ASAE Journal Article ## Improved Calibration of Frequency Domain Reflectometry Probes for Volumetric Water Content Measurements of Three Media

## Y. Ting, C. Chen

## Published in Transactions of the ASABE Vol. 55(1): 51-60 ( Copyright 2011 American Society of Agricultural and Biological Engineers ).

Submitted for review in August 2010 as manuscript number SW 8746; approved for publication by the Soil & Water Division of ASABE in November 2011.

The authors are

Yuchen Ting,Assistant, andChiachung Chen,Professor, Department of Bio-industrial Mechatronics Engineering, National Chung Hsing University, Taichung, Taiwan.Corresponding author:Chiachung Chen, Department of Bio-industrial Mechatronics Engineering, National Chung Hsing University, 250 Kuokuang Road, Taichung, Taiwan; phone: 886-4-22857562 fax: 886-4-22857135; e-mail: ccchen@dragon.nchu.edu.tw.

Abstract.The frequency domain reflectometry (FDR) meter is a convenient sensor used to measure the water content of soils and substrates. An adequate calibration equation evaluated by statistical techniques is important to ensure the performance of the FDR meter. In this study, the calibration equation was used to express the relationship between the volumetric water contents of substrates measured with the gravimetric method then calculated with bulk density and the values read from the FDR meter for three kinds of substrates. Two types of FDR probes were applied to illustrate the improvement with calibration equations for the volumetric water contents of three kinds of substrates. Seven calibration equations were selected to evaluate the fitting agreement. No unique FDR calibration equation could be proposed to fit all FDR probes and substrates. Each FDR probe had a specific calibration equation for the different substrates. The accuracy of an FDR probe could be improved significantly with the selection of an adequate calibration equation.

Keywords.Dielectric constant, Gravimetric method, Relative permittivity.Because of the water shortage in the world's agricultural production, accurately monitoring soil moisture content is important for sustainable agriculture. A water content meter with good performance for soils or substrates is useful. The accurate measurement of the substrate water content could help managers decide when and how much water is necessary. An adequate calibration equation is important to ensure the accuracy of these measurements.

Many kinds of water content meters have been commercialized for soil and substrates (Walker et al., 2004). There are two dielectric-based soil water content measurement methods (Yoshikawa et al., 2004). Both methods use electromagnetic signals to measure the relative permittivity of the surrounding medium. The first method is time domain reflectometry (TDR), which involves detecting the propagation speed of a signal along a transmission line through the surrounding medium. The relative permittivity of the medium is measured from the change in propagation speed. The other method is frequency domain reflectometry (FDR), which involves sending an oscillating signal through a transmission line. The dielectric properties of the surrounding medium influence the impedance of the transmission line, and the relative permittivity of the medium can be detected by the change in amplitude of the signal. The measurement principle of dielectric-based meters is based on the detection of the dielectric properties of the substrate indirectly and then transformation of the signal to volumetric moisture content using a calibration equation.

Topp et al. (1980) introduced the TDR theory and proposed the first calibration equation of a dielectric-based meter. Whalley et al. (1994) designed a two-wire TDR meter and combined it with tensionmetry to compare the performance. The authors confirmed a linear relationship between logarithmic specific hydraulic values and matrix potential values for sandy soils. Hilhorst (2000) first introduced the principle of the FDR meter and validated this theory with a prototype pure water meter. Noborio (2001) reviewed the effects of the geometry of TDR probes on measurement. The instrument characteristics of TDR probes were described by Blonquist and Robinson, (2005) and Jones et al. (2002). Woodhead et al. (2003) developed a general dielectric model and validated this model with measurement data of coarse soil, bentonite clay, and three kinds of sawdust. Souto et al. (2008) developed a new model to deal with the air-water relationship in stony soils.

The performance of TDR and FDR meters has been well studied. Hansen et al. (2006) evaluated the measurement performance of the Theta TDR HydroSense probe and the WET FDR meter with three statistics: bias, variance, and precision. The accuracy was 3% to 5% for the TDR and FDR instruments, and the standard deviations ranged from 1.2% to 2.1%. Ebrahimi-Birang et al. (2006) established a linear calibration equation to express the relationship between the soil water content and the recorded values of two types of TDR probes.

