Vadose Zone Journal 2:247-252 (2003)
© 2003 Soil Science Society of America
Near-Surface Water Content Estimation with Borehole Ground Penetrating Radar Using Critically Refracted Waves
Dale F. Rucker* and
Ty P. A. Ferré
Department of Hydrology and Water Resources, University of Arizona. P.O. Box 210110, Tucson, AZ 85721
* Corresponding author (druck{at}hwr.arizona.edu)
Received 3 December 2002.
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ABSTRACT
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Zero-offset profiling (ZOP) with borehole ground penetrating radar (BGPR) is a promising tool for profiling water contents in the subsurface to great depths with high spatial and temporal resolution. The ZOP method relies on determining the velocity of an electromagnetic (EM) wave that follows a direct path from the transmitter to the receiver. However, near the ground surface, critically refracted energy that travels along the ground surface at the velocity of an EM wave in air may arrive before direct waves that travel through the subsurface. If the critically refracted waves are mistakenly interpreted to be direct waves, the water content will be underestimated. As a result, the water content near the ground surface cannot be determined using standard BGPR analysis. We refer to the depth below which direct waves are the first to arrive as the refraction termination depth. An alternative analysis is presented to determine the water content above the refraction termination depth using the slope of the travel time vs. depth profile. Additionally, guidelines are presented to predict the refraction termination depth for known near-surface water contents.
Abbreviations: BGPR, borehole ground penetrating radar • EM, electromagnetic • TDR, time domain reflectometry • ZOP, zero-offset profiling
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INTRODUCTION
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CROSS-BOREHOLE ground penetrating radar is a high-frequency EM method that can be used to profile the water content in the subsurface (Binley et al., 2001, 2002a,b). In ZOP mode, a transmitting antenna and a receiving antenna are lowered to the same depth within nonmetallic access tubes and an EM wave is propagated between them. Assuming that losses are low enough to permit identification of transmitted energy, the travel time of the first-arriving energy, on the order of a few tens of nanoseconds, is measured. The velocity of the EM wave is calculated for a known antennae separation, assuming that the wave traveled along a direct path from the transmitter to the receiver. The unitless apparent relative dielectric permittivity, Ka, of the medium can be calculated directly from the EM velocity, assuming that the frequency-dependent dielectric loss is relatively small (Davis and Annan, 1989):
 | [1] |
where c is the speed of light in free space (0.3 m ns-1), which is assumed to be equal to the velocity in air, and v (m ns-1) is the velocity of propagation of the EM wave through the medium. The volumetric water content, 
(cm3 cm-3), can be determined from the apparent dielectric permittivity using the linearized form of the empirical relationship presented by Topp et al. (1980) as given by Ferré et al. (1996):
 | [2] |
The preceding analysis can be used to determine the water content at any depth if the direct wave is associated with the first energy to arrive at the receiver at that depth. However, the direct route does not always have the shortest travel time. Rather, under some conditions, a critically refracted wave may arrive before the direct arrival (Fig. 1). Critical refraction occurs at any interface across which the velocity increases (Sheriff and Geldart, 1995). The contrast between the EM wave velocity in air (vair = 0.3 m ns-1) and in soil (0.17 > vsoil > 0.05 m ns-1) gives rise to critical refraction (Bohidar and Hermance, 2002) when the antennae are located below the ground surface. Critically refracted energy arrives at the receiver through secondary waves generated at the surface (Parkin et al., 2000), where a head wave is created. The receiver can intersect the head wave at any depth.
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Fig. 1. Pathways of direct and critically refracted waves from a transmitting borehole ground penetrating radar (BGPR) antenna to a receiving BGPR antenna located a distance, z, below the ground surface and separated by a distance, x. The travel time of the refracted wave is ttx-surface + tair + tsurface-rx. The travel time of the direct wave is tdirect. The critically refracted wave approaches the interface at the critical angle from the normal to the interface, ic.
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In picking the first arrival, it is not possible to distinguish a critically refracted arrival from a direct arrival from initial inspection. This complicates the analysis of water content near the ground surface. If the travel time of a first-arriving critically refracted wave is assumed to correspond to a direct wave, the water content at the measurement depth will be underestimated. For example, near-surface travel times, such as those shown by Kuroda et al. (2002) can suggest low water contents in the shallow subsurface, even under ponded constant infiltration conditions. As a result, the water content profile can only be determined quantitatively below the depth at which direct waves arrive before critically refracted waves. We refer to this depth as the refraction termination depth, zrtd.
