THE ACCURACY OF PARAMETER-ESTIMATION FROM NOISY DATA, WITH APPLICATION TO RESONANCE PEAK ESTIMATION IN DISTRIBUTED BRILLOUIN SENSING

Distributed Brillouin sensing of strain and temperature works by makin g spatially resolved measurements of the position of the measurand-dep endent extremum of the resonance curve associated with the scattering process in the weakly nonlinear regime. Typically, measurements of bac kscattered Stokes intensity (the dependent variable) are made at a num ber of predetermined fixed frequencies covering the design measurand r ange of the apparatus and combined to yield an estimate of the positio n of the extremum. The measurand can then be found because its relatio nship to the position of the extremum is assumed known. We present ana lytical expressions relating the relative error in the extremum positi on to experimental errors in the dependent variable. This is done for two cases: (i) a simple non-parametric estimate of the mean based on m oments and (ii) the case in which a least squares technique is used to fit a Lorentzian to the data. The question of statistical bias in the estimates is discussed and in the second case we go further and prese nt for the first time a general method by which the probability densit y function (PDF) of errors in the fitted parameters can be obtained in closed form in terms of the PDFs of the errors in the noisy data.