2.2 Beach-face slope estimation algorithm
The objective of the automated algorithm developed here is to estimate the beach-face slope at any site worldwide without the requirement forin situ measurements (e.g., topographic surveys, tide gauges, etc) but instead relying exclusively on remotely-sensed data. This new technique fully leverages the capabilities of satellite remote sensing in the coastal zone, including the use of optical imagery for mapping shoreline changes and altimetry for measuring water level changes.
Recent developments in shoreline mapping now make it possible to extract instantaneous shorelines from publicly available satellite imagery (Bishop-Taylor et al., 2019b; Pardo-Pascual et al., 2018; Vos et al., 2019a). Note that these shorelines are referred to as ‘instantaneous’ because they are mapped on individual satellite images acquired at different and arbitrary stages of the tide. Consequently, time-series of cross-shore change obtained from these instantaneous shorelines also implicitly include the superposition of tidal excursion and sediment transport processes – i.e., beach erosion/accretion. In order to estimate the slope of the beach face, the fluctuations caused by tidal excursions must be isolated from the horizontal changes resulting from erosion and/or accretion of the beach. To do this a frequency-domain analysis is applied to isolate the high-frequency tidal signal in the cross-shore shoreline time-series from the typically lower-frequency morphological changes. The step-by-step methodology to estimate the typical beach-face slope from satellite-derived shorelines and modelled tide levels is illustrated in Figure 2 and described below.
Extract satellite-derived shorelines from Landsat imagery (Figure 2a):
Satellite-derived instantaneous shorelines are extracted usingCoastSat (Vos et al., 2019a), an open-source toolbox that enables users to obtain time-series of cross-shore shoreline position at any sandy coastline worldwide from 30+ years of publicly available satellite imagery (Landsat 5, 7, 8 and Sentinel-2) accessed via Google Earth Engine (Gorelick et al., 2017). A target sampling period of no more than 8 days was maintained by the use of all Landsat images between 1999 and 2019 (i.e., 16-day revisit with at least 2 satellites concurrently in orbit). Sentinel-2 images were excluded as the poor cloud masking algorithm hampered a fully automated shoreline extraction. The cross-shore accuracy of the mapped shorelines varies between 10 and 15 m depending on site characteristics, as was previously reported in Vos et al. (2019b). Time-series of cross-shore shoreline change were obtained by intersecting the mapped shorelines with shore-normal transects at each site. An example of the resulting raw time-series of shoreline change at Cable Beach is shown in Figure 2a.
Tide levels from a global tide model and peak tidal frequency (Figures 2b and 2d):
Once the satellite-derived shorelines have been mapped, the corresponding tide levels at the time of image acquisition are obtained from the FES2014 global tide model (Carrere et al., 2016). This model was chosen as it ranks amongst the best barotropic ocean tide models for coastal regions (Stammer et al., 2014). The next step is to identify in the tide elevation time-series (sub-sampled according to availability of satellite-derived shorelines) the frequency at which the tidal signal is the strongest. This frequency, hereafter referred to as ‘peak tidal frequency’, is determined by computing the Power Spectrum Density (PSD) of the tide level time-series. Importantly, the PSD cannot be computed with a traditional Fourier Transform (e.g., FFT) as the tide level time-series are unevenly sampled due to the presence of clouds within the associated satellite images. However, an alternative algorithm, the Lomb-Scargle transform (VanderPlas, 2018), widely used to analyse astronomical observations, is specifically suited to the calculation of the PSD from irregularly sampled time-series (see comparison with FFT in Supporting Information Figure S2).
The PSD of the tidal signal, depicted in Figure 2d, indicates how much tidal energy is contained at a given frequency, with the peaks revealing the frequency of the tidal harmonic constituents. Since the tide is sub-sampled at 8-day intervals, the higher-frequency semi-diurnal and diurnal components of the tide are completely missed, but some of the lower-frequency components can be resolved (e.g., spring-neap fortnightly cycle, monthly and annual cycles). Figure 2d shows the PSD of the sub-sampled tide time-series, indicating that the highest peak for this example is located at a period of 17.5 days. This energy corresponds to the spring-neap fortnightly cycle, which has a period of 14.76 days, but as the Nyquist limit is 16 days (twice the sampling period), the 14.76 days periodic signal is slightly aliased to 17.5 days. The aliasing of the tidal signal and the effect of the sampling frequency are further discussed in Supporting Information S3.
Tidal correction with a range of beach-face slope values (Figure 2c):
Tidal correction consists of the projection of individual instantaneous shorelines, acquired at different stages of the tide, to a standard reference elevation, for example Mean Sea Level (MSL). A simple tidal correction is applied by translating horizontally the shoreline points along a cross-shore transect using a linear slope:
\({x}_{\text{corrected}}=\ x+\ \frac{z_{\text{tide}}}{\text{tanβ}}\)(1)
where \({x}_{\text{corrected}}\) is the tidally-corrected cross-shore position, \(x\) is the instantaneous cross-shore position,\(z_{\text{tide}}\) is the corresponding tide level and tanβis the beach-face slope. Using Eq. (1) the raw time-series of cross-shore shoreline positions are tidally corrected using a range of potential slope values from 0.01 to 0.2, the latter considered “a universally relevant upper limit of sandy beach-face slopes” (Bujan et al., 2019).
Find the slope that minimises the tidal component of the shoreline time-series (Figures 2e and 2f):
As the final step in this automated process, the Lomb-Scargle transform is employed to compute the PSD of each of the tidally-corrected time-series. Figure 2e shows the PSD curves resulting from the tidally-corrected time-series depicted in Figure 2c. An inset on the peak tidal frequency band (17.5 days) demonstrates how the magnitude of this peak is modulated by the slope value used for tidal correction. In this example (Cable Beach), the 17.5 days peak is entirely suppressed when using a slope of 0.025 (as indicated by the blue dashed curve, Figure 2e).
Finally, the typical beach-face slope can be estimated by finding the slope value that, when used to tidally-correct the shoreline time-series, minimises the amount of tidal energy. Figure 2f shows the ‘tidal energy’ (i.e., the integral of PSD inside the peak tidal frequency band) as a function of the slope value used for tidal correction, indicating a distinct minimum for a slope of 0.025. For this site, this leads to the conclusion that tanβ = 0.025 is the temporal-average beach-face slope at this macro-tidal, fine sand grain size location.