2.3 TDR Principle and Analysis
2.3.2 TDR Travel Time Analysis
Definition and application of apparent dielectric constant
As Eq. [2-15] shows, the equivalent complex dielectric permittivity is a function of frequency, and the frequency dependence of dielectric property is called the dielectric dispersion. Therefore, the propagation velocity of electromagnetic wave in a transmission line is also a function of frequency, and it can be (Topp, et al., 1980):
( )
The dielectric property of material between the waveguide conductors affects the propagation behavior of electromagnetic wave in a transmission line. If the geometric factors of the sensing waveguide are fixed, according to the transmission line parameters from Eq.
[2-34] and [2-35], the reflections occur with system characteristic impedance mismatches as the material dielectric property differs. Methods for determining the dielectric dispersion are proposed in literatures (Fellner-Felldeg, 1969; Giese and Tiemann, 1975; Heimovaara, 1994a;
Weerts et al., 2001). These methods involve spectral analysis and will be further introduce in section 2.3.5.
Dielectric spectroscopy is complex, thus a simplified analysis of TDR waveform to capture apparent dielectric constant has been proposed. Topp et al. (1980) defined the apparent dielectric constant, Ka, as the quantity determined from the measured velocity of the electromagnetic wave travelling through a transmission line. The apparent propagation velocity, V, of an electromagnetic wave in a transmission line is related to the apparent
dielectric constant Ka, as
Ka
V = c [2-38]
in which Ka represent a velocity factor as imaginary part of εr∗ is much smaller than real part of εr∗. Topp et al. (1980) indicated that apparent velocity of the electromagnetic wave travelling through a transmission line is obtained by travel time analysis using a tangent line approximation to find the inflection points. The TDR device sends a step pulse down the cable that is reflected from both the beginning and end of the probe due to impedance mismatches.
The two reflections cause two discontinuities in the resulting signal. The time difference between these two discontinuities is the time (Δt) required by the signal to travel twice the length (L) of the probe in soil. So the apparent dielectric constant Ka can be formulated:
2
2 ⎟
⎠
⎜ ⎞
⎝
=⎛ Δ L
t
Ka c [2-39]
Topp et al. (1980) also proposed an experimental relation for determining the volumetric water content (θ) of soil from Ka:
θ =−5.3×10−2 +2.92×10−2Ka −5.5×10−4Ka2 +4.3×10−6Ka3 [2-40]
Equation [2-40] is a monumental development for the volumetric water content (θ ) of soil.
However the coefficients in the Eq. [2-40] may depend on soil type. For example, Eq. [2-40]
may not suitable for organic soils and fine-texture soils (Herkelrath et al., 1991; Dasberg and Hopmans, 1992).
Considering the effect of the bulk density γd of soil, Topp’s equation can be approximated by Ka =a+bθ and using gravimetric water content (θ = w·γd). The relation between Ka and gravimetric water content w can be described as (Siddiqui and Drnevich, 1995; Lin et al., 2000)
bw K a
w d
a γ = +
γ [2-41]
in which γw is density of water, a and b are calibration coefficients. Nevertheless, the relation between Ka and soil water content w in Eq. [2-41] also depends on the soil type, particularly in high plastic index (PI) soils.
Methodology for determining the travel time Δt
In order to calculate the round-trip travel time Δt in the measured probe, the reflection arrival should be first determined. One way is to locate the reflection arrival is located at the intersection of the two tangents to the reflection curve, marked as point A in Fig. 2-10(a) and called the “dual tangent method”. While the second tangent line can be drawn at the point of maximum gradient in the rising limb, the location to draw the first tangent line often lacks a clear definition. To facilitate automation, Baker and Allmaras (1990) used a horizontal line tangent to the waveform at the local minimum (or local maximum for the start reflection). The
as point B in Fig. 2-10(a) and called the “single tangent method”. The single tangent method appears to be less arbitrary than the dual tangent method because the points of the local minimum and the maximum gradient can be clearly defined mathematically.
