## 1 Introduction  
The need for adequate illumination in building interiors, building cores, deep offices or halls has escalated in the last few decades due to the rapid spread of densely built-up areas worldwide. The use of electric lights can solve the problem [1, 2], but typically results in wasting energy [3, 4], and in the loss of natural light that otherwise makes the interior environment more desirable for human activities and health. Light-guide systems can deliver natural light into rooms (Figure 1) and mimic the dynamics of external illumination conditions [5, 6]. Light pipe is a desirable technology because it does not impose an additional burden on energy sources and has no negative impact on the natural environment [7, 8]. The performance of such a passive system depends mainly on the transmission properties of the light-guide body, but also on the instantaneous meteorological conditions [9; 10; 11; 12]. The optical parameters of straight pipes have been intensively studied by many authors [13; 14; 15; 16; 17], while some have demonstrated that clear skies more than overcast ones produce complex illuminance patterns in the interior spaces. Illuminance levels, the angular distribution of light and also its dynamics altogether affect both the visual comfort and working conditions in a room. No doubt that interior illuminance is undergoing large changes during optically unstable days characterized by broken cloud arrays with bright sky windows between isolated clouds. Such heterogeneous cloud fields present the largest uncertainty in modeling the light-pipe efficiency and thus deserve a special attention. Nevertheless, no quantification of the light-pipe efficiency for in-lab simulated heterogeneous cloud distributions exists, and no fully controlled experiment is possible in the exterior. Such an experiment does not satisfy one (or even both) of the basic preconditions: (a) All atmospheric parameters must be a-priori known, and, (b) only a single parameter (e.g., cloud arrangement, or aerosol content) can be varied within a defined range, while all other parameters should be kept constant during the experiment. The models currently in use predict the optical effects from the CIE skies (sky types standardized by International Commission on Illumination), which are homogeneous in terms of luminance patterns. In these models any irregularities due to the presence of isolated clouds with distinct edges are ignored, so the luminance patterns are smooth and may largely differ from what we can observe under real conditions. However, the broken cloud arrays are recognized to have a great impact on the luminance distribution and thus a potentially great unknown impact on the light field at the lower interface of the light-pipe. Due to this reason the range of illuminance signatures we can expect from heterogeneous cloud fields remains unexplored. This problem requires a proper treatment to either improve the accuracy of numerical predictions of interior illuminance or to understand the limits of the models of homogeneous skies. For the first time, we treat the light field below a tubular pipe theoretically and numerically for random cloud arrangements taking into account inhomogeneous sky radiance patterns changing abruptly at the edges of each single cloud. We analyze a straight vertical pipe, which is a first step for the characterization of more complex systems. We address the principal question: What are the main differences between the light-guide optical signatures under a homogeneous sky and the fractional cloud cover with individual clouds distributed randomly over the sky. Among all homogeneous skies, the most known are [PERSON]'s system of sky standards [18] and [PERSON]'s model of sky luminance distributions [19]. It is still unclear how the light-pipe optics changes depending on the cloud coverage  
Figure 1: The schematic view of the configuration of the sky, light pipe and work plane. and sun position relative to the cloud field.