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Published in Vadose Zone Journal 3:1063-1071 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

SPECIAL SECTION: HYDROGEOPHYSICS

Measuring the Soil Water Content Profile of a Sandy Soil with an Off-Ground Monostatic Ground Penetrating Radar

S. Lambota,*, J. Rhebergenb, I. van den Boschc, E. C. Slobd and M. Vancloostera

a Department of Environmental Sciences and Land Use Planning, Catholic University of Louvain, Croix du Sud 2, Box 2, B-1348 Louvain-la-Neuve, Belgium
b TNO Physics and Electronics Laboratory, P.O. Box 96864, 2509 JG The Hague, The Netherlands
c Microwave Laboratory, Catholic University of Louvain, Place du Levant 3, B-1348 Louvain-la-Neuve, Belgium
d Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands
* Corresponding author (lambot{at}geru.ucl.ac.be)

Received 13 January 2004.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSIONS
 CONCLUSIONS
 REFERENCES
 
We explore the possibility of measuring a continuously variable soil moisture profile by inversion of a ground penetrating radar (GPR) signal. Synthetic experiments were conducted to demonstrate the well-posedness of the inverse problem for the specific case of identifying a soil moisture profile in hydrostatic equilibrium with a water table. In this case, the profile agrees with the water retention curve of the soil. The analysis subsequently extends to an actual case study in controlled outdoor conditions on a large tank filled with sand. Due to the presence of a discontinuity in the actual dielectric profile, inversion of the continuous model (Model 1) led to poor results. Only the surface soil moisture was well estimated. Including the observed discontinuity in the model (Model 2) led to a good estimation of the water content profile. Finally, we observed that the surface water content can be accurately estimated using a simplified three-layer model (Model 3). Generally, the observed confidence intervals on the estimated parameters are large, which denotes a lack of model sensitivity to the soil parameters. We attributed the low sensitivity to the high operating frequency range. Lower frequencies would have been required to obtain more information from the larger depths. Nevertheless, high frequencies allowed for an accurate estimation of the surface soil moisture, which offers particularly promising perspectives in humanitarian demining and agricultural applications.

Abbreviations: CMP, common midpoint • GMCS, global multilevel coordinate search • GPR, ground penetrating radar • NMS, Nelder–Mead simplex • SFCW, stepped frequency continuous wave • TEM, transverse electromagnetic • UWB, ultrawide band • VNA, vector network analyzer


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSIONS
 CONCLUSIONS
 REFERENCES
 
THE KNOWLEDGE of soil water dynamics is essential in agricultural, hydrological, and environmental engineering because these dynamics control plant growth, key hydrological processes, and the contamination of surface and subsurface water. Groundwater flow and contaminant transport are dominated by the soil hydraulic properties, including water content, and their inherent variability in the vadose zone (Dagan, 1982). It has become evident that modeling detailed spatial distributions of water and solutes in the heterogeneous subsurface requires extensive site characterization (Yeh, 1998). Characterizing this variability with conventional laboratory or downhole methods is invasive and, thus, time-consuming, costly, and subject to a large degree of uncertainty resulting from a lack of densely sampled in situ measurements.

Near-surface remote sensing can be used for nondestructive characterization of the hydrogeophysical properties of the subsurface. In that respect, GPR constitutes a promising high resolution characterization tool. However, despite considerable research devoted to GPR, its use for assessing quantitatively the subsurface properties has been constrained by a lack of appropriate GPR systems and signal analysis methods. Ground penetrating radar has been used to identify soil stratigraphy (Davis and Annan, 1989; Kung and Lu, 1993; Boll et al., 1996), to locate the water table (Nakashima et al., 2001), to follow wetting front movement (Vellidis et al., 1990), to measure soil water content (Greaves et al., 1996; Chanzy et al., 1996; van Overmeeren et al., 1997; Weiler et al., 1998; Huisman et al., 2001; Serbin and Or, 2003; Rucker and Ferré, 2003), to assist in subsurface hydraulic parameter identification (Hubbard et al., 1997; Gloaguen et al., 2001), to assess soil salinity (al Hagrey and Müller, 2000), and to support the monitoring of contaminants (Brewster and Annan, 1994; Darayan et al., 1998; Yoder et al., 2001). An excellent review of GPR principles and history was given by Annan (2002), and reviews of GPR applications for measuring soil water content in particular were given by Davis and Annan (2002) and Huisman et al. (2003).

