The Gandin optimal analysis scheme leaves open the problem of determining the covariance matrix on which it depends. The choice of an autocorrelation function for geopotential, from which geostrophically consistent cross-correlations—and thus covariances—for heights and winds may be derived, influences the performance of the analysis. Four related functions are considered for representation of height field autocorrelations. They are compared on the basis of spectral behavior, capacity to conform with geostrophy, and accuracy of analyses employing them. A multivariate objective analysis test, with both regular and irregular configurations of predictor stations, compares performance of analyses based on modeled covariances with performance of an analysis using observed sample covariances.