Pergamon J. Quanr. Specfrosc. Radiot. Transfer Vol. 59, Nos 112, pp. I-24, 1998 0 1998 Published by ElsevierScience Ltd. All rights reserved Printed in Great Britain PII: SOO22-4073(97)00129-5 0022-4073/98 $19.00 + 0.00 zyxwvu ADJOINT PERTURBATION METHOD A PPLIED TO TWO-STREAM RADIATIVE TRANSFER PHILIP GABRIEL, JERRY Y. HARRINGTON, GRAEME L. STEPHENS and TIMOTHY L. SCHNEIDER Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523. U.S.A. (Received May 1997) Abstract-This paper describes a computationally efficient method for solving the plane parallel equation of radiative transfer for the two-stream fluxes based on the adjoint pertur- bation formulation. Analytical results for the perturbed fluxes are presented for a single layer atmosphere containing both solar and thermal sources. Simple linear and exponential corrections to the base state fluxes are explored. For the solar radiative transfer problem, the exponential form of the perturbation correction can accommodate deviations exceeding 400% in the base state optical properties while maintaining accuracy to within a few per- cent. For thermal radiative transfer, the linear form of perturbation relation is the more accurate, but unlike the solar problem, deviations from the base state optical properties must remain relatively small (less than 20%) if the errors in the computed fluxes are to remain within a few percent of the true fluxes. The method is applied to the calculation of broadband solar fluxes in a layer of scatterers embedded in an absorbing gas, where the absorption is modeled via the k-distribution method. #Q 1998 Published by Elsevier Science Ltd. All rights reserved 1. INTRODUCTION Over the past few years, the demand for simple, robust and computationally fast methods for solving various radiative transfer problems has increased. This increase in demand is largely mo- tivated by the topic of climate and global change. In this context, problems of interest range from radiative transfer through clouds characterized by highly anisotropic scattering, radiative transfer through clear atmospheres involving mixes of Rayleigh scattering and molecular absorption, the effects of increased absorption associated with increases in selected greenhouse gases, scattering by optically thin atmospheres comprised of aerosols and cloud particles as well as the complex effects of cloud geometry on radiative transfer. The time required for radiative transfer computations is excessive for even the simplest of radiative transfer schemes when incorporated into Global Circulation Models (GCMs) that attempt to simulate the complex climate system. The problem stems from the nature of the cal- culation of broadband fluxes which involves repetitive summations of many radiative transfer solutions with input parameters that often have similar values. The desire is to develop highly efficient techniques to carry out these broadband calculations. By reducing significantly the com- putation time devoted to modeling of radiative transport while maintaining numerical accuracy, the potential of improving other critical processes can be realized. An approach that has historically been used to streamline this process of calculation is based on the use of the perturbation form of the radiative transfer equation. The Curtis matrix method is an example of a perturbation approach that is based on some form of simplification of the radiative transfer equation.’ There are, however, many other methods that are more ad hoc in nature. With the complexity of climate models and the demands for more physical and accurate radiative transfer models, perturbation methods have again emerged as a viable way of parameterizing radiative transfer. In this paper we extend the perturbation method introduced to the atmospheric science com- munity by Gerstl,* Box et a13*4 and others to the problem of broadband radiative transfer. The approach presented is based on a theoretical method successfully used in neutron transport pro-