Vision of a Clock-Based Network for Absolute Sea Level Monitoring

Full text

Vision of a Clock-Based Network for Absolute
Sea Level Monitoring
Asha Vincent and Jürgen Müller
Abstract
Global sea level shows an increasing trend for several decades driven mainly by climate
change. Absolute Sea Level (ASL) changes can only be extracted from Relative Sea Level
(RSL) measurements with proper reduction of vertical land movements of the bench marks.
Atomic clocks at those tide gauges can potentially provide the absolute, near real-time
physical height change. High-performance clocks with an uncertainty of 1018 enable a
height measurement with 1 cm accuracy. As RSL is related to regional tidal datums, one
has to account for the local variations to obtain a globally consistent measurement of
ASL. Hence, by incorporating land motion from clock observations, one can establish
a consistent and uniform reference datum for assessing geoid-based absolute sea level
changes worldwide.
Keywords
Absolute sea level Atomic clocks Physical height Relative sea level Relativistic
geodesy
1Introduction
The geoid, which represents the global mean sea level in rest,
serves as a reference surface in geodesy. Precise measure-
ments of the local relative sea level are typically carried out
by comparing the level of the sea surface at a specific location
to a fixed reference point on land (Wöppelmann and Marcos
2016). This reference point, often referred to as a benchmark
or tidal datum, serves as a baseline for determining how
the sea level at that location changes over time. At present
GNSS (Global Navigation Satellite System) stations nearby
tide gauge locations serve as benchmarks for estimating the
relative land movements (Larson et al. 2013) in order to
determine the Absolute Sea Level (ASL) (Peng et al. 2021).
A. Vincent () · J. Müller
Institute for Geodesy, Leibniz University Hannover, Hannover, Lower
Saxony, Germany
e-mail: vincent@ife.uni-hannover.de;mueller@ife.uni-hannover.de
The physical height at a point on the Earth surface
depends upon the geoid at the time of the measurement (Ihde
et al. 2017). As detailed in Dietrich (2014) ASL refers to
the height of the sea surface above a fixed global reference
without the effect of land motion. Unlike Relative Sea Level
(RSL), which is determined by tide gauges with respect to a
local or regional reference (tide gauge zero or a tidal datum),
absolute sea level is referenced to a mean ellipsoid when
using GNSS benchmarks as reference. Installation of high-
performance atomic clocks at the tide gauge locations can
provide the absolute physical height change which enables
to directly obtain ASL changes with respect to the geoid.
Philipp et al. (2020) give details on the relativistic definition
of geoid and gravity potential.
2 Clocks to Replace GNSS Benchmarks
The physical height at a point on Earth’s surface is contin-
uously varying mainly due to external tidal effects and non-
tidal mass distributions in the Earth system as explained in
International Association of Geodesy Symposia,
https://doi.org/10.1007/1345_2024_265, © The Author(s) 2024

A. Vincent and J. Müller
Fig. 1 Absolute sea level change (ASL) and relative sea level (RSL)
with respect to a tidal datum (TD) in the case of land uplift H vand
sea level (SL) rise
Voigtetal.(2016). Tidal effects include solid-earth tides
and ocean-load tides due to external bodies such as the
Sun, Moon and other planets, centrifugal effects of polar
motion result in pole tides and LOD (Length Of Day) tides
arise from the variations in the Earth’s angular velocity,
atmospheric tides, etc. Non-tidal effects include mass re-
distributions in the geosphere, hydrosphere, atmosphere and
biosphere. All these effects affect the geopotential at the
point of interest as a result of mass change and the associated
vertical displacements (Schröder et al. 2021). From terres-
trial clock observations, the physical height changes can
be inferred where GNSS receivers provide only ellipsoidal
height changes. Thus, we can compute ASL changes with
respect to the geoid by adding relative sea level changes with
time-variable land motion (H v) from clock observations:
ASL DRSL CH v:(1)
As RSL measurements are referenced to GNSS benchmarks
near tide gauges, it doesn’t necessarily guarantee consistency
in the vertical datum across different locations. Vertical
datums can vary regionally due to factors such as local
geoid variations, tectonic movements, and other geophysical
processes. Here, clocks offer an alternative by providing a
direct reference to an equipotential surface such as the geoid.
Using atomic clocks as benchmarks will be a more consistent
way of deriving the ASL change globally by referencing both
land and sea level measurements to a known equipotential
surface, e.g., geoid. Then, we do not need to consider the
regional tidal datum variations, as measurements will be
taken with respect to the global geoid that preserves the
uniformity of a reference surface without any local variations
(Fig. 1).
If connected to a stable reference clock that is related to
the geoid (W0), terrestrial clocks provide physical heights H,
which can also be represented in terms of the geopotential
Fig. 2 Loading and unloading effects on physical height which
depends upon geoid (G) height and vertical displacement of Earth
surface (T)
number, CpDW0Wp, where W0represents the potential
on the geoid and Wpis the gravity potential at point p (Torge
et al. 2023). Here, gravity potential changes can directly only
be observed by clocks. And keep in mind, a potential change
is independent of whether the mass change occurs above or
below the Earth’s surface, i.e. above or below the clock loca-
tion. The gravitational potential change or the corresponding
physical height difference between two clocks can be derived
from the fractional frequency difference (Bjerhammar 1986;
Müller et al. 2018; Wu and Müller 2019; Denker et al. 2018),
f
fW
c2gH
c2(2)
when using the GCRS (Geocentric Celestial Reference Sys-
tem) metric up to the first Newtonian order (orders of c4
are omitted) where W is the gravity potential difference
between the two clock sites, c is the speed of light and, g is
the mean gravity value. As illustrated in Fig.2, tidal effects
and non-tidal loading/unloading effects affect the physical
heights represented as H v
tidal and H v
nontidal respec-
tively. According to Eq. (2), these time-variable changes are
obtained by clocks (the static part cancels out for clocks at a
fixed location),
H vDH v
tidal CH v
nontidal:(3)
In order to get the time-dependent variations in physical
heights, there should be a stable reference clock somewhere
on ground or space that is related to the geoid (Wu and Müller
2020; Philipp et al. 2023). As shown by Lisdat et al. (2016),
the systematic uncertainty of the optical fibre link is of 1019.
For our purpose, we propose only a single link for each tide
gauge clock to the reference clock which is assumed to be in
a geostationary satellite. We assume a space link accuracy
in the 1018 level. The major influence on the space link
uncertainty is the clock error and velocity error as given in
Shen et al. (2023). A link accuracy of 1018 is sufficient to
achieve a 1 cm accuracy in the ASL estimation.

