Microwave Dielectric Properties of Rocks Fawwaz T. Ulaby, Tom Bengal, M. Craig Dobson, Jack East, Jim Garvin, and Diane Evans ABSTRACT A combination of several measurement techniques was used to investigate the dielectric properties of 80 rock samples in the microwave region. The real part of the dielectric constant, e', was measured in 0.1 GHz steps from 0.5 GHz to 18 GHz, and the imaginary part, e", was measured at five frequencies extending between 1.6 GHz and 16 GHz. In addition to the dielectric measurements, the bulk density was measured for all the samples and the bulk chemical composition was determined for 56 of the samples. This study shows that E' is frequency-independent over the range of 0.5-18 GHz for all rock samples, and that the bulk density Pb accounts for about 50% of the observed variance of e'. For silicate rocks, as much as 78% of the observed variance of E' may be explained by the combination of density and the fractional contents of various oxides determined by x-ray fluorescence when the silicates are subgrouped by genesis (volcanic, plutonic and sedimentary). In contrast, the loss factor ~" decreased with increasing frequency for most rock samples. It was not possible to establish*statistically significant relationships between E" and the measured density of the rock samples. However, in the case of silicate rocks, 60% of the variance in e" generally can be explained by bulk chemical composition when the silicates are subgrouped by genesis. F.T. Ulaby, Tom Bengal, M.C. Dobson, and J. East are with the Electrical Engineering and Computer Science Department, The University of Michigan, Ann Arbor, Ml 48109 J. Garvin is with NASA Goddard Space Flight Center, Greenbelt, MD 20771 D. Evans is with the Jet Propulsion Laboratory, Pasadena, CA 91109

1. INTRODUCTION The purpose of this study is to investigate the microwave dielecric properties of igneous and sedimentary rock, in support of radar investigations of the Earth's geology and of the future radar and radiometer missions to Mars. Several studies have been reported in the literature on the dielectric properties of rocks [1-6], but in most of these studies the reported experimental measurements had been made either at MHz or lower frequencies, or at one or very few microwave frequencies. Thus, no continuous microwave spectra of the relative dielectric constant e have been reported to date. Furthermore, the majority of the reproted data for the dielectric loss factor E" is of questionable accuracy. This is because e" of most rocks is between 0.01 and 0.1, and most dielectric measurement techniques do not have the accuracy required for measuring values that small. The relative dielectric constant e of a material is defined as e = C - j", where the real part E' is the perimittivity of the material (relative to that of free space) and the imaginary part e" is its dielectric loss factor (also relative to eo of free space). This study focuses on the spectral region extending from 0.5 GHz to 18 GHz. A combination of several measurement techniques was used to measure e over this frequency range. It included two probe techniques for measuring E' in steps of 0.1 GHz from 0.5 GHz to 18 GHz, and a resonant cavity perturbation technique for measuring e" at five frequencies extending from 1.6 GHz to 16 GHz. Because these cavity measurements are very time-consuming, it was not possible to make the measurements at more than five frequencies. The dielectric data reported in this study were generated from measurements performed for 80 rock samples. Each data point represents the average of several 2

measurements corresponding to spatially different parts of the rock sample. The variability among measurements made for a given rock sample is an indicator of the sample's spatial inhomogeneity. Such variations may be due to density variations or to variations in chemical composition and crystalline structure among mineral constituents. In addition to the dielectric measurements, the bulk density was measured for all of the samples and the chemical composition was determined for 56 of the samples by x-ray fluorescence. This paper provides statistical analyses relating e' and e" of a rock sample to its density and chemical contents. 2. DIELECTRIC PROBE MEASUREMENT TECHNIQUE The permittivity data reported in this study are based on measurements of the complex reflection coefficient of a coaxial probe terminated in the material under test. Two techniques were used. The first one is based on a third-order equivalent-circuit model that can be used for measuring the dielectric constant of any rock sample across the full frequency range of interest (0.5 - 18 GHz). The-second one is a simpler first-order equivalent-circuit model, but its validity range is limited to frequencies below 10 GHz when' is larger than 8. For all rock samples investigated in this study, e' was found to be approximately independent of frequency over the 0.5-18 GHz range. Because it is simpler to use and calibrate, the first-order technique was initially used to measure e' of a given sample, and if e' was found to exceed 8 over the 0.5-10 GHz range, the sample was remeasured using the more exact third-order technique. Brief descriptions of these two techniques are given next. 2.1 Third-Order Equivalent Circuit The dielectric probe system (Fig. 1) consists of a swept RF source, a network analyzer (HP 851 OA), and associated couplers and data processing instrumentation. Fig. 2(a) shows a cross-section of the probe tip and the dimensions of two of the 3

probes examined in this study. The operation of open-ended coaxial lines to measure the dielectric constant of unknown materials is well-documented in the literature [7-10]. The input reflection coefficient at the probe tip, p, is given by ZL-Z, Y,- YL p^^z Tz Y Y' (1) Z + Zo Y0 + YL where Y = 1/Z, Zo is the line impedance, and ZL is the load impedance, which is governed by the geometry of the probe tip and the dielectric constant of the material it is in contact with or immersed in (for liquid materials). In general, an open-ended coaxial line may be described by an equivalent circuit of the form shown in Fig. 2(b). When placed in contact with a homogeneous material whose thickness is sufficient to simulate a slab of infinite electrical thickness, an open coaxial line has an admittance YL (co, E) given by YL (co ) = Yi (o)+Ye (w, ), (2) where Yi (co) = jcoCi is the "internal" admittance corresponding to the fringing capacitance Cj that accounts for the fringing field in the Teflon region between the inner and outer conductors of the line. The "external" admittance Ye, which is a function of both co and the complex dielectric constant e of the material under test, consists of a frequency-dependent capacitor C(o, e) in parallel with a radiation conductance G(o, e) Ye (co, e) = jC (co, ) + G (co, ). (3) 4

