A new approach to fitting induced-polarization spectra

GEOPHYSICS, VOL. 73, NO. 6 NOVEMBER-DECEMBER 2008 ; P. F235 F245, 10 FIGS., 8 TABLES. 10.1190/1.2987412 A new approach to fitting induced-polarization spectra Sven Nordsiek 1 and Andreas Weller 1 ABSTRACT Best fitting of induced-polarization IP spectra by different models of Cole-Cole type evidences discrepancies in the resulting model parameters. The time constant determined from the same data could vary in magnitude over several decades. This effect, which makes an evaluation of the results of different models nearly impossible, is demonstrated by induced polarization measurements in the frequency range between 1.4 mhz and 12 khz on thirteen mixtures of quartz sand and slag grains. The samples differ in size and the amount of the slag grains. Parameters describing the IP spectra are derived by fitting models of the Cole- Cole type to the measured data. The fitting quality of the generalized Cole-Cole model, the standard Cole-Cole model, and the Cole-Davidson model is investigated. The parameters derived from these models are compared and correlated with mass percentage and grain size of the slag particles. An alternative fitting approach is introduced, using the decomposition of observed IP spectra into a variety of Debye spectra. Four integrating parameters are derived and correlated with parameters of the slag-sand mixtures and Cole-Cole parameters, respectively. The alternative approach generally enables a better fitting of measured spectra compared with Cole-Cole type models. It proves to be more flexible and stable, even for complicated phase spectra that cannot be fitted by single Cole-Cole type models. The integrating parameters are well correlated with characterizing parameters of the slag-sand mixtures. The total chargeability well indicates the mass percentage of slag grains, and the mean relaxation time is related to the grain size. The relaxation time distribution can be displayed by cumulative normalized chargeability versus relaxation time, similar to granulation curves. Anologous to the latter, a nonuniformity parameter characterizes the width of the relaxation time distribution. INTRODUCTION Since the first measurements at the beginning of the last century, the field of application of induced polarization IP has expanded considerably. IP is the classical tool for ore exploration van Voorhis et al., 1973; Pelton et al., 1978. Seigel et al. 2007 present an excellent historical review of the development of the IP method. Now this method, especially spectral induced polarization SIP, is applied to a variety of new problems. Permeability prediction of aquifer systems Börner et al., 1996; de Lima and Niwas, 2000; Slater and Lesmes, 2002; Hördt et al., 2007 and environmental studies Weller and Börner, 1996; Vanhala, 1997; Weller et al., 2000; Kemna et al., 2004 are common fields of activity. Laboratory SIP measurements on sandstone samples have revealed the membrane effect of the pore throat on the spectra Scott and Barker, 2003. Tong et al. 2006 use IP time-domain decay curves to predict capillary pressure and pore-size distribution of shaley sand. Slater et al. 2005 investigate the sensitivity of resistivity and IP measurements to physicochemical properties of columns containing zero-valent iron. Recently, IP measurements also were used at archaeological sites and in laboratory to study the response of wooden relics and slag material from smelting places Schleifer et al., 2002; Weller et al., 2006; Ullrich et al., 2007. Because not all IP spectra could be fitted appropriately by the same model, a quantitative interpretation was difficult. Until now, a well-known way to derive parameters from measured spectra of IP in the frequency domain is the fitting by a Cole- Cole model, or variations and combinations thereof, to the measured data Pelton et al., 1978. The generalized form of the Cole-Cole model Pelton et al., 1983 describes the frequency dependence of the impedance Z as a function of five independent parameters, Manuscript received by the Editor 11 December 2007; revised manuscript received 27 May 2008; published online 12 November 2008. 1 Clausthal University of Technology, Institute of Geophysics, Clausthal-Zellerfeld, Germany. E-mail: sven.nordsiek@tu-clausthal.de; andreas.weller@tuclausthal.de. 2008 Society of Exploration Geophysicists. All rights reserved. F235

