A Weak Constraint 4D-Var Assimilation System for the Navy Coastal Ocean Model Using the Representer Method - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 369

data space involving the representer matrix, the data error covariance matrix and
the innovation vector. The entire representer matrix need not be computed since the
linear system can be solved using an iterative algorithm (e.g. the conjugate gradient),
by taking advantage of the symmetry of each matrix involved. The representer
coefficients constitute the right hand side of the adjoint equation in the EL system.
Once the representer coefficients are computed, they are substituted in the adjoint
equation which is then solved and substituted in the forward linear equation for
the final solution. A background solution around which the model is linearized is
needed. Usually it is the solution of the nonlinear model. For the first guess solution,
one may consider either the background or the tangent linear solution around
the background. Also, the new optimal solution may replace the background for
another minimization process (i.e. outer loops) until formal convergence ( Bennett
et al. 1996 , 1998 , 2002 ; Ngodock et al. 2000 , 2007 , 2009 ).

15.4 Experiment Setup and Results

Assimilation experiments are carried out with two different data sets, and the results
shown below are primarily aimed at evaluating the 4D-Var system's ability to fit
both the assimilated and the non-assimilated observations.

15.4.1 MODAS Data

MODAS generates synthetic vertical profiles of temperature and salinity in the two
following steps: first, a subsurface temperature is computed at a given depth using a
regression from sea surface temperature and the steric component of the sea surface
height anomaly. Once the subsurface temperature is computed, a corresponding
subsurface salinity is computed using a climatology-based temperature/salinity
relationship, Fox et al. ( 2002 ). MODAS data are thus a combination of real sea
surface data (SSH and SST) and simulated sub-surface data derived from the real
surface data using regression and historical relationships.
MODAS synthetics are saved and utilized in the 4D-Var analysis at intervals of
6 h. There are approximately fifty-six uniformly distributed profiles of temperature
and salinity across the model domain. Each profile is represented on a vertical grid
of 46 layers that do not coincide with the model's vertical grid of 41 layers, but the
observation operator

in ( 15.10 ) handles the projection from the model grid to the
data grid. Temperature (salinity) observation errors are set to

H

0:2 ı C(0.1psu),and
held constant through the entire assimilation window. These observation errors are
purposefully set low, not because MODAS data are very accurate, but to test the
assimilation's ability to reduce large discrepancies with the model, i.e. to drive the
model with large errors to fit observations with small errors.