Variational Principle Applied to Numerical Objective Analysis of Urban Air Pollution Distributions

Abstract

The calculus of variations is used to derive formulations for optimally adjusting a numerical analysis scheme so as to minimize any analysis inconsistencies attending pollution concentration measurements that are obtained at different observation times. The assumption is made that the evolution of conditions between the objective analyses of network data at two different times will be subject to some dynamical constraint, in this case the diffusion equation. When the two analyses are at odds with the prediction model, the requirement for maintenance of dynamical consistency provides a criterion for optimizing a scheme for applying adjustments. A sample problem is solved for the functional expression of the corrections to be applied. The general method is also applicable to situations where data are obtained by seas across a region, rather than by simultaneous measurements from geographically fixed points within the region. Remote probing by orbiting satellites and by aircraft provides data inputs of the kind amenable to this method, and this fact assures that such techniques will continue to grow in importance.