F236 Nordsiek and Weller Z R 0 1 m 1 1 1 i c a, with R 0 being the resistance for direct current DC, m the chargeability, the time constant, and the two exponents c and a. The angular frequency is related to the measured frequency f by 2 f. The generalized Cole-Cole model forms the basis of several special cases for which certain parameters of this model become equal to one. The standard Cole-Cole model that was formulated originally to describe the behavior of dielectric properties Cole and Cole, 1941 is derived by putting the exponent a 1. If c is one, the resulting model is referred to as Cole-Davidson Davidson and Cole, 1951. In the case that exponents a and c become one, the model simplifies to Debye. For the Debye model, denotes the relaxation time of the exponential decay curve that is used to describe the discharge of a capacitor in an electrical circuit. Dias 2000 reports on problems to find a relation between Cole- Cole parameters and petrophysical properties. Grissemann et al. 2000 present correlations between the chargeability m and the amount of polarizable grains, the time constant, and the grain size, as well as between the frequency exponent c and the sorting of grains for IP spectra in the frequency domain. For IP measurements in time domain, Tarasov and Titov 2007 derive the grain-size distribution of pure sand mixtures by decomposition of the measured data into several exponential decay curves. In both time- and frequency-domain measurements, the relaxation time seems to be an important parameter to get information about the grain or pore size of the investigated material. A question that arises by looking at the variety of possible Cole-Cole type models is the comparability of different relaxation parameters. In this work, the resulting parameters of the generalized, the standard Cole-Cole model, and the Cole-Davidson model are compared. Table 1. Composition and origin of the investigated samples. no. Grain size mm Mass of slag Porosity Water saturation 1 Origin of the slag material 1 1 10 48 86 Pandelbach 2 1 20 48 87 Valley 3 1 30 48 88 4 1 2 30 48 75 5 2 4 30 43 89 6 4 6.3 30 43 86 7 0.1 30 40 79 Huneberg 8 0.1 0.5 30 45 85 9 0.5 1 30 41 97 10 1 2 30 43 87 11 2 4 30 39 99 12 4 6.3 30 40 90 13 10 30 36 95 In a second step, a method to derive relaxation times by decomposition of measured IP spectra into a superposition of Debye spectra is presented. IP spectra measured on slag-sand mixtures provide the experimental basis for both the comparison of Cole-Cole parameters and a test of the new Debye decomposition approach. INVESTIGATED MATERIAL, SAMPLE PREPARATION, AND MEASUREMENTS The slag material used in this study was taken from two old smelting places in the Harz mountains in central Germany, which have been partly excavated by archaeologists Weller, 2003. Table 1 compiles all samples that were prepared for IP measurements. Slag material of the two locations was chosen to investigate the influence of different grain-size distributions on the measured IP spectra. In addition, samples 1 through 3 were used for studying the influence of different slag contents in a slag-sand mixture on SIP results. Quartz sand with a grain size from 0.1 through 0.2 mm was used for the investigation of slag-sand mixtures. The slag was crushed and sieved into the grain size ranges shown in Table 1. A certain amount of sand and slag was mixed and filled in a sample holder and saturated with tap water with a conductivity of approximately 10 ms/m. The samples were compacted manually, resulting in slight differences in porosity and water saturation see Table 1. The sample holder was put in a climatic chamber to keep the sample at a constant temperature of 20 C. To monitor chemical processes that might result from contact between the tap water and slag-sand mixture, repeated measurements were performed for each sample until no significant changes could be observed between data measured on two succeeding days. The IP spectra were acquired with SIP-Fuchs equipment by Radic Research, Berlin, which covers a frequency range from 1.4 mhz through 12 khz. Because measurements in the upper frequency range are affected by electromagnetic coupling effects, reliable data with phase errors smaller than 1 mrad can be acquired only below 1000 Hz. The phase spectra of samples 3 through 6 and samples 7 through 13 are presented in Figure 1 and Figure 2, respectively. Both plots show that the frequency of phase maximum depends on the size of the slag grains. The frequency of the phase peak decreases for increasing grain size. FITTING RESULTS WITH COLE-COLE TYPE MODELS The IP spectra of samples compiled in Table 1 were fitted by the generalized Cole-Cole GCC model, the standard Cole-Cole CC model, and the Cole-Davidson CD model using an automatic curve-fitting program, which is based on a Marquardt-Levenberg algorithm and singular value decomposition. The best model of a regularized least-squares procedure is provided as fitting result. Because of the highest flexibility with five parameters, the GCC model works well in most cases and shows the best fitting results. Sometimes the CC model fits the measured data equally well. In only two cases, for samples 10 and 13, the CD model yields a better fit than the CC model. The fitting results depend to a large degree on the general behavior of the phase curve plotted in double logarithmic scale. Pelton et al. 1983 show that the CC model is characterized by a symmetric

