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JGeod
https://doi.org/10.1007/s00190-017-1075-1
ORIGINAL ARTICLE
Geodetic methods to determine the relativistic redshift at the level
of 10−18 in the context of international timescales: a review and
practical results
Heiner Denker1·Ludger Timmen1·Christian Voigt1,2·Stefan Weyers3·
Ekkehard Peik3·Helen S. Margolis4·Pacôme Delva5·Peter Wolf5·Gérard Petit6
Received: 20 December 2016 / Accepted: 4 October 2017
© The Author(s) 2017. This article is an open access publication
Abstract The frequency stability and uncertainty of the lat-
est generation of optical atomic clocks is now approaching
the one part in 1018 level. Comparisons between earthbound
clocks at rest must account for the relativistic redshift of
the clock frequencies, which is proportional to the corre-
sponding gravity (gravitational plus centrifugal) potential
difference. For contributions to international timescales, the
relativistic redshift correction must be computed with respect
to a conventional zero potential value in order to be con-
sistent with the definition of Terrestrial Time. To benefit
fully from the uncertainty of the optical clocks, the grav-
ity potential must be determined with an accuracy of about
0.1m
2s−2, equivalent to about 0.01 m in height. This con-
tribution focuses on the static part of the gravity field,
assuming that temporal variations are accounted for sepa-
rately by appropriate reductions. Two geodetic approaches
are investigated for the derivation of gravity potential val-
ues: geometric levelling and the Global Navigation Satellite
Systems (GNSS)/geoid approach. Geometric levelling gives
BHeiner Denker
denker@ife.uni-hannover.de
1Institut für Erdmessung, Leibniz Universität Hannover
(LUH), Schneiderberg 50, 30167 Hannover, Germany
2Present Address: GFZ German Research Centre for
Geosciences, Telegrafenberg, 14473 Potsdam, Germany
3Physikalisch-Technische Bundesanstalt (PTB), Bundesallee
100, 38116 Braunschweig, Germany
4National Physical Laboratory (NPL), Hampton Road,
Teddington, Middlesex TW11 0LW, UK
5SYRTE, Observatoire de Paris, PSL Research University,
CNRS, Sorbonne Universités, UPMC Univ. Paris 06, LNE, 61
avenue de l’Observatoire, 75014 Paris, France
6Bureau International des Poids et Mesures (BIPM), Pavillon
de Breteuil, 92312 Sèvres, France
potential differences with millimetre uncertainty over shorter
distances (several kilometres), but is susceptible to system-
atic errors at the decimetre level over large distances. The
GNSS/geoid approach gives absolute gravity potential val-
ues, but with an uncertainty corresponding to about 2 cm in
height. For large distances, the GNSS/geoid approach should
therefore be better than geometric levelling. This is demon-
strated by the results from practical investigations related to
three clock sites in Germany and one in France. The esti-
mated uncertainty for the relativistic redshift correction at
each site is about 2 ×10−18.
Keywords Relativistic redshift ·International timescales ·
Terrestrial Time ·Caesium and optical atomic clocks ·
Relativistic geodesy ·Chronometric levelling ·Zero level
reference gravity potential
1 Introduction
The accurate measurement of time and frequency is of fun-
damental importance to science and technology, with appli-
cations including the measurement of fundamental physical
constants, global navigation satellite systems and the geode-
tic determination of physical heights. Time is also one of the
seven base physical quantities within the International Sys-
tem of Units (SI), and the second is one of the seven base
units. Since 1967 the second has been defined in terms of the
transition frequency between the two hyperfine levels of the
ground state of the caesium-133 atom. The second can also
be realized with far lower uncertainty than the other SI units,
with many other physical measurements relying on it.
Since the successful operation of the first caesium atomic
frequency standard at the UK National Physical Laboratory
(NPL) in 1955, the accuracy of caesium microwave fre-
123
H. Denker et al.
quency standards has improved continuously and the best
primary standards now have fractional uncertainties of a few
parts in 1016 (Heavner et al. 2014;Levi et al. 2014;Guena
et al. 2012;Weyers et al. 2012;Li et al. 2011). However, fre-
quency standards based on optical, rather than microwave,
atomic transitions have recently demonstrated a performance
exceeding that of the best microwave frequency standards,
which is likely to lead to a redefinition of the SI second.
