Distribution of the maximum in air pollution fields

Air quality standards are set to protect public health. The values of the standards are often based on health effect studies, without any statistical considerations. In order to judge if a standard is met measurements of ambient air quality are taken at monitoring stations, and these measured values are used to decide whether or not the standard has been violated. In this paper we examine the statistical quality of some air quality standards by taking both measurement error and variability of the ambient field away from the monitoring sites into account. In particular we study the distribution of the maximum of the ambient field conditional on a measured monitoring value at the value prescribed by the standard. The distribution of the maximum is computed using Rice's method and relies on a generalization of upcrossings of a level in one dimension to two dimensions.