The effect of a priori knowledge on parameter estimation errors with applications to incoherent scatter

The dependence of the width of the a posteriori distribution (i.e. parameter estimation error) in a multiparameter fit on the a priori width of another parameter is studied in the framework of linear statistical inversion theory. An exact formula is given which makes it possible to estimate how accurate the a priori information on a given parameter must be in order to obtain the other parameter with a prescribed accuracy. These estimates are useful in those cases where some errors are strongly correlated, like the ion temperature and the O⁺ content in incoherent scatter measurements with the EISCAT UHF radar. The dependence of the solution on the accuracy of the power profile estimates is given. Likewise, the behaviour of the error when a linear combination of the parameters is known exactly is given. The formulae given have a very simple geometrical meaning, and together they give a complete solution to the problem of estimating the accuracies in a multiparameter fit when a linear combination of the variables is known with a prescribed accuracy. It is shown that under the conditions considered in this paper, oxygen content can be determined to an accuracy of 10 % when the ion temperature is known to within 7 %, if we can assume that the collision frequency is zero.