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We present a new application of a focusing regularization scheme for the inversion of resistivity and induced polarization ͑IP͒ data that supports large resistivity magnitude and phase contrasts. Similar approaches so far have only been used for the interpretation of gravity, magnetic, or seismic data sets. Unlike methods based on smoothness constraints, the approach is able to resolve sharp boundaries of bodies and layers, and it allows slight parameter variations within them. Therefore, it can be used in hydrogeologic applications where we need focused images to resolve high-contrast aquifer boundaries. Our approach is based on the minimum gradient support, which seeks to minimize the occurrence of parameter contrasts, independent of their magnitude. We study the effects of a variable control parameter on the reweighting optimization, allowing a continuous transition from smooth to sharp images. We also take the spatially varying sensitivity into account to allow focusing even where sensitivities are small. The implemented weighting leads to increased smoothing in well-resolved areas and a decrease in regions of lower sensitivity. The opposite approach is examined as well. This gradient-dependent sensitivity weighting is basically an extension of depth-dependent sensitivity weighting. We demonstrate the effectiveness and limitations of the approach and the influence of the control parameter using different synthetic models and field data from a hydrogeophysical test site. The technique has proven particularly suitable for revealing sharp parameter contrasts.

We have carried out spectral induced polarization (IP) measurements at three different hydrogeological test sites (Hasloh, Lüdingworth and Kappelen) and estimated hydraulic conductivity using empirical equations previously derived from laboratory measurements. We also reviewed previously published data from another site (Krauthausen). The intention was to explore the potential and practical limitations when applying the method at the field scale. The test sites cover a lithological spectrum from gravel to silt, with a variation in hydraulic conductivity (K) over three orders of magnitude. At each site, hydraulic conductivity was estimated from the real and imaginary conductivity resulting from 2D inversion. We applied the constant phase angle model, where only one frequency, typically around 1 Hz is being used. The uncertainty in K-estimates arising from inversion ambiguity was assessed by exploring the model space with a control parameter that permits a transition from smooth to blocky models and by using different starting models. At the Kappelen site, this uncertainty is larger than four orders of magnitude but a reasonable lower limit for K can be obtained. At the other three sites, the uncertainties are typically one order of magnitude.The IP-based hydraulic conductivity estimates were compared with K-values obtained from grain size analyses and pumping tests. At the Hasloh and Lüdingworth sites the results agree within one order of magnitude and at the Kappelen site the derived lower boundary for K is consistent with grain size information. At the Krauthausen site, the difference between IP-based data and the values derived from grain size and pumping tests is significantly larger than the estimated uncertainties, which is probably due to the non-uniform grain size distribution. The overall results indicate that order of magnitude K-estimates from IP data at the field scale are realistic targets. However, sites with significant deviations from the empirical equations can exist, emphasizing the recommendation to use a priori information whenever possible. (Mazac et al. 1985;Purvance and Andricevic 2000). Spectral induced polarization (IP) measures the complex, frequencydependent electrical conductivity. Physical considerations and laboratory measurements support the idea that the imaginary part of complex conductivity strongly depends on the geometrical characteristics of the pore space and thus might considerably improve quantitative hydraulic conductivity estimation.A number of laboratory measurements of unconsolidated sediments suggest that the phase of the complex conductivity is constant over a broad frequency range . The measured spectra may be described by the constant phase angle model and measurements of a single frequency (i.e., IP measurements) are sufficient to describe the spectra. Empirical equations

Geophysicists have tried for a long time to correlate induced polarization data with parameters of the pore geometry, often with the overall aim to estimate the important parameter of hydraulic conductivity. However, no correlation has been found so far that is applicable to more than just a few special cases. Using empirical relationships and equivalent circuits often neglects the description of the processes in the pore space. One reason is that the mechanisms controlling the low-frequency polarization are still not completely understood. Only a few existing models try to explain the processes and derive relationships with geometry and most models need strong assumptions. Here, we aim at a deeper understanding by numerical modelling of the main processes responsible for the IP effect. Our approach is based on a 1D solution of the late 1950s that gives an expression for the maximum frequency effect for simplified geometries. The models depend on the lengths of active and passive zones and the corresponding ion mobilities in each zone for anions and cations respectively. The theory describes the ion diffusion along concentration gradients, the influence of an external electric field and the coupling between the two.We first verify our numerical results by comparison with analytical solutions and then extend the approach to flexible geometry, including higher dimension. This constitutes a considerable progress from 1D models restricted to two alternating media with fixed lengths. Apart from arbitrary geometry, we can also simulate the full spectral behaviour. A relatively simple model is able to explain frequency-dependent magnitude and phase behaviour that is typically measured in the field. The Cole-Cole model can be considered as the result of a network of pores of varying lengths. From our modelling studies, we derive scaling laws for mobility and geometry and suggest an empirical equation for the relationship between the length of passive pores and the time of maximum phase shift. material, several approaches have aimed at characterizing the spectral behaviour of induced polarization (SIP) measurements (e.g., Olhoeft 1985). One of the most common models is given by the Cole-Cole equation (Pelton 1978):(1) Here, ρ denotes the complex resistivity, ρ 0 is the DC resitivity, m the chargeability, τ the time constant, c the frequency exponent and ω the angular frequency. The approaches often relate the spectral behaviour to equivalent circuits where the active or ion selective zones are implemented as a capacitance and passive zones are represented as resistances. Numerous combinations exist, some of which were set up to explain particular results measured at a small number of samples. A good overview is given by Dias (2000).Some studies were devoted to particular types of applications, e.g., on shaly sands (Vinegar and Waxman 1984) or the detection of oil contaminated sand or till (Vanhala 1997) without making