Freely available online through the author-supported open access option.

Abstract

The capacitive resistivity (CR) method is a time- and labor-saving alternative to traditional direct current (DC) resistivity methods. The line electrode variant of CR suffers from the absence of data inversion programs as available for the DC resistivity method. Direct current inversion programs were applied to determine the resistivity distribution from CR measurements using an approximately equivalent four-point dipole–dipole configuration. We optimized configurations to minimize the systematic error applying DC inversion programs to CR, using data based on the comparison of the two-dimensional sensitivities of the proposed DC approximations. The optimal four-point dipole–dipole geometry has a dipole length of 80%.

The capacitively coupled resistivity (CR) technique can outpace direct current (DC) methods in terms of measurement speed and data density but still lacks interpretation programs. We investigated the possibility of using established DC resistivity inversion programs on CR data to obtain more reliable subsurface resistivity models.

The CR method using line electrodes features quick and cheap data acquisition. It has been designed for highly resistive grounds, which are difficult to access with DC resistivity measurements. This study analyzed approximations of DC four-point dipole–dipole configurations for CR line electrode data to improve the applicability of DC inversion programs for CR data.

The CR method utilizes electrodes, which couple capacitively to the ground, to measure the subsurface resistivity. A realization of CR measurements uses line electrodes in a dipole–dipole configuration arranged in a straight line on the ground. Each dipole consists of two line electrodes with opposite polarity (see Fig. 1 ). The electrodes are electrically insulated to prevent galvanic current flow. By applying an alternating current (AC) voltage to the transmitter, the ground reacts with electrical current flow to counter the changing electrical potential on the cables. The displacement current (time-varying electrical field or potential) governs the energy transfer from the cable to the ground. The subsurface current flow eventually causes a potential field, which can be mapped with the receiver dipole. The potential is integrated along both receiver electrodes individually and ultimately provides the voltage. The introduced current and the measured voltage can be converted to an apparent ground resistivity in the same manner as for traditional DC resistivity methods by considering the geometric factor specific to the electrode configuration used (cf. Timofeev, 1974).

The major advantages of the CR method are the good applicability on highly resistive ground and the high measurement speed to data density ratio compared with DC methods. Thus, the method is of particular interest for studies in areas of permafrost, arid regions, and urban environments. The major drawbacks are, first, that the applicable frequency band limits the investigation depth and, second, that there are no suitable inversion programs. While the first is inevitable, the latter can be addressed by using the available DC inversion programs. When CR line data are subjected to inversion, the data geometry is approximated by the traditional four-point dipole–dipole geometry, where the point electrodes are positioned at the outer ends of the original line electrode dipole (cf. Geometrics, 2001).

The first known study on this method was performed by Timofeev in Russian and partly published in English (Timofeev, 1994; G. Rozenberg translated and summarized Timofeev's work in the framework of a Canadian–Russian cooperation in more detail). Later, Kuras performed a thorough study on the fundamental theory (Kuras, 2002; Kuras et al., 2006), suggesting an applicable frequency band for capacitively coupled resistivity: 
\[\frac{I}{2\mathrm{{\pi}}CU_{\mathrm{T}}}{<}f{<}\frac{\mathrm{{\rho}}}{\mathrm{{\pi}{\mu}}_{0}L^{2}}\]
[1]
where I is the introduced electric current, C is the system capacitance, UT is the transmitter voltage, f is the applicable frequency, ρ is the subsurface bulk resistivity, μ0 is the magnetic permeability, and L is the length of the measuring system. The lower limit stems from the limited power output of the current source. The constant current source must overcome the transmitter's total impedance to sustain a constant current; high currents favor the signal to noise ratio but require high voltages. The upper limit is imposed by the requirement to suppress electromagnetic induction effects. This upper limitation is known as quasi-static approximation. We must obey this frequency limitation to use DC resistivity interpretation schemes like, for instance, the static geometric factor (cf. Timofeev, 1974; Kuras, 2002).
Geometrics Inc. designed the OhmMapper based on Timofeev's pioneering work. It is a capacitively coupled resistivity meter with line electrodes (another CR method is the plate-wire combination). Groom (2008) described the principles, fields of application, advantages, and disadvantages of this device. The difference (compared with DC) in the physical coupling gives a very distinct geometric factor KL, which was determined by Timofeev (1974) as 
\[K_{\mathrm{L}}{=}\frac{\mathrm{{\pi}}l}{\mathrm{ln}\left\{\left(\frac{b^{2}}{b^{2}{-}1}\right)^{2b}\left[\frac{b^{2}{+}2b}{(b{+}1)^{2}}\right]^{b{+}2}\left[\frac{b^{2}{-}2b}{(b{-}1)^{2}}\right]^{b{-}2}\right\}}\]
[2]
where b = 2s/l, with s as the separation of the dipoles and l as the dipole length. For comparison, the geometric factor for DC dipole–dipole arrangements is 
\[K_{\mathrm{D}}{=}\mathrm{{\pi}}s(b^{2}{-}1)\]
[3]
This study was based on the OhmMapper specification by means of data modeling. Hauck and Kneisel (2006) and Ziekur and Grinat (2007) tested this device in the field (i) to verify its reproducibility and (ii) to compare its results with those of traditional DC resistivity meters. The field tests were performed at different sites and with different setups. In general, all tests were well reproducible and CR has been proven to be self-consistent; however, the CR data didn't match the measured DC data well and showed slight deviations. Large-scale features matched, but details were different (particularly the depth estimation) and, generally, the amplitude did not match well for identical geometric parameters.

