Application of a GEO + SA hybrid optimization algorithm to the solution of an inverse radiative transfer problem

Figures & data

Figure 1. The canonical variant of GEO.

Figure 2. Scheduling examples for γ.

Figure 3. Cyclic scheduling example.

Figure 4. GEO + SA algorithm.

Table 1. Exact values of the radiative properties.

Radiative property	Case 1	Case 2	Case 3
Optical thickness, τ0	1.0	2.0	0.5
Single scattering albedo, ω	0.5	0.8	0.3
Diffuse reflectivity, ρ1	0.2	0.1	0.1
Diffuse reflectivity, ρ2	0.2	0.8	0.8

Figure 5. Average of the best values of the objective function, as a function of the number of function evaluations for Case 1, without noise.

Figure 6. Average of the best values of the objective function, as a function of the number of function evaluations for Case 1, with noise.

Table 2. Worst, average and best estimates for Case 1.

Exact value			fx^a
SA	Without noise	Worst	6.07 × 10^−8
		Average^b	1.80 × 10^−8
		Best	1.84 × 10^−9
	With noise	Worst	4.03 × 10^−4
		Average^b	4.03 × 10^−4
		Best	4.03 × 10^−4
µGA	Without noise	Worst	2.50 × 10^−4
		Average	4.93 × 10^−5
		Best	2.79 × 10^−6
	With noise	Worst	2.2755
		Average	1.8205
		Best	5.58 × 10^−4
GEO	Without noise	Worst	2.63 × 10^−3
		Average	1.31 × 10^−4
		Best	9.00 × 10^−6
	With noise	Worst	9.46 × 10^−4
		Average	6.16 × 10^−4
		Best	4.37 × 10^−4
GEO + SA	Without noise	Worst	3.05 × 10^−4
		Average	2.15 × 10^−4
		Best	1.39 × 10^−4
	With noise	Worst	6.94 × 10^−4
		Average	5.58 × 10^−4
		Best	4.45 × 10^−4
Notes: ^afx is the value of the objective function at the end of the run. ^bThe SA stopping criterion was reached for all runs in <25,000 evaluations.

Figure 7. Average of the best values of the objective function, as a function of the number of function evaluations for Case 2, without noise.

Table 3. Worst, average and best estimates for Case 2.

Exact value			fx^a
SA	Without noise	Worst	1.72 × 10^−8
		Average^b	8.64 × 10^−9
		Best	0.92 × 10^−9
	With noise	Worst	7.76 × 10^−5
		Average^b	7.75 × 10^−5
		Best	7.75 × 10^−5
µGA	Without noise	Worst	1.45 × 10^−4
		Average	7.11 × 10^−5
		Best	7.63 × 10^−6
	With noise	Worst	4.01 × 10^−4
		Average	3.10 × 10^−4
		Best	2.50 × 10^−4
GEO	Without noise	Worst	4.28 × 10^−4
		Average	1.72 × 10^−4
		Best	1.80 × 10^−5
	With noise	Worst	5.21 × 10^−4
		Average	2.62 × 10^−4
		Best	9.90 × 10^−5
GEO + SA	Without noise	Worst	4.94 × 10^−4
		Average	1.86 × 10^−4
		Best	2.37 × 10^−5
	With noise	Worst	3.09 × 10^−4
		Average	1.80 × 10^−4
		Best	8.87 × 10^−5
Notes: ^afx is the value of the objective function at the end of the run. ^bThe SA stopping criterion was reached for all runs in <25,000 evaluations.

Figure 8. Average of the best values of the objective function, as a function of the number of function evaluations for Case 2, with noise.

Figure 9. Average of the best values of the objective function, as a function of the number of function evaluations for Case 3, without noise.

Table 4. Worst, average and best estimates for Case 3.

Exact value			fx^a
SA	Without noise	Worst	3.08 × 10^−8
		Average^b	1.57 × 10^−8
		Best	8.49 × 10^−10
	With noise	Worst	1.08 × 10^−4
		Average^b	1.08 × 10^−4
		Best	1.08 × 10^−4
µGA	Without noise	Worst	5.82 × 10^−4
		Average	2.08 × 10^−4
		Best	2.52 × 10^−5
	With noise	Worst	4.8 × 10^−3
		Average	8.01 × 10^−3
		Best	1.6 × 10^−3
GEO	Without noise	Worst	1.91 × 10^−2
		Average	4.32 × 10^−3
		Best	2.08 × 10^−4
	With noise	Worst	3.31 × 10^−2
		Average	6.82 × 10^−3
		Best	8.17 × 10^−4
GEO + SA	Without noise	Worst	7.86 × 10^−3
		Average	3.86 × 10^−3
		Best	4.54 × 10^−4
	With noise	Worst	3.72 × 10^−3
		Average	2.05 × 10^−3
		Best	1.05 × 10^−3
Notes: ^afx is the value of the objective function at the end of the run. ^bThe SA stopping criterion was reached for all runs in less than 25,000 evaluations.

Figure 10. Average of the best values of the objective function, as a function of the number of function evaluations for Case 3, with noise.