J Infrared Milli Terahz Waves (2013) 34:646–659
DOI 10.1007/s10762-013-0015-2

Prediction of the Thickness of a Thin Paint Filmby Applying a Modified Partial-Least-Squares-1 Method
to Data Obtained in Terahertz Reflectometry
Tetsuo Iwata & Shuji Yoshioka & Shota Nakamura &Yasuhiro Mizutani & Takeshi Yasui

Received: 27 June 2013 / Accepted: 31 July 2013 /Published online: 14 August 2013
# Springer Science+Business Media New York 2013

Abstract We applied a multivariate analysis method to time-domain (TD) data obtained interahertz (THz) reflectometry for predicting the thickness of a single-layered paint film
deposited on a metal substrate. For prediction purposes, we built a calibration model from
TD-THz waveforms obtained from films of different thicknesses but the same kind. Because
each TD-THz wave is approximate by the superposition of two echo pulses (one is reflectedfrom the air–film boundary and the other from the film–substrate boundary), a difference inthickness gives a relative peak shift in time in the two echo pulses. Then, we predictedunknown thicknesses of the paint films by using the calibration model. Although any
multivariate analysis method can be used, we proposed employing a modified partialleast-squares-1 (PLS1) method because it gives a superior calibration model in principle.
The prediction procedure worked well for a moderately thin film (typically, several toseveral tens of micrometers) rather than a thicker one.
Keywords Terahertz . Reflectometry . Partial least squares . Paint film . Thickness

1 Introduction
Recently, terahertz (THz) electromagnetic radiation, covering the frequency range of around
0.1–10 THz, has been used for determining the thickness of a thin coating. The major reason
for employing THz radiation in this application is that many coating materials are transparent in the THz frequency region but not in the visible and/or the mid-infrared wavelength
regions. Another reason is based on safety considerations: x-ray transmission, for example,
carries the risk of causing problems for human health. Although many promising
T. Iwata (*) : S. Yoshioka : S. Nakamura : Y. Mizutani : T. Yasui
Institute of Technology and Science, University of Tokushima, 2-1, Minami-Josanjima-cho, Tokushima,
Tokushima 770-8506, Japane-mail: iwata@tokushima-u.ac.jp
T. YasuiGraduate School of Engineering Science, Osaka University, 1-3, Machikaneyama-cho, Toyonaka,
Osaka 560-8531, Japan

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applications of THz techniques have been proposed [1–4], one of the most practicalapplications in industry is to determine the thicknesses of paint films deposited on metal
surfaces, such as car bodies and other objects [5–10], where a simple time-of-flight measurement technique in THz reflectometry is employed.
In this field, a crucial requirement is to lower the detection limit of the thickness
measurement. To achieve this, various efforts has been made, for example, reducing thewidth of the radiated THz pulses [11], introducing numerical deconvolution techniques indata processing [5], and using a practical parameter-fitting procedure [12]. Although suchtechniques work well to some extent, it is still difficult to achieve a detection limit less thanseveral tens of μm. The reason is that the two echo pulses (one is reflected from the air–filmboundary and the other from the film–substrate boundary) obtainable from THz reflectometry are temporally overlapped with each other by an amount that is too large to allow them
to be resolved. This difficulty is described in detail in the next section. If the sample issandwiched between two high-refractive-index media, the frequency dip due to Fabry–Perotoscillations could be utilized, and this might allow a simpler approach to measure thethickness, as pointed out by Koch et al. [13].
In the present paper, we propose a completely different approach for solving the problem:
we introduce a multivariate analysis technique in THz reflectometry. First, we gather a seriesof THz waveforms from many paint films whose thicknesses are known in advance anddistributed in a wide range. From these waveforms, we build a calibration model, which issometimes referred as a multivariate calibration (or working) curve. Then, by measuring theTHz waveforms of samples whose thicknesses are unknown, we predict their thickness
values by using the calibration model (or curve). In this scheme, the thickness of the paintfilm is a dependent variable (or objective variable), and discretized times in the TD-THz
waveform become independent variables (or explanatory variables). Although various
multivariate analysis methods are available, here we propose the use of a modified partialleast-squares-1 (PLS1) method because of the superiority of the calibration model
constructed. One of the aims of the present paper is to show the superiority of the modified
PLS1 method over the other methods when applied to THz reflectometry data.
As mentioned above, another point we emphasize in the present paper is that the
thickness of the paint film is set as the dependent variable. The thickness is correlated
directly with the time shift of the peak position of the two echo pulses. Therefore, information obtainable from the abscissa axis in the TD-THz waveform is set as the dependent
variable. Such a setting is in contrast with the conventional multivariate-analysis approach
used in spectroscopy. For example, in spectrochemical analysis, the concentration of thesample, which is related to the height of the peak (or the depth of the dip) in the spectrum, isset as the dependent variable, and information obtainable from the ordinate axis becomes thedependent variable. The reason why the prediction procedure in the proposed scheme works
well in spite of the “unusual” setting is that the amount of the peak shift is approximately(inversely) proportional to the change in height of the overlapped area of the two echopulses. Therefore, although it depends on the signal-to-noise ratio in the measurements, theprediction procedure works well only when the two echo pulses are overlapped by amoderate amount; without overlapping, it is obvious that the proposed scheme cannot work.
As a result, the prediction procedure can be carried out more successfully for a moderately
thin film rather than a thicker one.
One would think that adoption of such a setting or a scheme is obvious, but so far, wehave not been able to find such a scheme in the literature, especially in the field of THz
reflectometry. In the present paper, we focus on the application of this modified PLS1
method and show some numerical simulations and experimental results.

