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Calibration Functions for Estimating SoilMoisture from GPR Dielectric ConstantMeasurementsRemke L. Van Dam a ba Michigan State University, Department of Geological Sciences ,East Lansing , Michigan , USAb Queensland University of Technology, Science and EngineeringFaculty, Institute for Future Environments , Brisbane , Queensland ,AustraliaAccepted author version posted online: 04 Nov 2013.Publishedonline: 30 Jan 2014.

To cite this article: Remke L. Van Dam (2014) Calibration Functions for Estimating Soil Moisture fromGPR Dielectric Constant Measurements, Communications in Soil Science and Plant Analysis, 45:3,392-413, DOI: 10.1080/00103624.2013.854805

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Communications in Soil Science and Plant Analysis, 45:392–413, 2014Copyright © Taylor & Francis Group, LLCISSN: 0010-3624 print / 1532-2416 onlineDOI: 10.1080/00103624.2013.854805

Calibration Functions for Estimating Soil Moisturefrom GPR Dielectric Constant Measurements

REMKE L. VAN DAM

Michigan State University, Department of Geological Sciences, East Lansing,Michigan, USA, and Queensland University of Technology, Science andEngineering Faculty, Institute for Future Environments, Brisbane, Queensland,Australia

Ground-penetrating radar (GPR) is widely used for assessment of soil moisturevariability in field soils. Because GPR does not measure soil water content directly,it is common practice to use calibration functions that describe its relationship with thesoil dielectric properties and textural parameters. However, the large variety of mod-els complicates the selection of the appropriate function. In this article an overview ispresented of the different functions available, including volumetric models, empiricalfunctions, effective medium theories, and frequency-specific functions. Using detailedinformation presented in summary tables, the choice for which calibration function touse can be guided by the soil variables available to the user, the frequency of the GPRequipment, and the desired level of detail of the output. This article can thus serve as aguide for GPR practitioners to obtain soil moisture values and to estimate soil dielectricproperties.

Keywords Ground-penetrating radar, hydrogeology, soil water

Introduction

Fresh water is an essential commodity and a vital resource for many ecosystems on earth.However, with the demands for water increasing from all parts of society, the availability ofsufficient clean water is under threat from numerous environmental problems (Entekhabiet al. 1999; Vörösmarty et al. 2000). Although a large part of the available fresh wateris stored in subsurface aquifers, unsaturated zone processes such as recharge and evapo-transpiration make it actively interact with the atmosphere. This unsaturated, or vadose,zone forms a natural buffer between the groundwater and the atmosphere and is an impor-tant part of the interlinked global freshwater system. Because the vadose zone is directlyaffected by many human activities, it is of great importance to understand and monitor thespatial variability and temporal changes in the vadose zone on both global and local scales.

In recent decades, ground-penetrating radar (GPR) has become a popular method forthe study of soil water conditions and dynamics at a range of scales, including those relatedto contamination, water (re)distribution, permafrost, and climate change (Huisman et al.2003; Annan 2005; Rubin and Hubbard 2005; Mahmoudzadeh 2013). GPR offers differ-ent techniques for soil moisture estimation, such as normal move-out or common midpoint

Received 12 August 2012; accepted 26 July 2013.Address correspondence to Remke L. Van Dam, Michigan State University, Department of

Geological Sciences, 206 Natural Science, East Lansing, MI 48824, USA. E-mail: rvd@msu.edu

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Estimating Soil Moisture from GPR 393

measurements (Greaves et al. 1996; Turesson 2006; Mangel et al. 2012), vertical radar pro-filing and cross-borehole imaging (Hubbard, Rubin, and Majer 1997; Cassiani, Strobbia,and Gallotti 2004; Kuroda, Jang, and Kim 2009), common offset depth profiling (Lunt,Hubbard, and Rubin 2005; Moysey 2010; Schmelzbach, Tronicke, and Dietrich 2012),measurements for estimation of the groundwave velocity (Hubbard, Grote, and Rubin2002; Galagedara et al. 2005; Hamann et al. 2013), and surface reflectivity measurements(Redman et al. 2002; Weihermuller et al. 2007). Recent new approaches include the use ofmultichannel GPR (Bradford 2008; Gerhards et al. 2008) and analysis of early-time signalattributes (Pettinelli et al. 2007; Di Matteo, Pettinelli, and Slob 2013).

The behavior of the electromagnetic energy transmitted by GPR antennas is criti-cally dependent on the dielectric properties of the soil. The dielectric medium propertiescontrol factors such as the attenuation losses and propagation of electromagnetic energy(Miller, Hendrickx, and Borchers 2004; Annan 2005), as well as the reflective contrastbetween an object of study and the surrounding (soil) medium, and the surface reflectivity(Mätzler 1998). The dielectric properties of a material are a function of, among others,texture, bulk density, mineralogy, temperature, organic-matter content, soil water content,and fluid salinity (Hasted 1973; Topp, Davis, and Annan 1980; Wensink 1993; Van Damand Schlager 2000). A further complicating factor is that all these variables change spa-tially and sometimes also temporally in the soil system (Wilson et al. 2003; Van Dam et al.2005).

