An adaptive parametrized-background data-weak approach to variational data assimilation

We present an Adaptive Parametrized-Background Data-Weak (APBDW) approach to the variational data assimilation (state estimation) problem. The approach is based on the Tikhonov regularization of the PBDW formulation [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965], and exploits the connection between PBDW and kernel methods for regression. An adaptive procedure is presented to handle the experimental noise. A priori and a posteriori estimates for the L2 state-estimation error motivate the approach and guide the adaptive procedure. We present results for two synthetic model problems to illustrate the elements of the methodology. We also consider an experimental thermal patch configuration to demonstrate the applicability of our approach to real physical systems.