In this study, equations are derived to relate the refraction termination depth to the velocity of EM energy through the near-surface soil and to the antennae separation. This is similar to the development presented by Hammon et al. (2002). The analysis is extended here to determine the near-surface water content and to develop guidelines to predict the refraction termination depth.
It is important to note that these developments explicitly assume that the dielectric permittivity of the medium is both homogeneous and isotropic in the shallow subsurface. Using existing equipment to collect ZOP data, it is not possible to determine dielectric anisotropy. Similarly, ZOP measurements cannot be used to determine horizontal variations in dielectric permittivity between BGPR antennae. The precision with which vertical heterogeneity due to soil layering or water content distributions can be determined is a function of the sample depth interval. Presumably, there is some minimum thickness that can be resolved given the nature of signals generated by dipole antennae. In addition, we assumed that attenuation does not affect the ability to identify first arrivals on the BGPR trace. Under some conditions, refracted waves may experience greater attenuation than direct waves because of to their longer travel paths and transmission through different media. This could result in a termination depth that is controlled primarily by attenuation rather than relative velocities of adjacent layers.
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THEORY
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If an EM wave reaches a boundary between two media with different dielectric permittivities, a portion of the arriving energy will reflect from the boundary at the angle of incidence with respect to a line that is perpendicular to the interface, i1. The remaining energy will be transmitted across the boundary at a different angle, i2. The angle i1 is the angle of incidence, and the angle i2 is the angle of refraction. The relationship between these angles is described by Snell's Law:
 | [3] |
For the case of BGPR operated just below the ground surface, v1 and v2 are the velocities in the soil immediately below the ground surface and in the air, respectively, and the normal is a line that is perpendicular to the ground surface. For a given velocity contrast, only one incident angle will give rise to a wave that has an angle of refraction of 90° from the normal and travels along the ground surface at the velocity of air. This angle of incidence is referred to as the critical angle, ic, and the refracted wave is known as the critical refraction. The critical angle can be defined from Eq. [3] as
 | [4] |
Considering BGPR antennae located close to the ground surface (Fig. 1), the travel time of the critically refracted wave (trefr) is equal to the sum of the travel times of the propagating EM wave from the transmitter to the air–soil interface (ttx-surface), along the surface (tair), and down from the interface to the receiving antennae (tsurface-rx):
 | [5] |
From Huygen's principle, the angle of refraction from the interface to the receiving antennae is also equal to the critical angle. As a result, the distance that the wave travels to or from the ground surface is
 | [6] |
where z is the depth of the antennae below the air–soil interface.
The travel time over this distance is
 | [7] |
where v1 has been replaced by vsoil.
The distance that the wave travels in the air above the ground surface is
 | [8] |
where x is the antenna separation.
The travel time along the ground surface is
 | [9] |
where v2 has been replaced by vair.
Substituting Eq. [7] and [9] into Eq. [5] yields
 | [10] |
The direct travel time between the antennae at the same depth is
 | [11] |
Note that while tdirect is independent of z, trefr increases linearly with z. This linear increase in trefr with z is a useful characteristic for identifying critical refractions in ZOP profiles.
Critically refracted waves can arrive before direct waves when the antennae are near the ground surface (Fig. 2). However, below the refraction termination depth, zrtd, the travel time of the direct wave is shorter than that of the critically refracted wave. The refraction termination depth can be determined by equating the travel times of the critically refracted and direct waves (Equations 10 and 11) and solving for zrtd:
 | [12] |
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Fig. 2. Travel times as a function of antennae depth for direct and critically refracted waves for the case of a soil with a velocity of vsoil = 0.1 m ns-1 and an antennae separation of 3 m. The thin lines show the travel times of the direct and critically refracted waves. The first-arriving energy is highlighted as a thick line.
