Evaluation of Influence Factors

The wave phenomena in a TDR measurement include multiple reflections, dielectric dispersion, and attenuation due to conductive loss and cable resistance. A comprehensive TDR wave propagation model that accounts for all wave phenomena has been proposed and validated by Lin and Tang (2007). With the proved capability to accurately simulate TDR

effects of dielectric dispersion, electrical conductivity, and cable length on apparent dielectric constant and effective frequency. Synthetic TDR measurements (waveforms) were generated by varying the influential factors in a controlled fashion. The associated apparent dielectric constants and effective frequencies were calculated and compared.

The synthetic TDR measurement system is composed of a TDR device, a RG-58 lead cable, and a sensing waveguide. Possible mismatches due to connectors and probe head are neglected since the simplification will not affect the apparent dielectric constant. Tap water and a silt loam modeled by the Cole-Cole equation were used as the basic materials. It is understood that the Cole-Cole equation may not be perfect for modeling dielectric dispersion of soils, but it is used to parameterize the dielectric dispersion for the parametric studies. The transmission line parameters and dielectric properties used in the parametric study are listed in Table 3-1 and Table 3-2, respectively. Time interval dt = 2.69×10^-11 sec and time window 0.5N dt = 8192×40 dt = 8.8×10^-6 sec (slightly greater than the pulse length of 7×10^-6 sec in a TDR 100) were used in the numerical simulations. The corresponding Nyquist frequency and frequency resolution are 18 GHz and 60 kHz, respectively. The Nyquist frequency is well above the frequency bandwidth of TDR 100 and the long time window ensures that the steady state is obtained before onset of the next step pulse.

Table 3-1 TDR system parameters

Section Parameters Reference value Range

EC σ, S m^-1 0.01 0.005 ~ 0.1

Dielectric permittivityε_r Tap water, and Silt loam^†

Dielectric permittivity ε_r 1.95 1.95

Geometric impedance Zp , Ω 77.5 77.5

† Referring to the Cole-Cole parameters listed in Table 3-2

Table 3-2 Cole-Cole parameters for material used in numerical simulations (modified from Friel and Or, 1999)

Material ε_dc ε_∞ f _rel ξ

Silt loam 26.0 18.0 0.2e9 0.01

Tap water 78.54^† 4.22 17e9 0.0125

† Water temperature = 25^oC

Effect of electrical conductivity

The electrical conductivity is well known for having a smoothing effect on the reflected waveform and hence affecting the Ka determination. However, the degree of influence may depend on dielectric dispersion and the method of travel time analysis. Varying the value of electrical conductivity in water (as a non-dispersive case) and silt loam (as dispersive case), Fig.

3-1 shows the effects of electrical conductivity on Ka for different methods of travel time

electrical conductivity. Both the dual tangent method and derivative method are unexpectedly immune to changing electrical conductivity (see Fig. 3-1(a)). As the medium becomes dielectric dispersive within TDR bandwidth, the apparent dielectric constant becomes sensitive to changing electrical conductivity (see Fig. 3-1(b)). Among all methods, the dual tangent method is least affected by electrical conductivity. When EC is greater than 0.05 Sm^-1, the dual tangent method and derivative method suddenly obtains higher apparent dielectric constants as EC increases. The Ka may even become greater than DC electric permittivity due to significant contribution of EC at lower frequencies.

For each simulated waveform, the equivalent frequencies of different travel time analysis methods and the frequency bandwidth of the end reflection can be determined by Eq. [3-1] and Eq. [3-2], respectively. The equivalent frequencies and frequency band width associated with Fig. 3-1(b) (the dispersive case) is shown in Fig. 3-2. Only the dispersive case is shown since the equivalent frequencies in non-dispersive case is not meaningful. Against common perception, the frequency bandwidth is not significantly affected by electrical conductivity. The end reflection may appear smooth due do decreased reflection magnitude as electrical conductivity increases. The 10% to 90% rise time and hence the frequency bandwidth remains relatively constant. The equivalent frequencies decrease with increasing electrical conductivity as expected. The dual tangent method leads to the highest equivalent frequency while the derivative method, as also pointed out by Robinson et al. (2003), results in the lowest equivalent frequency, which is closer to the frequency bandwidth. The dual tangent is advantageous in this regard since, at higher frequency, the apparent dielectric permittivity is less affected by changing electrical conductivity. Unfortunately, its automation of data reduction is also most difficult.

