Investigation of a Tertiary maar structure using three-dimensional resistivity imaging

This paper presents the application of the electrical resistivity tomography (ERT) method to the investigation of the Tertiary maar structure of Baruth (Germany) known from previous gravimetric surveys. ERT was applied to support the optimum location for a palaeoclimatological drill hole.
  Special modifications of data acquisition, signal processing and inversion are introduced to adapt the method of ERT to the special requirements for the 3-D investigation of structures with horizontal extensions of 1  km or more. More than 5000 dipole–dipole combinations were recorded at three concentric circular electrode arrangements using stand-alone transient data acquisition systems (RefTek).
  We present a fast approximate imaging technique based on the simultaneous iterative reconstruction technique (SIRT). As the complete calculation of the inverse Frechét matrix is avoided, the algorithm is especially suitable for large data and model spaces, where complete inversion is beyond the limits of available computing hardware. The single-step method is applicable to arbitrary irregular electrode layouts. Synthetic tests show that the imaging procedure reconstructs the main features of the subsurface.
  A low-resistivity body could be interpreted as limnic sediments filling the interior of the Tertiary maar crater. Considering the horizontal resistivity gradient, estimates for the lateral and depth extents of the structure were made. An optimum position for a palaeoclimatological borehole was found, and was in good agreement with the gravimetric minimum.     

Two-dimensional non-linear inversion of VLF-R data using simulated annealing

The VLF-R (very low frequency-resistivity) data, i.e. the apparent resistivity ( ρ _a ) and phase ( φ ) data, were inverted individually and jointly using the VFSA (very fast simulated annealing) global inversion approach. Global inversion results for synthetic data without and with various amounts of random and normally distributed Gaussian noise reveal that the inversion of neither the ρ _a nor φ data alone yields the true parameters of the structures. However, the joint inversion of the ρ _a and φ data yields very good estimates of the model parameters. Five models, representing typical subsurface structures in the shield areas, are studied here. Various models achieved after 10 VFSA runs were used to compute the mean model and the corresponding covariance and correlation matrices, which were used to estimate the uncertainties in the mean model parameters and correlations between the model parameters. We observe that these correlations follow the physics associated with the problem. VLF-R field data due to a nearly vertical contact structure and a very thick dyke-like structure were also inverted to demonstrate the efficacy of the approach in the delineation of the parameters of 2-D structures.     

Frequency response modelling of seismic waves using finite difference time domain with phase sensitive detection (TD–PSD)

This paper describes an efficient approach for computing the frequency response of seismic waves propagating in 2- and 3-D earth models within which the magnitude and phase are required at many locations. The approach consists of running an explicit finite difference time domain (TD) code with a time harmonic source out to steady-state. The magnitudes and phases at locations in the model are computed using phase sensitive detection (PSD). PSD does not require storage of time-series (unlike a fast Fourier transform), reducing its memory requirements. Additionally, the response from multiple sources can be obtained from a single finite difference run by encoding each source with a different frequency. For 2-D models with many sources, this time domain phase sensitive detection (TD–PSD) approach has a higher arithmetic complexity than direct solution of the finite difference frequency domain (FD) equations using nested dissection re-ordering (FD–ND). The storage requirements for 2-D finite difference TD–PSD are lower than FD–ND. For 3-D finite difference models, TD–PSD has significantly lower arithmetic complexity and storage requirements than FD–ND, and therefore, may prove useful for computing the frequency response of large 3-D earth models.     

Forward modelling of direct current and low-frequency electromagnetic fields using integral equations

We present a semi-analytical, unifying approach for modelling the electromagnetic response of 3-D bodies excited by low-frequency electric and magnetic sources. We write the electric and magnetic fields in terms of power series of angular frequency, and show that to obey Maxwell's equations, the fields must be real when the exponent is even, and imaginary when it is odd. This leads to the result that the scattering equations for direct current fields and for fields proportional to frequency can both be explicitly formulated using a single, real dyadic Green's function. Although the underground current flow in each case is due to different physical phenomena, the interaction of the scattering currents is of the same type in both cases. This implies that direct current resistivity, magnetometric resistivity and electric and magnetic measurements at low induction numbers can all be modelled in parallel using basically the same algorithm. We make a systematic derivation of the quantities required and show that for these cases they can all be expressed analytically. The problem is finally formulated as the solution of a system of linear equations. The matrix of the system is real and does not depend on the type of source or receiver. We present modelling results for different arrays and apply the algorithm to the interpretation of field data. We assume the standard dipoledipole resistivity array for the direct current case, and vertical and horizontal magnetic dipoles for induction measurements. In the case of magnetometric resistivity we introduce a moving array composed of an electric dipole and a directional magnetometer. The array has multiple separations for depth discrimination and can operate in two modes. The mode where the predominant current flow runs along the profile is called MMR-TM. This mode is more sensitive to lateral variations in resistivity than its counterpart, MMR-TE, where the mode of conduction is predominantly perpendicular to the profile.     

The instantaneous apparent resistivity tensor: a visualization scheme for LOTEM electric field measurements

Long-offset transient electromagnetic (LOTEM) data have traditionally been represented as early- and late-time apparent resistivities. Time-varying electric field data recorded in a LOTEM survey made with multiple sources can be represented by an 'instantaneous apparent resistivity tensor'. Three independent, coordinate-invariant, time-varying apparent resistivities can be derived from this tensor. For dipolar sources, the invariants are also independent of source orientation. In a uniform-resistivity half-space, the invariant given by the square root of the tensor determinant remains almost constant with time, deviating from the half-space resistivity by a maximum of 6 per cent. For a layered half-space, a distance–time pseudo-section of the determinant apparent resistivity produces an image of the layering beneath the measurement profile. As time increases, the instantaneous apparent resistivity tensor approaches the direct current apparent resistivity tensor. An approximate time-to-depth conversion can be achieved by integrating the diffusion depth formula with time, using the determinant apparent resistivity at each instant to represent the resistivity of the conductive medium. Localized near-surface inhomogeneities produce shifts in the time-domain apparent resistivity sounding curves that preserve the gradient, analogous to static shifts seen in magnetotelluric soundings. Instantaneous apparent resistivity tensors calculated for 3-D resistivity models suggest that profiles of LOTEM measurements across a simple 3-D structure can be used to create an image that reproduces the main features of the subsurface resistivity. Where measurements are distributed over an area, maps of the tensor invariants can be made into a sequence of images, which provides a way of 'time slicing' down through the target structure.