  The use of Gaussian processes (GPs) is a common approach to account for
correlated noise in exoplanet time series, particularly for transmission and
emission spectroscopy. This analysis has typically been performed for each
wavelength channel separately, with the retrieved uncertainties in the
transmission spectrum assumed to be independent. However, the presence of noise
correlated in wavelength could cause these uncertainties to be correlated,
which could significantly affect the results of atmospheric retrievals. We
present a method that uses a GP to model noise correlated in both wavelength
and time simultaneously for the full spectroscopic dataset while avoiding the
use of a 'common-mode' correction. To make this analysis computationally
tractable, we introduce a new fast and flexible GP method that can analyse 2D
datasets when the input points lie on a (potentially non-uniform) 2D grid - in
our case a time by wavelength grid - and the kernel function has a Kronecker
product structure. This simultaneously fits all light curves and enables the
retrieval of the covariance matrix of the transmission spectrum. By testing on
synthetic datasets, we demonstrate that our new approach can reliably recover
atmospheric features contaminated by noise correlated in time and wavelength.
In contrast, fitting each spectroscopic light curve separately performed poorly
when wavelength-correlated noise was present. It frequently underestimated the
uncertainty of the scattering slope and overestimated the uncertainty in the
strength of sharp absorption peaks in transmission spectra. Two archival
VLT/FORS2 transit observations of WASP-31b were re-analysed, with our method
strongly constraining the presence of wavelength-correlated noise in both
datasets and recovering significantly different constraints on atmospheric
features such as the scattering slope and strength of sodium and potassium
features.
