Many industrial and commercial activities, by the nature of their operations, present inherent hazards to the safety of the general public in the surrounding area. Over the last decade or so, quantified risk assessment (QRA) has been increasingly used in making land use and other risk management decisions regarding hazardous facilities. Although the basic definitions of risk have been relatively well established, methods for quantifying individual and societal risk measures are not that well established, as evidenced by recent benchmark exercises in Europe and North America. It has been difficult to pinpoint exact reasons for the discrepancies between different studies, indicating a lack of common mathematical platform. The present paper puts forward a rigorous mathematical platform on which risk assessments can be built, especially for meteorology- and direction-dependent hazards and linear risk sources such as pipelines and dangerous goods transport corridors.