The optical clocks operate at frequencies about five orders of
magnitude higher than the caesium clocks and thus achieve
a higher frequency stability, approaching fractional stabili-
ties and uncertainties of one part in 1018 (Chou et al. 2010;
Hinkley et al. 2013;Bloom et al. 2014;Nicholson et al. 2015;
Ushijima et al. 2015;Huntemann et al. 2016). Reviews of
optical frequency standards and clocks are given in Ludlow
et al. (2015) and Margolis (2010). In this context, a thorough
definition of frequency standards and clocks is beyond the
scope of this contribution, and hence, for simplicity, both
terms are used interchangeably in the following.
One key prerequisite for a redefinition of the second is
the integration of optical atomic clocks into the interna-
tional timescales TAI (International Atomic Time) and UTC
(Coordinated Universal Time). This requires a coordinated
programme of clock comparisons to gain confidence in the
new generation of optical clocks within the international
metrology community and beyond, to validate the corre-
sponding uncertainty budgets, and to anchor their frequencies
to the present definition of the second. Such a comparison
programme has, for example, been carried out within the col-
laborative European project “International Timescales with
Optical Clocks” (ITOC; Margolis et al. 2013).
Due to the demonstrated performance of atomic clocks
and time transfer techniques, the definition of timescales and
clock comparison procedures must be handled within the
framework of general relativity. Einstein’s general relativ-
ity theory (GRT) predicts that ideal clocks will in general
run at different rates with respect to a common (coordi-
nate) timescale if they move or are under the influence of
a gravitational field, which is associated with the relativistic
redshift effect (one of the classical general relativity tests).
Considering the usual case of two earthbound clocks at rest,
the relativistic redshift effect is directly proportional to the
corresponding difference in the gravity (gravitational plus
centrifugal) potential Wat both sites, where one part in
1018 clock frequency shift corresponds to about 0.1m
2s−2
in terms of the gravity potential difference, which is equiv-
alent to 0.01 m in height. Hence, geodetic knowledge of
heights and the Earth’s gravitypotential can be used to predict
frequency shifts between local and remote (optical) clocks,
and vice versa, frequency standards can be used to deter-
mine gravity potential differences. The latter technique has
variously been termed “chronometric levelling”, “relativis-
tic geodesy”, and “chronometric geodesy” (e.g. Bjerhammar
1975,1985;Vermeer 1983;Delva and Lodewyck 2013).
It offers the great advantage of being independent of any
other geodetic data and infrastructure, with the perspective
to overcome some of the limitations inherent in the classi-
cal geodetic approaches. For example, it could be used to
interconnect tide gauges on different coasts without direct
geodetic connections and help to unify various national
height networks, even in remote areas.
In the context of general relativity, it is important to distin-
guish between proper quantities that are locally measurable
and coordinate quantities that depend on conventions. An
ideal clock can only measure local time, and hence, it defines
its own timescale that is only valid in the vicinity of the clock,
i.e. proper time. On the other hand, coordinate time is the
time defined for a larger region of space with associated con-
ventional spacetime coordinates. Time metrology provides
specifications for the unit of proper time as well as the rele-
vant models for coordinate timescales. In this context, the SI
second, as defined in 1967, has to be considered as an ideal
realization of the unit of proper time (e.g. Soffel and Lang-
hans 2013). Usually, the graduation unit of coordinate time
is also named the “second” due to the mathematical link to
the SI second as a unit of proper time, but some authors con-
sider relativistic coordinates as dimensionless; for a recent
discussion of this topic, see Klioner et al. (2010).
Furthermore, the notion of simultaneity is not defined a
priori in relativity, and thus, a conventional choice has to
be made, which is usually done by considering two events
in some (spacetime) reference system as simultaneous if
they have equal values of coordinate time in that system
(e.g. Klioner 1992). This definition of simultaneity is called
coordinate simultaneity (and is associated with coordinate
synchronization), making clear that it is entirely dependent
on the chosen reference system and hence is relative in nature.
Accordingly, syntonization is defined as the matching of cor-
responding clock frequencies.