We believe these discrepancies stem, at least partly, from the wrongly assumed CR data's geometry during the inversion process using a DC inversion program. Both methods (CR and DC) apply different mechanisms to couple the electric current into the ground and the measured response is different for each method. To use a DC geometry for CR data in a DC inversion program, one must be certain that both configurations would provide similar results for a given geology. Otherwise, this procedure may result in artifacts in the final model. In this study, we systematically analyzed the approximation of capacitive line electrodes by four-point dipole–dipole geometries with the aim of reducing such artifacts. Our analysis was based on the hypothesis that minimizing the difference between the spatial sensitivity distributions associated with either method should produce good resemblance between the resulting inverse models.

We started work with the sensitivity formula of Zhou and Greenhalgh (1999) to include the different requirements of CR line data. This formalism was then used to analyze suitable four-point approximations. To identify the optimal four-point geometry, we directly compared synthetic data for both the CR line data and DC four-point data. The comparison was done using four distinct, ideal subsurface models. Finally, we inverted the computed CR line data by using the optimal four-point approximation to confirm our methodology.

Theory and Methods

Sensitivity

In the resistivity inversion programs currently available, the sensitivity (also known as the Fréchet derivative or Jacobian matrix) is a very important parameter. It represents the perturbation of the measured variable across the variation of the model parameters. We derived an expression for the sensitivity of CR line electrode configurations. We know that configurations of point electrodes can be represented by superposition of single-point electrodes (the principle of superposition of electrostatic potentials), and we approximated the line electrodes with a finite number of point electrodes (geometric line representation). This idea is supported by the fact, that, within the quasi-static approximations, at each point along the line electrode the current input into the ground has been shown to be constant (Timofeev, 1974) and, therefore, represents a single independent point electrode (where independent means that its particular effect on the local potential field cannot be reproduced by any other point electrode).