Much of the research evaluation of FDR or TDR meters has focused on soils and other substrates. Recently, many valuable economic crops have been planted in soilless substrates. Nemali et al. (2007) tested the reliability of two FDR meters (ECH

_{ 2 }O and Theta) for the measurement of water content of custom-made substrates. Four types of substrates (peat, pine bark, perlite, and vermiculite) were mixed with different ratios. Overduin et al. (2005) compared three FDR meters and three TDR meters for volumetric water content measurement of feather moss and found that the FDR meters were suitable for measurement. Yoshikawa et al. (2004) evaluated the performance of three FDR meters and three TDR meters to detect the water content of moss (Sphagnumspp.) and selected a third-degree polynomial equation as the least systematic error (adequate) calibration equation.The criteria for the selection of adequate calibration equations are inconsistent. The coefficient of determination (R

^{ 2 }) was adopted to evaluate the performance of FDR and TDR meters of soils (Quinones and Ruelle, 2001; Regaladpo, 2004; Pumpanen and Livesniemi, 2005; Ebrabimi-Birang et al., 2006) and other substrates (Yoshikawa et al., 2004; Nernali et al., 2007). Regalado (2004) proposed a logarithmic TDR calibration equation to describe the relationship between volumetric water content and bulk dielectric constants, and the coefficient of determination (R^{ 2 }) was adopted as the criterion to evaluate the fitting ability. Pumpanen and Ilvesniemi (2005) described the relationship between volumetric water content and apparent dielectric constant of the TDR method with three empirical models. The coefficient of determination (R^{ 2 }) was the sole criterion for comparison. Maroufpoor et al. (2009) observed the effect of five types of soil textures on TDR calibration for water content measurement. The relationship between the values measured by TDR meter and gravimetric methods was assumed as a linear equation. The coefficient of determination (R^{ 2 }) and the root mean square error (RMSE) were used to evaluate TDR performance. However, the coefficient of determination should not be the sole criterion for judging accuracy. Residual plots and other criteria should be used to evaluate the fitting agreement of a regression equation (Myers, 1986; Kleinbaum et al., 1998).The standard error of estimation (SEE) has been used to express the accuracy of calibration equations and adopted as the source of the uncertainty calibration (Eurachem, 2000). The measurement uncertainty of a dielectric water content meter is important to express the meter's measurement performance. A method to evaluate TDR meter performance was introduced by Cataldo et al. (2009).

The objective of this study is to evaluate the performance of FDR meters for measuring water content in three kinds of soilless substrates. For evaluating the performance improvement of an FDR meter with an adequate calibration equation, the components of soilless substrates that have been treated with steam are simpler than those of soils because of the lack of microorganism activity.

## Materials and Methods

## Establishment of Calibration Equation

Measurement of soil moisture with an FDR or TDR meter involves two steps. First, the relative permittivity ( e

eff) of the substrate is measured. The relationship between the volumetric water content ( ? ) and the relative permittivity is proposed as the substrate calibration equation:(1)

The second step is to establish the relationship for the output signal of the FDR or TDR meter as a function of the relative permittivity, called the meter calibration equation. This equation is in nonlinear form because of the circuit design for signal conditioning and processing:

(2)

where

xis the reading of the FDR or TDR meter.Combining equations 1 and 2, the function of the volumetric water content ( ? ) and the displayed value of the FDR or TDR meter, is complex. The mathematical relationship between the relative permittivity of the FDR or TDR meter and the water content of soils or media has been reviewed by Woodhead et al. (2003) and Cerny (2009). However, their theoretical model was too complex to be applied.

Cataldo et al. (2010) considered establishing a standard method to incorporate the effect of components for measuring soil water content and found that individual calibration curves must be specified for each substrate. In this study, some empirical calibrations found most often in the literature are proposed and introduced.

## FDR Meters

Two types of FDR probes were adopted in this study to illustrate the effects of calibration equations on the performance of the FDR meter: the Delta-T WET probe and the SM200 moisture probe (Delta-T Devices, Ltd., Cambridge, U.K.). The probe is a sensing element. The probe was connected to a transducer to form a meter for measurement. The manufacturer's specifications for these probes are listed in table 1.

