Inversion of spectroscopic data, application on CO₂ radiation of flame combustion

Figures & data

Figure 1. The observation of radiation intensity.

Figure 2. Radiative intensity for Gaussian distributions, and some subsets of wave numbers.

Figure 3. The first six right singular vectors and their images by the sensitivity matrix.

Figure 4. Evaluation of N_r(1% ) corresponding to adimensionalized parameters for four subsets of wave numbers.

Figure 5. Inversion result of retrieving all discretization values of profiles.

Figure 6. The four right singular vectors and their images by the sensitivity matrix.

Figure 7. Evaluation of N_r(1% ) corresponding to the subparameterization Φ for four subsets of wave numbers.

Table 1. Averaged values and averaged relative errors for different noise levels.

	Noise level	Tm ×tmax	αt	Cm×cmax	αc
	(%)	(K)	(cm)	(%)	(cm)
Real values p0	–	750	10	2	10
Initial values	–	490	7	1.4	7
Averaged values	1	749.754	9.913	2.024	9.881
	10	748.933	9.738	2.144	9.762
	20	750.779	10.58	1.992	11.282
Averaged relative	1	0.0739	1.409	1.859	1.975
error (%)	10	0.6255	13.211	19.815	18.725
	20	0.8127	18.611	24.040	30.4019

Figure 8. Temperature and concentration profiles for real case inversion.

Figure 9. Inversion CO₂ spectrum of the exhaust of a motor.

Table

C=	Concentration (%)
h=	Length of a segment (cm)
JΣ=	Objective function
k=	Boltzman constant, k=1.38 e^−23 (J K^−1)
k1=	Constant in Planck's Law, k1=1.19 e^−8 (W cm^3)
k2=	Constant in Planck's Law, k2=1.438 (cm K)
Kσ =	Absorption coefficient for σ (cm^−1)
Lσ, 0=	Background Radiation (W m^−2,sr^−1,cm^−1)
Lσ =	Radiative intensity for σ (W m^−2,sr^−1,cm^−1)
(---219----40) =	Measured intensity for σ (W m^−2,sr^−1,cm^−1)
(---219----41) =	Vector of (---219----42) for all σ ∈ Σ
(---219----43) =	Computed intensity for σ (W m^−2,sr^−1,cm^−1)
(---219----44) =	Vector of (---219----45) for all σ ∈ Σ
(---219----46) =	Local Planck function for σ (W m^−2,sr^−1,cm^−1)
N=	Number of subsegments
P=	Pressure (bar)
T=	Temperature distribution (K)
u=	State variable
(---219----47) =	Adjoint state variable
Φσ =	Lagrangian function for σ
σ=	Wave number (cm^−1)
Σ=	A set of wave numbers

Figure 10. Convexity over the segment between (t₀,c₀) and (0.3,0) of radiative intensities for all wave numbers σ ∈ Σ.