On the direct assimilation of along-track sea-surface height observations into a free-surface ocean model using a weak constraints four-dimensional variational (4D-Var) method

The representer method is adopted for solving a weak constraints 4D-Var problem for the assimilation of ocean observations including along-track sea-surface height (SSH), using a free-surface ocean model. Direct 4D-Var assimilation of SSH observations along the satellite tracks requires that the adjoint model be integrated with Dirac impulses on the right-hand side (rhs) of the adjoint equations for the surface elevation equation. The solution of this adjoint model will inevitably include surface gravity waves, and it constitutes the forcing for the tangent linear model (TLM) according to the representer method. This yields an analysis that is contaminated by gravity waves. A method for avoiding the generation of the surface gravity waves in the analysis is proposed in this study; it consists of removing the adjoint of the free surface from the rhs of the free-surface mode in the TLM. The information from the SSH observations will still propagate to all other variables via the adjoint of the balance relationship between the barotropic and baroclinic modes, resulting in the correction to the surface elevation. Two assimilation experiments are carried out in the Gulf of Mexico: one with adjoint forcing included on the rhs of the TLM free-surface equation, and the other without. Both analyses are evaluated against the assimilated SSH observations, SSH maps from AVISO and independent surface drifters, showing that the analysis that did not include adjoint forcing in the free surface is more accurate. This study shows that when a weak constraints 4D-Var approach is considered for the assimilation of along-track SSH observations using a free-surface model, with the aim of correcting the mesoscale circulation, an independent model error should not be assigned to the free surface.