Table 1. Specifications of two types of FDR probes.

Parameter

WET Probe

SM200 Probe

Prongs

3

2

Measuring range

0.0 to 0.55 m

^{ 3 }m^{ -3 }0.0 to 0.50 m

^{ 3 }m^{ -3 }Accuracy

± 0.03 m

^{ 3 }m^{ -3 }± 0.03 m

^{ 3 }m^{ -3 }Frequency

20 MHz

100 MHz

Operating temp.

0°C to 40°C

-20°C to 60°C

Operating EC

0 to 200 mS m

^{ -1 }50 to 500 mS m

^{ -1 }Calibration

Individual sensor calibration

Individual sensor calibration

Output

Reading value indicated

0 to 1.0 V

## Substrates

Three kinds of substrates were used in this study. Moss (

Sphagnumspp.) samples and two types of bark were purchased from a local supplier. The moss was grown in Chilin, Chile. ThePinus radiatabark samples were from New Zealand (Pacific wide Ltd., New Zealand). The mixed bark samples were from The Netherlands. Each cubic meter of substrate was composed of 70% bark and 30% polyethylene flakes mixed with 2.5 kg m^{ -3 }moss, 2.0 kg m^{ -3 }Dolokal (commercial name of a natural fertilizer), and 0.5 kg m^{ -3 }PG (peat substrate) mix (Anthura, 2005).To obtain a range of water contents for each substrate, 12 kg of substrate was dried at 105°C for 108 h to ensure that all the water was removed. After drying, a standard volume of deionized water was added to each substrate to reach the predeterminated water content and mixed thoroughly. Treated substrates were stored in plastic containers (650 mL) and sealed to ensure uniform water distribution by vapor transfer effect (Shen and Chen, 2007).

## Test Procedures

The probe of the FDR meter was inserted into each container. The opening of the container was sealed with plastic to prevent water from evaporating. The WET probe was connected to an HH2 moisture meter (Delta-T Devices, Ltd., Cambridge, U.K.), and the output values could be read directly. The output of the SM200 probe was measured with a digital multimeter.

The measuring principle of the FDR meter is to send an electrical signal along the probe and then detect the dielectric properties of the medium around the probe. According to preliminary tests, the contact conditions between the substrates and the probe did not affect the measurement performance.

The weights of the plastic containers and the initial weight of the container filled with substrate were recorded before the measurement of water content. After the measurement, each container was weighed again to ensure that there was no measurable water loss during the test. After measurements, the substrates were removed and dried (105°C for 72 h) to again confirm the water content.

The volumetric water content is defined as:

(3)

where ? is the volumetric water content (m

^{ 3 }m^{ -3 }),Vwis the volume of water contained in the substrate (m^{ 3 }), andVtis the total volume of the substrate (m^{ 3 }).The gravimetric water content is defined as:

(4)

where ?

gis the gravimetric water content (g g^{ -1 }),WHis the mass of water in the substrate (g), andWsin the mass of the dry substrate (g).The volumetric water content could be converted from the gravimetric water content:

(5)

where ?

sis the bulk density of the substrate (g cm^{ -3 }), and ?wis the density of the water (g cm^{ -3 }).The bulk density is defined as the ratio of the mass of dry substrate to the bulk volume of the medium with the standard method (Warncke and Krauskopf, 1983). The bulk volume is the volume of substrate in the containers. The mass of dry substrate is determined after drying it at 105°C for 72 h. The accuracy of this determination was within 3% according to the method of Warncke and Krauskopf (1983).

The volumetric water content detected by the gravimetric method served as the standard value for establishing the calibration equations.