Because GPR does not measure soil water content directly, it is common practiceto estimate the volumetric water content (θ ) using calibration functions that describe itsrelationship with the soil dielectric properties (ε) and textural parameters. Over the pastseveral decades a large number of θ–ε models has been proposed, many of which areeither purely empirical in nature or are based on volumetric mixing laws. It is still a grandchallenge to select the best calibration function for each occasion, likely in part becausecurrent literature offers few overviews of available options. As a consequence, GPR usersoften resort to using one of only a few typical calibration functions, without consideringwhether other ones may be more appropriate.

This article presents a review of a number of calibration functions for the predictionof dielectric properties of field soils, based on information on water content and texturalvariables. Many of these functions can be easily inverted so that water content can beobtained from dielectric properties. This article is not intended as a theoretical overviewof porous media dielectrics; more on this topic can be found in recent review papers (e.g.,Chelidze and Gueguen 1999; Cosenza et al. 2009). Because of the large number of optionsin literature, this study presents an overview of some of the key calibration functions; it isnot intended as an all-encompassing list. Characteristics, applications, and advantages anddisadvantages are discussed. For easy comparison, key aspects of the different functions(input, output, frequency range) are presented in summary tables.

Background

The interaction of electromagnetic energy with matter is affected by the characteristicsof the material and by the frequency of the electromagnetic energy. Frequency-dependentdielectric properties can be characterized in terms of losses of energy due to relaxationmechanisms that operate at different frequencies. The relaxations are caused by differentforms of atomic- or molecular-scale resonance (Santamarina and Fam 1997). In a soil mix-ture the relaxation mechanisms may be attributed to the solid material and the pore water,as well as to interfacial phenomena. Hasted (1973) summarized some of the different typesof relaxation mechanisms that play a role in wet soils. Because most GPR systems operate

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394 R. L. Van Dam

in a frequency range between about 50 MHz and 1.2 GHz, bound water relaxation is themajor resonance mechanism of interest. Other relevant resonance mechanisms below andabove the typical GPR frequencies, respectively, are Maxwell-Wagner relaxation around1 MHz and free-water relaxation around 10 GHz (Hilhorst and Dirksen 1994). It must benoted, however, that although bound water has proven to be a very convenient fitting param-eter, it is a somewhat ill-defined concept. Indeed, many functions that use bound water toexplain θ–ε data could also be explained through structural and geometrical effects (e.g.,Sen 1984).

The dielectric permittivity is a complex function with real and imaginary compo-nents, defined as ε∗ = ε′ − jε′′, where j is the square root of –1. The real part of thepermittivity (ε′) is a measure for the total polarizability of the material, which is the ratioof the electric-field storage capacity to that of free space. The real part of the dielectricpermittivity is often expressed as the relative permittivity, dielectric constant, or apparentdielectric permittivity, which are denoted (interchangeably) by the symbols ε′, εr, or Ka.The relative permittivity is a frequency-dependent variable and decreases with increasingfrequency (Powers 1997). The imaginary part (ε′′) of the dielectric permittivity repre-sents the energy absorption by polarization (or relaxation) losses and the ionic conductionlosses. The dielectric losses include dispersive losses, as well as free-water relaxation andbound-water relaxation losses (Hasted 1973).

At frequencies between 0.1 and 1.5 GHz, ε∗ for free water is only weakly frequencydependent and dielectric losses are generally low (Topp, Davis, and Annan 1980; Davis andAnnan 1989). The situation is more complex in fine-textured soils containing clay minerals(Kelleners et al. 2005; Logsdon 2005), and in the presence of ferrimagnetic minerals (VanDam et al. 2013). Also, at these frequencies ε′ and ε′′ are very sensitive to changes insoil water conductivity above about 10 mS/m (Wensink 1993; Knoll and Knight 1994).At frequencies below around 50 MHz, ε∗ depends strongly on soil type (Campbell 1990;Roth et al. 1990). For these low frequencies, the relaxation losses are dominated by theconductivity (Hilhorst and Dirksen 1994). At frequencies above about 1 to 1.5 GHz thedielectric losses increase with increasing free water content, even for low conductivityvalues (Nguyen et al. 1997).

Accurate determination of the relationship between dielectric and hydrogeologicalproperties of soil material is essential for calibration of measurements and model data.Time-domain reflectometry (TDR) is one of the most common electromagnetic tech-niques for measurements of soil water content and soil electrical conductivity. Originally alaboratory-based technique, TDR has become a well-established and widely used methodfor water content determination of field soils at centimeters to decimeter scales (Topp andDavis 1985; Jones, Wraith, and Or 2002; Robinson et al. 2003; Cerny 2009). Several ofthe calibration functions discussed in this paper used TDR to derive or test their relation-ships. The TDR data collection is easily automated, which makes it an ideal technique formonitoring soil moisture variability at high temporal resolutions. However, because of thesmall measurement volume of TDR probes, a large number of probes are needed for goodspatial coverage (Huisman et al. 2002).