Timlin and Pachepsky (1996) and Klemunes et al. (1997) compared both methods and concluded that the latter provided a more accurate calibration equation for water content determination. However, Or and Wraith (1999) concluded that the dual tangent method is more accurate for conditions of high electrical conductivity. A second methodology is based on the apex of the derivative, as marked by point c in Fig. 2-10(b) and called the “derivative method”.
This relatively new method was proposed in research studied discussing the probe calibration (Mattei et al. 2005) and effective frequency of apparent dielectric constant (Robinson et al., 2005).
Furthermore, the electrical length L of the probe needs to be calibrated to convert the travel time to apparent velocity (and thereby apparent dielectric constant as shown in Eq.
[2-39]). Water is typically used for such a purpose since it has a well-known and high dielectric permittivity value as discussed in section 2.2.2. But the start reflection at the interface between probe head and soil typically may not be clearly defined as the start reflection due to the probe head mismatch. Heimovaara (1993) defined a consistent first reflection point and denoted the round-trip travel time as Δτ and the time difference between selected point and the actual start reflection point as t0, as shown in Fig. 2-10(a), and it can be written as:
c L t t
t0 s 0 2 εr* / τ = + = +
Δ [2-42]
where the ts is the true travel time in the measuring probe.
The probe length and t0 were then calibrated using measurements in air and water. The
the range of permittivity values in non-dispersive media. They also showed that calibration performed solely in water (i.e. only for probe length) using the apex of the first reflection as the first reference start point could introduce a small error at low permittivity values. Based on calibration of probe length only, Mattei et al. (2006) showed that the tangent line method (dual tangent) gives inconsistent probe length calibration in air and water while the derivative method can yield consistent probe length calibration. The anomalous result provided by the tangent line method was explained by dispersion effects. However, the dielectric dispersion of water is not significant in the TDR frequency range. The inconsistent probe length calibration may be attributed to error in defining the start reflection, as pointed out by Robinson et al. (2003) that the location of first time marker should be just to the right of the apex of the sensor head reflection.
The apparent dielectric constant traditionally determined by the travel time analysis using a tangent-line method does not have a clear physical meaning and is influenced by several system and material parameters. Lin (2003b) examined how TDR bandwidth, probe length, dielectric relaxation, and electrical conductivity affect travel time analysis by the automated single tangent method. The effects of TDR bandwidth and probe length could be quantified and calibrated, but the calibration equation for soil moisture measurements is still affected by dielectric relaxation and electrical conductivity, due to differences in soil texture and density.
Using the spectral analysis, Lin (2003b) suggested that the optimal frequency range, in which the dielectric permittivity is most invariant to soil texture, lies between 500 MHz and 1 GHz, as illustrated in Fig. 2-11. Robinson et al. (2005) investigated the effective frequencies, defined by the 10-90% rise time of the reflected signal, of the dual tangent and derivative methods, considering only the special case of non-conductive TDR measurements. Their results indicated that the effective frequency corresponds with the permittivity determined from the derivative method and not from the conventional dual tangent method. Nevertheless, Evett et al.
(2005) tried to incorporate bulk electrical conductivity and effective frequency defined primarily by the slope of the rising limb of the end reflection, into the water content calibration equation in a hypothesized form, and showed reduced calibration root mean square error (RMSE). However, the hypothesized form does not have a strong theoretical basis. The effects of dielectric dispersion, EC, cable length and effective frequency need further investigations.
65 70 75 80 85 90 95
-1 -0.5 0 0.5
ρ
65 70 75 80 85 90 95
-0.01 -0.005 0 0.005 0.01 0.015
ρ'
Traveltime (ns)
C A B
t0 t
s
(a)
(b)
Fig. 2-10 Illustration of various methods of travel time analysis: (a) locating the end reflection by the dual tangent (A point) and single tangent (B point) methods; (b) the derivative methods locates the end reflection by the apex of the derivative (C point) (modified after Robinson et al.,
2005)
Fig. 2-11 The optimal frequency range in which the dielectric permittivity is dominated by water content and least affected by electrical conductivity and dielectric dispersion due to
soil-water interaction. (modified after Lin, 2003b)