Distinguishing Capillary Fringe Reflection in a GPR Profile for Precise Water Table Depth Estimation in a Boreal Podzolic Soil Field

Abstract

Relative permittivity and soil moisture are highly correlated; therefore, the top boundary of saturated soil gives strong reflections in ground-penetrating radar (GPR) profiles. Conventionally in shallow groundwater systems, the first dominant reflection comes from the capillary fringe, followed by the actual water table. The objective of this study was to calibrate and validate a site-specific relationship between GPR-estimated depth to the capillary fringe (D_CF) and measured water table depth (WTD_m). Common midpoint (CMP) GPR surveys were carried out in order to estimate the average radar velocity, and common offset (CO) surveys were carried out to map the water table variability in the 2017 and 2018 growing seasons. Also, GPR sampling volume geometry with radar velocities in different soil layers was considered to support the CMP estimations. The regression model (R² = 0.9778) between D_CF and WTD_m, developed for the site in 2017, was validated using data from 2018. A regression analysis between D_CF and WTD_m for the two growing seasons suggested an average capillary height of 0.741 m (R² = 0.911, n = 16), which is compatible with the existing literature under similar soil conditions. The described method should be further developed over several growing seasons to encompass wider water table variability.

1. Introduction

Knowledge of the water table depth (WTD) is essential for understanding many environmental scenarios, including water management in agriculture. WTD, where hydraulic pressure equals atmospheric pressure, demarcates the saturated–unsaturated soil boundary. Naturally, a capillary fringe, which is quasi–saturated but still has negative water pressure, occurs above the water table [1]. The capillary fringe, sometimes called the transition zone, in the vadose zone mediates space, water, and nutrients for plants and soil organisms [2,3]. Both WTD and the depth to the top of the capillary fringe (D_CF) are subject to seasonal fluctuations, and thus, can affect agricultural water management practices throughout the growing season. WTD can be measured through a borehole, but this is generally unfeasible at larger scales, since it provides point scale measurements and is destructive.

High-resolution subsurface images of ground penetrating radar (GPR) can be used to measure shallow WTD, especially in coarse-grain soils [4,5,6]. Various GPR field techniques for water table studies have been developed over the decades. Annan et al. [7] stated that it was essential to have a sharp boundary between saturated and unsaturated zones in a GPR profile for precise WTD estimation. Larger wavelengths (i.e., lower frequencies) are recommended, even though the resolution of the radar profile decreases with increasing wavelength [7]. Loeffler and Bano [8] also found that GPR frequencies higher than 900 MHz do not identify the top of the saturated zone (water table) due to the effect from the capillary fringe and the transition zone. Therefore, most WTD studies have been carried out using GPR frequencies lower than 250 MHz [9]. Nakashima et al. [10] and Takeshita et al. [11] used common midpoint (CMP) data acquisition of GPR to explain the multiple reflections from the water table. CMP data can be used to estimate the variation of relative permittivity (ε_r), and therefore, to calculate the radar wave velocity (v) along the soil profile [10,12]. However, the common offset (CO) survey method using 100 MHz is the most commonly used method in water table studies [5].

The GPR technique was employed during pumping tests to measure the temporal fluctuation of WTD [13,14,15], and has been used for various groundwater studies [16,17,18,19,20,21,22,23,24,25,26,27,28,29]. However, it is challenging to interpret a shallow water table depth associated with closely spaced soil horizons with only the GPR outputs [30]. Advances in sophisticated sensor technology such as real-time WTD and soil moisture data would help to improve the GPR outputs [4]. Still, site-specific GPR data validation is needed to distinguish the accurate water table reflection from the capillary fringe or the transition zone. The objective of this study was to calibrate and validate a site-specific relationship between GPR-estimated D_CF and measured WTD data in an agriculturally managed podzolic soil under boreal climate conditions. We carried out GPR surveys along a 42 m transect over two growing seasons through the use of a borehole with a real-time WTD sensor and soil moisture probes installed in shallow soil.

2. Materials and Methods

2.1. GPR Theory

In GPR, a transmitter antenna (Tx) transmits high frequency (10 to 1200 MHz), short pulses of electromagnetic (EM) energy into the subsurface, and a receiver antenna (Rx) captures the transmitted energy [31,32]. Transmitted EM energy (henceforthreferred to as GPR wave) can be reflected, refracted, or attenuated [33,34,35]. GPR wave propagation through the subsurface is highly sensitive to soil moisture [32,33,36,37,38,39,40,41]. The water table reflects more than 40% of GPR wave energy in coarse-grained soils [42]. Accordingly, the water table can give continuous and mostly flat reflections with high amplitude in GPR radargrams [30,43,44,45,46,47,48]. Based on this advantage, GPR is instrumental and an essential method in shallow groundwater studies [15,49]. Early researchers, for example Johnson [50], Livari and Doolittle [51], and van Overmeeren [30,52], reported the ability of the GPR method to detect water tables.

