Instrumentation & Measurement Magazine 24-5 - 61

monitoring, the acquired
spectra did not show significant
changes, allowing
for the conclusion that the
environmental effects can
be considered negligible.
Following the procedure
shown in Fig. 4, the
data points from the spectra
seen in Fig. 5 were taken
around the wavelengths
of 1505, 1530, 1550, 1570
and 1590 nm. The 1505 and
1590 nm data was needed
because the attenuation
band was not included in
the original range of the
low-cost platform (1530
to 1570 nm). This also represents
a real possibility,
because the inclusion of
more DFBs is a possible
and already considered
update to the system.
Fig. 6 shows the behavior
of the wavelength
corresponding to the minimum
amplitude of the
attenuation band in an Fe
coated LPFG in the presence
of corrosion. Based
Fig. 7. Time evolution of the corrosion evidence based on the corrosion data collected with the Fe coated LPFG.
on this data, the performance of this curve fitting routine in
the determination of the minimum wavelength of the LPFG
attenuation band can be evaluated. Although in the first few
measurements the error is considerable (because the spectra
are very wide, as seen in Fig. 5), good results were obtained in
most measurements.
Given the available data, the models M1 and M2
fined in the following manner:
◗ Model M1
were de:
" The data does not indicate the presence of
corrosion. " This model is well represented by the following
parametrization:
 ;00
min
t
(3)
◗ Model M2: " The data indicates the presence of corrosion. " .
This model is well represented by the following
parametrization:
  

min
t

00
  
 
; , , ,
(
t A ut t
A exp t t
 1 u t t
 

  0
00
In this model, λ0 is the initial minimum wavelength (before
corrosion), A and γ are parameters related to the exponential
decay, t0
is the initial moment of the corrosion process, and u(t)
represents the Heaviside function. This model parametrizes
August 2021
(4)
p D MI p D Mx x x I p x M I dx (5)
In (5), the integration is done over all M independent param((
) ()
 ,, , , ,
i , )  
px M I represents the prior (it is used a


, and pD M x x x I
i 12
eters of each model, (),ji
uniform one) of the parameter xj (), 12 , , , ,iM
is the likelihood of the model assuming the parameters
1
xx . This likelihood is given by:
()
,, M
pD M x x x I
iM
 2
 
k0
N
12
/2
Nexp  dt  min k
t
N
k


22
In the calculation of the likelihood for each model, is assumed
a Gaussian error following the procedure of Gregory
[7], chapter 3. In further studies this assumption will be tested.
In (6), N is the number of measurements, σ the standard deviation
of the error, d(tk
) is the minimum wavelength at a time
tk (determined with the curve fitting routine). The integration
IEEE Instrumentation & Measurement Magazine
61
,, ,, ,


(6)
can be defined as:

M
M ji j
j
,
the minimum wavelength as a constant until a moment t0
where the corrosion process starts, in which λmin
decays exponentially
(Fig. 6).
To calculate the evidence of each model for the computation
of the odds ratio (2), one needs to marginalize over all the
parameters in both models. A general calculation method of
the likelihood of each model Mi
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