Zhou and Greenhalgh (1999) derived the formula for the sensitivity of the dipole–dipole array analytically. That formula can be reduced to the more fundamental pole–pole array expression 
\[S_{i}{=}\frac{{\partial}\mathrm{{\rho}}_{\mathrm{a}}}{{\partial}\mathrm{{\sigma}}_{i}}{=}{-}K_{\mathrm{P}}\mathrm{{\nabla}}G(\mathbf{\mathrm{r}}_{\mathrm{C}},\mathbf{\mathrm{r}}_{i}){\cdot}\mathrm{{\nabla}}G(\mathbf{\mathrm{r}}_{\mathrm{P}},\mathbf{\mathrm{r}}_{i})\]
[4]
where ρa denotes the measured apparent resistivity and σi the conductivity of the ith model cell (model parameters); KP is the geometric factor of this pole–pole geometry formed by the electrodes denoted by subscripts C and P; 
\[G(\mathbf{\mathrm{r}}_{\mathrm{q}},\mathbf{\mathrm{r}}_{i}){=}\frac{1}{2\mathrm{{\pi}}{\vert}\mathbf{\mathrm{r}}_{q}{-}\mathbf{\mathrm{r}}_{i}{\vert}}{=}G_{\mathrm{q}}\]
[5]
denotes Green's function for a point current source, where ri are the position vectors for the subscribed model cells i and rq is the position of the source.
This sensitivity expression for the pole–pole array can be extended by additional pole–pole arrays to form any kind of geometry. For example, by adding one more transmitter and one more receiver electrode with an opposite polarity to form a dipole–dipole array, we find 
\begin{eqnarray*}&&S_{i}{=}\\&&{-}K_{\mathrm{D}}[\mathrm{{\nabla}}(G_{\mathrm{C}1})\mathrm{{\nabla}}(G_{\mathrm{P}1}){+}\mathrm{{\nabla}}({-}G_{\mathrm{C}2})\mathrm{{\nabla}}(G_{\mathrm{P}1})\\&&{+}(\mathrm{{\nabla}}G_{\mathrm{C}1})\mathrm{{\nabla}}({-}G_{\mathrm{P}2}){+}\mathrm{{\nabla}}({-}G_{\mathrm{C}2})\mathrm{{\nabla}}({-}G_{\mathrm{P}2})]\end{eqnarray*}
[6]
where the first term inside the brackets stems from the original pole–pole array and the rest correspond to the interaction of the new electrodes. Zhou and Greenhalgh (1999) derived the same equation for the dipole–dipole array, but analytically. We extended this formula such that it includes a certain number of both transmitter electrodes (+c1, −c2) and receiver electrodes (+p1, −p2), still forming two dipoles. Such an array of physically equal electrodes would have the sensitivity 
\begin{eqnarray*}&&S_{i}{=}{-}K\left[{{\sum}_{\mathrm{c}1{=}1}^{\mathrm{C}1}}{{\sum}_{\mathrm{p}1{=}1}^{\mathrm{P}1}}\mathrm{{\nabla}}(G_{\mathrm{c}1})\mathrm{{\nabla}}(G_{\mathrm{p}1})\right.\ \\&&\left.\ {+}{{\sum}_{\mathrm{c}2{=}1}^{\mathrm{C}2}}{{\sum}_{\mathrm{p}1{=}1}^{\mathrm{P}1}}\mathrm{{\nabla}}({-}G_{\mathrm{c}2})\mathrm{{\nabla}}(G_{\mathrm{p}1})\right.\ \\&&\left.\ {+}{{\sum}_{\mathrm{c}1{=}1}^{\mathrm{C}1}}{{\sum}_{\mathrm{p}2{=}1}^{\mathrm{P}2}}\mathrm{{\nabla}}(G_{\mathrm{c}1})\mathrm{{\nabla}}({-}G_{\mathrm{p}2})\right.\ \\&&\left.\ {+}{{\sum}_{\mathrm{c}2{=}1}^{\mathrm{C}2}}{{\sum}_{\mathrm{p}2{=}1}^{\mathrm{P}2}}\mathrm{{\nabla}}({-}G_{\mathrm{c}2})\mathrm{{\nabla}}({-}G_{\mathrm{p}2})\right]\end{eqnarray*}
[7]
which, including the polarity in the sum, reduces to 
\[S_{i}{=}{-}K{{\sum}_{t{=}{-}\mathrm{C}2}^{\mathrm{C}1}}{{\sum}_{r{=}{-}\mathrm{P}2}^{\mathrm{P}1}}\mathrm{sign}{\,}(tr)\mathrm{{\nabla}}(G_{\mathrm{t}})\mathrm{{\nabla}}(G_{\mathrm{r}})\]
[8]
where Gt and Gr denote Green's function (Eq. [5]) for transmitter and receiver dipoles, respectively. The sign function (polarity change) includes the dipole character of both electrodes (1 and 2) for both dipoles (C and P). The index t consists of the former indices c1 and c2, whereby sign(t) indicates the polarity of the current electrodes. The index r is defined in analogy to t but consists of the potential electrodes p1 and p2. Using the line electrode geometric factor K = KL (Eq. [2]), choosing both rt and rr (Gq in Eq. [5]) equally distributed on their respective line electrodes and increasing (C1, C2, P1, P2) → ∞ (infinite point electrodes per line electrode), this expression yields the formulation of the CR line electrode dipole–dipole sensitivity. This expression for the sensitivity is only valid, however, for homogeneous ground; otherwise, the sensitivity calculation must account for subsurface heterogeneity.

Sensitivity Analysis

From Eq. [8], we can compute the sensitivity of the CR line electrodes. We compared CR with DC sensitivities to find the most adequate four-point configuration to represent the CR line electrodes in terms of sensitivity. The comparison was based on the linear first-order norm L1 of the sensitivity differences. The lower the discrepancies (and with it the norm), the better the quality of the approximation.

