vSmartMOM principles

This page summarizes the core ideas behind vSmartMOM (vector Smart Matrix-Operator Method), consolidated from the primary references.

What problem vSmartMOM solves

Vector (polarized) radiative transfer in layered media with molecular (Rayleigh) and aerosol scattering, plus gaseous absorption.
Efficient, accurate multiple scattering using the matrix-operator method (doubling/adding) with discrete-ordinate quadrature.
Modular treatment of polarization states (I, IQ, IQU, IQUV) and Fourier decomposition in azimuth.
Optional linearization with respect to aerosol optical properties to facilitate Jacobians and sensitivity studies.

Discretize angle with a quadrature (e.g., Gauss full-sphere, hemisphere, Radau) to get a finite set of streams.
Expand the vector phase matrix into Fourier series over azimuth and into Legendre polynomials over polar angle; use a truncation strategy.
Build per-layer single-scattering properties (optical thickness τ, single-scattering albedo ϖ, and polarized phase matrices Z⁺⁺, Z⁻⁺).
Compose layers via matrix-operator doubling/adding into a composite medium, preserving polarization.
Couple to surfaces via BRDF models (Lambertian, RPV, Ross–Li) in a vector-consistent way.

δ-m and δ-fit approaches are used to remove/approximate strong forward peaks in scattering, improving convergence at low order.
vSmartMOM applies these truncations consistently to the vector phase matrix, following Sanghavi & Stephens (2015).
Practical guidance:
- Choose L_trunc consistent with aerosol phase function complexity and desired accuracy.
- Use Radau or Gauss full-sphere quadrature depending on hemispheric symmetry and source configuration.

Supports Stokes vector subsets via fixed types: I; IQ; IQU; IQUV.
Phase matrix moments and the discrete-ordinate system are built per polarization type; this ensures type-stable kernels.

The matrix-operator formalism enables analytic derivatives with respect to aerosol properties (e.g., τ_ref, phase function parameters) by differentiating the composed operators.
vSmartMOM exposes aerosol optical property construction (Mie-based) and truncation, enabling efficient sensitivity calculations.

Per-layer quantities include:
- τλ (with gaseous absorption), ϖλ, τ, ϖ, and Fourier-expanded Z matrices.
- Rayleigh properties are computed from depolarization factor and wavelength; aerosol properties from Mie models at reference λ.
Composition:
- Added layer (A) and composite layer (C) operators are combined according to the standard matrix-operator algebra.
- Doubling logic computes appropriate optical thickness sub-steps for numerical stability.

Surfaces are modeled via BRDFs; interaction terms are included in the boundary condition and in HDRF post-processing.
The HDRF and VZA postprocessing utilities provide azimuthal averaging and RAMI-style outputs.

Accuracy knobs: quadrature choice, L_trunc, max Fourier moment m, and depolarization factor for Rayleigh.
Stability: small per-step optical depths via doubling; ensure consistent units (wavenumber bands in cm⁻¹).
Performance: choose architecture CPU/GPU; GPU accelerates batched operations and kernel calls.

Sanghavi, S., Davis, A.B., Eldering, A. (2014). vSmartMOM: A vector matrix-operator method-based radiative transfer model linearized with respect to aerosol properties. JQSRT 133: 412–433.
Sanghavi, S., Stephens, G. (2015). Adaptation of the δ-m and δ-fit truncation methods to vector radiative transfer: Effect of truncation on radiative transfer accuracy. JQSRT 159: 53–68.
Grant, I.P., Hunt, G.E. (1969). Discrete space theory of radiative transfer I. Proc. Roy. Soc. A 313(1513): 183–197.
Plass, G.N., Kattawar, G.W., Catchings, F.E. (1973). Matrix operator theory of radiative transfer. 1: Rayleigh scattering. Applied Optics 12(2): 314–329.