**Synthetic Studies to determine the Effects of Anomalous Magnetic Permeability on a New Electrical Resistivity Tomography/Magnetometric Resistivity Survey System**
Matthew A. Ludwig, University of Wisconsin, Madison, WI David Alumbaugh, University of Wisconsin, Madison, WI Douglas LaBrecque, Multiphase Technologies, LLC., Sparks, NV Roger Sharpe, Multiphase Technologies, LLC., Sparks, NV Gail Heath, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID
**Abstract**
Three-dimensional model studies are presented to determine the effects of electromagnetic induction and magnetic permeability on magnetometric resistivity measurements, as well as on images resulting from inversion of the data. This involves the use of two different forward modeling algorithms; a full electromagnetic code that includes anomalous magnetic permeability, and a steady-state algorithm that assumes free-space magnetic permeability. To determine the nature of problems caused by electromagnetic induction in the presence of permeable materials the two synthetic data sets are both compared directly, as well as inverted with a steady-state inversion algorithm. Including a magnetically permeable target significantly changes the synthetic data and produces noticeable artifacts in the inverted sections. However, despite the presence of artifacts, the responses are still dominated by conductivity effects and the extent of both conductive and resistive targets are well defined in the images.
**Introduction**
ERT is an adaptation of the traditional surface resistivity method where by tomographic resistivity images are produced through inversion of measured voltages generated by the grounded current source. ERT generated images have seen use for monitoring subsurface processes including the movement of fluids and contaminants in both the saturated and vadose zones (Daily et. al., 1992; Stubben and LaBrecque, 1998; Yang, and LaBrecque, 1999), the imaging of subsurface barrier emplacement (Pellerin et. al., 1998; Daily and Ramirez, 1999) and the monitoring of new, aggressive, environmental remediation processes (Ramirez, et. al., 1993; Ramirez et. al., 1995; LaBrecque et. al., 1998) The magnetometric-resistivity (MMR) method is seeing increasing attention in the environmental sciences where it is being used to augment traditional electrical resistivity tomography (ERT). The MMR method measures the magnetic fields generated by a known electric currents that are assumed to be steady state (Edwards and Nabighian, 1991). ERT inversion produces images exclusively of the ground resistivity distribution, and do not consider a distribution of magnetic permeability as this property does not affect the steady-state voltage measurements. Since the MMR method employs magnetic fields, the measured data will include contributions due to magnetic induction in magnetically permeable bodies. One possibility is that the permeability responses are small compared to the resistivity signal, and therefore these effects might be ignored during resistivity inversion. On the other hand if the responses are relatively strong they will affect the inversion and may result in artifacts, and possibly will prevent the inversion algorithm from converging. The questions addressed in this work, are: whether the inversion will be able to fit a ground resistivity distribution to the permeable responses, and if so, how will such a magnetic permeability-affected resistivity distribution compare to the actual resistivity distribution? Analyzing the ERT/MMR results of adding finite magnetic permeability to a buried target is described in this paper. This synthetic study is the final phase of a larger project involving INEEL, UW, and MPT. First, a prototype ERT/MMR survey system was designed and assembled at the INEEL. Preliminary numerical modeling studies completed by MPT provided the needed insight for successful construction of a field test site on the INEEL property. These studies also allowed for optimization of the survey protocol, including source size and configuration, (LaBrecque, 2003) at the INEEL site. The, UW group was included most recently to numerically investigate possible problems associated with electromagnetic coupling, and to determine the effect of magnetically permeable targets. The work began by comparing MMR responses computed with a DC solution (LaBrecque et. al., 2003) to those resulting from a full EM solution (Alumbaugh and Newman, 1996; Ludwig, 2003). Next the amount of electromagnetic coupling was determined by increasing the source frequency from DC to 8kHz A more detailed description of the initial work can be found in Ludwig (2003). The final study, which is described in detail here, includes targets of anomalous magnetic permeability. This synthetic study is comprised of three parts. First, verification of the accuracy of the full EM algorithm (em1d2d3d) with respect to modeling low-frequency magnetic fields in the presence of anomalous permeability contrasts is provided. Next, the synthetic responses computed with and without anomalous magnetic permeability are reviewed. Finally, the effect of magnetic permeability is examined by inverting the synthetic data sets with a joint ERT/MMR direct current inversion routine.
