Treating Nonlinearities in Data-Space Variational Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 228

10.1 Introduction

Data assimilation has become an important component of numerical modeling com-
bining numerical models with observational data to obtain an improved estimate of
the circulation. Variational methods aim to minimize the residual difference between
the model and observations via least-squares. Three-dimensional variational assim-
ilation (3D-Var) holds time constant and is valid for synoptic observations. Four-
dimensional variational assimilation (4D-Var) is constrained by the physics of the
model to preserve the dynamical relationships between the observations during a
time window. Much of the theoretical 4D-Var work is described in Le Dimet and
Talag r an d ( 1986 ), Talagrand and Courtier ( 1987 ), and Courtier et al. ( 1993 , 1994 ).
Lorenc ( 2006 ) provides a comparison of 3D-Var and 4D-Var.
The problem of minimizing the residuals can be accomplished in either the space
of the model or in the space defined by the observations referred to as model-space
and data-space, respectively. The focus of this work is on the data-space methods
that are well described in Courtier ( 1997 ), Bennett ( 2002 ), Chua and Bennett ( 2001 ),
Bennett et al. ( 2008 ), El Akkraoui et al. ( 2008 ), and El Akkraoui and Gauthier
( 2010 ).
Variational methods make assumptions of linearity for the time-scales over which
the assimilation occurs. For geophysical circulations that are nonlinear, a choice
must be made: limit the time window over which the assimilation is performed, or
occasionally update the nonlinear trajectory during the assimilation procedure. For
some applications, reducing the time window to ensure linearity would collapse
the problem to 3D-Var (in the limit as the assimilation time-window converges
to the model time-step, 4D-Var becomes 3D-Var). For many applications, it is
unacceptable to consider time-dependent observations synoptic. In 4D-Var, the goal
is to use the longest possible time window, incorporating as many observations as
are available; however, the growth of nonlinearities in the flow may disrupt the
convergence of the iteration scheme.
The purpose of this discussion is to examine how data-space variational methods
in weakly nonlinear regimes are properly constrained to prevent overfitting of
noisy observations. With a longer time window and more observational constraints,
a better estimate and prediction are produced. Furthermore, long time-windows
provide dynamically consistent circulations without frequent initialization shocks.
If the prior estimate is not the fixed reference as is often done in sequential 3D-Var
and the standard 4D-Var cost function, the residuals are minimized, but the model
structure is no longer physically consistent.
This discussion presents how outer-loops in variational data-space methods can
be updated to use a longer assimilation window to better estimate the state. First, the
standard data-space cost-function is derived and shown to be inappropriate when
applying multiple outer-loops. Two outer-loop methods (one constrained and one
unconstrained) and sequential 3D-Var are used to illustrate the insidious effects
of overfitting the observations. A number of posterior diagnostics are presented to
examine the consistency of the solution before concluding.