Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 232

Solving for the analysis increment reveals

k X

z k D G k 1 P 1 1 G k 1 R 1 G k 1 C R 1 d k 1 P 1 z j : (10.12)

j D 1

The constraint to minimize the total increment acts as an additional forcing on the
model-space solution as compared to ( 10.6 ). The additional background constraint
forcing from P k 1
j D 1

z j opposes the residuals force and prevents overfitting the obser-

vations. Once updating the prior nonlinear circulation no longer provides additional
increments, the right hand side of ( 10.12 ) vanishes and convergence is reached.
However, ( 10.12 ) remains in model-space, and the goal is to understand how
the additional constraint impacts data-space methods. This requires the additional
forcing term to be applied within data-space.
The solution is found by El Akkraoui et al. ( 2008 ), such that the analysis
increment for outer-loop

k is given by

z k D PG k 1 G k 1 PG k 1 C R 1

. d k 1 C G k 1 z k 1 /: (10.13)

This provides the constraint that each data-space outer-loop is constrained by its
prior loop. The prior increment is propagated through the tangent-linear model that
is now linearized about the previous outer-loop trajectory. It should be noted that if
the problem is fully linear, this method is identical to the typical approach because
G 1 D G 0 , and the additional term in ( 10.13 ) is negated.
This procedure can be carried forward for as many outer-loops as necessary
to approximate nonlinearities without violating the initial increment constraint.
It is important to note that with each outer-loop, a new minimization descent
begins from approximately zero using a new linearization, which will have negative
consequences on preconditioning schemes that improve their estimates during
subsequent outer-loop runs.

10.3 Experiments

As presented, there are two potential methods for handling multiple outer-loops in
data-space methods. The typical cost-function formulation ( 10.9 ) will be referred
to as the “overfit” method because it forgets the background constraint after more
than one outer-loop giving full weight to the observations. Overfitting observations
is never the goal of assimilation and should always be avoided, and the insidious
effects of overfitting are found in the results. The method derived by El Akkraoui
et al. ( 2008 ) is designated the “constrained” method as it respects the total
increment constraint. Because overfitting observations leads to solutions with highly
variable structure and poor prediction skill, the 4D-Var solutions are compared
along with sequential 3D-Var during both assimilation and prediction phases. The