This paper deals with source localization using a two-dimensional array of sensors whose locations are not known precisely. If only a single source is observed, uncertainties in sensor location increase errors in source bearing and range by an amount which is independent of signal-to-noise ratio and which can easily dominate overall localization accuracy. Major performance gains could therefore result from successful calibration of array geometry. The paper derives Cramer-Rao bounds on calibration and source location accuracies achievable with far-field sources whose bearings are not initially known. The sources are assumed to radiate Gaussian noise and to be spectrally disjoint of each other. When the location of one sensor and Manuscript

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This paper studies the minimum bearing error attainable with a linear passive array. Signal and noise are stationary Gaussian processes with arbitrary power spectra, and the noise is assumed to be statistically independent from hydrophone to hydrophone. The Cramér-Rao technique is used to set a lower bound on the rms bearing error and the results are compared with the bearing error of a slightly modified split-beam tracker. The latter reaches the lower bound for a two-element array and comes very close to reaching it for a linear array with an arbitrary number of equally spaced hydrophones. Thus, the modified split-beam tracker is very nearly optimal for the uniformly spaced array. Comparisons of split-beam tracker error with the Cramér-Rao lower bound are also obtained for nonuniform hydrophone spacings.

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The absolute minimum attainable values of incremental mean-square error in bearing and range estimation caused by random perturbations of sensor locations are determined and compared with the performance of conventional processors. At low signal-to-noise ratios the optimum estimator realizes no improvement over the conventional systems; at high signal-to-noise ratios it realizes only slight improvement.

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