## Calibration Equations

Seven calibration equations were introduced in which

yis the volumetric water content of the substrate as measured by the gravimetric method andxis the recorded value from the two FDR probes. The nonlinear distribution betweenyandxwas found in a previous study. Two of the equations were adopted from the literature, i.e., the second-degree polynomial equation (Nemali et al., 2007) and the third-degree polynomial equation (Overdium et al., 2005; Yoshikawa et al., 2004), and some nonlinear equations are proposed by us. These empirical equations could be applied to the TDR and FDR probes.Basic linear equation:

(6)

where

b_{ 0 }andb_{ 1 }are constants.Second-degree polynomial equation:

(7)

where

c_{ 0 },c_{ 1 }, andc_{ 2 }are constants.Third-degree polynomial equation:

(8)

where

d_{ 0 },d_{ 1 },d_{ 2 }, andd_{ 3 }are constants.Logarithmic linear equation:

(9)

where

e_{ 0 }ande_{ 1 }are constants.Logarithmic polynomial equation:

(10)

where

f_{ 0 },f_{ 1 }, andf_{ 2 }are constants.Logarithmic third-degree polynomial equation:

(11)

where

g_{ 0 },g_{ 1 },g_{ 2 }, andg_{ 3 }are constants.Nonlinear sigmoid equation:

(12)

where

a_{ 0 },a_{ 1 }, anda_{ 2 }are constants.## Statistical Analysis

The calibration equations were tested by regression analysis. SigmaPlot for Windows (ver. 10.0, SPSS, Inc., Chicago, Ill.) was used to estimate the parameters of the seven calibration equations at the 0.05 level of significance. The significance of each parameter was determined by Student's t-test. The criteria for evaluating the calibration equations are the coefficient of determination (R

^{ 2 }), the standard error of the estimation (SEE), the root mean square error (RMSE), and average absolute deviations (AAD).The error (

ei) is defined as:(13)

where

yiis the volumetric water content determined by the gravimetric method (m^{ 3 }m^{ -3 }), and is the predictive value of calibration equation (m^{ 3 }m^{ -3 }).The standard error of the estimation (SEE) is defined as:

(14)

where

nis the number of data, andpis the number of parameters for the calibration equation.The root mean square deviation (RMSE) is defined as:

(15)

The average absolute deviation (AAD) is defined as:

(16)

## Results and Discussion

## The WET Probe

The estimated parameters for five calibration equations for the recorded values of the WET probe are presented in table 2. The values of the third-degree terms (

d_{ 3 }x^{ 3 }in eq. 8, andg_{ 3 }x^{ 3 }in eq. 11) were found to be not significant and therefore were not considered for further discussion. The quantitative criteria of these calibrations are listed in table 3.Table 2. Estimated parameters of five equations for the WET probe with three types of substrate.

Model and Parameters

Moss

PinusBarkMixed Bark

Basic linear

b_{ 0 }0.168698

0.117362

0.188137

b_{ 1 }0.016753

0.024012

0.0146924

Second-degree polynomial

c_{ 0 }0.167911

0.89937

0.190502

c_{ 1 }0.016871

0.041944

0.017537

c_{ 2 }-2.92147 × 10

^{ -6 }-0.001692

-0.00061565

Logarithmic linear

e_{ 0 }-1.51384

-1.75655

-1.61986

e_{ 1 }0.036709

0.0692404

0.058036

Logarithmic polynomial

f_{ 0 }-1.71399

-2.20264

-1.66077

f_{ 1 }0.066683

0.218989

0.085645

f_{ 2 }0.00074337

-0.0104914

-0.0041632

Nonlinear sigmoid

a_{ 0 }1.0548

0.357864

0.330985

a_{ 1 }21.5213

2.41031

-1.47706

a_{ 2 }14.5684

2.69078

4.79192

Table 3. Criteria for evaluating five equations for the WET probe with three types of substrate.

