Measuring small-scale water surface waves: nonlinear interpolation and integration techniques for slope-image data

In observing oceans using synthetic aperture radar (SAR), researchers have shown that small-scale water surface waves can significantly affect the radar backscatter from an ocean surface. Furthermore, these small scale water surface waves are known to have a significant role in affecting the ocean's drag coefficients and gas-exchange rates. Unfortunately, the hydrodynamics of these small scale water waves are neither well understood nor easily measured. Key innovations that have allowed measurement of small-scale water waves currently include Jahne's refraction-based technique. Several instruments have now been built that have been based on Jahne's technique (including instruments at the Scripps Institute of Oceanography, the University of Heidelberg, and The University of Michigan). Unlike other techniques (like stereo imaging) that attempt to measure the topography of an ocean surface directly, Jahne's technique measures topography indirectly by imaging surface derivatives (i.e., slope-image data). In many cases, deriving topography from slope-image data is a linear problem and is amenable to numerical integration. However, for cases involving breaking waves and turbulence, the derivation becomes nonlinear because of discontinuities caused either by air entrainment or by instrument physics that limit the dynamic range of wave-slope angles that can be measured. Both types of discontinuities can significantly affect analyses that depend on either slope or topographic data. Consequently, the authors' investigation has focused on surface reconstruction methods using nonlinear interpolation and numerical integration techniques. This paper briefly describes their method and shows a few examples of reconstructed surfaces.