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Equation [12] can be used directly to estimate zrtd if vsoil is known. For the example homogeneous half-space considered in constructing Fig. 2; x is 3 m, vsoil is 0.1 m ns-1, vair is 0.3 m ns-1, and the critical angle is 19.5°. For these conditions, the direct wave arrives before waves that are critically refracted at the ground surface for measurement depths below 1.061 m. By rearranging Eq. [2], the EM wave velocity in a soil with a known volumetric water content is
 | [13] |
Alternatively, if the travel time profile resembles that shown in Fig. 2, then zrtd can be obtained directly from the profile. Then vsoil can be found by rearranging Eq. [12] to
 | [14] |
From the definition of sinic, the value of cosic can be expressed as
 | [15] |
Substituting Eq. [15] into Eq. [14] and solving for vsoil gives
 | [16] |
It should be noted that Eq. [16] requires a precise measure of zrtd. In practice, the sampling depth interval (depth that the antennae are lowered between readings) is commonly 0.25 m or more. The actual zrtd for the example system is 1.061 m. Typical measurement resolution gives rise to uncertainty as great as one-half the sampling depth interval: 1.061 ± 0.125. Water content estimates, considering these uncertainties, range from 0.08 to 0.33 cm3 cm-3, with the correct water content being 0.17 cm3 cm-3. Moreover, if the shallow subsurface is layered, or water content varies with depth, the calculated zrtd may be deeper than the base of the near-surface layer. In this case, the inferred water content applies only above a depth associated with an abrupt change in measured velocity.
Alternatively, the slope of the travel time profile near the ground surface, 
, can be used to estimate vsoil by taking the derivative of Eq. [10] with respect to depth:
 | [17] |
where tanic is
 | [18] |
Substituting Eq. [15] and [18] into Eq. [17] and solving for vsoil gives
 | [19] |
From Fig. 2, the slope of the travel time profile near the ground surface is 18.9 ns m-1 (20 ns/1.061 m), equating to a soil velocity of 0.10 m ns-1. Using Eq. [2] gives a water content of 0.17 cm3 cm-3. Actual measurements will give rise to some uncertainty of the slope. Choosing an arbitrary 5% underestimation ( = 18) or overestimation ( = 19.8), the calculated water contents range from 0.16 and 0.19, respectively. It is suggested that linear regression techniques be used to properly assess the slope error to give some measure of the accuracy of shallow water content measurements. Because zrtd may not be evident on a travel time profile plot, especially if the water content varies with depth in the shallow subsurface, we recommend using the slope analysis, which relies only on measurements made near the ground surface.
Because first-arriving critically refracted waves can complicate BGPR profiling applications, it is reasonable to ask whether a separation distance can be chosen that eliminates critically refracted waves as first arrivals. Equation [12] shows that zrtd increases with increasing antennae separation distance, x. Therefore, reducing the antennae separation will minimize the depth over which critically refracted waves arrive before direct waves. However, to allow for simple ray path analysis of travel times, the antennae separation distance must be greater than one-half of the wavelength of the center frequency in air. For example, a 100-MHz EM wave has a wavelength of 3 m in air, so the antennae separation distance when using 100-MHz antennae should be at least 1.5 m. Theoretically, a 900-MHz antenna has a minimum antennae separation of 0.18 m. However, this small separation is not practical for most field applications, given typical borehole annuli are on the order of tens of centimeters and vertical borehole deviations could be significant. The minimum zrtd is associated with a low water content, high velocity condition. Considering a medium with a volumetric water content of 0.05 cm3 cm-3, the minimum separation allowable for 100-MHz antennae results in a zrtd of 0.43 m. This result suggests that it is not possible to eliminate completely first-arriving critical refractions for measurements made very near the ground surface.
The preceding analysis only considered critical refraction at the air–soil interface. However, critical refraction can occur whenever EM waves traveling through a low-velocity medium contact a boundary with a higher-velocity medium. There are many examples of this below the ground surface. For example, the antennae may be located within a high water content, fine-textured layer that is in contact with a lower water content, coarse-textured layer. Or, the antennae may be located below the water table and the transmitted energy may refract critically at a boundary within unsaturated regions above the capillary fringe. Conclusions that are identical to those made for the impacts of refractions at the ground surface can be drawn regarding critical refractions at depth. That is, each layer will have a refraction termination depth below (or above) the layer boundary. Furthermore, if the critical refractions can be identified on the travel time profile, the slope of the travel time profile can be used to determine the water content in the high water content layers. However, given that the refraction termination depth increases with the contrast in velocities between adjacent layers, it is likely that the critical refractions will primarily affect measurements within layers that contrast sharply with adjacent layers.