For the construction and dissemination of international
timescales, the (spacetime) Geocentric Celestial Reference
System (GCRS) and the associated Geocentric Coordinate
Time (TCG) play a fundamental role. However, for an earth-
bound clock (at rest) near sea level, realizing proper time,
the sum of the gravitational and centrifugal potential gener-
ates a relative frequency shift of approximately 7 ×10−10
(corresponding to about 22 ms/year) with respect to TCG,
because the GCRS is a geocentric and non-rotating system.
In order to avoid this inconvenience for all practical timing
issues at or near the Earth’s surface, Terrestrial Time (TT)
was introduced as another coordinate time associated with
the GCRS. TT differs from TCG just by a constant rate,
which was first specified by the International Astronomical
Union (IAU) within Resolution A4 (1991) through a conver-
sion constant (denoted as LG)based on the “SI second on
the rotating geoid”, noting that LGis directly linked to a cor-
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
responding (zero) reference gravity potential value, usually
denoted as W0. However, due to the intricacy and problems
associated with the definition, realization and changes of the
geoid, the IAU decided in Resolution B1.9 (2000) to turn LG
into a defining constant with zero uncertainty. The numeri-
cal value of LGwas chosen to maintain continuity with the
previous definition, where it is important to note that the
corresponding zero potential W0also has no uncertainty (as
it is directly related to the defining constant LG). There-
fore, regarding the relativistic redshift correction for a clock
at rest on the Earth’s surface, contributing to international
timescales, the absolute gravity potential Wis required rel-
ative to a given conventional value W0, whereby only the
uncertainty of Wmatters (as W0has zero uncertainty), while
potential differences suffice for local and remote clock com-
parisons.
From the above, it becomes clear that TT is a theoretical
(conventional) timescale, which can have different realiza-
tions, such as TAI, UTC, Global Positioning System (GPS)
time, which differ mainly by some time offsets. For civil time-
keeping all over the Earth, the timescales TAI and UTC are of
primary importance and probably represent the most impor-
tant application of general relativity in worldwide metrology
today. While some local realizations of UTC are provided in
real time, TAI has never been disseminated directly, but is
constructed as a weighted average of over 450 free running
clocks worldwide. The resulting timescale is then steered to a
combination of the world’s best primary frequency standards
after doing a relativistic transformation of the (local) proper
time observations into TT; in other words, a relativistic red-
shift correction (with a relative value of about 1×10−13 per
kilometre altitude) is applied so that TAI is indeed a real-
ization of TT, associated with a virtual clock located on the
(zero) reference gravity potential surface (W0). The compu-
tation of the relativistic redshift is regularly re-evaluated to
account for progress in the knowledge of the gravity poten-
tial and in the standards, see, for example, Pavlis and Weiss
(2003,2017) for such work on frequency standards at NIST
(National Institute of Standards and Technology), Boulder,
Colorado, USA (United States of America), or Calonico et al.
(2007) for similar activities for the Italian metrology institute
INRIM (Istituto Nazionale di Ricerca Metrologica), Torino.
More information on time metrology, the general relativity
framework, and international timescales (especially TAI) is
given in the review papers by Guinot (2011), Guinot and
Arias (2005), Arias (2005), and Petit et al. (2014).
Beyond this, optical clock networks and corresponding
link technologies are currently established or are under dis-
cussion, from which distinct advantages are expected for the
dissemination of time, geodesy, astronomy and basic and
applied research; for a review, see Riehle (2017). Also under
investigation is the installation of optical master clocks in
space, where spatial and temporal variations of the Earth’s
gravitational field are smoothed out, such that the space
clocks could serve as a reference for ground-based optical
clock networks (e.g. Gill et al. 2008;Bongs et al. 2015;
Schuldt et al. 2016). However, clock networks as well as
their possible impact on science, including the establishment
of a new global height reference system, are beyond the scope
of this work.
