In shallow water systems like coral reefs, bottom friction is often a significant part of the overall momentum balance. The frictional effects of the bottom on the flow are in part determined by the structure of the topography, which varies over orders of magnitude in spatial scale. Predicting spatial and temporal patterns of water motion depends on adequately capturing the relevant properties of the topography. However, representing and quantifying the complex, heterogeneous structure of coral reefs using measures of roughness or geometry remains a challenge.Many roughness metrics have been proposed to relate seafloor structure to biological and physical processes. In Chapter 1, we assess the properties captured by one-dimensional roughness metrics previously proposed for the seafloor, as well as metrics developed to characterize other types of rough surfaces. We consider 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). We evaluate these metrics using idealized topography and natural seafloor topography of a reef lagoon system from airborne lidar measurements. The analyses illustrate that common metrics of bathymetric roughness (e.g., rugosity) can have the same value for topographies that are geometrically very different, thus limiting their utility. Application of the wavelet leaders technique to the reef dataset 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. While individual roughness metrics that capture specific topography properties relevant to a given process may be suitable for some studies, for many applications, adequately parameterizing bathymetric roughness will require a set of metrics.For reefs where the roughness layer takes up a large fraction of the water column, parameterizations of bottom friction require a representation of three-dimensional canopy geometry. In Chapter 2, we assess the implications of using obstacle- and surface-based representations to estimate geometric properties of coral colonies needed to parameterize drag. We collected high-resolution topography data using a scanning multi-beam sonar that resolved individual coral colonies within a set of 100 m2 reef patches primarily composed of mounding Porites corals. The topography measurements yielded 1-cm resolution gridded surfaces consisting of a single elevation value for each position in a regular horizontal grid. These surfaces were analyzed by (1) defining discrete obstacles and quantifying their properties (dimensions, shapes), and (2) computing properties of the elevation field (rms elevations, rms slopes, spectra). We then computed the roughness density (i.e., frontal area per unit plan area) using both analysis approaches. The obstacle and surface-based estimates of roughness density did not agree, largely because small-scale topographic variations contributed significantly to total frontal area. These results challenge the common conceptualization of shallow-water canopies as obstacle arrays, which may not capture significant contributions of high-wavenumber roughness to total frontal area. In contrast, the full range of roughness length scales present in natural reefs is captured by the continuous surface representation. Parameterizations of drag could potentially be improved by considering the distribution of frontal area across length scales.Collectively, the results presented in Chapters 1 and 2 show that coral reef topography is both multiscale and multifractal. However, there is a limited understanding of the effects of the structural complexity on water motion around individual and groups of corals. In Chapter 3, we present detailed hydrodynamic measurements from the same shallow reef sites for which we quantified reef geometry (Chapter 2). Using these measurements, we compare spatial and temporal variations in flow patterns across three sites: (1) a high relief site with waves; (2) a low relief site with waves; and (3) a high relief site without waves. Our observations suggest that the flow is likely unidirectional and current dominated over much of the backreef. These measurements also show that flow variations at different frequencies have different spatial patterns. At low frequencies, flow variations follow the spatial pattern of wakes. The lack of coherent structure in wave band variations can be explained by the distribution of orbital excursion length to colony diameter (Keulegan-Carpenter number), which is typically less than 2π, thus wakes do not form behind elements. Variations at high frequencies were up to two times larger in the canopy than upstream. In the future, these observations could be compared to computational models of flow at the sites, which would allow us to better understand mechanisms controlling frequency-dependent spatial patterns, as well as the importance of colony and patch-scale processes for reef and regional scale circulation.