Yu, P. (2006). *Development of New Techniques for Assimilating Satellite Altimetry Data into Ocean Models*. Ph.D. thesis, Florida State University, Tallahassee, FL.

State of the art fully three-dimensional ocean models are very computationally expensive and their adjoints are even more resource intensive. However, many features of interest are approximated by the first baroclinic mode over much of the ocean, especially in the lower and mid latitude regions. Based on this dynamical feature, a new type of data assimilation scheme to assimilate sea surface height (SSH) data, a reduced-space adjoint technique, is developed and implemented with a three-dimensional model using vertical normal mode decomposition. The technique is tested with the Navy Coastal Ocean Model (NCOM) configured to simulate the Gulf of Mexico. The assimilation procedure works by minimizing the cost function, which generalizes the misfit between the observations and their counterpart model variables. The “forward” model is integrated for the period during which the data are assimilated. Vertical normal mode decomposition retrieves the first baroclinic mode, and the data misfit between the model outputs and observations is calculated. Adjoint equations based on a one-active-layer reduced gravity model, which approximates the first baroclinic mode, are integrated backward in time to get the gradient of the cost function with respect to the control variables (velocity and SSH of the first baroclinic mode). The gradient is input to an optimization algorithm (the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is used for the cases presented here) to determine the new first baroclinic mode velocity and SSH fields, which are used to update the forward model variables at the initial time. Two main issues in the area of ocean data assimilation are addressed: 1. How can information provided only at the sea surface be transferred dynamically into deep layers? 2. How can information provided only locally, in limited oceanic regions, be horizontally transferred to ocean areas far away from the data-dense regions, but dynamically connected to it? The first problem is solved by the use of vertical normal mode decomposition, through which the vertical dependence of model variables is obtained. Analyses show that the first baroclinic mode SSH represents the full SSH field very closely in the model test domain, with a correlation of 93% in one of the experiments. One common way to solve the second issue is to lengthen the assimilation window in order to allow the dynamic model to propagate information to the data-sparse regions. However, this dramatically increases the computational cost, since many oceanic features move very slowly. An alternative solution to this is developed using a mapping method based on complex empirical orthogonal functions (EOF), which utilizes data from a much longer period than the assimilation cycle and deals with the information in space and time simultaneously. This method is applied to map satellite altimeter data from the ground track observation locations and times onto a regular spatial and temporal grid. Three different experiments are designed for testing the assimilation technique: two experiments assimilate SSH data produced from a model run to evaluate the method, and in the last experiment the technique is applied to TOPEX/Poseidon and Jason-1 altimeter data. The assimilation procedure converges in all experiments and reduces the error in the model fields. Since the adjoint, or “backward”, model is two-dimensional, the method is much more computationally efficient than if it were to use a fully three-dimensional backward model.