For the numerical evaluation of the line electrode sensitivity, we chose the same number N of point electrodes for each line electrode and equally distributed rt,r along the corresponding line electrodes (Eq. [8]). We will refer to the resulting line-electrode dipole–dipole configuration as a line array and to the four-point dipole–dipole configurations as four-point arrays. Four line electrodes (two positives and two negatives) are needed for one CR setup. The four-point arrays naturally come with four point electrodes (two positives and two negatives).

Both configurations are shown in the sensitivity plots in Fig. 1. The electrodes of the DC array are assumed to be at the outer ends of the line electrodes, so that the dipole length l is equal for both configurations. Currently, this DC four-point configuration is commonly used as a CR line electrode approximation. The sensitivity S was computed for a homogeneous ground with square unit cells of the size 1/40 l; each surface cell under the line electrode can be considered active, either in terms of current introduction or potential measurement. The physical dimensions of the unit cells do not contribute to the result because the calculations are naturally based on electrostatic Green's functions, which are dimensionless. The result was normalized so that all cells sum up to unity for comparability. The values range from −0.1 to 0.1. We chose to plot the sensitivity logarithmically. The resulting sensitivity strongly depends on the choice of the ratio n = r/l where r is the separation and l is the dipole length. Again, because of the electrostatic nature of our computations, only the ratio is of importance; equal ratios will provide an equal sensitivity distribution. We kept the dipole length constant and computed the sensitivity for a range of separations. The ratio n ranged from 0.25 to 2.50 in 10 steps, which covers the most common range in field application.

Figure 1 shows n = 0.25; the results with increasing ratios are similar. The sensitivities of the line and the four-point arrays are different. Even though they look similar at first glance, they differ in detail. The line electrode sensitivity seems overall smoother and blurred, whereas the four-point configuration clearly features high sensitivity peaks near the electrode stakes. Moreover, the polarity change is at different positions and has different inclination angles.

The four-point approximation shown in Fig. 1B is currently used to invert measured line electrode data. Apart from the measured data (resistivity and topography), the geometry usually is the only information provided to the program. A better approximation should directly improve the results.

Graphical comparison of the sensitivity distribution is subjective and cumbersome, therefore we need another, more objective way to compare the sensitivities. Figure 2 presents a cross-plot of all values of the four-point sensitivity Si4P and all line electrode sensitivity values Siline for all model cells i. Each dot represents one model cell and the axis indicates the sensitivity value. Model cells with equal values for both configurations fall near the diagonal. The distance to the diagonal measures the discrepancy between both values. The dotted lines represent a mismatch of one order of magnitude (e.g., S4P = 10−6 and Sline = 10−5).

Figure 2 shows that the discrepancy between the original and approximated configurations is on the order of half a magnitude. The sensitivity of the four-point approximation tends to be lower than that of the line electrode for most of the model cells. This is due to the fact that the regions near the electrodes are very sensitive and therefore have high values; on the other hand, the sensitivity of the bulk data is reduced because the matrix is normalized. This means that the very few extreme high values eventually reduce the significance of all other cells. To represent the discrepancy between the two configurations, we chose the L1 norm, defined by 
\[L^{1}(S_{i}^{4P},S_{i}^{\mathrm{line}}){=}{{\sum}_{i}}\left|\frac{S_{i}^{\mathrm{line}}{-}S_{i}^{4P}}{S_{i}^{\mathrm{line}}}\right|\]
[9]
The L1 norm sums the relative distances to the diagonal and provides a stable quantity to measure the discrepancy. With the discrepancy quantified, we can compare all plausible four-point geometry approximations for the line electrode configuration. We tested all geometries within the length of the line electrodes. Because of reciprocity (Kuras et al., 2006), we can safely assume that the transmitter and receiver dipoles are symmetric in terms of their sensitivity for homogeneous ground. The test was performed with the same model as the previous sensitivity computation (see Fig. 1). Each line electrode was divided into 20 unit cells. The sensitivities were computed for every four-point geometry (see Fig. 3 ). The result provides a field of sensitivities Si4P(X1, X2) with respect to the two dimensions X1 and X2 denoting the respective electrode move-outs: X1 represents the move of C1 and P2 and X2 the move of C2 and P1. In each case (X1 and X2), both electrodes are moved together and toward each other. For homogeneous ground, the reciprocity condition introduces a mirror plane for the sensitivity perpendicular to the dipole and centered between the transmitter and receiver. This mirror plane allows the simultaneous move of an electrode and its counterpart without losing any information. This scheme provided 400 four-point geometries, which were assessed with respect to their quality as a line electrode approximation (e.g., [1,20] represents the four-point geometry of Fig. 1B).
The assessment of the geometry approximations was quantified by the L1 norm (Eq. [9]). We calculated the L1 norm for n = 0.25 to 2.50 for all 400 geometries. For each geometry, we averaged across all 10 n values. The results are mapped in Fig. 4 with the dimensionless formulation of 
\[L(X_{1},X_{2}){=}\mathrm{log}_{10}\left\{\frac{L_{1}(S_{i,X_{1},X_{2}}^{4P},S_{i}^{\mathrm{line}})}{\mathrm{min}[L_{1}(S_{i,X_{1},X_{2}}^{4P},S_{i}^{\mathrm{line}})]}\right\}\]
[10]
so that the lowest value of L is set to zero. The L value has three distinct and approximately linear components in its matrix: (i) the diagonal X2 + X1 = 21 gathers all geometries that have the dipole center in common with the CR line geometry, hence, symmetry is favorable; (ii) the line X1 = 10 is the outer position of the polarity change of the CR line's sensitivity and the outer position of the polarity change of the four-point geometry's sensitivity, matching the position of the outer point electrodes (Fig. 1B), hence, matching the outer polarity change is favorable; and (iii) the line X2 = 16 − CX1 is the inner position of the polarity change of the line electrode's sensitivity and the same reasoning as in (ii) applies, so matching the inner polarity change is favorable, too. These three lines create a triangle of which the corners are of particular interest. Figure 4 shows four special points of interest with the corresponding L values:

Point 1 represents the result of the geometry that is currently used as an approximation for the line electrode. The coordinates are [1,20] and the sensitivity of this geometry is shown in Fig. 1B. The L value is L(1,20) = 0.128. We will refer to this geometry as 100pc (100% dipole length of the four-point array compared with the length of the line array).

Point 2 is the global minimum of L = 0.000. This point is the intersection of lines a and c. The coordinates [6,15] describe a four-point dipole–dipole configuration of l4P = 0.8lline with the same dipole center as the line electrodes. In the following, this geometry will be referred to as 80pc.

Point 3 is the second local minimum of L(10,10) = 0.063, where line a intersects line b. The coordinates [10,10] indicate that this geometry uses half the original CR dipole length. Note that there is no true midpoint in this 20 by 20 grid, so the dipole center of this geometry is virtually identical to that of the line electrodes. We will refer to this geometry as 50pc. Kuras et al. (2006) previously advocated this particular approximation. Their approach was based on the comparison of the geometric factors between DC and CR, which are most similar for this geometry. This geometry approximation is certainly best for comparing the apparent resistivity between DC and CR but cannot serve well in inversion programs, which compute the sensitivity from this geometry.

At first glance, Point 4 does not seem to be of any importance, but that is misleading. Numerically, the maximum line electrode sensitivity is located at exactly this point (see Fig. 1A, between the line electrodes of the transmitter and receiver). The maximum sensitivity of any four-point geometry is situated at these point electrodes (see Fig. 1B). With increasing order q, the Lq norm becomes more and more influenced by the misfit of the highest argument, which means that, for large q, we will find the lowest Lq matching only the highest argument. Already at the second order, this particular point becomes supervalued and the algorithm matches the highest line sensitivity value.

Multipoint Approximation

Although no inversion programs are available for DC configurations with more than four point electrodes, we also analyzed multipoint approximations. These approximations were characterized by N point electrodes per line electrode, whereby the polarity of the point electrodes did not change within one line electrode. For large N, the sensitivity of N point electrode approximations must converge to the line electrode sensitivity. Hence the L1 norm must vanish for N → ∞. For instance, a multipoint DC configuration with two points per line electrode would be [X1 = (5, 15); X2 = (5, 15)], referring to Fig. 3.

We analyzed the L1 norm for N = [1, 2, 3, 4, 6, 8, 12]. The line electrode reference was approximated by the N = 24 configuration. Each configuration consisted of N equally distributed and centered point electrodes per line electrode. In this context, centered means that the distances of the outermost point electrodes were equal at the beginning and the end of each respective line electrode. The single-point configuration was 50pc. The L1 norm was normalized to the N = 1 value.