Four methodologies are generally used for determining soil water content from GPR data (Huisman et al., 2003). Soil water content profiles can be determined from (i) reflected wave velocity and (ii) transmitted wave velocity between boreholes. A commonly used method is the surface common midpoint (CMP) technique (e.g., Garambois et al., 2002). With the CMP method, stacking velocity fields are extracted from multioffset radar soundings at a fixed central location. Common midpoint and borehole data can be processed using standard ray-tracing techniques (Fisher et al., 1992; Rucker and Ferré, 2003) or tomographic inversion (Goodman, 1994; Cai and McMechan, 1995). Yet, CMP-derived velocity estimates are generally characterized by low resolution and high uncertainty (Tillard and Dubois, 1995). The success of the measurements depends on the presence of clearly reflecting layers in the soil. Moreover, CMP and cross-hole tomographic GPR techniques are cumbersome because they require several measurements for a single profile characterization, and the antennas must be either in contact with the soil or lowered into wells. Therefore, they are not appropriate for real-time mapping. The soil water content can be obtained also from (iii) the ground wave velocity (Du and Rummel, 1994; Huisman et al., 2001; Grote et al., 2003) and (iv) the surface reflection coefficient (Chanzy et al., 1996; Serbin and Or, 2003). However, with these methods, only the surface soil skin is characterized. Furthermore, an important issue with these approaches is the effect of the variations of water content with depth.

The main limitations of the existing GPR characterization methods arise from the simplifying assumptions with respect to the antenna's radiative properties and electromagnetic wave propagation phenomena. As a result, only part of the information contained in the GPR signal is utilized, usually the propagation time. Additionally, commercially available GPR systems generally have a bandwidth of <1 GHz. A larger bandwidth is needed for a better spatial resolution (Daniels, 1996). For instance, Ulriksen (1982) observed a relation between GPR amplitude power reflection coefficients and water content. He showed, under laboratory conditions, that information about the vertical distribution of water content can be obtained by using multiple-frequency radar antennas, with the high-frequency signals sampling the shallowest layers and the lower-frequency signals sampling the deeper layers.

To circumvent the limitations of the existing methods, Lambot et al. (2004a)( 2004b) recently proposed a new promising integrated approach that optimizes the information gained from the subsurface from a single GPR measurement. It is based on an ultrawide band (UWB) stepped frequency continuous wave (SFCW) radar combined with a monostatic antenna to be used off ground. This radar configuration enables a high mobility, so one can acquire more information from the ground due to the large bandwidth, and allows for a more realistic and accurate forward and inverse modeling of the radar signal. The radar–antenna–subsurface system is modeled using linear system transfer functions and the exact solution of the three-dimensional Maxwell's equations for wave propagation in a horizontally multilayered medium. The inversion to identify the subsurface properties is formulated by the classical least squares problem and is performed iteratively using the global multilevel coordinate search optimization algorithm combined sequentially with the local Nelder–Mead simplex algorithm (GMCS-NMS) (Huyer and Neumaier, 1999; Lambot et al., 2002). By improving the radar–antenna–subsurface model of Lambot et al. (2004a), Lambot et al. (2004b) successfully validated the overall approach in simple laboratory conditions and obtained accurate estimates of the electric properties of a two-layered sand.