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2 Principle of Operation
2.1 THz Reflectometry
Figure 1 shows the principle of measuring the thickness of the paint film on the metal
substrate in THz reflectometry. As shown in Fig. 1(a), the THz pulse, Ein(t), is incident on
the surface of the single-layered paint film on the substrate with an incident angle θ = 0º,where E represents the electric field, and t time. The thickness and the group refractive index
of the paint film are d and ng, respectively. Then, as shown in Fig. 1(b), two THz echopulses, E1st(t) and E2nd(t), reflected from the air–film boundary and the film–substrateboundary are superimposed on each other with a relative time delay of Δt = t1 - t0 becauseof the discontinuity of ng. The synthesized waveform is therefore expressed by E1st(t) +E2nd(t). The time delay, or difference, between the two echo pulses is given by Δt = 2ngd/c.
Hence, if we know the value of ng, we can determine the geometric thickness d by measuring
Δt. When Δt is large in comparison with the pulse width of the THz waveform, we candetermine d easily. However, when d becomes small, precise measurement of Δt becomesdifficult because of overlapping of the two pulses. In order to solve this difficulty, we
employ the multivariate analysis method. In the multivariate analysis method, all uncertain
factors, such as noise, multiple reflections (even if they occur), and changes in the physicaland chemical conditions of the sample, are somehow modeled as the calibration curve,
although such factors might degrade the quality of the calibration model. In this sense, theprediction ability or limitation of the method is finally determined by the goodness of thecalibration model. Therefore, extension of the technique to the thickness measurement of amultilayered paint film is not difficult.
2.2 Modified PLS1 Method
Multivariate analysis methods can be classified into three categories: multiple linear regression (MLR), principal component regression (PCR), and partial least-squares (PLS).
Furthermore, the PLS method can be divided into two sub-methods: the PLS1 method andthe PLS2 method [14]. The difference between the PLS1 and PLS2 methods is the numberof dependent (objective) variables: in the PLS1 method, one dependent variable is predicted,
whereas in the PLS2 method, multiple variables are predicted at a time. Because the detailsof multivariate analysis methods have been described elsewhere [14–19], only a brief
explanation is given here. The aim of the present paper is to propose a suitable schemeand to show some experimental results.
We propose the use of a PLS method. The reason is that the PLS method is free from thecollinear problem suffered by the conventional MLR method [14]. In addition, the quality ofFig. 1 (a) Principle of THz reflectometry and (b) the corresponding THz waveforms