The capacitor C(o, e) represents the fringing field concentration in the dielectric medium (e) surrounding the probe tip, and the conductance G(co, e) represents the radiation into the dielectric medium. When the medium surrounding the probe tip is free space (i.e., an open-ended line), these two equivalent-circuit elements vary according to C (co, E) = Co + B2 (4) G (w, e0) = A4, (5) where Co, B, and A are constants for a given probe-tip geometry. If the radial dimensions of the coaxial line (namely, ri and r2) are small compared to the wavelength X, computations using the expressions given in Marcuvitz [11] yield values for A and B that are sufficiently small that the external admittance may be approximated as Ye (o, Eo) = jcCo. If the dielectric constant of the medium surrounding the probe tip is not the free space value Eo, however, the above simplification may lead to unacceptably large errors. Hence, in the general case we have Ye (co. o) = jo (Co + Bc + Aco4. (6) According to the theorem developed by Deschamps [12], the input admittance of an antenna immersed in a medium of complex dielectric constant ~ is related to the input admittance in free space through 5

Ye (W, c) = O O) (7) Y,.), o ~ The above expression is for materials characterized by g = go. If we regard the openended coaxial line as an antenna and henceforth abbreviate the relative dielectric constant ratio e/eo as simply e, we can write the following expression for the total input admittance of the probe when placed in contact with a material of relative dielectric constant ~. 32 4A25 YL (o, E) = joCi + jOoCOE + jBco + Aco. (8) With the line admittance Yo known, measurements of the amplitude and phase of p by the network analyzer system (Fig. 1) lead to a measurement of YL. The next step is to determine e from YL. This is accomplished by (1) calibrating the measurement probe in order to establish the values of the constants Ci, Co, B, and A, and (2) developing an iterative program for finding a value for e that minimizes the error between the measured value of YL and the value calcuated from the expression on the right-hand side of (8). Calibration entails finding the values of the constants Cj, Co, B, and A of (8) for each probe used in this study. Under ideal circumstances, one needs to determine these constants only once and at only one frequency. The equivalent-circuit model, however, is only approximate; hence, it is necessary to determine these constants at each frequency that the probe is intended to be used. Each dielectric probe was calibrated by measuring the complex reflection coefficient under four termination conditions: (1) short circuit, (2) open circuit, (3) probe immersed in distilled water, and 6

(4) probe immersed in methanol. Distilled water and methanol were used because their dispersion spectra are well known [13], [14]. 2.2 First-Order Equivalent Circuit If the diameter of the coaxial probe is much smaller than the wavelength in the material under test, the expression given by (6) and (7) for the equivalent-circuit admittance Ye (w, E) simplifies to only one term, jcoCoe, because the other two terms become negligibly small. For the 0.14-in probe used in this study, the conditoin f(GHz) <50/, (9) must be satisfied in order for the first-order model to yield accurate results. This condition was found by comparing measurements made with this technique to measurements made using the more-exact technique described in the previous section. 2.2.1 Reflection Measurement Technique For the first-order equivalent circuit, the admittance, YL (co, ) = jco C + j c e, (10) can be determined by measuring the reflection coefficient p. The constants Ci and Co can be determined by measuring YL (co, ) for two materials with known e. With the constants known, E of an unknown material may be computed directly from 7

e,1 Y(1p). C (11) CO C - 1 + P by measuring p. The coaxial line is a standard 50-ohm line (i.e., Yo = 1/50). 2.2.2 Group-Delay Measurement Technique As an alternative to measuring p (in order to determine E), a group-delay technique was developed which requires calibration against only one calibration material rather than two. For low loss materials with " << c', YL = j (Ci +' CO), (12) and the reflection coefficient P IPI ei (13) Y - YL (14) Yo + YL has a phase angle given by I = 2 cot [50 co (C +' C)]. (15) For the 0.141-in probe, the constants Ci and Co are on the order of 0.02 picofarads. Consequently, the entire quantity inside the square brackets is much smaller than 1 if f < 20 GHz and e' < 10. Hence, cot -1 ( ) may be expanded in a Taylor series 8

-1 ) = 2 - x + x3 cot (X) = 7t/2 -x+x /3-..., (16) and if we retain only the first two terms, we have I) =- X - 100o (C + eCo). (17) The group delay t is defined as the change in the phase 4 with angular frequency, = oa =-100(C +'Co). a o (18) If the group delay is measured with the probe in air (with e = 1) and not in contact with any other material, we get the reference group delay to, To = - 100 (C + Co). (19) The differential group delay is defined as At = t - to = -100Co(e'-1), (20) from which we obtain the expression' = 1- A1100 Co (21) 9

The constant Co may be determined by measuring AT for one material of known C'. The group delays t and T0 can be measured directly by the HP 851 OA network analyzer. Comparison of results using the group-delay technique with results obtained using the more-exact reflection coefficient technique has led to the conclusion that the condition f(GHz) < 30/^, (22) should be satisfied in order for the approximation made in going from (15) to (17) to be valid. 2.3 Sample Preparation For Permittivity Measurements When using the coaxial probe to measure the permittivity of a solid material, the following two conditions must be satisfied (in order for the measurements to produce accurate results): (1) The thickness of the sample must be at least equal to the probe diameter. For the 0.14-in probe, this condition is satisfied if the thickness is greater than 4 mm. (2) The surface of the sample in contact with the probe must be very smooth in order to insure good electrical contact. This was achieved by having each rock sample cut with a rock saw to obtain a flat surface and then the surface was smoothed using a table-top rotary sander. To avoid dielectric effects that may be caused by the possible presence of surficial water molecules on the sample, each sample was dried in an oven for 15 minutes at 105~C prior to performing the dielectric measurements. It was found, 10

