Apparent Dielectric Constant and EffecƟve Frequency of TDR Measurements: Influencing Factors and Comparison

Apparent Dielectric Constant and EffecƟve Frequency of TDR Measurements: Influencing Factors and Comparison

I for soil moisture

monitor-ing in a short time interval, the major technique for such a

purpose has become the measurement of soil dielectric

proper-ties as a surrogate for soil water content, including time domain

refl ectometry (TDR) (Topp et al., 1980; Topp and Ferre, 2002;

Robinson et al., 2003a) and capacitance methods (Dean et al.,

1987; Paltineanu and Starr, 1997) Time domain refl ectometry

is typically more accurate due to its higher eff ective frequency,

and often does not require a site-specifi c calibration It can also

provide accurate measurement of soil electrical conductivity in

the same sampling volume (Lin et al., 2007, 2008) Conventional

TDR probes using bifi lar or trifi lar TDR waveguides have limited

penetration depth, but new TDR penetrometers have been

devel-oped to overcome this limitation (Vaz and Hopmans, 2001; Lin

et al., 2006a,b) Despite the success of current TDR technology,

the travel time analysis algorithm that is used to extract Ka has

not been standardized, and there is room for further improvement

in the accuracy of water content determination h ree aspects

associated with the travel time analysis are: (i) determination of refl ection arrivals, (ii) probe calibration, and (iii) the physical meaning or eff ective frequency of travel time analysis h ese three aspects are briefl y reviewed

Diff erent methods have been proposed to determine the refl ection arrivals in travel time analysis h e fi rst methodology is based on the so-called “tangent method” (Topp et al., 1980) h e refl ection arrival is located at the intersection of the two tangents

to the curve, marked as Point A in Fig 1a 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 fi rst tangent line often lacks a clear defi nition To facili-tate automation, Baker and Allmaras (1990) used a horizontal line tangent to the waveform at the local minimum h e inter-section of this line with the second tangent line is determined

as the refl ection arrival, marked as Point B in Fig 1a and called the single tangent method h e 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 defi ned mathematically Timlin and Pachepsky (1996) and Klemunes et al (1997) compared both methods and concluded that the single tangent method provided a more accurate calibra-tion equacalibra-tion for water content determinacalibra-tion Or and Wraith (1999) concluded, however, that the dual tangent method is more accurate for conditions of high electrical conductivity h e second methodology is based on the apex of the derivative, as marked

by Point C in Fig 1b and is called the derivative method h is relatively new method was proposed in research studies discussing probe calibration (Mattei et al., 2006) and eff ective frequency

Apparent Dielectric Constant and Eff ec ve Frequency of TDR Measurements:

Infl uencing Factors and Comparison

C.-C Chung and C.-P Lin*

Dep of Civil Engineering, Na onal Chiao Tung Univ., Hsinchu, Taiwan

Received 8 May 2008 *Corresponding author (cplin@mail.nctu.edu.tw).

Vadose Zone J 8:548–556

doi:10.2136/vzj2008.0089

© Soil Science Society of America

677 S Segoe Rd Madison, WI 53711 USA.

All rights reserved No part of this periodical may be reproduced or transmi ed

in any form or by any means, electronic or mechanical, including photocopying,

recording, or any informa on storage and retrieval system, without permission

in wri ng from the publisher.

A : EC, electrical conductivity; TDR, time domain refl ectometry.

When measuring soil water content by me domain refl ectometry (TDR), several methods are available for

deter-mining the related apparent dielectric constant (Ka) from the TDR waveform Their infl uencing factors and eff ec ve frequencies have not been extensively inves gated and results obtained from diff erent methods have not been cri -cally compared The purpose of this study was to use numerical simula ons to systema -cally inves gate the eff ects of

electrical conduc vity, cable length, and dielectric dispersion on Ka and the associated eff ec ve frequency Not only does the dielectric dispersion signifi cantly aff ect the measured Ka, it also plays an important role in how Ka is aff ected

by the electrical conduc vity and cable length Three methods for determining Ka were compared, including the dual tangent, single tangent, and deriva ve methods Their eff ec ve frequencies were carefully examined with emphasis

on whether the eff ects of electrical conduc vity, cable length, and dielectric dispersion can be accounted for by the

es mated eff ec ve frequency The results show that there is no consistent trend between the change in Ka and the change in eff ec ve frequency as the infl uencing factors vary Compensa ng the eff ects of electrical conduc vity, cable length, and dielectric dispersion by the eff ec ve frequency seems theore cally infeasible To improve the accuracy of TDR soil water content measurements in the face of these infl uencing factors, future studies are recommended toward TDR dielectric spectroscopy or developing signal processing techniques for determining the dielectric permi vity near the op mal frequency range.