We extended the work reported by Lambot et al. (2004b) by considering a continuous variation of the electric properties through the subsurface. More specifically, we explored the potential of the technique to identify the dielectric profile within a sand that is in hydrostatic equilibrium with a water table. In this case, the dielectric profile corresponds to the water retention curve of the sand and can be described by a simple analytical function. We performed numerical experiments to investigate the theoretical feasibility (i.e., the well-posedness of the inverse problem); then we tested the method in partially controlled outdoor conditions. This raised additional complications compared with the laboratory experiments presented by Lambot et al. (2004a)(2004b). First, the bottom boundary condition in the electromagnetic model was not controlled, whereas in the laboratory experiments it was controlled by a perfect electric conductor, a metal sheet. Second, the presence of the water saturated zone will significantly attenuate the radar signal. Therefore, less information from the soil can be obtained. Finally, the outdoor sandy soil, being undisturbed, inherently contains heterogeneities.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSIONS
 CONCLUSIONS
 REFERENCES
 
Forward Modeling of GPR Signal
GPR System and Modeling Assumptions
For identifying the depth-dependent electric properties of the shallow subsurface, Lambot et al. (2004a)(2004b) proposed use of an UWB SFCW radar combined with a highly directive transverse electromagnetic (TEM) horn antenna to be used off ground in monostatic mode. This radar configuration possesses several advantages over commonly used systems, and the major value arises from the potential to model in a purely physical and efficient way the radar–antenna–subsurface system.

First, the antenna can simply be modeled by a single source and receiver point, instead of a spatially distributed source and receiver, since the subsurface can be situated in its far field (Balanis, 1997). Second, given the monostatic configuration, antenna modeling does not require the knowledge of the source radiation pattern because the picked-up signal has only propagated along the antenna axial direction when the subsurface is locally horizontally multilayered. Third, given the high directivity of the antenna, it results that the horizontal variability of the electric properties inherently encountered in environmental systems is expected to play a negligible role, and the ground can then be modeled realistically and efficiently using a horizontally multilayered configuration.

In the model, the effect of soil roughness is neglected. Considering the Rayleigh criterion (h < {lambda}/8, where h is the average height of surface irregularities and {lambda} is the wavelength), the upper limit of surface roughness below which the soil surface can be considered smooth is 0.047 m at 0.8 GHz and 0.011 m at 3.4 GHz (Chanzy et al., 1996; Huisman et al., 2003). Therefore, the approximation of a flat earth is valid in this study for which the sand surface is ideally flat. It is worth noting that the maximal value for h at 225 MHz is 0.167 m, which means that the approximation of a flat earth may be valid in a wide range of GPR environmental applications.

It is well known that in the operating frequency range of GPR, soil materials can exhibit significant dispersive properties; that is, the complex effective dielectric permittivity of the soil is function of frequency (Hipp, 1974; Hallikainen et al., 1985; Heimovaara et al., 1996; Teixeira et al., 1998). In the frequency range of interest (0.8–3.4 GHz), Lambot et al. (2004b) observed that the dielectric permittivity of the sand was almost independent of frequency, whereas the electric conductivity showed a clear frequency dependence. Since we are most interested in estimating the dielectric profile, we consider in this study an effective electric conductivity independent of frequency.

Antenna Equation in the Frequency Domain
Lambot et al. (2004b) modeled the antenna as a linear system composed of elementary model components in series and parallel, each accounting for a specific electromagnetic phenomenon. The block diagram of the antenna is presented in Fig. 1 , and the corresponding transfer function expressed in the frequency domain is given by

[1]
where S11({omega}) is the measured frequency-dependent complex ratio between the returned signal and the emitted signal; Y({omega}) and X({omega}) are, respectively, the received and emitted signals at the radar reference plane; Hi({omega}), Ht({omega}), Hr({omega}), and Hf({omega}) are, respectively, the complex return loss, transmitting, receiving, and feedback loss transfer functions of the antenna; Gxx is the transfer function of the air–subsurface system modeled as a horizontally multilayered medium; and {omega} is the angular frequency. This relationship is referred to as the antenna equation in the frequency domain.



