charles A. Williams

The effects of topography on magma chamber deformation models : Application to Mt. Etna and radar interferometry

We have used a three-dimensional elastic finite element model to examine the effects of topography on the surface deformation predicted by models of magma chamber deflation. We used the topography of Mt. Etna to control the geometry of our model, and compared the finite element results to those predicted by an analytical solution for a pressurized sphere in an elastic half-space. Topography has a significant effect on the predicted surface deformation for both displacement profiles and synthetic interferograms. Not only are the predicted displacement magnitudes significantly different, but also the map-view patterns of displacement. It is possible to match the predicted displacement magnitudes fairly well by adjusting the elevation of a reference surface; however, the horizontal pattern of deformation is still significantly different. Thus, inversions based on constant-elevation reference surfaces may not properly estimate the horizontal position of a magma chamber. We have investigated an approach where the elevation of the reference surface varies for each computation point, corresponding to topography. For vertical displacements and tilts this method provides a good fit to the finite element results, and thus may form the basis for an inversion scheme. For radial displacements, a constant reference elevation provides a better fit to the numerical results.

An accurate and efficient method for including the effects of topography in three‐dimensional elastic models of ground deformation with applications to radar interferometry

Topography has a large effect on the results predicted by elastic surface deformation models in regions of significant relief. In some cases, topography may have more influence on the predicted deformation field than do model source parameters. We have developed an approximate analytical technique for including topographic effects that retains most of the computational simplicity of elastic half-space models while providing an accurate representation of topographic effects. We use a series expansion of the elastic half-space solution with a small slope approximation, yielding a set of higher-order corrections. The integrated effect of these corrections is evaluated using Fourier methods. We investigate the effectiveness of our method by comparing predicted results for a tilted triaxial ellipsoid with those predicted by finite element models. The resulting displacements and displacement gradients are in good agreement with finite element results both for relatively smooth topography (synthetically generated) and for the greater relief in the vicinity of Mount Etna volcano. We then compare the results of our method with those predicted by traditional elastic half-space models using different reference elevations, and with a previously proposed method of estimating topographic effects. Our new method provides a significantly better fit to the finite element results than do the other methods. Our method is able to accurately portray both the number and the horizontal pattern of fringes in a synthetic interferogram when compared with finite element results. In particular, the method accurately represents the broadening of fringes that is observed in regions of high relief, as well as reproducing the location of the fringe center and the topographically induced deviation of the fringes from a regular pattern. Elastic half-space models typically show a fringe center that is displaced with respect to the finite element results, indicating that parameter inversions based on such a model would provide incorrect estimates of the horizontal source position. The lack of topographically generated fringe distortions in elastic half-space results would also likely lead to inaccuracies in the predicted magnitude and orientation of proposed deformation sources. Our new technique over-comes these limitations and should be useful when evaluating either traditional geodetic results or the greater areal coverage provided by interferometric observations.

A RHEOLOGICALLY LAYERED THREE-DIMENSIONAL MODEL OF THE SAN ANDREAS FAULT IN CENTRAL AND SOUTHERN CALIFORNIA

Three-dimensional kinematic finite element models of the San Andreas fault in central and southern California have been used to estimate the effects of rheological parameters and fault slip distribution on the horizontal and vertical deformation in the vicinity of the fault. The models include the effects of vertically layered power law viscoelastic rheology, and isostatic forces are considered in calculations of vertical uplift. Several different rheological layering schemes are used, using laboratory results on rock rheology to define the properties of the various layers. The depth to which the fault remains locked between earthquakes (D) is held constant at 20 km for the entire locked portion of the fault between Cholame and the Salton Sea. Between Hollister and Cholame the entire fault is assumed to slip at a rate consistent with a relative plate velocity of 35 mm/yr along a direction striking N41°W. Steady aseismic slip corresponding to plate velocity is imposed below the fault locking depth to a depth H on the locked section of the fault. The depth to which aseismic slip occurs (H) is assigned a value of either 20 km or 40 km, resulting in two versions of each rheological model. Variations in the model parameters are found to produce distinctive deformation patterns, providing a means for differentiating between models. Specifically, lower effective viscosities near the surface result in increased strain rates and uplift rates at all times during the earthquake cycle. Lower effective viscosities also produce subsidence near the creeping portion of the fault. Models that do not include aseismic slip below the fault locking depth (H = 20 km) display greater time dependence in both horizontal and vertical deformation than those including aseismic slip below the locking depth (H = 40 km). These differences are due, in part, to the time-invariant nature of the imposed slip condition. The differences are more pronounced as the effective viscosity close to the surface is increased. The vertical uplift rate is particularly sensitive to the depth of aseismic slip (H) at the two bends in the fault, especially for models with high effective viscosities below the surface. For models in which the effective viscosity near the surface is relatively low, measurements of total uplift at the two bends in the fault could provide sufficient resolution to distinguish between models with and without aseismic slip over time periods of 10 to 20 years or more with current abilities to measure vertical uplift. Among our San Andreas fault models, the one most consistent with current strain rate data includes aseismic slip between 20 and 40 km (H = 40 km) and uses assumed rheological properties from the surface to 100 km depth consistent with laboratory results for wet rock samples. The rheological parameters for this model are based on laboratory results for the following rock types wet granite in the upper crust (0 to 20 km), wet diabase in the lower crust (20 to 40 km), wet dunite in the upper mantle (40 to 100 km), and dry olivine below 100 km. These modeling results are preliminary, however, and several additional factors should be considered prior to constructing a comprehensive model. Furthermore, it should be emphasized that the present models represent a small subset of possible rheological models, and numerous other models may provide similar or better fits to the data. The field of possible models will continue to narrow with further knowledge of the variations in Earth composition and temperature with depth, with more information on rock rheology, and with further observations of the earthquake cycle.

