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In eigenstructure methods for direction of arrival estimation of signal wavefronts, the additive noise is assumed to be spatially white, i.e., of equal power and uncorrelated from sensor to sensor. When the noise is nonwhite but has a known covariance, we can still handle the problem through prewhitening. However, there are no techniques presently available to deal with completely unknown noise fields. In this paper, we propose a solution to this unknown noise covariance problem for the case when the noise field is invariant under two measurements of the array covariance; situations where this assumption is valid are not uncommon in sonar applications. In fact, the idea has been used in certain so-called "despoking" algorithms for conventional beamformers. Results of computer simulations carried out to compare the performance of the new algorithm to earlier methods are also presented.