Calibration Functions

Volumetric Mixing Models

Volumetric mixing models describe the dielectric properties of a soil based on the relativeamounts of the different soil constituents and their individual dielectric characteristics.The basic input parameters to all models include solid matter, pore space, and volumetric

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Estimating Soil Moisture from GPR 395

water content. Depending on the model, additional input variables such as organic matterand bound water may provide improved detail and fitting options for specific conditions(Table 1). Usually, frequency-dependent behavior is not taken into account. Most modelshave been calibrated by TDR, which uses a wide frequency range. Over the years differentvolumetric mixing models have been proposed; the basic Liechtenecker-Rother mixingmodel for a material with n components can be written as

εαm =

n∑i=1

viεαi , (1)

where vi is the volume fraction of the ith soil constituent, and α is a fitting parame-ter. Figure 1 demonstrates the important effect of the scaling factor α on the correlationbetween water content and dielectric properties for a water-saturated two-phase medium.The α parameter can theoretically vary from –1 for an electrical field perpendicular to lay-ering to +1 for an electrical field parallel to layering (see insets in Figure 1). For α = 0.5,Eq. (1) becomes the widely used complex refractive index (CRI) or exponential model,which is assumed to be representative of a true isotropic soil medium (Birchak et al. 1974;Roth et al. 1990). Other empirical values and ranges for α reported in the literature include0.33 (Landau and Lifshitz 1960; Zegelin and White 1994), 0.4 to 0.8 for multiphase mix-tures (Jacobsen and Schjonning 1995), 0.46 for three-phase systems (Roth et al. 1990),0.65 for four-phase systems including bound water (Dobson et al. 1985), and 0.67 to 1.0 forpure clays (illite, benthonite) and a vertisol with 84% clay (Dirksen and Dasberg 1993).Based on measurements on quartz–kaolinite mixtures and silt loam soils, Ponizovsky,Chudinova, and Pachepsky (1999) present a mixing model with multiple adjustable param-eters. They found that a model with three adjustable parameters (α, vbw, εb) fitted data forall tested samples (Table 1).

Although the scaling factor α is largely empirical, several attempts have been madeto give it a more physical basis (e.g., Zakri, Laurent, and Vauclin 1998). The value of α ispartly related to the dominant orientation of the sediment grains and pore geometry (e.g.,Brovelli and Cassiani 2008). As shown by Jones and Friedman (2000), the presence ofaligned ellipsoidal particles, for example, in bedding planes of sedimentary deposits, hasan effect on the effective permittivity. It has been shown that the value of α also (inversely)correlates with the measurement frequency (Hilhorst 1998).

Volumetric mixing models can consist of as many components as desired, but usuallycontain three (solids, liquid, air) or four in case of (1) the presence of significant amounts ofclay or organic matter or (2) the presence of a mixture of two liquids such as oil and water.The addition of a fourth component can account for water that is physically and chemi-cally bound to the particles, which has a significantly lower relative permittivity than freewater. In the case of materials with a large specific surface area, such as clay and organicmatter, the bound water component is important to correctly predict the relation betweenthe dielectric properties and total water content of a soil (Figure 2). In four-phase dielectricmixing models, the relative permittivity of the bound water component is represented by asingle value (usually around 3.2 to 5.5). The bound water volume can be calculated usingθbw = lδρbSSA, where l is the number of molecular water layers, each with thickness δ, ρb

is the dry bulk density of the soil, and SSA is the specific surface area of the soil (Dirksenand Dasberg 1993).

In reality, the relative permittivity of the bound water layer is a function of distancefrom the solid surface (Conway, Bockris, and Ammar 1951) and ranges from a minimum(εb,min = 3.2–5.5) near the solid-phase surface toward a maximum where the permittivity

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Estimating Soil Moisture from GPR 397

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1

App

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α = 0.3

α = –1

α = 0.8

α = 1

Figure 1. Diagram illustrating the effects of the value of empirical parameter α, ranging from −1 to1, on the results of a two-phase dielectric mixing model. The relationship was calculated for amedium with 40% porosity. The curves for α values 0.3 and 0.8 (common extremes reported inthe literature for soil material) have been indicated. The insets show the electric equivalents of thetwo extremes of the complex refractive mixing (α = −1 and α = 1), illustrating the effect of grainorientation.

equals that of free water (εb,max = 80). The effective permittivity (εeff) of the mixture ofbound and free water can be estimated using (Friedman 1998)

εeff = dwεb,max

dw + 1

λln

(εb,max − (

εb,max − εb,min)

e−λdw

εb,min

) , and (2)

0.1

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20

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0 0.1 0.2 0.3 0.4

App

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Figure 2. Diagram showing the effect of the amount of bound water on a four-phase dielectricmixing model as in Eq. (1), with α = 0.5. The modeled medium consists of 60% grains (relativepermittivity = 4) and 40% porosity, filled with air (1), free water (80), and bound water (5.5). Themaximum volumetric bound water content (cm3/cm3) ranges in four incremental steps from 0 to 0.1.

