Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 227

Chapter 10
Treating Nonlinearities in Data-Space
Variational Assimilation
Brian S. Powell
Abstract One goal of four-dimensional variational (4D-Var) state estimation is to
utilize the longest time window that maximizes the observational constraints to
improve predictive skill; unfortunately, nonlinearities are present in geophysical
flows and limit the time in which the linear approximation is valid. For weakly
nonlinear flows, updating the background trajectory, relinearizing, and repeating
the minimization is a way to lengthen the time window. This so called “outer-
loop” requires special consideration when minimizing the solution in data-space.
This discussion provides a review of the relevant theory and presents two data-
space cost functions: the standard cost-function that becomes unconstrained during
additional outer-loops and a modified function that preserves the original con-
straint. Experiments with the Lorenz (J Atmos Sci 20:130-141, 1963) model
show that unconstrained outer-loops perform similarly to sequentially applied 3D-
Var assimilations by overfitting the observations and producing state estimates
with poor predictive skill. Evaluating the posterior error covariances, the analysis
error, and minimum cost function illustrate how overfitting degrades the solution.
This is an important lesson for assimilation schemes: minimizing the model
data residuals without proper constraint does not provide the optimal solution.
By properly constraining the data-space outer-loop, adjoint-based methods will
be well constrained over time windows that are longer than those required by
linearity.