Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) 3 – 34 www.elsevier.com/locate/jqsrt Photon propagation in heterogeneous optical media with spatial correlations: enhanced mean-free-paths and wider-than-exponential free-path distributions Anthony B. Davis a ; ∗ , Alexander Marshak b a Los Alamos National Laboratory, Space & Remote Sensing Sciences Group (NIS-2), Los Alamos, NM 87545, USA b NASA’s Goddard Space Flight Center, Climate & Radiation Branch (Code 913), Greenbelt, MD 20771, USA Received 26 November 2001; received in revised form 29 December 2002; accepted 27 February 2003 Abstract Beer’s law of exponential decay in direct transmission is well-known but its break-down in spatially variable optical media has been discussed only sporadically in the literature. We document here this break-down in three-dimensional (3D) media with complete generality and explore its ramications for photon propagation. We show that eective transmission laws and their associated free-path distributions (FPDs) are in fact never exactly exponential in variable media of any kind. Moreover, if spatial correlations in the extinction eld extend at least to the scale of the mean-free-path (MFP), FPDs are necessarily wider-than-exponential in the sense that all higher-order moments of the relevant mean-eld FPDs exceed those of the exponential FPD, even if it is tuned to yield the proper MFP. The MFP itself is always larger than the inverse of average extinction in a variable medium. In a vast and important class of spatially-correlated random media, the MFP is indeed the average of the inverse of extinction. We translate these theoretical ndings into a practical method for deciding a priori when 3D eects become important. Finally, we discuss an obvious but limited analogy between our analysis of spatial variability and the well-known eects of strong spectral variability in gaseous media when observed or modeled at moderate resolution. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Beer’s law; Photon propagation; Mean-free-path; Three-dimensional radiative transfer; Spatially correlated heterogeneous media; Cloudy atmospheres * Corresponding author. Tel.: +1-505-665-6577 (voice); fax: +1-505-667-9208. E-mail address: adavis@lanl.gov (A.B. Davis). 0022-4073/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0022-4073(03)00114-6