This paper proposes a near-optimal procedure to localize a single stationary source in a two-path underwater acoustic environment. The investigation is for an M-element vertical array with omnidirectional sensors. The range and depth estimators are developed using a linear least-squares technique when a set of auto- and cross-correlators is used for time difference of arrival (TDOA) estimates. A weighting matrix is derived to achieve the approximate maximum likelihood (ML) performance of the weighted least-squares range and depth estimators. The expressions for error variances and covariances of the range and depth estimates are derived with a small error analysis technique. It is verified analytically that the error covariance matrix of the weighted least-squares solutions reaches the Cramer-Rao lower bound in the small error region. The correlation of the range and depth estimation errors is investigated. Results show that the range and depth estimation errors are highly correlated in a multipath environment. The accuracy properties of the proposed multipath localization procedure are analyzed using different array configurations. The results show that the performances of the range and depth estimators are significantly improved if the linear-dependent TDOA estimates are included for localizing and that the unweighted range and depth estimators, using the entire set of TDOAs, are approximately optimal for most of the applications. The theoretical development of error variance and covariance expressions of the range and depth estimates, which incorporates the correlation in the TDOA estimates, is corroborated with Monte Carlo simulations.
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