Vision of a Clock-Based Network for Absolute Sea Level Monitoring
Fig. 3 Tide gauge locations considered in this study: NEWL (Newlyn),
AND1 (Andenes), REYK (Reykjavik), IBIZ (Ibiza), GENO (Genova),
and, 0NYB (Kalix)
3 Methodology
Tide gauge locations along the European coast are consid-
ered for this study (Fig. 3). A total of 6 sites – Newlyn (UK),
Andenes (Norway), Reykjavik (Iceland), Ibiza (Spain), Gen-
ova (Italy), and Kalix (Sweden) are chosen based on the
availability of nearby GNSS stations with longer time-series.
The simulations are carried out for the time period 2006-
2016 based on the availability of the needed datasets. The
mean over the time period is reduced from monthly RSL
RLR (Revised Local Reference) data obtained from PSMSL
(Permanent Service for Mean Sea Level) (Holgate et al.
2013; Permanent service for mean sea level 2023) to derive
the RSL change. Clock observations are simulated by com-
bining the physical height change due to tidal and non-tidal
effects (Eq. 3). Addition of RSL changes with the simulated
effective vertical displacements or the clock observations
generate the corresponding ASL changes at the selected tide
gauge locations (Eq. 1).
4 Results and Discussion
4.1 Simulation of Tidal Signals as Clock
Observations
Potential variations due to solid-earth tides, pole tides and
LOD tides are computed on the deformable Earth surface
using modified ETERNA34 (PREDICT program) Earth tide
data processing package (Wenzel 2022) for the chosen time
period. The HW95 (Hartmann and Wenzel 1995) tidal poten-
tial catalogue is used for deriving the hourly tidal poten-
tial values which are subsequently averaged to produce
monthly values. Similarly, monthly averages of potential
variations due to ocean tidal loading are determined with the
SPOTL3.3.0.2 package (Agnew 2012) using the Empirical
Ocean Tidal model (EOT11a) (Savcenko and Bosch 2012).
Both ETERNA34 and SPOTL calculate the effective poten-
tial variations due to mass changes and land motion. Figure 4
shows the simulated major tidal values at a high-latitude
site and a low-latitude site. The LOD tidal values are of
the order of 0:001 m2/s2which can be neglected. Similarly,
atmospheric tidal values are also much less than the range
of 1018 (= 0.1 m2/s2) clock sensitivity which again can be
neglected (Voigt et al. 2016).
The monthly averages of solid-earth tides range from
approximately 10 cm at high-latitude sites, making it an
important factor in total land motion. All the three major tidal
contributions (Fig. 4) are combined to generate the total tidal
effects (second subplots of Fig. 5). The observed negative
long-term trend in the tidal values is caused by the 18.61 year
lunar nodal cycle. According to Rochlin and Morris (2017),
the declination of Moon’s orbit is smallest from 2005 to 2015
which results in increasing tidal amplitudes that correspond
to negative potential variations on the Earth surface.
4.2 Simulation of Non-Tidal Signals as
Clock Observations
Non-tidal mass distributions include loading and unloading
effects arising from various effects such as changes in Ter-
restrial Water Storage (TWS), atmospheric pressure, ocean
bottom pressure, solid earth processes like GIA (Glacial
Isostatic Adjustment), etc. (Schröder et al. 2021). The time
series of geoid height variation due to the variation in the
TWS can be determined from GRACE (Gravity Recovery
And Climate Experiment) fully normalised monthly spheri-
cal harmonic coefficients (Kvas et al. 2019). Similarly, using
the fully normalised monthly spherical harmonic coefficients
(GAC) from Atmosphere and Ocean De-aliasing product
(AOD1B RL06) (Dobslaw et al. 2017), the effect of atmo-
sphere and ocean mass variability can be well estimated.
The surface deformations associated with these non-tidal
mass distributions are given by NGL (Nevada Geodetic Lab-
oratory) GNSS time-series solutions (Blewitt et al. 2018). A
loading effect leads to a downward vertical displacement and
vice versa. The effective gravitational potential variations
can be computed by combining the potential variations due
to mass changes and surface displacements (Vincent and
Müller 2023). The separate effects of geoid height variations
(GRACE+AOD) and the corresponding vertical displace-
ments of the Earth surface (GNSS) are shown by the first
subplots of Fig. 5.