(Robinson et al., 2005) A calibration equation based on such a

travel time defi nition has not been found

h e electrical length of the probe needs to be calibrated

to convert the apparent travel time to apparent velocity (and

thereby Ka) Water is typically used for such a purpose since it

has a well-known and high dielectric permittivity value h e

starting refl ection at the interface between the probe head and

sensing rods typically cannot be clearly defi ned, however, due

to mismatches in the probe head Heimovaara (1993) defi ned a

consistent fi rst refl ection point and denoted the round-trip travel

time as tp and the time diff erence between a selected point and

the actual starting refl ection point as t0, as shown in Fig 1a h e

probe length and t0 were then calibrated using measurements

in air and water h e air–water calibration method was

dem-onstrated by Robinson et al (2003b) to be accurate across the

range of permittivity values in nondispersive media h ey also

showed that the calibration performed solely in water (i.e., only

for probe length) using the apex of the fi rst refl ection as the

fi rst reference starting point could introduce a small error at low

permittivity values Locating the starting refl ection by the dual

tangent method and calibrating along the probe length, Mattei

et al (2006) showed that the dual tangent method (for locating

the end refl ection) gives inconsistent probe length calibration in

air and water while the derivative method can yield consistent

probe length calibration h e anomalous result provided by the

dual tangent method was explained by dispersion eff ects; however,

the dielectric dispersion of water is not signifi cant in the TDR

frequency range We believe that the inconsistent probe length

calibration with the dual tangent method should be attributed

to error in defi ning the starting refl ection point h e approach

proposed by Heimovaara (1993) using the air–water calibration

is supported and used in this study

h e apparent dielectric constant traditionally determined by the travel time analysis using a tangent method does not have a clear physical meaning and is infl uenced by several system and material parameters Logsdon (2000) experimentally demon-strated that cable length has a great eff ect on measurement in high-surface-area soils and suggested using the same cable length for calibration and measurements Neglecting cable resistance, Lin (2003) examined how TDR bandwidth, probe length, dielectric relaxation, and electrical conductivity aff ected travel time analysis

by the automated single tangent method h e eff ects of TDR bandwidth and probe length could be quantifi ed and calibrated, but the calibration equation for soil moisture measurements is still aff ected by dielectric relaxation and electrical conductivity due to diff erences in soil texture and density Using spectral analy-sis, Lin (2003) suggested that the optimal frequency range, the range in which the dielectric permittivity is most invariant to soil texture, lies between 500 MHz and 1 GHz, as illustrated in Fig

2 Robinson et al (2005) investigated the eff ective frequencies, defi ned by the 10 to 90% rise time of the refl ected signal, of the dual tangent and derivative methods, considering only the special case of nonconductive and lossless TDR measurements h eir results indicated that the eff ective 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

eff ective frequency, defi ned by the slope of the rising limb of the end refl ection, into the water content calibration equation in a hypothesized form, and showed a reduced calibration RMSE h e hypothesized form, however, does not have a strong theoretical basis h e eff ects of dielectric dispersion, electrical conductivity, and cable length on the apparent dielectric constant and eff ective frequency need further investigation

Several methods have been proposed for determining Ka

from a TDR waveform h eir infl uencing factors have not been extensively investigated and the apparent dielectric constant and

F 1 Illustra on of various methods of travel me analysis: (a)

loca ng the end refl ec on by the dual tangent (Point A) and single

tangent (Point B) methods; (b) the deriva ve methods locates the

end refl ec on by the apex of the deriva ve (Point C) (modifi ed

a er Robinson et al., 2005); ts is the actual travel me in the

sens-ing waveguide, t0 is a constant me off set between the reference

me and the actual start point, ρ is the refl ec on coeffi cient of a

me domain refl ectometry waveform, and ρ′ is the deriva ve of ρ.

F 2 Dielectric dispersion of a soil depends on the soil texture

(parameterized by the specifi c surface As) The dielectric vity is aff ected by the interfacial polariza on at low frequencies and by the free water polariza on at high frequencies The op mal frequency range in which the dielectric permi vity is dominated

by water content and least aff ected by electrical conduc vity and dielectric dispersion due to soil–water interac on lies between 500 MHz and 1 GHz (modifi ed a er Lin, 2003); θ is soil moisture content, and ε r′ is the real part of the permi vity due to energy storage.

eff ective frequency obtained from diff erent methods have not

been critically compared h e objectives of this study were

two-fold: (i) to examine the eff ects of electrical conductivity, dielectric

dispersion, and cable length on Ka and the eff ective frequency,

and (ii) to investigate whether the eff ects of those factors on Ka

can be accounted for by the eff ective frequency

Materials and Methods

h e wave phenomena in a TDR measurement include

mul-tiple refl ections, dielectric dispersion, and attenuations 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) In the

context of TDR electrical conductivity measurement, Lin et al

(2007, 2008) utilized the TDR wave propagation model to show

the correct method for taking cable resistance into account and

presented guidelines for selecting the proper recording time With

the proven capability to accurately simulate TDR measurements,

the TDR wave propagation model can be used to systematically

investigate the eff ects of dielectric dispersion, electrical

conductiv-ity, and cable length on Ka and the eff ective frequency Synthetic

TDR measurements (waveforms) were generated by varying the

infl uential factors in a controlled fashion h e associated

appar-ent dielectric constants and eff ective frequencies were calculated

and compared

Synthe c TDR Measurements (Waveforms)