however, that there was very little difference in e', if any, between the results obtained after drying the samples and those obtained on the basis of the measurements made prior to drying the samples. An entirely different conclusion was reached for the measurements of the dielectric loss factor e"; for some rocks, the values measured prior to drying the sample were as much as twice the values measured for the samples after drying. 2.4 Measurement Accuracy and Precision The measurement accuracy of the probe technique was evaluated by comparing the permittivity measured by the probe with the permittivity of standard materials. The reference materials are homogeneous, thick blocks of solid materials, such as teflon, whose dielectric constants had been carefully measured using waveguide techniques. Based on such comparisons, the probe measurement accuracy was found to be better than ~0.03 of the measured value. By accuracy, we refer to the absolute level of e', whereas by precision, we refer to the variability associated with the spatial inhomogeneity of the sample. For all rock samples, E' was measured by applying the probe to at least 16 spatially different locations on the polished surface of the rock sample. In each case, we computed the mean value of e', the associated standard deviation s and the ratio s / e'. For all samples, the ratio s / e' was found to be smaller than 0.12, and for most the ratio was smaller than 0.05. Figure 3 shows typical permittivity spectra of four rock samples. The absence of a discernible dependence on frequency was characterisitic of all samples measured in this study and is in agreement with previous conclusions reached by Olhoeft et al. [2]. Hence, in all forthcoming discussions and analyses, e will be treated as frequency independent and will be represented by the average value measured over the 0.5-18 GHz range. 11

3. RESONANT-CAVITY PERTURBATION TECHNIQUE A resonant cavity is a closed volume. The diagram in Fig. 4(a) shows a cylindrical cavity with two magnetic loop couplers protruding slightly into the cavity volume on the inside walls and connected to SMA connectors on the outside walls at a height midway between the top side (the lid) and the cavity floor. Figure 5 shows the measurement system. With the cavity empty, if one were to connect a signal generator (HP 8350B in Fig. 5) to one of the connectors and a network analyzer to the other and then sweep the generator frequency across the resonance region of the cavity, the output power would be a Gaussian-like function of frequency (Fig. 4(b)). This power spectrum is characterized by fo, the frequency at which the power is a maximum, and by Qo, the quality factor, fo Qo - Af (23) where Af is the half-power width of the power spectrum. If we insert a dielectric material into the cavity, the spectrum will change in two ways: (1) the resonant frequency decreases to a lower value, which we shall call fs, and (2) the quality factor decreases to a lower value Qs. In order to maintain Qs large (i.e., maintain a resonant-like spectrum), the volume of the material inserted into the cavity must be kept small relative to the cavity volume. When this is the case, the resonant-cavity perturbation technique [15] may be used to determine e' and e" of a dielectric material from measurement of fo, fs, Qo, and Qs. 12

For a cylindrical cavity with radius a and height d<2a oscillating in the TMo1o mode and containing a needle-shaped dielectric material oriented along the vertical axis of the cylinder, the shift in the resonant frequency is fs' fo ~ = -1.855 V(e'-1) (24) f S if the volume fraction of the cavity occupied by the sample is small. Solving for E' we get fo - fs E' -1 + 25) C'= 1 +'1.855 fsV (25) If the material has a dielectric loss factor E", it can be shown [14, p. 373] that (e'- 1) f 1 1 ) C 2 -Q (26) These expressions are valid only if V is very small, and (24) is valid only if the dielectric material is approximately needle shaped and oriented vertically. One of the major problems associated with using this method to determine ~' and E" is the need to know V very accurately (e" depends on (E' - 1) which, in turn, depends directly on 1/V). In our case, however, we did not need to know V exactly becuase we already know E' from the probe measurements discussed in the previous section. Hence, e" could be determined from (26) without the need to measure V. This procedure of using dielectric probes to measure e' and resonant cavities to measure e" proved extremely effective because the errors associated with the handling and the 13

measuring of the weight and volume of very small rock samples are intolerably high. As will be discussed below, a desirable value for V is about 0.5 percent. For a cavity volume of 2.5 cm3 (which was the volume of one of the cavities used in this study), V would have to be about 1.25 x 10-2 cm3 and the corresponding weight would be about 31 mg (for a typical density of 2.5 g/cm3). 3.1 Measurement Accuracy and Presicion By way of evaluating the measurement technique as well as establishing the range of validity of (26) as a function of V, we conducted a carefully designed experiment in which e" of plexiglass was measured as a function of V for values of V extending from 10-3 percent to 10.1 percent. We chose plexiglass because its complex dielectric constant is well known (e = 2.55 - j 0.0165) and its dielectric loss factor is small. The results of the experiment are shown in Fig. 6. The measurement technique predicted the correct value for e" within an rms error of 0.001 for the range 0.01 percent <V< 1 percent, and with a slightly larger error for V up to 5 percent. A detailed anlaysis of the errors associated with the measurements of the quantities fo, fs, Qo, and Qs led to the conclusion that the optimum range of V is between 0.5 percent and 1 percent, and that if V is in this range the minimum measurable value of'" is around 0.002. For each rock sample, the e" data reported in this study are averages of measurements conducted for five small sub-samples of the (parent) sample. For a few of the measured samples, e" was observed to exhibit no discernible dependence on frequency. For most samples, however, E" decreased with increasing frequency over the 1.6 GHz - 16 GHz range. Typical examples of these two types of spectra are shown in Fig. 7. 3.2 Cavity Characteristics 14

