Tide gauges (TG from now on) are used in this study to compare the results obtained from other data sources. High-frequency (hourly) data, similar to those distributed by the Global Extreme Sea Level Analysis (GESLA-3, https://www.gesla.org, last access: February 2025, ), are used. Since the GESLA-3 data available online end in 2022, a specific update was requested, and data for 2023 were obtained through personal communication with Ivan Haigh (University of Southampton, UK) and Ben Hague (Bureau of Meteorology, Australia).

Stations in Eastern Australia within the rectangle visible in Fig.  were selected. Stations were retained if they ensured a minimum spacing of 50 km between them and were at least 25 km away from a river mouth. Based on data availability and these criteria, the following stations were used: Bermagui, Crookhaven_Heads, Port_Macquarie, Coffs_Harbour, and Brunswick_Heads.

To make the sea level retrievals from TGs comparable to other sources of sea level data, the following steps were performed. Firstly, the TG data were detided using a 40 h LOESS filter. Previous literature has shown this to be the most effective method for reducing tidal variance for periods shorter than two days . Subsequently, the atmospheric contribution to sea level variability was removed by applying the same correction used for altimetry data, the Dynamic Atmosphere Correction from . Finally, the hourly and sub-hourly data collected by each TG were averaged to a daily rate to match the data rate of the altimeter product.

The approach of is followed, which studied CTWs in the same region using data from the Bluelink ReANalysis (BRAN) experiment (referred to simply as BLUELINK in the following sections). The latest version, BRAN2020, is used. It simulates the period from 1993 to 2023 using a near-global ocean model with a 10 km spatial resolution and daily temporal resolution. Through data assimilation, the model integrates observations of temperature, salinity, and sea surface height (from satellite altimetry) to refine the ocean state. Details of the model and an analysis of its performance can be found in .

The altimetry data used in this study come from two distinct products:

CMEMS: the operational daily sea level grids distributed by the Copernicus Marine Service, based solely on nadir altimeters.

All altimetry data used are provided at a daily rate. The latest generation, named DT2024 and available since November 2024 in the Copernicus Marine Service (CMEMS), applies for the first time a different approach compared to the optimal interpolation adopted in previous versions (e.g., ): the Multiscale Inversion of Ocean Surface Topography (MIOST) technique .

The MIOST technique solves the mapping problem by integrating all components within a reduced space and utilizing a preconditioned conjugate gradient method . This iterative process continuously refines the solution until convergence. Once the final reduced solution is obtained, it is projected back onto the full spatial grid using wavelet-based transformations specific to each component.

The MIOST technique is also applied to an experimental product distributed by AVISO (called MIOST_SWOT+nadirs in this study), representing the first attempt to combine traditional Level-3 along-track sea level anomalies with the two-dimensional Level-3 product from the SWOT mission . Besides the integration of the swath-altimeter data, minor methodological differences include a different selection of the components of the wavelet-based transformations (Maxime Ballarotta, personal communication, 2025).

Three months of data (from September 2023 to November 2023) are considered during the science phase of SWOT. In this period, both CMEMS and MIOST_SWOT+nadirs blend along-track measurements from the following nadir altimeters: SARAL/AltiKa, Cryosat-2, HaiYang-2B, Jason-3, Sentinel-3A, Sentinel-3B, Sentinel-6A, and SWOT nadir.

For each latitude-longitude point of all the data sources listed above, a time series is produced and stored at a daily rate, which is referred to as unfiltered sea level. Subsequently, the time series is band-pass filtered using a Butterworth bandpass filter, filtering out components longer than 29 d (0.035 cycles per day in frequency) and shorter than 7 d (0.15 cycles per day in frequency).

A correlation analysis is performed to study the consistency of the dataset with the ground truth represented by the TG data. The computation is based on the Pearson correlation coefficient to measure the linear relationship between time series x and y: 1r=∑i=1n(xi-x‾)(yi-y‾)∑i=1n(xi-x‾)2∑i=1n(yi-y‾)2 where xi and yi are the individual sample points of the two time series, x‾ and y‾ are the mean values of x and y, respectively, and n is the number of observations.

To observe the along-shore coherency of intervals, the lag-correlation is also computed. In this case, the correlation is computed between xt and a time-shifted version of yt, denoted as yt+τ, where τ represents the lag (in days): 2r(τ)=∑i=1n-τ(xi-x‾)(yi+τ-y‾τ)∑i=1n-τ(xi-x‾)2∑i=1n-τ(yi+τ-y‾τ)2

The average propagation speed of the CTWs can be estimated from the Hovmöller diagrams. These diagrams are obtained by representing the temporal evolution of sea level anomalies along equally spaced points along the coastline, which are associated with the closest grid point of each dataset. The propagation speed of the observed contours is then computed based on the slopes that can be derived using image processing methods. The method based on the Radon transform, as described in , is followed. First, the Hovmöller diagrams are normalized to the range 0–1. The Radon transform works by considering straight lines in the 2D field, which are rotated by an angle θ and shifted by a distance s. It is defined as: 3Rf(s,θ)=∫-∞∞f(scos⁡θ-tsin⁡θ,ssin⁡θ+tcos⁡θ)dt, where Rf(s,θ) is the Radon transform of f(x,y), s is the perpendicular distance of the line from the origin, θ is the angle of the line relative to the ordinate, and t is a parameter along the line L, integrating over all points on that line. The transform sums the squares of the 2D field along each straight line to quantify the energy of the features aligned at each angle: 4E(θ)=∫-∞∞Rf2(s,θ)ds.

