
GRM07 & GRM15: Error Covariance Products
The error covariance matrix is a dataset that specifies the correlations in the observation errors between all possible pairs of vertical levels. It is given as a twodimensional array, of size NxN , where N is the number of vertical levels in the sounding data products.
Specifically, the ECM for variable X
is given by E[(X
 E[X])(X  E[X])^{T}],
where in practice the expectation value is estimated by an average
over a representative sample. It is a matrix because levels
are treated independently: X
is a vector of length equal to the number of vertical levels.
X could be an observation (refractivity N or bending angle α), in which case the ECM is sometimes called O,
or a
background state ({T, q, p*} for ECMWFlike models; {p, q} for Met
Officelike models), in which case the ECM is sometimes called B. Both need to be specified in the minimisation of the usual 1dvar cost function
J(x) = ½(xb)^{T}B^{1}(xb) + ½(H(x)y)^{T}O^{1}(H(x)y))
which is needed to produce a retrieved analysis x.
ECMs are positive semidefinite matrices which can be factorised as
ECM = D^{T} C D
where the diagonal matrix D comprises the standard deviations of X, and defines the "size" of the variance, while the correlation matrix C describes the "shape" of the variance of X. By definition of standard deviation, the diagonal terms of C are all equal to 1.
The ROPP user has a choice of whether to use fixed standard deviations D,
read from an auxiliary covariance file, or to allow them to vary
profilebyprofile, by including them in the observation or background
file. At present, correlations C can only be read from the same auxiliary file. It is the content of these auxiliary covariance files which is described here.
In ROPP, the correlations in the covariance files are stored in packed,
"triangular" format, so that successive elements of the variable "corr"
hold the following matrix elements:
1 
2 
4 
7 
... 

3 
5 
8 
... 


6 
9 
... 



10 
... 




... 
By exploiting the symmetry of the correlation coefficients, only ½N(N+1) of them need to be stored.
The correlation matrices can be binned (in the same file) according to
latitude. In principle this could be extended to seasonality. Standard
deviations can also be held in the same file. These, rather than the
profilebyprofile SDs, wil be used in the retrieval if
bg/obs_covar_method = FSFC or FSDC.
Further details on all the above can be found in the ROPP FM and 1DVAR User Guide.
There are two basic errorcovariance matrices:
 Covariance matrix for NRT sounding data products (GRM07 NEM)
 Covariance matrix for Reprocessed products
 Covariance matrix for Offline sounding data products (GRM15 OEM)
Each matrix can be provided in several versions reflecting variations with geographical areas (e.g. latitude) and with season.
NRT STATUS:
Expected dissemination and availability for download: Q4 2011
Description 
Image
(click to expand) 
File 
Background
Suitable for 137L ECMWF background state.
Currently, the T and p* standard deviations are the ones
shown in the file. The standard deviation of q is
calculated profilebyprofile, using the background q
multiplied by a fixed <dq/q>. These
standard deviations are then used to calculate the
covariances using the given correlation matrix as ECM
= D C D.
The block diagonal structure of correlation matrix arises
from use of a 3 element state vector: {T, q, p*}.


ropp_bg_ecmwf_error_corr_L60.nc
ropp_bg_ses_ecmwf_error_corr_L137.nc

Observations
Suitable for 247L refractivity retrievals. (Bending
angle correlation matrices are usually assumed to be
diagonal.)
Note that there is a minimum σ(N) of 0.02 Nunits above
about 25km.


ropp_ob_refrac_error_corr_247L.nc 
Reprocessed STATUS:
Description 
Image
(click to expand) 
File 
Background
Suitable for 60 L ERAI background profiles. Average of
ERAI error of first guess resolved on 5 deg latitude
intervals. Pressure standard deviation has been inflated to
compensate for bias of ERAI forecast with respect to ERAI
analysis.


ropp_bg_ecmwf_error_corr_L60.nc
bg_ef_ei_error_corr_sigma_PsBias_LatB5d_L60.nc

Offline STATUS:
Some example ECMs are available in the ropp_1dvar/errors directory
of the ROPP distribution, for the benefit of the users. They
are descibed below.
Description 
Image
(click to expand) 
File 
Background
Suitable for (early) 60L ECMWF background state {T, q, p*}.
Note
that correlation matrix assumes level 1 at the top of
the atmosphere.


ropp_bg_ecmwf_error_corr_L60.nc 
Background
Suitable for (current) 91L ECMWF background state {T, q,
p*}.
Note
that correlation matrix assumes level 1 at the bottom of
the atmosphere.


ropp_bg_ecmwf_error_corr_L91.nc 
Background
Suitable for (early) 50L Met Office background state
{p, q}.
Three latitude bands: 90 to 20, 20 to 20, 20 to 90N.


ropp_bg_meto_error_corr_L50.nc 
Background
Suitable for (current) 70L Met Office background
state {p, q}.
Three latitude bands: 90 to 20, 20 to 20, 20 to
90N.


ropp_bg_meto_error_corr_L70.nc 
Observation
Suitable for 300L bending angles. (Note that C is the
identity matrix).


ropp_ob_bangle_error_corr.nc 
Observation
Suitable for standard 247L refractivities.


ropp_ob_refrac_error_corr.nc 