Model and Parameters

Moss

PinusBarkMixed Bark

Basic linear

R

^{ 2 }0.9490

0.8633

0.8166

RMSE

0.04518

0.02951

0.03114

SEE

0.04540

0.03346

0.03121

AAD

0.03498

0.02324

0.02806

Residual plots

^{ [a] }Pattern

Pattern

Pattern

Second-degree polynomial

R

^{ 2 }0.9490

0.8974

0.8811

RMSE

0.04615

0.02120

0.02091

SEE

0.04551

0.02142

0.02122

AAD

0.03491

0.01622

0.01768

Residual plots

Pattern

UD

UD

Logarithmic linear

R

^{ 2 }0.9273

0.7390

0.8221

RMSE

0.04913

0.029380

0.03122

SEE

0.04898

0.03187

0.03155

AAD

0.04633

0.02651

0.02790

Residual plots

Pattern

Pattern

Pattern

Logarithmic polynomial

R

^{ 2 }0.9605

0.8821

0.8284

RMSE

0.0346

0.0259

0.03094

SEE

0.03356

0.02294

0.03135

AAD

0.02431

0.01823

0.02775

Residual plots

UD

Pattern

Pattern

Nonlinear sigmoid

R

^{ 2 }0.9569

0.8976

0.5412

RMSE

0.03516

0.02117

0.03092

SEE

0.03469

0.02150

0.03132

AAD

0.02516

0.01637

0.02769

Residual plots

UD

UD

Pattern

^{ [a] }Residual plots are the plots of the relationship between the residual values and predicted values of the calibration equations. "Pattern" indicates a clear pattern of residuals, and "UD" indicates a uniform distribution of residuals.## Moss Substrate

The relationship between the measured volumetric water content and the recorded values from the WET probe for moss water content measurement is presented in figure 1. For the residual plots of the basic linear equation (eq. 6) and the second-degree polynomial equation (eq. 7), a uniform error distribution did not occur, and a funnel-shaped distribution of residuals occurred instead (figs. 2a and 2b, respectively). These results indicate that the basic linear equation and the second-degree polynomial equation were not adequate for the WET probe because the variance of the volumetric water content was not kept constant. This non-constant variance should be overcome by data transformation, such as a logarithmic transform (Myers, 1986).

A clean systematic pattern of residuals was found for the logarithmic linear equation (eq. 9). However, a more uniform distribution of residuals was found for the logarithmic polynomial equation (eq. 10) (figs. 2c and 2d, respectively). Therefore, the latter equation was adequate for the WET probe.

The nonlinear sigmoid equation presented a uniform distribution of residuals (fig. 2e). The quantitative criteria in table 3 indicate that this equation has the highest R

^{ 2 }value and the lowest RMSE, SEE, and AAD values among all five equations for the moss media. The equation has good fitting ability.In summary, for the water content measurement of moss with the WET probe, the adequate calibration equations were found to be the logarithmic polynomial equation (eq. 17) and the nonlinear sigmoid equation (eq. 18):

(17)

(18)

These two equations could be considered adequate calibration equations. Compared with the quantitative criteria, the logarithmic polynomial equation had a higher R

^{ 2 }value and smaller SEE, RMSE, and AAD values than the nonlinear sigmoid equation (table 3).Figure 1. Relationship between volumetric water contents and recorded values of the WET probe for measurement of moss water content.

Figure 2. Residual plots of calibration equations for the recorded values of the WET probe for the measurement of moss water content: (a) basic linear equation, (b) second-degree polynomial equation, (c) logarithmic linear equation, (d) logarithmic polynomial equation, and (e) nonlinear sigmoid equation.

PinusBarkThe relationship between the measured volumetric moisture content and the recorded values from the WET probe for

Pinusbark water content measurement is a curve (fig. 3). According to the distributions of the residuals (figs. 4a to 4d), the basic linear equation had a clear pattern (fig. 4a). The second-degree polynomial equation (fig. 4b) and the nonlinear sigmoid equation (fig. 4c) had uniform distributions and could serve as adequate calibration equations forPinusbark. The other calibration equations all had clear patterns of residuals. The adequate calibration equation is:(19)

and (20)

The SEE values for the second-degree polynomial and nonlinear sigmoid equations were 0.02142 and 0.02150, respectively, and the AAD values were 0.01622 and 0.01637, respectively (table 3). The other five calibration equations all had the clean systematic patterns of residuals. The SEE values for the basic linear, logarithmic linear, and logarithmic polynomial equations were 0.03346, 0.03187, and 0.02294, respectively, and the AAD values for these equations were 0.02324, 0.02651, and 0.01823, respectively (table 3). The SEE and AAD values were 0.03346 and 0.02324, respectively, for the basic equation and 0.02142 and 0.01612, respectively, for the second-degree polynomial equation (table 3). Thus, an adequate equation could significantly improve the accuracy of the WET probe.