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398 R. L. Van Dam

0

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60

80

0 0.1 0.2 0.3 0.4

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Figure 3. Effects of specific surface area and bulk density on the effective permittivity of the waterphase, calculated using Eqs. (2) and (3). Gray lines represent coarse-grained soils with a bulk densityof 1.6 g/cm3 and specific surface areas of 1, 10, 25, and 50 m2/kg, respectively. Black lines representfine-grained soils with a bulk density of 1.2 g/cm3 and specific surface areas of 50, 100, 250, and500 m2/kg, respectively (after Friedman 1998).

dw = θ/ρbSSA. (3)

Here, ρb is the dry bulk density (g/cm3), SSA is the specific surface area (cm2/g) of thematerial, λ is an empirical decay factor taken to be 108 cm−1, and dw is the thickness of thewater film.

Figure 3 shows how the effective permittivity of the total water phase changes asa function of volumetric water content and textural characteristics. With an increase inspecific surface area, the larger contribution of bound water leads to a decrease in εeff.Also, it is apparent that for soils with equal SSA (and equal particle density), an increasein bulk density decreases the effective permittivity (Figure 3). Figure 4 compares the θ–ε

relationship for loamy sand (ρb = 1.47 g/cm3 and SSA = 5.2·105 cm2/g) and a silty claysoil (ρb = 1.19 g/cm3 and SSA = 2.52·106 cm2/g) with that of a three-phase mixturewithout bound water. It shows that not taking the bound water into account leads to anoverestimation of the relative permittivity.

Empirical Functions

Empirical calibration functions are mathematical descriptions of the relationship betweendielectric properties and other characteristics of a medium, especially volumetric watercontent and texture information. There is no physical basis for such functions and theymay therefore be valid only for the soil types that were used to develop the relationship.Many empirical functions have originated in the fields of hydrology and crop and soilsciences and are typically based on network-analyzer measurements using time-domainreflectometry (TDR), waveguides, or coaxial transmission lines.

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0

5

10

15

20

25

0 0.1 0.2 0.3 0.4

App

aren

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lect

ric p

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ittiv

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Volumetric water content (cm3/cm3)

Figure 4. Apparent dielectric permittivity as a function of volumetric water content for a sandyloam (solid black line) and silty clay (dashed black line) (from Dobson et al. 1985). The effectivepermittivity of water was calculated using Eq. (2) and a three-phase dielectric mixing model as inEq. (1), with α = 0.5. The modeled medium consists of 60% grains (εr = 4) and 40% porosity,filled with air (εr = 1), free water (εr = 80), and bound water (εb,min = 5.5). The result of a simplethree-phase mixing model (solids, air, free water), without the effect of bound water, is shown forcomparison (solid gray line).

The calibration function that has seen the most widespread use is the classic Toppequation, which uses a third-order polynomial to describe the relation between soilvolumetric water content (θ ) and bulk or apparent relative permittivity (Ka) for measure-ments taken below the relaxation frequency of free water (Topp, Davis, and Annan 1980).In its classic form θ is estimated from the apparent relative permittivity; here it is presentedinverted for easier comparison with other calibration functions:

Ka = 3.03 + 9.3θ + 146θ2 − 76.7θ3 (4)

The regression is an average of TDR measurements integrated over a frequency range ofaround 1 MHz to 1 GHz for four mineral soils ranging from clay to sandy loam (Table 2).The regression has been proven to be a reliable estimator of water content or dielectricproperties for mineral soils. Figure 5 shows the good correlation between the Topp equationand an empirical function derived from TDR measurements of nine mineral soils with claycontents ranging from 2 to 46% (Roth, Malicki, and Plagge 1992).

The bulk density and a large clay or organic-matter content can all have a profoundeffect on the relation between θ and Ka (Roth et al. 1990; Dirksen and Dasberg 1993;Malicki, Plagge, and Roth 1996; Curtis and Narayanan 1998). The increase in specificsurface area and bound water associated with most clays leads to a reduction in the bulkdielectric properties of clay-rich samples. Several authors have therefore presented empir-ical functions similar to Eq. (4) for situations when the Topp equation does not performwell. Lines 3 and 4 in Figure 5 represent alternatives to the Topp equation for a marinesilty-clay soil with 45% illite–kaolinite clay and a Vertisol with 86% smectite clay, respec-tively (Dirksen and Dasberg 1993). Likewise, soils high in organic matter usually havea large specific surface area, so that the Topp equation does not predict the volumetricwater content correctly (Malicki, Plagge, and Roth 1996). Empirical functions derived

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Tabl

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ean

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fere

nce

fra

nge

(GH

z)Te

xtur

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ther

Out

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nge

(GH

z)N

o.of

soils

No.

ofsa

mpl

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ilty

pes

Her

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ath

etal

.(19

91)

Om

––

θK

a∫ T

DR

rang

e1

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Sc

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ieu

etal

.(19

86)

linea

r–

–θ

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∫ TD

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nge

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1996

)B

D–

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1834

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SDA

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02)

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a∫ T

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rang

e5

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,Lsa

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oth

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.(19

92)

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e9

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:Sa,

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,LSa

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.(19

80)

––

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a∫ 0.

001–

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ean

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%;S

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%;O

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icm

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;ρb,d

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ater

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ittiv

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dU

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ure

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atio

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inTa

ble

1.