Figure 5 shows the L1 values. The result of the sensitivity calculation indicates that the L1 norm already decreased from N = 1 to N = 2. All approximations with N > 1 show an L1 of less than one-half. Note that this calculation used predefined configurations and represents the trend if the number of point electrodes used to substitute for the line electrode is increased. To find an optimal configuration, a more detailed analysis must be performed.

Data Modeling

We have shown that the sensitivities differ between the line and the four-point array geometry. Therefore, measurements with those different geometries must be dissimilar for a general inhomogeneous ground as well. If we want to evaluate line data with state-of-the-art DC inversion programs, we need to approximate this data with a suitable four-point geometry. We tested the validity of three four-point approximations by comparing synthetic data from well-structured subsurface models for four-point and line arrays. The better the obtained four-point data matched the line data, the more likely it was that the inversion program would recover the original subsurface model from the CR data.

The four-point and line array geometries were used to compute data with four two-dimensional subsurface models. The models represent (i) horizontal layers with the interface at 2 m, (ii) upside-down horizontal layers with the interface at 2 m, (iii) a vertical fault, and (iv) an infinitely long tube with a 2-m square cross-section at a depth of 1 m (Fig. 6 ). Each model consisted of two resistivity regions, 100 and 1000 Ω m. For the three-dimensional computation of the line data, we extended these models in the third dimension to simulate the two-dimensional character of the final data.

We computed the line data using the three-dimensional finite element program COMSOL Multiphysics (COMSOL, Burlington, MA). The module Quasi-Electrostatics was set up with a constant frequency of f = 16,500 Hz, which is the operating frequency of the OhmMapper. The size of the finite element mesh of our three-dimensional model space was much larger (1:1000) than the actual dipole length of 5 m. We used Dirichlet boundary conditions (zero potential) far away from the line electrodes (|x|/l = y/l = 1000) and Neumann boundary conditions [∂U/∂(z,y) = constant] at the ground surface (z = 0 m) and at y = 0 m. The latter planes were symmetrical for the potential because of the dipole orientation in the x direction, the homogeneity in the y direction, and the nonconducting ceiling (air). The transmitter dipole introduced the current I in an area of small width compared with the length of the antennae (1:100). The current in- and outflow was constant per unit area. This directly resulted from the applied frequency, which must be chosen so that the line electrode reactance governs the total system impedance. If not, the current experiences larger resistance at the outer ends of the dipole than in the middle and therefore more current will flow in the dipole center than at the outer ends. Timofeev (1974) assumed a uniform current flow to derive the line electrode geometric factor; hence, we must obey this assumption when using his geometric factor. Each of the receiver electrodes was defined as a single line. On each line, the potential was integrated along the line. Two adjacent receiver lines, j and j + 1, provided the measured voltage Uj, which is the difference between their potential line integrals. Thus, the apparent resistivity ρa results from the voltage-to-current ratio and the geometric factor for line electrodes KL (see Eq. [2]): 
\[\mathrm{{\rho}}_{j}^{\mathrm{a}}{=}K_{\mathrm{L}}\frac{U_{j}}{I}\]
[11]
For calibration, we used a homogeneous earth model with 27 adjacent receiver lines. The obtained resistivity was, on average, 0.32% too low with a standard deviation of 0.041% (for 26 separations). The final data were computed for the five shortest separations.

For the computation of the four-point data, we used the DC inversion and modeling program DC2DResInv (Günther, 2004). For comparison, we chose three four-point array geometries with a common dipole center but varying dipole length: (i) full dipole length ([1,20], 100pc), (ii) 80% of the dipole length ([6,15], 80pc), and (iii) half the dipole length ([10,10], 50pc); (i) is the state-of-the-art approximation and is shown in Fig. 1B; (ii) has been found to match the line sensitivity best (see global minimum above), and (iii) has been found to match the line geometric factor well (see Kuras et al. [2006] and local minimum above). The three four-point data sets were computed in the same measurement layout as the line data, so that they represented genuine line data approximations, although the scheme is untypical for DC resistivity surveys. The survey was laid out with the constant dipole length of 5 m and an electrode spacing of only 0.25 m. Each computation provided the results of five transmitter–receiver separations.

Each separation was characterized by the n factor, defined by 
\[n{=}\frac{\mathrm{transmitter-receiver}{\,}\mathrm{dipole{\,}separation}}{\mathrm{dipole}{\,}\mathrm{length}}\]
[12]
The transmitter–receiver dipole separation is the distance between the inner ends of the transmitter and receiver dipole. We chose a line dipole length of lD,line = 5 m and separations starting from 1.25 m, increasing four times by 2.5 m. By definition, the n factor differed for all four geometries; hence, for convenience, we refer only to the n factor of the line electrodes (nline = [0.25, 0.75, 1.25, 1.75, 2.25]). The four-point data are always meant to be the respective n value.

