Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 238

Fig. 10.5 The total cost
function improvement factor
of the constrained versus
overfit ( black ) methods and
constrained versus 3D-Var
( grey ) as a function of
number of observations. The
assimilation ( dashed )and
forecast ( solid ) periods are
shown. Values near 1 show a
significant error in the overfit
or 3D-Var cases

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

3 6 9 12 15 18 21

Observations

poor forecast results. These two methods may exhibit extremely small residuals at
individual times; however, this comes at the expense of adding noisy model structure
with a loss of dynamical consistency throughout the time window. For the remainder
of the discussion, posterior analysis of the two 4D-Var cases are examined.

10.4 Posterior Statistics

The overfit method exhibits worse predictive skill for every case examined. By
ignoring the total increment constraint, it emphasizes the noisy observations. There
are a number of posterior diagnostics available to quantify the performance of each
outer-loop method, including: the final analysis error, the consistency of the prior
and posterior errors, and the true minimum cost functions.
The analysis error ( E a ) of the assimilation is provided by the inverse term in
( 10.6 ). For most geophysical applications, direct calculation of this matrix is not
possible due to the size; however, it can be computed directly for the Lorenz63
system. Because these are twin experiments, the true error between the analysis and
the truth is computed and compared against the diagonal of E a . The ratio between
the true error, E t , and the analysis error, (diag ), is compared to examine how

well each cost-function evaluates the true statistics. By ignoring the background
error, it is expected that the overfit method will underestimate the analysis error.
For all valid ensemble members, the mean E t =.

ΠE a

was 1.075 for the overfit
method. This is an underestimate of the true error by 7.5 %. For the constrained
method, the mean ratio was 0.984, which is an overestimate of the true analysis
error by 1.6 %.
The initial background state and analysis state are compared to the true initial
state for all valid ensemble members tested and the overfit method increases the
initial error by 8.8 % on average, while the constrained method improves the initial
error by 3.2 %. Without a fixed constraint, the overfit method tends to push away

diag ΠE a /