Radar-Ray Refraction Associated with Horizontal Variations in the Refractivity

Abstract

The relative bending effects on a radio ray of the horizontal and vertical gradient of refractivity are investigated. With the use of ray-curvature formulas developed by Wong, convenient differential equations governing the three-dimensional ray path are derived. It is also shown that the ratio of 'horizontal to vertical' bending effects is maximum when the ray is propagated in the vertical plane containing the horizontal gradient of refractivity. In a two-dimensional example, computations of path height as a function of horizontal distance are obtained by numerical integration. For this purpose, a regional space-time averaged exponential model is drawn from climatological studies of Bean and Thayer has been used. Even though this smoothed model and an extreme version of it derived synthetically by increasing the horizontal gradient everywhere by a factor of 10 have comparatively strong horizontal gradients, the 'horizontal bending' effect is virtually negligible. It is not implied, however, that this conclusion is necessarily applicable to an atmosphere exhibiting small-scale fluctuations in the refractivity pattern. The ray paths corresponding to the nonhomogeneous model are significantly different from those in a horizontally uniform atmosphere. The problem of replacing a nonhomogeneous model by an equivalent horizontally uniform one is also investigated. The choice of the best uniform atmosphere depends upon minimizing the rms height error over a designated horizontal path length, and this in turn requires the specification of several ray-propagation parameters in addition to the distribution of refractivity.