Collapsing Complexity: Quantifying Multiscale Properties of Reef Topography

Abstract

Seafloor topography affects a wide range of physical and biological processes; therefore, collapsing the three‐dimensional structure of the bottom to roughness metrics is a common challenge in studies of marine systems. Here we assessed the properties captured by metrics previously proposed for the seafloor, as well as metrics developed to characterize other types of rough surfaces. We considered three classes of metrics: properties of the bottom elevation distribution (e.g., standard deviation), length scale ratios (e.g., rugosity), and metrics that describe how topography varies with spatial scale (e.g., Hölder exponents). The metrics were assessed using idealized topography and natural seafloor topography data from airborne lidar measurements of a coral reef. We illustrate that common roughness metrics (e.g., rugosity) can have the same value for topographies that are geometrically very different, limiting their utility. Application of the wavelet leaders technique to the reef data set demonstrates that the topography has a power law scaling behavior, but it is multifractal so a distribution of Hölder exponents is needed to describe its scaling behavior. Using principal component analysis, we identify three dominant modes of topographic variability, or ways metrics covary, among and within reef zones. Collectively, the results presented here show that coral reef topography is both multiscale and multifractal. While individual metrics that capture specific topography properties relevant to a given process may be suitable for some studies, many applications will require a set of metrics that includes statistics that capture how topography varies with spatial scale.

On the small‐scale fractal geometrical structure of a living coral reef barrier

“…The present methodology gave access to decimetric length scales, therefore extending the recent analysis of the Moorea reef presented by Duvall et al, (Duvall et al, 2019). Our results confirm and extend, at small scale, the Moorea reef observations (Duvall et al, 2019) that topographical structure shows strong spatial variations which cannot be captured by some of the common metrics.…”

“…As most studies are based on satellite or airborne measurements, current knowledge of reef geometry usually involves metric to kilometric scales. The highest resolution reported to date, to the best of the authors' knowledge, was obtained by an airborne lidar topographic survey of the Moorea Reef Barrier, French Polynesia (Duvall et al, 2019). These observations demonstrated the multiscale and multifractal nature of the coral reef crestography.…”

“…This resolution reached by the present dataset enables topographical analysis down to decimetric scales, i.e. at least ten times more accurate than lidar-based measurements (Duvall et al, 2019).…”

“…A lowpass frequency cutoff was applied at 0.1m for each of the presented spectra. A complementary wavelet analysis was applied, based on Haar wavelets over scales logarithmically spaced from 0.1 to 10 3 m. The present spectral analysis is focused on a monofractal approach in order to ensure robust comparisons with future research works (Duvall et al, 2019).…”

“…A series of recent studies have established the statistical fractal self-similar structure of several coral reef environments from metric to kilometric scales; see e.g. (Purkis and Kohler, 2008;Schlager and Purkis, 2013;Purkis et al, 2015;Purkis et al, 2016;Duvall et al, 2019). Synthetic terrain models based on fractional Brownian motion can accurately reproduce the statistical properties of complex natural reef seabeds (Purkis and Kohler, 2008).…”