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Estimating Soil Moisture from GPR 401

0

10

20

30

40

0 0.2 0.4 0.6

App

aren

t die

lect

ric p

erm

ittiv

ity

Volumetric water content (cm3/cm3)

1

2 3

4

5

6

Figure 5. Effects of soil type on empirical models describing the relationship between apparentbulk dielectric permittivity and volumetric water content. Line 1 (black dashed line) represents theTopp-model (Topp et al. 1980). Line 2 represents measurements of mineral soils (Roth et al. 1992).Lines 3 and 4 represent empirical models for a Dutch marine silty-clay soil with 45% illite-kaoliniteclay and a Kenyan vertisol with 86% smectite clay, respectively (Dirksen and Dasberg 1993). Lines5 and 6 are empirical models derived for organic-rich soils by Roth et al. (1992) and Herkelrath et al.(1991), respectively.

from dielectric measurements of samples high in organic-matter content (curves 5 and6 in Figure 5) show that Eq. (4) may overpredict the apparent dielectric permittivity by upto about 50% (Herkelrath, Hamburg, and Murphy 1991; Roth, Malicki, and Plagge 1992).

An alternative means for the empirical determination of the relationship between watercontent and bulk relative permittivity of soil is via artificial neural networks (ANNs),although ANNs do not produce a universal predictive function and may need to be recal-ibrated for each new sample set. Using 10 samples (sand, loamy sand, sandy loam, sandyclay loam) from five different soils in Denmark, Persson et al. (2002) demonstrated thatANNs could improve the accuracy of predicting the relationship between the soil bulkrelative permittivity and soil water content.

Effective Medium Approach

Effective medium theory assumes that natural sample heterogeneity can be representedby a basic geomorphological unit. The unit is an elementary cell filled with water (matrix)and contains a subspherical heterogeneity that corresponds to solids and/or air (Sen, Scala,and Cohen 1981; Friedman 1998; Miyamoto, Annaka, and Chikushi 2005; Cosenza et al.2009). As a consequence of this conceptualization, effective medium theories specificallyaddress inclusion (heterogeneity) shape rather than size. By combining the effects of elec-tric fields in each of the cell components, the effective property of the cell is calculated. Thesimplest effective medium theories (or composite spheres models) are those where the het-erogeneity is contained in a cell filled with the matrix fluid, neglecting any electromagneticinteractions between the two (Maxwell–Garnett rule). Another approach is the Bruggemanrule where the heterogeneity and matrix fluid are symmetrical inclusions in cells filled withthe unknown effective medium (Cosenza et al. 2009). A disadvantage of the composite

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402 R. L. Van Dam

spheres model is that it is only accurate for known geometries and is difficult to implementfor heterogeneous and multiple-phase materials (Nguyen et al. 1997). Here, calibrationfunctions based on the effective medium theory have been grouped with volumetric mixingmodels (Table 1), as some of their characteristics are similar.

The Maxwell–De Loor model (De Loor 1968) assumes disc-shape inclusions with ran-dom distribution and orientation. This model has been used to describe dielectric propertiesof four-phase mixtures using (Bohl and Roth 1994; Dobson et al. 1985):

εm = εh +3∑

i=1

vi

3(εi − εh)

3∑j=1

(1

1 + Aj εi/εb − 1

). (5)

Here, εh, εi, and εb are the relative permittivity of the host medium (solids), the permittivityof the inclusions, and the effective permittivity near boundaries, respectively; vi representsthe volume fraction of the inclusions; and Aj refers to so-called depolarization ellipsoidfactors.

A different equation is based purely on the depolarization factors of different soil con-stituents (Hilhorst et al. 2000). In this approach the measured relative permittivity is relatedto the volume-weighted sum of the permittivities of the individual material constituents.The depolarization factor (S) is introduced to account for electric-field refractions at thematerial interfaces:

ε =n∑

i=1

εiSivi (6)

Here, vi is the volume fraction of the ith soil constituent. The depolarization factor S isrelated to the electric field refraction in soil, which is in turn a function of the shape andsurface roughness of the grains. Theoretically, the depolarization factor can be calculatedfor all materials. However, in practice this remains difficult for all but homogeneous mate-rials with regularly shaped grains. For randomly oriented spheroids and a frequency abovethat of bound water relaxation effects, the depolarization factors for air and solid matter Sa

and Ss can be approximated by 1 (Hilhorst 1998), while Sw can be estimated by

Sw (θ) = 1

3 (2φ − θ)(7)

where φ is the porosity. A frequency domain sensor was used to measure the dielectricproperties of glass beads and fine sands (Hilhorst and Dirksen 1994; Hilhorst et al. 2000).Both measured values and the calculated θ–ε relationship [using Eq. (6)] show reasonableto good overlap with data presented by Topp, Davis, and Annan (1980).