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the calibration model built by the PLS method is superior to that constructed by the PCR
method. In particular, we use the PLS1 method, which targets one dependent variable,
because it is suited for application to spectral datasets [15]. Furthermore, we introduce twotechniques into the PLS1 method. One is called the uninformative-variable-elimination(UVE) technique [16], and the other is the uninformative-sample-elimination (USE) technique [17]. UVE is a procedure by which independent variables (discretized times) thatcannot contribute to the calibration model are eliminated, and only those that can contributeare retained [18]. USE is a procedure by which calibration sample data (TD-THz waveforms
themselves) that cannot contribute to the model are eliminated [19]. The word
“uninformative” means “not relevant to building the calibration model”. The reason for
introducing these two techniques in the PLS1 method is to build a calibration model that ismore robust against noise than a model that does not use these techniques [20]. We call the
PLS1 method combined with the two techniques the “modified version of the PLS1
method”. We tried two versions: UVE-PLS1 (a combination of UVE and the PLS1 method)and UVE-USE-PLS1 (a combination of UVE, USE, and the PLS1 method). In the UVEPLS1 method, the UVE procedure is carried out before applying the PLS1 method.
Similarly, in the UVE-USE-PLS1 method, the UVE and USE procedures are carried out
before applying the PLS1 method. In the present paper, we show the results of comparing
PLS1, UVE-PLS1, and UVE-USE-PLS1.
2.3 RMSEP Criterion
For the purpose of conducting a quantitative comparison of the three calibration models, weused a criterion known as the root-mean-square error of prediction (RMSEP), which isdefined as [14, 15]vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
uX
u N
N 
⌢ 2 u Xuuud−djΔd i j2
i
i
t
t
i¼1
i¼1
¼
;
ð1Þ
RMSEP ¼
NN
where N is the total number of calibration datasets (the total number of TD-THz waveforms
or the total number of paint films whose thicknesses are known in advance), and di is thereference value of the i-th dependent variable (thickness of the i-th paint film). We measured
the reference values with an eddy-current thickness meter (Coating Thickness Tester LZ300C, Kett Electric Laboratory) and/or from microscope observations. d i is the i-th predicted thickness value obtained by using the calibration model built from N-1 calibration
datasets, except for the i-th one. Therefore, Δd i ¼ jd i −d i j gives an absolute valueof the prediction error for the i-th sample. This prediction procedure is repeated N times from i =1 to N. The RMSEP value obtained from such a “leave-one-out” procedure gives asort of standard deviation in prediction and can be used as a measure of the goodnessof the calibration model. The smaller the RMSEP value, the better the calibration
model obtained.
The UVE procedure was carried out by adding random noise to the calibration dataset as
independent variables. The number of noise variables was made equal to that of the original
independent ones. Then, in the course of the conventional PLS1 procedure, based on the
RMSEP criterion, we identified the independent variables that contributed to the model morerather than the noise ones. The USE procedure was carried out by judging the prediction
error: the sample data that gave large prediction errors were eliminated in the course of the
USE-PLS1 procedure [15–18].

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3 Numerical Simulations
3.1 Highly-Overlapped Gaussian Waveforms
First, we have to verify whether the proposed scheme worked well. Specifically, we have to
confirm whether the abscissa information could be extracted successfully from highly
overlapped waveforms by using the PLS1 method. To this end, we carried out numericalsimulations first by using a simple Gaussian waveform:()
ðx−pÞ2;
ð2Þ
f ðxÞ ¼ Aexp −2σ2where A is the peak height, p the peak position, and σ the standard deviation. The numericalsimulation using a “slightly complicated” synthesized THz waveform is presented in thenext section.
We prepared two datasets. One consisted of fifteen waveforms (N = 15), as shown in theupper part of Fig. 2(a), where p was varied from 730 to 870 with a step interval of 15 for
fixed values A = 10 and σ = 80. The total number of data points was M = 1,600. Then, we
superimposed uniformly distributed noise having an amplitude of 1.0 % with respect to thepeak value of each waveform. The other dataset is shown in the upper part of Fig. 2(b),which is the same as Fig. 2(a) except that the peak height A was varied from A = 10 to 2/3 insteps of 2/3.
The results of applying the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods to thedataset shown in Fig. 2(a) are summarized in Table 1(a), where bpi is the predicted value ofthe peak position of the i-th sample. The number of retained independent informative
variables used to build each calibration model is also shown in Table 1(a) and is depicted

Fig. 2 (a) Calibration dataset (N = 15 and M = 1,600), in which each waveform was a single Gaussian
waveform, where the peak position p was varied from 730 to 870 with a step of 10 for fixed values of A = 10
and σ = 80. Uniformly distributed noise with amplitude of 1.0 % with respect to the peak value wassuperimposed on each waveform. The lower part of the figure shows informative independent variables aslevel one and uninformative ones as level zero. (b) The same as (a), but the peak height was varied from A = 10
to 2/3 in steps of 2/3

(b)

(a)