The best estimate of the westward propagation speed is then the tangent of the angle at which the sum of the squares reaches its maximum: 5v=tan⁡(θmax⁡), where v represents the westward propagation speed.

To determine whether CTWs represent a dominant pattern of sub-monthly coastal sea level variability, a Complex Empirical Orthogonal Function (CEOF) analysis is performed using all time series of filtered sea level within the shelf domain, characterized by bathymetry shallower than 500 m.

The CEOF analysis is a well-known technique to identify variability patterns in time and space and has also been used for this purpose in previous studies of CTWs such as . In this analysis the data are arranged as a complex matrix, where the real part is the original data, and the imaginary part is the Hilbert-transformed data. The coastal time series can be represented as a matrix X, where each row corresponds to a spatial location and each column corresponds to a time step. An anomaly matrix X′ is then computed by subtracting the mean at each spatial location to remove the temporal trend. To identify the dominant spatial patterns, the complex anomaly matrix using the Hilbert transform is defined: 6Xc=X′+i⋅H(X′) where H(X′) is the Hilbert transform of X′, and i is the imaginary unit.

The complex covariance matrix is computed as: 7C=1NXc′Xc′T. where N is the number of time steps. The CEOF modes are obtained by solving the eigenvalue problem: 8Cek=λkek where ek are the complex eigenvectors (the CEOF spatial modes), and λk are the eigenvalues, representing the variance explained by each mode. Finally, the Principal Component (PC) time series describes how the CEOF modes vary over time: 9PCk=ekTXc′.

In Fig. , an example of the time series from all data sources closest to one TG is presented, which can be qualitatively discussed before a quantification of the correlation is provided. The time series are shown in both their unfiltered and filtered versions. A visual inspection of the TG data reveals oscillations with a period of approximately 10 d and amplitudes ranging from 5 to 10 cm. The strongest one is found around day 50 and corresponds to a maximum in the SLA. For the purposes of this section, it is assumed that the observed oscillations represent the signature of CTWs, which are known to dominate sub-monthly variability in the study region (see Introduction). This assumption will be further examined in the following sections.

The BLUELINK time series matches very well the TG data, capturing all the CTWs and showing a good match in terms of phase as well. Only the peak of the strongest event is slightly underestimated. The CMEMS data fail to capture most of the sub-monthly oscillations and underestimate the amplitude peak. The MIOST_SWOT+nadirs data show much better agreement with the TG and BLUELINK, corresponding to at least five of the CTWs correctly observed, with amplitude and phase similar to the BLUELINK and TG estimations. However, events that almost superimpose cannot be correctly distinguished, as clearly seen between days 40 and 60.

To quantify the agreement, the correlation of BLUELINK, MIOST_SWOT+nadirs, and CMEMS against each TG for the unfiltered and filtered time series was computed and the results are reported in Fig. a and b. BLUELINK has the best score, showing higher agreement with the TGs at every station for both the unfiltered and filtered time series. Notably, the correlation coefficient is never lower than 0.7, and there is no strong sign of worse performance in the filtered version. This confirms the findings of and allows us to use the reanalysis as a validation of the altimetry-based dataset in this study, in addition to the TGs, which have the obvious limitation of providing only point-wise time series at a limited number of locations and therefore contain limited spatial information.

In the unfiltered time series, MIOST_SWOT+nadirs performs significantly better than CMEMS, with a correlation coefficient never lower than 0.6. Most notably, while a drop in correlation is observed for both altimetry data sources when considering the filtered time series, a strong improvement is seen in MIOST_SWOT+nadirs compared to CMEMS. The correlation between the filtered MIOST_SWOT+nadirs and TGs is higher than 0.5 in four out of five stations, while for CMEMS the correlation exceeds 0.5 only in one station. The largest drop in correlation between unfiltered and filtered signals in the altimetry dataset occurs in Bermagui, the station closest to the shelf break, which in that region is located 30–40 km from the coast. The drop in correlation at high frequencies is therefore likely due to the influence of open ocean processes in the estimation of the coastal grid points.

Figure c and d presents the spectral analysis of the unfiltered and filtered time series, respectively, performed using the Power Spectral Density (PSD) estimated with Welch's method. For this analysis, a Hann window and a segment length of 45 data points were used, with the PSD averaged across the five selected tide gauges. The shaded areas represent the uncertainty, computed as the standard deviation of the PSD across these gauges at each period. The tide gauges and BLUELINK exhibit a robust energetic local maximum at around 10 d, while the energetic content of the altimetry datasets is shifted toward longer periods. All datasets exhibit significant energy at sub-monthly periods, although a clear characterisation is hindered by the shortness of the time series.