Frequency-Specific Empirical Functions

Many studies have documented frequency-dependent (complex) dielectric properties ofsoil samples (Table 3), highlighting the difficulty of using one single model to describethe θ–ε relationship for GPR data. As these studies indicate, using different GPR antennafrequencies may require that the frequency-dependent nature of dielectric properties andloss mechanisms be taken into account. Although the Topp equation [Eq. (4)] often per-forms very well for mineral soils over the GPR frequency spectrum, frequency-specific

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Estimating Soil Moisture from GPR 403

Table 3Studies of frequency-dependent dielectric soil properties

ReferenceFrequency range

(GHz)

No. ofsamples[soils] Soil types

Curtis et al. (1995) 0.45–26.5 30[12]a USDAb: Sa, SaL, Si, SiCl,SiClL, Cl

Dirksen and Dasberg (1993)∫

0.01–1 11 USDAb: SiL, L, Cl, SiClHeimovaara et al. (1994) 0.001–0.15 3 USDAb: SiL, LSa, SiClLKnoll and Knight (1994) 0.0001, 0.001, 0.01 11 Artificial mixtures of sand

and clayNguyen et al. (1997) 1–0.75 1 SandWensink (1993) 0.001–3 11 Clay, silt, peat

aJ.O. Curtis, personal communication.bUSDA texture classification, as in Table 1.

measurements by Curtis (2001) using a network analyzer and a large number of samplesin a coaxial transmission line demonstrate that only at 100 MHz can the Topp equationbe replicated (line 4 in Figure 6). At both lower and higher frequencies the Topp equa-tion loses accuracy. At 50 MHz, measurements of the θ–ε relationship for six soils showa reasonable overlap (line 3 in Figure 6) with the Topp equation (Campbell 1990). Belowaround 50 MHz both ε′ and ε′′ depend strongly on soil type, soil water temperature, andmeasurement frequency. Based on measurements of 6 soils at 1 and 5 MHz, Campbell(1990) shows that Eq. (4) overestimates the volumetric water content at those frequencies

0

10

20

30

40

0 0.2 0.4 0.6

App

aren

t die

lect

ric p

erm

ittiv

ity

Volumetric water content (cm3/cm3)

1

5

1

4

2

5

3

Figure 6. Effects of measurement frequency on the relationship between apparent bulk dielectricpermittivity and volumetric water content. Line 1 is the Topp model (black dashed line). Lines2 and 3 represent polynomial fits to measurements on a broad range of soil types, conducted at1 and 50 MHz, respectively (Campbell 1990). Line 4 represents a polynomial fit to measurementsconducted at 100 MHz (Curtis 2001). Line 5 was derived from Curtis (2001) for measurementsconducted at 1 GHz.

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404 R. L. Van Dam

(results for measurements at 1 MHz given by line 2 in Figure 6). At frequencies of 1 GHz(line 5 in Figure 6), Eq. (4) underpredicts the volumetric water content (overpredicts therelative permittivity) by up to around 15% (Curtis 2001).

Relaxation-Time Functions. The Cole–Cole model relates characteristic relaxation times tofrequency-dependent behavior of a material and describes the induced polarization effectsof a mixture as a function of frequency, f (Cole and Cole 1941). The complex dielectricpermittivity can be described as

ε∗ (f ) =[ε∞ + εs − ε∞

1 + (jf/

frel)1−β

]− jσdc

2π f ε0(8)

where εs and ε∞ are the static value of the dielectric permittivity and the high-frequencylimit of the real dielectric permittivity, respectively; ε0 is the dielectric permittivity of freespace (8.854·10−12 F/m); frel is the dielectric relaxation frequency of the material; σ dc isthe electrical conductivity; and β is an empirical parameter (0–1) to describe the spreadin relaxation frequencies. For distilled water or other pure liquids with a single relaxationfrequency, β is zero, resulting in the original Debye model (Debye 1929).

An adaptation to the Debye model by Hilhorst et al. (2000) combines Eqs. (6) and (7)with a relationship between soil matric pressure and dielectric relaxation frequency. Wateris bound to the soil matrix by a number of forces, some of which reduce its energy status(Koorevaar, Menelik, and Dirksen 1983). The degree of bonding decreases with distancefrom the particle surface. The soil matric pressure (pm) can be used as a measure for thetightness of bond between the soil particles and the water molecules and can be used todetermine the relaxation frequency of soil water according to (Hilhorst 1998)

frel = frel,0epmVRT (9)

where V is the partial molar volume of water, R is the universal gas constant, and T isabsolute temperature. This equation confirms that the relaxation frequency of bound wateris several orders of magnitude smaller than that of free water under atmospheric pressure(frel,0 ≈ 10 GHz). By substitution in the Debye equation, the permittivity of a water layerwith volume dθ and with matric pressure pm can be written as

ε∗(f ) =[

Sw (θ)εs − ε∞

1 + jf /frel (pm)+ Mf→∞

]dθ (10)

where Mf→∞ accounts for high-frequency electric field discontinuities. As in a compositespheres model, the total permittivity can be found by integrating over all water layers withinthe soil, with soil matric pressures ranging from dry [pm(θ = 0)] to saturated [pm(θ )].The model can in principle be used to calculate complex dielectric soil properties for awide range of frequencies. However, the complex and difficult-to-obtain input variablessuch as the matric pressure and relaxation frequency as well as the empirical nature ofthe depolarization factor S make this model less suitable for routine estimations of soildielectric properties.

