Parameter mixing errors within a measuring volume with applications to incoherent scatter

The effect on parameter error estimates resulting from parameter variations within the measuring volume under consideration is studied in the framework of linear statistical inversion theory. It is shown that using estimates for the parameter averages is equivalent to having the theory corrected by the covariances of the variables coupled with the second derivatives of the theory function. If the parameter distributions were known exactly, this would only introduce a bias in the linear theory and hence a systematic error in the parameter centre point estimates. When the distributions are not known exactly, there is another source of error consisting of the uncertainties in the parameter variation estimates. This leads to new error bounds on the allowed parameter variability within the volume under consideration if some prescribed accuracy in the parameter average estimates is required. These considerations are applied to incoherent scatter (IS) radar measurements, where it is important to be able to estimate the effect of integrating both in space and time over a volume with varying parameters in order to obtain spectra or autocorrelation functions. Numerical examples are given in the case of the O⁺ content estimates in measurements with the EISCAT UHF radar, when for example the ion temperature varies over the integration ranges. The results obtained may be used in the design of experiments when high resolution composition measurements are required.