  Attaining reliable profile gradients is of utmost relevance for many physical
systems. In most situations, the estimation of gradient can be inaccurate due
to noise. It is common practice to first estimate the underlying system and
then compute the profile gradient by taking the subsequent analytic derivative.
The underlying system is often estimated by fitting or smoothing the data using
other techniques. Taking the subsequent analytic derivative of an estimated
function can be ill-posed. The ill-posedness gets worse as the noise in the
system increases. As a result, the uncertainty generated in the gradient
estimate increases. In this paper, a theoretical framework for a method to
estimate the profile gradient of discrete noisy data is presented. The method
is developed within a Bayesian framework. Comprehensive numerical experiments
are conducted on synthetic data at different levels of random noise. The
accuracy of the proposed method is quantified. Our findings suggest that the
proposed gradient profile estimation method outperforms the state-of-the-art
methods.