Figure 3. Relationship between volumetric water contents and recorded values of the WET probe for the measurement of

Pinusbark water content.Figure 4. Residual plots of calibration equations for the recorded values of the WET probe for the measurement of

Pinusbark water content: (a) basic linear equation, (b) second-degree polynomial equation, and (c) nonlinear sigmoid equation.## Mixed Bark

The relationship between the measured volumetric moisture content and the recorded values from the WET probe for mixed bark water content measurement is presented in figure 5. As indicated by the largest R

^{ 2 }value and the smallest SEE, RMSE, and AAD values (tables 2 and 3), the second-degree polynomial equation provided the best estimated ability. This calibration equation is:(21)

Figure 5. Relationship between volumetric water contents and recorded values of the WET probe for the measurement of mixed bark water content.

## SM200 Probe

The estimated parameters of the seven calibration equations for the SM200 probe are listed in table 4, and the quantitative criteria for these calibration equations are presented in table 5.

Table 4. Estimated parameters of seven equations for the SM200 probe with three types of substrate.

Model and Parameters

Moss

PinusbarkMixed bark

Basic linear

b_{ 0 }0.0728054

0.022201

0.0231

b_{ 1 }0.000833858

0.00231

0.0024

Second-degree polynomial

c_{ 0 }0.070537

-0.03005

-0.00174

c_{ 1 }0.00128505

0.0041

0.0041

c_{ 2 }-5.64875 × 10

^{ -2 }-9.451 × 10

^{ -6 }-7.1661 × 10

^{ -6 }Basic third-degree polynomial

d_{ 0 }-2.020982

-0.0312

0.0123

d_{ 1 }0.001992

0.0042

0.0037

d_{ 2 }-2.07985 × 10

^{ -6 }-1.0421 × 10

^{ -5 }-1.7933 × 10

^{ -6 }

d_{ 3 }8.08578 × 10

^{ -10 }2.7721 × 1

^{ -9 }-1.4526 × 10

^{ -8 }Logarithmic linear

e_{ 0 }-1.07007

-3.7207

-3.8485

e_{ 1 }7.9598 × 10

^{ -4 }0.0206

0.0201

Logarithmic polynomial

f_{ 0 }-2.10295

-4.6284

-4.8019

f_{ 1 }0.00430622

0.0529

0.0597

f_{ 2 }-2.31388 × 10

^{ -6 }-0.00021

-0.0021

Logarithmic third-degree polynomial

g_{ 0 }-3.99517

-5.2396

-5.2018

g_{ 1 }0.0194557

0.09301

0.09369

g_{ 2 }-2.8331 × 10

^{ -5 }-0.00061

-0.00578

g_{ 3 }0.2227 × 10

^{ -8 }1.3715 × 10

^{ -6 }1.1489 × 10

^{ -6 }Nonlinear sigmoid

a_{ 0 }0.802667

0.3640

0.5236

a_{ 1 }291.078

60.2178

75.3707

a_{ 2 }191.089

19.1046

26.5485

Table 5. Criteria for evaluation of seven equations for the SM200 probe with three types of substrate.