740

741

740

730

737

723

723

730

729

729

723

pi

PLS1

UVE-PLS1

UVE-USE-PLS1

pi

PLS1

UVE-PLS1

UVE-USE-PLS1

741

738

738

741

740

2

1

Sample number i

750

750

749

750

750

750

748

750

3

759

760

760

760

759

759

758

760

4

770

771

771

770

770

770

767

770

5

782

781

781

780

782

782

777

780

6

793

791

791

790

793

793

788

790

7

800

800

802

800

800

800

799

800

8

810

811

812

810

810

810

812

810

9

820

821

822

820

820

820

823

820

10

831

830

831

830

831

831

833

830

11

840

841

840

840

840

840

841

840

12

850

850

850

850

850

850

852

850

13

856

859

858

860

856

856

860

860

14

876

867

865

870

876

876

864

870

15

1.00

1.00

1.81

2.54

2.54

3.05

RMSEP

213

213

1,600

960

960

1,600

Number of retained
variables

Table 1 (a) Predicted values of bpi for the i-th sample pi, RMSEP values, and the number of retained variables for the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods. The
calibration dataset is shown in Fig. 2(a). (b) The same as (a) but showing those obtained from the calibration dataset shown in Fig. 2(b)J Infrared Milli Terahz Waves (2013) 34:646–659
651

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graphically in the lower part of Fig. 2(a). In the figure, the parts having level one are relevant
for building the calibration model for the thickness extraction (informative), whereas theparts having level zero are not relevant (uninformative). The independent variables aroundthe overlapped peak region in the waveforms were retained as informative variables. The
number of retained variables with the USE-UVE-PLS1 method was the same as that with the
UVE-PLS1 method. This means that no sample was eliminated in the USE-UVE-PLS1
method for this case. The prediction procedure, therefore, worked well even for the case of
the UVE-PLS1 method.
The results of applying the methods to the dataset shown in Fig. 2(b) are summarized inTable 1(b). In this case, prediction errors and RMSEP values lower than those obtained in
the previous case. The reason is that two kinds of information, peak height and peak
position, were effectively used to build the calibration model. The number of retained
independent informative variables used to build each calibration model is also shown in
the lower part of Fig. 2(b). In general, the RMSEP value increased with decreasing degree ofoverlap of the calibration waveforms.
In summary, we found that the proposed scheme worked quite well for datasets with
Gaussian waveforms.
3.2 Synthesized THz Waveforms
Next, we carried out numerical simulations by using synthesized THz waveforms. We madea calibration dataset, E(t), from a reference waveform, Er(t), according to the equation E(t)=Er(t)+ΔEr(t+Δt), where Er(t) is a TD-THz pulse in free space. We used the actually measured
waveform, Er(t), reflected from bare aluminum metal. To generate the synthesized THz
waveforms, we set α = 1 and varied Δt from 0.0125 to 0.5 ps in steps of 0.0125 ps. The
reference values of the thicknesses, d, of the paint films were calculated from the equationd= cΔt/(2ng), with ng = 1.66. For each synthesized waveform, we superimposed 1.0%
uniformly distributed noise. Figure 3 shows the synthesized waveforms together with the
waveform of Er(t), where M = 3,960. For the sake of clarity, we show only three waveforms(i = 13, 28, and 40) listed in Table 2. The synthesized waveforms were severely overlapped

Fig. 3 Numerically synthesized THz waveforms (see text). For the sake of clarity, we show only three
waveforms (i =13, 28, and 40, listed in Table 2) and the reference waveform Er(t). In the lower part of the
figure, informative variables are shown as level one

1

1.1

1.9

1.1

1.1

Sample number idi (μm)PLS1

UVE-PLS1

UVE-USE-PLS1

4.7

4.7

4.6

4.5

4

7.9

7.9

7.8

7.9

7

11.2

11.2

11.0

11.3

10

14.4

14.4

14.2

14.7

13

18.0

18.0

17.7

18.1

16

21.5

21.5

21.4

21.5

19

24.9

24.9

24.9

24.9

22

28.3

28.3

28.4

28.3

25

31.7

31.7

31.8

31.6

28

35.2

35.2

35.3

35.0

31

38.5

38.5

38.5

38.4

34

41.5

41.5

41.6

41.8

37

45.2

45.2

44.4

45.2

40

0.19

0.19

0.33

RMSEP

699

699

3,960

Number ofretained variables

Table 2 Predicted values of d i for the i-th sample di, RMSEP values, and the number of retained variables for the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods. The
calibration dataset is shown in Fig. 3, where N = 10 and M = 3,960

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Fig. 4 Calibration waveforms obtained from various thickness of oil-based, acrylic, black paint films on analuminum substrate, where N = 12 and M = 900. In the lower part of the figure, informative variables areshown as level one

in time when d < 20 μm. Predicted values of b
d i for the i-th sample and RMSEP values
obtained from the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods are listed in Table 2 for
i = 1, 3, 6,. . . . , 37, and 40. In the lower part of Fig. 3, informative independent variables areshown. In spite of the high degree of overlapping of the waveforms, a film thickness of lessthan 10 μm could be predicted with a prediction error Δdi < 2 μm. In this numericalsimulation, the error might be due to the artificially added noise.