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  		Fig. 1. Block diagram representing the vector network analyzer–antenna–multilayered medium system modeled as linear systems in series and parallel.

 
Due to the variations of impedance between the antenna feed point, antenna aperture, and air, multiple wave reflections occur within the antenna. The return loss transfer function Hi({omega}) represents the part of this ringing, measured at the reference plane, that is independent of the backscattered electromagnetic field Gxx. Hi({omega}) can thus be measured directly by performing S11({omega}) measurements in free space conditions for which Gxx = 0. The transmitting and receiving transfer functions describe the antenna gain and phase delay between the measurement point and the source and receiver virtual point. Equation [1] can be simplified by defining H({omega}) = Ht({omega})Hr({omega}), in this way reducing the number of transfer functions to be determined. The positive feedback loop with transfer function Hf({omega}), similarly to Hi({omega}), accounts for the variations of impedance between the antenna feed point, antenna aperture, and air, which creates a part of the backscattered field to be reflected again toward the subsurface. This leads to multiple wave reflections between the antenna and the subsurface. The transfer functions H({omega}) and Hf({omega}) can be determined by solving a system of two equations as Eq. [1], pertaining to two different model configurations for which S11({omega}) is measured and the corresponding Gxx is computed, to two unknowns.

Horizontally Multilayered Medium
The model is three-dimensional and consists of N horizontal layers separated by N – 1 interfaces (Fig. 2) . Consideration of a three-dimensional model is essential to take into account the spherical divergence (geometric spreading) in wave propagation. The medium of the nth layer is homogeneous and characterized by the dielectric permittivity {epsilon}n, electric conductivity {sigma}n, and thickness hn. The magnetic permeability µ is assumed to be constant and equal to the permeability of free space, namely, µ0 = 4{pi}10–7 Hm–1. The relative dielectric permittivity is defined as {epsilon}n = {epsilon}/{epsilon}0, with {epsilon}0 = 8.85 x 10–12 Fm–1 being the permittivity of free space. The source and receiver point is located in the upper half-space. Following Lambot et al. (2004a), the emitting part of the TEM horn antenna is approximated by an infinitesimal horizontal x-directed electric dipole, whereas the receiving part of the antenna is emulated by recording the horizontal x-directed component of the backscattered electric field. For this model configuration, closed-form analytical expressions can be derived for the exact solution of the three-dimensional system of Maxwell's equations (Tai, 1994; Michalski and Mosig, 1997; Peterson et al., 1998). Following the approach of Slob and Fokkema (2002) and Lambot et al. (2004a), we compute the air–subsurface transfer Green function Gxx (i.e., the solution of Maxwell's equations) by computing recursively the transverse electric and magnetic global reflection coefficients of the multilayered system in the two-dimensional spatial Fourier domain. A continuous electric profile can be obtained by considering layers with hn -> 0.



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  		Fig. 2. Three-dimensional N-layered medium with a point source and receiver S. Each layer is characterized by the dielectric permittivity {epsilon}, electric conductivity {sigma}, and thickness h.

 
Soil Electric Properties
In this study, the soil electric properties are assumed to vary continuously with depth. The sand is subject to a hydrostatic equilibrium with a water table, and consequently, the dielectric profile is determined by the water retention curve of the sand, since the dielectric permittivity is strongly correlated to the water content (Topp et al., 1980). The water retention curve of the sand is assumed to be described by the commonly used van Genuchten model (van Genuchten, 1980):

[2]
where {theta} is the volumetric water content (m3 m–3); h is the pressure head (m); {theta}r and {theta}s are the residual and saturated water contents (m3 m–3); {alpha} (m–1) and n (dimensionless) are curve shape parameters that are inversely related to the air-entry value and the width of the pore size distribution, respectively; and m = 1 – 1/n is restricted by Mualem's condition with n > 1. In hydrostatic equilibrium with a water table located at position zw (see Fig. 2 for the coordinate system), the pressure head h is related to the position z by h = z – zw.

