SSSAJ: Volume 72: Number 5 • September–October 2008 1401

Soil Sci. Soc. Am. J. 72:1401-1409doi:10.2136/sssaj2007.0319Received 28 Aug. 2007.*Corresponding author (andrewmarkthomas@yahoo.com).© Soil Science Society of America677 S. Segoe Rd. Madison WI 53711 USAAll rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

This research was based on the need to obtain data on the dispersive nature of soils, i.e., soils such as wet clays whose

electromagnetic properties vary with frequency. It was consid-ered important that this need be met by a system suitable for use both in the laboratory and in situ, across a wide frequency range, using a relatively simple and robust methodology. The options available for achieving these aims are limited, particu-larly if the same equipment and data processing techniques are to be used for both scenarios. For instance, laboratory testing generally relies on coaxial sample containers and frequency-domain measurement equipment, whereas fi eld measurements are normally undertaken using multiwire probes and time-domain measurement equipment, the former providing data with a higher measurement confi dence level and the latter uti-lizing probes suitable for insertion into undisturbed soils.

Although dielectric probes exist for use with frequency-domain measurement equipment, they generally only measure the properties of a material within a small depth below the

surface and so may not be representative of the bulk material properties. Time-domain refl ectometry probes have the advan-tage of measuring average properties across the whole probe length, but provide only a single value of apparent permittiv-ity based on measurement of the time taken for an electro-magnetic pulse to be refl ected from the end of a cell or probe. This single measurement of apparent permittivity does little to provide data on the dispersive nature of soils and there is diffi culty in determining the frequency at which the measure-ment is valid.

Regardless of these limitations, TDR has one overriding advantage over other measurement systems, this being the wide range of uses that a combination of portable equipment and simple probes allows. Therefore, our study focused on an alternative method of obtaining frequency-domain data from TDR cells and probes and, in so doing, developed a robust methodology for obtaining such data. The method adopted is a quarter-wavelength technique, sometimes referred to as resonant frequency analysis, based on variations that occur in transmission line voltages at discrete frequency intervals and that are considered to provide accurate values of signal velocity and hence apparent permittivity.

This technique (together with related non-TDR fre-quency-domain probe techniques) has been used successfully by researchers such as Logsdon (2005) in conjunction with a short coaxial cell and Heimovaara (1994) and Heimovaara et al. (1996) for characterization of a seven-conductor TDR probe in air and water (based on the low-frequency impedance probe design of Campbell, 1990). Logsdon and Hornbuckle (2006) also described the use of a short 12-conductor coaxial probe

A. M. Thomas*D. N. ChapmanC. D. F. RogersN. MetjeP. R. AtkinsH. M. LimSchool of EngineeringUniv. of BirminghamEdgbastonBirmingham B15 2TTUK

SOIL PH

YSIC

S

Broadband Apparent Permittivity Measurement in Dispersive Soils Using Quarter-Wavelength Analysis

Time-domain refl ectometry (TDR) has, since the early 1980s, become a widely used and robust method for measuring the electromagnetic properties of soils. Although it allows use of relatively simple probes and measurement equipment that free users from complex analysis of small soil samples in the laboratory, it provides little information on the dispersive nature of fi ne-grained soils. Therefore, we developed a quarter-wavelength analysis (QWA) methodology that allows apparent permittivity spectra to be obtained for soils, while retain-ing the same range of cells and probes and the simplicity of data analysis associated with TDR. We tested the technique on clay soils across a wide water content range and found that it provided useful data for frequencies ranging from approximately 100 MHz to >1 GHz—the accuracy being considered no worse than two units of apparent permittivity. The results show that the methodology is capable of providing broadband dispersion data that give signifi cantly greater detail on soil electromagnetic properties than can be expected from single-frequency TDR data. This has a number of important advantages including the poten-tial to provide accurate water content measurements in clay soils, a means to extend TDR soil monitoring data, and the opportunity to validate research into the measurement frequency associated with simple TDR measurements and time- to frequency-domain inversions.

Abbreviations: QWA, quarter-wavelength analysis; TDR, time-domain refl ectometry; VNA, vector network analyzer.

Published September, 2008

1402 SSSAJ: Volume 72: Number 5 • September–October 2008

for fi eld soil measurements, although this does not appear to have been based on quarter-wavelength data and provided low-frequency data up to approximately 100 MHz. Some research-ers have also made limited use of the technique to validate methods for the inversion of TDR probe data to the frequency domain, including Cataldo et al. (2007) using a coaxial liquid testing probe and Jones and Or (2004) using a three-conduc-tor TDR soil probe. Both identifi ed quarter wavelengths from inverted data, however, rather than by directly measuring them in the frequency domain.