Fitting induced-polarization spectra F237 phase spectrum with an equal slope c or c at both sides of the phase peak. The CD model can be used only for fitting phase curves with a fixed slope of about one on the low-frequency side of the phase maximum. On the high-frequency side of the phase peak, an arbitrary slope a is possible. If the slope at the left side does not fulfill the strong requirement, the fitting quality remains rather poor. The GCC model provides the largest flexibility, because it allows different slopes at the left c and right ac sides of the phase peak. Figure 1. Phase spectra of samples numbered 3 through 6. Figure 2. Phase spectra of samples numbered 7 through 13. In contrast to other samples, the slag grains for sample 13 were not crushed. The slag grains were selected manually according to predetermined size from a larger volume of uncrushed slag material. For this reason, the surface of these slag grains is smoother than that of the crushed grains. In these experiments, we noted that the phase maximum broadened with increasing grain size, except for sample 13. We assume that this effect is caused by the fractal geometry of the rough grain surfaces of the crushed slag material in comparison with the smooth natural surface of the grains of sample 13. Figure 3 shows the amplitude and phase spectra of sample 13, along with curves of the best fitting Cole-Cole type models. Leroy et al. 2008 investigated water-saturated packs of glass beads in which roughness of the surfaces of the glass beads results in a second phase peak. A similar observation is presented in Figure 4. 6 exhibits a phase curve that contains not only a single phase peak at approximately 50 Hz, but also a slight indication of a second phase maximum at 0.7 Hz.Asingle CC model cannot achieve an accurate fit of this curve. A model with two Cole-Cole terms should be used to fit this phase curve appropriately. Table 2 compiles the values of chargeability m and time constant determined with models of the Cole-Cole type for all samples. For most of the thirteen samples, chargeability values determined with the GCC and CC models are close to each other. Differences occur at the time constant determined with both models. For only five samples, the time constant is similar for the GCC and CC models. Fitting with the CD model fails for ten samples. The determined chargeability values are greater than one, which contradicts the original definition of chargeability Seigel, 1959; Pelton et al., 1983. The chargeability value is acceptable only for samples 8, 12, and 13. For sample 13, the GCC and CD models show similar chargeability values of 0.34 and 0.35, respectively. The time constants for these samples are 46.6 s for the GCC model and 62.9 s for the CD model. The appropriate exponents c and a are listed in Table 3. Except for samples 10 and 13, there is a good agreement between the frequency exponents c determined for the GCC and the CC models. The exponent a of the CD model shows huge differences compared with exponent a determined with the GCC model. Only the exponent a of sample 13 is within the same range as the one calculated with the GCC model. The factors of discrepancy between the parameters determined by different models of Cole-Cole type are compiled in Table 4. The factors for chargeability, time constant, and a vary in a wide range. Only the ratio of c GCC to c CC is limited in an interval between 0.92 and 1.77. It becomes obvious that chargeability and time constant resulting from different models cannot be compared directly.

F238 Nordsiek and Weller a) b) Figure 3. Measured spectra of sample no. 13 and fits of Cole-Cole type models. a Spectra of resistivity amplitude. b Spectra of phase shift between current and voltage signal. Figure 4. Measured spectra of sample no. 6 and fits of Cole-Cole type models. a Spectra of resistivity amplitude. b Spectra of phase shift between current and voltage signal. Table 2. Compilation of the values of chargeability m and time constant determined with the models of the Cole-Cole type. no. GCC CC CD m in 10 3 s m in 10 3 s m in 10 3 s 1 0.070 0.113 0.051 0.124 17.754 22.147 2 0.149 0.184 0.119 0.040 662.541 3.469 3 0.135 0.124 0.130 0.182 824.926 10.523 4 0.172 0.022 0.204 0.731 781.239 114.266 5 0.223 34.183 0.192 3.744 359.223 930.945 6 0.225 74.518 0.198 5.934 1025.163 3994.107 7 0.902 0.701 0.096 0.315 48.648 6.454 8 0.100 0.234 0.115 1.564 0.272 15.058 9 0.302 1.002 0.305 1.011 603.215 88.362 10 0.453 338.053 0.291 17.217 1.629 985.541 11 0.414 314.246 0.371 41.439 262.556 13826.049 12 0.373 1332.990 0.343 153.384 0.816 22745.306 13 0.337 46606.892 0.341 8054.410 0.345 62885.317

