Inaccuracies in Doppler Radar Navigation Systems Due to Terrain Directivity Efects, Nonzero Beamwidths and Eclipsing

A study is made of inaccuracies due to environmental beam-distorting effects in precision Doppler radar navigation systems and specifically of ¿interrupted continuous wave¿ (ICW) Doppler systems which simultaneously determine altitude and velocity by the employment of several beams, all of them produced by a single transmitter. The three causes of errors considered include terrain directivity (back-scattered power varying with angle of incidence); range effects (power backscattered by individual infinitesimal patches of terrain varying as R-4, where the range R is not constant over the area illutminated by any single beam); and nonconstant eclipsing [the degree of eclipsing of the energy arriving from any direction varying as a function of the angles (¿, ¿) specifying the direction]. All of these effects tend to cause the ¿effective¿ Doppler frequency fTj6 indicated by the jth tracker to be different from the frequency fTj0 corresponding to zero beamwidth [i.e., fTj0 = -(2/¿)(V · bj0) where bj0 is the unit vector along the beam axis]. A related cause of inaccuracy is the noise-like character of the return, caused in part by the nonzero bandwidth BJ of the Doppler spectrum corresponding to beam #j (the Doppler frequency of the return from any direction varying as a function of the angles ¿, ¿). Formulas, in the form of integrals, are derived for fTj6 and Bj. The velocity components Va1, Va2, Va3 relative to antenna axes, or any other convenient set, are linear combinations of the effective tracker frequencies, begin{equation*} V_ai = sum_{i} C_{ij}{f_{Ti}}^e.