Despite this previous research, QWA appears not to have been developed fully as a soil measurement system in its own right, even though it is based on a relatively simple method and is capable of providing broadband data. In fact, insofar as we have been able to ascertain, the only use of quarter-wavelength frequencies, directly measured in the frequency domain using TDR probes, is that of Heimovaara (1994) and Heimovaara et al. (1996), and this appears to have been limited to measure-ments in air, water, and a number of sandy soils for validation of inverted TDR data.

Therefore, this study investigated QWA as a simple, robust, and reliable methodology for soil electromagnetic characteriza-tion using both cells and TDR probes. In so doing, the central objective was to provide details of an electromagnetic mea-surement system that allows TDR-compatible soil data to be collected, while signifi cantly extending the usefulness of these data through incorporation of additional data on the effects of frequency due to dispersion.

APPARENT PERMITTIVITYCompatibility between QWA and TDR data is dependent on

there being a common measurement unit. As TDR provides data in the form of apparent permittivities, which can be derived from the propagation velocity, it has been adopted for this study.

In a soil, the propagation velocity of an electromagnetic signal can be determined from knowledge of the permittivity and magnetic permeability, both of which are complex parameters describing charge storage and loss mechanisms, respectively (see Thomas et al., 2006). As most soils are considered to have low and noncomplex magnetic permeabilities, the signal velocity can be calculated using (after Hayt and Buck, 2006)

( )1/2

2

1

2 1 1V

ωβ με ε ε

= =⎡ ⎤′ ′′ ′+ +⎢ ⎥⎣ ⎦

[1]

where V is the velocity (m s−1), ω is the radian frequency (ω = 2πf where f is the frequency in Hertz), β is the phase constant (i.e., the phase change per unit distance expressed in radians m−1), μ is the magnetic permeability and ε′ and ε″ are, respectively, the real and imaginary parts of the permittivity (i.e., the electromagnetic charge storage and loss mechanisms). It should be noted that the phase constant includes the radian frequency, which cancels the radian fre-quency in the numerator.

Loss mechanisms in soils, at high frequency, are generally consid-ered to comprise those associated with dispersion, i.e., dipolar losses associated with the inability of water molecules to rotate in time with the applied electromagnetic fi eld, and those associated with the conductivity. Therefore, for the purposes of Eq. [1], losses are often included through the use of

DCP

σε ε

ω′′′′= + [2]

where σDC is the static (direct current) conductivity of the material (S m−1) and εp″ represents dipolar losses. These losses, when divided by the real part of the permittivity as shown in Eq. [1], form the loss tangent associated with the soil. This loss tangent is an important fac-tor determining variations in both signal velocity and attenuation in soils tested using QWA.

The measurement of apparent permittivity (εr*) has become widespread due to the popularity of TDR and provides a single value to describe all of the electromagnetic parameters. It is based on cal-culation of the real relative permittivity of a lossless material, which corresponds to the velocity obtained in Eq. [1]:

2

r *c

Vε ⎛ ⎞=⎜ ⎟⎝ ⎠

[3]

where c is the velocity of light in a vacuum (2.9979 ×10−8 m s−1). From this, it can be seen that the apparent permittivity is a repre-sentation of a number of parameters in one simple value and can be determined from any measurement method that can provide velocity data, rather than being limited solely to TDR.

MATERIALS AND METHODSQuarter-Wavelength Analysis

When a sinusoidal signal, generated by a network analyzer, is injected into a coaxial measurement cell, the magnitude of the refl ected signal returning to the analyzer will be a function of two sinusoidal voltages: the forward-traveling (incident) wave and the backward-traveling (refl ected) wave refl ected from the end of the cell, as illustrated in Fig. 1. The combined wave, also known as the volt-age standing wave, therefore varies in amplitude and phase as a func-tion of the incident and refl ected waves and will exhibit maxima and minima spaced at multiples of a quarter wavelength. Therefore, if the voltage is measured at the point marked “signal in,” the maxima and minima of the measured signal will occur only at frequencies where the change in phase during one-way transit of the signal along the cell length is a multiple of 90°. For this reason, they are known as the quarter-wavelength frequencies.

The fact that these quarter-wavelength points can be used to calculate the velocity, and hence apparent permittivity, of a material stems from the nature of the phase constant as a measure of radian phase change per meter. Therefore, if the phase change per meter is divided into the phase change per second (i.e., the radian frequency),

Fig. 1. Incident and refl ected signals in the measurement cell, which interfere to create a combined wave exhibiting maxima and minima that move as the frequency is varied, causing voltage changes measurable at the point “signal in.”

SSSAJ: Volume 72: Number 5 • September –October 2008 1403

the result is the velocity in meters per second. Therefore, based on Eq. [1] and [3], the velocity can be described using the progression

Rr

2

*

c LV

ωφ φε

= =− [4]

where L is the cell length (m), φ is the total phase change in radians (i.e., across the full two-way transit of the signal through the soil) and φR is the phase change that occurs on refl ection at the cell end (zero for an open circuit termination and 180°, i.e., π radians, for a short-circuited termination).