Fitting induced-polarization spectra F239 The fitting quality of the resistivity amplitude spectra is excellent for all samples and models. Table 5 shows the rms error for the amplitude and phase curve of Cole-Cole type models. The maximum fitting error is less than 0.7 percent. The unacceptable fitting of several models can be recognized only in the phase spectra. If the fitting rms errors are larger than 1 mrad, which corresponds to the accuracy of the laboratory procedure, the used model should be regarded to be Table 3. Compilation of the exponents c and a determined with the models of the Cole-Cole type. GCC CC CD no. c a c a in 10 3 1 0.420 0.680 0.484 0.215 2 0.469 0.455 0.436 0.019 3 0.439 1.079 0.461 0.018 4 0.331 3.383 0.361 0.023 5 0.401 0.446 0.350 0.043 6 0.322 0.476 0.283 0.013 7 0.600 0.060 0.609 0.279 8 0.576 3.051 0.639 91.227 9 0.419 1.005 0.415 0.051 10 0.633 0.180 0.437 20.696 11 0.375 0.496 0.320 0.106 12 0.391 0.481 0.319 38.668 13 0.739 0.321 0.417 194.016 not appropriate. This limit is exceeded by two samples for GCC 6 and 13, six samples for CC samples 2, 6, 8, 10, 12, and 13 and twelve of thirteen samples for the CD model. The poor fitting quality affects the reliability of determined parameters. ALTERNATIVE APPROACH: DEBYE DECOMPOSITION Assuming that polarization effects of natural material are caused by different charging and discharging processes of some polarizing cells such as grain surface, pore throat, membrane, electrical double layer, a decomposition of the polarization effect into several separate relaxation processes seems to be useful. This approach is recommended already for time-domain IP data Tong et al., 2006; Tarasov and Titov, 2007. Because time- and frequency-domain responses are related to each other by Laplace transform, a decomposition of IP spectra into a sequence of relaxation processes seems to be possible. A transformation of frequency-domain into time-domain data can be performed easily by linear digital filters Guptasarma, 1982 ; but it should be considered that the measured complex impedance spectra cover only a limited frequency range. Assumptions must be made to predict the behavior of the spectra to higher and lower frequencies. An extrapolation of several decades at both ends of the SIP frequency range is needed to apply the digital filters. On the other hand, only the real part of the measured impedance spectrum is considered in the filtering procedure. The imaginary part, in which changes with frequency are more significant, is neglected completely. Thus, the accuracy of the transformation from frequency-domain to time-domain data is limited. Table 4. Factors of discrepancy between the parameters determined by different models of Cole-Cole type; x indicates samples with chargeability m CD 1. no. m GCC m CC m CD m GCC GCC CC CD GCC c GCC c CC a GCC a CD 1 1.37 x 0.91 x 0.87 x 2 1.25 x 4.60 x 1.08 x 3 1.04 x 0.68 x 0.95 x 4 0.84 x 0.03 x 0.92 x 5 1.16 x 9.13 x 1.15 x 6 1.14 x 12.56 x 1.14 x 7 9.40 x 2.23 x 0.99 x 8 0.87 2.72 0.15 64.35 0.90 33.44 9 0.99 x 0.99 x 1.01 x 10 1.56 x 19.63 x 1.45 x 11 1.12 x 7.58 x 1.17 x 12 1.09 2.19 8.69 17.06 1.23 12.44 13 0.99 1.02 5.79 1.35 1.77 1.65 Table 5. Compilation of the rms error for the amplitude and phase curve of the Cole-Cole type models. no. GCC CC CD mrad mrad mrad 1 0.56 0.18 0.53 0.29 0.51 0.56 2 0.16 0.97 0.19 1.05 0.21 1.71 3 0.08 0.52 0.08 0.56 0.31 1.98 4 0.14 0.62 0.14 0.73 0.32 3.81 5 0.20 0.66 0.22 0.91 0.22 2.16 6 0.17 1.42 0.18 1.58 0.22 1.72 7 0.07 0.58 0.11 0.63 0.15 1.20 8 0.16 0.58 0.22 1.04 0.40 2.79 9 0.12 0.33 0.14 0.25 0.64 4.53 10 0.11 0.50 0.33 2.76 0.23 2.66 11 0.31 0.45 0.36 1.00 0.64 4.69 12 0.34 0.78 0.33 1.24 0.59 4.85 13 0.15 1.09 0.42 3.84 0.23 2.78

F240 A different approach is suggested to derive relaxation time distributions from SIP data. It is similar to the procedure of Morgan and Lesmes 1994, who use a least-squares inversion algorithm to invert the real part of dielectric data for a spectrum of relaxation times. In our approach, both the spectra of resistivity amplitude and phase are regarded as a superposition of a certain number n of different Debye spectra. These spectra are characterized by a specific chargeability m k and a relaxation time k, considering that the Debye formula represents the frequency analog of an exponential decay function in the time domain. The frequency-dependent impedance Z that describes the IP spectrum can be presented by the equation n 1 Z R 0 1 m k 1 2 1 i k, k 1 which is equivalent to the time-domain voltage decay normalized to the constant strength of current I: Z t U t I n R 0 k 1 m k exp t k. The n pairs of relaxation time k and chargeability m k, and the value of the DC resistance R 0, are the resulting parameters of this model. The DC resistance R 0 is approximated by extrapolating the impedance amplitude to lower frequencies. In an additional step after the determination of all other parameters, the value of R 0 is adjusted using only the amplitude spectrum. Multiplying the DC resistance R 0 by the geometric factor results in the DC resistivity 0. To express the problem as a system of linear equations, a simple transformation must be done. The impedance is normalized in the following way: 3 Z norm R 0 Z R 0. 