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FIELD EXPERIMENT
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A field experiment was conducted to test the use of the slope of the travel time profile for determining near-surface water content. Zero-offset profiling BGPR travel time measurements were made as a wetting front moved downward through a soil profile during a constant infiltration experiment. The ZOP BGPR measurements were collected to 15 m depth with a sampling interval of 0.25 m. The separation distance between the transmitting and receiving antennae was 3.1 m.
The infiltration experiment was conducted at the Western Campus Agricultural Center, an experimental farm along the bank of the Santa Cruz River in Tucson, AZ. During the experiment, water was applied to the ground surface at a constant rate of 0.0136 m h-1 for 66 h (Rucker and Ferré, 2002). For this study, BGPR measurements were made using two access tubes located within the infiltration gallery (Fig. 3). A Sensors and Software PulseEkko100 system with 100-MHz antennae (Sensors and Software, Mississauga, ON) was used for all BGPR measurements. Postprocessing of first-arrival picking was accomplished through an automated picking software, Picker, provided by Sensors and Software.
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Fig. 3. Schematic diagram of field site used for infiltration experiment showing the locations of borehole ground penetrating radar (BGPR) access tubes (circles), buried time domain reflectometry (TDR) probes (stars), and multilevel TDR probe (square) within the infiltration gallery (solid line).
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The top 15 cm of soil at the experimental site consisted of dry, unconsolidated loose sand and silt. The surface material is underlain by a thick (approximately 3.3 m) sandy layer with interspersed lenses of organic material and clay. A black, peat-like organic layer is located approximately 1 m below surface. The material between 3.3 and 6.5 m depth is a gravely sand, which turns to cobbles, gravel, and sand to 8.5 m. From 8.5 to 13 m the medium varies from a 1.5-m-thick clay lens to a rich clayey sand. The remainder of the profile to 15 m is comprised of fine sand with small amounts of clay.
In addition to measurements with BGPR, two buried time domain reflectometry (TDR) probes were located within the infiltration gallery. One TDR probe was located outside of the gallery, and one multilevel TDR sensor was located within the gallery (Fig. 3). The buried TDR probes were placed perpendicular with the ground surface within hand-augered holes that were 10 cm in diameter and backfilled with native material. The multilevel TDR (Environmental Sensors Inc., Victoria, BC, Canada) was installed at the surface and extended to 1.2 m below ground surface. The multilevel TDR is discretized into five segments, with the segments having lengths of 0.15, 0.15, 0.3, 0.3, and 0.3 m from top to bottom. Before infiltration began, the TDR-measured volumetric water contents for these depth intervals were 0.163, 0.123, 0.125, 0.13, and 0.17 cm3 cm-3. These values give a length-weighted water content of 0.145 cm3 cm-3. Once infiltration began, the multilevel TDR malfunctioned, so measurements with this instrument were not available for comparison with later travel time profiles.
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RESULTS
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Figure 4 shows three profiles collected before, immediately after, and 1 wk after the infiltration experiment. At 0 m depth, the travel time is equal to the calculated travel time in air, 10.33 ns. As the antennae were lowered across the ground surface, the travel time slowly increased from that in air to a value representative of the velocities in the shallow subsurface. Figure 4 also includes an expanded view of the travel times from 0 to 2.75 m depth. Straight lines are fitted to the shallowest sections of each depth vs. travel time profile. The slope of the regression line fitted to the preinfiltration profile has a value of 1 = 19.10 ± 0.09 ns m-1, where the uncertainty represents one standard deviation above and below the mean. This yields a mean velocity of 0.0989 m ns-1, with velocities ranging from 0.0985 to 0.0993 m ns-1 over one standard deviation of the value of 1. The mean volumetric water content determined from Eq. [2] using this slope was 0.170 cm3 cm-3, which is in good agreement with that measured using the multilevel TDR. The range of water contents based on the range of slopes measured was 0.173 to 0.176 cm3 cm-3. Similarly, high resolution was found for other measurements based on the uncertainty of the fitted slope. The velocity measured just before infiltration ceased (see "After Infiltration" in Fig. 4) was 0.056 m ns-1, which equates to a volumetric water content of 0.44 cm3 cm-3. The profile measured 1 wk after infiltration ceased yielded a volumetric water content of 0.25 cm3 cm-3. The value of zrtd calculated using Eq. [12] is reported in Table 1 for each travel time profile.
