Range difference multilateration for obtaining precision geodetic and trajectory measurements

Abstract

The theoretical aspects of a new multilateration technique suitable for precision geodesy and orbit determination applications are examined. The multilateration technique considered herein makes use of the differential time of arrival of signals at an ensemble of ground stations from a spacecraft or aircraft as the fundamental data type. It will be demonstrated that simultaneous measurements give rise to a system of equations which upon solution permits the determination of the three-dimensional vehicle coordinates plus the three-dimensional coordinates of the station net relative to an arbitrarily adopted origin (which may be taken to be one of the stations). A solution to these equations can be obtained without any a priori knowledge of the locations of the stations and vehicle. The necessary conditions for obtaining all of these coordinates in the same solution are discussed, and it is indicated that at least five stations are required in the station ensemble. A subsequently performed study of the mathematical degeneracies indicates that the five-station configuration becomes singular if the stations are in the same plane. As a result, stations spaced closely together on the Earth's surface will yield ill-conditioned solutions. This degeneracy can be overcome by the addition of one more station. Hence, the system recommended herein will be comprised of six stations simultaneously receiving radio pulses emitted from an arbitrary vehicle. The formulation of the mathematical equations is undertaken along with the appropriate method of solution. An error analysis is also developed. Some preliminary system considerations such as data type synchronization and ionospheric and tropospheric corrections to the data types are examined. It is demonstrated that synchronization of the data types presents no problem and that the radio data types can be calibrated to accuracies of better than five centimeters if suitable instrumentation is available. The system proposed herein is a convenient alternate to a range-only multilateration system because obtaining radio range as opposed to differential range (not range-rate) will always be less accurate. The pulsed laser range multilateration system already has been discussed by the authors in another paper.