3b · Mie, Rayleigh, and the Phase Matrices

For: users adding aerosols to their scene; method developers extending the scattering codebase; retrieval developers reasoning about polarization sensitivity.
Prev: 3a · Gas Absorption · Next: 3c · Mixing & δ-M Truncation

This page covers three of the four layer-optics arrays plus the aerosol contribution to the optical depth: the Fourier-moment phase matrices and , the aerosol single-scattering albedo , and the extinction cross-section (which maps microphysics to ).

What this page produces

For each aerosol type and each Fourier moment :

Symbol	Meaning	Used for
,	phase matrix Fourier moments	layer scattering
	aerosol single-scattering albedo	layer SSA
	extinction cross-section	aerosol optical depth from microphysics
	δ-M forward-truncation factor	consumed in Concepts/03c

For Rayleigh (and Cabannes), the same are computed from the Rayleigh / Cabannes Greek source.

The Mie pipeline

   Aerosol(size_dist, n_r, n_i)
        │
        ▼
    MieModel
        │
        ▼
   compute_aerosol_optical_properties
        │
        ▼
   AerosolOptics(GreekCoefs, ω̃, k, fᵗ)
        │
        ▼
   compute_Z_moments  (per Fourier moment m)
        │
        ▼
   (Z⁺⁺, Z⁻⁺) per (μ', μ, m)

The user-facing entry is compute_aerosol_optical_properties(model::MieModel) in src/Scattering/. The dispatcher branches on the decomposition algorithm (NAI2() or PCW()) carried in MieModel.

The Greek matrix `B_l`

Polarized Mie scattering is fully captured by the Greek expansion of the phase matrix (Sanghavi 2014 App. A; Sanghavi 2014 Mie-Fourier Eq. 16):

The six coefficients fully determine the polarized scattering pattern — is the phase function expansion (the only one a scalar code uses); the other five carry polarization coupling. They live in the GreekCoefs struct at src/Scattering/types.jl:231–244:

julia

struct GreekCoefs{FT}
    α::Vector{FT}
    β::Vector{FT}
    γ::Vector{FT}
    δ::Vector{FT}
    ϵ::Vector{FT}
    ζ::Vector{FT}
end

The output of compute_aerosol_optical_properties packages this with the scalar bookkeeping (src/Scattering/types.jl:281–292):

julia

struct AerosolOptics{FT}
    greek_coefs::GreekCoefs
    ω̃::Union{FT, Array{FT}}      # single-scattering albedo
    k::Union{FT, Array{FT}}      # extinction cross-section
    fᵗ::Union{FT, Array{FT}}     # δ-M truncation factor
end

Phase matrix `Z` from Greek + μ

Once the Greek coefficients are in hand, the per-Fourier-moment phase matrices come from , the generalized spherical-function matrices (Sanghavi 2014 Mie-Fourier Eq. 12–14):

The coded entry point is compute_Z_moments(pol_type, μ, greek, m, arr_type=arr_type) in src/Scattering/compute_Z_matrices.jl. It returns of shape (NquadN, NquadN) (or the pre-broadcast scalar-spectral version that gets expanded across nSpec in expandOpticalProperties).

NAI-2 vs PCW — two ways to compute the Greek expansion

Sanghavi 2014 (the Mie-Fourier paper) compares two decomposition methods:

NAI-2 (Numerical Angular Integration, version 2; Siewert's formalism): computes the scattering matrix on a Gauss-Legendre quadrature in scattering cosine , then projects to Greek coefficients via integrals against associated Legendre functions. The quadrature roots needed for accurate integration are determined upfront. Default path; entry at src/Scattering/compute_NAI2.jl:44.
PCW (Pre-computed Wigner; Domke's formalism corrected): computes Greek coefficients directly from products of size-averaged Mie coefficients etc. weighted by precomputed Wigner-3j symbols. No reconstruction of the angular scattering matrix. Entry at src/Scattering/compute_PCW.jl:28.

Tradeoffs (Sanghavi 2014 Mie-Fourier §5):

Regime	Faster method
Single particle, (size parameter > 228)	NAI-2 (no angular reconstruction needed)
Polydispersion over a finite size range up to ~80 μm	PCW (6–8× faster than NAI-2)
Single small particle	PCW (precomputed Wigner-3j makes individual particles cheap)

NAI-2 is the safer default. PCW pays off for big polydisperse populations that dominate aerosol retrievals over ocean. Selection is via the DECOMP_MAP in src/IO/Parameters.jl ("NAI2" or "PCW").

