  Gaussian processes (GPs) are a class of Kernel methods that have shown to be
very useful in geoscience and remote sensing applications for parameter
retrieval, model inversion, and emulation. They are widely used because they
are simple, flexible, and provide accurate estimates. GPs are based on a
Bayesian statistical framework which provides a posterior probability function
for each estimation. Therefore, besides the usual prediction (given in this
case by the mean function), GPs come equipped with the possibility to obtain a
predictive variance (i.e., error bars, confidence intervals) for each
prediction. Unfortunately, the GP formulation usually assumes that there is no
noise in the inputs, only in the observations. However, this is often not the
case in earth observation problems where an accurate assessment of the
measuring instrument error is typically available, and where there is huge
interest in characterizing the error propagation through the processing
pipeline. In this letter, we demonstrate how one can account for input noise
estimates using a GP model formulation which propagates the error terms using
the derivative of the predictive mean function. We analyze the resulting
predictive variance term and show how they more accurately represent the model
error in a temperature prediction problem from infrared sounding data.