Five cylindrical cavities were used in this study, with center frequencies ranging from 1.6 GHz to 16 GHz. Table 1 provides a listing of their pertinent characteristics. Of particular note is the cavity volume, ranging from approximately 1000 cm3 for the 1.6 GHz cavity to only 1 cm3 for the 16 GHz cavity. 4. MEASURED DATA The measured data has been tabulated in a technical report [16]. For each sample, the entries include (1) rock type, (2) rock # and source, (3) density, (4) C (average value over the 0.5 - 18 GHz range), (5) s/e', the standard derivation-to-mean ratio of the measured value of ~', (6) e" at 1.6, 5.0, 7.8, 11.4, and 16.0 GHz, and (7) the weight fractions of 10 chemical compounds (for only 56 of the 80 rock samples). The information on chemical composition was obtained using x-ray fluorescence. The distribution of the 80 samples by rock type is given in conjunction with Fig. 8. 5. ANALYSIS OF PERMITTIVITY DATA Among the 80 rock samples, the measured value of e' ranged between 2.5 and 8.3. The ranges for individual rock groups are presented in a horizontal bar-chart format in Fig. 8. 5.1 Dependence of e' on Density According to previous studies [1-6], the density p of the rock is the single most important parameter governing the magnitude of e'. One of the commonly used formulas relating e to p is based upon the geometric mean formula where loge = Vi log ~. (27) 15

where Vi is the volume fraction of the ith component of the material and is related to density by Vi = Pb / Ps (28) where Pb and ps are the bulk and specific densities of the rock material, respectively. Thus, a two component mixture of solid rock and air is described by Pb log p - log. (29) Ps Ps Hence, at an arbitrary density p Pb/P b = Ep (30) Campbell and Ulrichs [1] conducted measurements for a large number of powdered rocks, all at a density of Pb = 1 g/cm3, and found that E'Pb varied over the narrow range between 1.9 and 2.1 for most of the 25 different types of powdered rocks measured and that the mean value is around 2.0. Upon setting p = 1 g/cm3 and e'p = 2 in (30), we find e' = 2Pb. This result is in close agreement with the formula used by = (1.96 0.14)Pb (31) Pb 16

This function is shown in Figure 9 along with the measured data. The linear correlation coefficient between the values predicted by (31) and the measured values of e' is R = 0.72. Hence, about 50% of the variance in the data can be attributed to the density of the rock samples; the remaining data scatter about the regression curve is attributed to dependence of e' on the mineral composition of the rocks. The variation of e' with Pb is also shown in Figure 9 for individual rock types. The carbonates exhibit the narrowest density range, followed in order by igneous, volcanic and sedimentary silicates. The igneous volcanic silicates are found to have the strongest sensitivity of e' to Pb. 5.2 Dependence of' on Chemical Composition The geometric mean formulation given by (27) can be used to investigate the dependence of e' on bulk mineral composition of rock. Ideally, the volume fraction Vi of each discrete constituent mineral would be known for use in the analysis. However, the rock chemistry of the samples used in the present study is ascertained by x-ray fluorescence which provides the mass percent Wi of various oxides for the bulk material. Consequently, the following analysis evaluates the dependence of E' on bulk chemical composition. X-ray fluorescence of the silicate samples yields values for the mass percent WI of 10 oxides plus the percent loss on ignition of the sample (LOI). The volume fraction of each oxide can be approximated by Vi = PbWi/Mi (32) 17

where Mi is the molecular weight of the ith oxide constituent. Hence, for the ten measured oxides plus loss on ignition (LOI) 1 Pb Wi 11 log E' = S M log ~'i 11 W =Pb M log e'i (33) An empirical evaluation of the dependence of e' on bulk chemical composition is possible for the silicate rock samples which had been measured by x-ray fluorescence. A multiple linear regression analysis of the form given in (33) utilized measurements of c', Pb and Wi/Mi to yield estimates of log e'i. In order to gain some understanding of the role of mineralogy and crystal size and structure in determining ~', the analysis is conducted for all silicate rocks treated as a single class and as subdivided by genesis into volcanic, plutonic and sedimentary subgroups. Since x-ray fluorescence provides measurements of 10 oxides plus loss on ignition (LOI) and the smallest subclass, plutonic silicates, contains only 14 samples, a stepwise approach is used in the analysis wherein at each step an oxide or (LOI) is added to the regression on the basis of its F-ratio. In this analysis it is presumed that LOI is dominated by the liberation of bound water during sample ignition and hence the molecular weight of H20 is assumed. For all silicate rock samples combined (54 samples), the regression analysis is truncated after the inclusion of 6 variables with 8.44 as the F to enter of the 6th variable and a resultant standard error of the estimate (SEE) of 0.025. The regression equation is 18

l.1 31 (~.007) Ws log' = Pb M.517 (_.093)W 0 M51.517 (~.093) WFe2 03 Fe2 03.1 (~.021)WM 0 +...... + MMg 0.637 (~.062) WAI, 0 MAI2 03.11 (~.031)WLOI + M MLOI.626 (~.216) WK MKK2 0 K2 0 (34) The variables in (34) are listed in decreasing order of significance. The estimates of e provided by (34) are related to the measured values of e' by e' = 0.643 + 0.891 eCa calculated with a linear correlation coefficient of r = 0.835 for the 54 silicate samples. For each silicate subclass, the regression analysis is truncated after inclusion of four variables and yields: a) volcanic silicate rocks, 20 samples, SEE =.02,.174 (~.01) Wsi 0 log e' = Pb M si 02.62 (~.06) Wca o Mca o 0 14.808 (~5.364) Wpo 0 -+ - K ——!A / 1.334 (~.256) WK2'2 + (35) MK20 19

b) plutonic silicate rocks, 14 samples, SEE =.02 /.165 (~.011) Wsi log' = b + M S O Si 02 1.995 (~.197) WFe0 Fe 2 03 2 +.855 (~.139) WNa 0 MNaO Na2 0 1.687 (~.439) WT 2 ) MTi 02 (36) c) sedimentary silicate rocks, 20 samples, SEE =.028,.111 (+.016)WFe20 log c = Pb MF MFe2 03.157 (~.045) WTi 2 M+ o2.982 (~.353) Wsi 2 + M. Msi o2 2.561 (~.677) WNa0) MN2 Na2 0 (37) The values of c' estimated by (35) to (37) are related to the measured values of e' by e = 0.864 + 0.841 calculated with a linear correlation coefficient of r = 0.886 as shown in Figure 10. The most significant oxides are found to be Si 02 and Fe2 03. However, the variability in the oxides selected for each class and the differences of their respective regression coefficients between subclasses indicates that bulk chemical composition and density alone are not accounting for all of the observed sample variance in a robust fashion. 20

