1Leibniz Institute for Applied Geophysics, Hannover, Germany ... In Geophysics, electrical resistivity tomography (ERT) is one of the most popular techniques for.
Electrical Resistivity Tomography (ERT) in geophysical applications - state of the art and future challenges Thomas G¨unther1 and Carsten R¨ucker2 1
Leibniz Institute for Applied Geophysics, Hannover, Germany 2 Berlin University of Technology, Germany
Abstract: Electrical imaging in geophysics is wide-spread and used for very different applications that have all their specific demands. Differently from medical imaging the area of investigation can be very large and is mostly approached only from one side. Measurement accuracy can vary significantly with one data or over time. Very often the forward problem is three-dimensional, but for the inverse problem a model reduction to fewer dimensions is needed. In most cases ERT is just one of several techniques that need to be applied jointly to reduce ambiguity. The coupling of different parameters, but also structures, is one of the most challenging tasks. The incorporation of any prior information available can significantly improve the quality of the result. Moreover, resistivity does often not represent the target value, instead water content or flow velocity are used. Either petrophysical relations or even partial differential equations need to be coupled to achieve an electrical imaging.
1
Introduction
In Geophysics, electrical resistivity tomography (ERT) is one of the most popular techniques for the shallow subsurface and is applied for hydrogeological, engineering, or agricultural questions. Applications cover a wide range of scales, from mm/cm scales at laboratory samples, dm to m scale in soils, m to decametres for groundwater questions, but can reach several hundreds m or even km for deep geological structures. Instruments have been rapidly developing in the past decades. In most cases fully automated instruments with multiple receiver channels are used, but in large-scale applications independent dipoles for current injection and potential difference registration are used. Whereas the first imaging experiments done by the brothers Schlumberger aimed at discovering electrical conductivity as a one-dimensional function of depth only, measuring techniques, numerical modelling approaches and inversion algorithms made rapid process since the 1970ies and the 1990ies so that 2D imaging became standard for industrial investigations. Whereas the algorithms for 3D inversion were developed almost at the same time, three-dimensional investigations are still not used routinely for surface measurements, mostly due to limitations in measuring effort and time. However, there are many applications using borehole electrodes or even borehole casings as extended electrodes. Getting closer to the area of investigation drastically improves resolution. Additional to the amplitude of the voltage, the phase shift to the injected voltage can be measured and reconstructed as a material property. The so gained induced polarisation is measured over a wide frequency range and can, due to its origin at the matrix-fluid interface, reveal intrinsic lengths of material. In the past years, a strong trend to repeated monitoring experiments can be noticed that aim at imaging processes such as water transport. Time-lapse approaches go from individual inversion
over reference techniques [5] to fully discretised all-at-once solutions. Noise is a very important issue in geophysical ERT since it determines the degree of how accurate given data should be explained by the imaging. It can vary over large magnitudes and over time and must be estimated, e.g. by normal-reciprocal data analysis [4], to avoid over- or under-fitting or loosing resolution. Although the mathematical problem and the general solutions are identical for ERT and EIT (electrical impedance tomography), there are some differences: 1. the unbounded earth requiring separation of forward and inverse domain, 2. electrodes (and man-made installations) are within the area of investigation, 3. sources and conductivity are generally three-dimensional, anisotropic and complex-valued, 4. a complete and optimal survey design does generally not exist and a trade-off between resolution power, noise susceptibility and measuring speed must be searched. Specific challenges for imaging techniques are the following: • Main problem is to overcome the ambiguity, i.e. to choose the model that is least in contradiction with knowledge or expectations of very different kinds. • Discretisation for both forward and inverse problem has to be chosen such that minimum computer power for a given accuracy is needed while taking resolution power and memory requirements into account. • An optimum survey design has to be chosen to maximise resolution power under noise [2]. • There is a scale problem, e.g. lab samples are not representative since material properties are statistically distributed and show macroscopic anisotropy.
2 2.1
Techniques Numerical modelling and discretisation
For a long time, Finite Differences have dominated due to their simplicity, before Finite elements codes had their break-through [6, e.g.]. Modern direct solvers yield solutions for many right hand side vectors efficiently. The numerical effort caused by the singular current sources can be avoided using a secondary field approach [6]. If the primary fields are not known analytically, they can be computed by a highly refined mesh [3] or the boundary element method [1]. Irregularly structured triangular and tetrahedral meshes bear the greatest flexibility to describe surface topography, structural information or a resolution-dependent discretisation. Since the modelling domain is unbounded but the area of resolution is only beyond the electrodes, most codes use some sort of conductivity prolongation to an outer box. Another way of overcoming the unbounded domain is using infinite elements [1].