There are three characteristics to be considered when interpreting WTD from a GPR profile (Figure 1). Firstly, saturation decreases from the bottom to the top of the capillary fringe (within the transition zone), consequently increasing the radar wave velocity [53]. The thickness of this zone depends on the amount, size (diameter), and interconnectivity of soil pores [4,54]. Secondly, oscillations of the reflected radar pulse due to the transition zone result in a series of bands representing the water table in the radar profile [4]. The top of the capillary fringe gives an earlier and more robust reflection than the actual water table [55,56]. The wetting front also has the same phase reflection as the water table [57]. Thirdly, the two-way travel time (TWTT) correspondence with the maximum absolute amplitude of the airwave (t_air) is the opposite of that of a reflection event (t_reflect) [55] (Figure 1). In addition to the above characteristics, the maximum absolute amplitude occurs at the second half of the respective GPR wavelet [58].

Equation (1) gives the radar signal velocity (v) of a nonmagnetic and low-loss geological material:

$$v=\frac{c}{\sqrt{{\mathsf{\epsilon }}_{\mathrm{r}}}}$$

(1)

where ε_r is the relative permittivity and c is the electromagnetic wave propagation velocity in free space (=0.3 m/ns) [25,31,59]. In addition, the depth to a known reflector method [34] can be used to calculate the GPR reflection wave velocity (v_rw) of a monostatic antenna using Equation (2):

$$v_{rw}=\frac{2D}{t}$$

(2)

where t is the TWTT of the reflected GPR wave from a reflection boundary and D is the depth to the boundary (ASTM D6432-11). ε_r of the soil just below the ground surface can be calculated using the GPR direct groundwave method, using Equation (3) [32]:

$${\mathsf{\epsilon }}_{\mathrm{r}}={\left[\frac{c\left(t_{GW}-t_{AW}\right)+x}{x}\right]}^2$$

(3)

where t_GW and t_AW are the direct groundwave and airwave arrival times, respectively, from the Tx to the Rx antenna, and x is the antenna separation. ε_r can be derived from the measured volumetric soil moisture content, θ_v, using an empirical model. Topp et al. [60] suggested the first model as given in Equation (4) [61].

$${\mathsf{\epsilon }}_{\mathrm{r}}=3.03+9.3{\theta }_v+146.0{\theta }_v^2-76.7{\theta }_v^3$$

(4)

2.2. Study Area

The experimental site included a silage-cornfield and a grass field at the Pynn’s Brook Research Station (PBRS), managed by the Department of Fisheries and Land Resources of the Government of Newfoundland and Labrador, located in Pasadena (49.073 N, 57.561 W), Newfoundland and Labrador (NL), Canada. The area is characterized by a 2–5% slope, and the depth to the bedrock is >1 m from the surface [62]. Details of the observed shallow soil profile are given in

Table 1. Details of different soil layers.

Soil Layer	Depth Range (m)	Layer Thickness (m)	Relative Permittivity
Topsoil	0–0.05	d1 = 0.05	εr1
Loamy sand	0.05–0.35	d2 = 0.30	εr2
Sand (unsaturated)	0.35—top of the capillary fringe	d3 = WTDm 1 − CF 2 − 0.35	εr3

¹ Measured water table depth; ² Height of the capillary fringe.

. The topsoil (ε_r1, d₁) is an organic soil layer with gravel. Immediately below the top layer is the Ap horizon (ε_r2, d₂), classified as loamy sand (sand = 82.0 ± 3.4%; silt = 11.6 ± 2.4%; clay = 6.4 ± 1.2%) [63]. The average bulk density and porosity of the loamy sand layer (n = 28) are 1.31 g cm⁻³ (±0.07) and 51% (±0.03), respectively [63]. A well-sorted sandy soil layer (ε_r3, d₃) was observed between the depths of 0.35 and 3.47 m, which is likely to continue further down and act as a shallow aquifer. The average annual precipitation at the site is 1113 mm per year, with 410 mm falling as snow, and the annual mean temperature is 4 °C. Both parameters are as recorded at the nearest weather station, Deer Lake, NL, for last 30 years [64].

2.3. Data Collection and Basic Proccessing

The following materials and instruments were used for data acquisition, processing, and interpretation in this study:

• PulseEKKO^® Pro GPR system (Sensors and Software Inc., Mississauga, ON, Canada) with 100 and 250 MHz center frequency antennas
• Em50 data logger, and water level, electrical conductivity-, temperature- and SM-probes (METER group Inc. (former Decagon Devices), Pullman, DC, USA)
• EKKO Project V3 R1 and IcePicker V3 R7 GPR data processing Software (Sensors and Software Inc., Mississauga, ON, Canada)

The experimental site was ~50 m wide and 200 m long. The main GPR survey line of 42 m (along the width of the field) was marked between the silage-cornfield and the grass field using wooden pegs (Figure 2a). A shallow groundwater-monitoring borehole (3.47 m deep) was drilled at 19 m along the main GPR survey line. The perpendicular distance between the borehole and the survey line was 0.5 m (Figure 2b). A water level, electrical conductivity, and temperature sensor, connected to a data logger (Em50—Meter Group Inc. (former Decagon Devices), Pullman, DC, USA), was installed at the bottom of the borehole. The water level sensor measured the height of the water column in the borehole. Three soil moisture probes of 5 cm in length (ECH₂O EC-5 of Meter Group Inc. (former Decagon Devices), Pullman, DC, USA) were installed horizontally at depths of 0.1 m, 0.2 m, and 0.3 m from the soil surface and connected to the same data logger (Figure 3a). An additional temperature sensor was installed with the soil moisture probe at a depth of 0.2 m. GPR surveys were carried out between the soil moisture probes and the borehole (Figure 3b). The soil moisture probes were oriented perpendicular to the GPR survey direction.