As the quarter-wavelength frequencies occur at total phase changes that are a multiple of 180° (i.e., 90°, or 0.5π, for each of the incident and refl ected wave transits), Eq. [4] can be simplifi ed for the fi rst quarter-wavelength frequency as detailed in the progression

qw qwqw

r

2 44

*

L f Lcf L

ω π

π πε= = = [5]

where ωqw is the fi rst quarter-wavelength radian frequency and fqw is the corresponding frequency in Hertz.

Therefore, rearrangement of Eq. [5] gives the apparent permit-tivity of the material within the cell:

( )2

rqw

2 1*

4

n c

f Lε

⎡ ⎤−= ⎢ ⎥⎢ ⎥⎣ ⎦

[6]

where n is an index of the quarter-wavelength frequency (n = 1, 2, 3, ...). The term (2n − 1) is added to refl ect the fact that the quarter-wavelength frequencies being used are only those associated with one-way phase changes of 90 and 270°.

The quarter-wavelength frequencies are sometimes described in the literature based on the relationship between frequency, wavelength, and velocity. Therefore, this alternative defi nition is provided in the Appendix. Also, it should be noted that the voltage measured is, in fact, the sum of the combined wave and variations in voltage refl ected from the interface between cable and cell or probe due to impedance mismatch between them. As QWA is based only on the measurement of the frequency of the maxima and minima of voltage variations, however, this can be ignored. Therefore, a more complex measure-ment method would be required for calculating apparent permittivity for frequencies not associated with 90 and 270° quarter wavelengths.

Measurement of Resonant FrequenciesTo verify the QWA theory, a coaxial cell was constructed, as

depicted in Fig. 2. The cell was manufactured from an aluminum tube, 23-mm internal diameter and 91-mm internal length, with a 3-mm-diameter stainless steel rod acting as the central conductor, which was connected to the end plate to provide a short-circuited transmission line. Impedance matching, to minimize spurious refl ec-tions occurring in the transition between cable and soil within the cell end plate, was achieved by simply extending the coaxial geometry of the N-type connector’s inner conductor and insulation through the cell end plate.

Measurement of the voltage magnitude in a transmission line—of which the measurement cell, and the probe described below, are types—is generally undertaken using S11 data (also known as the scat-tering parameter) obtained from a vector network analyzer (VNA). The S11 parameter is a complex value representing both the magni-tude and phase of a signal based on measuring the output signal, rela-tive to the input signal, at a single test point, hence the one–one in the name, representing input at Port 1 and, similarly, output at Port 1.

The VNA used was a portable handheld spectrum analyzer (Rohde & Schwarz, Munich), with voltage standing wave ratio bridge and vector measurement options installed. Rohde & Schwarz produce a number of these devices, the FSH3 (3-GHz) model having been used for initial prototyping and the FSH6 (6-GHz) model having been used for the development and testing described here. It should be noted, however, that other models, and those produced by other manufacturers, may be equally appropriate for QWA. While a rela-tively “low end” instrument in terms of cost, the VNA used had a signifi cant advantage over most alternative analyzers in being highly portable and able to operate on battery power, allowing it to be used for fi eld measurements. The VNA equipment was connected to the cell or probe by a fl exible coaxial cable with N-type connectors, and via USB to a computer for data acquisition.

The FSH6 VNA provides measurements of the voltage magni-tude and phase angle, whereas QWA is generally based on measure-ment of the real part of S11 values. The S11 values are essentially a pha-sor representation of the refl ected voltage derived from Euler’s identity and can be determined using (Shang et al., 1999)

dB/20 dB/2011M 10 cos 10 sinS jφ φ= + [7]

where S11M is the measured scattering parameter, dB is the return loss in decibels, φ is the phase angle of the returned signal relative to that of the injected signal, and j is √(−1).

The data used for QWA is therefore the real, left-hand part of Eq. [7]. While the quarter-wavelength frequencies could be estimated directly from the phase data, the use of real S11 values was preferred due to the cosine function providing a more intuitive representation of the voltage maxima and minima and because the resulting graphs provide an indication of the degree of signal loss occurring, in differ-ent materials, in the cell.

Calibration of the VNA was performed by successively connect-ing an open, short, and load termination to the end of the N-type cable used to connect to the measurement cell. Subsequent analyzer measurements were, therefore, automatically corrected for the effects of the cable. This aspect of the testing procedure should not be under-estimated as the cable forms a coaxial transmission line and so the voltage measured by the VNA will incorporate quarter-wavelength maxima and minima, together with some signal loss associated with the cable geometry and the dielectric material within it. These effects are specifi c to the cable being used and must be removed by the VNA if

Fig. 2. The measurement cell used in this study, illustrating the N-type connector for connection of a vector network analyzer and screw to short circuit the central conductor at the far end.