Model and Parameters

Moss

PinusBarkMixed Bark

Basic linear

R

^{ 2 }0.8576

0.8768

0.9126

RMSE

0.10983

0.04610

0.04509

SEE

0.11003

0.04657

0.04554

AAD

0.092558

0.03632

0.03461

Residual plots

^{ [a] }Pattern

Pattern

Pattern

Second-degree polynomial

R

^{ 2 }0.9689

0.9719

0.9781

RMSE

0.05106

0.02201

0.02259

SEE

0.04873

0.02334

0.02281

AAD

0.04305

0.01910

0.01734

Residual plots

Pattern

Pattern

UD

Basic third-degree polynomial

R

^{ 2 }0.9912

0.9720

0.9794

RMSE

0.02530

0.02201

0.0255

SEE

0.02458

0.02314

0.0223

AAD

0.02046

0.01903

0.01714

Residual plots

UD

UD

UD

Logarithmic linear

R

^{ 2 }0.8220

0.6075

0.5197

RMSE

0.12766

0.09805

0.1322

SEE

0.12902

0.09908

0.1323

AAD

0.09155

0.07159

0.0805

Residual plots

Pattern

Pattern

Pattern

Logarithmic polynomial

R

^{ 2 }0.8497

0.8547

0.8262

RMSE

0.1192

0.1138

0.09705

SEE

0.1213

0.1156

0.09711

AAD

0.09636

0.0681

0.0775

Residual plots

Pattern

Pattern

Pattern

Logarithmic third-degree polynomial

R

^{ 2 }0.8525

0.9327

0.8969

RMSE

0.1205

0.03763

0.05433

SEE

0.1213

0.03821

0.05548

AAD

0.09519

0.02209

0.03235

Residual plots

Pattern

UD

Pattern

Nonlinear sigmoid

R

^{ 2 }0.9421

0.9796

0.9607

RMSE

0.0626

0.01874

0.03023

SEE

0.0636

0.01903

0.03069

AAD

0.04894

0.01441

0.02585

Residual plots

Pattern

UD

UD

^{ [a] }Residual plots are the plots of the relationship between the residual values and predicted values of the calibration equations. "Pattern" indicates a clear pattern of residuals, and "UD" indicates a uniform distribution of residuals.## Moss

The relationship between the volumetric water content and the recorded values for the SM200 probe is shown in figure 6. Only the basic third-degree polynomial equation was adequate, as shown by the residual plots:

(22)

Figure 6. Distribution of data between the volumetric water content and the recorded values of the SM200 probe for the measurement of moss water content.

PinusBarkThe relationship between the volumetric water content of

Pinusbark and the recorded values of the SM200 probe is shown in figure 7. From the distributions of residuals, three calibration equations could be recognized as adequate:Basic third-degree polynomial equation:

(23)

Logarithmic third-degree polynomial equation:

(24)

Nonlinear sigmoid equation:

(25)

In comparing the quantitative criteria of the three adequate calibration equations, the nonlinear sigmoid equation was most adequate for the water content measurement of

Pinusbark with the SM200 probe.Figure 7. Distribution of data between the volumetric water content and the recorded values of the SM200 probe for measurement of

Pinusbark water content.## Mixed Bark

The relationship between the standard water content values and the recorded values for the SM200 probe for mixed bark is shown in figure 8. Two calibration equations, the second-degree polynomial equation and the nonlinear sigmoid equation, had uniform distributions of residuals. The basic third-degree polynomial equation did not have better fitting ability than the second-degree polynomial equation, so it was not considered for further analysis:

Second-degree polynomial equation:

(26)

Nonlinear sigmoid equation:

(27)

In comparing the quantitative criteria in table 5, the second-degree polynomial equation had better fitting ability than the nonlinear sigmoid equation.

Figure 8. Distribution of data between the volumetric water content and the recorded values of the SM200 probe for measurement of mixed bark water content.

## Calibration Equations for FDR or TDR Probes in the Literature

In a study of calibration equations for different custom-made substrates with two FDR probes (ECH

_{ 2 }O-10 and Theta ML2X), a polynomial equation was recommended as the calibration equation because the R^{ 2 }values of these fitting equations ranged from 0.95 to 0.96 (Nemali et al. 2007). In this study, the R^{ 2 }values of some calibration equations were higher than 0.96. However, the residual plots indicated that these equations could not be recognized as adequate. Overduin et al. (2005) proposed a linear equation to describe the relationship between the water content of feather moss and the recorded values of three FDR probes (WET, ECH_{ 2 }O, and CS615) and three TDR probes (TDR100, GroPoint, and HydraVitel). The accuracy of these meters was ± 0.06 of the measurement range. Yoshikawa et al. (2004) recommended a basic third-degree polynomial equation (y=d_{ 0 }+d_{ 1 }x+d_{ 2 }x^{ 2 }+d_{ 3 }x^{ 3 }) as the calibration equation for moss water content measurement using FDR and TDR probes, as did Overduin et al. (2005). The criterion of the model selection was the high correlation coefficients (R^{ 2 }> 0.99). In this study, the second-degree polynomial equation (y=c_{ 0 }+c_{ 1 }x+c_{ 2 }x^{ 2 }) was the adequate calibration equation for moss with the WET probe, and the third-degree polynomial equation was the best equation for the SM200 probe. The results of this study differ from those of Yoshikawa et al. (2004), perhaps because of the different evaluation criteria for the calibration equations.Maroufpoor et al. (2009) studied linear calibration equations for water content measurement using Trase system I TDR probes with five kinds of soils. The RMSE values ranged from 0.026 to 0.061 m

