  Hausdorff dimension results are a classical topic in the study of path
properties of random fields. This article presents an alternative approach to
Hausdorff dimension results for the sample functions of a large class of
self-affine random fields. We present a close relationship between the carrying
dimension of the corresponding self-affine random occupation measure introduced
by U. Z\"ahle and the Hausdorff dimension of the graph of self-affine fields.
In the case of exponential scaling operators, the dimension formula can be
explicitly calculated by means of the singular value function. This also
enables to get a lower bound for the Hausdorff dimension of the range of
general self-affine random fields under mild regularity assumptions.
