Achieving comparable uncertainty estimates with Kalman filters or linear smoothers for bathymetry data

Abstract

This paper examines and contrasts two estimation methods, Kalman filtering and linear smoothing, for creating interpolated data products from bathymetry measurements. Using targeted examples, we demonstrate previously obscured behavior showing the dependence of linear smoothers on the spatial arrangement of the measurements, yielding markedly different estimation results than the Kalman filter. For bathymetry data, we have modified the variance estimates from both the Kalman filter and linear smoothers to obtain comparable estimators for dense data. These comparable estimators produce uncertainty estimates that have statistically insignificant differences via hypothesis testing. Achieving comparable estimation is accomplished by applying the “propagated uncertainty” concept and a numerical realization of Tobler's principle to the measurement data prior to the computation of the estimate. We show new mathematical derivations for these modifications. In addition, we show test results with (a) synthetic data and (b) gridded bathymetry in the area of the Scripps and La Jolla Canyons. Our tenfold cross‐validation for case (b) shows that the modified equations create comparable uncertainty for both gridding algorithms with null hypothesis acceptance rates of greater than 99.95% of the data points. In contrast, bilinear interpolation has 10 times the amount of rejection. We then discuss how the uncertainty estimators are, in principle, applicable to interpolate geophysical data other than bathymetry.

Computationally efficient variable resolution depth estimation

“…A technique that straddles both camps is the cube algorithm (Calder and Mayer, 2003), which is designed to compute the best estimate of depth at any given point in the area of interest, taking into account measurement uncertainty, which is often repeated over a regular grid to reconstruct the measurand. Modifications of the cube algorithm for sparse data have also been proposed (Bourgeois et al, 2016;Zambo et al, 2015). The cube algorithm has become a widely accepted approach for bathymetric data processing, being incorporated into a large majority of the software packages used for this purpose.…”

Airborne Geophysical Investigation beneath Antarctica's Ross Ice Shelf

“…While we show this to be true with our synthetic models, a quantitative method of predicting this uncertainty has yet to be implemented here. The field of conventional bathymetry surveying has developed uncertainty analysis tools that may be applicable to this (Bourgeois et al, 2016).…”

“…Quantify the uncertainty of the regional separation process. This will likely include techniques used in conventional bathymetry surveying (Bourgeois et al, 2016;Calder & Elmore, 2017), or a sequential Gaussian simulation (Perozzi et al, 2021).…”