In this book we share our work with those who are faced with the challenging problem of studying the earth's atmosphere and the interactions between the atmosphere and the earth's surface. While there are some excellent books on this topic written from the physical point of view, those discussing the modeling and computational aspects are few and far between. Our book is intended to bridge this gap so that students as well as investigators will be able to understand and apply practical ways of determining solutions. Radiative transfer theory, on which this book is based, is elegant, and great minds have contributed to its richness. Instead of duplicating the clas sical references, we have taken a different approach: We have developed the in variant imbedding approach, both analytically and computationally, because of its attractiveness for producing numerical solutions. Having witnessed the transition to the computer age, we know that a new attitude to mathemati cal formulation is required. The one that we endorse is a model stated in the form of a Cauchy problem: a system of ordinary differential equations with a complete set of initial conditions. We chose this approach because it is well suited to implementation on digital computers.