1404 SSSAJ: Volume 72: Number 5 • September–October 2008

the refl ected signal strengths are to be reduced accurately to those only associated with the properties of the cell or probe and the soil within.

An important aspect of the calibration to note, however, is that it takes place at the end of the N-type cable rather than at the start of the soil sample, which will cause measurement errors if the change in phase angle across this additional distance is not accounted for. Therefore, the S11M values require adjustment to the phase angle and this may be accomplished using (after Shang et al., 1999)

( )11M

11 2exp

SS

j z cω=

− [8]

where S11 is the adjusted scattering parameter, S11M is the measured scattering parameter, ω is the radian frequency, c is the speed of light in a vacuum, and z is the difference in length between the cable end and the start of the soil sample. It should also be noted that care was taken during the calibration and measurement to ensure that movement of the cable was minimized and the calibration was performed immedi-ately before taking the corresponding measurement in order that the calibration data would not be affected by temperature variations.

The S11 values were calculated from VNA data fi les obtained using Rohde & Schwarz FSH View software, using a program written using LME (Calerga, 2007), based on the above equations, ported to MATLAB (Mathworks, 2007), and the quarter-wavelength fre-quencies were obtained by inspection of the resulting graph. Initial attempts to automate this process in MATLAB have also been undertaken and have proved largely successful; however, identifi ca-tion of the exact quarter wavelengths in soils with high loss tangents requires careful inspection of the S11 data, as attenuation can cause noise to become more signifi cant. This diffi culty was partially over-come through the use of a 15-point moving average fi lter, but further research is necessary before the QWA methodology described here can be fully automated.

As the VNA used has a limited memory store, it was necessary to take a number of measurements across a fraction of the total desired frequency coverage to ensure suffi cient resolution and hence to iden-tify the quarter-wavelength frequencies to an acceptable accuracy. The data were, therefore, collected across spans of 200 MHz, with the VNA recalibrated at the end of each span. This allowed the frequency to be determined to a maximum accuracy of 333 kHz, which was found during initial prototyping and calibration to provide an accept-able balance between measurement duration and accuracy—the cali-bration indicating that the quarter-wavelength frequencies were being identifi ed within 1 MHz of their true frequencies.

Soils Used in the StudyThe soils used in this study were selected to represent the wide

range of dispersive properties across which the measurement meth-odology would be required to operate. As both the degree of disper-sion and the liquid limit of a soil are known to relate to the specifi c surface area of the soil particles, it was decided that tested soils should exhibit a wide range of liquid limits. The liquid limit was considered an important parameter to measure, as it represents the maximum saturated water content of a soil expected in the fi eld, fi eld values gen-erally being expected to lie somewhere between the plastic limit (WP, the maximum water content at which the soil remains friable) and the liquid limit (WL, the water content at which the soil starts to fl ow as a fl uid) (see Craig, 1997). For a soil with signifi cant fi nes content (i.e., particles having a nominal diameter of <425 μm), the saturated volumetric water content will vary linearly between these values. Since the liquid limit represents the highest expected fi eld water content, it represents a particularly challenging state for measurement of electro-magnetic properties. For clarity, the term maximum saturated water content can therefore be considered synonymous with the liquid limit for this study.

The function of the soils was to allow testing of the measure-ment methodology, rather than for the purposes of geotechnical or taxonomic study. Therefore, characterization of the soils was limited to measurement of the maximum saturated water content, together with linear shrinkage as an indicator of the potential volume change that will occur in the soil with increased water content—higher values indicating a greater propensity for the soil to expand and shrink with changes in water content. Also, as an indication of soil texture, the per-centage by mass of sand in each sample, together with the combined silt + clay content, were measured by passing the soil through a 63-μm sieve. Measurements of these parameters were undertaken in accordance with British Standard 1377 (British Standards Institution, 1990).

The conductivity of each soil was also determined by placing the soil between two parallel metal plates and measuring the resis-tance using a Hewlett Packard low-frequency impedance analyzer, the measurement frequency being set at 200 Hz to minimize the effects of electrode polarization. These data were not, however, used as part of the QWA analysis and were obtained only for characterization of the con-ductivity range across which the measurement system testing occurred. Summary data for soil characterization are provided in Table 1.

Of the seven soils tested, three were processed clay materials pro-duced for use in clay product manufacture, these being the weathered Mercia mudstone, the Oxford clay, and the English China clay. The two samples of London clay were obtained directly from site excava-

Table 1. Some properties of the soils used for testing the cell and probe.

