The major topic of this paper is the resolvable spatial scales that can be analyzed by statistical interpolation of an undersampled dataset. The inquiry was motivated by the need to design the most appropriate procedures for spatial analysis of the upper-air sounding data from the GARP Atlantic Tropical Experiment. A reliable representation of horizontal scales in the analyzed wind fields was a matter of utmost concern, since the derived fields of vorticity, divergence and vertical motion were also of vital interest. To achieve our goal, it was found that the traditional promise of statistical interpolation had to be reexamined. The main conclusions of this theoretical inquiry are (i) resolvable scales are determined by the geometrical distribution of observing stations; (ii) precise knowledge of the second-moment statistics improves the analysis by de-aliasing the amplitudes of resolvable scales, but has no effect on the definition of resolvable scales; (iii) residual effects of unresolvable signals in the data are removable by a spatial filter and must be so removed., and (iv) spatial phases of the de-aliased resolvable scales may still be in error. On the basis of these findings the objective analysis procedures we have developed are targeted on the best achievable analysis of resolvable scales. The procedures include the following: an adequate estimation of “true” statistical fields from the given ensemble of data, a search for the optimum spatial filter by monitoring the targeted error variance, and a rational method of desensitizing the analysis to statistically errant data. In order to reduce the spatial phase error of propagating disturbances, the procedures are extended to the analysis of the timewise Fourier-transformed dataset (actually in the frequency-band analog). Since the wind is a physical vector, the entire procedure for the wind analysis is given in the tensor-invariant form, which is decidedly advantageous for very practical reasons. For example, the tensor approach eliminates the notorious ambiguity in normalization that is encountered in the multivariate approach. The paper also describes, in the Appendix, a method of filtered mechanical interpolation, which is specifically designed, with a variety of optional boundary conditions, for application to analysis in a finite domain.