**Anomalous Magnetic Permeability Modeling Verification**
**Method** An important step in any numerical modeling process is verification of the forward algorithm (Anderson and Woesner, 1992) by comparing with results from either some type of exact solution or measurement over a known structure. To verify the accuracy of the EM routine used here to calculate the scattered field responses due to magnetically permeable bodies, we compare the numerically computed responses for a cube-shaped anomalous body with analytically derived responses for a uniformly magnetized sphere of equal volume. The finite difference (FD) scenario includes a cube 1.5m on a side centered at (0.0, 0.0, 2.25). The cube is buried in a 30 -m half-space, has magnetic permeability equal to 1.1 times that of free space, and no anomalous resistivity. The transmitter is a 1m long, vertically oriented source (VES), centered at (0.0, -2.0, 2.25). The source is operating with a current of 1 amp and frequency of 8 kilohertz. The standard 0.5-meter grid magnetometer array (Ludwig, 2003;and described in detail below) was deployed 35 centimeters above the ground surface. The primary magnetic field resulting from the source in free space was calculated at the center of the cube to provide the inducing field for the analytical solution. This field had a value of -22.5832941 nT/A and is exclusively x-aligned. Blakely (1996) describes an analytical solution (AS) (Equation 1), for a uniformly magnetized sphere as ** (1)** where **(2)**
and is the magnetic field strength, is a unit correction constant, is the magnetic moment of the sphere, is the distance from the center of the sphere to an observation point (i.e. the magnetic sensor), a is the radius of the sphere, and , a vector, is the inducing field. We simplified the solution somewhat by centering the VES at the same depth as the anomalous cube. Thus, the inclination and declination of the inducing field are equal to zero as the resulting magnetic field is entirely x-aligned. The diameter of the sphere was determined to make its volume identical to that of the cube in the FD calculations. This results in the magnetic moments of the two bodies being equal per Equation 3:
**(3)**
where is the magnetic moment, is the magnetizing field, is the susceptibility contrast between the body and the half-space, and V is the volume of the anomalous body (Blakely, 1995).
**Figure 1** Comparison of em1d2d3d to the analytical sphere solution. The left column presents 2D contour plots of each magnetic field component computed with the analytic sphere solution, and the middle column are results from em1d2d3d. B_{x}, B_{y}, and B_{z} are (a) and (b), (c) and (d), and (e) and (f), respectively. Plots (g), (h), and (i) are x-aligned profiles of B_{x}, B_{y}, and B_{z,} respectively, for y= -1.0.
**Verification Results**
Figure 1 shows the results of the verification test. Qualitatively, it is apparent that the character of the response within each field component is very similar. A more detailed presentation is made in the far right column of Figure 1. In these plots each field component x, y, and z is contained in profiles (g), (h), and (i) respectively. The worst agreement occurs furthest from the target body in the horizontal field components (Figure 1 (g) & (h)) and directly over the target in the vertical component (Figure 1 (i)). It is obvious from this set of profiles that em1d2d3d is computing accurate magnetic field responses with respect to magnetic permeability contrast. In addition, these results also reinforce the earlier conclusion that an operating frequency of 8kHz does not increase the level of EM coupling to a point that comparisons to DC results cannot be made.
**Modeling Targets with Anomalous Magnetic Permeability**
**The Synthetic Modeling Study Description** The 3D electromagnetic property distribution shown in Figure 2 was used throughout this synthetic study. Targets modeled include a low contrast conductive target in a half-space, a high contrast resistive target in a half-space, a magnetically permeable target in a half-space, and combined permeable and resistive, and permeable and conductive targets in a half-space. In each case the target is 2 m x 1 m x 1.5 m, and the depth to the top of the target is 1.5 m. The half-space is defined with a resistivity of 30 -m and permeability equal to free-space. Magnetically permeable targets were defined with =1.1_{0}. Resistive and conductive target bodies have electrical resistivities of 1x10^{+8} -m, and 10 -m respectively. **Figure 2 **Spatial distribution of resistivity (and/or permeability). (a) Green represents the homogeneous 30 Ohm-m half-space. Blue indicates the buried target. Location of each borehole deployed VES, numbered 1-4, is indicated; (b) plan view of the survey site showing the sensor array of magnetometers
Magnetic fields due to various electrical transmitters were numerically calculated for the array of three-component magnetometer locations shown in Figure 2. The array extends 4m on a side. Each sensor is 0.5m distant from adjacent sensors 0.35 m above the surface. The transmitter electrodes are distributed in four boreholes at the corners of the magnetometer array. The boreholes are 6m deep and electrodes are distributed at 1 m intervals from the surface to bottom of the well. Both vertical and horizontal transmitters are modeled. Vertical transmitters are 2m in length at 0-2m depth, 2-4m depth and 4-6m depth in each borehole except well #2. Horizontal transmitters employed electrodes in wells 1 and 4, 4 and 3 and 1 and 3. Due to the arrangement of the electrode arrays, the horizontal dipoles are 4m long when