This implies that mineralogy per se and crystalline structure also exert a strong residual influence on'. The significance of mineral composition is illustrated by a sample of gabbro in Figure 11. The average value of ~' for this sample is 6.6. However, olivine inclusions are found to have much larger values of ~' ranging from 7.2 to 13.2. 5.3 Dependence of Dielectric Loss on Density and Chemical Composition The measured dielectric loss factor E" ranged between <0.002 to 0.245 as shown in Figure 12 and, unlike e', the loss factor is found to generally decrease with frequency. Prior to measurement in the resonant cavities, all samples were oven dried for 15 minutes at 105~C in order to liberate potential layers of moisture absorbed from handling or from the laboratory atmosphere. The effects of this drying process on E are shown in Figure 13 for samples of basalt and siltstone. In general, the drying process had little effect upon e" of igneous samples and more pronounced effect for sedimentary samples. However, even with the low temperature drying (at 105~C), the vast majority of rock samples display a decrease of c" with frequency. This is illustrated by Figure 14 which shows the average loss of 72 rock samples as a function of frequency. This behavior may be partially attributable to the effects of bound water within the rock samples. The frequency behavior of the average dielectric loss factor for each gereralized rock type is shown in Figure 15. E" is the lowest for carbonates and the highest for volcanic silicates. The decrease of C with frequency is found to be the greatest for sedimentary and plutonic silicates. The loss factor e" may be modeled as the sum of a conductive component Cc" and a frequency-independent residual component Er" 21

" (f) =;' + c (f) r EC = e + r 2Xf E0 P + P2/ f (38) where a is the conductivity, and P1 and P2 are abbreviations for Er" and (o/2xno, respectively. For each of the measured rock samples, the values of P1 and P2 are determined by fitting the data obtained at five frequencies (1.6, 5.0, 7.8, 11.4, and 16.0 GHz) to the linear function given by (38). Figure 16 presents the values of P1 and P2 determined for each of the rock samples. The constant P2 of a rock sample is proportional to its conductivity o and is given in units of GHz-1. The values of P1 and P2 are shown be be uncorrelated. Unlike e', which exhibits a strong dependence on bulk density, the loss factor e" is not significantly correlated with Pb at any of the measured frequencies. This same conclusion applies to P1 and P2 as shown in Figure 17. The role of bulk chemical composition in the determination of the loss factor is evaluated in a fashion similar to that presented for'. The conductive and frequencyindenendent components of e" are each empirically related to the bulk chemistry by multiple linear regression analyses on P1 and P2 using a stepwise approach, P1 = 1 V P. ijg (39) P2 = Vi P2 i,g (40) 22

where Vj = Pb Wj / Mj in moles / cm3 and the subscripts i and g refer to the specific oxide (from x-ray diffraction analysis) and rock sub-category, respectively. For the frequency-independent component P1 of the loss factor, the regression results for treatment of all 54 silicate rock samples as a single class yield.19 (~.114) WFe0 ~I=(M F3+ P1 Pb \ M- Fe2 03.051 (~.018) WLOI MLOI.082 (~.033) WAi 0 MAI 03 1.132 (~.806)WMo 0 MMn 0 Mn 0 (41) with SEE =.019. The estimates of P1 provided by (41) are found to be very poorly correlated with the values of P1 determined for each of 54 samples (r =.42) and with a regression P1 = 0.007 + 0.86 P1 calculated Consideration of each silicate subclass seperately yields a) volcanic silicate rocks, 20 samples, SEE =.015 1.563 (~.434) WF2 3 P1=pb( M MFe2 03.167 (+.099) Wm MMg O.232 (~.107) WCa 0 Mca o 1 86 (~.067) WLO MLOI (42) 23

b) plutonic silicate rocks, 14 samples, SEE =.004 P = Pb (.62 (+.076) WFe MF Fe203.465 (~.105) WT: O 2MTi2.027 (~.006) WM 0 MMg 0.071 (~.024) Wca ) Ma o (43) c) sedimentary silicate rocks, 20 samples, SEE =.017.537 (~.127) WAi2 0.952 (~.421) WK2 Pi = Pb M Al 203 MK20 4.348 (~2.058) WT: 0 1.407 (~.685) W MTi 02 MMg O 2~~~~~~ (44) The estimates of P1 provided by (42 to 44) are found to be moderately well correlated with the values of Pi determined for each of 56 samples (r =.80) as shown in Figure 18 with a regression of P = 1.007 Pcalc 1 i calc (45) For the frequency-dependent, conductive component P2 of the loss factor, the stepwise linear regresion results for all silicate rock samples treated as a single class yield 24