The relation between the volumetric water content and the relative dielectric permittivity is assumed to be described by (Ledieu et al., 1986):

[3]
where a and b are soil specific empirical parameters. Substituting Eq. [3] into Eq. [2] leads to the function describing the dielectric profile corresponding to the hydrostatic equilibrium with the water table:

[4]
where {epsilon}r,r and {epsilon}r,s are, respectively, the relative dielectric permittivity of the dry and water saturated soil, and z1 is the position of the soil surface.

The electric conductivity is derived from the dielectric permittivity using Eq. [3] and Rhoades' model expressed as (Rhoades et al., 1976):

[5]
where {sigma}w is the soil solution electric conductivity (S m–1), {sigma}r is the electric conductivity of dry soil (S m–1), and a and b are soil specific empirical parameters.

In this study, the sand specific relation established by Lambot et al. (2004b) is used (see Fig. 3) , with a = 0.1264 and b = –0.1933 in Eq. [3]. In Eq. [4], parameters {epsilon}r,r, {epsilon}r,s, z1, and zw are assumed to be known. In Eq. [5], we set a = 1.85, b = 3.85 x 10–2, and {sigma}r = 5.89 x 10–4 S m–1, which are characteristic values for sandy soils. The thickness of the layers in the model configuration (Fig. 2) is set to 1 cm. It can be shown that a finer resolution is not required to simulate a continuously variable profile for frequencies lower than 3.4 GHz (results not presented).



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  		Fig. 3. Sand specific relation between the volumetric water content and the GPR derived relative dielectric permittivity ({epsilon}r). Topp's model is represented for comparison.

 
Model Inversion
Given the forward model and its parameterization, model inversion consists of finding the parameter vector b = [{alpha}, n, {sigma}w], so an objective function {phi}(b) is minimized. In the particular case where no prior information on the parameters is taken into account and assuming observation errors to be independent, homoskedastic, and normally distributed with mean zero and covariance matrix C, the maximum likelihood theory reduces to the weighted least squares problem. The objective function to be minimized is defined accordingly as follows:

[6]
where Gxx* = Gxx* and Gxx* = Gxx* are the vectors containing the observed and simulated Green functions, respectively. Since these response functions are complex functions, the difference between observed and modeled data is expressed by the amplitude of the errors in the complex plane.

Objective function Eq. [6] indirectly relates the response function of the multilayered medium to its constitutive parameters. However, as in most electromagnetic inverse problems, this function is highly nonlinear and is characterized by an oscillatory behavior and a multitude of local minima. This complex topography necessitates the use of a robust global optimization algorithm. Following the approach of Lambot et al. (2002)(2004a), we used the global multilevel coordinate search (GMCS) algorithm (Huyer and Neumaier, 1999) combined sequentially with the classical Nelder–Mead simplex algorithm (NMS) (Lagarias et al., 1998).


   RESULTS AND DISCUSSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSIONS
CONCLUSIONS
 REFERENCES
 
Numerical Experiments
For analyzing the well-posedness of the inverse problem dealt with in this study, synthetic Green's functions were generated for five different scenarios, that is, five different hypothetical dielectric profiles. The benefit of this numerical approach is that the true parameter values characterizing the profiles are known and that the response function is perfectly described by the electromagnetic model, without measurement and modeling errors. The hypothetical dielectric profiles are represented in Fig. 4 , as well as the corresponding Green's functions, and cover a large range of possible cases. For each scenario, a realistic electric profile (not presented) has been determined from the dielectric profile using Rhoades' model (Eq. [5]) with the parameterization described above and {sigma}w = 0.075 S m–1. Inversions were performed considering the frequency range of 0.8 to 1.8 GHz, with a frequency step of 8 MHz. The lower frequency corresponds to the lower cut-off frequency of the radar system used in this study.