Reassessment of pore shapes in microstructurally equilibrated rocks, with implications for permeability of the upper mantle

[1]xa0With the goal of constraining fluid and melt transport rates in the upper mantle and lower crust, various permeability–porosity relations have been proposed for deep-seated, microstructurally equilibrated rocks. Two relations that are in close agreement include one that is based on numerical modeling for an idealized rock and another that is based on direct measurement of permeability on synthetic quartzite analogs. Each relation describes a continuous increase in permeability as a function of porosity, raised to some power. The empirical relation is displaced to lower permeabilities, reflecting the more complex (nonuniform) pore geometry of the quartzites, which resembles that expected in natural, deep-seated rocks. Despite differences, the numerical models assume, and the quartzites display, a similar pore geometry that is consistent with thermodynamic equilibria: grain boundaries are “dry,” with pores largely confined to joins of three or more grains. In contrast, a fundamentally different pore microstructure, in which most of the melt occupies grain boundaries, has been proposed for partially molten dunite. This has led to the suggestion of a unique, noncontinuous, or “threshold” permeability relation for upper mantle melts. A more critical analysis of pore microstructure in dunite indicates, however, that pore shapes had been misinterpreted: very little melt is present along grain boundaries. Melt instead occupies a triple-junction network closely resembling that of the quartzite analog. Consequently, a relation similar to that for synthetic quartzites can describe upper mantle grain-scale permeabilities, because permeability is a function of pore geometry alone.

Effect of the brittle‐ductile transition on the topography of compressive mountain belts on Earth and Venus

The Coulomb critical taper model has been very successful in explaining the large-scale topography of a number of terrestrial accretionary wedges; however, this model is limited to cases of purely brittle-frictional deformation. In this paper we extend the range of applicability of the critical taper model by explicitly including the effects of temperature-dependent ductile deformation. The new model includes temperature-dependent power law flow, an assumed velocity field, and linear thermal gradients in the atmosphere and within the crust. Flexural isostasy is also incorporated so that the decollement geometry is computed as a response to the applied load of the wedge material. We assume that ductile deformation within the decollement zone is controlled primarily by diffusion flow, whereas ductile deformation within the wedge itself is controlled by dislocation creep. The topographic profiles predicted by the model are very similar to those of a number of fold-and-thrust belts on both Earth and Venus. A typical wedge profile includes three distinctive topographic regions: a narrow taper toe, where both the wedge and the decollement zone deform in a brittle-frictional manner; a region of relatively steep slope, where the wedge base deforms ductilely and the decollement zone is still frictional; and a flat plateau region, where both the wedge base and the decollement zone are deforming by ductile flow. We have applied the model to two fold-and-thrust belts on Venus (Maxwell Montes and Uorsar Rupes) and to the Andes on Earth, and we find good agreement between observed and predicted topography using reasonable parameter values. The model accounts for the observed positive correlation between relief and elevation of Venusian fold-and-thrust belts on the basis of different thermal environments at different elevations. It is also able to explain the first-order differences between terrestrial and Venusian fold-and-thrust belts; fundamentally, this difference is due to a combination of the lower temperatures and the presence of water on Earth.

Static stress changes induced by the magmatic intrusions during the 2002–2003 Etna eruption

[1]xa0The shallow intrusive processes that occurred during 2002–2003 Etna eruption, as well as the complex interaction between the magma intrusive events and the tectonic response of the volcanos eastern flank, are investigated with numerical deformation modeling and the estimation of changes in the static Coulomb stress. Ground deformation and volcanologic evidence clearly indicate a composite mechanism of intrusion on both the southern and northeastern flanks of the volcano. Geodetic data inversions have been based on a homogeneous elastic half-space model, although geological data and seismic tomography indicate that Mt. Etna is elastically inhomogeneous and that rigidity layering and heterogeneities are likely to affect the magnitude and pattern of the deformation field. To account for topographic effects, as well as a complicated distribution of material properties, we use the finite element method (FEM) to provide a more realistic model. The presence of medium heterogeneity strongly affects the amplitudes of the static stress changes. Seismicity matches well the areas of positive increase in the static stress caused by the intrusive events along the southern and northeastern flanks. The changes in the state of stress generated by the southern dike produce an extensional stress field that favors magma propagation along the north-east Rift. The highest seismic releases were associated with the activation of two fault systems, the Timpe Fault System and the Pernicana Fault. The static stress changes resolved onto these faults indicate that the magma intrusions on the southern and northeastern flanks encouraged these seismogenic structures to slip.

Post-emplacement lava subsidence and the accuracy of ERS InSAR digital elevation models of volcanoes

Repeat-pass synthetic aperture radar interferometry (InSAR) using data acquired by the ERS platforms is an attractive method for acquiring topographic data of volcanoes. Caution is advised, however, when using this technique in regions covered by young, thick lava flows. In this study, the magnitude of post-emplacement subsidence associated with the 1991-93 lava flow at Mount Etna, Sicily, was measured using differential radar interferometric techniques, and it was found that the rates of subsidence are large enough to contribute a significant component to the measured phase shift, even in ERS data acquired on consecutive orbits. It demonstrates the detrimental effect that such phase shifts have on the accuracy of digital elevation models derived by repeat-pass radar interferometry.

Stress rates in the central Cascadia subduction zone inferred from an elastic plate model

GPS vectors and surface tilt and uplift rates from northwestern Oregon and southwestern Washington are inverted to estimate rates of stress changes along the Cascadia thrust fault and base of the overriding plate using a finite thickness elastic plate model. The data are fit by elevated shear stress and Coulomb Failure Function (CFF) rates within 80 km of the trench. By contrast, an elastic half-space dislocation model does not fit as well and predicts significant amounts of locking and elevated CFF rates near the coast.