h e behavior of electromagnetic wave propagation in the

frequency domain can be characterized by the propagation

con-stant (γ) and the characteristic impedance (Zc) h e propagation

constant controls the velocity and attenuation of electromagnetic

wave propagation and the characteristic impedance controls the

magnitude of the refl ection Taking into account dielectric

disper-sion, electrical conductivity, and cable resistance, γ and Zc can be

written as (Lin and Tang, 2007)

r

2

*

j f

A

c

π

p

c

r*

Z

Z = A

p

1 1

Z f

⎛η ⎞⎟α

⎜ ⎟

⎜

= + − ⎜⎜ ⎟⎟

⎟

where c is the speed of light, εr* = εr − jσ/(2πfε0) is the complex

dielectric permittivity (including the eff ect of dielectric

permit-tivity εr and electrical conductivity σ, in which ε0 is the dielectric

permittivity of free space), Zp is the geometric impedance (the

characteristic impedance in air), A is the per-unit-length

resis-tance correction factor, j is the complex unit, η0 = √(μ0/ε0) ?

120π is the intrinsic impedance of free space (in which μ0 is

the magnetic permeability of free space), αR (s−0.5) is the

resis-tance loss factor (a function of the cross-sectional geometry and

surface resistivity due to the skin eff ect), and f is the frequency

Each uniform section of a transmission line is characterized by

its length, cross-sectional geometry, dielectric property, and cable

resistance h ese properties are parameterized by the length (L),

Zp, εr*, and αR Once these parameters are known or calibrated,

TDR waveforms can be simulated using Eq [1] and the modeling framework proposed by Lin (2003) h e propagation constants and characteristic impedances of each uniform section are fi rst determined by Eq [1] h e input impedance at location z = 0 (the

source end), Zin(0), represents the total impedance of the entire nonuniform transmission line It can be derived recursively from the characteristic impedance and the propagation constant of each uniform section, starting from the terminal impedance Z L: ( )

in

c,

c,

tanh tanh

tanh ( )tanh

tanh 0

tanh

Z z Z

Z z Z

Z z Z

−

=

=

=

=

where Zc,i, γi, and l i, are the characteristic impedance, propaga-tion constant, and length of each uniform secpropaga-tion, respectively A typical TDR measurement system uses an open loop (Z L = ∞)

h e frequency response of the TDR sampling voltage, V(0), can

then be written in terms of the input impedance as

( )

in

0 0

0

Z

where V(0) is the Fourier transform of the TDR waveform (v t); Vs

is the Fourier transform of the TDR step input; Zs is the source impedance of the TDR instrument (typically Zs = 50 Ω), and H =

Zin (0)/(Zin (0) + Zs) is the transfer function of the TDR response

h e TDR waveform is the inverse Fourier transform of V(0).

h e synthetic TDR measurement system is composed of

a TDR device, an RG-58 lead cable, and a sensing waveguide Possible mismatches due to connectors and probe head are neglected since this simplifi cation will not aff ect Ka 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 equa-tion may not be perfect for modeling the dielectric dispersion

of soils, since additional relaxations at lower frequencies might exist and multiple Cole–Cole relaxations would be more accurate Although multiple Cole–Cole relaxations might be mandatory for dielectric spectroscopy, the simple Cole–Cole equation was used to parameterize the dielectric dispersion for the parametric study of the dispersion eff ect h e transmission line parameters and dielectric properties used in the parametric study are listed in Tables 1 and 2, respectively Time interval Δt = 2.5 × 10−11 s and time window T = 8.2 × 10−6 s (slightly greater than the pulse length of 7 × 10−6 s in a TDR100 [Campbell Scientifi c, Logan, UT]) were used in the numerical simulations h e correspond-ing Nyquist frequency (half the samplcorrespond-ing frequency, sometimes called the cut-off frequency) and frequency resolution are 20

GHz and 60 kHz, respectively h e Nyquist frequency is well above the frequency bandwidth of the TDR100 and the long time window ensures that a steady state is obtained before onset

of the next step pulse

As shown in Table 2, two dielectric permittivity values rep-resenting water and a silt loam soil were used in the parametric

study to show how Ka and the eff ective frequency are aff ected

by electrical conductivity (EC), cable length, and dielectric

dispersion A similar study was done by Robinson et al (2005),

but their study was limited to nonconductive materials and a

lossless cable To compare our results with the results of

previ-ous work, the same permittivity range (dielectric permittivity

at zero frequency [εdc] values of 10, 25, 50, 75, and 100; and

dielectric permittivity at infi nite frequency [ε∞] values of 1.44,

2.18, 3.40, 4.63 and 5.85) with two diff erent relaxation

fre-quencies (0.1 GHz and 10 GHz) were used to reproduce Fig

3b in Robinson et al (2005) h e transmission line parameters

used were the same as the parametric study’s reference case

listed in Table 1 Diff erent EC and cable length values were

used to show their infl uence and importance

Travel Time Analysis and Eff ec ve Frequency

An arbitrary time in the refl ection waveform was chosen

as the reference time h e arrival time of the end refl ection was

determined by diff erent methods including the single tangent,

dual tangent, and derivative methods, as shown in Fig 1 h e

time between these two points is denoted as tp, which is a

combination of the actual travel time in the sensing waveguide

(ts) and a constant time off set (t0) between the reference time

and the actual starting point h e travel time tp is related to

the Ka by the following relationship:

a

t t t t L

c

where L is the electrical length of the probe As suggested by

Heimovaara (1993), t0 and L were calibrated by taking

mea-surements in air and water with known values of permittivity

It should be noted that diff erent values of system parameters (t0

and L) may be obtained when diff erent methods of travel time

analysis are used

Two methods have been used to investigate the “eff ective

frequency” of the Ka measurement One method compares the

Ka from the travel time analysis with the permittivity obtained

from the frequency domain dispersion curve (Or and Rasmussen,

1999; Lin, 2003) h e other method is based on the 10 to 90%

rise time of the end refl ection (Logsdon, 2000; Robinson et al.,

2005)

To avoid confusion, the fi rst approach is termed equivalent

frequency, feq It is determined by matching Ka estimated from

travel time analysis methods to the frequency-dependent apparent

dielectric permittivity εa(f ) (Von Hippel, 1954):

( )

( )

1/2 2

r eq

r eq

2 1

2

f

f

=ε

⎜ ⎪⎢ε + ⎥ ⎪ ⎟

′

ε ⎜⎜ ⎪⎪⎢⎢ π ε ⎥⎥ ⎪⎪ ⎟⎟

⎟

= ⎜⎜⎜ +⎨⎪⎢ ε′ ⎥ ⎬ ⎟⎪ ⎟

⎟

[5]

where ε r′ is the real part of the permittivity due to energy storage

and εr″ is the imaginary component due to dielectric loss For

determining equivalent frequencies in the parametric study, the

real and imaginary permittivity as functions of frequency were

known a priori from model parameters listed in Tables 1 and 2

Unlike Or and Rasmussen (1999), we used the apparent dielectric

permittivity εa(f ) instead of the real part of the dielectric

permit-tivity to take into account the eff ects of dielectric loss and EC on the phase velocity

h e second approach is termed frequency bandwidth, fbw It

is defi ned by the 10 to 90% rise time (tr) of the end refl ection as (Strickland, 1970)

( )

bw

ln 0.9 0.1 0.35 2

f

=

where tr is measured in seconds

In actual TDR measurements, the equivalent frequency cannot be uniquely determined since the real and imaginary per-mittivities in Eq [5] are also unknown h erefore, the frequency bandwidth was defi ned in the hope that it can represent the equivalent frequency In this study, both the equivalent frequency and the frequency bandwidth as functions of the infl uencing fac-tors were examined and compared

Results and Discussion

Importance of Electrical Conduc vity and Cable Length Robinson et al (2005) investigated the frequency band-width (defi ned by Eq [6]) of the dual tangent and derivative methods h eir results (Fig 3b in Robinson et al., 2005) indi-cated that Ka of the derivative method is equivalent to the calculated permittivity by substituting the frequency bandwidth for the equivalent frequency in Eq [5], providing physical

T 1 Time domain refl ectometry system parameters used in the numerical simula ons.

value Range Sensing waveguide electrical conduc vity (σ), S/m 0.01 0.005 ? 0.1

dielectric permi vity (εr) tap water

and silt loam†

with varying

frel

geometric impedance (Zp), Ω 300 300

resistance loss factor (αR), s−0.5 0 0 Lead cable (RG-58) electrical conduc vity (σ), S/m 0 0

dielectric permi vity (εr) 1.95 1.95

geometric impedance (Zp), Ω 77.5 77.5

resistance loss factor (αR), s−0.5 19.8 19.8

† Referring to the Cole–Cole parameters in Table 2.

T 2 Cole–Cole† parameters for the materials used in the numerical simula ons.

† Cole–Cole equa on: εr(f) = ε∞ + (εdc − ε∞)/{1 + [j(f/frel)] 1−β }, where εr is the

dielectric permi vity, f is frequency, ε∞ is the dielectric permi vity at infi nite frequency, εdc is the dielectric permi vity at zero frequency, j is a complex unit, frel is the relaxa on frequency, and β is a parameter charac-terizing a spread of the relaxa on frequency.

‡ From Friel and Or (1999).