0 0.02 0.04 0.06 0.08 0.1 0.12 75

76 77 78 79 80

K a

0 0.02 0.04 0.06 0.08 0.1 0.12

15 20 25 30 35 40

EC (Sm^-1)

K a

Single Tangent method Dual Tangent method Derivative method

ε

dc

ε

dc

ε

_∞

(a)

(b)

Fig. 3-1 The apparent dielectric constants as affected by electrical conductivity in (a) the non-dispersive case and (b) the dispersive case.

0 0.02 0.04 0.06 0.08 0.1 10⁷

10⁸ 10⁹

f eq (Hz)

EC (Sm^-1) Single Tangent method Dual Tangent method Derivative method fbw

Fig. 3-2 The equivalent frequency and frequency bandwidth corresponding silt loam case Effect of cable resistance

The per-unit-length parameters that govern the TDR waveform include capacitance, inductance, conductance, and resistance. The first three parameters are associated with electrical properties of the medium and cross-sectional geometry of the waveguide. The per-unit-length resistance is a result of surface resistivity and cross-sectional geometry of the waveguide (including cable, connector, and sensing probe), which is often ignored in early studies of TDR waveform by assuming a short cable. The cable resistance is practically important since significantly long cable is often used in monitoring (Lin and Tang 2007). Not only does it affect the steady-state response and how fast the TDR waveform approaches the steady state, the cable resistance also interferes with the transient waveform related to the travel time analysis, as shown in Fig. 3-3 for measurements in water with different cable lengths. The

“significant length” in which cable resistance becomes unnegligible depends on the cable type,

which could range from lower quality RG-58, medium quality RG-8, to higher quality cables with solid outer conductor used in cable television (CATV) industry. The RG-58 cable is used for simulation in this study to manifest the effect of cable resistance and since it has been widely used for its easy handling.

The measurements of water and the silt loam with various cable lengths were simulated.

As an attempt to counteract the effects of cable length, the system parameters (i.e. t0 and L) were obtained by air-water calibration for each cable length. The calibrations of system parameters indicated that the L increases as cable length increases as shown in Table 3-3, except the case of using dual tangent method with 50m cable. Fig. 3-4 shows the effects of cable length on Ka for different methods of travel time analyses. In the non-dispersive case (Fig. 3-4 (a)), all methods are not affected by cable length if air-water calibrations are performed each cable lengths. As the medium becomes dielectric dispersive within TDR bandwidth, the apparent dielectric constant becomes quite sensitive to changing cable length (see Fig. 3-4 (b)), in particularly for the derivative method, even though the probe parameters have been calibrated by the air-water calibration procedure for each cable length. Fig. 3-4 suggests that the empirical relationship between Ka and soil water content would depends on cable length if the soil is significantly dielectric-dispersive. This is in agreement with the finding by Logsdon (2000). When studying the effect of cable length on TDR calibration for high surface areas soils (or called “dispersion material” in this study), Logsdon (2000) concluded that high surface area samples should be calibrated using the same cable length used for measurements. This is even more imperative if the derivate method is used.

Both the equivalent frequency and frequency bandwidth decreases with increasing cable length, as shown in Fig. 3-5. The single tangent and dual tangent methods have similar trends, while the derivative method is most sensitive to and results in the lowest effective frequency and frequency bandwidth. Therefore, the derivative method can leads to a K greater than DC

dielectric permittivity due to existence of electrical conductivity and low effective frequencies.

It should be noted that, for the simulated RG-58 cable, the equivalent frequency corresponds to the frequency bandwidth only valid for cable length around 10~15 m. Although Robinson et al.

(2005) concluded that the permittivity determined from the derivative method corresponds to the frequency bandwidth, but they made few account of the cable length effect, which may decrease the frequency bandwidth as the cable length increases. According to Fig. 3-5, this conclusion holds only for limited range of cable length.

Table 3-3 The calibrated probe length (m) obtained from the air-water calibration for different cable lengths and methods of travel time analysis

Cable Length Methods

1 m 10 m 25 m 50 m

Single tangent

method 0.2935 0.2968 0.3020 0.3049

Dual tangent

method 0.2934 0.2968 0.3015 0.2993

Derivative

method 0.3025 0.3062 0.3129 0.3352

Fig. 3-3 Measurements in water with various cable lengths.

0 10 20 30 40 50 60 75

76 77 78 79 80

K a

0 10 20 30 40 50 60

15 20 25 30 35 40

Cable length (m)

K a

Single Tangent method Dual Tangent method Derivative method

ε

dc

ε

dc

ε

_∞

(a)

(b)

Fig. 3-4 The apparent dielectric constants as affected by cable length in (a) the non-dispersive case and (b) the dispersive case.