2.2
Inversion and regularization
Most algorithms use either conjugate gradient or Gauss-Newton based minimization methods, others are Kalman filters, stochastic processes. The latter show considerably faster convergence, but do need a lot of memory for storing the Jacobian matrix explicitly. For 3D data sets, 100,000 data and model parameters can be frequently found. Alternatives could be artificial condensing to sparse matrices [3] or real quasi-Newton algorithms. Since the problem is ambiguous, i.e. a variety of models is able to satisfy the data within error bounds, regularisation procedures are
necessary. In most cases, smoothness constraints are used, i.e. the smoothest model of all equivalent ones is searched. In other cases, smoothness is avoided and the difference to a reference solution is minimized, e.g. in time-lapse schemes. The inverse problem can be stated as quadratic programming min kCmk22
= Φm subject to Φd =
N X di − fi (m) 2 i
i
=N,
(1)
where di are the N data with errors i , fi is the corresponding forward response and C is a regularization operator for the model m. Lagrangian methods are used to minimize Φd + λΦm .
2.3
Transformations and constraints
Traditionally, model parameters have been the logarithms of the cell resistivities to ensure positive values. In order to further constrain the resistivity range for one or more cells, logarithmic functions can be used, e.g. m = log(ρ − ρl ) − log(ρu − ρ) with lower and upper boundary ρl and ρu , respectively. Another way is to transform the resistivity into a target quantity such as water content, using an assumed petrophysical relationship. The latter can be further constrained using logarithmic functions. In time-lapse experiments, this allows for global constraints as the amount of infiltrated water, by appending an additional line in the Lagrange equation. Generally, transformations and global constraints improve the ill-posedness of the inverse problem, reduce ambiguity and increase resolution.
2.4
Region technique for flexible control
Only in few cases the subsurface is a black-box. There is prior information from direct measurements, borehole logs or descriptions, or from structural imaging techniques as seismics and GPR. Different regions have different parameter ranges, statistical distributions and behaviour in time. A region technique allows for choosing preferential directions and degree of smoothing, parameter range and temporal behaviour for different part of the model. These regions can be decoupled leading to sharp contrasts or to an arbitrary degree coupled with each other, e.g. for coupling point-wise parameter information with their neighbourhood.
2.5
A complex example from hydrogeophysics
In order to investigate the exchange between ground water and river water, a cross-hole monitoring experiment was designed by a group of ETH Zurich [2]. Aim is the fluid conductivity as a function of time for deriving transport velocities. The inversion domain, in the vicinity of 18 boreholes, is embedded in a three-dimensional topography with an artificial river water body of known conductivity (Fig. 1). There are several geological units: a heterogeneous, resistive unsaturated zone, the aquifer and a homogeneous, conductive clay. Its layer boundaries are known by boreholes and GPR and form the three regions. As the water table changes, the regions must be adapted in time. Additionally, the fluid in the boreholes is monitored and taken into account by own regions. In the time-lapse inversion, only changes in the aquifer are allowed and constrained. Systematic error sources are compensated by a difference inversion scheme [5]. Only doing so, slight changes caused by less conductive river water can be imaged accurately.
3
Conclusions and Outlook
ERT deserves an growingly larger attention since it is a low-cost and high-resolution technique that can rapidly image states and processes. Modern numerical algorithms make even large data
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Figure 1: Example for region-based control (from [2]): modelling domain with topography, discretised inversion domain with geological units and borehole regions (yellow). sets computable on standard computers. In order to meet the requirements of applications, various kinds of prior information needs to be integrated. This requires a flexible formulation and regularization of the inverse problem by combination of region technique with model transformations. Until now, measuring and inversion are separate steps. However, the development of intelligent, self-adaptive instruments that combine the processes of measurement, inversion and resolution analysis can significantly enhance the performance of ERT [2]. Since ERT is increasingly used for process monitoring, a coupling with different physical processes such infiltration or mass transport is needed to derive the target properties in one step. Mutual learning between the ERT and EIT communities is highly favourable to improve algorithms and practicability of solutions.
References [1] Blome, M., Maurer, H. and Schmidt, K., Advances in three-dimensional geoelectric forward solver techniques. Geophys. J. Int. 176: 740-752, 2009. [2] Blome, M., Maurer, H. and Greenhalgh, S., Geoelectric experimental design – Efficient acquisition and exploitation of complete pole-bipole data sets. Geophysics 76: F15-F26, 2011. [2] Coscia, I., Greenhalgh, S., Linde, N., Doetsch, J., Marescot, L., G¨unther, T., Vogt, T. and Green, A., 3D crosshole ERT for aquifer characterization and monitoring of infiltrating river water, Geophysics 76(2): G49–G59, 2011. [3] G¨unther, T., R¨ucker, C. and Spitzer, K., 3-d modeling and inversion of DC resistivity data incorporating topography - Part II: Inversion. Geophys. J. Int. 166: 506-517, 2006. [4] LaBrecque, D. J., Miletto, M. , Daily, W., Ramirez, A. and Owen, E., The effects of noise on Occam’s inversion of resistivity tomography data, Geophysics 61; 538? 548, 1996. [5] LaBrecque, D. J., and Yang, X., Difference Inversion of ERT Data: a Fast Inversion Method for 3-D in Situ Monitoring: J. Environ. Eng. Geophys. 6: 83, 2001. [6] R¨ucker, C., G¨unther, T. and Spitzer, K., 3-d modeling and inversion of DC resistivity data incorporating topography - Part II: Inversion. Geophys. J. Int. 166: 495-505, 2006.