117 (~.04) WAI2 0 P = 2 + 2 Pb M + Al2 03 3.447 (~2.349) Wpo Mp PPO.043 (~.021) WLOI MLOI.003 (~.005) Wsi O2 ~+ M-M — Si 2 2Si 0 (46) The oxides in (46) are listed in decreasing order of significance. Similar to P1, the esteimates of P2 given by the single class regression (46) are poorly correlated to the values of P2 determined for each sample by (38) with r = 0.41 for the 54 samples. P2 = 0.009 + 0.863 P2 cac. Regression analyses of P2 seperately for each silicate class yields a) volcanic silicate rocks, 20 samples, SEE =.013,.232 (~.081) WAI2 P2 =Pb M MA'* Al.299 (~.18) WNa MNa2 0 3.216 (+.116) WLOI MLOI.011 (~.007) Wsi2 MSi o2 (47) 25

b) plutonic silicate rocks, 14 samples, SEE = 0.12, = ( 26.24 (~5.219) WLOI 2 Pb \ MLOI MLOI.05 (~.015) Wsi 0 Msi'+ Si 02 13.268 (~2.604) WNa2 MNa 0.055 (~.039) WAI 0 MAI2 03 A2 03 (48) c) sedimentary silicate rocks, 20 samples, SEE = 0.15,.087 (+.024) WLOI P2= Pb \ MLOI MLOI 1.565 (~.707) WM C MM O MMg 0 1.098 (~.264) WNa2 0 Na2 0.146(~.064) Wao \ Mca (49) Also similar to the results found for P1, the estimates of P2 provided by each class treated seperately using (47) to (49) are moderately correlated with the values of P2 determined for each sample using (38) with r = 0.75. As shown by Figure 19, P2 = 0.004 + 0.839 P2 calc (50) 26

Not suprisingly, LOI and Na2 0 are generally the most significant determinants of the conductive component of the loss factor. The quantity of LOI is presumed to be largely determined by the bound water content of a sample. The aggregate effect of using bulk rock chemistry to explain the measured variance in e" as a function of frequency via (38) is shown in Figure 20. Linear A correlation between the measured loss factor and e" estimated from bulk chemistry using estimates of P1 from (42) to (44) and P2 from (47) to (49) for each silicate class yields correlation coefficients in excess of 0.77 at each of the five frequencies. Thus, the empirical formulations generally account for better than 60% of the observed variance in c". 6.0 CONCLUSIONS The real part of the permittivity c' is found to be independent of frequency between 1.6 GHz and 16GHz. The bulk density of rock accounts for 50% of the observed variance in e', and the geometric mean formulation adequately describes this relationship in confirmation with prior studies. For silicate rocks, the bulk chemical composition can account for an additional 28% of the variance in e provided that each silicate rock class (volcanic, plutonic, and sedimentary) are treated as distinct classes. This implies that mineralogical differences between the various silicate classes are significant with respect to e'. The most significant oxide constituents in determination of e' are SiO2, Fe203, Ti 02, and Na2 0. The dielectric loss factor " << e'. In general, c" decreases with frequency and can be modeled as the sum of a frequency-dependent conductive term P2 and a frequency-independent residual term P1. The loss factor e" is very poorly correlated with the bulk density of rock as are the component terms P1 and P2. Bulk chemical composition as given by x-ray fluorescence can account for 60% of the variance in E at frequencies from 1.6 GHz to 16 GHz when silicates are subgrouped by genesis 27

(i.e., volcanic, plutonic and sedimentary). Hence, mineralogy and crystal structure are presumed to be important determinants of dielectric loss. The measured loss is greatest for volcanic silicates and least for carbonates. For silicate rocks, the most important oxide consituents determining e" are Fe203, TiO2, LOI, Al203, Na20, and MgO. The significance of LOI is greatest for sedimentary silicates and least for plutonic silicates, and may be related to the role of bound water in influencing dielectric loss. The empirical relationships derived in this study yield highly satisfactory estimates of permittivity as functions of bulk density and bulk chemical composition for the frequency range from 1.6 GHz to 16 GHz. In applying these results to a terrestrial evnironment, the role of both adsorbed and absorbed water must be kept in mind as this will influence both e' and the loss factor. Finally, in light of these results it is noted that the role of mineralogy and crystal structure remains to be examined analytically. 28

REFERENCES 1. Campbell, M.J. and J. Ulrichs, "Electrical properties of rocks and their significance for lunar radar observations," Journal of Geophysical Res., Vol. 74, pp. 5867-5881, 1969. 2. Olhoeft, G.R., A.L. Frisillo, and D.W. Strangway, "Electrical properties of lunar soil sample 15301, 38," Journal of Geophysical Res. Vol. 79, pp. 1599-1604, 1974. 3. Strangway, D.W., G.R. Olhoeft, W.B. Chapman, and J. Carnes, "Electrical properties of lunar soils: dependence upon frequency, temperature and moisture," Earth Planet Sci. Lett., Vol. 16, pp. 275-281, 1972. 4. Olhoeft, G.R. and D.W. Strangway, "Dielectric properties of the first 100 meters of the moon," Earth Planet Sci. Lett., Vol. 24, pp. 394-404, 1975. 5. Parkhomenko, E.I., Electrical Properties of Rocks, trans. G.V. Keller, Plenum Press, New York, pp. 314, 1967. 6. von Hippel, A.R., Dielectric Materials and Applications," John Wiley, New York, pp. 405, 1954. 7. Athey, T.W., M.A. Stuchly, and S.S. Stuchly, "Measurement of radio frequency permittivity of biological tissues with an open-ended coaxial line: Part I," IEEE Trans. Microwave Theory Tech., vol. MTT-30, no. 1, pp. 82-86, Jan. 1982. 8. Stuchly, M.A., M.M. Brady, S.S. Stuchly, and G. Gajda, "Equivalent circuit of an open-ended coaxial line in a lossy dielectric," IEEE Trans. on Instrumentation and Meas., Vol. IM-31, No. 2, June, 1982. 9. Burdette, E.C., F.L. Cain, and J. Seals, "In Vivo probe measurement technique for determining dielectric properties at VHF through microwave frequencies," IEEE Trans. Microwave Theory Tech., vol. MTT-28, no. 4, p. 414-427, April, 1980. 10. EI-Rayes, M.A., and F.T. Ulaby, "Microwave dielectric spectrum of vegetation Part 1: experimental observations," IEEE Trans. Geoscience and Remote Sensing, Vol. GE-25, No. 5, pp. 550-557, 1987. 11. Marcuvitz, N., Waveguide Handbook, New York, Dover, 1965. 12. Deschamps, G.A., "Impedance of antenna in a conducting medium," IRE Trans. Antennas Propagat., pp. 648-650, Sept. 1962. 29

