Compressive sensing based reconstruction of the sea free surface elevation on a vertical wall

Measuring the free surface displacement on a vertical wall of a marine structure is not a trivial problem. In this context, the efficacy of ultrasonic probes is affected by the interaction between the signal emitted by the sensor and the vertical wall, while image based techniques are computationally demanding especially if long time series are utilized. Considering these difficulties, this paper proposes a novel approach for measuring the sea surface elevation on vertical breakwaters. The proposed methodology involves the use of pressure measurements and a reconstruction algorithm based on a compressive sensing (CS) technique in conjunction with a generalized harmonic wavelet (GHW) basis. In particular, a “constrained” CS optimization approach is proposed by utilizing the known values of the free surface data to reconstruct all other missing data, while adhering at the same time to prescribed upper and lower bounds at all time instants. The reliability of the methodology is assessed against field data pertaining to a vertical wall equipped with pressure transducers recorded at the Natural Ocean Engineering Laboratory of Reggio Calabria. It is shown that a direct application of an unconstrained GHW based CS optimization approach yields physically inconsistent minima and maxima values; and thus, it is inadequate for reconstructing reliably the free surface. These drawbacks are removed by the constrained GHW based CS. Further, the examination of the reconstructed sea surface profiles in the vicinity of extremely high wave crests or wave troughs shows that they are in agreement with pertinent theoretical data obtained by employing the nonlinear Quasi-Determinism theory.