Background GPR surveys were carried out: (i) before construction of the borehole, (ii) after the construction of the borehole, but before installation of the water level sensor, and (iii) after installation of the water level sensor. Sixteen 250 MHz GPR CO surveys (42 m in length, antenna separation = 0.38 m, sampling interval = 0.05 m, temporal sampling interval = 200 ps) and 16 CMP surveys (near the borehole) were performed in 2017 and 2018. Three 100 MHz GPR CO surveys (~30 m in length, antenna separation = 1.0 m, sampling interval = 0.25 m, temporal sampling interval = 800 ps) were also conducted under wet, median, and dry conditions in 2018 along the same GPR line.

Three basic GPR data processing steps were applied using the EKKO Project V3 R1 Software (Sensors and Software Inc., Mississauga, ON, Canada), as listed below.

• Edit the first break (time-zero correction)
• Apply dewow and SEC2 (Spreading and Exponential Compensation) gain
• Background subtraction—applied to the full length of the trace

The dataset did not require extra processing such as bandpass filters, and the processing was intentionally simplified to be applicable for every GPR profile in the dataset due to the low noise level observed. After completing this basic processing, GPR files were exported to the IcePicker V3 R7 software (Sensors and Software Inc., Mississauga, ON, Canada) for automatic time picking.

2.4. Data Analysis

CMP surveys were carried out using the borehole as the common midpoint at every field campaign to calculate the average v_rw from the surface down to the top of the capillary fringe. GPR CO survey traces nearest to the borehole were only considered for the D_CF estimation. Accordingly, the mean TWTT to the capillary fringe reflection (mean t_CF) was obtained from twelve GPR traces. The subjective error was high when picking the leading edge of the wavelet [39]. Therefore, TWTT related to the absolute maximum amplitude of the airwave (t_air) and the reflection event (t_reflect) were picked. This procedure was similar to the direct groundwave analysis by Grote et al. [65]. The actual t_CF of every single trace was determined using Equation (5).

$${\mathrm{t}}_{\mathrm{C}\mathrm{F}}={\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}}-{\mathrm{t}}_{\mathrm{a}\mathrm{i}\mathrm{r}}$$

(5)

D_CF was then calculated from the mean t_CF and the average v_rw using Equation (2). D_CF of eight GPR surveys in 2017 were plotted in a linear regression model against the measured WTD (WTD_m) using the water level sensor at the same time as the GPR surveys. The regression equation obtained from 2017 data was used to predict WTD (WTD_p) for eight survey days in 2018. The WTD_p and WTD_m for 2018 were compared using a 1:1 plot and root mean square error (RMSE). In a second step, WTD_m and D_CF for all 16-survey days were plotted in a linear regression plot to estimate the average capillary height. The slope and the intercept of the regression line were compared statistically with those of the 1:1 line. In the same manner, the prediction line was statistically compared with the 1:1 line.

A Pearson’s correlation test was performed at α = 0.05 (95% confidence level) for 33 variables using 12 out of 16 survey days (n = 12) to evaluate the results. The variables used for the correlation analysis were WTD, groundwater temperature (GW–Temp), electrical conductivity of groundwater (GW–EC), soil moisture at 30 cm depth (SM30), 20 cm depth (SM20), and 10 cm depth (SM10), soil temperature at 20 cm depth (Temp20), daily precipitation (Daily P), daily evapotranspiration (Daily E), daily P minus daily E (P–E 1), and cumulative daily P minus daily E (P–E 2 to P–E 20 from 2 days to 20 days cumulative, respectively), ε_r−avg, ε_r1, ε_r2, and ε_r3 [66].

3. Results and Discussion

3.1. GPR Survey Outputs

GPR profiles from 250 MHz provided high-resolution radar images; therefore, multiple reflections near the water table can be clearly observed (Figure 4). The elevations have not been corrected; therefore, the reflections in Figure 4 can be observed as undulating boundaries. The GPR velocity is independent of the frequency (for frequencies above 100 MHz to 1500 MHz) and dependent only on the relative permittivity and the magnetic permeability [67]. Thus, the same velocity derived from 250 MHz could be used to analyze 100 MHz data on the same day; 100 MHz gives low-resolution radar images with a relatively flat and clear boundary for the water table zone, and with fewer multiple reflections (Figure 5a). The results from this study indicate that 250 MHz is suitable to examine shallow D_CF, whereas 100 MHz is more suitable to examine deeper WTD.