The main objectives of this contribution are to review the
scientific background of international timescales and geode-
tic methods to determine the relativistic redshift, to discuss
the conditions and requirements to be met, and to provide
some practical results for the state of the art, aiming at
the geodetic community on the one hand and the physics,
time, frequency, and metrology community on the other
hand. Section 2starts with an introduction to the relativistic
background of reference systems and timescales, includ-
ing proper time and coordinate time, the relativistic redshift
effect, as well as TT and the realization of international
timescales. Section 3describes geodetic methods for deter-
mining the gravity potential, considering both the geometric
levelling approach and the GNSS/geoid approach, together
with corresponding uncertainty considerations. In Sect. 4,
some practical redshift results based on geometric level-
ling and the GNSS/geoid approach are presented for three
clock sites in Germany (Braunschweig, Hannover, Garch-
ing near Munich) and one site in France (Paris). Section 5
contains a discussion of the results obtained and some con-
clusions. Further details on some fundamentals of physical
geodesy, regional gravity field modelling, the implementa-
tion of geometric levelling and GNSS observations in general
and specifically for the clock sites in Germany and France, as
well as a list of the abbreviations used are given in “Appen-
dices 1–5”.
2 Relativistic background of reference systems and
timescales
2.1 Spacetime reference systems
Measurement techniques in metrology, astronomy, and space
geodesy have reached accuracies that require routine mod-
elling within the general relativity rather than the Newtonian
framework. Einstein’s GRT is founded on spacetime as a
four-dimensional manifold, the equivalence principle, and
the Einstein field equations, including the postulate of a finite
invariant speed of light. The GRT is a metric theory with an
associated metric tensor, which explains gravitation via the
curvature of four-dimensional spacetime (although occasion-
ally also called a “fictitious force”); spacetime curvature tells
matter how to move, and matter tells spacetime how to curve
(e.g. Misner et al. 1973). Based on the equivalence principle,
no privileged reference systems exist within the framework
123
H. Denker et al.
of GRT, but local inertial systems may be constructed in any
sufficiently small (infinitesimal) region of spacetime; in other
words, all local, freely falling, non-rotating laboratories are
fully equivalent for the performance of physical experiments.
In contrast to this, Newtonian mechanics describe gravitation
as a force caused by matter with an associated Newtonian
potential and presume the existence of universal absolute
time and three-dimensional (Euclidean) space, i.e. globally
preferred inertial (Galilean) coordinate systems with a direct
physical meaning exist (where Newton’s laws of mechan-
ics apply everywhere), and idealized clocks show absolute
time everywhere in space. However, considering the most
accurate measurements today (e.g. in time and frequency
metrology), Newtonian theory is unable to describe fully the
effects of gravitation, even in a weak gravitational field and
when objects move with low velocities in the field.
Spacetime is a four-dimensional continuum, in which
points, denoted as events, can be located by their coordi-
nates; these can be in general four real numbers, but usually
three spatial coordinates and one time coordinate are consid-
ered. A world line is the unique path that an object travels
through spacetime, and the world lines of freely falling point
particles or objects are geodesics in curved spacetime. Usu-
ally, spacetime coordinates are curvilinear and have no direct
physical meaning, and according to the principle of covari-
ance, different reference systems may be chosen to model
observations and to describe the outcome of experiments.
This freedom to choose the reference system can be used to
simplify the models or to make the resulting parameters more
physically adequate (Soffel et al. 2003). With regard to the
terminology, it is fundamental to distinguish between a “ref-
erence system”, which is based on theoretical considerations
or conventions, and its realization, the “reference frame”, to
which users have access, e.g. in the form of position cata-
logues. Furthermore, it is important to distinguish between
proper and coordinate quantities (Wolf 2001); proper quan-
tities (e.g. proper time and length) are the direct result of
local measurements, while coordinate quantities are depen-
dent on conventional choices, e.g. a spacetime coordinate
system or a convention for synchronization (Petit and Wolf
2005). In addition, due to the curvature of spacetime, the
relation between proper and coordinate quantities is in gen-
eral not constant and depends on the position in spacetime,
in contrast to Newtonian theory.
