Shoreline seasonality measurements from Landsat satellite imagery for California beaches, 2000-2022

The first processing step involved the Seasonal Trend decomposition with LOESS (STL) techniques of Cleveland and others (1990) to measure the magnitude and timing of shoreline seasonal patterns. For each CoastSat transect, the 2000-2022 raw shorelines were used to compute monthly median shoreline positions for each year of the record. Standard STL techniques were then used to remove the time-dependent trend in the resulting data using the single-pass LOESS function with the standard 2nd order polynomial fit and tri-cube weight function (Cleveland and others 1990). The detrended data were then used to compute the median monthly shoreline positions of the entire record, from which the range of these median values was assigned to be the ‘shoreline seasonal excursion distance’ and the months of the minimum and maximum shoreline positions were assigned to be ‘tmin’ and ‘tmax.’ The twelve median monthly shoreline position values were used to conduct k-mean clustering as noted below. The complete set of shoreline positions for the months of tmin and tmax were compared using a student’s t-test p-values, the results of which were used to assess whether the range of values from these two months were significantly different. These statistical results were used to assess whether the STL seasonality results were statistically significant as noted below.

The second processing step involved spectral analyses of the 22-year monthly median shoreline position values for each transect. First, we computed the periodograms of the monthly shoreline positions over a range of frequencies associated with periodicity of 61 days to 42 years to assess whether annual frequency cycles were statistically significant using the Fischer’s g-statistic. We computed p-values from each g-statistic using the first term of the Fischer’s series, and these p-values were used to assess the significance of the seasonality patterns as discussed below. Second, spectra peaks and a spectra-derived seasonal shoreline excursion distances were computed using a fast Fourier transform (FFT) of the monthly shoreline position time-series for the purpose of having an independent excursion distance measurement to compare with the STL results. We did not use zero padding because there were enough samples to adequately characterize the annual frequency results, and windowing techniques were unnecessary because the time series length (exactly 22-yr) was chosen to be an integer multiple of the primary, i.e., the annual, frequency. The FFT-derived seasonal shoreline excursion distances were computed as twice the single-sided FFT amplitude for the annual frequency, where the amplitude was calculated to be the absolute value of the complex FFT magnitude for the annual frequency divided by the number of frequencies assessed in the FFT, which was 132 in our analyses.

The third processing step consisted of combining the STL and spectral analyses to evaluate whether the seasonal patterns were statistically significant for each transect. The p-values from both the STL t-test and the periodgrams (see above) were used to evaluate statistical significance at p<0.05. If either of these p-values were greater than this threshold, the transect was determined to be ‘nonseasonal’ and the STL-derived values of seasonal excursion distance were changed to zero meters, and the tmin and tmax were changed into ‘NaN’ values (not a number) to denote that a significant seasonal cycle could not be measured.

The fourth processing step was an unsupervised classification of the STL results using k-means clustering. For this analysis, the twelve-monthly values of median shoreline position for each transect were used. Transects deemed ‘nonseasonal’ by the statistical tests denoted above were given shoreline position values of zero for each of the twelve months. Prior to clustering, the monthly median shoreline positions were standardized by subtracting the mean and dividing by the standard deviation for each transect. The number of k-means clusters was defined using the gap statistic technique that characterizes the change in within-cluster dispersion with respect to changes that would occur from a reference null distribution, which was assigned to be a random distribution over the range of the source values. This analysis showed peaks in the gaps at 5 and 11 clusters over a range of zero to 50 clusters, and the larger of these values (11) was used for our analyses to capture a broader range of seasonal timing groups. Cluster distances were computed as squared Euclidean distances, and the final cluster groups were organized by the month of the narrowest beach conditions, starting in the early winter (cluster 1) and ending in the summer (cluster 10), with the final cluster given to the nonseasonal group (cluster 11).

The fifth processing step included summary and geographical information about each transect. First, the mean position and orientation of the shoreline were computed using the initial and final transect points defined and tabulated in Vos (2023). The ‘back-beach position’ was derived from a non-erodible shoreline for California mapped and used by Vitousek and others (2023) that was modified in places to reflect the differences in perspectives of our goals of defining the recent active beach and Vitousek and others’ goals of defining the inland limit of future coastal erosion under sea-level rise scenarios. These changes consisted primarily of altering the back-beach lines of broad coastal plains and dune settings from the full inland extent of the low elevation landscape to the initial densely vegetated areas. The positions of this back beach and the active beach widths were found by intersecting the CoastSat transects with the back beach location.

The citation for the accompanying journal article was added to the Cross Reference section of this metadata. No data were changed (mau@usgs.gov)