§ From Lin et al (2007) and water temperature = 25°C.

meaning to the derivative method h eir study, however, was

limited to zero EC and lossless cables

To see whether the neglected EC and cable resistance matter,

the same procedure was followed but additionally bringing in the

eff ect of EC and cable resistance Figure 3, similar to Fig 3b of

Robinson et al (2005), shows the Ka of the derivative method

vs the calculated permittivity from the frequency bandwidth for

various conditions Figures 3a and 3b reveal the eff ect of EC

for the reference cable length h e relationship between Ka of

the derivative method and the calculated permittivity from the

frequency bandwidth falls on the 1:1 line in nondispersive

materi-als (with relaxation frequency greater than the TDR bandwidth)

regardless of the EC value As the material becomes dispersive and

conductive, the relation deviates from the 1:1 line Figures 3c and

3d reveal the eff ect of the cable length for zero EC

Similarly, cable resistance becomes an infl uencing factor

when the material is dispersive h ese results show that both EC

and cable resistance play important roles for dispersive materials,

and the fi nding of Robinson et al (2005) that the frequency bandwidth corresponds with the Ka of the derivative method holds only for limited EC and cable length values In the context

of soil moisture determination, whether Ka is the same as the calculated permittivity from the eff ective frequency is not criti-cal; it is of more concern how Ka varies with infl uencing factors while the actual water content remains the same It is also of interest whether the eff ective frequency can provide useful infor-mation for compensating the eff ects of the infl uencing factors

h erefore, the subsequent discussions focus on the variation of

Ka and the eff ective frequency as functions of EC, cable length, and dielectric dispersion

Eff ect of Electrical Conduc vity

h e electrical conductivity is well known for having a smoothing eff ect on the refl ected waveform and hence aff ect-ing the Ka determination; however, the degree of infl uence may depend on dielectric dispersion and the method of travel time analysis Varying the value of EC in water (as a nondispersive case) and silt loam (as a dispersive case), Fig 4 shows the eff ects

of EC on Ka for diff erent methods of travel time analysis In the nondispersive case, only the single tangent method is slightly

aff ected by the EC Both the dual tangent method and deriva-tive method are unexpectedly immune to changing EC (see Fig 4a) As the medium becomes dispersive within the TDR bandwidth, Ka becomes sensitive to changing EC (see Fig 4b) Among all the methods, the dual tangent method is the least

aff ected by EC When EC is >0.05 S m−1, the single tangent method and derivative method suddenly obtain higher Ka values

as EC increases h e Ka may even become greater than the direct current electrical permittivity due to the signifi cant contribu-tion of EC at low frequencies

F 3 The rela on between the apparent dielectric constant Ka

from the deriva ve method and Ka calculated from the frequency

bandwidth: (a) and (b) show results as aff ected by electrical

con-duc vity (EC) for nondispersive (relaxa on frequency frel = 10 GHz)

and dispersive (frel = 0.1 GHz) cases, respec vely; (c) and (d) show

results as aff ected by cable length for nondispersive (frel = 10 GHz)

and dispersive (frel = 0.1 GHz) cases, respec vely.

F 4 The apparent dielectric constant Ka as aff ected by electrical conduc vity (EC) in (a) the nondispersive case and (b) the disper-sive case; εdc is the dielectric permi vity at zero frequency, ε∞ is the dielectric permi vity at infi nite frequency.

For each simulated waveform, the equivalent frequencies of

diff erent travel time analysis methods and the frequency

band-width of the end refl ection were determined by Eq [5] and [6],

respectively h e equivalent frequencies and frequency bandwidth

associated with Fig 4b (the dispersive case) is shown in Fig 5

Only the dispersive case is shown since the equivalent frequencies

in the nondispersive case were not meaningful Against common

perception, the frequency bandwidth is not signifi cantly aff ected

by EC h e end refl ection may appear smoothed due to decreased

refl ection magnitude as EC increases h e 10 to 90% rise time,

and hence the frequency bandwidth, remains relatively

con-stant h e equivalent frequencies decrease with increasing EC

as expected In this particular case, the frequency bandwidth is

close to the equivalent frequency of the derivative method in

the middle range of EC h e dual tangent method leads to the

highest equivalent frequency, while the derivative method, as

also pointed out by Robinson et al (2005), results in the lowest

equivalent frequency, which is closer to the frequency bandwidth

h e dual tangent is advantageous in this regard since, at higher

frequency, the apparent dielectric permittivity is less aff ected by

changing EC But unfortunately, its automation of data reduction

is also most diffi cult

Eff ect of Cable Resistance The per-unit-length parameters that govern the TDR

waveform include capacitance, inductance, conductance, and

resistance h e fi rst three parameters are associated with the

elec-trical properties of the medium and cross-sectional geometry of

the waveguide h e per-unit-length resistance is a result of

sur-face resistivity and the cross-sectional geometry of the waveguide

(including the cable, connector, and sensing probe), which was

often ignored in early studies of TDR waveform by assuming a

short cable h e cable resistance is practically important since

a signifi cantly long cable is often used in monitoring (Lin and

Tang, 2007; Lin et al., 2007) Not only does it aff ect 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 6

for measurements in water with diff erent cable lengths h e “sig-nifi cant length” in which cable resistance becomes nonnegligible depends on the cable type, which could range from lower quality RG-58, to medium quality RG-8, to the higher quality cables with solid outer conductors used in the cable TV industry h e RG-58 cable was used for simulation in this study to manifest the eff ect of cable resistance and since it has been widely used for its easy handling

h e measurements of water and the silt loam soil with vari-ous cable lengths were simulated As an attempt to counteract the eff ects of cable length, the system parameters (i.e., t0 and