Soil Soil class† Liquid limit (volumetric water content) Sand Silt + clay Linear shrinkage Conductivity at 200 Hz

—————————— % —————————— mS m−1

Weathered Mercia mudstone CL 47 29(45)‡ 71(55) 10 58

Colliery spoil clay ML 53 29 71 9 94

Oxford clay CH 58 70 30 13 231

English China clay CH 63 0 100 10 53

London clay 1 CH 67 5 95 17 127

London clay 2 CH 70 6 94 18 99

Na bentonite CH 87 11 89 45 621Birmingham soil ML 38 (85) (15) 2 96

† C, clay; M, silt; L, low plasticity; H, high plasticity.

‡ Numbers in parentheses represent values for additional samples with maximum particle sizes of 2 mm, as used for comparison of quarter-wave-length analysis and time-domain refl ectometry. All other measurements were based on particle sizes ≤425 μm.

SSSAJ: Volume 72: Number 5 • September –October 2008 1405

tions, and the colliery spoil clay was taken from a spoil heap in the Midlands region of England. The Na bentonite sample was obtained as a commercial product. To ensure that the apparent permittivity data could be correlated to the measured maximum saturated water content, the weathered Mercia mudstone, colliery spoil clay, and Oxford clay samples, which incorporated some sand and gravel, were fi rst sieved to remove the size fraction >475 μm.

Expected AccuracyUnderstanding potential accuracy is central to any measure-

ment system and the importance of fully understanding the effects of frequency resolution, which includes the measurement equipment frequency accuracy and quarter-wavelength frequency measurement indeterminacy due to noise, is apparent from Eq. [6]. This illustrates that an inability to determine accurately the quarter-wavelength fre-quency results in variation of the calculated apparent permittivity. Also, it should be noted that the apparent permittivity of the tested material affects the spacing between quarter-wavelength frequencies, and so measurement errors increase with an increase in the appar-ent permittivity. This is illustrated in Fig. 3 for a range of frequency resolutions for the cell and probe described below, calculated as the difference between the true apparent permittivity and that calculated for a frequency different by the value of the resolution.

Furthermore, because the loss tangent of the tested soils will cause a continuous reduction in the amplitude of the quarter-wavelength frequencies, as frequency increases, higher frequency maxima and minima may become diffi cult to identify clearly and so the accuracy associated with them will diminish. Also, some noise will be inherent in all electromagnetic measurement systems, and it was found that the effects of noise became more pronounced as the amplitude of the maxima and minima became small, i.e., the signal/noise ratio increased with increasing attenuation. Therefore, in such attenuating soils, it is advisable to ignore the higher frequency data or adjust the cell or probe dimensions to reduce attenuation to ensure that acceptable accuracy levels are achieved.

RESULTSMeasurements in a Coaxial Cell

Initial tests, using air and distilled water, yielded the real S11 values shown in Fig. 4. As can be seen from this fi gure, the

data contained the expected maxima at the relevant quarter-wavelength frequencies for both air and water and, for air, no signifi cant signal loss was evident; however, attenuation of the signal in distilled water prevented measurements above 1.5 GHz. While this could be addressed through the use of a shorter cell, it is obvious from Eq. [6] that this would cause an increase in spacing between the quarter-wavelength frequen-cies. Therefore, as dispersion affects the electromagnetic prop-erties of dispersive soil mostly at frequencies below 1 GHz, the cell length used was considered to give an acceptable balance between the needs of data resolution and frequency coverage.

The cell was tested using air and distilled water due to their nondispersive nature and well-documented apparent permittiv-ities, the dispersive nature of water being considered insignifi -cant below 2 GHz. The tests with water were used to calibrate the cell, i.e., by using them to calculate an apparent length (92 mm) for use in Eq. [6]. The concept of apparent length is well known in TDR, and provides a simple means of removing any minor errors in the measurement arrangement, thus allow-ing the calibration data to match theory. The simplicity of this method was therefore adopted for QWA to account for factors such as potential phase changes due to impedance mismatch between cable and cell, the accuracy of the VNA frequency generation, and potential deviations of the VNA signal from a true sinusoidal voltage variation.

It was also considered appropriate to carry out an initial test with a dispersive material, for which ethanol was used. The results of these series of tests on each of these three materi-als are shown in Fig. 5 and are in close agreement with the expected values (see Heimovaara [1994] for full details) shown as dashed lines in the fi gure. It can be noted that the great-est variation occurs for water, which can be explained by the increase in potential error associated with the measurement sys-tem at higher apparent permittivities, as described above. Tests using dry sand and penetrating oil yielded apparent permittivity values of 2.8 and 2.5, respectively, which also accorded with doc-umented values—2.6 for dry sand reported by Mätzler (1998) and 2.3 for penetrating oil reported by Robinson et al. (2003).

Since a particular need for the cell was to measure appar-ent permittivity in wet clays due to the high level of dispersion they exhibit, the cell was used to measure the electromagnetic

Fig. 3. Potential apparent permittivity errors associated with quar-ter-wavelength measurement resolution for the cell and probe. Resolution refers to the difference between actual and measured quarter-wavelength frequencies caused by the spacing of mea-surements in the frequency domain and electrical noise.

