Calculation of Jacobians for inverse radiative transfer: An efficient hybrid method

We present an accurate and numerically efficient procedure of calculating Jacobians by finite difference that consists of two components: (1) a method employing the saving of atmospheric layers that accelerates the solution to the equation of radiative transfer for solvers that use the Discrete Space formulation and (2) a method of perturbing the eigenmatrix spectrum associated with a reduced attenuation matrix. The procedure eliminates the need to call the eigenmatrix package, here, LAPACK a second time and provides insights into the fundamental properties of the attenuation matrix, useful for characterizing the accuracy of the derivatives calculated by finite difference methods. The computational complexity of the perturbation method is 8n3+22n2, where n is one half the number of streams in the radiance field as opposed to 16n3 using LAPACK. The method is not limited to the calculation of base state radiances I(ω) and those associated with an 'infinitesimal' perturbation I(ω+δω) (from which the numerical derivative of I(ω+δω) with respect to δω may be approximated), but is also useful in the calculation of radiances associated with a 'finite' perturbation I(ω+Δω) from which a sensitivity can be calculated.