L) were obtained by air–water calibration for each cable length

h e cable resistance signifi cantly distorted the TDR waveform Consequently, the calibrated probe length increased with increas-ing cable length, as shown in Table 3 Figure 7 shows the eff ects

of cable length on Ka for diff erent methods of travel time analy-ses In the nondispersive case (Fig 7a), none of the methods are

aff ected by the cable length if air–water calibrations are performed for each cable length As the medium becomes dispersive within the TDR bandwidth, the apparent dielectric constant becomes quite sensitive to changing cable length (see Fig 7b), in particular for the derivative method, even though the probe parameters have been calibrated by the air–water calibration procedure for each cable length Figure 7 suggests that the empirical relation-ship between Ka and the soil water content depends on the cable length if the soil is signifi cantly dielectric dispersive h is is in agreement with the results of Logsdon (2000) When studying the eff ect of cable length on Ka–water content calibration for

F 5 The equivalent frequency for various methods of travel me

analysis and frequency bandwidth as aff ected by electrical

vity in the dispersive case.

F 6 Time domain refl ectometry (TDR) waveforms in water with various cable lengths, in which waveforms of 25 and 50 m are shi ed in me such that the refl ec ons from the TDR probe can be compared for diff erent cable lengths; ρ is the refl ec on coeffi cient

of a TDR waveform.

T 3 The calibrated probe length (m) obtained from the air– water calibra on for cable lengths from 1 to 50 m and diff erent methods of travel me analysis.

————————————— m ————————————— Single tangent method 0.2935 0.2968 0.3020 0.3049 Dual tangent method 0.2934 0.2968 0.3015 0.2993 Deriva ve method 0.3025 0.3062 0.3129 0.3352

surface-areas soils, Logsdon (2000) concluded that

high-surface-area samples should be calibrated using the same cable

length used for measurements h is is even more imperative if

the derivate method is used

The equivalent frequencies and frequency bandwidth

associated with Fig 7b (the dispersive case) is shown in Fig

8 Both the equivalent frequency and frequency bandwidth

decrease with increasing cable length h e single tangent and

dual tangent methods have similar trends, while the derivative

method is most sensitive to the cable length and results in the

lowest equivalent frequency h erefore, the derivative method

can yield a Ka greater than the direct current dielectric

permit-tivity due to the existence of EC and low equivalent frequency

In this particular case, the equivalent frequency of the deriva-tive method corresponds to the frequency bandwidth only for

a cable length of around 10 to 15 m

Eff ect of Dielectric Relaxa on Frequency

h e apparent dielectric constant does not have a clear physi-cal meaning when the dielectric permittivity is dispersive and conductive Based on the Cole–Cole equation, the eff ects of dielectric relaxation frequency frel on Ka were investigated by vary-ing frel in Table 2 while keeping the other Cole–Cole parameters constant h e water-based cases represent cases with a large dif-ference between ε∞ and εdc (defi ned as Δε = εdc − ε∞), and the silt loam cases represent cases with relatively small Δε values h e apparent dielectric constants as aff ected by frel are shown in Fig

9 h e frel seems to have a lower bound frequency below which the dielectric permittivity is equivalently nondispersive and equal

to ε∞, and a higher bound frequency above which the dielectric permittivity is equivalently nondispersive and equal to εdc As frel

increases from the lower bound frequency to the higher bound frequency, the apparent dielectric constant goes from ε∞ to εdc

In these relaxation frequencies, the derivative method yields a higher Ka than tangent methods because its equivalent frequency

is always lower than that of the tangent methods Comparing Fig 9a with Fig 9b, the lower bound frequency seems to decrease as

Δε increases h at is, the higher the Δε, the wider the relaxation frequency range aff ected by the dielectric dispersion

Also depicted in Fig 9 are the associated frequency band-widths as affected by the relaxation frequency When the relaxation frequency is outside the frequency range spanned by the aforementioned lower and upper bounds, the dielectric per-mittivity does not show dispersion in the TDR frequency range, and hence the corresponding frequency bandwidth is relatively

F 9 The apparent dielectric constant Ka and frequency band-width obtained by changing the dielectric relaxa on frequency while keeping other Cole–Cole parameters constant in (a) water and (b) silt loam; εdc is the dielectric permi vity at zero frequency,

ε∞ is the dielectric permi vity at infi nite frequency.

F 7 The apparent dielectric constant Ka as aff ected by cable

length in (a) the nondispersive case and (b) the dispersive case; εdc

is the dielectric permi vity at zero frequency, ε∞ is the dielectric

permi vity at infi nite frequency.