0 10 20 30 40 50 10⁶

10⁷ 10⁸ 10⁹ 10¹⁰

f eq (Hz)

Cable length (m)

Single Tangent method Dual Tangent method Derivative method fbw

Fig. 3-5 The equivalent frequency and frequency bandwidth corresponding to silt loam case

Effect of dielectric relaxation frequency, frel

The apparent dielectric constant does not have a clear physical meaning when the dielectric permittivity is dispersive. Based on the Cole-Cole equation, the effects of dielectric relaxation frequency frel on Ka were investigated by varying frel in Table 3-2, while keeping other Cole-Cole parameters constant. The water-based cases represent cases with large difference between ε∞ and εdc (defined as Δε = εdc - ε∞), and the silt loam-based cases represent cases with relatively small Δε. The apparent dielectric constants as affected by frel are shown in Fig. 3-6. The frel seems to have a lower bound frequency below which the dielectric permittivity is equivalently non-dispersive and equal toε∞, and a higher bound frequency above which the dielectric permittivity is equivalently non-dispersive and equal to εdc. As frel increases from the lower bound frequency to higher bound frequency, the apparent dielectric constant goes from

methods because its equivalent frequency is always lower than that of tangent methods, as shown in Fig. 3-2 and Fig. 3-5. Comparing Fig. 3-6a with Fig. 3-6b, the lower bound frequency seems to decrease as Δε increases. That is, the higher the Δε, the wider the relaxation frequency range is affected by the dielectric dispersion.

Also depicted in Fig. 3-6 are the associated frequency bandwidths as affected by the relaxation frequency. When the relaxation frequency is outside the frequency range spanned by the aforementioned lower bound and higher bound, the dielectric permittivity does not show dispersion in the TDR frequency range, and hence the frequency bandwidths are similar. The frequency bandwidth decreases as the relaxation frequency becomes “active” and reaches the lowest point near the middle of the “active” frequency range spanned by the lower bound and higher bound.

0

Ka FrequencyBandwidth (Hz)

ε

_dc

Ka FrequencyBandwidth (Hz)

Singal Tangent method

Fig. 3-6 The apparent dielectric constants and frequency bandwidth by changing the dielectric relaxation frequency while keeping other Cole-Cole parameters constant in (a) water

and (b) silt loam.

Apparent Dielectric Constant vs. Frequency Bandwidth

The effects of electrical conductivity, cable resistance, and dielectric dispersion were systematically investigated. These factors can significantly affect the measured apparent dielectric constant. The equivalent frequency would give some physical meaning to the measured apparent dielectric constant, but no method is available for its direct determination.

not correspond to the optimal frequency range for water content measurement, as shown in Fig.

2-11. The frequency bandwidth, sometimes referred to as the effective frequency, can be determined from the rise time of the end reflection. It was anticipated to correspond to the equivalent frequency of certain travel time analysis (i.e. the derivative method). However, this correspondence is not generally true. Besides, the derivative method is quite sensitive to electrical conductivity and cable resistance, and hence would not be a good alternative to the conventional tangent line methods. Nevertheless, the frequency bandwidth of the TDR measurement offers an extra piece of information. An idea has been proposed to incorporate frequency bandwidth into the empirical relationship between apparent dielectric constant and soil water content (Evett et al. 2005). To examine whether this idea is generally feasible, the relationship between apparent dielectric constant from the dual tangent method and frequency bandwidth is plotted in Fig. 3-7 using the data obtained from previous three parametric studies.

The electrical conductivity, cable length, and dielectric dispersion apparently have distinct effects on the Ka-fbw relationship. In fact, the change in apparent dielectric constant vs. the change in frequency bandwidth as the influencing factors vary is divergent. When measuring soil water contents, the same water content may measure different apparent dielectric constant due to different electrical conductivity (e.g. from water salinity), cable length, and dielectric dispersion (e.g. from soil structure). Since there is no consistent trend between the change in apparent dielectric constant and the change in frequency bandwidth, compensating the effects of electrical conductivity, cable length, and dielectric dispersion by the frequency bandwidth seem theoretically infeasible.

10⁸ 15

20 25 30

Frequency Bandwidth, f

bw (Hz)

K a

Reference case

2*10⁷ 4*10⁸

Changing cable length

Changing f

rel

Changing EC

Fig. 3-7 The relationship between Ka from the dual tangent method and frequency bandwidth due to different influences

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