Rayleigh and the Cabannes choice

For the molecular contribution (no aerosols, just air molecules):

Pure-elastic runs (noRS) use the full Rayleigh Greek expansion. In this case rotational Raman is rolled into the effective depolarization ratio ; the phase matrix is the standard Rayleigh form with depolarization of ~0.028 for air at 532 nm.
Raman-aware runs (RRS, VS_*) use the Cabannes Greek expansion. Rotational Raman is then handled explicitly by the inelastic kernel (Concepts/08), and the elastic Cabannes path uses a lower depolarization (~0.007).

The selector at src/CoreRT/LayerOpticalProperties/compEffectiveLayerProperties.jl:8–9:

julia

_rayleigh_greek_source(::Union{noRS, noRS_plus}, greek_rayleigh, greek_cabannes) = greek_rayleigh
_rayleigh_greek_source(::AbstractRamanType, greek_rayleigh, greek_cabannes) = greek_cabannes

Mismatch — using Cabannes greek with noRS, or full Rayleigh greek with RRS — produces a ~1% bias on Stokes and a ~3% bias on , because the polarization-sensitive coefficients (, ) shift by the depolarization difference.

Aerosol microphysics → optics

The size distribution and refractive index are inputs to the Mie calculation. vSmartMOM ships with two microphysics schemes (in the Aerosols module):

TOMAS15Scheme — TOMAS-15-bin sectional scheme used in CESM/GEOS-Chem.
TwoMomentScheme — two-moment lognormal scheme (median radius + width).

Plus a refractive-index database for common aerosol species (sulfate, dust, black carbon, organic carbon, sea salt). See Aerosols (microphysics) for the schemes (status: API still being stabilized — WIP).

Code anchors

Concept	Source
Mie entry (NAI-2)	`src/Scattering/compute_NAI2.jl:44`
Mie entry (PCW)	`src/Scattering/compute_PCW.jl:28`
Mie helper (a_n, b_n)	`src/Scattering/mie_helper_functions.jl`
GreekCoefs type	`src/Scattering/types.jl:231–244`
AerosolOptics type	`src/Scattering/types.jl:281–292`
Polarization types	`src/Scattering/types.jl:92–143`
Z from Greek + μ	`src/Scattering/compute_Z_matrices.jl::compute_Z_moments`
Truncation entry	`src/Scattering/truncate_phase.jl` (used by `createAero`)
Rayleigh / Cabannes selector	`src/CoreRT/LayerOpticalProperties/compEffectiveLayerProperties.jl:8–9`
GPU Mie kernel	`src/Scattering/gpu_mie_kernels.jl:30–100`
YAML decomposition map	`src/IO/Parameters.jl::DECOMP_MAP`

Hands-on tutorials

Runnable examples with Plotly figures:

References

Sanghavi (2014), Revisiting the Fourier expansion of Mie scattering matrices in generalized spherical functions, JQSRT 136:16–27, doi:10.1016/j.jqsrt.2013.12.015. NAI-2 vs PCW comparison; corrected Domke formalism.
Sanghavi et al. (2014), JQSRT 133:412–433, doi:10.1016/j.jqsrt.2013.09.004, App. A.
Hansen, J. E. & Travis, L. D. (1974), Light scattering in planetary atmospheres, Space Sci. Rev. 16:527. (Background on Mie + Greek matrix.)
Siewert, C. E. (1981, 1982), On the equation of transfer relevant to the scattering of polarized light, On the phase matrix basic to the scattering of polarized light. (NAI formalism.)
Domke, H. (1974), The expansion of scattering matrices for an isotropic medium in generalized spherical functions, Astrophys. Space Sci. 29:379. (PCW basis.)
de Rooij, W. A. & van der Stap, C. C. A. H. (1984), Expansion of Mie scattering matrices in generalized spherical functions, A&A 131:237. (Domke corrections that PCW further fixes.)
Crib sheet: docs/dev_notes/theory_references.md §G.

3b · Mie, Rayleigh, and the Phase Matrices ​

What this page produces ​

The Mie pipeline ​

The Greek matrix B_l ​

Phase matrix Z from Greek + μ ​

NAI-2 vs PCW — two ways to compute the Greek expansion ​

Rayleigh and the Cabannes choice ​

Aerosol microphysics → optics ​

Code anchors ​

Hands-on tutorials ​

References ​

3b · Mie, Rayleigh, and the Phase Matrices

What this page produces

The Mie pipeline

The Greek matrix `B_l`

Phase matrix `Z` from Greek + μ

NAI-2 vs PCW — two ways to compute the Greek expansion

Rayleigh and the Cabannes choice

Aerosol microphysics → optics

Code anchors

Hands-on tutorials

References