Enhancement of solar irradiance (IE) has been observed in many locations under partly cloudy sky conditions. Despite its notoriety, few studies have attempted to systematically examine the underlying mechanism for the event through simulation. As a result, there is no consensus regarding the causes of IE, nor an understanding of the effects of cloud optical depth and/or solar elevation on the event. Using a 2D Monte Carlo radiative transfer model (RTM), we show that IE through a homogenous square cloud is caused by the superposition of a direct irradiance beam with diffuse light scattered through the bottom and out of the edge of a cloud, through Mie and diffusion scattering. Using the RTM, we investigated the effect of optical depth up to 40 and solar zenith angles ranging from 0 degrees to 80 degrees on the IE magnitude and spatial extent. For the overhead zenith case the IE magnitude is maximum between optical depths of 2 and 3, with a value of 1.27 times the clear sky value. IE magnitude increases with increasing solar zenith angle, with the maximum occurring at a higher optical depth. The greatest magnitude overall occurred for a solar zenith angle 80 degrees (our maximum) and optical depth of 20, with a value of 1.46 times the clear sky value. The simulations show that for optically thick clouds and small solar zenith angles forward scattering is the dominate contributor to large IE, but at increased zenith angles outward scattering leads to larger IE's. Differences between simulated IE maximums and measured values in literature were resolved with a simple two cloud model to show mechanisms behind the production of IE extremes. (C) 2016 Elsevier Ltd. All rights reserved.