13. Ulaby, F.T., R.K. Moore, and A.K. Fung, Microwave Remote Sensing, vol. 3, Dedham, MA: Artech, 1986, Appendix E. 14. Jordan, B.P., R.J. Sheppard, and S. Szwarnowski, "The dielectric properties of formamide, ethanediol, and methanol," J. Phys. D: AppI. Phys., vol. 11, pp. 695701, 1978. 15. Harrington, Roger F., Time-Harmonic Electromagnetic Fields, New York, McGraw-Hill Book Company, pp. 317-326, 1961. 16. Ulaby, F.T., T. Bengal, J. East, M.C. Dobson, J. Garvin, and D. Evans, "Microwave dielectric spectrum of rocks," Radiation Lab. Tech. Rep. No. 023817-1-T, Department of Electrical Eng. and Computer Sci., Univ. of Michigan, Ann Arbor, March, 1988. 30

HP8530B SWEEP OSCILLATOR 0.5 TO 18 GHz CONNECTOR (APC-7) FREQUENCY SWEEP/ INTERCONNECT / CO 2 LU cn,0 0'::: S11 HP8514A S-PARAMETER TEST SET PROBE IF TEST SET INTERCONNECT ROCK CO Co a. I Fig. 1. Block diagram of the dielectric-probe measurement system.

-2r2 - 2rl Plane Longitudinal Cross Section High Frequency Probe Low Frequency Probe (a) Dielectric Probes 4 Yi Ye (b) Probe Equivalent Circuit Fig. 2. The dielectric probe; (a) cross-sectional views, and (b) equivalent circuit.

8 7 6 5. m -*' m m m..w *9 9 * I -W aIm a a a _,____ m I m U U -_ U 0 G-14 G - PW-33 * J - WRB85-7 x J - 7-25-84L 4 3 n\ I 0.5 0.5 5.0 9.5 14.0 14.a0 18.5 Frequency (GHz) Figure 3. Typical examples of measured permittivity spectra.

Pin -' Pout I 1 _1 I L mm %Ikl - ---- Resonant Cavity (a) Cavity with Sample Pout Pin fo (b) Transmission Spectra Figure 4. When a sample is inserted in the cavity, its transmission spectrum changes: the resonant frequency shifts from fo to fs and the spectrum becomes broader.

HP8530B SWEEP OSCILLATOR 0.5 TO 18 GHz FRE UENCYSWEEP/ CONNECTOR (APC-7) FREQUENCY SWEEP/' INTERCONNECT SMA - 1 COAN 1 S11 HP8514A S-PARAMETER TEST SET S22 CAVITY IF TEST SET INTERCONNECT 0.141 INCH SMA CABLE HP8510A NETWORK ANALYZER 03 CD Q. Figure 5. Block diagram of the system used to measure the cavity transmission spectrum.

0.2 LM 0._ U) U) 0 -O 0 C) C) 0 Q) 0.1 1 e" for plexiglass 776VA 999wam Hm /r 0.0.002.010.031.057.180.424.640 1.50 5.06 10.1 Volume Fraction (%) Figure 6. The vertical bars denote the measured dielectric loss factor c" at the indicated volume fraction. The test material is plexiglass e" = 0.0165.

0.3 C ( C C & C C C! 0.2 - I) 0 L 0.0 0.0 - ~ I' - I, - I ~ i, - i 0 3 6 9 12 15 18 Frequency (GHz) Figure 7. Typical examples of the measured spectra of the dielectric loss factor e".

Sedimentary Silicates (22) Igneous Plutonic Silicates (16) Igneous Volcanic Silicates (26) All Igneous Silicates (42) Carbonates (10) Sulfates (1) 2 4 6 8 10 Permittivity Figure 8. Range of e' for individual rock classes.

10.0 8.0 6.0 Pb = 1.96 c c c c v c / c s c* pp sv sp p sv sp/ p V )v p p p A co 0 E.S S so vy 0 oo s s pv / s v P V S S.V S V S 4.0 s s S S S s s s 2.0 0.0 v = volcanic p = plutonic s = sedimentary c = carbonates o = other } silicates I I I I I I I I I I 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 density Pb (g/cm3) Figure 9. Variation of the real part of the dielectric constant e' with rock bulk density.

10 0 1 2 3 4 5 6 7 8. 9 10 c' calculated Figure 10. Comparison of measured e' with the permittivity estimated on the basis of density and bulk chemistry by (35) to (37) for volcanic, plutonic and sedimentary silicate rocks.

Nw P. - 5.5 cm. I I (a) Sketch of rock surface (b) Measured permittivity Effect of mineral inclusions on c' for a sample of gabbro. Figure 1 1

G-6 G- PW-14 J 7-27-84L-7J G-8-1 G- PW-30 J-WRB 85-13 J-WRB 85-2 G -8' J 7-21-84L-6 J-WRB 85-12 G-15 G - 5 G - PW-39 G-5 G-14 G- PW-22 G - PW-125 G-1 I G - PW-33 J-WRB 85-6 G-PW-4 J-WRB 85-14 G-9 G-10 G - PW-127 J-WRB 85-18 J-WRB 85-19 G-18 G-17 G- PW-36B G - PW-28 G - PW-36A G-2 J-WRB 85-11 G- PW-40 J 7-27-84L-81 0.00 rlrwm~~~ I I - M E ~ (1.6 GHz) Ea (16 GHz) G - PW-76 G - PW-101 J 7-27-84L-5A G-3 J-WRB 85-1 G- PW-17 G- PW-41 J-WRB 85-8 J-WRB 85-20 G- ELGY J-WRB 85-15 J-WRB 85- 7 J 7-23-84L-7 J 7-27-84L-6A J-WRB 85-3 J 7-25-84L-4 J 7-23-84L-4 J 7-25-84L-8 J 7-27-84L-6G G - PW-19 J 7-21-84L-4 G-12 J-WRB 85-4 J 7-27-84L-2 J 7-23-84L-14 G - PW-32 J-WRB 85-7 J7-23-84L-11 J-WRB 85-5 G-7-1 G-PW-7 G -4-2 G - PW-24 G-13 J 7-25-84L-6 G- PW-25 11.1m -f Jr xffxx -4 * ~E(1.6 GIz) a c (16 GHz) IlIIiIIII -- M 0I * 0 0.I1 0.05 0.10 0.15 0.20 I0. 0.25 0.00 0.01 0.02 0.03 0.04 0 05 0 06 Is Figure 12. Measured dielectric loss factor at 1.6 GHz ahd 16 GHz.

