Vertical land motion (VLM) at the coast is a substantial contributor to relative sea level change. In this work, we present a refined method for its
determination, which is based on the combination of absolute satellite altimetry (SAT) sea level measurements and relative sea level changes
recorded by tide gauges (TGs). These measurements complement VLM estimates from the GNSS (Global Navigation Satellite System) by increasing their spatial
coverage. Trend estimates from the SAT and TG combination are particularly sensitive to the quality and resolution of applied altimetry data as well as
to the coupling procedure of altimetry and TGs. Hence, a multi-mission, dedicated coastal along-track altimetry dataset is coupled with
high-frequency TG measurements at 58 stations. To improve the coupling procedure, a so-called “zone of influence” (ZOI) is defined, which confines
coherent zones of sea level variability on the basis of relative levels of comparability between TG and altimetry observations. Selecting 20 %
of the most representative absolute sea level observations in a 300

Coastal vertical land motion (VLM) significantly contributes to relative sea level change (SLC). VLM is in many places of the same order of magnitude
(1–10

VLM is caused by the superimposition of natural processes and anthropogenic influences in the Earth system and operates on manifold spatial and
temporal scales

In response to the substantial impact on relative sea level and the large spectrum of VLM sources, several strategies have been developed to estimate
VLM. The ability to capture the diversity of VLM processes, however, strongly depends on the method and geodetic technique used in the VLM
estimation. Furthermore, the coverage and associated accuracy of VLM estimates differ across the methods. Given that the global absolute sea level
trend during the altimeter era is of the order of 3

The majority of global sea level studies have utilized geodynamic GIA models to correct, for example, tide gauge records for secular land motion trends or
to extrapolate future relative SLC based on climate projections

For more than a decade, these direct geodetic estimates (GNSS, such as GPS, GLONASS and GALILEO) have been exploited to determine vertical velocities

For the latter use a necessary working hypothesis is that GNSS VLM represents the same movements as experienced at the tide gauge

To extend the number of VLM estimates, several studies advanced the application of combining satellite altimetry (SAT) and tide gauge (TG)
observations

While all three sources of information, GIA models, GNSS and “satellite altimetry minus tide gauge” (SAT–TG) techniques, have individual
merits, their combined use is valuable to further substantiate VLM estimates. GNSS observations are necessary to validate both GIA models and the
SAT–TG approach

In this study, we focus on enhancing the application of the SAT–TG difference for VLM detection. Our investigations not only further develop the latest
progress of the method, but also gain a new perspective on sea level trend and uncertainty characterization as well as quantification in coastal zones. The
next section recapitulates the latest state of the VLM

The combination of SAT and TG observations for VLM determination was steadily improved over the last 2 decades and is elaborated in the latest
review by

Notwithstanding the weaker performance of the along-track product (from GSFC) achieved in WM16,

Based on

Applied models and geophysical corrections for estimating sea level anomalies.

These reasons motivate further improvements of both components: the quality of the data and the practice of combining altimetry and TG data. We aim to understand how dedicated along-track coastal altimetry can outperform standard gridded products. We also seek to generalize an optimal selection of SLAs, underpinned by the local dynamical features of measured sea level variability.

In this work, we present a new approach of combining SAT and TG observations to improve VLM estimates. In contrast to previous attempts, we exploit TG
and SAT data at the highest available temporal and spatial scale for globally distributed stations. We couple advanced coastal altimetry data with
high-frequency TG records from the Global Extreme Sea Level Analysis (GESLA). Implementation of these high-frequency TG records constitutes a further
innovation for VLM estimation. So far such data have only been applied in local studies

Sections

We use different altimetry products in order to assess the impact of special coastal products on associated VLM

The coastal altimetry product is constructed from 1 Hz multi-mission altimetry measurements processed by DGFI-TUM with OpenADB
(

We map all altimetry records on 1 Hz nominal tracks consistent with the CTOH nominal paths (Center for Topographic studies of the Ocean and
Hydrosphere,

SLAs along the same track and cycle are then averaged over predefined areas as described in Sects. 2.5 and 2.6. We built a time series by considering
all averaged SLAs from the along-track multi-mission dataset for the study period. To check for outliers in each SLA time series, we exclude values
exceeding absolute values of 3

The gridded Ssalto/Duacs altimeter product was produced and distributed by the Copernicus Marine Environment Monitoring Service (CMEMS;

We use monthly mean TG data from the datum-controlled PSMSL

In addition to monthly mean PSMSL TG data, we exploit the GESLA dataset

In contrast to PSMSL data used in WM16, GESLA TGs feature no rigorous outlier rejection by default except that of the primary data providers

To obtain a uniform temporal resolution, we resample this outlier-free TG set to hourly records by cubic interpolation. The records are then corrected
for the tidal signal and for the ocean response to atmospheric wind and pressure forcing. The tidal variability is suppressed by using a 40 h
loess (locally estimated scatter plot smoothing) filter

To understand the sensitivity of the VLM estimations to the (1) quality and resolution of the data and (2) the selection procedure, we analyze the performances of four different dataset combinations: ALES–PSMSL-250 km, ALES–GESLA-250 km, AVISO–PSMSL-250 km and ALES–GESLA–ZOI.