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  		Fig. 4. (a) Dielectric profiles pertaining to the five synthetic scenarios, z being the depth and {epsilon}r being the relative dielectric permittivity; (b) corresponding Green's functions.

 
Figure 5 represents response surfaces of the objective functions pertaining to Scenarios S2, S3, S4, and S5 (S1 leads to similar results). The considered parameter space includes simultaneously all scenarios and was divided into 100 discrete values, resulting in 10000 objective function calculations for each contour plot. A star represents the known global minimum of the objective function, which corresponds in every case exactly to the solution obtained by inversion of the synthetic Green functions using GMCS–NMS. These results demonstrate the identifiability of the model parameters and the uniqueness of the inverse solution for this model configuration. That means that sufficient information is theoretically contained in the Green function to estimate the dielectric profile. Parameters {alpha} and n are generally negatively correlated. For Scenarios S2, S4, and S5, the model is less sensitive to n than to {alpha}, in contrast to Scenario S3. The presence of local minimum regions in the objective functions imposes the use of a global optimization approach. These are due to the nonlinear relation between the Green function and the soil electromagnetic properties.



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  		Fig. 5. Objective function logarithm [log10({phi})] for Scenarios (a) S2, (b) S3, (c) S4, and (d) S5. A star represents the global minimum of the objective function, which is accurately found by GMCS–NMS in every case.

 
It is worth noting that the amplitude and phase of the Green's functions represented in Fig. 4 are not very much different for some scenarios. That means that the inverse problem may suffer from a lack of stability when subject to measurements and modeling errors. Moreover, their frequency dependence is almost linear in this frequency range. This indicates that most of the information in the Green function comes from a single reflector, namely, the air–soil surface interface. The linear dependence vanishes progressively for lower frequencies (see in particular Scenario S1) since the electromagnetic waves then penetrate deeper into the soil.

Outdoor Experiment
Description
From a mathematical point of view, the objective of the outdoor experiment was to investigate the stability properties of the inverse problem, that is, the sensitivity of the inverse solution in relation to actual modeling and measurement errors. Outdoor experiments were conducted at the TNO facilities in The Hague, The Netherlands. Measurements were performed on a 1.5-m-deep and 3 x 10 m2 area tank filled with sand (Fig. 6) . The benefit of this outdoor test site, compared with laboratory conditions, is that there is no metallic and a few non-metallic objects in the surroundings of the tank, minimizing ambiguous scattering from extraneous objects. Moreover, the dimensions of the tank make the approximation of an infinite horizontally multilayered medium much more realistic compared with laboratory-scale experiments.



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  		Fig. 6. TNO outdoor experimental setup including the sand tank (3 by 10 m), the air-launched TEM horn antenna, the saturated sand, and the unsaturated sand in hydrostatic equilibrium with the water table.

 
The sand was subject to a hydrostatic equilibrium with a water table imposed at a depth of 0.67 m. Evaporation was prevented using a plastic sheet on the soil surface. The saturated zone extended from the 1.50- to 0.67-m depths. Given the operating frequency range (0.8–3.4 GHz), the saturated zone constitutes the lower half-space in the electromagnetic model, since electromagnetic waves are expected to be completely absorbed in this layer. Apart from soil roughness, the experimental conditions at TNO are closely representative of field conditions in partially saturated sandy soils.