3.2. Estimation of D_CF

The D_CF derived using Equation (2), and CMP derived radar velocities for all 16 GPR survey days, are given in

Table 2. Ground penetrating radar (GPR) estimated depth to the top of the capillary fringe (D_CF) which is derived from the mean two-way travel time (TWTT) to the capillary fringe reflection (t_CF) and common mid point (CMP) derived radar wave velocity (v_rw) using Equation (2) for all GPR surveys. Standard error (SE) of mean, and minimum (Min), median, and maximum (Max) of t_CF time picks are also given.

Date	tCF (ns)					CMP Derived vrw (m/ns)	DCF (m) (= vrw × tCF/2)
	Min	Median	Max	Mean	SE
23 June 2017	32.65	35.08	38.04	35.07 ± 1.70	0.43	0.091	1.60
6 July 2017	27.97	28.51	34.80	29.21 ± 1.93	0.58	0.112	1.64
28 July 2017	27.71	28.82	39.78	32.40 ± 5.39	1.44	0.117	1.90
18 August 2017	32.11	37.81	39.67	36.43 ± 2.87	0.70	0.117	2.13
29 August 2017	30.21	32.29	39.16	33.59 ± 2.90	0.70	0.130	2.18
15 September 2017	27.70	35.23	38.83	34.31 ± 3.86	1.11	0.125	2.14
3 October 2017	32.80	36.26	42.15	36.84 ± 3.69	1.02	0.107	1.97
7 November 2017	33.09	39.45	44.36	40.23 ± 4.53	1.26	0.090	1.81
1 June 2018	35.64	39.35	40.59	38.08 ± 2.06	0.47	0.082	1.56
20 June 2018	30.18	31.38	40.54	33.35 ± 3.82	0.83	0.100	1.67
29 June 2018	30.07	30.73	34.40	31.18 ± 1.35	0.33	0.103	1.61
20 July 2018	28.07	35.20	37.56	32.89 ± 3.33	0.77	0.113	1.86
9 August 2018	28.15	28.80	32.69	30.25 ± 2.01	0.44	0.125	1.89
7 September 2018	27.78	29.29	36.47	30.05 ± 2.67	0.67	0.129	1.94
2 October 2018	34.33	35.42	41.31	36.93 ± 2.65	0.61	0.091	1.68
31 October 2018	27.93	31.27	35.56	31.64 ± 2.78	0.61	0.074	1.17

. Two examples of semblance analyses for CMP velocity estimation are illustrated in Figure 6. The fluctuation of WTD_m related to GPR survey days (n = 16) varied from 1.85 m to 2.91 m. For the entire study period covering growing seasons in 2017 and 2018 (496 days), the WTD_m varied between 1.58–2.95 m. The shallowest WTD_m of 1.58 m was observed in the spring of 2018 (30 April 2018), while the deepest WTD_m of 2.95 m was found in the early fall of 2017 (10–12 September 2017). Throughout the studied period, the average WTD_m was 2.48 m, with an annual average of 2.69 m in 2017 and 2.34 m in 2018 during the growing season (from May to the end of October). These data imply that the 2017 growing season was relatively dry, which was also confirmed by weather data collected onsite (Appendix A).

3.3. Site-Specific Relationship for WTD_m vs. D_CF

The WTD_m at the same time as the GPR survey and corresponding D_CF obtained using GPR data are plotted in Figure 7a. A strong linear regression with an R² of 0.98 was found (Equation (6)) between WTD_m and D_CF for GPR surveys in 2017:

$${\mathrm{W}\mathrm{T}\mathrm{D}}_{\mathrm{m}}=0.6956{\mathrm{D}}_{\mathrm{C}\mathrm{F}}+1.3884$$

(6)

The WTD_p for 2018 based on the GPR measured D_CF and the regression model (Equation (6)) was plotted against the WTD_m in a 1:1 plot (Figure 7b). The slope of the prediction line (1.6) and that of the 1:1 line (1.0) are significantly different at α = 0.05 (p-value = 0.004, df = 12, t_critical = 2.179 < t_calculated = 3.536) (Appendix B,

Table A1. Comparison of regression line and 1:1 line.

	Regression	1:1 Line	sb1-b2	0.085
n	16	16	t	0.146
b	1.012	1.000	df	28
sy.x	0.086	0.000	α	0.050
sx	0.264	0.264	p-value	0.885
sb	0.085	0.000	t-critical	2.048
			significant	No

Since t < t-critical and p-value > α, two slopes are not significantly different at α = 0.05.

