Reflections and the focusing effect from an ideal three-dimensional rough surface

Abstract

An analytical expression for higher-order reflectances from a shallow-water homogeneous ocean bottom modeled as an egg-carton surface is presented. Roughness of this ideal surface is expressed as the amplitude-to-length ratio of its basic sinusoidal function. Any real surface that can be approximated by an egg-carton function will effectively have a comparable roughness metric. Incidence and reflection directions are considered in full azimuthal variation. The detector is located just below the water surface so that only in-water reflections are considered and there are no air-water transmission effects. Furthermore, this setup allows for an understanding of reflections that occur in media with any index of refraction or absorption coefficient. Fixing the detector footprint but adjusting its field-of-view enables the observation of the same bottom surface area as the depth varies while keeping the roughness and the number of waveforms viewed constant.

First-order reflectance decreases as the roughness increases, as was shown in the two-dimensional case. This is true as the roughness varies, regardless of the bottom reference level chosen. Focusing effects are expected from (but are not limited to) second-order reflectance and are due to parts of the bottom whose angles maximize both incoming light and the reflections toward the detector. Along a plane about the vertical axis, the roughness ratio for a fixed-length waveform that returns the highest reflectance can be found. In three dimensions, this phenomenon is complicated by reflections from all hemispherical directions. Shadowing and obscuration behave similarly as in the two-dimensional case although shadowed areas will have an increased potential to reflect light from other directions (than the plane defined by the source incidence and the vertical directions). This is expected to cause higher order reflections to increase as the roughness increases.