F 8 The equivalent frequency for various methods of travel me

analysis and frequency bandwidth as aff ected by cable length in

the dispersive case.

independent of frel h e 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 and upper bounds

Apparent Dielectric Constant vs Frequency Bandwidth

h e eff ects of EC, cable resistance, and dielectric dispersion

were systematically investigated h ese factors can signifi cantly

aff ect the measured Ka h e equivalent frequency would give

some physical meaning to the measured Ka, but no method is

available for its direct determination from the TDR measurement

Even if the equivalent frequency of Ka can be determined, it may

not correspond to the optimal frequency range for water

con-tent measurement, as shown in Fig 2 h e frequency bandwidth,

often referred to as the eff ective frequency in the literature, can be

determined from the rise time of the end refl ection It was

antici-pated that it would correspond to the equivalent frequency of the

derivative method h is correspondence, however, is not generally

true Besides, the derivative method is quite sensitive to EC 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 off ers an extra piece of

information An idea has been proposed to incorporate frequency

bandwidth into the empirical relationship between Ka and the

soil water content (e.g., Evett et al., 2005) To examine whether

this idea is generally feasible, the relationship between Ka from

the dual tangent method and frequency bandwidth is plotted in

Fig 10 using the data obtained from three previous parametric

studies h e EC, cable length, and dielectric dispersion

appar-ently have distinct eff ects on the Ka–fbw relationship In fact, the

change in Ka vs the change in fbw as the infl uencing factors vary

is divergent When measuring soil water content, the same water

content may measure diff erent apparent dielectric constants due

to diff erent EC (e.g., water salinity), cable length, and dielectric

dispersion (e.g., soil texture) Since there is no consistent trend

between the change in Ka and the change in fbw, compensating

the eff ects of EC, cable length, and dielectric dispersion by the

frequency bandwidth seems theoretically infeasible As shown in

Fig 2, Lin (2003) suggested that there is an optimal frequency

range in which the dielectric permittivity is most invariant to

soil texture (dielectric dispersion) To improve the accuracy of

TDR soil water content in light of the existence of the infl

uenc-ing factors, the actual real part of the dielectric permittivity near

the optimal frequency range should be measured and used to

correlate with water content Dielectric spectroscopy

(measure-ment of the frequency-dependent dielectric permittivity) based

on the full waveform model that takes into account the EC and

cable resistance can be used for such a purpose Dielectric

spec-troscopy, however, is still not at a state of general practice due to

its complex computation and system calibration Future studies

are suggested to simplify TDR dielectric spectroscopy or develop

signal processing techniques for determining the dielectric

per-mittivity near the optimal frequency range

Conclusions

h e Ka derived from various travel time analyses (e.g., duel

tangent, single tangent, and derivative methods) does not have

a clear physical meaning Although an earlier study showed that

the Ka of the derivative method corresponds with the eff ective

frequency determined from the refl ection rise time, this fi nding is true only for limited EC and cable length values Using numerical simulations, this study systematically investigated the infl uencing factors, including EC, dielectric dispersion, and cable resistance, and the associated eff ective frequencies

h e material is perceivably dispersive in a TDR measure-ment when the dielectric relaxation frequency (frel) is within

a frequency range Within this frequency range, the apparent dielectric constant and frequency bandwidth (determined from the rise time of the end refl ection) are sensitive to frel Dielectric dispersion also plays an important role on how EC and cable length aff ect Ka In nondispersive cases, Ka is not aff ected by

EC, and the eff ects of cable length on Ka can be accounted for

by adjusting the probe parameters (i.e., the probe length and a constant time associated with the arrival time of the incident wave) using air–water calibration for each cable length In dis-persive cases, Ka becomes dependent on EC, particularly at high

EC, and cable length, regardless of the air–water calibration for each cable length

Comparing methods of travel time analysis, the dual tangent method, although most diffi cult to automate, yields a Ka with the highest equivalent frequency (i.e., a frequency at which the Ka

is equal to the frequency-dependent dielectric permittivity) and

is least sensitive to EC and cable length h e derivative method has the lowest equivalent frequency and is quite sensitive to EC and cable length for dispersive materials h us it is not a good alternative to the conventional tangent line methods

h ere is no general correspondence between the frequency bandwidth and equivalent frequencies from various travel time analyses Nevertheless, the frequency bandwidth of the TDR measurement does off er an extra piece of information Simulation results were examined to see whether the eff ects of EC, cable length, and dielectric dispersion on the Ka can be refl ected on and accounted for by the frequency bandwidth h e results show that there is no consistent trend between the change in Ka and the change in frequency bandwidth as the infl uencing factors

F 10 The rela onship between the apparent dielectric constant

Ka determined by the dual tangent method and frequency

band-width; EC is electrical conduc vity, frel is the relaxa on frequency.

vary h erefore, compensating the eff ects of EC, cable length, and

dielectric dispersion by the frequency bandwidth seems

theoreti-cally infeasible To improve the accuracy of soil water content

measurement by TDR, future studies are suggested on TDR

dielectric spectroscopy or the development of signal processing

techniques for determining the dielectric permittivity within the

optimal frequency range between 500 MHz to 1 GHz

References

Baker, J.M., and R.R Allmaras 1990 System for automating and multiplexing

soil moisture measurement by time-domain refl ectometry Soil Sci Soc

Am J 54:1–6.

Dean, T.J., J.P Bell, and A.B.J Baty 1987 Soil moisture measurement by an

improved capacitance technique: I Sensor design and performance J

Hy-drol 93:67–78.

