The influence of bottom morphology on reflectance: Theory and two-dimensional geometry model

The reflectance of the bottom is of importance when interpreting optical data in shallow water. Closure studies of radiative transfer, interpretation of laser line scanner data, lidar, and remote sensing in shallow waters require understanding of the bottom reflectance. In the Coastal Benthic Optical Properties experiment (CoBOP), extensive measurements of the material reflectance (reflectance very close to the bottom) were made. Far field reflectance will be needed in carrying out closure of the radiative transfer model and observed radiometric and inherent optical properties. The far field reflectance is the bottom reflectance that includes the effect of bottom morphology (such as sand ripples) as well as the material reflectance. We present here a first-order analytical model to derive the relationship between the material and far field reflectances. We show that the effective reflectance of the bottom is proportional to the average cosine of the bottom slope. Using a simple two-dimensional geometry without scattering and absorption, we show that errors in ignoring the bottom morphology can lead to overestimations of the far field reflectance on the order of 30%. Shallow water optical signals are influenced by the bottom reflectivity. Closure of radiative transfer calculations, determination of the contrast of objects with the bottom such as measured by laser line scanners, inversion of remotely sensed radiance for bathymetry, and diver visibility can all be improved with proper knowledge of the bottom reflectance. Many bottoms are nearly Lambertian surfaces, surfaces for which the detected radiance is independent of the viewing angle (Mobley 1994). The radiance reflected from a bottom is not independent of the irradiance impinging on the bottom, however. Therefore, a bottom with topography has a reflectance that is different from a flat, horizontal bottom. Bottom reflectances are usually measured on scales of centimeters (Voss et al. 2000). We call this the material reflectance, although it includes small-scale morphology such as individual grain size. Larger scale morphology, e.g., Wheatcroft (1994), is not included in direct measurements of the bottom reflectance. Radiance and irradiance detectors at larger distances from the bottom will thus see the effect of bottom morphology. Hapke (1993) has analyzed the reflectance of randomly rough surfaces. He assumed that the distribution function of the facet orientations is independent of azimuth angle and that the two-dimensional azimuth-independent distribution function can be described as a Gaussian distribution multiplied by the sine of the zenith angle of the facets. For vertical illumination and detection, Hapke found that the reflectance relative to a flat bottom was the average cosine of the bottom facets. For different illumination and detection angles, the results are far more complicated. The assumptions made by Hapke do not apply to the regular bottom features we discuss in this paper. Hapke’s analysis should be considered, how-