THE BROAD APPLICATION OF A DEPTH INVERSION ALGORITHM BASED ON THE DYNAMIC MODE DECOMPOSITION

INTRODUCTION Mapping coastal bathymetries from remote sensing is an attractive alternative to in situ measurements due to the large spatial coverage and relatively low costs. The idea to derive bathymetries from video of a wave field stems from the 1980’s and has since lead to the development of various depth inversion algorithms (DIAs). Originally developed for stationary beach cameras and/or radar, these algorithms were built for accuracy on specific locations and high computational speeds and broad applications were not a primary concern. The technological boost of drones and satellites now offers new and more flexible platforms for video-based DIAs; it thereby signals a desire to generate depth estimates on-the-fly, requiring more flexibility and high computational speeds of the DIAs. For this purpose, a novel algorithm was developed, which is fast enough to be used for on-the-fly depth and surface current estimation at a broad range of application areas.


INTRODUCTION
Mapping coastal bathymetries from remote sensing is an attractive alternative to in situ measurements due to the large spatial coverage and relatively low costs. The idea to derive bathymetries from video of a wave field stems from the 1980's and has since lead to the development of various depth inversion algorithms (DIAs). Originally developed for stationary beach cameras and/or radar, these algorithms were built for accuracy on specific locations and high computational speeds and broad applications were not a primary concern. The technological boost of drones and satellites now offers new and more flexible platforms for video-based DIAs; it thereby signals a desire to generate depth estimates on-the-fly, requiring more flexibility and high computational speeds of the DIAs. For this purpose, a novel algorithm was developed, which is fast enough to be used for on-the-fly depth and surface current estimation at a broad range of application areas.

METHOD
To reach high computational speeds we make use of a dimensionality reduction technique called a Dynamic Mode Decomposition (DMD) (e.g. Schmid, 2010). It describes a wave field in terms of dynamic modes, which in this case resemble the significant wave components. Their wave periods are combined with wavenumber estimates from localized two-dimensional FFTs (2D-FFTs), which can then be used to estimate water depths, but also vector fields of wave propagation and surface currents (Gawehn et al. 2019). For quick convergence of the depth and current estimates, we successively analyze small time-blocks of video frames and use an on-line Kalman filter for quality control during each iteration.
In essence, the new method is a "lean and mean" approach to computationally more expensive 3D-FFT based DIAs. To study the robustness and the flexibility of the DMD-based DIA, we performed experiments to video footage from various sites around the world, amongst which the Netherlands, Australia and the US. Both fixed stations and drones are used. Moreover, we use a modern variant of the DMD (Askham & Kutz, 2018), which is superior in both accuracy and flexibility.

RESULTS
Experiments on 2 Hz video of the field-site of Duck, (North Carolina, USA) implied that time-blocks of 16 s of video were enough to produce dynamic modes that are stable between iterations (Gawehn et al. 2019), forming a solid basis for depth inversion (Figure 1). However, it was unclear whether these settings were specific to the case or generally applicable. Analyses of 2 Hz video footage collected with drones at the field-sites of Scheveningen (Netherlands) and Narrabean (Australia) suggest that a universal setting for the DMD can be found in using time-blocks of 32s. After 1 min of video, root mean square errors (RMSE) of the corresponding depth estimates are 0.8 m (Scheveningen) (Figure 1), 2.6 m (Narrabean), 0.5 m (Duck). The larger errors at Narrabean are suspected to be caused by inaccuracies in the orthorectification of the video. Nevertheless, after 10-15 Kalman iterations the errors reduce to 0.6 m(Scheveningen), 1.3 m (Narrabean) and 0.4 m (Duck).

CONCLUSION
Depth inversion from video using dynamic modes has proven to be applicable to derive coastal morphology at different locations using images from drones and fixed stations. The DMD significantly reduces data complexity, leading to fast and accurate computation of depths and surface currents. The analysis of video from three different field-sites in the Netherlands, Australia and US, suggests that 32 s of video are sufficient to correctly capture the significant wave components in the wave-field. It means that DMD-based depth inversion is suited for fast coastal reconnaissance.