Parallel computation of optimized arrays for 2-D electrical imaging surveys

Modern automatic multi-electrode survey instruments have made it possible to use non-traditional arrays to maximize the subsurface resolution from electrical imaging surveys. Previous studies have shown that one of the best methods for generating optimized arrays is to select the set of array configurations that maximizes the model resolution for a homogeneous earth model. The Sherman–Morrison Rank-1 update is used to calculate the change in the model resolution when a new array is added to a selected set of array configurations. This method had the disadvantage that it required several hours of computer time even for short 2-D survey lines. The algorithm was modified to calculate the change in the model resolution rather than the entire resolution matrix. This reduces the computer time and memory required as well as the computational round-off errors. The matrix–vector multiplications for a single add-on array were replaced with matrix–matrix multiplications for 28 add-on arrays to further reduce the computer time. The temporary variables were stored in the double-precision Single Instruction Multiple Data (SIMD) registers within the CPU to minimize computer memory access. A further reduction in the computer time is achieved by using the computer graphics card Graphics Processor Unit (GPU) as a highly parallel mathematical coprocessor. This makes it possible to carry out the calculations for 512 add-on arrays in parallel using the GPU. The changes reduce the computer time by more than two orders of magnitude. The algorithm used to generate an optimized data set adds a specified number of new array configurations after each iteration to the existing set. The resolution of the optimized data set can be increased by adding a smaller number of new array configurations after each iteration. Although this increases the computer time required to generate an optimized data set with the same number of data points, the new fast numerical routines has made this practical on commonly available microcomputers

Array optimisation for multichannel electrical resistivity tomography instruments

In recent years there has been considerable research into the selection of near-optimal arrays of electrode configurations that enhance the resolution of electrical resistivity tomography (ERT) images. Several algorithms have been developed that select resistivity measurements based on their contribution to the cumulative sensitivity of the array (Furman et al., 2004; Hennig and Weller, 2005) or its model resolution matrix (Stummer et al. , 2004; Wilkinson et al., 2006a; 2006b). Homogeneous subsurface resistivity distributions were assumed for these studies, although better results can be obtained using the same algorithms if the resistivity distribution is known a priori (Anthansiou, 2006). When compared to standard arrays, such as dipole-dipole or Wenner- Schlumberger, optimised arrays can substantially improve the resolution of the ERT image for the same number of measurements (Wilkinson et al., 2006b). The driver for researching array optimisation techniques has been the development of computer controlled ERT systems that can address arbitrary combinations of current and potential electrodes. Unfortunately all the published optimisation algorithms share a problem that is likely to impede their wider use: the arrays that they produce are inherently ‘single channel’ (SC). Since they do not take advantage of the multichannel (MC) capability of modern ERT instruments, the optimised arrays that they produce are rather inefficient to use compared to many standard arrays that are well suited to MC operation. However, we have developed a simple extension that constrains our previous algorithm to choose near-optimal configurations that also fit well into a MC measurement scheme. This extension could easily be adapted to work with the other optimisation schemes cited above

The detection and tracking of mine-water pollution from abandoned mines using electrical tomography

Increasing emphasis is being placed on the environmental and societal impact of mining, particularly in the EU, where the environmental impacts of abandoned mine sites (spoil heaps and tailings) are now subject to the legally binding Water Framework and Mine Waste Directives. Traditional sampling to monitor the impact of mining on surface waters and groundwater is laborious, expensive and often unrepresentative. In particular, sparse and infrequent borehole sampling may fail to capture the dynamic behaviour associated with important events such as flash flooding, mine-water break-out, and subsurface acid mine drainage. Current monitoring practice is therefore failing to provide the information needed to assess the socio-economic and environmental impact of mining on vulnerable eco-systems, or to give adequate early warning to allow preventative maintenance or containment. BGS has developed a tomographic imaging system known as ALERT ( Automated time-Lapse Electrical Resistivity Tomography) which allows the near real-time measurement of geoelectric properties "on demand", thereby giving early warning of potential threats to vulnerable water systems. Permanent in-situ geoelectric measurements are used to provide surrogate indicators of hydrochemical and hydrogeological properties. The ALERT survey concept uses electrode arrays, permanently buried in shallow trenches at the surface but these arrays could equally be deployed in mine entries or shafts or underground workings. This sensor network is then interrogated from the office by wireless telemetry (e.g: GSM, low-power radio, internet, and satellite) to provide volumetric images of the subsurface at regular intervals. Once installed, no manual intervention is required; data is transmitted automatically according to a pre-programmed schedule and for specific survey parameters, both of which may be varied remotely as conditions change (i.e: an adaptive sampling approach). The entire process from data capture to visualisation on the web-portal is seamless, with no manual intervention. Examples are given where ALERT has been installed and used to remotely monitor (i) seawater intrusion in a coastal aquifer (ii) domestic landfills and contaminated land and (iii) vulnerable earth embankments. The full potential of the ALERT concept for monitoring mine-waste has yet to be demonstrated. However we have used manual electrical tomography surveys to characterise mine-waste pollution at an abandoned metalliferous mine in the Central Wales orefield in the UK. Hydrogeochemical sampling confirms that electrical tomography can provide a reliable surrogate for the mapping and long-term monitoring of mine-water pollution

