Differentiation of discrete data with unequal measurement intervals and quantification of uncertainty in differentiation using Bayesian compressive sampling

Calculation of derivatives on discrete measurement data with unequal intervals is often required in geotechnical engineering, such as interpretation of stiffness reduction curve of soil from pressuremeter test data, pile lateral responses from inclinometer data. Such a task is however tricky and challenging, because a small error or noise in the measurements may amplify and lead to huge fluctuations in the derivatives obtained. The amplification becomes increasingly significant as the order of derivative increases. It is therefore of great importance to evaluate reliability of the derivatives obtained and quantify the uncertainty associated with the derivative calculation. A Bayesian compressive sampling-based method is proposed in this paper to address this problem. It not only provides high-order derivatives on discrete measurement data, even at un-sampled locations, but also quantifies the uncertainty associated with the derivatives obtained and offers an index to evaluate reliability of the derivatives obtained. The proposed approach is illustrated using both real-life pressuremeter data and numerical example of pile lateral responses. A comparison is also made between the proposed method and several existing methods in geotechnical literature. It shows that the proposed method performs better than existing methods and it is applicable to problems with both elastic and plastic soil responses.