Analysis of Synthetic Data

We evaluated the computed data by means of the total difference between the line data and all three four-point approximations for each n factor. The difference was calculated as 
\[\mathrm{misfit}(\%){=}\frac{\mathrm{data}_{4\mathrm{P}}{-}\mathrm{data}_{\mathrm{line}}}{\mathrm{data}_{\mathrm{line}}}100\]
[13]
The position of the array midpoint defines the x axis reference.

Horizontal Models

Neither horizontal layer model varied along the profile. Each separation provided one distinct apparent resistivity value for each geometry. The differences between those values are plotted in Fig. 7 and 8 over the transmitter–receiver separation, quantified as the n factor. The absolute deviation decreased with increasing separation (depth profiling), probably because the interface depth was fixed and the lower half space was homogeneous. For increasing separation, all geometries must converge to the same apparent resistivity; therefore, the misfit must converge to zero. Considering the entire range of separations, the 80pc geometry performed best. Only for large separations when a highly resistive layer covered a conductive one (upside-down horizontal layering) was the geometry with half the dipole length better. The misfit for the common approximation (100pc) was usually three times higher than that for the 80pc array, and the 50pc array always underestimated the line data when both of the others overestimated it and vice versa; however, the 80pc array was closest to the line data.

Vertical Model

The vertical fault model provided a varying apparent resistivity profile for each separation (n factor). The data are plotted for n = 0.25 and the data deviations compared with the line data for n = 0.25 and n = 2.25, the shortest and longest computed separation (Fig. 9 ). The data were very similar at the outer limits at ±15 m. The samples deviated in the region of one to two dipole lengths around the edge, meaning whenever one of the dipoles passed the anomaly. The absolute deviation was largest for the 100pc, intermediate for the 50pc, and smallest for 80pc geometry. Both of the first variants slightly improved their performance for larger separations, but none of them seemed to converge to the line electrode data for separations this large. The 50pc array always deviated in the opposite direction to both of the other two. The maximum absolute misfit of the 80pc array was half the value of for the 50 and 100pc geometries.

For all geometries, the deviation was larger when the receiver crossed the edge. At this point, we cannot explain this observation entirely but probably this was due to the difference in line and point dipoles. The long dipole, for instance, feels the new ground resistivity immediately when one receiver electrode crosses the interface, while the line dipole reacts more slowly and virtually linearly increases the apparent resistivity to the maximum. Thus, the misfit naturally scales with the resistivity change across the interface. When the transmitter crossed, we observed a spike in the line electrode data. This spike was due to a numerical anomaly in the computation that always appeared when the dipole center plane (perpendicular to the dipole) is equal to the model resistivity interface.

Tube Model

The infinitely long tube with square 2-m cross-section is an example for a small anomaly in the ground (compared with the dipole length). The apparent resistivity profile is plotted in Fig. 10 and shows the data for n = 0.25 and the deviations from the line data for n = 0.25 and n = 2.25. Again, the data were similar for large distances to the anomaly for all computations, although the computed value at the bounds was systematically too low (cf. the homogeneous model), because the scale of the overall deviation is much smaller in this example. The maximum amplitude was <140% of the background value for all data. Therefore, the misfit must be evaluated with respect to this maximum amplitude. The highest data deviation was for the 50pc geometry, the smallest for the 80pc geometry; the 50 and 100pc arrays deviated up to 10 and 12%, respectively, with a short separation and up to 5 and 10%, respectively, with a long separation. Considering the low anomaly amplitude, a misfit of 4% represents a data interpretation error of 10%.

Inversion Results

The computations of the line electrode and four-point geometries have distinct apparent resistivity profiles. The resistivity profiles form the input for inversion programs. Using the wrong input must produce numerical artifacts. We needed to apply the DC inversion routines to CR measurements because CR inversion tools do not yet exist. We showed which of the four-point configurations has the most similar sensitivity distribution compared with the CR line electrodes. Then we compared artificial data for the different configurations and found that the four-point configuration with the most similar sensitivity also provided the most similar apparent resistivity profiles.

