A Synthesis and Comparison of Approaches for Quantifying Coral Reef Structure

The complex physical structures of coral reefs provide substrate for benthic organisms, surface area for material fluxes, and have been used as a predictor of reef-fish biomass and biodiversity. Coral reef topography has a first order effect on reef hydrodynamics by imposing drag forces and increasing momentum and scalar dispersion. Despite its importance, quantifying reef topography remains a challenge, as it is patchy and discontinuous while also varying over orders of magnitude in spatial scale. Previous studies have quantified reef structure using a range of 1D and 2D metrics that estimate vertical roughness, which is the departure from a flat geometric profile or surface. However, there is no general mathematical or conceptual framework by which to apply or compare these roughness metrics. While the specific calculations of different metrics vary, we propose that they can be classified into four categories based on: 1) vertical relief relative to a reference height; 2) gradients in vertical relief; 3) surface contour distance; or 4) variations in roughness with scale. We apply metrics from these four classes to idealized reef topography as well as natural reef topography data from Moorea, French Polynesia. Through the use of idealized profiles, we demonstrate the potential for reefs with different morphologies to possess the same value for some scale-dependent metrics (i.e. classes 1-3). Due to the superposition of variable-scale roughness elements in reef topography, we find that multi-scale metrics (i.e. class 4) can better characterize structural complexity by capturing surface roughness across a range of spatial scales. In particular, we provide evidence of the ability of 1D continuous wavelet transforms to detect changes in dominant roughness scales on idealized topography as well as within real reef systems.