Normalisation

Consider a scalar multiple of a field. If this scalar is constant and independent of position, then the covariance of the field scales with the constant squared. For instance, let $S(\ensuremath{\mathbf{x}}) = k F(\ensuremath{\mathbf{x}})$, then

$$E \{ S(\ensuremath{\mathbf{x_1}}) S(\ensuremath{\mathbf{x_2}... ... F(\ensuremath{\mathbf{x_1}}) F(\ensuremath{\mathbf{x_2}}) \}.$$ (29)

The importance of this result is that by choosing an appropriate scaling factor, the covariance of the new smoothed field, $S$, can be set to any constant value -- in agreement with the experimentally normalised results.