Practical Considerations with Radar Data Height and Great Circle Distance Determination

Owing to their ease of use, "simplified" propagation models, like the equivalent Earth model, are commonly employed to determine radar data locations. With the assumption that electromagnetic rays follow paths of constant curvature, which is a fundamental assumption in the equivalent Earth model, propagation equations that do not depend upon the spatial transformation that is utilized in the equivalent Earth model are derived. This set of equations provides the true constant curvature solution and is less complicated, conceptually, as it does not depend upon a spatial transformation. Moreover, with the assumption of constant curvature, the relations derived herein arise naturally from ray tracing relations. Tests show that this new set of equations is more accurate than the equivalent Earth equations for a "typical" propagation environment in which the index of refraction n decreases linearly at the rate dn/dh = -1/4a, where h is height above ground and a is Earth's radius. Moreover, this new set of equations performs better than the equivalent Earth equations for an exponential reference atmosphere, which provides a very accurate representation of the average atmospheric n structure in the United States. However, with this n profile the equations derived herein, the equivalent Earth equations, and the relation associated with a flat Earth constant curvature model produce relatively large height errors at low elevations and large ranges. Taylor series approximations of the new equations are examined. While a second-order Taylor series approximation for height performs well under "typical" propagation conditions, a convenient Taylor series approximation for great circle distance was not obtained.