We used DC2DResInv (Günther, 2004) to invert the data sets. We then compared the CR resistivity data using the 80 and 100pc four-point geometries as well as the 100pc DC data. Figure 11 shows the inverted data for the vertical fault and tube models. To obtain the CR resistivity data modified by the four-point geometries, we changed the geometry information in the CR data file into the desired four-point geometry, e.g., when using OhmMapper, the exported data file—ready for inversion—contained the 100pc four-point geometry. The inversion of the data was performed using the default options, the regularization was kept constant, and a data error of 3% was assumed.

Vertical Fault Model

Figures 11A to 11C show the results for the vertical fault model for three different data-geometry modifications. The black line at 0 m is the model interface. The inversion of the 100pc data resulted in a very distorted image (Fig. 11A). The correct position of the resistivity interface is not well defined. The regions near the interface are strongly over- or underestimated. Moreover, the distortion does not disappear toward the boundary, where the model suggests variability of up to one-third magnitude.

On the other hand, the inversion of the 80pc data performed well and provided a clear image (Fig. 11B). Very small distortion can be found near the surface close to the interface. The interface is very well resolved and the resolution naturally decreases with depth.

The last data set contained four-point DC resistivity data. The inversion resulted in a very clear model with a well-resolved interface (Fig. 11C). The resolution of the interface decreases with depth but less than for the previous data. Very little distortion is spread across the model, which could be explained by the allowed data error of up to 3%. Comparable to the previous data set, we found small variations near the surface and close to the interface.

Tube Model

As for the vertical fault model, we inverted three data sets with the same geometry modifications: the CR data with 100pc and 80pc approximations and the DC data.

Beginning with the long dipole geometry, which is incorporated in the CR data, we found a large body with a slightly increased resistivity (Fig. 11D). On top of the body is a resistive volume near the surface. Regions with lower resistivity flank the volume at both the surface and the body. The center of the main body is identical with the center of the anomaly but it extends about twice as wide; the image seems blurred. The inversion program introduced the low-resistivity regions and the resistive spot at the surface to achieve a good fit.

With the 80pc CR data, the anomaly is more distinguishable and more focused on the original position of the body (Fig. 11E). The resistivity amplitude was higher than in the previous inversion but artificially low resistivity regions were also observed.

In contrast, the reference inversion of the DC data resolved the body very well (Fig. 11F).

Discussion

For quantitative interpretation of line dipole–dipole array data, we still need to apply inversion codes originally designed for four-point arrays. In this study, we evaluated three four-point array geometry approximations and demonstrated that the best performing array has the dipole center in common and a dipole length that is reduced to 80pc. For the possibility that inversion programs will allow more than two electrodes per dipole in the future, we demonstrated that an approximation with more than one point substitute per line electrode further reduces the sensitivity contrast to the line array; however, the improvement still needs to be quantified and a rule for the optimal choice needs to be investigated. In our analyses, we found that minimizing the difference between the spatial sensitivity distributions associated with either method appears to result in a good resemblance between inversion models. To date, no inversion programs are available to process these multipoint approximations but research in this direction could prove useful because the sensitivity analysis promises an even higher model recovery (compared with the four-point approximation) at the cost of more electrodes to compute. This way the computationally expensive inversion of real line electrode data could be avoided. The more points per line electrode that are chosen, the more of the character of the line electrode should be preserved in the approximation.

To search for an optimal dipole length, we divided the line electrode into 20 parts and computed the sensitivity for each possible four-point geometry within the bounds of the line array. For a more precise approximation, the line electrode could be discretized with a finer mesh. Our code and software solution did not support finer discretization because we had a 32-bit limitation; however, the 80pc geometry approximation appeared to deviate equally around the line data. The other approximations appeared to be less balanced. It is unlikely that a perfectly matching four-point approximation exists for line data and the best match is likely to be the one that is most balanced. To take advantage of the inversion software for four-point geometries and the hardware solution using the capacitively coupled line array, however, we suggest using an 80pc dipole length equivalent to minimize the systematic data error in DC inversion interpretation schemes. Using more than one point per line electrode is likely to improve the inversion further but still needs a more detailed analysis.

We wish to thank Doug Groom from Geometrics, Inc., San Jose, CA, for his support with the understanding of the fundamental OhmMapper principle during our early work. We are indebted to Oliver Kuras of the British Geological Survey, Nottingham, for kindly sharing his literature on capacitive resistivity, his suggestion of the hypothesis, and his intense review; and Carsten Rücker of the University of Leipzig for his appreciated opinion regarding computation and model design.