wells parallel to the axis are employed, and 5.65 m long when diagonal wells are used. Current injection is modeled at seven depths spaced one meter apart (0 m, 1 m, 2 m, 3 m, 4 m, 5 m, and 6 m) for each well pair. The lead wires going from the transmitter to the electrodes are routed away from the central survey area as described for the Mud Lake site by (Ludwig, 2003). Due to numerical convergence problems at lower frequencies, a moderately high frequency (8 kHz) alternating current was employed. Application of an alternating magnetic field ensures that any static natural-earth magnetic field effects should cancel during stacking in a field situation, and thus only permeability responses due to the transmitter wires are considered. Further, comparing em1d2d3d results with those from the static field solution (Ludwig, 2003) revealed that EM coupling in the magnetic field data is essentially negligible in the study environment so that 8 kHz responses are effectively identical to 8 Hz and DC data. The insignificance of EM coupling is supported by comparing the scattered field results from the em1d2d3d code at 8kHz to results from the analytical solution for a uniformly magnetized sphere. These tests are important because the inversion algorithm expects DC data, but the EM forward modeling routine used to generate the magnetic fields converged better at higher operating frequencies.
**Magnetic Permeability Effects**
**Forward Modeling Results** The effects of adding magnetic permeability to the simple resistivity distribution presented in Figure 2 are analyzed using two methods. First, several examples of scattered magnetic field responses computed by the forward modeling code for the permeable target scenarios are described. More importantly, the results of inverting the synthetic data sets to produce 3D conductivity distributions are presented. It is important to note that the effects of including magnetic permeability in the conductive and the resistive targets were found to be additive (Figure 3). For example, synthetic data sets were computed for conductive targets, conductive and permeable targets, and targets with only anomalous magnetic permeability. The results of the conductive and permeable model (the solid lines Figure 3) were found to be nearly identical to the results of manually summing the conductive model and the permeable model (the dashed lines in Figure 3). A similar relationship is apparent with the resistive models. In fact, the permeability effect was found to be independent of the conductivity of the body. This is reinforced by the results presented in Figure 4. Here, the diamonds show the results of subtracting the conductive only responses from the conductive and permeable results. The squares present the difference of the resistive permeable and the resistive only, and finally, the results of a permeable only target are displayed at triangles. This leads to a very important statement about MMR that at least for the contrasts analyzed here, the MMR responses to the different properties are completely separable, i.e., are not coupled. This is to be expected in the absence of electromagnetic coupling, as we are looking at two separate static effects.
**Figure 3 **Shown are magnetic field profiles parallel to the y-axis at x=-0.5 m due to x-aligned HES positioned at the depths indicated. The ‘y’ oriented magnetic field (By) is shown. Dashed lines are the result of manually summing the response from a conductive target and a permeable target. Solid lines are the numerically computed responses due to a conductive and permeable target
**Figure 4 **Y aligned profile of the scattered magnetic field “permeability effect” produced with an x-aligned source. The em1d2d3d results presented are the outputs of the conductive and permeable target minus the conductive only target (diamonds), resistive and permeable minus the resistive only target (squares), and finally the results of a permeable only target (triangles) A summary of the raw responses computed with the em1d2d3d code is presented in Figure 5 and Figure 6. Figure 5 contains data from a moderately conductive target. Whereas, Figure 6 displays the results of a very resistive buried body. Two generalizations can be made about all of the plots in Figure 5 and Figure 6. First, the effects of magnetic permeability are significantly smaller in magnitude than the conductivity contrast responses. Note that there are two dependent axes. The right side is the magnetic field strength due to the conductive anomaly, and the left is a result of the permeability contrast only. Next, the permeability profiles representing the horizontal field components are always opposite in sign to the conductive only responses (Figure 5 plots (a), (b), (d), and (e)) and the same sign over the resistive target body (Figure 6 again plots (a), (b), (d), and (e)). In addition, looking at the horizontal field components resulting from the vertical source (plots (d) and (e) in Figure 5 and Figure 6), the maxima and minima are coincident with the location of the anomalous body. Finally, the vertical field component of the conductive only profiles (Figure 5f) changes sign over the target. Looking specifically at the HES profiles (Figure 5(a-c) and Figure 6(a-c)), a phenomenon unique to targets with magnetic permeability is noted. For the conductive only targets, the magnitude of the scattered field tends to increase as the source is lowered, peaking when the source and target are at the same depth, and then decrease as the source is lowered to the bottom of the boreholes. A similar phenomenon, manifesting in a ‘crossover’ effect, is encountered with VES’s.