In order to adequately describe modern observations in
astronomy, geodesy, and metrology, several relativistic ref-
erence systems are needed. This was first recognized by the
IAU with Resolution A4 (1991). The corresponding IAU
Resolution B1 (2000) extended and clarified the relativistic
framework, providing the relativistic definition of the BCRS
(Barycentric Celestial Reference System) and the GCRS
with origins at the solar system barycentre and the geocen-
tre of the Earth, respectively, and including the choice of
harmonic coordinates, the definition of corresponding met-
ric tensors and timescales, and the relevant four-dimensional
spacetime transformations. The BCRS with coordinates (t,
x), where tis the Barycentric Coordinate Time (TCB), is
useful for modelling the motion of bodies within the solar
system, while the GCRS with coordinates (T,X), with T
being TCG, is appropriate for modelling all processes in the
near-Earth environment, including the Earth’s gravity field
and the motion of Earth’s satellites. The BCRS can be con-
sidered to a good approximation as a global quasi-inertial
system, while the GCRS can be regarded as a local system
(Müller et al. 2008). While the BCRS and GCRS provide the
general conceptual (relativistic) framework for a barycentric
and a geocentric reference system, the absolute (spatial) ori-
entation of either system was first left open and then fixed
by employing the International Celestial Reference System
(ICRS), as recommended in the IAU Resolutions B2 (1997)
and B2 (2006). The ICRS is maintained by the International
Earth Rotation Service (IERS) and realized by the Interna-
tional Celestial Reference Frame (ICRF) as a catalogue of
adopted positions of extragalactic radio sources observed by
Very Long Baseline Interferometry (VLBI).
The GCRS also provides the foundation for the Interna-
tional Terrestrial Reference System (ITRS), and both systems
differ by just a (time-dependent) spatial rotation (or a series
of rotations); accordingly the ITRS coordinate time coincides
with TCG (e.g. Soffel et al. 2003;Kaplan 2005). The spatial
rotation involves the Earth orientation parameters (EOP), and
according to the IAU 2000 and 2006 resolutions, the transfor-
mation between both systems is based on an “intermediate
system” with the Celestial Intermediate Pole (CIP, pole of
the nominal rotation axis), as well as precession, nutation,
frame bias, Earth Rotation Angle (ERA), and polar motion
parameters; for details, see the IERS Conventions 2010 (Petit
and Luzum 2010). The ITRS is an “Earth-fixed” system,
co-rotating with the Earth in its diurnal motion in space,
in which points at the solid Earth’s surface undergo only
small variations with time (e.g. due to geophysical effects
related to tectonics); it is therefore a convenient choice for
all disciplines requiring terrestrial positions, such as geodesy,
navigation, and geographic information services. The ITRS
origin is at the centre of mass of the whole Earth includ-
ing its oceans and atmosphere (geocentre), the scale unit of
length is the metre (SI), the scale is consistent with TCG,
the orientation is equatorial and initially given by the Bureau
International de l’Heure (BIH) terrestrial system at epoch
1984.0, and the time evolution of the orientation is ensured by
using a no-net-rotation condition with regard to the horizontal
tectonic motions over the whole Earth. The Z-axis is directed
towards the IERS reference pole (i.e. the mean terrestrial
North Pole), and the axes Xand Yspan the equatorial plane,
with the X-axis being defined by the IERS reference meridian
(Greenwich), such that the coordinate triplet X,Y,Zforms
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
a right-handed Cartesian system. The International Union
of Geodesy and Geophysics (IUGG) formally adopted the
ITRS at its General Assembly in 2007, with the IERS being
the responsible body.
The ITRS is realized by the International Terrestrial Refer-
ence Frame (ITRF), which consists of the three-dimensional
positions and velocities of stations observed by space geode-
tic techniques, where the positions are regularized in the
sense that high-frequency time variations are removed by
conventional corrections. These corrections are mainly geo-
physical ones, such as solid Earth tides and ocean tidal
loading (for full details, see the IERS Conventions 2010,
Petit and Luzum 2010); the purpose of these corrections is
to obtain positions with more regular time variation, which
better conform to the linear time-variable coordinate mod-
elling approach used and thus improve the transformation
to a certain reference epoch (to obtain a quasi-static state).
The most recent realization of the ITRS is the ITRF2014
(Altamimi et al. 2016); however, all results in the present
paper are based on the previous realization ITRF2008 with
the reference epoch 2005.0. The uncertainty (standard devi-
ation) of the geocentric Cartesian coordinates (X,Y,Z)is at
the level of 1 cm or better; for further details, see Petit and
Luzum (2010).
