  In the past few decades, fibre Bragg grating (FBG) sensors have gained a lot
of attention in the field of distributed point strain measurement. One of the
most interesting properties of these sensors is the presumed linear
relationship between the strain and the peak wavelength shift of the FBG
reflected spectra. However, subjecting sensors to a non-uniform stress field
will in general result in a strain estimation error when using this linear
relationship. In this paper we propose a new strain estimation algorithm that
accurately estimates the mean strain value in the case of smooth non-uniform
strain distributions. To do so, we first introduce an approximation of the
classical transfer matrix model, which we will refer to as the approximated
transfer matrix model (ATMM). This model facilitates the analysis of FBG
reflected spectra under arbitrary strain distributions, particularly by
providing a closed-form approximation of the side-lobes of the reflected
spectra. Based on this new formulation, we derive a maximum likelihood
estimator of the mean strain value. The algorithm is validated using both
computer simulations and experimental FBG measurements. Compared to
state-of-the-art methods, which typically introduce errors of tens of
microstrains, the proposed method is able to compensate for this error. In the
typical examples that were analysed in this study, mean strain errors of around
60 microstrains were compensated.
