Using the Lomb-Scargle method for wave statistics from gappy time series

Sandwich Town Neck Beach in Sandwich, MA, has experienced substantial erosion and has been the subject of efforts by the town and private landowners to limit the sand loss. Erosion has been particularly dramatic in the past five years with the loss of dwellings. Sandwich's nourishment efforts presented a unique opportunity for scientists at the U.S. Geological Survey Woods Hole Coastal and Marine Science Center to monitor beach morphology and to test new technologies and techniques such as geo-referenced drone imaging. Two bottom lander deployments were performed in Cape Cod Bay at a location that was key to model the fate of waves at Sandwich Town Neck Beach and to support the study of beach morphological evolution. The study period was after the town nourished the beach and during a time when several intense winter storms reshaped the beach and removed much of the nourished sand. A TRDI Workhorse Sentinel V ADCP was used for both deployments. For wave bursts, the instruments collected 2048 samples at 2 Hz every hour. The first deployment during the winter of 2016 returned good quality data. The second deployment during the following winter had gaps throughout the time series from a wiring problem in the external battery pack. The timing of the gaps was random, the duration approximately 100 s. While most of the bursts started at the top of each hour, many had 1–3 gaps within. Time series data with random gaps are problematic for computing spectral density, and thus, wave statistics. This kind of situation is familiar in other scientific disciplines such as astrophysics [1], where techniques exist to find stationary signals in sparse data. One of these methods is the Lomb-Scargle technique for computing periodograms. The most useful feature of the Lomb-Scargle (LS) method is that it allows the spectral analysis of incomplete records, without having to manipulate the record to extrapolate from or replace missing data. We compared the effectiveness of LS against common methods of averaging Fourier transforms such as a simple un-windowed Fast Fourier transform (FFT), Welch's method, and TRDI's Wavesmon software; methods that are commonly used in oceanography for non-gappy data. Synthetic data series that have been artificially modified to introduce gaps were used to evaluate the performance of each method. The LS approach was able to recover spectral density even with several 100-s gaps present. The method was applied here to the gappy and non-gappy data from both Sandwich deployments, and wave statistics were obtained and compared to the wave-buoy data. LS was used to process data that contains gaps that was rejected by Wavesmon, which was approximately 39% of the dataset. Significant wave height and peak period from LS compared well with buoy data. Mean period computed on gappy data using LS produced values biased low, compared with other methods when gaps were filled with the mean value. The LS technique has potential to uncover low-frequency signals such as infragravity waves from gappy records where the non-gappy segments are not long enough to resolve them. It has potential to unlock new information from older data sets.