Various data sources and methodologies have been applied in the study and are described in detail in the following sections.
2.2. Step by Step Methodology
The analytical steps involved in estimating extreme sea levels at Sydney for design and planning purposes over differing planning horizons are detailed in the following sections. All analysis and graphical outputs have been developed by the author from customised scripting code within the framework of the R Project for Statistical Computing [
30].
Step 1: Determination of MSL trend. This is a critical initial step as the fitting of a GPD function to estimate extreme values is based on the statistical principles of stationarity. The key task with this step is to isolate the comparatively small, non-stationary, non-linear MSL signal from the significant and substantial dynamic inter-decadal (and other) influences and noise. This is achieved through the application of Singular Spectrum Analysis (SSA) techniques adapted specifically for MSL research [
29,
37] which have been further refined and applied to large regional studies of long tide gauge records [
4,
38]. The process uses annual average MSL time series data from the PSMSL, where the data time series has been adjusted to reflect the midpoint of each year.
The SSA procedure requires a complete time series, therefore the three missing annual average values at Fort Denison (1930, 1941 and 2000) have been filled using an iterative SSA procedure [
39] which has an (assumed) advantage in preserving the principal spectral structures of the complete portions of the original data set in filling the gaps.
The complete time series is then decomposed using one-dimensional SSA to isolate components of slowly varying trend (i.e., MSL) from oscillatory components with variable amplitude, and noise. A technique known as frequency thresholding [
40] is then used to aggregate components from the SSA decomposition that have “trend-like” characteristics. In general, the trend of MSL associated with external climate forcing can be considered to comprise components in which more than 50% of the relative spectral energy resides in the low-frequency band between 0 and 0.02 cycles per year, which accord with the findings of Mann et al. [
41].
The isolated MSL signal is then fitted with a cubic smoothing spline model to permit prediction of MSL at each hourly time step over the time span of the data (1915.5 to 2020.5). More specific detail on the parameterisation of SSA and spline fitting adopted can be found in Watson (2021) [
4].
Step 2: Detrending of hourly tide gauge measurements. The available hourly tide gauge measurements span the timeframe from 1914 to present, split across two overlapping timeframes and sources (NOC and UHSLC). The overlapping portion of the data (January 1965 to December 1994) have firstly been compared and confirmed to be consistent, having used the same datum (tide gauge zero). Thus, the more recent portion of available data (UHSLC) have been added to the NOC data to provide a continuous time series of hourly measurements from 10 June 1914 (0400 h) to 30 June 2021 (2300 h). These measurements have been converted to millimetres and adjusted to the same datum as the MSL data for Fort Denison from the PSMSL (used in Step 1).
The actual detrending of the hourly tide gauge measurements is a very straightforward step involving the subtraction of the MSL value (Step 1) at every common time step. Only time steps containing hourly tide gauge measurements are retained for EVA.
Step 3: Declustering input data. Declustering procedures (i.e., making use only of the single highest exceedance within a cluster) are routinely employed in applications of the POT approach to avoid the effects of dependence [
42]. Along with stationarity, statistical independence is the other key requirements for fitting a GPD to data for estimating extremes. The clustering influence of hourly tide gauge measurements were examined by Arns et al. (2013) [
27], concluding no discernible influence from clustering on return interval water levels when the time span between specified events was set at >24 h. It has been demonstrated that with either the BM or POT approach there is strong dependency on the clustering time, leading to significant over-estimation of the extreme return water level if the clustering time is short (<24 h) [
27].
The ‘decluster’ function in the ‘extRemes’ package has been used to decluster the data, by setting 25 h as the minimum time span between successive peaks above the notional threshold [
20,
32]. Hourly measurements of extremes separated by fewer than 25 non-extremes are therefore considered to belong to the same cluster (or event). The sensitivity of the setting of the time span between events in declustering the data is considered in further detail in the Discussion section, confirming the utility of the 25 h selected.
Step 4: Selection of threshold level for EVA. This is a central consideration for the POT approach, occupying much of the literature with accepted conventions ranging from expert model fitting judgement to more analytically derived metrics. The approach advocated by Arns et al. (2013) [
27], which is based on significant optimised testing, recommends the threshold be set at the 99.7th percentile of hourly measurements. Other approaches rely on fitting GPD models over a range of thresholds using various optimisation techniques for parameter estimation (e.g., Maximum Likelihood Estimation (MLE), Generalised Maximum Likelihood Estimation (GMLE), linear combinations of order statistics (L-Moments) and Bayesian).
The optimum-fitted GPD model is considered the one in which all points from a plot of empirical vs. modeled outcome sit on a 45-degree line [
42]. Other approaches suggest adopting the lowest threshold for which the predicted extreme outputs remain stable and consistent [
20], noting a threshold set to low will encompass significant irrelevant data whilst a threshold that is too high produces unstable results from too few data points to train a GPD model.
The approach adopted in this study uses elements of all the above. Firstly, a central point for threshold testing is established around the 99.7th percentile of hourly measurements. A threshold range of ±200 mm of the 99.7th percentile is tested using 5 mm increments across the range.
At each threshold increment, the detrended hourly measurements are then declustered (Step 3) and fitted with a GPD based on separate parameter optimisation using MLE, GMLE, L-Moments, and Bayesian approaches using the ‘extRemes’ extension package in R [
30,
31,
32]. For each of the four parameter optimisation approaches at each threshold value, the root mean square error (RMSE) is separately calculated for the fitted model and the predicted extreme values are determined for 1-, 10-, 100- and 1000-year return periods. With all the results plotted, the optimum threshold and fitted model can be readily identified from the stability and consistency in predicted extreme values in combination with the lowest RMSE.
Step 5: Estimated prediction of extreme heights above MSL. Predicted extreme water level heights above MSL are readily determined from the optimum threshold and fitted model (Step 4) for a range of return periods (i.e., 1, 2, 5, 10, 20, 50, 100, 200, 500 and 1000 years).
Step 6: Integration of extreme predictions with sea level rise projections. Global averaged sea level rise projection data from Figure SPM8 [
43] have been normalised to a start date of 2020 for each of the modelled SSP scenarios (SSP1-1.9, SSP1-2.6, SSP2-4.5, SSP3-7.0 and SSP5-8.5).
These normalised projections of sea level rise for each SSP scenario are simply added to the predicted extreme water level heights above MSL (Step 5) to derive relevant extreme predictions for future planning horizons in 2050, 2070, and 2100.
Step 7: Predicted extreme SWLs at 2050, 2070, and 2100 corrected to Australian Height Datum (AHD). The last estimated MSL data point from Step 1 (which occurred in 2020) is simply converted from the PSMSL datum to AHD and then added to the values derived in Step 6.