A Closed-Form Method With Low Noise Sensitivity for Locating a Moving Source on Earth at a Known Altitude

This article proposes a closed-form method that does not require an initial guess to determine the geographical location and velocity of a moving source at a known altitude. The primary objective of this method is to overcome the main limitations of the two-step weighted least squares (TWLS) method and its related approaches, by exploiting the prior knowledge that the source is on earth with known altitude such as in geolocation. For this reason, our proposed approach refrains from assuming simplifications or linearizations in the TWLS relationships by utilizing the earth model and known source altitude in locating the source with the measurements. Unlike the TWLS method that does not use any prior knowledge of the source location, which becomes less effective beyond a noise threshold, our method consistently demonstrates improved performance. The main idea of the proposed method is to use a different regressor in the formulation from the measurement models of the localization problem. The regressor matrix in the proposed method is noise-free and depends only on the 2-D coordinates of the receivers. The proposed approach in the formulation not only solves the singularity issue of the regressor caused by factors like noise variance and receiver arrangement (such as being in a line, a plane, or close to each other) but also reduces the required minimum number of receivers from five to three. Finally, for the purpose of evaluating and comparing the proposed method with other recent approaches, the constrained Cramer–Rao lower bound (CCRLB) has been evaluated. The CCRLB is obtained with two constraints, both of which pertain to the World Geodetic System 1984 model assumed for the earth. Simulation results show that the proposed method performs better than other methods in challenging scenarios.