A modified DOI-based method to statistically estimate the depth of investigation of dc resistivity surveys
Introduction
Nowadays, direct current (dc) resistivity surveys are frequently used to map soil electrical properties in two to three dimensions in a wide range of engineering, hydrogeological and risks management applications and to monitor dynamic subsurface processes through time-lapse measurements.
Ill-posed problems including electrical resistivity tomography (ERT) inversion are inherently difficult. The general idea behind data inversion is to find a model that fits to the data given a defined error criterion which is generally estimated from noise estimates (see for example LaBrecque et al., 1996). However, this model is non-unique since an infinite number of models can fit to the data equally. Moreover, ERT inversion is an ill-conditioned inverse problem that leads to instability of the solution. This means that a small change in the data (e.g., a small increase of noise) can lead to severe changes in the resulting model. To overcome the non-uniqueness problem as well as to introduce stability in the inversion process, regularization is added (e.g. Tikhonov and Arsenin, 1977). Regularization consists of adding some model constraints that bias the solution, generally in a way to satisfy available a priori information of the subsurface.
ERT data inversion consists then of finding the model that fits to the data subject to a model constraint such as the widely used smoothness constraint (Constable et al., 1987, deGroot-Hedlin and Constable, 1990). The resulting image must then be appraised in order to find which parts are well resolved and therefore physically interpretable. One aspect of appraising electrical images consists of finding its depth of investigation (DOI).
The depth of investigation is defined by the depth to which the data are particularly sensitive to the physical properties of the subsurface (Oldenburg and Li, 1999) or in other terms, the depth below which the data are no longer influenced by the electrical resistivity, in our case. This means that below this depth, inverted structures are not related to surface data anymore and should be interpreted with caution.
Several resolution indicators can be used to estimate the depth of investigation or to identify possible artifacts in the electrical structures. The model resolution matrix (e.g. Alumbaugh and Newman, 2000, Friedel, 2003, Hilbich et al., 2009, Oldenborger and Routh, 2009, Ramirez et al., 1995) is the approximation for non-linear problem of the resolution matrix defined theoretically for linear problems. Ideally, the matrix should be close to the identity matrix to achieve good resolution. The cumulative sensitivity matrix (e.g. Christiansen and Auken, 2012, Kemna, 2000, Nguyen et al., 2009, Park and Van, 1991, Robert et al., 2012) quantifies the sensitivity of all measurements subject to changes in the electrical structures. Generally, areas with poor sensitivity or resolution are considered as less reliable. The main difficulty consists of finding an objective and quantitative way to select a threshold value which will define “well” and “poorly” resolved areas, especially since ground truth information is not always available.
The depth of investigation (DOI) index was introduced by Oldenburg and Li (1999) in order to discriminate the electrical structures related to surface data from the ones related to the choice of a reference model introduced during inversion. The idea of Oldenburg and Li (1999) is to force the inverted model to resemble a reference model. Analyzing resulting models by using different reference models (put mathematically in an index) allows finding the electrical structures that strongly depend on the choice of the reference model, and therefore are not constrained by surface data. The analysis of the DOI index was successfully used by several authors such as Hilbich et al. (2009), Marescot et al. (2003) or Robert et al. (2011) to appraise electrical images. However, the threshold value remains difficult to define. According to Oldenburg and Li (1999) and Marescot et al. (2003), DOI index values of 0.1 or 0.2 are recommended.
All these methodologies provide information on how well the regions of the subsurface are covered by the data. Recently, Caterina et al. (2013) compared quantitatively different resolution indicators to detect artifacts, to estimate the depth of investigation, to address parameters resolution and to appraise ERT-derived geometry. They compared numerical benchmark models to their corresponding ERT images to define the errors on inverted parameters and to evaluate them against resolution indicator values allowing the definition of threshold values. In the case of field applications, the errors on inverted parameters are not known except at borehole locations and the threshold value definition is uneasy. Caterina et al., 2011, Caterina et al., 2013 proposed a methodology to overcome this problem by defining a synthetic model that resembles the inverted model to appraise. Simulated data related to this synthetic model are then inverted (all acquisition and inversion parameters being the same) and compared to the true synthetic model. The errors on inverted parameter can then be evaluated against a resolution indicator as long as there is a correlation between both resolution indicators (the one related to the real data inverted model and the one related to the synthetic data inverted model). This methodology is time-consuming but works well for the analysis of the diagonal values of the resolution matrix or the cumulative sensitivity matrix.
Except for a few work such as the tracer experiment of Robert et al. (2012) where authors had access to ground truth information to objectively select a cut-off value, the depth of investigation is commonly estimated using a recommended but arbitrarily chosen cut-off value on the selected resolution indicator (diagonal of the resolution matrix, sensitivity or DOI index). Therefore, different practitioners could obtain significantly different depth of investigation.
In this paper, we present a new methodology to statistically determine the depth of investigation. This method is based on a modified DOI index approach where statistical distributions of DOI index values are computed to discriminate between reliably- and poorly-reconstructed cells. This is a quick and easy-to-use methodology allowing the identification of statistically poorly-reconstructed areas in depth. The efficiency of the methodology is tested on synthetic and field data where the statistically determined depths of investigation are compared to the depths estimated using recommended cut-off values.
Beyond this introduction, this paper is organized as follows. First, we define the three resolution indicators considered (resolution matrix, sensitivity matrix and DOI index). Then, we present our statistical approach based on a modified DOI index approach. This method is then tested and compared to other approaches of depth of investigation estimation. This comparison, based on synthetic and real-world data will then be discussed. Finally, we draw conclusions and recommendations for depth of investigation issues.
Section snippets
Inversion of dc resistivity data
The inversion of resistivity and IP data was carried out by using the 2D inversion program RES2DINV (Loke and Barker, 1996, Loke and Dahlin, 2002). The inversion software allows the application of least-squares or robust inversions. The Gauss–Newton smoothness-constrained least-squares algorithm is commonly used to determine smooth changes in the model parameters that would minimize the sum of the squares errors between the model response and the observed data values (deGroot-Hedlin and
Sensitivity and resolution matrix analysis
The Jacobian matrix defines the sensitivity of predicted data to changes in model parameters. The cumulative sensitivities are often defined as the sum of the absolute sensitivities of the arrays used, since negative sensitivity values also give information about the subsurface. Based on this definition, the cumulative sensitivity of each model block is a measure of the amount of information or “coverage” about the resistivity of the block that is contained in the measured data set. If the
Results and discussions
We benchmarked the statistical DOI methodology on synthetic examples and tested its robustness with field data. The results obtained are compared to the depths of investigation estimated using frequently used cut-off values on model resolution matrix and scaled DOI index values.
Conclusions
Compared to the existing image appraisal tools, where the depth of investigation is commonly defined using an arbitrarily chosen cut-off value, we proposed a statistical method which allows the identification of well-constrained cells of the model without using any arbitrary threshold. This method is based on the principles of the DOI index method. This method is thus data-driven and sensitive to inversion errors. It can easily be implemented and automated. Even if we only considered 2D surveys
Acknowledgments
This work was partly conducted under the auspices of the Walloon Region Ministry (FIRST SPIN–OFF GRANSPE visa no 091/6974). It was also a part of a Ph.D. thesis funded by the Research Institute for the Science and Management of Risks at University of Mons (Faculty of Engineering, Geology and Applied Geology Department). The authors thank the Editor and two anonymous reviewers for their relevant comments and suggestions that improve the overall quality of the manuscript.
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