Bioinspired low-frequency material characterisation

New-coded signals, transmitted by high-sensitivity broadband transducers in the 40–200 kHz range, allow subwavelength material discrimination and thickness determination of polypropylene, polyvinylchloride, and brass samples. Frequency domain spectra enable simultaneous measurement of material properties including longitudinal sound velocity and the attenuation constant as well as thickness measurements. Laboratory test measurements agree well with model results, with sound velocity prediction errors of less than 1%, and thickness discrimination of at least wavelength/15. The resolution of these measurements has only been matched in the past through methods that utilise higher frequencies. The ability to obtain the same resolution using low frequencies has many advantages, particularly when dealing with highly attenuating materials. This approach differs significantly from past biomimetic approaches where actual or simulated animal signals have been used and consequently has the potential for application in a range of fields where both improved penetration and high resolution are required, such as nondestructive testing and evaluation, geophysics, and medical physics

Optimized arrays for 2-D resistivity survey lines with a large number of electrodes

Previous studies show that optimized arrays generated using the ‘Compare R’ method have significantly better resolution than conventional arrays. This method determines the optimum set of arrays by selecting those that give the maximum model resolution. The number of possible arrays (the comprehensive data set) increases with the fourth power of the number of electrodes. The optimization method faces practical limitations for 2-D survey lines with more than 60 electrodes where the number of possible arrays exceeds a million. Several techniques are proposed to reduce the calculation time for such survey lines. A single-precision version of the ‘Compare R’ algorithm using a new ranking function reduces the calculation time by two to eight times while providing results similar to the double-precision version. Recent improvements in computer GPU technology can reduce the calculation time by about seven times. The calculation time is reduced by half by using the fact that arrays that are symmetrical about the center of the line produce identical changes in the model resolution values. It is further reduced by more than thirty times by calculating the Sherman–Morrison update for all the possible two-electrode combinations, which are then used to calculate the model resolution values for the four-electrode arrays. The calculation time is reduced by more then ten times by using a subset of the comprehensive data set consisting of only symmetrical arrays. Tests with a synthetic model and field data set show that optimized arrays derived from this subset produce inversion models with differences of less than 10% from those derived using the full comprehensive data set. The optimized data sets produced models that are more accurate than the Wenner–Schlumberger array data sets in all the tests

The robustness and general applicability of Optimal Resistivity Surveys designed by maximising model resolution

Most optimal survey design algorithms for resistivity imaging have not incorporated prior knowledge of the resistivity of the subsurface. The resulting surveys are optimal for a homogeneous earth, but little investigation has yet been carried out to test whether they are robust, i.e. that they remain optimal when applied to imaging heterogeneous subsurface resistivity distributions. This paper compares a generic survey, which is designed to maximise the estimated model resolution evenly across a homogeneous earth, with specific surveys similarly designed for a number of heterogeneous resistivity distributions. In terms of both the average estimated model resolution and the correlations between the inverted and true resistivity models, the generic and heterogeneous survey designs give near-identical results. This suggests that surveys designed using homogeneous earth approximations are robust in the presence of resistivity heterogeneities and are therefore generally applicable. Traditional dipole-dipole surveys with the same number of measurements do not give such good inverted images, and their degree of optimality (measured either by average resolution or image correlation) is less robust in the presence of heterogeneity

