![Sample our Engineering & Technology journals, sign in here to start your access, latest two full volumes FREE to you for 14 days]({"type":"image" , "src" : "/sda/5933/TOC-Subject-Sample-2021-Engineering-Tech.jpg"})
ABSTRACT
Inspired by the particle swarm optimization algorithm, an improved ant colony optimization algorithm based on probability density function (IPDF-ACO) was developed for solving the inverse problem of frequency domain radiative transfer in a one-dimensional participating medium. The plural hemispherical reflectance and transmittance simulated by the finite-volume method served as input of the inverse analysis. The extinction coefficient, scattering albedo, and asymmetry parameter were retrieved simultaneously by using the amplitude and phase information as measured values. Through five test cases, the IPDF-ACO proved to be more accurate and efficient than the basic PDF-ACO even with noisy data.
Nomenclature
| c | = | speed of light, m/s |
| c1, c2 | = | two positive acceleration coefficients in Eq. (24) |
| Fobj | = | objective function |
| Gi,j | = | Gaussian-function probability density distribution |
| G | = | scattering asymmetry factor |
| I | = | radiative intensity, W/(m2 · K) |
| = | ||
| L | = | length of the media, m |
| M | = | measured value |
| N | = | total number |
| n | = | refractive index |
| Pi,j(t) | = | probability of the ith inverse parameter with the jth rank at the tth iteration |
| q | = | heat flux, W/m2 |
| r1, r2 | = | two uniform random numbers in [0, 1] |
| randn | = | standard normal distributed random number |
| t | = | time or iteration in ACO algorithm, s |
| tp | = | laser pulse width, s |
| v | = | velocity |
| w | = | inertia weight coefficient |
| Wj | = | weight of the jth rank |
| x | = | position |
| Y | = | inverse parameter |
| Greeks symbols | = | |
| α | = | factor of pheromone value |
| β | = | factor of heuristic information or extinction coefficient, m−1 |
| ϵd, ϵ0 | = | tolerance for minimizing the standard deviation and objective function |
| ϵrel | = | relative error, % |
| Φ | = | scattering phase function |
| γ | = | measurement errors, % |
| μ | = | direction cosine |
| μi,j (t) | = | retrieval value of the ith inverse parameter with the jth rank at the tth iteration |
| θ, θ′ | = | phase angle or outgoing and incoming directions |
| ρ | = | hemispherical reflectance or transmittance |
| σi,j (t) | = | standard deviation of ith inverse parameter with the jth rank at the tth iteration |
| σl | = | standard deviation of the measured value |
| τ | = | optical thickness |
| τj | = | pheromone value of the jth rank |
| ω | = | single scattering albedo or frequency, Hz |
| Subscripts | = | |
| c | = | collimated |
| d | = | diffuse |
| R | = | reflectance |
| T | = | transmittance |
| Superscripts | = | |
| E | = | estimated |
| g | = | global best |
| l | = | local best |
| M | = | measured |
| * | = | dimensionless term or the exact value |
| ∼ | = | exact measured value |
| – | = | mean retrieval value |
| ^ | = | temporal Fourier transform operator |
Nomenclature
| c | = | speed of light, m/s |
| c1, c2 | = | two positive acceleration coefficients in Eq. (24) |
| Fobj | = | objective function |
| Gi,j | = | Gaussian-function probability density distribution |
| G | = | scattering asymmetry factor |
| I | = | radiative intensity, W/(m2 · K) |
| = | ||
| L | = | length of the media, m |
| M | = | measured value |
| N | = | total number |
| n | = | refractive index |
| Pi,j(t) | = | probability of the ith inverse parameter with the jth rank at the tth iteration |
| q | = | heat flux, W/m2 |
| r1, r2 | = | two uniform random numbers in [0, 1] |
| randn | = | standard normal distributed random number |
| t | = | time or iteration in ACO algorithm, s |
| tp | = | laser pulse width, s |
| v | = | velocity |
| w | = | inertia weight coefficient |
| Wj | = | weight of the jth rank |
| x | = | position |
| Y | = | inverse parameter |
| Greeks symbols | = | |
| α | = | factor of pheromone value |
| β | = | factor of heuristic information or extinction coefficient, m−1 |
| ϵd, ϵ0 | = | tolerance for minimizing the standard deviation and objective function |
| ϵrel | = | relative error, % |
| Φ | = | scattering phase function |
| γ | = | measurement errors, % |
| μ | = | direction cosine |
| μi,j (t) | = | retrieval value of the ith inverse parameter with the jth rank at the tth iteration |
| θ, θ′ | = | phase angle or outgoing and incoming directions |
| ρ | = | hemispherical reflectance or transmittance |
| σi,j (t) | = | standard deviation of ith inverse parameter with the jth rank at the tth iteration |
| σl | = | standard deviation of the measured value |
| τ | = | optical thickness |
| τj | = | pheromone value of the jth rank |
| ω | = | single scattering albedo or frequency, Hz |
| Subscripts | = | |
| c | = | collimated |
| d | = | diffuse |
| R | = | reflectance |
| T | = | transmittance |
| Superscripts | = | |
| E | = | estimated |
| g | = | global best |
| l | = | local best |
| M | = | measured |
| * | = | dimensionless term or the exact value |
| ∼ | = | exact measured value |
| – | = | mean retrieval value |
| ^ | = | temporal Fourier transform operator |
Acknowledgements
A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.