A Robust Quantitative Comparison Criterion of Two Signals based on the Sobolev Norm of Their Difference

TL;DR

This paper introduces a Sobolev norm-based method for quantitatively comparing two surfaces across spatial or temporal domains, providing a normalized error metric that ranges from perfect agreement to complete disagreement.

Contribution

The work presents a novel, normalized surface similarity parameter based on Sobolev norms for quantitative surface comparison, applicable to analytical, numerical, and experimental data.

Findings

The parameter ranges from 0 (perfect match) to 1 (complete mismatch).

Validated with water wave experiments and numerical simulations.

Effectively distinguishes surface differences in various canonical cases.

Abstract

In this manuscript we present a method for the quantitative comparison of two surfaces, applicable to temporal and/or spatial extent in one or two dimensions. Often surface comparisons are simply overlaid graphs of results from different methodologies that are qualitative at best; it is the purpose of this work to facilitate quantitative evaluation. The surfaces can be analytical, numerical, and/or experimental, and the result returned by the method, termed surface similarity parameter or normalized error, has been normalized so that its value lies between zero and one. When the parameter has a value of zero, the surfaces are in perfect agreement, whereas a value of one is indicative of perfect disagreement. To provide insight regarding the magnitude of the parameter, several canonical cases are presented, followed by results from breaking water wave experimental measurements with numerical simulations, and by a comparison of a prescribed, periodic, square-wave surface profile and the subsequent manufactured surface.