A 4D-Var Analysis System for the California Current: A Prototype for an Operational Regional Ocean Data Assimilation System - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 346

Observation errors were assumed to be uncorrelated in space and time, resulting
in a diagonal observation error covariance matrix, R . The variances along the main
diagonal of R were assigned as the sum of measurement error and the error of
representativeness. Measurement errors were chosen with the following standard
deviations: 0.02-0.04 m for dynamic topography depending on the instrument;
0:3

1 ı C for SST depending on the satellite platform; 0:5 ı C for in situ

- 0:1 - T ;and

0.01-0.1 for in situ .

The background error standard deviations for the initial condition components
of the control vector (i.e. the elements of the diagonal matrix

S

) were estimated
based on the variance of a long run of the model subject only to surface forcing and
boundary conditions (i.e. no data assimilation). The surface forcing and boundary
condition fields used depend on the application as described above. However, in the
case of salinity, past experience has revealed that the background errors computed
using this method are too large, so the standard deviations for

˙

were capped at
0.1. The temporal variability of the surface forcing fields for the appropriate period
was used as the variance for the background surface forcing error, and the open
boundary condition background error variances were chosen to be the variances of
the appropriate data (as described above) at the boundaries.
As noted in Sect. 14.2.4 , each block diagonal component of the background
error covariance matrix D was modeled using the diffusion operator approach of
Weaver and Courtier ( 2001 ). The capability to have spatially varying correlation
lengths is a fairly recent addition to the ROMS 4D-Var code, so in the calculations
and applications presented in Sects. 14.4 and 14.5 , the horizontal and vertical
correlation lengths were held constant over the model domain. The decorrelation
length scales used to model the prior errors of all initial condition control variable
components of B x were 50 km in the horizontal and 30 m in the vertical. Horizontal
correlation scales chosen for the background surface forcing error components of
B f were 300 km for wind stress and 100 km for heat and freshwater fluxes. The
correlation lengths for the background open boundary condition error components
of B b were chosen to be 100 km in the horizontal and 30 m in the vertical. No
explicit account is taken of temporal correlations in any of the background errors
in the current version of ROMS 4D-Var, although this capability is currently under
development. However, the surface forcing, and boundary condition increments
were only computed daily and interpolated to each intervening model time step, a
procedure which effectively introduces some temporal correlation of the errors. The
correlation lengths chosen for the prior errors are typically estimated using semi-
variogram techniques that have traditionally been applied to observational data (e.g.
Banerjee et al. 2004 ; Milliff et al. 2003 ; Matthews et al. 2011 ). However, some
level of subjective tuning of the correlation lengths is also typically required to
optimize the performance of the 4D-Var algorithm. A discussion of the choice of
the aforementioned background error covariance parameters for the CCS can be
found in Broquet et al. ( 2009a , 2009b , 2011 )and Moore et al. ( 2011b ). In all of the
calculations presented here, the multivariate balance operator was not used.

S