Gaussian processes (GPs) are widely used in statistical modeling as the distribution of a random effect in a mixed linear model. The GP's unknowns are commonly estimated using the restricted likelihood (RL) or the closely related Bayesian analysis. It is unclear how the error variance and the GP's variance and range parameters are fit to features in the data because the RL does not have a closed form. To clarify this, we need a simple, interpretable form of the RL. We use the spectral approximation to obtain a simple approximate RL, which is identical to the likelihood arising from a gamma-errors generalized linear model (GLM) with the identity link. We use this GLM to conjecture about how GP parameters are fit to data and investigate those conjectures by introducing features into simulated data, e.g., outliers and mean shifts, and observing how those features affect parameter estimates. We describe briefly how this representation can be used to derive diagnostic tools to identify potential covariates, to examine whether and how the data support their inclusion, and to assess how their inclusion will affect the fit of the GP and error parts of the mixed linear model.