[1] Appropriate regularizations of geophysical inverse problems and joint inversion of different data types improve geophysical models and increase their usefulness in hydrogeological studies. We have developed an efficient method to calculate stochastic regularization operators for given geostatistical models. The method, which combines circulant embedding and the diagonalization theorem of circulant matrices, is applicable for stationary geostatistical models when the grid discretization, in each spatial direction, is uniform in the volume of interest. We also used a structural approach to jointly invert cross-hole electrical resistance and ground-penetrating radar traveltime data in three dimensions. The two models are coupled by assuming, at all points, that the cross product of the gradients of the two models is zero. No petrophysical relationship between electrical conductivity and relative permittivity is assumed but is instead obtained as a by-product of the inversion. The approach has been applied to data collected in a U.K. sandstone aquifer in order to improve characterization of the vadose zone hydrostratigraphy. By analyzing scatterplots of electrical conductivity versus relative permittivity together with petrophysical models a zonation could be obtained with corresponding estimates of the electrical formation factor, the water content, and the effective grain radius of the sediments. The approach provides greater insight into the hydrogeological characteristics of the subsurface than by using conventional geophysical inversion methods.

SUMMARY Using local earthquakes and explosions recorded on the 3‐D seismic network operating in Southwest Iceland, the 3‐D velocity structure has been modelled to depths of 10–15 km. The tomography algorithm simultaneously inverts for both P‐ and S‐wave velocities and hypocentral locations. Major tectonic features within the 224 × 112 km2 rectangular study region include the South Iceland Seismic Zone, the Hengill volcanic system and the Reykjanes Volcanic Zone. Reduced velocities from the surface to as deep as we can resolve, or about 9 km, are associated with the Hengill central volcano. As low Vp/Vs ratios prevail in the entire anomalous region, we suggest supercritical fluids within the volcanic fissure system cause the reduced velocities, rather than any large regions of partial melt. Along the Reykjanes Volcanic Zone, relatively low velocities down to depths of 6–8 km are observed in the centre of the zone. Normal velocities are observed in the South Iceland Seismic Zone with a slight reduction in Vp/Vs ratio. The thickness of the brittle crust, defined as the depth above which 90 per cent of the earthquakes occur, increases from about 5 km in the relatively young crust of the Reykjanes Volcanic Zone to about 12 km in the eastern end of the South Iceland Seismic Zone. These depths correspond to temperatures in the range of 580–750°C, estimated from borehole heat flow measurements.

SUMMARY A recent seismic refraction study across southern Norway has revealed that the up to 2469 m high Southern Scandes Mountains are not isostatically compensated by a thick crust. Rather, the Moho depths are close to average for continental crust with elevations of ∼1 km. Evidence from new seismic data indicate that beneath the highest topography Moho depths are around 38–40 km. These measurements are ∼2 km deeper than early estimates interpolated from coarsely spaced refraction profiles, but up to 3 km shallower than Receiver Function estimates for the area. Moho depth variation beneath the mountains roughly correlates with changes in surface topography indicating that topography is, at least to the first order, controlled by crustal thickness. However, the highest mountains do not overlie the thickest crust and additional support for topography, for example from flexural strength in the lithosphere, low densities in the upper‐mantle or mantle dynamics, is likely. The relationship between topography and Moho depth breaks down for the Oslo Graben and the Fennoscandian Shield to the east and north. High density lower crustal rocks below Oslo Graben and increasing crust and lithospheric thicknesses below the Fennoscandian Shield may produce a negative correlation between topography and Moho depth.

Summary In seismic tomography, the LSQR algorithm is commonly used for solving the inverse problem. LSQR belongs to the family of conjugate gradient methods, so the generalized inverse is not solved explicitly. Consequently, neither the covariance nor the resolution matrix are provided by LSQR, which limits one’s ability to obtain estimates of uncertainty and errors in the computed models. In this paper we present a method, demonstrated by synthetic examples, for calculating the resolution and covariance matrices via the general inverse of LSQR. The extra computational effort is limited and only a few lines of computer code in the original LSQR routine are needed to produce the required output. The resolution matrices produced demonstrate that great care must be taken to ensure that a sufficient number of iterations are used when applying LSQR inversion. The relationship of the LSQR‐based resolution estimates to those produced using other methods is briefly discussed.

Abstract:The structural approach to joint inversion, entailing common boundaries or gradients, offers a flexible way to invert diverse types of surface-based and/or crosshole geophysical data. The cross-gradients function has been introduced as a means to construct models in which spatial changes in two models are parallel or anti-parallel. Inversion methods that use such structural constraints also provide estimates of non-linear and non-unique field-scale relationships between model parameters. Here, we invert jointly crosshole radar and seismic traveltimes for structurally similar models using an iterative non-linear traveltime tomography algorithm.Application of the inversion scheme to synthetic data demonstrates that it better resolves lithological boundaries than the individual inversions. Tests of the scheme on observed radar and seismic data acquired within a shallow aquifer illustrate that the resultant models have improved correlations with flowmeter data than with models based on individual inversions.The highest correlation with the flowmeter data is obtained when the joint inversion is combined with a stochastic regularization operator, where the vertical integral scale is estimated from the flowmeter data. Point-spread functions shows that the most significant resolution improvements of the joint inversion is in the horizontal direction.2