^{ 3 }m^{ -3 }. In this study, the RMSE values were less than 0.033 m^{ 3 }m^{ -3 }and the AAD values were below 0.02 for the adequate calibration equations. Thus, the performance of the FDR probes could be improved significantly by selecting an adequate calibration equation.The accuracy of the calibration equations can be expressed as the standard error of the estimation (SEE) (Eurachem, 2000). In this study, the accuracy of the volumetric water content measurement for moss,

Pinusbark, and mixed bark for the WET probe with the adequate calibration equations was 0.045, 0.0214, and 0.0212, respectively. The accuracy for the SM200 probe with the adequate calibration equations was 0.0254, 0.0190, and 0.0223, respectively.From the results of this study, no unique form of the calibration equation could be applied with both types of FDR probes for water content measurement of the three kinds of substrates. Each FDR probe had a specific calibration equation for the different substrates. An adequate calibration is important to ensure the accuracy of the FDR probes for water content measurement.

Calculation of the measurement uncertainty is important to express the performance of FDR and TDR meters. However, the components of uncertainty involve replicates of measurement, particle size distribution, moisture retention (at different suctions), standard deviations of the calibration equations, the precision of the dielectric water content meters, and the uncertain source of standard values of the tested substrates. Detailed calculation of the uncertainty measurement will be introduced and discussed in a further study.

## Summary

Because of its real-time, rapid response, and non-destructive operation, the frequency domain reflectometry (FDR) meter is suitable to measure the water content of substrates consisting of sphagnum moss and two types of bark. The relationship between the volumetric water content of the substrates and the recorded values of the FDR probe need to be established. An adequate calibration equation is important to ensure the performance of the FDR meter. In this study, two types of FDR probes were used to detect the volumetric water contents of three kinds of substrates. Seven calibration equations were selected to evaluate the fitting agreement of the measurement data. The evaluation criteria of the equations included residual plots and four quantitative criteria. Each FDR probe had a specific calibration equation for the different substrates. The accuracy of the adequate calibration equations for moss,

Pinusbark, and mixed bark was 0.033, 0.0214, and 0.0212, respectively, for the WET probe and 0.0254, 0.0190, and 0.0223, respectively, for the SM200 probe. Thus, the accuracy of FDR probes could be improved significantly with adequate calibration equations. This method also could be used to improve the performance of TDR meters.## REFERENCES

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a_{ 0 },a_{ 1 },a_{ 2 }= constantsAAD = average absolute deviation

b_{ 0 },b_{ 1 }= constants

c_{ 0 },c_{ 1 },c_{ 2 }= constants

d_{ 0 },d_{ 1 },d_{ 2 },d_{ 3 }= constants

e_{ 0 },e_{ 1 }= constants

ei= error

f_{ 0 },f_{ 1 },f_{ 2 }= constants

g_{ 0 },g_{ 1 },g_{ 2 },g_{ 3 }= constants

k= constant

n= number of data

p= number of the parameters for the calibration equationsSEE = standard error of the estimation

RMSE = root mean square error (m

^{ 3 }m^{ -3 })R

^{ 2 }= coefficient of determination

Vt= total volume of the substrate (m^{ 3 })

Vw= volume of water contained in the substrate (m^{ 3 })

WH= mass of water in the substrate (g)

Ws= mass of the dry substrate (g)

wi= volumetric fraction of the particular phase

x= recorded values of the FDR probe

y= volumetric water content of substrates measured with gravimetric method (m^{ 3 }m^{ -3 })? = volumetric water content (m

^{ 3 }m^{ -3 })?

g= gravimetric water content (g g^{ -1 })?

s= bulk density of the substrate (g cm^{ -3 })?

w= density of the water (g cm^{ -3 })= predictive values of calibration equation (m

^{ 3 }m^{ -3 })e

eff= relative permittivitye

_{ 1 }= relative permittivity of theiphase