## 1. Introduction

[2] Atmospheric absorption in the oxygen A band (≈770 nm) has been studied extensively in remote sensing for retrieval of surface pressure and cloud top heights from satellite measurements [*Grechko et al.*, 1973; *Fischer and Grassl*, 1999a, 1991b; *O'Brien and Mitchell*, 1992]. More recently, efforts have been made to use ground-based A band measurements at moderate to high spectral resolution to infer the distribution of photon path lengths, and to relate these to column cloud properties [*Harrison and Min*, 1997; *Pfeilsticker et al.*, 1998a; *Veitel et al.*, 1998; *Min and Harrison*, 1999; *Portmann et al.*, 2001; *Min et al.*, 2001; *Funk and Pfeilsticker*, 2003; *Min and Clothiaux*, 2003; *Min et al.*, 2004]. Simultaneously, there is a renewed interest in using the A band to assess the internal variability of clouds from spaceborne instruments [*Stephens and Heidinger*, 2000; *Heidinger and Stephens*, 2000, 2002; *Stephens et al.*, 2005].

[3] Using oxygen A band spectrometry with sufficient resolution, information on the cloudy-sky photon path length probability density distribution (briefly called photon path pdf in the following) is drawn from the fact that photons in optically thin and thick spectral interval wavelengths travel on average different long paths during their random cruise in the atmosphere owing to their different probability of being absorbed [e.g., *van de Hulst*, 1980]. In mathematical notation, this leads to a Laplace transformation with the measured intensity ratio *I*(λ)/*I*_{0}(λ) being the Laplace transform ((*k*)) of the desired path length distribution *p*(*L*) with respect to the wavelength-dependent and gas density (*n*)–dependent extinction coefficient (*k*(λ) = σ(λ) · *n*), where σ(λ) is the absorption cross section per molecule and *n* the absorber density. In mathematical terms, we have

where the impact of the dependence of *k* on pressure and temperature (hence height) is discussed further on. Here, *I*(λ) is the measured intensity and *I*_{0}(λ) is the extraterrestrial solar intensity. The latter goes back to the work by *Kurucz et al.* [1984] from which residual atmospheric absorption have been removed [*Funk and Pfeilsticker*, 2003].

[4] Unfortunately, the inverse Laplace transformation of the measured intensity ratio *I*(λ)/*I*_{0}(λ) results in a mathematically ill-posed problem since no information is made available by the measurement process on the complex part of the Laplace transformation. This dilemma is frequently resolved by prescribing the mathematical form of the photon path pdf *p*(*L*), e.g., by a Gamma, lognormal or any other suitable distribution on the positive real axis normalized to unity.

[5] Another feature of the oxygen A band technique is the information content (IC) of an individual measurement i.e., the number of independent pieces of information that can be drawn from the measurements [i.e., *Stephens and Heidinger*, 2000; *Heidinger and Stephens*, 2000; *Funk*, 2000; *Heidinger and Stephens*, 2002; *Min et al.*, 2004]. These studies revealed that the IC is primarily a function of the out-of-band rejection (OBR) and the spectral resolution of the spectrometer. State-of-the-art instruments are known to provide as much as four independent pieces of information, such as the first 4 moments of the path length distribution, or the first 2 moments and additionally some information the height distribution of the tropospheric aerosol, cloud cover, etc.

[6] A straightforward application of oxygen A band spectrometry is to investigate low-order moments of the photon path length distribution and to relate them to cloud column properties [*Pfeilsticker et al.*, 1998b; *Veitel et al.*, 1998; *Pfeilsticker*, 1999; *Min and Harrison*, 1999; *Min and Clothiaux*, 2003; *Min et al.*, 2004]. In particular, *Pfeilsticker* [1999] investigated the relation of the mean in-cloud photon path 〈*L*_{c}〉 as a function of the rescaled cloud optical depth τ*_{c} = (1 − *g*) · τ_{c}, where angular brackets 〈⋯〉 denote an average over all paths, τ_{c} the cloud depth and *g* the asymmetry factor for Mie scattering. Following the suggestion of *Davis and Marshak* [1997], he found for optically thick cloud covers (〈τ*_{c}〉 ≳ 1) a dependency

where ℓ_{tr} is the rescaled or “transport” photon mean free path (MFP). The so-called Lévy index α in *Pfeilsticker* [1999] data spans the expected range of 1 ≤ α ≤ 2, with the precise value depending on the structure of the cloud cover. We refer readers interested in Lévy flights and related topics to *Shlesinger et al.* [1995] and, for applications to fractal clouds, to *Lovejoy and Mandelbrot* [1985].

[7] Likewise, even though not explicitly stated in their study, *Min et al.* [2001] found joint distributions of τ_{c} (not rescaled) and 〈*L*_{c}〉 (in air mass units) that follow roughly a law in 〈*L*_{c}〉/Δ*H* ∼ (τ_{c})^{α−1} with α in the range 1.4 to 1.7. This is compatible with the above equation (2) since in their Figures 4, 5, 6, and 7 we can identify 〈*L*_{c}〉/Δ*H* with their mean path estimated in air masses, Δ*H* being the physical thickness of the cloudy part of the atmospheric column, and recalling that τ*_{c} = Δ*H*/ℓ_{tr}.

[8] The regime α = 2 is the classical diffusion case where the path length 〈*L*_{c}〉 of photons diffusing via long convoluted random walks through a uniform medium of large rescaled optical depth τ*_{c} behaves like 〈*L*_{c}〉 ∼ ℓ_{tr}(τ*_{c})^{2}, equivalently 〈*L*_{c}〉 ∼ Δ*H*(1 − *g*)τ_{c}. In sharp contrast, a Lévy index of α = 1 describes a predominance of direct (straight) transmission through the “cloud” layer, which is only possible for a negligible small probability of Mie scattering, i.e., clear skies with at most sparse clouds. Accordingly, the diffusion regime 1 ≤ α < 2 is called “anomalous”, which *Davis and Marshak* [1997] proposed as a model for paths of photon transmitted by optically thick but inhomogeneous (multilayered and/or broken) cloud covers. More generally, anomalous diffusion by so-called Lévy walks is known to occur for physical entities transported in unlimited (boundary-free) inhomogeneous media. Lévy walks in unlimited media support very large but rare jumps between the visited sites (here, clusters of Mie scattering events inside clouds), which leads to infinite higher-order moments of the path lengths, specifically, moments of the order ≥α [*Samorodnitsky and Taqqu*, 1994]. Because of absorption at the ground and escape at the top of the atmosphere (TOA). The cloudy-sky photon transport can be thought of has having “2.5 dimensions” (i.e., it unfolds in a horizontally infinite but vertically finite medium). Thus Lévy walks confined to a finite slab result [*Buldyrev et al.*, 2001], and these will have finite moments of any order.

[9] In a recent study, *Davis and Marshak* [2002] revisited the problem of classical (α = 2) diffusion through a homogeneous optically thick slab. Therein they provided full analytical expressions for the first two moments *L*_{c}〉 and as a function of the slab (cloud) vertical extension Δ*H* and optical depth τ_{c}. These relations for classical diffusion are tested in our study, in particular, as to whether they hold true for real cloud covers (hence Lévy indices α < 2).

[10] Accordingly the study is organized as follows. Section 2 describes the employed methods in our observations. Section 3 reports on the observation. Section 4 discusses the theoretical relations between cloud properties and photon paths. Section 5 discusses our results with respect to the theoretical information in the *Davis and Marshak* [1997, 2002] studies, and section 6 concludes our study with the wider implications for photon transport in cloudy skies.