[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.

The full gradient tensor is presently not measured routinely onboard airplanes or on land. This paper describes some improvements that can be made in strategies of data collection and in processing of potential field maps if such tensor measurements were available. We suggest that, in addition to producing for example standard total field anomaly maps, the invariants of the tensor be mapped. Strikes of magnetic or gravimetric structures may be determined from minimizing the power in the first row and column of the tensor. Invariants can be looked upon as nonlinear filters enhancing sources with big volumes. Their lateral resolution is superior to that of the field proper and, for a given resolution, the flight altitude and separation between flight lines can be increased compared with the standard mode of operation. In airborne surveys the distance between flight lines is normally much larger than the height above the ground. This may introduce severe aliasing effects in the direction perpendicular to the flight lines. By increasing the flight altitude, aliasing effects are reduced at the expense of lateral resolution which, however, may be improved by mapping the tensor invariants in addition to the magnetic field. The estimated gradient tensor from total field magnetic data over the Siljan impact region shows that the magnetic properties of the area are very nonuniform even from a height of 430 m above the topography. The nonlinear filters discriminate major anomalies into separate units.

The distortion of the magnetotelluric impedance tensor by complex “near‐surface” structure leads to leakage between the elements of the tensor. The magnetotelluric impedance tensor for our principal model, which has both a local and a regional strike, can be written in the long‐period limit as a sum of the regional, undistorted impedance and a perturbed impedance. The latter can be written as a product of a local distortion (which can be regarded as thin‐sheet distortion in the long‐period range) tensor and the regional impedance. Local and regional strikes are found by rotating the impedance tensor into directions in which diagonal elements are proportional and column elements are proportional, respectively. The regional impedance tensor is calculated assuming that the strikes are uniquely defined. An example from a crystalline area with well conducting fracture zones illustrates the model concepts. A weighted least‐squares procedure is used for the estimation of distortion parameters.

We have developed a new method to locate geologic bodies using the gravity gradient tensor. The eigenvectors of the symmetric gravity gradient tensor can be used to estimate the position of the source body as well as its strike direction. For a given measurement point, the eigenvector corresponding to the maximum eigenvalue points approximately toward the center of mass of the causative body. For a collection of measurement points, a robust least-squares procedure is used to estimate the source point as the point that has the smallest sum of square distances to the lines defined by the eigenvectors and the measurement positions. It’s assumed that the maximum of the first vertical derivative of the vertical component of gravity vector [Formula: see text] is approximately located above the center of mass. Observation points enclosed in a square window centered at the maximum of [Formula: see text] are used to estimate the source location. By increasing the size of the window, the number of eigenvectors used in the robust least squares and subsequently the number of solutions increase. As a criterion for selecting the best solution from a set of previously computed solutions, we chose that solution having the minimum relative error (less than a given threshold) of its depth estimate. The strike direction of the source can be estimated from the direction of the eigenvectors corresponding to the smallest eigenvalue for quasi 2D structures. To study the effect of additive random noise and interfering sources, the method was tested on synthetic data sets, and it appears that our method is robust to random noise in the different measurement channels. The method was also tested on gravity gradient tensor data from the Vredefort impact structure, South Africa. The results show a very good agreement with the available geologic information.

S U M M A R YFor the first time, a comparative analysis of the resolution and variance properties of 2-D models of electrical resistivity derived from single and joint inversions of dc resistivity (DCR) and radiomagnetotelluric (RMT) measurements is presented.DCR and RMT data are inverted with a smoothness-constrained 2-D scheme. Model resolution, model variance and data resolution analyses are performed both with a classical linearized scheme that employs the smoothness-constrained generalized inverse and a non-linear truncated singular value decomposition (TSVD). In the latter method, the model regularization used in the inversion is avoided and non-linear semi-axes give an approximate description of the non-linear confidence surface in the directions of the model eigenvectors. Hence, this method analyses the constraints that can be provided by the data. Model error estimates are checked against improved and independent estimates of model variability from most-squares inversions.For single and joint inverse models of synthetic data sets, the smoothness-constrained scheme suggests relatively small model errors (typically up to 30 to 40 per cent) and resolving kernels that are spread over several cells in the vicinity of the investigated cell. Linearized smoothness-constrained errors are in good agreement with the corresponding most-squares errors. The variability of the RMT model as estimated from non-linear semi-axes is confirmed by TSVD-based most-squares inversions for most model cells within the depth range of investigation. In contrast to this, most-squares errors of the DCR model are consistently larger than errors estimated from non-linear semi-axes except for the smallest truncation levels.The model analyses confirm previous studies that DCR data can constrain resistive and conductive structures equally well while RMT data provide superior constraints for conductive structures. The joint inversion can improve error and resolution of structures which are within the depth ranges of exploration of both methods. In such parts of the model which are outside the depth range of exploration for one method, error and resolution of the joint inverse model are close to those of the best single inversion result subject to an appropriate weighting of the different data sets.