). The error of WTD prediction was high during the wet survey days and overestimated from the 1:1 line (Figure 7b). This behaviour could be due to the fact that the capillary fringe would not fluctuate uniformly with the WTD fluctuation. As Bentley and Trenholm [55] stated, the capillary height is greater when the WTD is increasing (during discharging) and lower when it is decreasing (during recharging). However, the present study determines the capillary height based on the WTD_m through a regression equation (Equation (6)). The regression equation is a generalized form of all measured data throughout a growing season. Therefore, the regression model might not be suitable when there is a sudden decrease in WTD, like during heavy or long duration rain events. The rain event at the end of the growing season of 2018 was unexpectedly heavy (Appendix A); consequently, it produced a maximum error in the WTD prediction on 31 October 2018 (Figure 8). However, it is worth noting that the RMSE of 0.194 m between WTD_m and WTD_p can be considered acceptable for the scale of application in most agricultural practices.

In general, the spatial variation of D_CF cannot be measured directly under heterogeneous field conditions [68]. The proposed method provides a noninvasive approach to estimate D_CF, which is more beneficial in agricultural fields during the growing season since the capillary zone provides the best soil moisture conditions for plants. The advantage of the proposed method is that both WTD and D_CF can be estimated in real time. The results would have been improved if a broader range of measured data were available under different soil moisture and variable water table conditions.

As seen in Figure 9, WTD_m and D_CF for all survey days have a linear relationship (WTD_m = 1.0123 D_CF + 0.741) with an R² of 0.91. The slopes of the regression line (1.01) and the slope of the 1:1 line (1.00) are not significantly different at α = 0.05 (p-value = 0.89, df = 28, t_critical = 2.048 > t_calculated = 0.146) (Appendix B,

Table A2. Comparison of prediction line and 1:1 line.

	Regression	1:1 Line	sb1-b2	0.169
n	8	8	t	3.536
b	1.597	1.000	df	12
sy.x	0.086	0.000	α	0.050
sx	0.171	0.171	p-value	0.004
sb	0.169	0.000	t-critical	2.179
			significant	Yes

Since t > t-critical and p-value < α, two slopes are significantly different at α = 0.05.

). Therefore, the intercept of the regression line (0.74 m, n = 16) can be considered as the average capillary height within the growing seasons in 2017 and 2018. As a result, an average capillary height of 0.74 m was suggested for the particular site throughout the growing season. The average value agrees with the value of ~0.70 m capillary height for similar soil conditions described by Liu et al. [69]. However, in contrast, Saintenoy and Hopmans argued that D_CF cannot be estimated directly because the maximum reflection is not related to the capillary fringe, but rather, to the inflection point of the transition curve [70]. It is worth noting that in the present study, we believe that transition zone reflections have no effect due to their relatively larger wavelength (i.e., low frequency) when compared to the transition zone thickness, as described by Bano [9].

3.4. Evaluation of Results

3.4.1. Calculation of ε_r

There are three soil layers in the experimental plot, as described in Section 2.2. The first layer occurs at 0–0.05 m soil depth (d₁ = 0.05 m, ε_r1), the second layer between 0.05–0.35 m (d₂ = 0.30 m, ε_r2), and the third from 0.35 m to the top of the capillary fringe (d₃, ε_r3). The average ε_r from the surface to the top of the capillary fringe (ε_r−avg) can be obtained from Equation (1), because v_rw is known from the CMP surveys. It is worth noting that CMP analysis might be suitable for layer velocity estimation as well; however, in the present study, the resolution was not sufficient to extract the layer velocities separately, due to low soil layer thickness. Instead, ε_r1 can be calculated using the GPR direct groundwave arrival time (Equation (3)), while ε_r2 can be calculated using soil moisture data (Equation (4)). Once ε_r−avg, ε_r1, and ε_r2 are known, ε_r3, which has the key contribution to the ε_r−avg, can be calculated.

The calculated ε_r3 was compared with the literature and onsite weather data to evaluate the results of this study. The average WTD_m in all GPR survey days (n = 16) was 2.55 m. When the average capillary height was approximated as 0.70 m (according to [69] and the present study), the average d₃ was calculated as 1.50 m (2.55 − 0.70 − (0.05 + 0.30)). Figure 10 illustrates the overlapping of these three soil layers with the sampling volume geometry of the GPR CO survey, as suggested by Illawathure et al. [71]. Twelve GPR traces collected near the borehole and the soil moisture probes (i.e., the same as in the calculation of mean t_CF) were considered for the illustration and calculations below.

The soil moisture data logging interval was 60 min. Therefore, each daily mean soil moisture data point had 24 replicated measurements. Each soil moisture probe had a cylindrical sampling volume (radius ~ 0.05 m, volume = 0.715 L) that covered 0.05 m soil heights, both above and below the respective probe (Figure 3b) [72,73]. Daily mean soil moisture values at three depths were converted to daily mean ε_r values using Equation (4); an average ε_r for the soil layer between 0.05–0.35 m (ε_r2) was obtained from those ε_r values.

If the weighted average of ε_r1, ε_r2, and ε_r3 was considered as ε_r−avg, the percentage sample area of each layer should be calculated with respect to the total GPR sample area (Figure 10 and

Table 3. Percentage sample area of each soil layer out of the total ground penetrating radar (GPR) sample area related to twelve GPR traces collected from the top of the soil moisture probes on the vertical plane of the GPR survey (refer to Figure 10).