**Figure 5** Y-aligned profiles (x=-0.5m) of magnetic responses due to x-aligned horizontal sources in the left column and vertical source in well #4 on the right at depths indicated. Solid lines are responses due to the conductive target; dashed lines are responses due to the permeability contrast. Each field component, x, y, and z, is represented in (a), (b), and (c) and (d), (e), and (f) respectively. I n contrast, the responses due to magnetic permeability continue to increase with increases in source depth (Figure 5(a-c) and Figure 6(a-c)), This is because the current running through the wires, and not that being injected into the half-space, causes the permeability responses. More specifically, the magnetic field produced by the current running through the wire leads to the electrodes produces a magnetic field or H-field. Any anomalous magnetic permeability will amplify the magnetic flux density, or B- field associated with the wire-line H-field. Therefore, as the sources are lowered, the length of the vertical wire continues to increase resulting in larger inducing fields and hence bigger scattered fields due to permeability contrasts. **Figure 6** Y-aligned profiles (x=-0.5m) of magnetic responses due to x-aligned horizontal sources in the left column and vertical source in well #4 on the right at depths indicated. Solid lines are responses due to the resistive target; dashed lines are responses due to the permeability contrast. Each field component, x, y, and z, is represented in (a), (b), and (c) and (d), (e), and (f) respectively.
**Inversion results** Inversion results are presented in order of increasing complexity. First, the results of inverting a comprehensive ERT survey are presented. Next, the product of a reduced set of ERT data is described where borehole number 2 is not employed; this was actually encountered at the test site due to a lack of electrode contact. Third, a complete MMR data set is jointly inverted with the reduced ERT synthetic data. To conclude, inverse images resulting from data sets created by varying the resistivity of the target body, with and without anomalous magnetic permeability, are described. Besides the synthetic data sets, the inverse routine requires the two error parameters (LaBrecque, 2003). The error controls employed in this study were chosen to relate the em1d2d3d modeling code to the DC modeling code based on a comparison of forward models (not shown) that plots misfit squared (the difference in data values for each point) against the data values squared. The percent data error is determined by finding square root of the slope of the straightest line that can be fit through this plot, and the constant data error is the low data value where the fit line flattens significantly with respect to the data axis. Special considerations occur when noise is not random, as would typically be seen with field data, but systematic as it is in this case where forward codes are being compared. A constant data error of 0.005 nT/A was chosen along with a percent data error of 1 %. To start, electric fields due to a standard skip-3 dipole-dipole (DPDP) ERT survey were numerically determined using the forward scalar-vector FD solution (LaBrecque et. al. 2003). The skip-3 DPDP configuration utilizes VES electrode pairs. The active electrodes are 2m apart; hence, the 3 electrodes in between are skipped. Two-meter long dipoles are used to measure potential differences in both boreholes defining an ERT plane. It is important to note that in the field, the source/sink electrodes would be swapped and the electric potential measurements repeated to check for reciprocity. Finally, field measurements would be repeated and stacked to reduce uncorrelated noise. Note, the data employed in these schemes were computed using two different codes; magnetic field data were computed using em1d2d3d, while electric potential data were generated by the same forward algorithm as employed in the inversion code. In addition no random noise was added to the synthetic data in order to represent a best-case scenario. Initially an ERT data set that included all borehole pairs and common-well data was inverted to produce Figure 7b. For the conductive case, the complete ERT data were able to resolve an over-smooth, but correctly positioned target (Figure 7b,). The reduced ^{2} fit of 0.416 is well below one indicating that this resistivity distribution fits the synthetic data very well. Next, the ERT data set was decimated to remove any data involving electrodes in borehole #2 and to exclusively use even numbered electrodes. Decimation resulted in the target being undetectable by inversion of ERT data alone (Figure 7c), even though the final ^{2} value 1.002 indicates that the data have been adequately fit. However, as shown in Figure 7d the problem is remedied when the MMR data are included within the inversion. For this inversion a reduced ^{2} fit of 0.999 was achieved in an image that distinctly shows a blurred target in the correct location. Similar to the complete ERT inversion, this resulting target is more resistive than the input model, with a minimum value just below 20 -m. Varying the resistivity along with including magnetic permeability of the target were the next steps in the study. The results of increasing the magnetic permeability of the target body by 10% are shown in Figure 8 and. MMR data for the conductive permeable model were added to the reduced ERT data (Figure 8a). For this result a ^{2} fit of 4.92 suggests that the inversion is not able to find a resistivity model that adequately fits the data. However, the images indicate that the target is identified in its correct location. There are some artifacts in the form of scattered resistivity highs, though they do not detract significantly from the identification of the target location
**Figure 7 **Model resistivity distribution (a), Tomographic images showing inversion results for the complete ERT data set (b), the reduced ERT source set data (c) and the combined reduced ERT/complete MMR data (d). **Figure 8 **Tomograms resulting from inverting synthetic data produced from conductive and permeable (a), resistive (b), resistive and permeable (c), and permeable only (d) models.
