  Approximating significance scans of searches for new particles in high-energy
physics experiments as Gaussian fields is a well-established way to estimate
the trials factors required to quantify global significances. We propose a
novel, highly efficient method to estimate the covariance matrix of such a
Gaussian field. The method is based on the linear approximation of statistical
fluctuations of the signal amplitude. For one-dimensional searches the upper
bound on the trials factor can then be calculated directly from the covariance
matrix. For higher dimensions, the Gaussian process described by this
covariance matrix may be sampled to calculate the trials factor directly. This
method also serves as the theoretical basis for a recent study of the trials
factor with an empirically constructed set of Asmiov-like background datasets.
We illustrate the method with studies of a $H \rightarrow \gamma \gamma$
inspired model that was used in the empirical paper.