Soil Layer	Polygon (Refer to Figure 9)	Area (m2)	Percentage Out of Total GPR Sample Area (%)
Topsoil	pqsr	0.0462	3.4
Loamy sand	rsut	0.2667	19.5
Sand (unsaturated)	tuwv	1.0561	77.1
Total	pqwv	1.3690	100.0

). If w is the weight and x the data number, then the weighted average ($\overline{x}$) equals:

$$\overline{x}=\frac{{{\displaystyle \sum }}_{i=1}^n{w}_i\times x_i}{{{\displaystyle \sum }}_{i=1}^n{w}_i}$$

(7)

e.g., estimation of the ε_r3 on 6/23/2017 (first data row in

Table 4. Assumed depth of unsaturated sandy layer (d₃), and calculated relative permittivities.

Date	WTD (m)	d3 (= WTD − 0.7 − d1 − d2) (m)	εr−avg	εr1	εr2	εr3
23 June 2017	2.47	1.42	10.9	23.0	10.5	10.5
6 July 2017	2.55	1.50	7.1	4.0	12.2	6.0
28 July 2017	2.74	1.69	6.5	13.2	7.5	6.1
18 August 2017	2.85	1.80	6.5	10.9	7.8	6.0
29 August 2017	2.90	1.85	5.3	19.5	8.1	4.0
15 September 2017	2.91	1.86	5.7	14.0	11.2	4.2
3 October 2017	2.77	1.72	7.9	6.2	11.6	7.0
7 November 2017	2.63	1.58	11.2	13.1	13.7	10.5
1 June 2018	2.24	1.19	13.3	15.2	12.2	13.5
20 June 2018	2.33	1.28	9.1	14.0	12.5	8.0
29 June 2018	2.31	1.26	8.4	19.5	12.1	7.0
20 July 2018	2.54	1.49	7.0	20.0	8.6	6.3
9 August 2018	2.61	1.56	5.8	6.5	12.8	4.0
7 September 2018	2.75	1.70	5.4	4.0	12.8	4.1
2 October 2018	2.56	1.51	10.8	8.0	12.4	10.5
31 October 2018	1.86	0.81	16.5	23.3	13.1	17.0

)

$$\begin{array}{c}{\mathsf{\epsilon }}_{\mathrm{r}-\mathrm{avg}}=\frac{\left(3.4\%\times {\mathsf{\epsilon }}_{{\mathrm{r}}_1}\right)+\left(19.5\%\times {\mathsf{\epsilon }}_{{\mathrm{r}}_2}\right)+\left(77.1\%\times {\mathsf{\epsilon }}_{{\mathrm{r}}_3}\right)}{100\%}\\ {\mathsf{\epsilon }}_{{\mathrm{r}}_3}=\frac{{\mathsf{\epsilon }}_{\mathrm{r}-\mathrm{avg}}-\left(0.034{\mathsf{\epsilon }}_{{\mathrm{r}}_1}+0.195{\mathsf{\epsilon }}_{{\mathrm{r}}_2}\right)}{0.771}\\ {\mathsf{\epsilon }}_{{\mathrm{r}}_3}=\frac{10.9-\left(0.034\times 23.0+0.195\times 10.5\right)}{0.771}\\ {\mathsf{\epsilon }}_{{\mathrm{r}}_3}=10.5\end{array}$$

3.4.2. Correlation Analysis

The results of the Pearson’s correlation test are shown in Figure 11. The dataset and the detailed correlation analysis are given in Appendix C (

Table A3. Dataset for correlation analysis.

