Case Study CS-LDTC10 Rev. A

Page 4

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RESULTS

Simulation ReSultS

The results of the simulation of the improved concentration

inversion model were compared to another simulation of

the traditional concentration inversion model. The scatter

plots of Δx versus m and k

1(m) versus m are consistent with

the theoretical plots shown in

Figure 3 and Figure 4. Using

Equation 3 and Equation 5, the parameters can be fitted

as seen below:

m = 0.605 × Δx – 0.558

(

6)

C = 9.783 × 10-3m3 – 0.1144m2 + 0.3665m + 0.06627

P

2f – 0

(

7)

The scatter plot of inversed values versus true values of the

modulation depth is shown in

Figure 6.

Figure 6. Scatter plot of inversed values versus true

values of modulation depth m in simulations.1

To compare these expressions and values to those of the

traditional concentration inversion model, the root-mean-

square error (RMSE) is used. For the improved model

the fitted RMSE of modulation depth is 5.563 × 10-3, and

R2 is 99.97%. The calculated RMSE of concentration C is

5.766 × 10-5, and R2 is 1.1

The calculated RMSE of the traditional model concentration

C is 3.179 × 10-4, and R2 is 99.96%. This shows that the

RMSE of the new improved model is almost an order

of magnitude lower than the traditional model, and the

absolute value of relative error of the improved model

(<0.35%) is almost five times lower than that of the

traditional model (<1.54%).1 The relative error can be seen

in Figure 7.

Figure 7. Scatter plot of the absolute value of relative

error of inversed concentration C in simulations.1

expeRiment ReSultS

The same methods were used, as in the simulations, to

obtain and calculate the parameter fitting dataset and

scatter plots. Expressions obtained are below.

m = 0.5991 × Δx – 0.5178

(

8)

C = –0.5884 × m3 + 2.261 × m2 – 1.375 × m + 18.53

P

2f – 0

(

9)

Scatter plots of both the inversed values compared to the

true values of m and the relative error of both models can

be seen in

Figure 8 and Figure 9, respectively.

Figure 8. Scatter plot of inversed values versus true

values of modulation depth m in experiments.1

The RMSE between the inversed values and the true

values of the CO concentration using the improved

model is 5.468 × 10-5 while the traditional model RMSE

is 2.178 × 10-4. Relative error for the improved model is

<0.37%, and the relative error for the traditional model is

<1.70%. Once again, the experimental data and results

show almost an order of magnitude improvement with the