SUMMARY We consider deformation due to sill‐like magma intrusions using a model of a horizontal circular crack in a semi‐infinite elastic solid. We present exact expressions for vertical and horizontal displacements of the free surface of a half‐space, and calculate surface displacements for a special case of a uniformly pressurized crack. We derive expressions for other observable geophysical parameters, such as the volume of a surface uplift/subsidence, and the corresponding volume change due to fluid injection/withdrawal at depth. We demonstrate that for essentially oblate (i.e. sill‐like) source geometries the volume change at the source always equals the volume of the displaced material at the surface of a half‐space. Our solutions compare favourably to a number of previously published approximate models. Surface deformation due to a ‘point’ crack (that is, a crack with a large depth‐to‐radius ratio) differs appreciably from that due to an isotropic point source (‘Mogi model’). Geodetic inversions that employ only one component of deformation (either vertical or horizontal) are unlikely to resolve the overall geometry of subsurface deformation sources even in a simplest case of axisymmetric deformation. Measurements of a complete vector displacement field at the Earth's surface may help to constrain the depth and morphology of active magma reservoirs. However, our results indicate that differences in surface displacements due to various axisymmetric sources may be subtle. In particular, the sill‐like and pluton‐like magma chambers may give rise to differences in the ratio of maximum horizontal displacements to maximum vertical displacements (a parameter that is most indicative of the source geometry) that are less than 30 per cent. Given measurement errors in geodetic data, such differences may be hard to distinguish.

[1] Field observations of pseudotachylites and experimental studies of high-speed friction indicate that melting on a slipping interface may significantly affect the magnitude of shear stresses resisting slip. We investigate the effects of rock melting on the dynamic friction using theoretical models of shear heating that couple heat transfer, thermodynamics of phase transitions, and fluid mechanics. Results of laboratory experiments conducted at high (order of m/s) slip velocities but low (order of MPa) normal stresses suggest that the onset of frictional melting may give rise to substantial increases in the effective fault strength, presumably due to viscous effects. However, extrapolation of the modeling results to in situ conditions suggests that the efficiency of viscous braking is significantly reduced under high normal and shear stresses. When transient increases in the dynamic fault strength due to fusion are not sufficient to inhibit slip, decreases in the effective melt viscosity due to shear heating and melting of clasts drastically decrease the dynamic friction, resulting in a nearly complete stress drop (''thermal runaway''). The amount of energy dissipation associated with the formation of pseudotachylites is governed by the temperature dependence of melt viscosity and the average clast size in the fault gouge prior to melting. Clasts from a coarse-grained gouge have lower chances of survival in a pseudotachylite due to a higher likelihood of nonequilibrium overheating. The maximum temperature and energy dissipation attainable on the fault surface are ultimately limited by either the rock solidus (via viscous braking, and slip arrest) or liquidus (via thermal runaway and vanishing resistance to sliding). Our modeling results indicate that the thermally activated fault strengthening and rupture arrest are unlikely to occur in most mafic protoliths but might be relevant for quartz-rich rocks, especially at shallow (<5-7 km) depths where the driving shear stress is relatively low.

SUMMARY We investigate surface deformation associated with currently active crustal magma bodies in Socorro, New Mexico, and Long Valley, California, USA. We invert available geodetic data from these locations to constrain the overall geometry and dynamics of the inferred deformation sources at depth. Our best‐fitting model for the Socorro magma body is a sill with a depth of 19 km, an effective diameter of 70 km and a rate of increase in the excess magma pressure of 0.6 kPa yr−1. We show that the corresponding volumetric inflation rate is ∼6×10−3 km3 yr−1, which is considerably less than previously suggested. The measured inflation rate of the Socorro magma body may result from a steady influx of magma from a deep source, or a volume increase associated with melting of the magma chamber roof (i.e. crustal anatexis). In the latter case, the most recent major injection of mantle‐derived melts into the middle crust beneath Socorro may have occurred within the last several tens to several hundreds of years. The Synthetic Interferometric Aperture Radar (InSAR) data collected in the area of the Long Valley caldera, CA, between June 1996 and July 1998 reveal an intracaldera uplift with a maximum amplitude of ∼11 cm and a volume of 3.5×10−2 km3. Modelling of the InSAR data suggests that the observed deformation might be due to either a sill‐like magma body at a depth of ∼12 km or a pluton‐like magma body at a depth of ∼8 km beneath the resurgent dome. Assuming that the caldera fill deforms as an isotropic linear elastic solid, a joint inversion of the InSAR data and two‐colour laser geodimeter data (which provide independent constraints on horizontal displacements at the surface) suggests that the inferred magma chamber is a steeply dipping prolate spheroid with a depth of 7–9 km and an aspect ratio in excess of 2:1. Our results highlight the need for large radar look angles and multiple look directions in future InSAR missions.

[1] Analysis of the trace element contents in kimberlites from various provinces around the world, including South Africa, India, and Yakutia (Siberia, Russia), reveals remarkable similarity of the maximum abundances. In addition, we find that abundances of the rare earth elements (REE) in the South African kimberlites are highly coherent between individual elements. We suggest that the observed similarity of the trace element patterns may result from a common physicochemical process operating in the kimberlite source region, rather than from peculiar source compositions and magmatic histories. The most likely candidates for such a process are (1) partial melting at very low melting degrees and (2) porous melt flow and diffusive exchange with the host rocks. These two processes can produce the same maximum trace element abundances and similar undersaturated patterns. We argue that the porous flow, and the associated chromatographic enrichment, is preferred because it allows high saturations at relatively large melt fractions of $1%. Observations of enrichment of the xenolith grain rims due to an exchange with metasomatizing melts of quasi-kimberlitic composition imply that the melt percolated beyond the source region, in agreement with basic assumptions of the percolation model. We demonstrate that the saturated REE patterns are in a good agreement with the maximum observed REE abundances in kimberlites from different provinces. The theoretical patterns are independent of the melt fraction and only weakly (if at all) depend on the source modal composition. Characteristic diverging fan-like patterns of trace elements predicted by the percolation model are identified in kimberlites from South Africa. We propose that a high coherency of the REE patterns in the South African kimberlites results from a general dependence of all REE abundances on the calcium content. According to this interpretation, the overall depletion of the source rocks in REE with temperature (and depth) postulated by our model is a natural consequence of a decrease in the calcium content along the lherzolite trend.