4 Experimental Results and Discussion
4.1 Dry Paint Film
We prepared twelve calibration samples (N = 12) with different thicknesses. The samples
were oil-based, acrylic, black dry paint films deposited on aluminum substrates. The
experimental setup was the same as that described in the previous paper [6, 12], where
typical THz time-of-flight measurement in a reflection geometry was used. The grouprefractive index of the film in the THz wavelength region was ng = 1.66, which was obtained
Table 3 Predicted values of d i for the i-th sample di, RMSEP values, and the number of retained variables for
the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods. The calibration dataset (N = 12 and M = 900) is shownin Fig. 4
Samplenumber i

1

2

3

4

5

6

7

8

9

10

11

12

RMSEP Number ofretainedvariablesdi (μm)

6.5 6.8 9.4 11.0 12.2 14.3 15.2 15.5 18.2 18.9 21.0 21.2

PLS1

5.6 8.9 9.0 10.2 11.9 12.9 14.4 14.1 17.8 20.1 22.3 20.4 1.11

900

UVE-PLS1 6.6 9.3 8.8 10.3 12.2 14.1 14.1 14.5 18.4 19.9 21.7 20.4 0.99

134

UVE-USE- 7.3 PLS1

588

9.7 11.0 12.6 14.6 14.3 14.4 17.6 19.3 21.4 20.3 0.71

1

2.5

7.4

5.8

5.8

Sample number idi (μm)PLS1

UVE-PLS1

UVE-USE-PLS1

6.2

6.2

6.0

5.0

2

7.4

7.4

7.1

6.0

3

6.4

6.4

7.0

6.5

4

7.9

7.9

7.6

7.0

5

8.0

8.0

6.3

8.0

6

8.7

8.7

7.6

9.0

7

9.9

9.9

9.4

9.5

8

10.6

10.6

11.1

11.0

9

11.6

11.6

12.5

13.0

10

11.0

11.0

11.7

13.5

11

16.8

16.8

17.6

16.0

12

14.6

14.6

14.2

17.0

13

19.4

19.4

19.1

18.0

14

19.4

19.4

17.9

20.0

15

1.47

1.47

1.85

RMSEP

699

699

900

Number of retained
variables

Table 4 Predicted values of d i for the i-th sample di, RMSEP values, and the number of retained variables for the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods. The
calibration dataset (N = 15 and M = 900, not shown) was obtained from white polyester resin paint films deposited on galvanized steel substrates

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656
Fig. 5 Rescaled andreconstructed THz waveforms
obtained from a wet paint film foreight elapsed times from 0 to 8
minutes with a step interval of one
minute after painting. In the lower
part of the figure, informative
variables are shown as level one

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Table 5 Predicted values of d i for the i-th sample di, RMSEP values, and the number of retained variables for
the PLS1, UVE-PLS1, and USE-UVE-PLS1 methods. The calibration dataset (N = 9 and M = 5,000) is shownin Fig. 5
Sample number i

1

2

3

4

5

6

7

8

9

RMSEP Number ofretained variables

Elapsed time (min) 0
1
2
3
4
5
6
7
8
143 115 98.0 88.0 83.0 79.0 78.0 77.0 75.0
di (μm)PLS1

