Paramter Based Particle Swarm Optimization Reproducibility Test
This program examines the reproducibility of retrieved aerosol size distribution parameters by generating 10 sets (populations) of optimized global parameters. The salient features are:

• aerosol optical depths (AODs) at 13 wavelengths are optimized
• swarm size is 200 particles
• velocity weighting factor w=0
• individual and group acceleration coefficients c₁=c₂=1.8
• boundary conditions: particles moving outside solution domain are destroyed and a new random particle is generated
• 10 random scouts are generated each generation to limit falling into a local minimum
• options to hold α and/or β constant
• stop conditions:
  1. fractional change in all global parameters is ≤1.0E–5 for 100 consecutive generations or
  2. 5000 generations reached

Synthetic Data Sets Used

Name	Size Distribution	Parameters	AOD file
Junge	nJ(r)=N0βr–(α+1)	N0=2.5E6, α=3, β=0.3	junge-aod.txt
Modified Gamma	nG(r)=N0rβe–αrγ with γ=1	N0=2.5E9, α=0.1, β=0.35	mgamma-aod.txt
Lognormal	nL(r)=(2π)–1/2N0(βr)–1exp[–ln(r/α)2/2β2]	N0=1.0E7, α=0.5, β=0.307	lognormal-aod.txt
Bimodal	nB(r)=(2π)–1/2N0(β0r)–1exp[–ln(r/α0)2/2β02]         +(2π)–1/2N1(β1r)–1exp[–ln(r/α1)2/2β12]	N0=1.0E7, α0=0.1, β0=0.307 N1=2.5E5, α1=2.0, β1=0.307	bimodal-aod.txt

AOD file format
The 14 entries are the fractional day at the time of creation followed by AODs at 412, 440, 463, 479, 500, 520, 556, 610, 675, 750, 778, 870, and 1020 nm calculated using Bohren-Huffman Mie extinction coefficients for a refractive index of η=1.5+0i and the size distribution with specified parameters.
WMO standard wavelengths: 412, 500, 610, 675, 778 nm
WMO standard wavelengths shifted out of absorption regions: 463, 870, 1020 nm
Additional wavelengths (avoiding absorption regions): 441, 479, 520, 556, 750 nm

Results for Synthetic Data Sets

Name	Solution Space			Results file
Junge	1.0E3 ≤ N0 ≤ 1.0E9	2.0 ≤ α ≤ 5.0	0.1 ≤ β ≤ 0.5	junge-results.txt
Modified Gamma	1.0E6 ≤ N0 ≤ 1.0E12	0.05 ≤ α ≤ 0.50	0.1 ≤ β ≤ 0.5	mgamma-results.txt
Lognormal	1.0E4 ≤ N0 ≤ 1.0E10	0.05 ≤ α ≤ 1.0	0.1 ≤ β ≤ 0.5	lognormal-results.txt
Bimodal	1.0E5 ≤ N0 ≤ 1.0E11 1.0E3 ≤ N1 ≤ 1.0E9	0.05 ≤ α0 ≤ 0.50 0.50 ≤ α1 ≤ 5.0	0.1 ≤ β0 ≤ 0.5 0.1 ≤ β1 ≤ 0.5	bimodal-results.txt

Results file format

1. header line
2. whether α and/or β will vary
3. if scouts helped find global minumum along with generation number (possibly multiple entries)
4. global best N₀, α, β
5. global best N₁, α₁, β₁ if bimodal
6. number of generations required, population number
7. the next 13 lines are target (measured) AODs, calculated AODs for each wavelenth (412–1020 nm)
8. reduced chi-squared of expected and calculated AODs for this solution
9. repeat 3-8 for 10 populations
10. repeat 1-9 for each test

Plots of size distribution parameters within the explored solution domain that were retrieved by PBPSO can be seen here.