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,

(13) |

where

With this, we can write

(14) |

Applying (13) to the definition of ,
gives

(15) |

If we suppose that *the actual state and the noise are
uncorrelated*, then the terms, *C*_{xv} and *C*_{sv} are each zero.

So now we have

(16) |

and,

(17) |

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,

(18) |

and,

(19) |

If the noise is white noise, then *C*_{v} 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.