Spatial variances of wind fields and their relation to second-order structure functions and spectra

Kinetic energy variance as a function of spatial scale for wind fields is commonly estimated either using second-order structure functions (in the spatial domain) or by spectral analysis (in the frequency domain). Both techniques give an order-of-magnitude estimate. More accurate estimates are given by a statistic called spatial variance. Spatial variances have a clear interpretation and are tolerant for missing data. They can be related to second-order structure functions, both for discrete and continuous data. Spatial variances can also be Fourier transformed to yield a relation with spectra. The flexibility of spatial variances is used to study various sampling strategies, and to compare them with second-order structure functions and spectral variances. It is shown that the spectral sampling strategy is not seriously biased to calm conditions for scatterometer ocean surface vector winds. When the second-order structure function behaves like r(p), its ratio with the spatial variance equals (p+1)(p+2). Ocean surface winds in the tropics have p between 2/3 and 1, so one-sixth to one-fifth of the second-order structure function value is a good proxy for the cumulative variance.