Besides their temporal signature, we expect the sub-monthly CTWs identified in the previous section to also have a spatial signature along the shelf. The sea level oscillation caused by the CTWs can typically be observed by computing lag correlations. Moreover, and our results give us confidence that the BLUELINK data can serve as a reference for the altimetry dataset. Using the filtered time series, we therefore compute the correlation coefficient between each grid point and the Bermagui TG at different lags of 0, 2, and 4 d. The results are shown in Fig. . Bermagui is chosen because it is the southernmost location in the domain, in order to highlight, through lag-correlation, the spatial footprint of coastally trapped waves traveling northward. However, the same statistics have been produced for every TG station, and the corresponding figures can be found in Appendix B (Figs. to ).

Taking BLUELINK as the reference, it is possible to observe how the correlation pattern is very well constrained within the shelf, which is defined by the detached line showing the bathymetric contour at -500 m. Shelf locations north of 31° S are anticorrelated with Bermagui at lag 0, although some of these correlations are not statistically significant, while the same region shows correlations well above 0.5 with a lag of 4 d. In MIOST_SWOT+nadirs, the same pattern as in BLUELINK can be traced, although the average correlation with the TG is generally lower than in BLUELINK, and the areas of high lag-correlation are slightly more spread outside the shelf. CMEMS also shows a similar pattern, but the level of correlation, especially at Lag 2 and Lag 4, is much lower than in MIOST_SWOT+nadirs and therefore less distinguishable from the random patterns in the open ocean. Observing the correlations at Lag 0, it is possible to better understand the drop in performance for Bermagui seen in Fig. : this is due to an eddy-shaped uncorrelated feature that is not visible in the model and that MIOST_SWOT+nadirs and CMEMS localize on both sides of the shelf break.

After subdividing the coastline into equally spaced points, we analyze the Hovmöller diagram of the filtered signals in the three datasets and compute the average phase speed of the CTWs based on the Radon transform. The results are shown in Fig. . The pattern of the CTWs traveling anticlockwise along the Australian coastline is very clearly modeled by BLUELINK, but can also be recognized in MIOST_SWOT+nadirs and, partially, in CMEMS. The filtered altimetry time series in MIOST_SWOT+nadirs are good enough that it is possible to measure exactly the same phase speed of the CTWs as in BLUELINK (4.01 m s⁻¹), while the phase speed estimated for CMEMS is still very close (4.33 m s⁻¹). This is a striking result, considering that we are using the time series closest to the coastline, meaning the points for which the quality of the altimetry data is supposed to be the lowest.

As the next step, we analyze the results of the CEOF analysis applied to the filtered signal. The reconstructed signals from the first CEOF are shown as Hovmöller diagrams in Fig.  using the same coastal locations as previously. As expected from previous studies, BLUELINK shows a dominant first mode (71 % of the variance of the filtered signal) characterized by an oscillating pattern typical of CTWs. A similar first mode is seen with different degrees of similarity also in MIOST_SWOT+nadirs and CMEMS, although the degree of variance explained is less in the altimetry dataset (55 % for MIOST_SWOT+nadirs, 52 % for CMEMS).

The use of CEOF analysis enables the examination of the signal's phase in both spatial and temporal dimensions. Both MIOST_SWOT+nadirs and BLUELINK exhibit a clear linear phase progression along the coastline (Fig. , lower panels), characteristic of traveling waves, whereas this pattern appears noisier in CMEMS. By analyzing the slope of the spatial phase and the corresponding spatial distance, the spatial wavelength of the signal can be estimated. The mean spatial wavelength is approximately 2365 km for BLUELINK and 3142 km for MIOST_SWOT+nadirs. These values are challenging to validate: reported a mean spatial wavelength of 4000 km in the same domain for 2009 using BLUELINK, while in-situ campaigns from the 1980s cited in the same study reported wavelengths around 2500 km. For the purposes of this study, it is important that the average spatial wavelength is within a realistic range and that the spatial phase decreases linearly in both BLUELINK and MIOST_SWOT+nadirs. In contrast, the average wavelength of 1211 km derived from CMEMS appears unrealistic due to the irregularity of its spatial phase slope. The temporal phase of the leading mode increases linearly across all three datasets, indicating a sinusoidal oscillation with a dominant period. The average period of the first mode is approximately 10 d for BLUELINK, 14 d for MIOST_SWOT+nadirs, and 18 d for CMEMS.

A confirmation of this evaluation is evident in the temporal and spatial evolution of the reconstructed signal from the first CEOF, presented in Fig.  for three days during the strongest CTW event observed in the second half of October 2023. MIOST_SWOT+nadirs and BLUELINK exhibit a consistent pattern of anticlockwise oscillating amplitudes, although in BLUELINK the decreasing amplitude of the CTW from onshore toward the shelf is more clearly visible. BLUELINK displays the highest CTW amplitudes, while these are strongly damped in CMEMS. Similar conclusions can be drawn from the filtered signal shown in Fig.  in Appendix C.