Fig. 4. Air and water real scattering parameter (S11) values measured using the cell, representing the signal strength measured at the cell input. The maxima and minima correspond to constructive and destructive interference of signals within the cell at the quarter-wavelength frequencies.

1406 SSSAJ: Volume 72: Number 5 • September–October 2008

properties of a number of soil fi nes (i.e., ≤425 μm in size) close to their maximum saturated water contents. The results of these tests are illustrated in Fig. 6, which shows each exhib-iting low-frequency dispersion effects associated with the bound water phase, and higher frequency, more stable values associated with the free water phase, as expected from theory (Saarenketo, 1998). In such theory, the low-frequency disper-sion is a result of the water molecules being unable to rotate in time with the electromagnetic signal due to the attractive infl uence of charged soil boundaries, causing a reduction in the permittivity centered around a characteristic “relaxation frequency” (see Debye, 1929). For the free water phase, this frequency is around 17 GHz, resulting in much less variation in its apparent permittivity within the frequency range being covered by QWA.

The importance of measuring the apparent permittivity at the maximum saturated water content is illustrated by Fig. 7. This shows the apparent permittivity of London Clay 2 across a

wide range of frequencies and volumetric water contents using QWA in the measurement cell. The range of water contents between the points marked WP (the water content below which the soil ceases to exhibit plasticity) and WL (herein termed the maximum saturated water content) are the saturated water con-tents expected to be found under normal fi eld conditions for this soil (Craig, 1997). The WL value therefore provides an upper bound for the volumetric water contents that need to be measured for practical geotechnical engineering purposes.

The cell was also tested using TDR equipment (TDR100, Campbell Scientifi c, Logan, UT) to allow comparison between QWA and TDR data using the volumetric water content of the soils shown in Fig. 6. The end refl ection distances for air and distilled water were used to calibrate the cell TDR measure-ments and the following equation was used to relate apparent permittivity to end refl ection distances:

2

TDRr

PROBE

*L

Lε

⎛ ⎞=⎜ ⎟⎝ ⎠

[9]

where LTDR is the probe length calculated from the distance to end refl ection measured using the TDR equipment and LPROBE is the apparent probe length derived from the TDR calibration process.

To allow comparison between the two data sets, the QWA data were reduced to a single value representing the average of data points at the higher frequencies, where apparent permit-tivity had leveled off. This averaging was intended to reduce variations in apparent permittivity that may occur due to the small variations in expected error. These data, together with the TDR data, are plotted against volumetric water contents in Fig. 8. Also presented in the fi gure is the Topp model (Topp et al., 1980), which is a widely used method for relating TDR data to water content. It is apparent from these data that QWA values vary signifi cantly from both the TDR values and the Topp model in these dispersive soils, the Topp model being unrepresentative of highly dispersive soils.

Fig. 5. Results of three initial apparent permittivity tests, after calibra-tion, for the measurement cell. The dashed lines represent ex-pected values for distilled water, ethanol, and air (Heimovaara, 1994) using the Debye equation (Debye, 1929). The markers represent values measured during calibration.

Fig. 6. Frequency-domain effects of dispersion in a range of wet clays, measured at their maximum saturated water contents. Although curve fi ts can be applied to the data, straight lines join the dis-crete data points in this fi gure to illustrate the level of detail obtained using quarter-wavelength analysis in this study.

Fig. 7. The apparent permittivity of London Clay 2 measured across a wide range of volumetric water contents (VWC) and frequen-cies. WP and WL represent the minimum and maximum satu-rated water contents generally expected for this soil in the fi eld. The graph was prepared by fi tting fi fth-order polynomial curves to discrete quarter-wavelength analysis data points.

SSSAJ: Volume 72: Number 5 • September –October 2008 1407

This variation between the QWA and TDR data is not unexpected, however, as the power level of the frequency band generated by the electrical pulse in TDR equipment has been shown to be signifi cantly biased to frequencies below 500 MHz (Friel and Or, 1999), leaving higher frequencies to be attenuated in conductive soils to the extent that they have little infl uence on time-domain measurements. Therefore, the TDR data were compared with the frequency-domain data of Fig. 6 and it was found that the TDR values intersect with the corre-sponding data lines within a frequency range of approximately 200 to 400 MHz.