The first three combinations are constructed to compare the performances of the along-track (ALES) against the gridded altimetry product (AVISO)
combined with monthly TG observations. With ALES–GESLA-250 km we also investigate the possible advantage of using the GESLA high-rate TG product. For
all these experimental sets, SLA time series are merged or averaged within a 250

To produce the ALES–GESLA-250 km dataset, we derive differences of the merged, nonuniformly sampled SLAs and the hourly sampled GESLA TG records through cubic interpolation of the latter and a maximum allowed time lag of 3 h between the measurements. We downsample these high-rate differenced time series to monthly means. For ALES–PSMSL-250 km, on the other hand, we first compute monthly means from SLAs and subsequently subtract these monthly SLAs from the monthly sampled relative SLAs from PSMSL. Finally, we directly compute the differenced SAT–TG time series from the averaged monthly AVISO and the PSMSL data, which yields the AVISO–PSMSL-250 km dataset.

Using these combinations, we investigate the mere changes from differences formed using along-track data at high or at low frequency (ALES–GESLA-250 km and ALES–PSMSL-250 km) or using monthly gridded data (AVISO–PSMSL-250 km). Here, “high frequency” refers to daily timescales of variability and “low frequency” to monthly timescales. The dataset ALES–GESLA–ZOI incorporates further SLA selection schemes, which are explained in the following section.

We aim to develop a new SLA selection scheme, which accounts for the observed coherency of sea level variability. However, due to the diversity of the
underlying physical mechanisms and their complex interplay with the coast, the spatial coherency of sea level dynamics is highly variable in coastal
regions

The key concept of our approach is to capture the extent to which coastal altimetry measurements are similar to the in situ TG observations. To do so,
we extend the methodology proposed by

We exploit combinations of along-track ALES data and high-frequency GESLA records to identify regions of sea level variability that show maximum
coherency with TG observations, which we hereinafter call the zone of influence (ZOI). With this approach, our objective is to decrease noise of the
differenced, high-frequency VLM

To define the ZOI, we investigate different statistical criteria

To define the ZOI, we select subsets of the data containing the best-performing statistics (i.e., highest correlation, lowest rms

Zone of influence: different coherent zones of sea level variability are identified by different statistical criteria

Note that in contrast to the 250

We identify coherent zones of sea level variability represented by different selection criteria in Fig.

The obtained coherent structures reveal notable dependencies on the local bathymetric and coastal properties. Figure

Comparing these three examples, we also observe that absolute values of the statistics differ from site to site. Correlations of along-track data near
the Australian coastline, for instance, outperform the ones in the example in Fig.

Shown are “SAT minus TG” time series for different datasets and configurations for the TG in Fig.

A correct choice of the ZOI based on a subset of high-performance SLAs can significantly reduce the SAT–TG residuals as exemplified in
Fig.

While using relative thresholds can reduce the noise of VLM

We fit the differenced time series to a combination of a deterministic model and stochastic noise models with the maximum likelihood estimation (MLE)
method. Parameters of the deterministic model are comprised of a constant offset

When combining altimetry and TGs for VLM estimation, several sources can contaminate the differenced time series and inflate the actual “red” noise
(low-frequency) content in the residuals, which generates autocorrelated signals in the data. The SLA computation is affected by the instrumental
errors of the range estimation and of each of the geophysical corrections

To validate SAT–TG-based trend estimates, we use the ULR6a GPS solution provided by the GNSS data assembly center SONEL (Systeme d'Observation du
Niveau des Eaux Littorales,

The TG locations and record lengths differ among the presented experimental datasets (Sect.

We compute the rms

We also analyze the median of the absolute value of differences (

Statistics of different SAT–TG combinations.

We compare the performances of the three datasets which are constructed from 250

For both combinations the median of the VLM differences (ALES–PSMSL-250 km:

Scatter and box plots comparing estimated SAT–TG trends and GNSS trends, as in WM16 Fig. 14.

In comparison with the low-frequency datasets (ALES–PSMSL-250 km and AVISO–PSMSL-250 km), the high-rate setup ALES–GESLA-250 km improves the
rms

We investigate how the ZOI selection of SLAs fosters quality SAT–TG VLM estimates. As addressed in Sect.

