In order to account for this important distinction we need to make some assumptions about how the measurements are related to the actual state of the system. We will assume that the observations are a linear function of the actual state plus random noise,
With this, we can write
Applying (13) to the definition of ,
If we suppose that the actual state and the noise are uncorrelated, then the terms, Cxv and Csv are each zero.
So now we have
If we consider the case where the measurements are the same quantity as what we are estimating x (i.e. we are using density data to estimate the density field, as opposed to using salinity and temperature to estimate the density field), then H is just the identity matrix, so our estimator is,
If the noise is white noise, then Cv is a diagonal matrix and we see that the effect of not having the true state correlations, but estimates of it based upon the observations, is to increase the diagonal elements of the matrix to be inverted by the measurement noise variance.