Smoothing and Interpolating Noisy GPS Data with Smoothing Splines

Abstract

A comprehensive methodology is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers which are typical in GPS signals. We demonstrate the effectiveness of our methods on GPS trajectory data obtained from oceanographic floating instruments known as drifters.This work has not yet been peer-reviewed and is provided by the contributing author(s) as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis.

Fast and Accurate Computation of Vertical Modes

“…The primary advantage to using B‐splines in this scenario is that B‐splines can be created for arbitrary grids without suffering from Runge's phenomena at lower orders. We fit the data to sixth‐order interpolating spline (with five nonzero derivatives) using the methodology and numerical implementation described in Early and Sykulski ().…”

“…This decision, and how to deal with measurement noise in general, is beyond the scope of this manuscript. Additional smoothing of the density data can be done using many techniques, including the constrained smoothing splines described in Early and Sykulski ().…”

Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories

“…For the work here it is necessary to use map coordinates {x(t), y(t)} with a projection that locally preserves area and shape. Following [11] we use the transverse Mercator projection with central meridian placed between the minimum and maximum longitude of the drifter experiment and add a false northing and easting to shift the origin to the southwest corner. The total velocity u total of a drifter is then two-dimensional and assumed to represent the velocity at the depth of the drifter drogue.…”

“…At degree S = 1, B-splines are triangle functions that span two knot points, thus providing continuity in time as well as a piecewise first derivative. This generalises to higher degrees, where a B-spline of degree S has S non-zero derivatives, as reviewed in [11]. The key benefit to this approach is that we can allow for time variation in the parameters while simultaneously choosing an effective window length-all while adding only a few coefficients to the model.…”

“…Choosing the appropriate model from Figure 9 proceeds exactly as in Section 3.3, but with the additional caveat that one must choose the spline degree S and the number of splines M. With the restriction that the spline degree S < M, a reasonable upper bound is S = 3, the cubic spline. The number of splines M can be chosen by assuming a minimum window length (as discussed in Section 4.2), treating the centre of each window as a data point, and then applying the formula for the canonical interpolating spline in [11]. To compute this explicitly, assume a time series of length T, with minimum window length W, then this results in a total of M = max( T/W , 1) evenly sized windows of minimum length.…”