4 The separation of the normalized impedance Z norm into real and imaginary parts results in the equations Z norm Z norm n m k k 2 1 k 2 1, k 1 n m k k 1 k 2 1. k 1 Assuming that at p discrete angular frequencies the normalized impedance values are determined, a combination of equations 5 and 6 provides the following system of linear equations, consisting of 2p equations and n parameters m k to be determined: Nordsiek and Weller 5 6 1 1 2 1 1 1 2 1 n 2 1 1 n 2 ] ] p 1 2 1 p 1 2 p n 2 1 p n 2 1 1 1 n 1 1 1 2 1 1 n 2 ] ] p 1 p n 1 p 1 2 ] Z Znorm Z norm 1 1 n n 2 m 1 ] n m p 7 1 ] p. Z norm The n values of relaxation time k within the matrix are predetermined. Considering the results of numerical tests, n 700 equally distributed values at a logarithmic scale from 10 4 through 10 3 s have proved to be a good choice regarding computation time and required resolution. The overdetermined system of linear equations is solved under the constraint that no negative chargeability value m k is allowed. The lsqnonneg algorithm for solving this problem was developed by Lawson and Hanson 1974 and is implemented in the MAT- LAB programming environment. The algorithm yields the n values m k, each belonging to a predefined relaxation time k. In comparison with the procedure presented by Morgan and Lesmes 1994, in this algorithm both the real and imaginary parts of the impedance spectrum are inverted simultaneously, and a considerably larger amount of free parameters is used. The only constraint is the restriction to nonnegative chargeability values. RESULTS OF THE INVERSION WITH DEBYE DECOMPOSITION The algorithm of Debye decomposition was applied to fit all measured IP spectra of slag-sand mixtures. To give a graphic impression of the fitting quality, Figures 5 and 6 present the IPspectra of samples 13 and 6 fitted with the Debye decomposition and the GCC model. It can be seen that the curve of sample 13 is fitted better with the Debye decomposition than with the GCC model. For sample 6, the GCC model does not fit the phase curve, but the Debye decomposition yields an accurate fit. A further result of the Debye decomposition is the relaxation time distribution, which relates the chargeability values to the time scale between 10 4 and 10 3 s. To get an evident presentation that widely ignores the number of predefined relaxation times, all chargeability

Fitting induced-polarization spectra F241 values belonging to a relaxation time within a time decade are added. The sum for each decade is normalized to the total chargeability m and gives the loading coefficient of relaxation times for the decade. Figure 7 presents the decade loadings for samples 8, 9, 10, 11, and 13 as histograms. The distribution becomes flatter with increasing grain size, except for sample 13. This corresponds to the widening of the phase spectra described above. It can be observed that the loading in the decade from 10 3 through 10 2 s decreases with increasing grain size, but it does not vanish, even for the largest grain size. Consequently, structures of smaller size, which cause fast relaxation processes, are supposed to be present in the mixtures of larger grains. These structures can be related to the surface roughness of the slag grains, according to the investigations of Leroy et al. 2008. Figure 8 shows the normalized chargeability plotted versus relaxation time as cumulative curves for samples 7 through 13. In this plot, the relation between relaxation times and the size of the slag grains becomes obvious. For smaller slag grains, most of the chargeability originates from lower relaxation times. For bigger slag grains, the majority of the chargeability is related to higher relaxation times. This presentation reveals the similarity to a grain-size distribution. To compress the amount of data, four integrating parameters are proposed: 1 The total chargeability m 2 The mean relaxation time 3 The non-uniformity parameter U 4 The DC resistivity 0 The total chargeability m sums up all chargeability values m k determined as solution of the system of linear equations 7. The mean relaxation time describes the logarithmic average value of the relaxation times weighted by their chargeability. It is determined by the formula exp km k ln k k m k. 8 The formula of the nonuniformity parameter U 60 / 10 is defined in analogy to the degree of nonuniformity of a grain-size distribution curve. Expressions 10 and 60 mark those relaxation times whereby, in a cumulative curve, 10% and 60% of the total chargeability is reached. The parameter U characterizes the width a) b) Figure 5. Measured spectra of sample no. 13 and fits of the generalized Cole-Cole model and Debye decomposition. a Spectra of the resistivity amplitude. b Spectra of the phase shift between current and voltage signal. Figure 6. Measured spectra of sample no. 6 and fits of the generalized Cole-Cole model and Debye decomposition. a Spectra of the resistivity amplitude. b Spectra of the phase shift between current and voltage signal.