A. Vincent and J. Müller
Fig. 4 Monthly averages of solid-earth tides (SET), ocean-load tides (OTL) and pole tides (POL) with their linear trends at Andenes (a)and
Genova (b)
The simulated clock observations by combining the tidal
and non-tidal potential variations (Eq. 3) at the chosen tide
gauge locations are given by the third subplots of Fig.5.
1018 clocks can provide absolute land motion of 1 cm
(i.e. corresponding gravity potential variations of 0.1 m2/s2)
between the clock sites in near real-time. When one considers
monthly observations over several years – as done in this
study – the resulting trend accuracy is less than 1 mm/yr
(Fig. 5).
4.3 Estimation of ASL Changes
As mentioned in Sect. 2, geoid-based ASL changes can be
derived by combining clock observations with RSL data
from tide gauges (Eq. 1). Clock observations of height
variations provide a globally consistent and uniform refer-
ence for deriving ASL changes. High-performance clocks
with fractional frequency uncertainty of 1018 (McGrew
et al. 2018; Takamoto et al. 2022) with same level of link
accuracy can be utilised well in deriving ASL changes with
high accuracy of 1 cm. The monthly variations of RSL are
determined by reducing the mean over the chosen time-
span. The clock observations which are simulated in terms
of potential variations (m2/s2) are transferred into physical
height variations (cm) by multiplying with g to combine
with the RSL changes. Figure 6shows the graphs of the
estimated ASL changes with the long-term trend, clock
observations in terms of physical height, and RSL changes.
The measurement noise of RSL affects the estimation of
ASL.
Present-day land uplift, e.g., at Kalix (0NYB) can fake a
sea level fall. So, the land motion must be taken into account
when deriving actual sea level changes. As seen in Figs.5
and 6, the offsets from the monthly means of land motion
affect the monthly averages of ASL change at the clock sites.
On analyzing the linear trends at the tide gauge sites for the
estimated time period, there is an overall increase in ASL
with given accuracy.
5 Conclusions
Till now, land-based absolute sea level measurements are
estimated with respect to GNSS benchmarks or local tidal
datums that give ASL changes with respect to the reference
ellipsoid. Further calculations are needed in order to derive
a globally consistent measure of ASL. High-performance
atomic clocks at tide gauge locations with better link uncer-
tainty can provide more accurate physical height changes
and thus, geoid-based ASL changes can be directly obtained.
Hence, clock networks can be used for monitoring vertical
land movements and providing uniform and consistent ASL
changes world-wide.
We have estimated the ASL changes at some tide gauge
locations along the European coast. The obtained long-
term trend values indicate the geoid-based ASL changes
per year for the given time-span. A maximum value of
0.71˙0.49 cm/year was observed at Andenes (AND1) for
the time period 2008–2016. All sites other than Ibiza (IBIZ)
and Newlyn (NEWL)show an increasing trend. Significant
land uplift at sites Kalix (0NYB), Andenes (AND1), and
Reykjavik (REYK) should be effectively reduced for the
accurate estimation of ASL changes. Also, for all the sites,
tidal effects play an important role as even the monthly-
averages can be as high as 10–20 cm. Thus, the accurate

Vision of a Clock-Based Network for Absolute Sea Level Monitoring
Fig. 5 Simulated clock observations (CLOCK) by combining the potential variations due to non-tidal effects (mass changes in TWS (GRACE),
atmosphere and ocean (AOD) and associated vertical deformations (GNSS)) and tidal effects (TIDAL=SET+OTL+POL) with their linear trends
(m2/s2yr) at the tide gauge sites

A. Vincent and J. Müller
Fig. 6 Estimated monthly ASL changes (ASL) by reducing the clock observations (CLOCK) from RSL changes (RSL) at the tide gauge sites.
The long-term trends of ASL changes (cm/yr) are also given

Vision of a Clock-Based Network for Absolute Sea Level Monitoring
estimation of globally consistent geoid-based ASL changes
favours the realization of clock-based networks in the near
future.
Acknowledgements This study has been funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC 2123 Quantum Frontiers –
Project-ID 90837967 and the SFB 1464 TerraQ – Project-ID 434617780
within project C02.
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