Characterising sand and gravel deposits using electrical resistivity tomography (ERT) : case histories from England and Wales

Electrical Resistivity Tomography (ERT) is a rapidly developing geophysical imaging technique that is now widely used to visualise subsurface geological structure, groundwater and lithological variations. It is being increasingly used in environmental and engineering site investigations, but despite its suitability and potential benefits, ERT has yet to be routinely applied by the minerals industry to sand and gravel deposit assessment and quarry planning. The principal advantages of ERT for this application are that it is a cost-effective non-invasive method, which can provide 2D or 3D spatial models of the subsurface throughout the full region of interest. This complements intrusive sampling methods, which typically provide information only at discrete locations. Provided that suitable resistivity contrasts are present, ERT has the potential to reveal mineral and overburden thickness and quality variations within the body of the deposit. Here we present a number of case studies from the UK illustrating the use of 2D and 3D ERT for sand and gravel deposit investigation in a variety of geological settings. We use these case studies to evaluate the performance of ERT, and to illustrate good practice in the application of ERT to deposit investigation. We propose an integrated approach to site investigation and quarry planning incorporating both conventional intrusive methods and ERT

Measurement and modelling of moisture—electrical resistivity relationship of fine-grained unsaturated soils and electrical anisotropy

A methodology for developing resistivity-moisture content relationships of materials associated with a clayey landslide is presented. Key elements of the methodology include sample selection and preparation, laboratory measurement of resistivity with changing moisture content, and the derivation of models describing the relationship between resistivity and moisture content. Laboratory resistivity measurements show that the techniques utilised (samples and square array) have considerable potential as a means of electropetrophysical calibration of engineering soils and weak rock. Experimental electrical resistivity results show a hierarchy of values dependent on sample lithology, with silty clay exhibiting the lowest resistivities, followed by siltstones and sands, which return the highest resistivities. In addition, finer grained samples show a greater degree of anisotropy between measurement orientations than coarser grained samples. However, suitability of results in light of issues such as sample cracking and electrical conduction must be identified and accounted for if the results are to be accurately up-scaled to inverted model resistivity results. The existence of directional anisotropy makes model calibration curve selection more difficult due to variability in the range of measured laboratory resistances. The use of larger measurement array size means that experimental data will be more representative of bulk lithological properties. In addition, use of electrodes with a relatively high surface area (wide diameter) help maintain low contact resistances and repeat measurement error, relative to narrow electrodes. Variation exists between the fit of experimental data and petrophysical models. Model fit is best for clay-dominated samples but fits less well for sand-dominated samples. Waxman–Smits equation is appropriately applied in this investigation as all samples have considerable clay mineral content, as is shown in non-negligible CEC results. The incorporation of pressure plate suction measurements on samples, allows suction dissipation to be quantified and evaluated alongside moisture content and electrical resistivity

Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz

We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum. We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible representations of $U_q(sl_2)$ with definite parity). In this paper we consider the approximation of noninteracting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum.Comment: revtex, 30 pages, 6 Figs, 8 postscript files are enclosed, important references are adde

3D ground model development for an active landslide in Lias mudrocks using geophysical, remote sensing and geotechnical methods

A ground model of an active and complex landslide system in instability prone Lias mudrocks of North Yorkshire, UK is developed through an integrated approach, utilising geophysical, geotechnical and remote sensing investigative methods. Surface geomorphology is mapped and interpreted using immersive 3D visualisation software to interpret airborne light detection and ranging data and aerial photographs. Subsurface structure is determined by core logging and 3D electrical resistivity tomography (ERT), which is deployed at two scales of resolution to provide a means of volumetrically characterising the subsurface expression of both site scale (tens of metres) geological structure, and finer (metre to sub-metre) scale earth-flow related structures. Petrophysical analysis of the borehole core samples is used to develop relationships between the electrical and physical formation properties, to aid calibration and interpretation of 3D ERT images. Results of the landslide investigation reveal that an integrated approach centred on volumetric geophysical imaging successfully achieves a detailed understanding of structure and lithology of a complex landslide system, which cannot be achieved through the use of remotely sensed data or discrete intrusive sampling alone