F242 Nordsiek and Weller of relaxation time distribution, which reflects the different scale lengths of structures involved in the relaxation processes. The fourth parameter is the DC resistivity 0, which is known already from the Cole-Cole models. a) The four parameters of all observed spectra and the appropriate rms errors are compiled in Table 6. It should be noted that not only the mean relaxation time, but also the nonuniformity parameter U, generally increase with rising grain size. The increase of the latter might be explained by the superposition of grain size and surfaceroughness effects, which consequently widen the relaxation time distribution. DISCUSSION b) c) d) To evaluate the empirical relations between properties of the slagsand mixture mass percentage and size of the slag grains and the parameters provided by the Cole-Cole type models or the Debye decomposition approach, the correlation coefficients R are regarded. Table 7 compiles the correlation coefficients for the relations suggested by Grissemann et al. 2000. For the Cole-Cole type models, there is a good correlation between the mass percentage of slag grains and the chargeability m. In addition, a relation between the grain size and time constant is noticeable for all models. In the same way, we determined the correlation coefficients of the integrating parameters m and of Debye decomposition with the described parameters of the slag-sand mixture. For both investigated parameters, a significant correlation can be observed. The mass percentage of slag correlates to total chargeability, and the grain diameter d well correlates to the mean relaxation time. The latter correlation is illustrated in Figure 9. The fit for the Debye decomposition DD is indicated by the solid line and can be expressed by the equation e) DD d 0.008 d 1.54. The fits for the CC model, CC d 0.007 d 1.97, and the CD model, CD d 0.278 d 2.23, 9 10 11 Figure 7. Decade loadings of relaxation time for samples numbered 8 through 11 and 13. a 8, grain size 0.1 0.5 mm. b 9, grain size 0.5 1.0 mm. c 10, grain size 1.0 2.0 mm. d 11, grain size 2.0 4.0 mm. e 13, grain size approximately 10 mm. Figure 8. Normalized chargeability plotted versus relaxation time as cumulative curves for samples numbered 7 through 13. Table 6. Parameters derived from Debye decomposition of the 13 slag-sand mixture samples and the appropriate rms errors. no. m 10 3 s U 0 m mrad 1 0.039 0.796 5.370 141.659 0.56 0.02 2 0.063 0.379 5.888 120.147 0.04 0.32 3 0.092 2.666 5.623 112.931 0.45 0.05 4 0.176 11.218 85.114 136.578 0.48 0.10 5 0.166 8.881 165.959 105.614 0.10 0.04 6 0.160 17.232 870.964 98.894 0.05 0.00 7 0.147 0.327 1.000 242.287 0.27 0.04 8 0.101 4.891 6.166 103.050 0.31 0.84 9 0.260 3.018 21.380 87.603 0.10 0.03 10 0.321 11.404 281.838 194.242 0.15 0.09 11 0.349 39.756 954.993 172.712 0.23 0.26 12 0.312 121.127 426.580 147.254 0.30 0.20 13 0.358 842.766 7762.471 283.001 0.11 0.09

Fitting induced-polarization spectra F243 are plotted as dashed lines. Because the resulting correlation of the GCC model is only weak, the fitting line is not shown. Pelton et al. 1978 and Olhoeft 1985 found a similar power law between the time constant of the CC model and the diameter d of polarizing particles with an exponent of approximately 2. The electrochemical model developed by Wong 1979 confirms this relationship, but only for grains with a radius greater than 0.1 mm. The exponent decreases toward 1 for smaller grains. If the sample with the smallest grain sizes 0.1 mm is ignored, the exponent of d in equation 9 increases to 1.74. It can be assumed that surface-roughness effects reduce the mean relaxation time and consequently the exponent of d. Regarding the polarization of a smooth sphere in an electrolyte solution theoretically, the exponent of d is expected to be 2 Schwarz, 1962. Correlating the exponent c of the GCC and CC models with the logarithm of the parameter U leads to correlation coefficients of 0.025 and 0.724, respectively. The negative correlation coefficient resulting from the CC model confirms the expectation that a smaller exponent c causes a wider distribution of relaxation times and consequently a higher value of the nonuniformity parameter U. Because the values of the exponent c determined with the CC model vary only between 0.28 and 0.64, an additional test with synthetic data was performed to verify the correlation between exponent c and U. Ten CC models, each with a DC resistivity 0 of 100 m, a time constant of 0.5 s, and a chargeability m of 0.3, were generated. The exponent c increases from 0.1 through 1.0 in steps of 0.1. The parameters U resulting from the fitting of the synthetic models are compiled in Table 8. Correlating the exponent c with the logarithm of the width parameter U determined for the synthetic data sets yields a correlation coefficient of 0.9881. The strong correlation between both parameters shows that the width parameter U is an adequate alternative to the exponent c of the Cole-Cole model. In addition to the correlation between model parameters and petrophysical properties of the sample, it is important to consider the deviation between data predicted by the model and the measured values. To demonstrate both issues for the discussed IP models, Figure 10 shows the mean rms error of the phase spectra fitted by the investigated models see Tables 5 and 6 plotted versus the correlation coefficient R from Table 7. For each model, two groups are displayed: The open symbols represent the correlation between the total chargeability and the mass of slag for samples 1 through 3. The filled symbols show the correlation of the time constant with the maximum grain size of the polarizing particles for samples 7 through 13. The Cole-Davidson model shows high correlation coefficients, but the mean rms errors are unacceptable. The best result regarding Table 8. Ten synthetic Cole-Cole models, each with a DC resistivity 0 of 100 m, a time constant of 0.5 s, and a chargeability m of 0.3. The parameter U of the synthetic IP data was determined with Debye decomposition. Table 7. Correlation coefficient of chargeability m with the mass of slag for samples 1 through 3 and of relaxation time with the grain size for samples 7 through 13. GCC CC CD Debye decomposition Mass of slag 1-3 and m 0.772 0.922 0.945 0.997 Grain size 7-13 and 0.813 0.809 0.962 0.870 Exponent c Parameter U Exponent c Parameter U 0.1 1.096 10 4 0.6 2.570 10 1 0.2 2.042 10 3 0.7 7.079 10 0 0.3 5.129 10 2 0.8 6.310 10 0 0.4 4.266 10 2 0.9 2.455 10 0 0.5 5.623 10 1 1.0 1.023 10 0 Figure 9. Time constant and mean relaxation time plotted versus maximum diameter of the slag grains for samples numbered 7 through 13. The quality of the fitting is indicated by the coefficient of determination R 2. Figure 10. Mean rms error of the phase plotted versus the correlation coefficient for samples numbered 1 through 3 correlation between mass percentage of slag grains and chargeability open symbols and samples numbered 7 through 13 correlation between the grain size and time constant filled symbols.

