  Gaussian Processes (GPs) are a class of kernel methods that have shown to be
very useful in geoscience applications. They are widely used because they are
simple, flexible and provide very accurate estimates for nonlinear problems,
especially in parameter retrieval. An addition to a predictive mean function,
GPs come equipped with a useful property: the predictive variance function
which provides confidence intervals for the predictions. The GP formulation
usually assumes that there is no input noise in the training and testing
points, only in the observations. However, this is often not the case in Earth
observation problems where an accurate assessment of the instrument error is
usually available. In this paper, we showcase how the derivative of a GP model
can be used to provide an analytical error propagation formulation and we
analyze the predictive variance and the propagated error terms in a temperature
prediction problem from infrared sounding data.
