Are Equilibrium Shoreline Models Just Convolutions?

Abstract

Yes. Equilibrium shoreline models, which simulate wave-driven cross-shore erosion and accretion, are mathematically equivalent to a discrete convolution (i.e., a weighted, moving average) of a time series of wave-forcing conditions with a parameterized memory-decay kernel function. The direct equivalence between equilibrium shoreline models and convolutions reveals key theoretical aspects of equilibrium behavior. Convolutions (representing quasi-low-pass filter operations) provide an intuitive theoretical description of shoreline erosion and accretion behavior in response to waves: that is, shoreline position often mirrors the weighted moving average of wave time series. Model-convolution equivalence also provides a conceptual basis to interpret, evaluate, and construct data-driven Machine-Learning/Deep-Learning (ML/DL) models that use convolutions to extract features from data and then apply them for prediction (e.g., Convolutional Neural Networks (CNNs)). Finally, our findings provide a methodological pathway (based on Fourier transforms) for future understanding of wave-driven shoreline change, which can be used to interpret the coherence between the frequency spectrum of the processes of waves and shoreline change and construct more computationally efficient and effective shoreline-modeling approaches.