F244 Nordsiek and Weller both criteria is achieved by the Debye decomposition approach, yielding high correlation coefficients and low rms errors simultaneously. CONCLUSIONS Cole-Cole type models fit IP spectra measured on slag-sand mixtures with different quality. The generalized Cole-Cole model provides one parameter more than the standard Cole-Cole model and the Cole-Davidson model. This additional degree of freedom allows a better fit to the data, whereas the standard Cole-Cole model and the Cole-Davidson model are restricted to a distinct shape of the phase spectra. The comparison shows that a high fitting quality of experimental data is important, but it is not a sufficient criterion to identify which model is most suitable to describe the phenomenon of induced polarization. The generalized Cole-Cole model provides the best fitting curves but otherwise the worst correlation to the petrophysical characteristics of the medium. Because the Cole-Davidson model fails for most samples to fit the phase spectra with the required accuracy, this model proves to be of limited value. Just for a few cases, it provides physically possible values for the fundamental parameter of chargeability. Adrawback of all models with only one Cole-Cole term is that the fit fails if there is more than one phase peak in the observed spectrum. In this case, the superposition of several Cole-Cole terms is necessary, which increases the number of parameters and makes the interpretation and comparison of different spectra more complicated. Because large discrepancies between the same parameters of different Cole-Cole type models fitted to identical spectra occur, it becomes obvious that the parameters are not comparable. An alternative approach uses the decomposition of IP spectra into a sequence of Debye models. The suitability of this approach is demonstrated by its application to the spectra of slag-sand mixtures. The Debye decomposition proves to be more flexible and stable because it guarantees an excellent fitting quality, even for complicated phase spectra. The alternative approach yields a relaxation time distribution that can be compared with granulation curves. Four integrating parameters can be derived from the relaxation time distribution: DC resistivity, total chargeability, mean relaxation time, and a nonuniformity parameter. Reliable correlations between slag mass percentage and grain size with the integrating parameters of the Debye decomposition were found. The correlation coefficients are slightly better compared with those of the Cole-Cole model parameters. Although parameters of the Cole-Davidson model are well correlated, this model cannot be recommended because of the poor fitting quality of phase spectra. The experiment with slag-sand mixtures has shown that for particles with electronic conduction, such as in ores and slags, the relation between the mean relaxation time and the grain size can be described by a power law with an exponent of approximately 1.5, which is slightly lower than the theoretically predicted value of 2 for smooth spherical particles. ACKNOWLEDGMENTS The work was carried out within two projects WE 1557/8 and WE 1557/10 supported by the German Research Foundation DFG. We are grateful to Carlos Alberto Dias and two anonymous reviewers, whose constructive comments and remarks helped to improve the manuscript. REFERENCES Börner, F. D., J. R. Schopper, and A. Weller, 1996, Evaluation of transport and storage properties in the soil and groundwater zone from induced polarization measurements: Geophysical Prospecting, 44, 583 601. Cole, K. S., and R. H. Cole, 1941, Dispersion and absorption in dielectrics: Journal of Chemical Physics, 9, 341 351. Davidson, D. W., and R. H. Cole, 1951, Dielectric relaxation in glycerol, propylene glycol and n-propanol: Journal of Chemical Physics, 19, 1484 1490. de Lima, O. A. L., and S. Niwas, 2000, Estimation of hydraulic parameters of shaly sandstone aquifers from geoelectrical measurements: Journal of Hydrology, 235, 12 26. Dias, C. A., 2000, Developments in a model to describe low-frequency electrical polarization of rocks: Geophysics, 65, 437 451. Grissemann, C., D. Rammlmair, C. Siegwart, and N. Fouillet, 2000, Spectral induced polarization linked to image analyses: A new approach, in D. Rammlmair, J. Mederer, T. Oberthür, R. Heimann, and