
Error Covariance MatricesThe error covariance matrix (ECM) 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:
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, will 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. Below, there are errorcovariance matrices for the following:
Covariance matrices for NRT products
Covariance matrices for Reprocessed climate data records
Other covariance matrices (included in ROPP)Some example ECMs are available in the ropp_1dvar/errors directory of the ROPP distribution, for the benefit of the users. They are described below.


