Geometric Realism Without Angular Resolution Structural Classification of Multilayer Kubelka-Munk Theory within Radiative Transport

TL;DR

This paper rigorously connects Kubelka-Munk theory to the radiative transfer equation, showing it as a low-resolution, rank-2 Galerkin projection that explains its accuracy and limitations in layered media.

Contribution

It establishes a mathematical foundation for KM theory as a specific projection of the RTE, clarifying its scope and accuracy in multilayer media.

Findings

KM coefficients are hemispherical moments of the transport operator.

Projection error depends on reduced optical thickness tau* = tau(1-g).

KM provides a low-resolution approximation valid for certain media.

Abstract

Kubelka-Munk (KM) theory provides a two-flux description of radiative transport in layered scattering and absorbing media. Despite its wide use in the coatings, paper, paint, and textile industries, the theory has often been regarded as a phenomenological model whose connection to the full radiative transfer equation (RTE) remains unclear. Under the standard steady-state, plane-parallel, azimuthally symmetric assumptions, we show that multilayer KM theory is exactly a rank-2 Galerkin projection of the RTE onto hemispherical basis functions. The projection is idempotent with an infinite-dimensional kernel, and its rank is preserved under multilayer composition -- so no amount of layer stacking can recover angular information discarded by the projection. We derive the KM coefficients as hemispherical moments of the transport operator and compute the projection error for representative scattering media (g from 0 to 0.85), finding that the reduced optical thickness tau* = tau(1-g) governs KM accuracy. The projection-error framework explains the well-documented accuracy of compositional multilayer models in printed media and shows where higher-order methods become necessary. The result places KM theory on rigorous footing as a legitimate -- if low-resolution -- transport approximation rather than an ad hoc phenomenology.