An optimal Savitzky–Golay derivative filter with geophysical applications: an example of self‐potential data

Abstract

A B S T R A C TWe propose a strategy in designing an optimal set of filter parameters, such as the order of interpolating polynomial and the filter length for a Savitzky-Golay derivative filter. The proposed strategy is based on the 'principle of parsimony' while satisfying the optimality conditions. The optimality conditions are based on the Durbin-Watson lag-1 test statistic and the Derringer-Suich desirability function. While the former checks for an appropriate data fitting, the latter, on the other hand, ensures minimal shape distortion of the reconstructed response. The proposed strategy of designing filter parameters is developed and validated through numerical experiments using Gaussian pulse as a test function which is contaminated with additive white Gaussian noise. In the numerical tests, the polynomial orders used were 3, 5 and 7, but the filter length for each polynomial was varying until the optimality conditions were satisfied. The Savitzky-Golay derivative filtering is used in obtaining the robust reconstruction of noisy geophysical anomaly and the robust estimation of its first-and secondorder derivatives. We validated the proposed technique on the published self-potential anomaly data using a data-based interpretation technique where the reconstructed anomaly and its first-and second-order derivatives were used in estimating model parameters. The data-based interpretation using the proposed technique of Savitzky-Golay derivative filtering provides a close agreement with the published results.

Applications of Hilbert transform and Newton's root finding algorithm in delineating buried fracture from magnetic anomaly data

“…The SGDF, although appearing almost six decades ago (Savitzky and Golay, 1964) and immediately found a huge interest in the field of analytical chemistry, is getting increasing interests among geoscientists and engineers as well in the recent times (Roy, 2013c, 2017, 2020; Baba et al ., 2014; Liu et al ., 2016). The SGDF is essentially a type‐I finite impulse response (FIR) filter that filters out noise in data by replacing each data point with a locally interpolated value of a low‐order polynomial.…”

“…In this paper, I demonstrate that the Hilbert transform, well‐known Newton's root finding algorithm, and simple standard method of estimating derivatives are all one needs to delineate a buried fracture from the measured magnetic data, provided the measured data demonstrate a sufficiently smooth anomaly response. Since such a requirement is difficult to achieve in practice, I provide a solution by reconstructing the noisy measurements and estimating derivatives in a robust manner via the Savitzky–Golay derivative filter (SGDF) (Roy, 2020). I also propose using the Hilbert–Noda transformation (HNT) (Roy, 2018a) in estimating the Hilbert transform.…”

Digital Filtering of Railway Track Coordinates in Mobile Multi–Receiver GNSS Measurements

“…The filter requires parametrization, i.e., assuming the degree n of the approximating polynomial and the window dimension m . Automatic parametrization of the filter is possible [ 38 ].…”

“…This process is repeated for all points, thus obtaining a smoothed signal and its differences (playing the same role as derivatives for continuous functions). The filter requires parametrization, i.e., assuming the degree n of the approximating polynomial and the window dimension m. Automatic parametrization of the filter is possible [38].…”

“…All this information was used in the algorithm for filtering the disturbed GNSS coordinates. For this purpose, the Whittaker filter was used, which belongs to the group of discrete penalized least square filters and is resistant to relatively long intervals of data absence in measuring data sequences [ 37 , 38 ]. It is noteworthy that this is the first application of this type of GNSS data filtering to precisely determine the railway track axis coordinates.…”