Date				WTD	GW-Temp (°C)	GW-EC (mS/cm)	SM30 (m3/m3)	SM20 (m3/m3)	Temp 20 (°C)	SM10 (m3/m3)	Daily P (mm)	Daily E (mm)
28 July 2017				2.74	7.4	0.278	0.1972	0.1449	21.0	0.1029	0.3	2.3
18 August 2017				2.85	8.4	0.281	0.2153	0.1424	17.5	0.0996	0.8	3.1
29 August 2017				2.89	8.8	0.280	0.2294	0.1394	16.5	0.0993	0.0	3.7
15 September 2017				2.91	9.2	0.290	0.2822	0.2162	14.0	0.1384	0.0	3.1
03 October 2017				2.77	9.4	0.287	0.2808	0.2344	9.4	0.1407	0.0	2.3
07 November 2017				2.63	9.2	0.301	0.2966	0.2803	8.0	0.1662	15.2	0.4
01 June 2018				2.24	4.7	0.270	0.2890	0.2469	11.8	0.1501	0.0	3.9
20 July 2018				2.54	6.4	0.259	0.2260	0.1668	20.8	0.1138	0.0	5.7
09 August 2018				2.65	8.8	0.252	0.2923	0.2606	20.7	0.1627	13.7	3.6
07 September 2018				2.75	9.1	0.266	0.2932	0.2605	16.1	0.1598	0.0	2.6
03 October 2018				2.54	9.6	0.267	0.2850	0.2534	8.5	0.1480	0.0	1.0
31 October 2018				1.85	9.3	0.264	0.2960	0.2756	6.8	0.1560	0.3	0.0
Date	P-E 1	P-E 2	P-E 3	P-E 4	P-E 5	P-E 6	P-E 7	P-E 8	P-E 9	P-E 10	P-E 11	P-E 12
28 July 2017	−2.1	−6.8	−12.0	−16.3	−20.5	−25.5	−28.9	−32.0	−31.6	−36.0	−41.5	−46.6
18 August 2017	−2.4	−4.1	−5.4	−10.0	−13.6	−8.7	−5.9	−10.7	−15.5	−19.3	−22.7	−6.0
29 August 2017	−3.7	−7.6	−11.1	3.7	0.3	1.4	3.2	1.9	0.7	2.1	−1.5	−3.9
15 September 2017	−3.1	−5.8	8.1	24.2	23.1	21.0	18.4	19.8	27.0	24.9	21.6	25.1
03 October 2017	−2.3	−4.1	−6.4	−7.8	−8.1	2.9	8.9	33.9	32.6	31.6	31.3	29.0
07 November 2017	14.9	22.2	20.9	29.7	30.6	29.4	27.8	27.8	27.1	26.2	27.3	31.3
01 June 2018	−3.9	−7.8	−10.1	7.1	3.4	0.8	−0.5	−0.8	1.2	7.3	5.5	2.8
20 July 2018	−5.7	−9.1	−10.5	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7
09 August 2018	10.2	25.6	25.1	22.5	23.1	20.6	17.1	16.8	11.7	6.8	5.5	42.1
07 September 2018	−2.6	−2.6	−2.6	−2.6	−2.6	−2.6	−2.6	−2.6	−2.6	−5.2	−8.3	−11.7
03 October 2018	−1.0	−2.7	−3.3	−5.2	−2.0	15.7	25.5	24.7	22.6	20.9	21.6	27.7
31 October 2018	0.2	−0.3	23.7	42.0	41.5	41.0	48.9	51.9	51.2	50.9	79.0	89.3
Date	P-E 13	P-E 14	P-E 15	P-E 16	P-E 17	P-E 18	P-E 19	P-E 20	εr−avg	εr1	εr2	εr3
28 July 2017	−51.8	−56.9	−62.3	−67.9	−71.2	−75.5	−80.9	−83.9	6.5	13	7.5	6.1
18 August 2017	−9.2	−13.8	−18.0	−22.4	−24.2	−8.4	−12.1	−-15.0	6.5	11	7.8	6
29 August 2017	−5.6	−7.0	−11.5	−15.1	−10.2	−7.4	−12.3	−17.1	5.3	20	8.1	4
15 September 2017	21.7	20.6	24.4	22.9	19.1	15.4	11.5	8.0	5.7	14	11	4.2
03 October 2017	26.2	40.2	41.0	38.3	36.6	40.5	37.3	34.7	7.9	6.2	12	7
07 November 2017	29.9	28.4	26.8	25.5	24.7	26.5	29.9	28.5	11.2	13	14	11
01 June 2018	5.8	2.6	0.1	0.3	−0.6	2.7	−0.5	−3.1	13.3	15	12	14
20 July 2018	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7	−14.7	7	20	8.6	6.3
09 August 2018	38.7	38.5	34.8	33.3	34.3	42.4	41.7	36.9	5.8	6.5	13	4
07 September 2018	−15.0	−18.6	−22.6	−24.4	−26.8	−30.6	−33.9	−37.9	5.4	4	13	4.1
03 October 2018	26.0	25.4	23.7	36.9	48.6	49.0	45.8	42.8	10.8	8	12	11
31 October 2018	91.0	99.6	99.6	114.3	115.3	116.7	116.8	116.9	16.5	23	13	17

and Figure A3, respectively). The present results reveal that most of the variables tested were not correlated with WTD (also see Figure A1). However, the WTD had strong negative correlations with ε_r−avg (r = −0.9041, p = 0.0001) and ε_r3 (r = −0.9019, p = 0.0001). Dry sand had lower ε_r than wet sand [74,75]. According to Equations (1) and (2), ε_r and WTD are inversely proportional, based on the fact that water table recharging makes unsaturated soil wetter, and water table discharging makes unsaturated soil drier [73]. It was also clear that ε_r−avg and ε_r3 were strongly positively correlated (r = 0.9961, p = 0.0000), as ε_r3 was dominant in the ε_r−avg calculation (Figure 10 and Table 3). WTD had negative (but moderate) correlations with P–E 11 through P–E 20, though the p-value of WTD vs. P–E 12 was slightly above the α-level (p = 0.0534). The highest correlation among P–E vs. WTD was for P–E 16 (r = −0.6285, p= 0.0286). This suggested that the WTD does not quickly respond to variables like daily P, daily E, or daily P–E, and that there is a long time-lag between WTD response and P–E timing (Figure A2). This could be due to high runoff, relatively thicker aquifer, etc. [76,77]. Since it is a sandy aquifer, low infiltration was not a factor in the long responsive time of the WTD to the P–E [78,79].

