Figure 2 . Step-by-step description of the algorithm developed to estimate typical beach-face slope from satellite-derived shorelines. The time-series of shoreline change and tide levels shown here are from Cable Beach (Western Australia). a) Raw (i.e., non-tidally corrected) time-series of shoreline change along the southern shore-normal transect at Cable Beach (see Figure S1 in Supporting Information). b) Modelled tide levels associated with the satellite-derived shorelines (black line). The grey shaded area indicates the overall tidal fluctuations, noting that the vertical bias here is due to the sun-synchronous orbit of Landsat satellites (refer to Bishop-Taylor et al., 2019a). c) Ensemble of tidally-corrected time-series of shoreline change using slope values ranging from 0.01 (red) to 0.2 (green). d) Power Spectrum Density (PSD) of the sub-sampled tide level time-series. The peak tidal frequency band (grey shaded area) is centred at a frequency of 17.5 days and stretches 10-8 Hz each side. e) PSD of the ensemble of tidally-corrected shoreline time-series. The inset zooms on the peak tidal frequency band and shows how the magnitude of the peak at this frequency is entirely suppressed when using a slope of 0.025 (blue dashed line). f) Energy in the peak tidal frequency band for the range of slopes tested. The slope that minimises the energy inside the peak tidal frequency band is selected as the best estimate of the beach-face slope.
3 Results
The beach-face slope estimation algorithm described in the previous section was applied to the eight test sites along the transects depicted in Figure S1 (39 transects in total) and compared to in situmeasurements. The measured beach-face slope at each transect was computed as the average of all the available surveys (calculated from MSL to MHWS), except at Cable Beach where only two known surveys were conducted (Masselink & Pattiaratchi, 2000; Wright et al., 1982). Figure 3a shows a 1:1 plot comparing the satellite-derived beach-face slopes (tanβsatellite) to the in situ beach-face slopes (tanβin situ). For the in situ data (x-axis), a horizontal bar indicates one standard deviation around the average to highlight the degree of temporal variability in beach-face slope observed at each location.
Overall, there is a strong correlation between satellite-derived estimates and in situ averages, with a coefficient of determination (R2) of 0.93 and no systematic under- or over-estimation observed. While the slope estimation algorithm performed very well along the gentle sloping profiles (tanβin situ< 0.05) of Cable Beach, Ensenada and Torrey Pines, as well as along the steeper (tanβin situ > 0.12) profiles at Slapton Sands and Tairua, relatively more scatter is observed at the more intermediate sites (Duck, Moruya/Pedro and Narrabeen). These intermediate sites (0.5 < tanβin situ < 0.12) are also characterised by a larger temporal variability in beach slope as indicated by the width of the standard deviation bars. In terms of accuracy, the standard deviation of the errors was 0.01 with 90% of the errors falling below 0.015.
While this new capability may become highly valuable for a range of applications as it does not rely on any field measurements, there are limitations. Firstly, this method relies on the existence of a measurable tidal excursion signal in the satellite-derived shoreline time-series. Since the horizontal accuracy of the satellite-derived shorelines is ~10m, in order to capture the tidal excursion signal, the amplitude of this fluctuation needs to be significantly larger than 10m – e.g., a tidal excursion of 20 m has a signal-to-noise ratio of 2. In turn, the amplitude of the tidal excursion depends on the tidal range and on the angle of the intertidal zone.
To identify the range of tidal regimes and beach-face slopes over which this method is applicable, additional synthetic time-series of shoreline change were generated for a planar beach with specified slope and tidal range (the details on how these time-series were generated are included in Supporting Information S4). Figure 3b summarises the accuracy of the estimated beach-face slopes - i.e., Normalised Mean Absolute Error (MNAE) based on 100 synthetic time-series - as a function of tidal range (TR) and beach-face slope (tanβ). As anticipated, the accuracy declines with decreasing tidal range and increasing beach-face slope, as a decrease in the ratio TR/tanβ is equivalent to reducing the amplitude of the horizontal tidal excursions. Based on these synthetic data, the errors can be as much as 30% for TR/tanβ ratios smaller than 10 (i.e., TR < 1 m and tanβ > 0.1). Consequently, in order to obtain accurate slope estimates, it is recommended that the technique be applied at any site where TR/tanβ is larger than 10. This is further emphasised by the fact that every one of the eight test sites, for which the tidal ranges and average in situ beach-face slopes are also included in Figure 3b, are situated at locations where TR/tanβ > 10 (exact ratios reported in Table 1).