Several authors have proposed simpler frequency-specific empirical functions to pre-dict dielectric properties of soils (Table 4). These approaches often use a volumetric mixingmodel as their base and include information of physical background of dielectric behavior.As previously for the empirical calibration functions, these models have been developedand calibrated for a specific set of soils and thus may only be valid for the data that wereused to develop the relationship.

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Tabl

e4

Ove

rvie

wof

freq

uenc

y-sp

ecifi

cem

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calc

alib

ratio

nfu

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ns

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taC

alib

rate

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r

Nam

ean

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fere

nce

fra

nge

(GH

z)Te

xtur

eO

ther

Out

putb

fra

nge

(GH

z)N

o.of

soils

No.

ofsa

mpl

esSo

ilty

pes

Bri

sco

etal

.(19

92)

0.45

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3,9.

3–

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45,1

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33

USD

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056

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lay

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rel,

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eff

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1–1

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520

0–25

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929)

∞–

εs,

ε∞

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rel

ε′ ,

ε′′ ,

σef

f∞

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–D

obso

net

al.(

1985

)1.

4–18

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,ρε

′ ,ε

′′ ,σ

eff

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185

5U

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allik

aine

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1985

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(9in

cr.)

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Saθ

ε′ ,

ε′′

1.4–

185

5U

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,L,S

iL,

SiL

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0.00

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0.02

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01–1

711

USD

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)an

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)Pe

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tal.

(199

5)0.

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θ,ρ

b,ρ

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′ ,ε

′′ ,σ

eff

0.3–

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44

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cont

ent;

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nin

soil;

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ume

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soil;

v s,v

olum

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soil;

εs,

stat

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ittiv

ity;ε

0,d

iele

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ivity

ofva

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real

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ive

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ity.

c USD

Ate

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ion,

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Tabl

e1.

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urtis

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sona

lcom

mun

icat

ion.

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406 R. L. Van Dam

Above 1.4 GHz. Based on network analyzer measurements of soils over a frequency rangeof 1.4 to 18 GHz, Hallikainen et al. (1985) published a general empirical function thatcalculates the real (ε′) and imaginary (ε′′) parts of the permittivity as a function of clay(C), sand (S), and volumetric water content (θ ):

ε′orε′′ = (a + bS + cC) + (d + eS + fC)θ + (g + hS + iC)θ2 (11)

In this equation the coefficients a to i vary with frequency. For all frequencies, an increasein clay content leads to an increase in the ratio θ over ε′. For three soil types with sand/clayratios of 3.8, 2.3, and 0.1, respectively, a frequency increase from 4 to 18 GHz correlatespositively with the θ–ε′ ratio and inversely with the θ–ε′′ ratio. For dry soils, however,there is virtually no effect of frequency on either ε′ or ε′′ (Hallikainen et al. 1985; Mätzler1998).

Another high-frequency model, by Dobson et al. (1985), was developed from experi-mental observations between 1.4 to 18 GHz on four different soils (Table 4) with a varietyof water contents. This model calculates the real and imaginary parts of the permittivityaccording to

ε′ =[

1 + ρb

ρs

(εα

s − 1)+ θβ′ε′α

fw − θ

]1/α, and (12)

ε′′ =[θβ ′′

ε′′fw

α]1/α

, (13)

where the real and imaginary parts of the free water permittivity are defined as

ε′fw = εw∞ + εw0 − εw∞

1 + (2π f τw)2 , and (14)

ε′′fw = 2π f τw(εw0 − εw∞)

1 + (2π f τw)2 + σeff

2πε0f

ρs − ρb

ρsθ. (15)

In these equations, α = 0.65, which was based on an empirical fit to experimental data,and Tw is the relaxation time of water. Variables β ′ and β ′′ and the dielectric permittivityof solid material were empirically determined as β ′ = 127.48 − 0.519S − 0.152C, β ′′ =1.33797 − 0.603S − 0.166C, and εs = (1.01 + 0.44ρs)2 − 0.062. The effective dielectricconductivity (σ eff) is given as

σeff = −1.645 + 1.939ρb − 2.013S + 1.594C (16)

Using this pedotransfer function, the real (ε′) and imaginary (ε′′) parts of the dielectriccoefficient (ε) of a soil are found using the known fractions of sand (S) and clay (C), thebulk density (ρb), and the particle density (ρs) for different soil water contents.

Below 1.3 GHz. The model of Peplinski, Ulaby, and Dobson (1995) is complementary tothe model described previously and has been developed from experimental observationsbetween 0.3 to 1.3 GHz on a variety of artificial mixtures of sand, silt, clay, and watercontents. It is identical to the higher frequency model (Dobson et al. 1985) except that thereal part of the dielectric permittivity is given by

ε′ = 1.15

[1 + ρb

ρs(εα

s − 1) + θβ′ε′αfw − θ

]1/α

− 0.68 (17)

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Estimating Soil Moisture from GPR 407

and the conductivity by

σeff = 0.0467 + 0.2204ρb − 0.4111S − 0.6614C (18)

The Dobson–Peplinski models have been successfully tested for different soil types(Miller, Hendrickx, and Borchers 2004) and use inputs that are often readily available insoil databases. The poor overlap between the two models, however, illustrates a significantlevel of uncertainty in the 1.3- to 1.4-GHz range.