GW–EC or GW–Temp were not correlated with any of other variables tested. The Temp20 had a strong negative correlation with P–E 8, P–E 9, P–E 10, and P–E 11, with the maximum at P–E 10 (r = −0.8478, p = 0.0005). Further, Temp20 had moderate correlations with SM20 and SM30 as well. To support these observations, the collected TDR data were examined. First, the SM10, SM20, and SM30 variables had strong positive correlations with each other (r > 0.95, p = 0.00), since all three values mostly represented the same soil layer. Next, SM10 had no correlation with P, E, or P–E, while both SM20 and SM30 had no correlation with P, E, P–E, P–E 1, or P–E 2. The highest correlations between SM and cumulative P–E were as follows: between SM10 and P–E 6 (r = 0.7121, p = 0.0094), between SM20 and P–E 6 (r = 0.7486, p = 0.0051), and between SM30 and P–E 10 (r = 0.7920, p = 0.0021). From SM10 to SM30, the time lag of P–E increased, as did the strength of the correlation. These results imply that cumulative P–E affected the soil moisture as a function of depth and time. Cumulative P–E also significantly affected the soil temperature at 20 cm, but not the temperature of the groundwater. The same idea was reflected from the correlation analysis between ε_r and other variables as well [80,81].

ε_r1 had no correlation with any of the variables tested. It is worth noting that the depth of this first soil layer was only 5 cm. It is obvious that ε_r2 had a strong positive correlation with SM10, SM20, and SM30 (r > 0.98, p = 0.00), since ε_r2 was calculated from TDR data. Also, ε_r2 had a negative but moderate correlation with Temp20 (r = −0.6551, p = 0.0208). There are positive moderate correlations between ε_r2 and from P–E 3 to P–E 20. ε_r3 also had a negative but moderate correlation with Temp20 (r = −0.7226, p = 0.0079), and a moderate positive correlation with P–E 11 (r = 0.6516, p = 0.0217) and beyond, except P–E 12 and P–E 15. This was because part of the cumulative P–E infiltrates through the soil, increasing soil moisture and lowering soil temperature [81,82]. The soil moisture governs the dielectric properties of the soil; therefore, the correlation analysis suggested that the behavior of the estimated and calculated data was supported by the measured and onsite weather data.

4. Summary

In GPR, the Rx records different amplitudes of the receiving signals with respect to wave travel time. The GPR interpreter observes the reflected or direct radar waves at Rx in a radargram and obtains relevant TWTTs. Without knowing the v_rw, it is impossible to derive D_CF (e.g., using Equation (2)). Under these circumstances, there are two challenges to estimating the D_CF in a GPR radargram: First, picking the TWTT of the capillary fringe reflection correctly; and second, knowing or properly assuming the average v_rw from the surface down to the top of the capillary fringe reflection.

The procedures described in the literature were combined to overcome the challenges of picking the correct TWTT. The standard error of the mean t_CF was > 1 except in 4 cases in 2017: 1.44 ns (28 July), 1.26 ns (7 November), 1.11 ns (15 September), and 1.02 ns (3 October). The standard deviation (SD) of the mean t_CF was above 3 ns for those four survey days. The error associated with the surveys was mostly due to a high signal-to-noise ratio in the data acquisition. The lowest SDs of the mean t_CF were 1.70 ns and 1.35 ns for 23 June 2017 and 29 June 2018, respectively. The inconsistency of the GPR reflection amplitude under wet conditions, as observed by Lunt et al. [21], could be a reason for the error of time picking observed in our data set.

The challenge of defining v_rw during GPR data analysis is associated with determining the average ε_r of the material above the capillary fringe. ε_r controls the v_rw and the reflection coefficients at the interfaces. The common v_rw values suggested in the literature may not be accurate for heterogeneous soil profiles in highly variable field conditions under different agricultural practices. In addition, seasonal fluctuation of WTD and capillary height can significantly change the average ε_r. However, v_rw can be measured by using multi-offset GPR survey methods such as CMP and WARR (wide-angle reflection and refraction), even though it is time- and labor- consuming to carry out multi-offset surveys in every field campaign [5,32]. In the present study, three different ε_r values were assumed for different soil layers above the capillary fringe. A weighted average method using CMP data, ground wave analysis, and TDR data based on the GPR sampling geometry was used to calculate ε_r of unsaturated soil layers in order to evaluate the velocity estimations.

These types approaches might help in addressing natural barriers of inherent soil properties and the effect of their variability on agriculture. Maintaining the required soil moisture in the root zone will lead to optimum crop yields, and implementing the best possible water management strategies will reduce groundwater pollution threats due to intensive agriculture. Nevertheless, the average capillary height considered (0.70 m) based on the study of Liu et al. [69] to estimate the ε_r3 is closer to the average capillary height obtained from this study (0.74 m). It should be mentioned that taking an average capillary height would be reasonable under static conditions, but this would not always be suitable if the seasonal fluctuation of WTD is significant during the study period.