We examine how modal aberration measurements degraded by turbulence-induced anisoplanatism may be used to optimally conjugate atmospheric phase aberrations. By examining the form of the aperture-averaged mean square residual phase error, we show that atmospheric compensation is suboptimal when the measuredcoefficients from off-axis or finite-altitude guide stars are applied directly. The optimal compensation is obtained only when conjugate phase coefficients are estimated, given the guide-star measurements and knowledge of the spatial correlation of the on-axis and measured phase coefficients, by use of a minimum-mean-square-error (MMSE) estimator. The form of this estimator is outlined, thus motivating the need to quantify the spatial cross correlation of the Zernike coefficients of the phase aberrations. With a knowledge of the medal cross correlation, we show that wave-front compensation performance can be enhanced by use of the MMSE estimator over use of the beacon measurements directly for all orders of correction. For high-order off-axis natural-guide-star correction, equivalent imaging performance is obtained at a beacon offset 10% larger than when beacon measurements are used directly. For high-order laser-guide-star correction, equivalent imaging performance is obtained at laser-guide-star altitudes 20% lower when the MMSE estimator is employed. # 1998 Optical Society of America [S0740-3232(98)01808-0] OCIs codes: 010.1080, 010.1290.
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