Statistical properties of a 3D field from spatial averages at different scales

I'm trying to estimate the statistical properties of a 3D vector field (the magnetic field vector in the solar atmosphere) over a 2D field of view from observations. I will neglect for the moment that the problem is complex because I cannot measure directly the vector field but only some quantities (the Stokes profiles in some spectral lines) that are non-linearly proportional to the components along the line of sight and on the perpendicular plane. So, I will assume that I can measure $B_z$ (the component along the vertical) and $B_x$ and $B_y$ (in the perpendicular plane). The main difficulty is that the spatial resolution of our best instruments integrates all the information in a square of side $L$, so I get the average of the components on the resolution element $\langle B_z \rangle_L$, $\langle B_x \rangle_L$ and $\langle B_y \rangle_L$.

My question is if there is any way in which I can relate the properties of such averages at different sizes $nL$ (multiples of the smallest resolution element) with the statistical properties of the vector field at full resolution?

-
 This is known as the "change of support" problem: Google it for some good introductory references. – whuber♦ Aug 13 '12 at 17:48