Because the spectral index (for ALES–GESLA–ZOI) is slightly lower (

Performance of VLM

The rms

When setting this optimal threshold to 20 %, the ALES–GESLA–ZOI setup outperforms the other investigated configurations. Figure

Figure

The integration of the ZOI primarily reduces the uncertainty of VLM

The results presented in Fig.

Figure

VLM

In contrast to these examples, we find very low local optima for some stations (Fig.

To further shed light on the relationships between dynamical sea-level-based SLA selection and spacial coherence of trends and uncertainties, we show
trend and uncertainty maps (Fig.

Bathymetric and coastal properties can cause large discrepancies in responses of coastal sea level variability as they modify the character of the
impact of large-scale atmospheric forcing and remote variability from the deeper ocean

Next to site-dependent physical factors, the spatial scales of trend coherency might also depend on the time span of the observations
themselves. Global maps of sea level trends, for example, even when derived from 2 decades of observations, still show distinct patterns of
natural and forced variability and thus overshadow signals of ocean mass or steric contributions

Time and space dependencies of trend uncertainty and accuracies:

To investigate this timescale dependency, we truncate the VLM

Mean rms

The same evaluation for the dependency of uncertainty on time and level of comparability

VLM estimates from different datasets (e.g., AVISO–PSMSL-250 km and ALES–GESLA–ZOI) are biased compared to trends inferred from GNSS observations. Based
on Monte Carlo simulations (see Appendix Fig.

Next to the record length (see Sect.

Moreover, as mentioned before, the multi-mission calibration applied (MMXO) reduces intermission biases and regionally coherent systematic
errors but does not feature a calibration against TG. The median bias identified for ALES–GESLA–ZOI could be affected by a drift of the mission used
as a reference. In contrast, the AVISO dataset does not include time-dependent intermission biases and might therefore be additionally influenced by
systematic effects of, e.g., Envisat or Sentinel-3a

Next to altimeter bias drift, nonlinear VLM from contemporary mass redistribution (CMR) changes was shown to cause differences between
VLM

Based on optimal relative thresholds, we estimated an rms

Accuracies of estimated VLM

We highlight the fact that the SAT–TG estimates are not only limited by the broad spectrum of error sources, ranging from systematic to correction errors, such as the residual long-period tides remaining in the TG time series, which all contribute to the error budget of the estimates. Another factor is the possible nonlinearity of the VLM itself, which strongly hampers the comparability with measurements from other geodetic techniques when sampled over different time spans. Thus, addressing this issue in SAT–TG time series could represent a further crucial improvement of the application.

We investigate potential improvements of combining altimetry and TGs for coastal vertical land motion estimation. The innovations of our approach are twofold: (1) for the first time, we exploit a global network of high-frequency TG data (GESLA) and dedicated coastal altimetry (ALES) to determine VLM at a variety of colocated GNSS stations. Secondly, (2) we define a zone of influence to identify coherent zones of coastal SL variability, which optimizes the combination of altimetry and TGs. We rate improvements of both innovations against various SAT–TG datasets, which are comprised of along-track and gridded altimetry, as well as high-frequency (daily) and low-frequency (monthly) TG combinations.

Combining high-frequency TG with coastal altimetry data (ALES–GESLA-250 km) yields modest improvements of trend accuracies, compared to a monthly
gridded or monthly along-track combination, when averaging SLAs in a radius of 250

In Fig. 6 we map linear VLM

To gain a better understanding of when the VLM

Figure

Histogram of median values of randomly sampled subsets. The subsets consist of 52 samples (according to the number of TGs in AVISO–PSMSL-250 km) and are randomly drawn from normally distributed values with zero mean and a standard deviation of 1.5

ULR6a GNSS trend estimates are obtained from the data assembly center SONEL (Systeme d'Observation du Niveau des Eaux Littorales,

JO and MP conceptualized and designed the study. JO wrote the paper and is the author of the full software code used in this study. MP is the author of the ALES retracking algorithm and mentored the work of JO; CS and DD are responsible for the altimetry database organization and the data structure. LS provided assistance in the use of GNSS data. FS provided the basic resources making the study possible and coordinates the activities of the institute. All authors read and commented on the final paper.

The authors declare that they have no conflict of interest.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) (grant agreement 411072120) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program. We thank the data providers GESLA, PSMSL, SONEL and AVISO for the opportunity to use their products. We thank Sergiy Rudenko, Ashwita Chouksey and Michael Hart-Davis for their help and comments. We are very grateful for the comments of the reviewers, Alvaro Santamaría Gómez and Christopher Watson, which strongly improved the paper.

This research has been supported by the Deutsche Forschungsgemeinschaft (DFG) (grant no. 411072120).This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Joanne Williams and reviewed by Alvaro Santamaría-Gómez and Christopher Watson.