Next, the results of increasing the resistivity of the target with anomalous magnetic permeability are presented in Figure 8b. The synthetic MMR data for a resistive model were jointly inverted with a reduced ERT dataset that was generated for a homogenous background with the resistive target. A reduced ^{2} fit of 0.9868 indicates the data are sufficiently fit. Adding the MMR data for a resistive and permeable body to the reduced ERT dataset for the resistive target described above produces the results in Figure 8c. In this scenario, the size of the resolved body is overestimated, particularly in the vertical direction. A reduced ^{2} fit of 3.328 is similar to the fits obtained with the conductive permeable model described above. The last 3D image, Figure 8d, demonstrates the effect of a target with only anomalous permeability. The MMR data for a permeable only model was added to a reduced ERT dataset that was generated for a homogenous background with no anomalous resistivity (Figure 8d). For the last inversion a reduced ^{2} fit of 4.352 is very similar to the fit obtained with the other permeable models. This suggests that the main reason for the misfit in all three models containing anomalous permeability models is the magnetization that is occurring. The tomogram in Figure 8d shows that the anomalous permeability is interpreted as a dipolar resistivity with a high centered on the actual location of the target. Thus the location of the interpreted artifact is consistent with the actual location of the causative source, even though it is imaging the wrong property. However, due to the increasing scattered magnetic field amplitude with increasing horizontal source depth, the anomalous resistivity extends to a much greater depth.
**Conclusions**
Comparing the numerical responses computed by em1d2d3d to scattered magnetic fields computed with an analytical, static solution confirmed that the numerical code is producing accurate results with respect to targets including anomalous magnetic permeability. In addition, the analytical comparison supported previously published result that EM coupling effects for the environment described and operating frequencies up to 8kHz are negligible. A suite of synthetic models that included varying electrical conductivity and magnetic permeability were computed and the results analyzed by comparing the numerical data sets with and without magnetic effects. Analysis of the synthetic data sets led to several important conclusions. First, for the contrasts studied, the magnetic field responses due to the magnetic permeability anomaly were much smaller in magnitude than the electrical conductivity response for both resistive and conductive bodies. Next, the character of the permeability effect was determined to be unique in comparison to the electrical conductivity responses. This could lead to the determination of the presence or absence of magnetically permeable bodies at a survey site. Finally, the two effects were determined to be independent phenomena and were linearly additive. Comparing the three permeable models (Figure 8a,c,d), it is apparent that the permeable effect is consistent and additive to the true resistivities of the models. Since the response due to the target permeability is small relative to the response due to the target resistivity the target permeability does not significantly detract from the determination of an accurate resistivity distribution. Since the permeability response occurs coincident with the body, and since it is modest with respect to responses due to the resistivity distribution, it is concluded that moderate magnetic permeability within an MMR/ERT survey area is not immediately significant to the development of the joint MMR/ERT technology. This is especially true in a monitoring situation where permeable bodies will remain static. However the study leaves little doubt that permeability effects must be dealt with in order to absolutely determine resistivity distributions. Further, if strong permeabilities are present, especially if in small or low conductivity contrast bodies whose resistivity would not typically impact a survey, responses to these bodies may significantly distort results or prevent convergence all together. There are several ways that the effects of magnetic permeability might be mitigated in joint ERT/MMR surveys such as those presented here. These include preliminary or concurrent inversion for magnetic permeability, mapping and modeling of known permeable objects, performance of a magnetic survey using the earth’s magnetic field, performance of a magnetic survey using another known magnetic field or a combination of the these methods.
**Acknowledgements**
Multi-Phase Technologies, LLC, funded development of the MMR forward and inverse modeling code internally. Additional funding was provided through the Department of Energy, Idaho National Environmental and Engineering Laboratory DOE contract number DE-AC07-99ID13727.
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