Increasing the spatial resolution of wave fields when the amount of available instruments is limited

A relatively high spatial resolution is often desirable when capturing the spatial variability of evolving wave fields. Some of these situations where a high spatial resolution is needed may be, for example, studing the shape evolution of breaking waves, or studing the directional distribution of the energy of wind-wave spectra, where computing the frequency-wavenumber spectra is critically needed. In both cases, an ideal space-time representation of the wave field is the one whose spatial and time resolutions are high enough (∆x and ∆t being very small), or to some extend, comparable in relative terms (∆x/λ ≈ ∆t/T), i.e., ∆x and ∆t representing a tiny fraction of the characteristic wavelength (λ) and period (T), respectively. In this case, the wave field in the space-time domain looks like a continuous 2D function. However, reaching such high spatial resolutions is not very common in experimental or field works. In some cases, such high resolution is not needed. In some other cases, such high resolution is not possible due to unavoidable experimental, technical and cost constraints, and that results in a limited number of available instruments. To overcome this limitation, we present a relatively simple procedure called S-interp, which is freely available at https://github.com/EMPadilla/Sinterp.
 
S-interp is to interpolate wave fields at spatial locations where no measurements are available. S-interp uses a Modified Akima cubic Hermite interpolation along points in the wave field that are in phase. The main hypothesis of S-interp is that the wave field follows a linear-like evolution along points being at the same phase. Therefore, along these points, differences between the interpolated and the actual wave fields are minimal. Some factors for these differences to rise are: (i) The spatial distribution of the instruments, (ii) the nonlinear effects that modify the wave geometry increasing its asymmetry and skewness and (iii) the interpolation method used. 

We assess the performance of S-interp by reconstructing missing areas of experimental non-breaking wave conditions, gathered in SIREN-NB data set. These are 33 non-breaking focused wave events designed using a NewWave-type spectra for peak periods ranging from 1.0 s to 1.7 s with a peak enhancement factor set to 2. The wave conditions are recorded by video cameras and the wave fields are measured by video-image detection. The experiments were conducted in the Wind-Wave-Current flume at the Hydrodynamics Lab - Imperial College London. Preliminary results suggest that S-interp seems to be more sensitive to the spacing between the instruments than to the nonlinear effects of the wave fields.