Environmental Engineering Reference
In-Depth Information
where 2.26 is the
t
-score associated with a confidence level of 95% and nine degrees
of freedom. Once
L
app
and
U
L
app
were both known for each of the ten-measurement
sets, referring to
U
L
as the expanded uncertainty of the probe electrical length in
air, the measurement uncertainty was evaluated using the uncertainty propagation
theory [34], according to the following equation:
⎡
L
U
L
2
⎤
2
∂ε
∂ε
i
i
⎣
⎦
=
U
ε
i
=
U
L
app
(
i
)
+
∂
L
app
(
i
)
∂
⎡
⎣
2
⎤
⎦
⎛
⎝
2
⎞
⎠
2
L
app
(
i
)
L
2
2
2
L
app
(
i
)
=
U
L
app
(
i
)
+
U
L
(5.7)
L
3
In the above equation,
U
ε
i
is the propagated uncertainty associated to the
i
-th mea-
surement of the
ε
app
value. Finally, data with corresponding uncertainty bars were
fitted in a
app
curve.
To determine the
θ
−
ε
relationship, the experimental data were fitted, thus
obtaining the optimal calibration curve. As also reported in many related papers,
third order polynomial curves prove to be a good fitting of experimental data:
ε
−
θ
app
app
app
θ
=
+
+
+
B
0
B
1
ε
B
2
ε
B
3
ε
(5.8)
app
where
B
0
,
B
1
,
B
2
and
B
3
are the regression coefficients. This way, (5.8) represents
the calibration curve: in correspondence of each measured dielectric constant value
ε
app
, the curve provides the corresponding
θ
.
from a metrological point of view,
the associated uncertainty was evaluated through the non-linear regression theory
[34]. Hence, the variance analysis for the single values expected from the previous
equation was conducted according to
To characterize the extrapolated value of
θ
2
1
(5.9)
∂θ
∂
2
∂θ
∂
cov
1
n
+
∑
i
∂θ
∂
ij
var
[
θ
]=
σ
+
var
[
B
i
]+
2
[
B
i
,
B
j
]
B
i
B
i
B
j
2
is the variance between ex-
perimental and fitted data;
n
is the number of experimental points;
i
where
var
[
θ
]
is the variance of the moisture level;
σ
,
j
= 0,1,2,3; and
cov
is the covariance between
B
i
and
B
j
parameters.
For a confidence level of 95%, the equations associated to the lower and upper
confidence limits for the regression curve are given by the following equations:
alpha
[
B
i
,
B
j
]
=
B
0
app
−
t
n
−
4
,
1
−
2
var
app
L
low
+
B
1
ε
+
B
2
ε
+
B
3
ε
[
θ
]
(5.10)
app
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