Introduction

The task is to find a statistic that will estimate the smoothing parameters of a Gaussian Random Field. Such a field can be constructed by taking an uncorrelated, zero-mean (white-noise) field, $F_W$, and smoothing it spatially with a Gaussian filter. That is, the smoothed field $F_S$ is given by:

$$F_S(\ensuremath{\mathbf{x}}) = G(\ensuremath{\mathbf{x}}) \o... ...f{p}}) F_W(\ensuremath{\mathbf{p}}) \, d\ensuremath{\mathbf{p}}$$ (1)

where $\otimes$ denotes convolution and

$$G(\ensuremath{\mathbf{x}}) = G_{\sigma_x}(x) \, G_{\sigma_y}(y) \, G_{\sigma_z}(z)$$ (2)

$$G_\sigma(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp(\frac{- x^2}{2 \sigma^2})$$ (3)

is the spatial filter. Note that all integrals in this report are definite integrals over all space (that is, from $-\infty$ to $+\infty$ in 1D).

In addition, the auto-correlation of $F_W$ is:

$$E\{ F_W(\ensuremath{\mathbf{x_1}}) F_W(\ensuremath{\mathbf{x... ...= \delta(\ensuremath{\mathbf{x_1}}- \ensuremath{\mathbf{x_2}})$$ (4)

where $\delta(\cdots )$ is the Dirac delta function.