The Welch-Satterthwaite equation (Wikipedia) giving approximate degrees of freedom is used when the populations don't have equal variances. This result is obtained by equalizing the moments. However, according to Scheffé [H.Scheffé, The Analysis of Variance, 1959] and some more recent texts, the degree of freedom can and should be understood as the dimension of spaces spanned by canonical estimable functions (linear parametric functions with unbiased estimate).
My question is how to interpret the degree of freedom given by Welch-Satterthwaite approximation in terms of basis of vector spaces? It is usually non-integral and could not be understood as a common dimension of some vector space. It seems very unnatural. Or is there any reference that treats this problem?