An UWB SFCW radar system was emulated using a vector network analyzer (VNA) (ZVRE, Rohde & Schwarz, Munich, Germany) with an excellent dynamic range (>130 dB). The antenna system consisted of a linear polarized double ridged broadband TEM horn (BBHA 9120 A, Schwarzbeck Mess-Elektronik, Schönau, Germany). The antenna is 22 cm long with a 14 x 24 cm2 aperture area. Its nominal frequency range is 0.8 to 5 GHz. The antenna was connected to the reflection port of the VNA via an N type 50-{Omega} impedance coaxial cable. We calibrated the VNA at the connection between the antenna feed point and the cable using a 50-{Omega} OSM (Open, Short, Match) series of a high precision standard calibration kit (ZVZ21-N, Rohde & Schwarz), thus establishing a reference calibration plane to which the frequency-dependent complex ratio S11({omega}) between the returned signal and the emitted signal is measured. Parameter S11 was measured sequentially at 1601 stepped frequencies across the range 0.8 to 4 GHz, with a frequency step of 2 MHz. However, only the range 0.8 to 3.4 GHz with a frequency step of 32 MHz was used for the inversions. Measurements were performed with the antenna aperture situated at about 0.41 m above the sand surface. Above 3.4 GHz, the quality of the measured Green function is poor because the far-field approximation is no longer valid. Following to the radar measurements, 300-cm3 soil samples were collected along the soil profile for determining the actual volumetric water content by means of the standard oven-drying method at 105°C for at least 24 h.

Water Content Profile: Model 1
Figure 7 represents the observed and modeled Green function (Model 1), in both the frequency and time domains. Contrary to expectations for this frequency range, the frequency dependence of the Green function amplitude is far from linear. This nonlinearity stems from the presence of a discontinuity in the sand dielectric profile. This discontinuity is visible in the time domain representation of the measurement (Fig. 7b). The reflection from the soil surface occurs at about 3.3 ns. Then a second reflection occurs at about 3.9 ns. Although this discontinuity is not accounted for in the electromagnetic model, the modeled Green function fits relatively well to the measurement. The observed differences may originate from different error sources: (i) the different hypothesis underlying the electromagnetic antenna and subsurface models, (ii) the characterization of the antenna frequency response functions, (iii) the fact that the actual dielectric profile is not as continuous as assumed because of the inherent presence of heterogeneities in the sand tank and in particular due to the observed discontinuity, and (iv) the presence of ambiguous clutter. In the study of Lambot et al. (2004b), it was shown that the model was very accurate for describing the radar signal, so the two first reasons should have a negligible effect. It is worth noting that the water table interface is not visible in the time domain Green function. This means that the electromagnetic waves are almost totally attenuated in the unsaturated zone. The approximation of an infinite lower half-space to describe the saturated zone in the electromagnetic model is therefore justified.



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  		Fig. 7. Measured and modeled Green's function represented in both the (a) frequency and (b) time domains.

 
Figure 8 represents the inversely estimated water content profile compared with the ground truth measurements and the fitted van Genuchten model [{theta}r = 0, {theta}s = 0.33, {alpha} = 6.63 (m–1), and n = 2.57 in Eq. [2]) representing the water retention curve of the sand. Except for the surface soil moisture, which is well estimated, the inversely estimated water content profile is significantly different from the actual one. A sharp variation occurs at the 6-cm depth, which emulates the discontinuity. Then the water content is largely overestimated due to the constraints on the shape of the dielectric profile.



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  		Fig. 8. Directly measured and GPR derived water content profile. The dash line represents van Genuchten's model fitted to the ground truth data.

 
Water Content Profile: Model 2
To inverse the measured Green function properly, the observed discontinuity must be included in the electromagnetic model. Accordingly, we redefine the dielectric profile as (Model 2):

[7]
where {epsilon}r,2 is the relative dielectric permittivity of the surface layer, which extends to depth z2.

In this case we observe that the inversely estimated water content profile agrees remarkably well with the directly measured one (see Fig. 8). The observed differences are to be attributed to errors in the gravimetric measurements, to the presence of heterogeneities in the sand (local variations of the hydraulic properties), and also to the decreasing sensitivity of the Green function with depth. Indeed, we can observe in the time domain representation of the Green function that the major part of the energy picked up by the antenna originates from the two first interfaces.

