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

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.

💡 Research Summary

The paper introduces a novel quantitative metric for comparing two signals or surface profiles, termed the “surface similarity parameter” or normalized error, which is based on the Sobolev norm of the difference between the signals. Traditional approaches to signal comparison—such as visual overlay, root‑mean‑square (RMS) error, or correlation coefficients—often fail to capture subtle discrepancies, especially those involving high‑frequency content, phase shifts, or localized structural differences. By employing the Sobolev space (H^{s}), the authors are able to weight both low‑ and high‑frequency components in a single, mathematically rigorous framework.

Theoretical foundation
Given two functions (f_{1}(x)) and (f_{2}(x)) (or their discrete counterparts), the difference (\Delta(x)=f_{1}(x)-f_{2}(x)) is measured using the Sobolev norm
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