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Fig. 4. Three borehole ground penetrating radar travel time profiles taken before infiltration, immediately after infiltration, and 1 wk after infiltration. The calculated refraction termination depth is shown with a horizontal arrow for each time. The antennae separation was 3.1 m, and the vertical sampling interval was 0.25 m.
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Only the early time profile has a predicted zrtd that corresponds with an observed change in the slope of the travel time profile in Fig. 4. At a depth of 1 to 1.25 m below ground surface, the slope decreases to a negative value, indicating that the water content is decreasing. Above 1 m, the slope is increasing and the zrtd suggests that the slope increases to a depth of 1.05 m. The predicted water content from the slope analysis is valid from the surface to 1m. The "After Infiltration" profile shows a slightly deeper refraction termination depth, and is much deeper than the depth at which a change in slope occurs. The calculated zrtd is 1.22 m, whereas the change in slope occurs at 0.75 m. In this case, the heterogeneity in water content precludes the use of the slope method below 0.75 m, and it can be shown that the measured travel times at this depth and below are from direct arrivals.
During the infiltration experiment, the volumetric water content was also measured using three buried TDR probes (Fig. 5). The measurement depths were 1.2, 1.6, and 1.67 m below ground surface. All of the TDR time series showed similar response as the wetting front moved through the soil profile. At early time, the water content was constant with time at each depth, although it differs among probes. The wetting front reached all of the TDR probes at approximately the same time. After this time, the water content increased to a maximum value and remained constant until infiltration ceased. Throughout the remaining measurement time, the water content decreased with time to a constant value. The two TDR probes located within the infiltration gallery, at depths of 1.6 m and greater, showed lower maximum water contents (0.28 cm3 cm-3) than the TDR probe buried just outside of the gallery at 1.2 m depth (0.37 cm3 cm-3).
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Fig. 5. Volumetric water content measured using buried time domain reflectometry (TDR) probes (lines), using direct borehole ground penetrating radar (BGPR) waves at 1.5 m depth (circles), and using the slope of the BGPR travel time profile at the ground surface (squares). TDR4 is located outside of the infiltration gallery at a depth of 1.2 m. TDR2 and TDR3 are located within the gallery at depths of 1.62 and 1.67 m, respectively.
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The water contents determined using BGPR direct arrivals at 1.5 m depth (black circles) and those determined from the slope of the travel time profile near the surface (gray boxes) are included in Fig. 5. The BGPR-determined water contents showed good agreement with the TDR water content time series, with both methods showing an increase in water content due to infiltration followed by a decrease after infiltration ceased. Specifically, the BGPR-measured water contents at 1.5 m depth were lower than those measured with BGPR in the upper 1 m. The difference between the BGPR-measured water contents at 1.5 m depth and the TDR-measured water contents measured at 1.6 m depth at the same time was no greater than the difference among the TDR measurements. The BGPR measurements in the upper meter showed similar agreement with the shallow TDR measurements. The good agreement at both depths is surprisingly good considering the differences in the locations of measurement and the different sample volumes of the two methods.
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CONCLUSIONS
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Zero-offset profiling borehole ground penetrating radar shows promise as a method for water content profiling throughout the vadose zone. However, the effects of first-arriving critically refracted waves must be considered when interpreting travel time profiles. Measurements made near the ground surface will be affected by first-arriving critically refracted waves (Hammon et al., 2002). The depth below which first arrivals can be associated with direct waves, referred to here as the refraction termination depth, depends on the volumetric water content of the shallow subsurface and the antennae separation. However, even under ideal conditions (very dry soil), this depth will be approximately 0.5 m for 100-MHz antennae. Given that critical refractions at the ground surface cannot be eliminated, we present a method whereby an estimate of the shallow water content can be obtained from the travel time profile. This method showed good agreement with independent measurements of water content collected near the ground surface during a controlled infiltration experiment. Critical refractions are not limited to the ground surface; they can occur whenever an EM wave contacts a boundary with a higher velocity medium (Bohidar and Hermance 2002). Therefore, the conclusions presented regarding the influence of first-arriving critical refractions from the ground surface may also have implications for measuring the water content across layer boundaries within the vadose zone or across the water table.
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ACKNOWLEDGMENTS
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This material is based on work supported by the National Science Foundation under Grant no. 0097171.
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REFERENCES
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TOP
ABSTRACT
INTRODUCTION