Water Content Profile: Model 3
Since most of the information contained in the GPR signal comes from the two first interfaces, we also performed the inversion by considering a simplified three-layered dielectric profile defined as (Model 3):

[8]

The electric conductivity and thickness of the layers were also considered as unknown in the inversion procedure.

In this case, the measured Green function is also well reproduced (Fig. 7). We observe in Fig. 8 that the surface soil moisture (first layer) is very well estimated, but larger errors are observed for the second and third layers. This is to be attributed to the high noise in the measured Green function, and to a lesser extent, to the simplifying assumptions in this model. Additionally, the inversion of Model 3 is not constrained by the knowledge of the depth of the water table and its dielectric properties.

Estimated Parameters and Confidence Intervals
The inversely estimated parameters for the three models and corresponding confidence intervals are presented in Table 1. For Model 1, as observed from the estimated water content profile, parameters are not well estimated. For Model 2, parameters {alpha} and n are close to the van Genuchten parameters ({alpha} = 6.63 and n = 2.57) characterizing the water retention curve of the sand. However the 95% confidence intervals are very large, denoting to some extent a weak sensitivity of the Green function in this frequency range. The large confidence intervals are also due to the important ambiguous clutter in the measured Green function, and also mainly to the high correlation between these parameters, which approaches –1. The surface soil dielectric permittivity and layer thickness are accurately determined, with small confidence intervals. These parameters are not correlated (r = –0.07). The position of the discontinuity in the soil profile is 6 cm. The presence of this discontinuity is either due to a layering in the sand, or to the fact that this part of the profile could not reach a state of perfect hydrostatic equilibrium. Indeed, the very low hydraulic conductivity in the sand at this suction and the very low pressure head gradient have probably hampered the upward movement of water at this level of the profile. Model 3 led to a good estimation of the surface soil moisture and layer thickness, but with larger confidence intervals than for model 1. It is worth noting the increasing confidence interval values with depth.


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  		Table 1. Inversely estimated parameters for the three models and corresponding confidence intervals.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSIONS
 CONCLUSIONS
REFERENCES
 
We tackled the issue of estimating a continuous dielectric profile using GPR. In particular, we investigated the feasibility of measuring a dielectric profile in a sandy soil in hydrostatic equilibrium with a water table. In this case, the dielectric profile corresponds to the water retention curve of the sand. Compared with other existing GPR characterization methods, the benefit of the approach presented here is that information from the ground is obtained from a single GPR measurement with a monostatic antenna to be used off ground. This ensures a high degree of mobility, which is necessary for low-cost, rapid, and high resolution mapping of the subsurface hydrological properties.

As shown with the synthetic examples in this study, the radar signal contains sufficient information to estimate with uniqueness such a continuous dielectric profile. However, the outdoor experiment demonstrated that the inverse solution may not be stable when the operating frequency range is too high. Only the surface soil moisture was accurately determined and large confidence intervals were obtained for the rest of the profile. Note that a discontinuity had to be included in the continuous model to represent more realistically the actual profile.

This study offers promising perspectives in mapping the hydraulic properties of the shallow subsurface, which is particularly relevant in environmental and agricultural applications (e.g., for bridging the gap between airborne or spaceborne remote sensing and ground truth measurements of soil moisture), as well as in humanitarian demining. Further research will focus on the joint use of low- and high-frequency antennas to improve the estimation of a whole water content profile.


   ACKNOWLEDGMENTS
 
This work was supported by a research grant from the Fonds pour la formation à la Recherche dans l'Industrie et dans l‘Agriculture (Belgium), the Fonds National de la Recherche Scientifique (Belgium), the TNO Physics and Electronics Laboratory (The Netherlands), the Delft University of Technology (The Netherlands), the Royal Military Academy (Belgium), and the Catholic University of Louvain (Belgium).


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSIONS
 CONCLUSIONS
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