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Freak/rogue waves are considered to be the causes of marine accidents and their generation mechanism is closely related to the formation of wave groups. However, observations that capture the spatiotemporal evolution of coherent wave groups in directional windsea are rather limited. The paper presents a new technique known as the surface wave reconstruction by ensemble adjoint-free data assimilation (SWEAD) method that enables reconstruction of a spatiotemporal wave field covering a large area from wave records limited in observational density and spatial extent. We reconstructed spatiotemporal profiles of nonlinear surface gravity waves from virtual observational data using the adjoint-free four-dimensional variational data assimilation (a4DVar) scheme. The higher-order spectral method (HOSM) is used as a forward deep-water nonlinear wave model in a realistic sea state. The a4DVar scheme uses perturbed ensemble simulations to calculate the cost function gradient and Hessian; thus, construction of an adjoint model is not needed. A few extensions of the a4DVar scheme are proposed in this study. For efficient wave reconstruction, perturbed ensemble simulation results are reused by increasing the searching direction dimension at each iteration while assuring conformity to the perturbed model’s linearity. For regularization, Fourier coefficient magnitudes are constrained by a known power spectrum from the phase-averaged wave model. Twin experiments were conducted for a unidirectional wave with virtual wave gauge data and a multidirectional wave with virtual stereo camera imaging data. For both unidirectional and multidirectional cases, nonlinear freak wave–related wave groups were well reproduced, which is impossible using a linear model.

There exists considerable disagreement among the observed values of the drag coefficient CD. To develop a model of CD, the wind stress generally will be calculated from the eddy correlation method. A buoy is suitable to measure the wind stress in many sea surface conditions. However, the motion correction is very difficult because the anemometer measures the wind components, including the motion of the buoy. In this study, as a first approach, the motion of a prototype buoy system with a three-axis sonic anemometer and a six-axis motion sensor installed in the small-size GPS observation buoy was investigated. The wave tank is in the ocean engineering basin of the Institute of Industrial Science, University of Tokyo, Japan. The imposed conditions were wave periods from 1.1 to 2.5 s; wind speeds of 0, 2, and 5 m s−1; and the wave spectrum was either regular or irregular. The motion of the buoy was measured in 120 cases. For all the wave periods and without wind, the wind velocity measured by the sonic anemometer and the velocity of the anemometer motion calculated from the motion sensor data showed good agreement. Also, in the condition with wind speeds of 2 and 5 m s−1, the motion-corrected wind velocity, obtained by deducting the velocity of the anemometer motion from the wind velocity measured by the anemometer, yielded the true wind velocity with better-than-average (4.3%) accuracy. The friction velocity from corrected wind velocity components shows agreement with the friction velocity measured from a fixed sonic anemometer within expected intrinsic error. The buoy system is expected to be able to measure the wind stress in the field. The next stage is to do comprehensive field tests.

The second generation of a new approach to data assimilation where wavelet analysis is used for error estimation is presented here. The first generation is known as EEWADAi. This modified and optimized method uses wavelet analysis to not only estimate numerical error but to also acquire an estimate of the variation at various scales of the model simulation. In the original EEWADAi, wavelet analysis on the finest scale was used to estimate numerical error. In the second-generation version, called SUgOiWADAi, wavelet analysis is used on a variety of scales to not only obtain an estimate of numerical error, finest-scale information, but to also obtain an estimate of model variation, information from coarser scales. This new algorithm is computationally very inexpensive and is very effective.

A new method is presented for estimating numerical errors in simulations as a function of space and time. This knowledge of numerical errors can provide critical information for the effective assimilation of external data. The new method utilizes wavelet analysis for the detection of deviation from low-order polynomial structure in the computational data indicating regions of the domain where relatively large numerical errors will occur. This wavelet-based technique has a very low computational cost, and in practice the cost can be considered negligible compared to the computational cost of the simulation. It is proposed here that this be used in the field of data assimilation for fast and efficient assimilation of external data, and a numerical example illustrating that the new method performs better than the existing method of optimal interpolation is given.