Discussion

With the abundance of calibration functions available to correlate soil dielectric propertieswith volumetric water content and other physical material properties, it is hard to choosethe best function for a given situation. Indeed, because of the large number of differentparameters used in these calibration functions, comparing them in a quantitative way is adifficult task. Only the most basic variables θ and ε are part of all functions. The choicefor which calibration function to use depends on (1) the type of input information availableto the user, (2) the frequency of the GPR equipment, and (3) the desired level of detail ofthe output. In the following, these three items are discussed in more detail for the differentfunctions that were presented in the summary tables.

Input

To find the dielectric properties of a soil, some functions will not require more inputthan the volumetric water content [e.g., Eq. (4)], whereas others require input that can berelatively easily acquired using field or laboratory measurements or from soil databases.Common input parameters are the relative volumes and dielectric properties of air, solids,and clay fractions in a soil sample. Other functions require input that is empirical innature, which cannot easily be determined experimentally. In such cases, it is critical that acalibration function is selected that has been tested on relevant sample material and at fre-quencies comparable to the user’s GPR equipment All empirical functions (Table 2) andmost volumetric mixing models (Table 1) require relatively few input parameters. Valuesfor the spread in relaxation frequencies, necessary to describe the behavior of heteroge-neous mixtures in some models (e.g., Debye 1929; Hilhorst 1998), need to be measured foreach specific sample. As a result, the use of such models to describe frequency-dependentdielectric properties of field soils may require recalibration for each sample.

Frequency Range

Because dielectric properties and, thus, the θ–ε relationship, are frequency dependent(Table 3), it is of great importance to consider for what frequency range each calibrationfunction was developed. Frequency is an input variable in only very few models, becausemost were developed for a wide frequency range using TDR instrumentation. Those thatdo include frequency as an input parameter thus enable prediction of frequency-dependentbehavior. Although most of the functions in Tables 1, 2, and 4 give only apparent dielectricpermittivity as output, some do predict both the real and imaginary parts of the permittiv-ity. The relaxation-based models are well suited for describing all frequency-dependentrelaxation phenomena, but their complexity and difficult-to-obtain input variables aredrawbacks.

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408 R. L. Van Dam

Empirical functions have been mostly developed from controlled laboratory TDR andcoaxial transmission line experiments, the results of which are most accurate for the low100s of MHz. For higher and lower frequencies alternative empirical functions have beenproposed by Hallikainen et al. (1985) and Campbell (1990), respectively. Volumetric mix-ing and empirical models usually do not incorporate frequency-dependent behavior of soildielectric properties. The empirical function by Hallikainen et al. (1985), the “De Loor”volumetric mixing model in Dobson et al. (1985), and frequency-specific calibration func-tions by Dobson et al. (1985) and Peplinski, Ulaby, and Dobson (1995), among a fewothers, are exceptions. The frequency-specific empirical model by Hilhorst (1998) canbe used to calculate complex dielectric soil properties for a wide range of frequencies.However, the complex and difficult-to-obtain input variables make this model less suitablefor routine estimations of soil dielectric properties.

Output

Most empirical functions have been developed for a specific frequency range and oftena limited number of soils (Table 2). Nevertheless, despite these limitations, it has beenshown that if care is taken regarding the type of material that the model has been calibratedfor, these functions can be used with satisfying accuracy (Van Dam et al. 2002; Lunt,Hubbard, and Rubin 2005). The output of volumetric mixing models is limited by theuse of the poorly defined empirical factor α. Both Dobson et al. (1985) and Bohl andRoth (1994) compare a number of mixing models for predicting the relationships betweensoil water content and dielectric soil properties. They concluded that simple three- andfour-phase CRI mixing models were adequate to describe mineral soils. For organic soilsonly four-phase mixing models and the Maxwell–De Loor model provided good results.Frequency-specific calibration functions describe the θ–ε relationship of soils in significantdetail. However, the empirical nature of these models makes them potentially inaccuratefor soil types not used to develop the relationships. Also, these models should be used withcaution when operating at the extremes of their frequency ranges.

Conclusions

This article presented a review of methods for the prediction of dielectric properties of fieldsoils with the goal of providing users of ground-penetrating radar for soil moisture mea-surements with an overview of calibration functions. The methods have been grouped intovolumetric, empirical, effective medium, and frequency-specific approaches. The majorcharacteristics of each model, such as the input and output variables, the range of fre-quencies the model can be used at, and the soil types the models have been calibrated for,have been presented in tables and should provide the reader with information to choosebetween models. This review has shown that among the many different calibration func-tions available, none are suitable for all situations. Many calibration functions are verylimited in their number of input variables, whereas others are too complicated for routineuse. Frequency-specific empirical functions for the prediction of soil dielectric propertiescombine the advantages of empirical and physical models to deal with behavior for whichno good theoretical models are available.

Funding

The author acknowledges U.S. Army Research Office grant (DAAD19-02-1-027).

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Estimating Soil Moisture from GPR 409

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