H. Pentinghaus, eds., Applied mineralogy: Balkema, 561 564. Guptasarma, D., 1982, Computation of the time-domain response of a polarizable ground: Geophysics, 47, 1574 1576. Hördt, A., R. Blaschek, A. Kemna, and N. Zisser, 2007, Hydraulic conductivity estimation from induced polarization data at the field scale The Krauthausen case history: Journal ofapplied Geophysics, 62, 33 46. Kemna, A., A. Binley, and L. D. Slater, 2004, Crosshole IP imaging for engineering and environmental applications: Geophysics, 69, 97 107. Lawson, C. L., and R. J. Hanson, 1974, Solving least squares problems: Prentice-Hall. Leroy, P., A. Revil, A. Kemna, P. Cosenza, and A. Ghorbani, 2008, Complex conductivity of water-saturated packs of glass beads: Journal of Colloid and Interface Science, 321, 103 117. Morgan, F. D., and D. P. Lesmes, 1994, Inversion for dielectric relaxation spectra: Journal of Chemical Physics, 100, 671 681. Olhoeft, G. R., 1985, Low-frequency electrical properties: Geophysics, 50, 2492 2503. Pelton, W. H., W. R. Sill, and B. D. Smith, 1983, Interpretation of complex resistivity and dielectric data Part 1: Geophysical Transactions, 29, 297 330. Pelton, W. H., S. H. Ward, P. G. Hallof, W. R. Sill, and P. H. Nelson, 1978, Mineral discrimination and removal of inductive coupling with multifrequency IP: Geophysics, 43, 588 609. Schleifer, N., A. Weller, S. Schneider, and A. Junge, 2002, Investigation of a Bronze Age plankway by spectral induced polarization: Archaeological Prospection, 9, 243 253. Schwarz, G., 1962, A theory of the low-frequency dielectric dispersion of colloidal particles in electrolyte solution: Journal of Chemical Physics, 66, 2636 2642. Scott, J. B. T., and R. D. Barker, 2003, Determining pore-throat size in permo-triassic sandstones from low-frequency electrical spectroscopy: Geophysical Research Letters, 30, 1450. Seigel, H. O., 1959, A theory of induced polarization effects for step-function excitation, in J. R. Wait, ed., Overvoltage research and geophysical applications: Pergamon Press, 4 21. Seigel, H. O., M. Nabighian, D. S. Parasnis, and K. Vozoff, 2007, The early history of the induced polarization method: The Leading Edge, 26, 312 321. Slater, L. D., J. Choi, and Y. Wu, 2005, Electrical properties of iron-sand columns: Implications for induced polarization investigation and performance monitoring of iron-wall barriers: Geophysics, 70, no. 4, G87 G94. Slater, L. D., and D. P. Lesmes, 2002, Electric-hydraulic relationships observed for unconsolidated sediments: Water Resources Research, 38, 1213. Tarasov, A., and K. Titov, 2007, Relaxation time distribution from time domain induced polarization measurements: Geophysical Journal International, 170, 31 43. Tong, M., L. Li, W. Wang, and Y. Jiang, 2006, Determining capillary-pressure curve, pore-size distribution and permeability from induced polarization of shaley sand: Geophysics, 71, no. 3, N33 N40. Ullrich, B., C. Meyer, and A. Weller, 2007, Geoelektrik und Georadar in der archäologischen Forschung: Geophysikalische 3D-Untersuchungen in Munigua Spanien, in G. A. Wagner, ed., Einführung in die Archäometrie: Springer-Verlag, 75 93. Vanhala, H., 1997, Mapping oil-contaminated sand and till with the spectral induced polarization SIP method: Geophysical Prospecting, 45, 303 326. van Voorhis, G. D., P. H. Nelson, and T. L. Drake, 1973, Complex resistivity spectra of porphyry copper mineralization: Geophysics, 38, 49 60. Weller, A., 2003, Spektrale Induzierte Polarisation in der Archäometrie, in

Fitting induced-polarization spectra F245 Freiberger Forschungshefte C 496 Geophysik bei unterschiedlichen Skalierungen der geologischen Strukturen: TU Bergakademie Freiberg, 14 29. Weller, A., and F. D. Börner, 1996, Measurements of spectral induced polarization for environmental purposes: Environmental Geology, 27, 329 334. Weller, A., W. Frangos, and M. Seichter, 2000, Three-dimensional inversion of induced polarization data from simulated waste: Journal of Applied Geophysics, 44, 67 83. Weller, A., S. Nordsiek, and A. Bauerochse, 2006, Spectral induced polarization Ageophysical method for archaeological prospection in peatlands: Journal of WetlandArchaeology, 6, 105 125. Wong, J., 1979, An electrochemical model of the induced-polarization phenomenon in disseminated sulfide ores: Geophysics, 44, 1245 1265.