Moment expansion for ionospheric range error

On a plane earth, the ionospheric or tropospheric range error depends only on the total refractivity content or zeroth moment of the refracting layer and the elevation angle. On a spherical earth, however, the dependence is more complex; so for more accurate results it has been necessary to resort to complex ray-tracing calculations. A simple, high-accuracy alternative to the ray-tracing calculation is presented. By appropriate expansion of the angular dependence in the ray-tracing integral in a power series in height, an expression is obtained for the range error in terms of a simple function of elevation angle, E, at the expansion height and of the mth moment of the refractivity, N, distribution about the expansion height. The rapidity of convergence is heavily dependent on the choice of expansion height. For expansion heights in the neighborhood of the centroid of the layer (300-490 km), the expansion to N = 2 (three terms) gives results accurate to about 0.4% at E = 10 deg. As an analytic tool, the expansion affords some insight on the influence of layer shape on range errors in special problems.