Evett, S.R., J.A Tolk, and T.A Howell 2005 Time domain refl ectometry

labo-ratory calibration in travel time, bulk electrical conductivity, and eff ective

frequency Vadose Zone J 4:1020–1029.

Friel, R., and D Or 1999 Frequency analysis of time-domain refl ectometry

with application to dielectric spectroscopy of soil constituents Geophysics

64:707–718.

Heimovaara, T.J 1993 Design of triple-wire time domain refl ectometry probes

in practice and theory Soil Sci Soc Am J 57:1410–1417.

Klemunes, J.A., W.W Mathew, and A Lopez, Jr 1997 Analysis of methods used

in time domain refl ectometry response Transp Res Rec 1548:89–96.

Lin, C.-P 2003 Frequency domain versus travel time analyses of TDR

wave-forms for soil moisture measurements Soil Sci Soc Am J 67:720–729.

Lin, C.-P., C.-C Chung, J.A Huisman, and S.-H Tang 2008 Clarifi cation and

calibration of refl ection coeffi cient for TDR electrical conductivity

mea-surement Soil Sci Soc Am J 72:1033–1040.

Lin, C.-P., C.-C Chung, and S.-H Tang 2006a Development of TDR

pen-etrometer through theoretical and laboratory investigations: 2

Measure-ment of soil electrical conductivity Geotech Testing J 29(4), doi:10.1520/

GTJ14315.

Lin, C.-P., C.-C Chung, and S.-H Tang 2007 Accurate TDR measurement of

electrical conductivity accounting for cable resistance and recording time

Soil Sci Soc Am J 71:1278–1287.

Lin, C.-P., and S.-H Tang 2007 Comprehensive wave propagation model to

improve TDR interpretations for geotechnical applications J Geotech

Geoenviron Eng 30(2), doi:10.1520/GTJ100012.

Lin, C.-P., S.-H Tang, and C.-C Chung 2006b Development of TDR

pen-etrometer through theoretical and laboratory investigations: 1

Measure-ment of soil dielectric constant Geotech Testing J 29(4), doi:10.1520/

GTJ14093.

Logsdon, S.D 2000 Eff ect of cable length on time domain refl ectometry

cali-bration for high surface area soils Soil Sci Soc Am J 64:54–61.

Mattei, E., A Di Matteo, A De Santis, and E Pettinelli 2006 Role of

dis-persive eff ects in determining probe and electromagnetic

param-eters by time domain refl ectometry Water Resour Res 42:W08408,

doi:10.1029/2005WR004728.

Or, D., and V.P Rasmussen 1999 Eff ective frequency of TDR traveltime-based

measurement of soil bulk dielectric permittivity p 257–260 In Worksh

on Electromagnetic Wave Interaction with Water and Moist Substances,

3rd, Athens, GA 11–13 Apr 1999 USDA-ARS, Athens, GA.

Or, D., and J.M Wraith 1999 Temperature eff ects on soil bulk dielectric

per-mittivity measured by time domain refl ectometry: A physical model

Wa-ter Resour Res 35:371–383.

Paltineanu, I.C., and J.L Starr 1997 Real-time water dynamics using

multisen-sor capacitance probes: Laboratory capacitance probes Soil Sci Soc Am

J 61:1576–1585.

Robinson, D.A., S.B Jones, J.M Wrath, D Or, and S.P Friedman 2003a A

review of advances in dielectric and electrical conductivity measurements

in soils using time domain refl ectometry Vadose Zone J 2:444–475.

Robinson, D.A., M Schaap, S.B Jones, S.P Friedman, and C.M.K Gardner

2003b Considerations for improving the accuracy of permittivity

mea-surement using TDR: Air/water calibration, eff ects of cable length Soil

Sci Soc Am J 76:62–70.

Robinson, D.A., M.G Schaap, D Or, and S.B Jones 2005 On the eff ective

measurement frequency of time domain refl ectometry in dispersive and non-conductive dielectric materials Water Resour Res 40:W02007, doi:10.1029/2004WR003816.

Strickland, J.A 1970 Time-domain refl ectometry measurements Tektronix Inc., Beaverton, OR.

Timlin, D.J., and Y.A Pachepsky 1996 Comparison of three methods to ob-tain the apparent dielectric constant from time domain refl ectometry wave traces Soil Sci Soc Am J 60:970–977.

Topp, G.C., J.L Davis, and A.P Annan 1980 Electromagnetic determination

of soil water content: Measurements in coaxial transmission lines Water Resour Res 16:574–582.

Topp, G.C., and P.A Ferre 2002 Water content p 417–421 In J.H Dane and

G.C Topp (ed.) Methods of soil analysis Part 4 Physical methods SSSA Book Ser 5 SSSA, Madison, WI.

Vaz, C.M.P., and J.W Hopmans 2001 Simultaneous measurement of soil pen-etration resistance and water content with a combined penetrometer–TDR moisture probe Soil Sci Soc Am J 65:4–12.

Von Hippel, A.R 1954 Dielectrics and waves John Wiley & Sons, Hoboken, NJ.