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Accurate localization of nodes in a sensor network is a crucial step before the sensor network can be utilized for various applications. This paper proposes a successive estimation method to self-localize the sensor nodes using time of arrival (TOA) measurements, through the aid of a few anchor nodes whose positions are known a priori. The successive nature of the proposed algorithm makes it attractive for distributed computation in a resource constrained environment. The proposed technique uses subsets of TOAs to obtain coarse sensor node estimates. Depending on available computation and transmission resources, the node positions are refined to improve their accuracies. The solution of the proposed node localization algorithm is algebraic and closed-form, which simplifies computation and avoids possible local convergence or divergence problems as in the traditional iterative approaches. A main benefit of the proposed algorithm is that although it is closed-form and performs successive estimation of the node locations, it is able to reach the Cramer-Rao lower bound (CRLB) accuracy under white Gaussian measurement noise with sufficient SNR. This is confirmed by the theoretical analysis and corroborated by simulations.