Theory

This section describes the mathematics of the adding–doubling solver. It follows Plass, Hansen & Kattawar (1973) and Wiscombe (1975, 1976, 1977), with the extensions used in adrt:

  1. a linear-in-\(\tau\) thermal source (Wiscombe 1976);

  2. separate tracking of the direct solar beam through the doubling and adding steps;

  3. delta-M scaling for forward-peaked phase functions (Wiscombe 1977);

  4. a diffusion-approximation lower boundary condition for stellar atmospheres;

  5. analytic temperature Jacobians obtained by re-running the forward operators.

Throughout, the solver uses Gauss–Legendre quadrature on \([0, 1]\) with \(M\) streams per hemisphere. All matrices are \(M \times M\) and all source vectors have length \(M\). The azimuthally averaged (\(m = 0\)) problem is solved, which is sufficient for hemispheric fluxes and mean intensities.