  We present fixed domain asymptotic results that establish consistent
estimates of the variance and scale parameters for a Gaussian random field with
a geometric anisotropic Mat\'ern autocovariance in dimension $d>4$. When $d<4$
this is impossible due to the mutual absolute continuity of Mat\'ern Gaussian
random fields with different scale and variance (see Zhang \cite{zhang:2004}).
Informally, when $d>4$, we show that one can estimate the coefficient on the
principle irregular term accurately enough to get a consistent estimate of the
coefficient on the second irregular term. These two coefficients can then be
used to separate the scale and variance. We extend our results to the general
problem of estimating a variance and geometric anisotropy for more general
autocovariance functions. Our results illustrate the interaction between the
accuracy of estimation, the smoothness of the random field, the dimension of
the observation space, and the number of increments used for estimation. As a
corollary, our results establish the orthogonality of Mat\'ern Gaussian random
fields with different parameters when $d>4$. The case $d=4$ is still open.