To investigate these differences between the QWA and TDR data, the QWA apparent permittivity values were also compared with other research relating to the permittivity of soils at high water contents. One such study by Wensink (1993) showed good correlation between the permittivity of wet dispersive soils at 1 GHz and the volumetric water con-tent. The Wensink model has therefore been included in Fig. 8, and it can be seen that the QWA data show good correlation with this model, although it should be noted that the Wensink model is a polynomial curve fi t to measured data on a small number of soils, and so should not be expected to fi t precisely to the QWA data. Figure 8, however, can be considered to illus-trate that QWA data are more appropriate than TDR data for analysis of soil electromagnetic properties at frequencies where dispersion has minimal effect. Furthermore, it indicates that QWA, combined with models such as that of Wensink, could, subject to further research, provide improved water content measurement accuracy in wet clay soils than is achieved cur-rently by TDR.

Measurements Using Time-Domain Refl ectometry Probes

While many methods already exist for measurements of soil electromagnetic properties in coaxial cells, it is the mea-surement of broadband apparent permittivity values of soils in situ that largely motivates this study, since it allows consid-eration of the dispersive nature of soils in the fi eld. For this reason, QWA use with TDR probes has been investigated, as TDR probes are basically simple, quasi-coaxial transmission lines (see, e.g., Robinson et al., 2003). A three-pin TDR probe design was chosen as a basis for this arrangement, as it is easily

inserted into soil with minimal disturbance. The length and diameter of the stainless steel rods was 100 and 3 mm, respec-tively, with a separation of 19 mm between rod centers.

For connection to a VNA, the head of the TDR probe was constructed using the same N-type connectors and imped-ance matching method used for the cell. In all other respects, however, there is very little difference between it and most alternative TDR probes. Initial tests were performed in air and distilled water, the real S11 values being presented in Fig. 9. Several features are evident when comparing this fi gure to the S11 values recorded in the cell, the most evident being that the resonant frequencies exhibit themselves as minima, due to the absence of the 180° phase change associated with refl ections at the short-circuited end of the cell. Also, multiple refl ections are obvious in the water data, associated with radiation losses past the end of the probe refl ecting back from the base of the ves-sel containing the calibration fl uid. As the vessel used allowed 300 mm of water to be contained beyond the end of the probe, it was evident that multiple refl ections would be a feature of any calibration using relatively lossless dielectrics, within prac-tical limitations. It is possible, however, that further research may identify a container design that will allow reductions in the effect of multiple refl ections by, for example, incorporat-ing interface materials that absorb or disperse electromagnetic signals. Multiple refl ections were not found in the air data, as it was not necessary to contain it within a vessel, or in soil tests due to the higher attenuation associated with them. The prob-lem of multiple refl ections was, therefore, only associated with the need for the use of water for calibration.

To calibrate the probe in water, the effects of multiple refl ections were reduced by taking the average of four mea-surements and calculating an apparent probe length—105 mm, which correlated with the TDR apparent length—that allowed the apparent permittivity variations to be centered around a value of 80. The results of this calibration are detailed in Fig. 10. It can be seen from this fi gure that variations in the values for water reduce with frequency due to multiple refl ections suf-fering increased attenuation in comparison with that occurring

Fig. 8. High-frequency apparent permittivity values for the studied soils, at maximum saturated water contents, using quarter-wave-length analysis (QWA) and time-domain refl ectometry (TDR). The Topp (Topp et al., 1980) and Wensink (Wensink, 1993) models are shown for comparison with TDR and QWA data, respectively.

Fig. 9. Air and water scattering parameter (S11) apparent permittiv-ity values, representing the signal strength measured at the cell input, after calibration, for the probe. Additional refl ections in the water data due to the base of the container are also il-lustrated. The maxima and minima correspond to constructive and destructive interference of signals within the probe at the quarter-wavelength frequencies.

1408 SSSAJ: Volume 72: Number 5 • September–October 2008

at lower frequencies. It was found, however, that once a suit-ably large vessel was used to contain the water, the variations were generally within two units of apparent permittivity of the theoretical values.

Following calibration, the probe was tested in a number of wet soils, including an undisturbed sample of London clay, a silty sand from the University of Birmingham campus, and a sample of weathered Mercia mudstone having a low water con-tent. These soils provided a range of dispersive properties, the results being included in Fig. 10. The higher frequency appar-ent permittivity values of these tests were found to correlate with the Topp and Wensink models, which are similar at low water contents, and, as can be seen from the inclusion of values for the Birmingham soil measured at a similar water content in the cell, the results of the cell and probe tests are comparable. Also, while increased water content resulted in a reduction of the highest frequency at which the quarter wavelengths could be identifi ed, data were obtained with the probe, even in wet clay, across a useful range approaching 1 GHz.

Finally, as it was intended that QWA data be compatible with TDR data, at least when testing with the probe, it was considered necessary to take measurements in soils exhibit-ing low dispersion, in which case high-frequency QWA and TDR values should correlate, using both techniques. This was accomplished using the probe, a silty soil from the University of Birmingham campus, and a sample of weathered Mercia

mudstone, which was shown in the cell tests to exhibit limited dispersion. In this case, the size fraction up to 2 mm was used to reduce specifi c surface area and hence dispersion. The sam-ples were tested across a range of water contents. The results of this exercise are shown in Fig. 11, in which the slope of the linear regression can be seen to be almost unity, indicating that the two techniques provided comparable data.

