Beyond Plane Sailing: Solving the Range-Doppler equations in a reduced geometry

An important part of geometry computations for synthetic aperture radar (SAR) carried on a satellite platform is to convert between the instrument-specific coordinate system and a geocentric coordinate system such as Cartesian Earth-centered, Earth-fixed (ECEF) coordinates, geodetic coordinates (latitude/longitude) or a projection of these. The conversion involves solving the range-Doppler equations. The solutions for points on or near ellipsoid height typically involves iteration over geodetic coordinates, which means performing the transformation from geodetic to ECEF and its 6 partial derivatives in every iteration step. We present a method for solving the equations in the satellite's zero Doppler plane, which is a common choice for coordinate plane for SAR systems with moderate squint angles. Solving the system in this plane means one of the constraints is satisfied implicitly, and allows solution which satisfies the other constraints (correct range from satellite and geodetic height) using a rank-1 Newton method. The method is simple to implement, fast and accurate. For targets on the ellipsoid, the solution can be made as accurate as machine precision allows. For targets with non-zero height above the ellipsoid, a small correction step is necessary, and we show how this step is computed.