Inverse modelling techniques for snow and ice thickness retrievals from satellite altimetry &#160;
Authors: Perez Ferrer Joel , Tsamados Michel , Fox Matthew , Suciu Tudor , Heorton Harry , Nab Carmen 
Published: March 2022

&lt;p&gt;We have recently applied an objective mapping type approach to merge observations from multiple altimeters, both for enhancing the temporal/spatial resolution of freeboard samples and for analyzing crossovers between satellites (Gregory et al, 2021). This mapping provides optimal interpolation of proximal observations to a location in space and time based on the covariance of the observations and &lt;em&gt;a priori&lt;/em&gt; understanding of their spatiotemporal correlation length scales. This offers a best linear estimator and error field for the observation (radar freeboard or snow depth), which can be used to better constrain pan-Arctic uncertainties.&amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;In addition we will explore here a newly developed inverse modelling framework &amp;#160;to synchronously retrieve the snow and ice thickness from bias corrected or calibrated radar freeboards from multiple satellite retrievals. The radar equations expressed in section can be rearranged to formulate the joint forward model at gridded level relating measured radar freeboards from multiple satellites (and airborne data) to the underlying snow and ice thickness. In doing so we have also introduced a penetration factor correction term for OIB radar freeboard measurements. To solve this inverse model problem for &amp;#160;and &amp;#160;we use the following two methodologies inspired from Earth Sciences applications (i.e. seismology): &amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Space &amp;#8216;uncorrelated&amp;#8217; inverse modelling.&lt;/strong&gt; The method is called `space uncorrelated' inverse modelling as the algorithm is applied locally, for small distinct regions in the Arctic Ocean, multiple times, until the entire Arctic ocean is covered. To sample the parameter space &amp;#160;we use the publicly available Neighbourhoud Algorithm (NA) developed originally for seismic tomography of Earth&amp;#8217;s interior and recently by us to a sea ice dynamic inversion problem (Hoerton et al, 2019). &amp;#160;&amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Space &amp;#8216;correlated inverse modelling.&lt;/strong&gt; For the second method of inverse modelling, we used what we call a `space correlated' approach. Here the main algorithm is applied over the entire Arctic region, aiming to retrieve the desired parameters at once. In contrast with the previous approach, in this method we take into account positional correlations for the physical parameters when we are solving the inverse problem, the output being a map of the Arctic composed of a dynamically generated a tiling in terms of Voronoi cells. In that way, regions with less accurate observations will be more coarsely resolved while highly sampled regions will be provided on a finer grid with a smaller uncertainty. The main algorithm used here to calculate the posterior solution is called `reverse jump Monte Carlo Markov Chain' (hereafter referred to as rj-MCMC) and its concept was designed by Peter Green in 1999 (Green, 1995). Bodin and Sambridge (2009) adapted this algorithm for seismic inversion, which is the basis of the algorithm used in this study. &amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;