120 112 99.0 87.8 80.9 84.0 77.2 75.9 78.6 8.12

UVE-PLS1

128 114 99.2 87.5 80.8 82.5 77.1 75.9 78.8 5.34

2,298

UVE-USE-PLS1

-

444

-

95.7 88.9 83.3 80.2 76.7 76.1 77.1 1.43

5,000from the literature [6, 12]. The reference thickness values of the paint films were obtainedusing an eddy-current meter. The precision of the thickness measurements at 20 μm wasaround 3.0 % according to the manufacturer’s data sheet. The relative standard deviation ofrepetitive measurements of the same film was about 10 %. The relatively large deviation
might be due to the spatial non-uniformity of the film thickness.
The upper part of Fig. 4 shows the twelve THz waveforms used as the calibration dataset(M = 900). Table 3 summarizes predicted values of b
d i and RMSEP values obtained from the
PLS1, UVE-PLS1, and USE-UVE-PLS1 methods. With the USE-UVE-PLS1 method,
sample #2 was eliminated as uninformative, resulting in a smaller value of RMSEP. From
this experimental result, the USE-UVE-PLS1 method produced a better calibration model
than the UVE-PLS1 or PLS1 method. For reference, we show informative variables in thelower part of Fig. 4. Thus, the USE-UVE-PLS1 method enables us to predict the film
thickness with a prediction error of Δdi < 0.9 μm for di < 20 μm.
Next, we measured the thickness of a white polyester resin paint film on a galvanizedsteel substrate, which is used as a pre-painted steel sheet in many steel products. We
prepared fifteen calibration samples (N = 15 and M = 900) whose thickness were from 2.5
to 18.0 μm (waveforms not shown). Table 4 summarizes predicted values of b
d i for the i-th
sample and RMSEP values obtained from the PLS1, UVE-PLS1, and USE-UVE-PLS1
methods. The USE-UVE-PLS1 method gave a relative prediction error of around 10 %,except for d1 = 2.5 μm, whose prediction error was the lower detection limit in this case.
4.2 Wet Paint Film
One of the advantages of the THz paint meter over the conventional one is the ability tomonitor the drying process of a wet film. Such an ability was already confirmed in theprevious paper by using a parameter fitting method [12]. In the present paper, in order todemonstrate the usefulness of the proposed scheme, we applied a modified PLS1 method tothe same dataset (Fig. 8 in Ref. 12). The THz waveforms corresponding to elapsed time from0 to 8 minutes with a step interval of one minute after painting (0 min) are shown in Fig. 5,
where M = 5,000. As reference thickness values for building the calibration model, we
directly used estimated values of di obtained from the fitting method (Table II in Ref. 13).
The predicted values of b
d i for the i-th sample obtained by the proposed method aresummarized in Table 5. As was discussed in the previous paper [12], we found that the
thickness of the film reduced with the elapsed time. From this we can judge whether thepaint is dried, although the change in refractive index (and absorption coefficient) during thedrying must affect the obtained thicknesses. The informative variables are shown in the

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lower part of Fig. 5. Again, the USE-UVE-PLS1 method gave the best calibration model for
prediction. In this case, sample #1 and #2 were eliminated as uninformative in building themodel.
In the proposed scheme, a major factor that determines the precision in predicting the
thickness of the film is the uncertainty in detecting the zero-point position of time in the THz
waveforms. The uncertainty is mainly due to the poor reproducibility of the mechanical error
in attaching the sample: which might be around a few micrometers. In such a case, asystematic error is introduced when building the calibration model. In addition, for thicker
paint films, the leading edge of the first echo pulse appears earlier in time, and the trailingedge of the second echo pulse appears later, even if the zero position could be determined
precisely. Taking these facts into account, we defined the zero position as a 10%-constantfraction point on the leading edge of the first echo pulse. Because only the relative time shift
is important in building the calibration model, this procedure worked well. Another factor
that affects the precision and accuracy in prediction is errors due to the reference thickness
values used for building the calibration model. However, the overall precision and accuracy
in the proposed scheme might be improved if the number of calibration samples is increased,
thanks to employing the multivariate analysis method.

5 Conclusions
We demonstrated that the application of a multivariate analysis method to a dataset obtainedin THz reflectometry is effective for predicting the thickness of a single-layered paint film.
Also, we found that the modified PLS1 method is useful. The prediction procedure worked
well, provided that an accurate calibration model is built in advance, even for a thickness ofless than 10 μm. Although we applied the method to a single-layered film in the present
paper, the proposed scheme is applicable to data obtained from multilayered paint films
merely by increasing the number of the dependent variables.
Acknowledgements This work was supported by Grants-in-Aid for Scientific Research 21650111 and23656265 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. We also
gratefully acknowledge financial support from the Renovation Center of Instruments for Science Education
and Technology at Osaka University, Japan. The authors thank Mr. Hiroyasu Furukawa of Nippon Steel and
Sumitomo Metal Corporation, Japan, for preparation of the paint samples and Drs. Hideyuki Ohtake and Jun
Takayanagi of AISIN SEIKI Co. Ltd., Japan, for conducting the THz reflectometry measurements.

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