0.30 II 0.25 0 LL oC 0) 0 -.i U., U CD 7F) a 0.200.15-, G.PW-368 - G- PW-368 Before Drying After Drying 4 I U.1 - 1 r 5 9 13 Frequency (GHz) (a) Sample PW-36B (basalt) 17 k Before Drying 0 U et TU 0 J._ 0 - h. U 0.25 0.20 0.15 After Drying \ _t- J.WRB 8514 -. J.WRB8S5-14 I i _LF~ 0.10 T 1 5 9 13 17 Frequency (GHz) (b) Sample WRB-85-14 (siltstone) Figure 13. Effect of drying surficicial water on e".

0.07 mt LL U, 0 -J 0'C n "3 0.060.050 A% O.[ 4 _4 I W II I 6rT 0 3 6 9 12 Frequency (GHz) I, 15 18 Figure 14. Average dielectric loss factor of all 72 rock samples versus frequency.

, 0.06LL. (n C, (3 0 a= 0.04' a 18 0 3 6 9 Frequency (GHz) Variation in average loss spectrum for various rock classes. Figure 15.

P1 C - PW 32 G P'W 14 j WHII 8s 5 J /,'1 8-1-4 J Will 85-7 J 7-21-84L-6G G-18 G- PW-24 J 1 23 84L-14 J-WFRL 85-11 G- PW-25 J 1-23 84L- 11 J WIRB 85-6 G -PW-7 J /-25 84L-6 G-5 / -2/-84L- 7J G- PW-39 G - fW-127 G - 14 J WiRlt 85-2 G- 10 G - PW-22 J WHB 85-13 G -PW-33 G -8' G - 2 G -2 G-6 G -5' (G -PW-28 ( -'W- 125 (; - PW-4 J / 21 84L 81.I Wil 85 19 J WiHil85 18 ( PW 40 I *I G - PW-32 G - PW-14 J WRB 85-5 J 7-21-84L-4 J-WRB 85-7 J 7-27-84L-6G G-18 G- PW-24 J 7-23-84L-14 J-WRB 85-11 G- PW-25 J 7-23-84L-11 J-WRB 85-6 G-PW-7 J 7-25-84L-6 G-5 J 7-27-84L-7J G- PW-39 G-PW-127 G-14 J-WRB 85-2 G-10 G- PW-22 J-WRB 85-13 G- PW-33 G-8' G-2 G-6 G-5' G- PW-28 G- PW-125 G -PW-4 J 7-27-84L-81 J-WRB 85-19 J-WRB 85-18 G- PW-40 I I I I I I r I% - I N_ -I -I-- -III- - -I r L-~~~~~~~~~~~~~~ I 1 -1 ) 20 )O 0.05 0.10 0.15 0.20 () O.C ------— I 0 00 0.05 0.10 P1 0.15 0 P2 Figure 16. Values of P1 and P2 calculated from the measured loss spectrum for each rock sample by (34).

0.2 I - 0.10.0 0.0 - 0 0 0 a o a 0 a a Q 0G Qa A. a FA Pe a.,a I 1 2 3 DENSITY (g/cm3) 4 0.3 0.2 NO 0.1 0.0 0.0.0.1 1 2 3 4 DENSITY (g/cm3) Figure 17. Variation of P1 and P2 as calculated by (34) with rock density.

0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 P1 0.00.r.b' 0.00 0.02 0.04 0.06 0.08 010 0.12 0.14 0.16 0.18 0.20 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 P1 calculated Figure 18. Comparison of observed P1 with that estimated on the basis of density and bulk chemistry for each silicate rock class (volcanic, plutonic and sedimentary) using (42) to (44), respectively.

0.30 0.25 0.20 0.15 P2 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P calculated Figure 19. Comparison of observed P2 with that estimated on the basis of density and bulk chemistry for each silicate rock class (volcanic, plutonic and sedimentary) using (47) to (49) respectively.

0.30 0.25 - 0.20 0.15 0.10 0.05 0.00 I0.00 0.05 0.10 0.15 0.20 0.25 0.30 " calculcated a) Frequency = 1.6 GHz Figure 20. Relationships of measured loss factor to the loss factor estimated on the basis of density and bulk chemistry at (a) 1.6 GHz, (b) 7.8 GHz, and (c) 16 GHz.

0.30 0.25 * sedimentary 0.20 - 0.15- 6, EI 0.10- 0 * 0 0 o 0.05 - 0. 00 0.00 0.02 0.04 0.06 0.08 0.10 E" calculated 0.12 0.14 0.16 0.18 0.20 0.12 0.14 0.16'0.18 0.20 b) Frequency = 7.8 GHz

0.20 0.18 0.16 0.14 0.12 0.12 - i 0.10 ~" 0.08 o ~. *I 0.06- o o0~ / ~e'~"= -.002 +.921l" 0.04 N= 56 * * o*0' * r=.774 0.02- c 0 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 E" calculated c) Frequency = 16 GHz