DISCUSSION AND CONCLUSIONSThis study has shown that coaxial cells and TDR probes

can be used for soil electromagnetic measurements using both TDR and QWA techniques. In addition, it has been shown that broadband velocity measurements using TDR probes offer a viable alternative to coaxial cells for both laboratory and fi eld use. This offers a signifi cant benefi t in circumstances where coaxial cells are impractical, such as measurements on undis-turbed samples and monitoring of soil electromagnetic proper-ties during consolidation of a soil.

Rather than replacing TDR, however, which relies on cheaper equipment more suited to long-term exposure to fi eld conditions, it is intended that the described use of QWA will complement that of TDR. For instance, TDR probes are often calibrated to the water content of a soil they are to monitor by taking measurements of apparent permittivity across a wide range of water contents. By incorporating QWA into this cali-bration procedure, it is intended that TDR measurements can be used to provide broadband velocity data for such applica-tions as ground-penetrating radar test sites.

Furthermore, conversion of time-domain data to the fre-quency domain is complicated and highly dependent on the underlying electromagnetic modeling of the probe, cables, and TDR equipment used. Verifi cation of such inverted data, and other novel attempts to gain measurement-frequency data from the time domain (e.g., Robinson et al., 2005), is diffi cult if comparison cannot be made to VNA data directly measured in the frequency domain, using the same probe used for the time-domain measurements. Quarter-wavelength analysis offers the potential to provide such verifi cation and could, therefore, be used to calibrate inverted data and thus provide increased con-fi dence in the inversion methodology. It also offers a means of validating attempts to gain continuous data—i.e., data at all measured frequencies—from TDR probes using VNAs, as well as providing a basis to obtain such data through parameter matching with dispersion models.

In terms of direct application of QWA, it has been high-lighted that the accuracy of the measurement can be expected to be variable; however, for the apparent permittivity val-ues associated with even wet clays, and for shorter cells and probes, good levels of accuracy can be achieved. For the longer probes required for low-frequency measurements, accuracy can be expected to be reduced if measurement resolution is not improved by way of compensation; however, this diffi culty can be considered offset by the ability of QWA to provide broad-band apparent permittivity values for in situ soils and poten-tially, using higher frequency data, the ability to accurately measure water content in dispersive soils.

Overall, it can be concluded that QWA provides a rela-tively simple and accurate method for broadband velocity mea-surements in soils, both in the laboratory and in situ. As with

Fig. 10. Apparent permittivity vs. frequency, after calibration, for the modifi ed time-domain refl ectometry probe, testing a number of soils at different volumetric water contents (VWC).

Fig. 11. Comparison between apparent permittivity measured by quarter-wavelength analysis (QWA) and time-domain refl ecto-metry (TDR) techniques for two soils exhibiting low dispersion.

SSSAJ: Volume 72: Number 5 • September –October 2008 1409

TDR, when combined with the use of relatively inexpensive and portable VNA equipment, these measurements become accessible to a wide sector of the geophysical community, rather than being limited to those practitioners with extensive knowledge of electromagnetic measurements and access to more expensive and sophisticated frequency-domain measure-ment equipment.

APPENDIXAlternative Defi nition of the Quarter-Wavelength Frequencies

Quarter-wavelength analysis is sometimes explained in terms of the relationship between the frequency of a signal, its wavelength, and its velocity. Also, as described above, the veloc-ity can be determined from the apparent permittivity. These relationships are described by Eq. [3] and

Vf

λ= [10]

The actual wavelength associated with each quarter-wavelength frequency is easily defi ned from the apparent length of the cell or probe and the index (N = 1, 2, 3, 4, ...) of all quarter-wave-length frequencies:

4

0.25

L LN N

λ= = [11]

Combining Eq. [3], [10], and [11] leads to the defi nition of the quarter-wavelength frequencies (Heimovaara et al., 1996):

( )( )qw

r qw

2 1

4 *

n cf

f Lε

−= [12]

where fqw is the quarter-wavelength frequency (Hz), n is an index (n = 1, 2, 3, 4, ...) used to reduce the quarter-wavelength frequencies to just those associated with 90 and 270° phase changes, and εr*(fqw) is the apparent permittivity at the mea-sured quarter-wavelength frequency.

To determine the apparent permittivity of the material, Eq. [12] can then be rearranged to give Eq. [6].

ACKNOWLEDGMENTSWe gratefully acknowledge the fi nancial and other support provided by the UK’s Engineering and Physical Sciences Research Council (EPSRC) under grant reference EP/C547365 and UK Water Industry Research (UKWIR).

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