Besides the ITRS and its frames (ITRF), various other
national, regional, and global systems are in use. Of some
relevance are the World Geodetic System 1984 (WGS84) for
GPS users, and the European Terrestrial Reference System
(ETRS) for European users. While WGS84 is intended to
be as closely coincident as possible with the ITRS (the lat-
est realization, i.e. “Reference Frame G1762”, agrees with
the ITRF at the level of 1 cm; NGA 2014; NGA—National
Geospatial-Intelligence Agency, USA), the ETRS89 refer-
ence system is attached to the stable part of the Eurasian plate
in order to compensate for the movement of the Eurasian tec-
tonic plate (of roughly 2.5 cm per year). The latter reference
system and corresponding frames (ETRF, European Terres-
trial Reference Frame) are implemented in most European
countries, as this results in much smaller station velocities.
However, for global work in connection with international
timescales, the ITRS and corresponding reference frames
should be employed.
2.2 Proper time and coordinate time
For time metrology, the most fundamental quantities are the
proper time, as observed by a local ideal clock, and the coor-
dinate time of a conventional spacetime reference system.
The relation between both quantities can be derived in general
from the relativistic line element dsand the coordinate-
dependent metric tensor gαβ , whose components are in
general not globally constant. Hence, the measured proper
quantities between two events (e.g. proper time observed by
an ideal clock) depend in principle on the path followed by a
particle (e.g. a clock) between these two events. Considering
spacetime coordinates xγ=(x0,x1,x2,x3)with x0=ct,
where cis the speed of light in vacuum, and tis the coor-
dinate time, the line element along a time-like world line is
given by
ds2=gαβ (xγ)dxαdxβ=−c2dτ2,(1)
where τis the proper time along that world line. In this con-
text, Einstein’s summation convention over repeated indices
is employed, with Greek indices ranging from 0 to 3 and
Latin indices taking values from 1 to 3. The relation between
proper and coordinate time is obtained by rearranging the
above equation, resulting in
dτ
dt2
=−g00 −2g0i
1
c
dxi
dt−gij
1
c2
dxi
dt
dxj
dt
=−g00 −2g0i
vi
c−gij
vivj
c2,(2)
where vi(t)is the coordinate velocity along the path xi(t).
Inserting the GCRS metric, as recommended by IAU Res-
olution B1 (2000), and using a binomial series expansion lead
to
dτ
dT(TCG)=1−1
c2V+v2
2+O(c−4)
=1−1
c2VE+Vext +v2
2+O(c−4). (3)
In this equation, Vis the usual gravitational scalar potential
(denoted as “W” in the IAU 2000 resolutions, a general-
ized Newtonian potential), which is split up into two parts
arising from the gravitational action of the Earth itself (VE)
and external parts due to tidal and inertial effects (Vext =
Vtidal +Viner), and vis the coordinate velocity of the observer
in the GCRS, while terms of the order c−4are omitted. In this
context, it should be noted that all potentials are defined here
with a positive sign, which is consistent with geodetic prac-
tice, but in contrast to most physics literature, where usually
the opposite sign (conceptually closer to potential energy) is
employed (Jekeli 2009). The above equation corresponds to
the first post-Newtonian approximation and is accurate to a
few parts in 1019 for locations from the Earth’s surface up
to geostationary orbits, which is fully sufficient, as in prac-
tice the limiting factor is the uncertainty with which VEcan
be determined in the vicinity of the Earth. Consequently,
contributions from V2/c4and the so-called gravitomagnetic
vector potential (with the notation chosen in formal analogy
to classical magnetism theory) have been neglected in the
above equation as they do not exceed a few parts in 1019,
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H. Denker et al.
while the inertial terms in Vext remain below 2 parts in 1020;
for further details, see Soffeletal.(2003).
Regarding the (local) gravitational potential of the Earth
(VE)within an Earth-fixed system, for many applications it is
advantageous to utilize series expansions (with basically con-
stant coefficients), which usually converge for points outside
the Earth’s surface. Within the relativistic context, multipole
expansions, which have great similarities with corresponding
Newtonian series, are very useful. These presently have more
than sufficient accuracy and lead in the end to the well-known
spherical harmonic expansion; however, in contrast to the
classical theory, all relevant parameters have to be interpreted
within a relativistic scope. A summary of this approach,
including a detailed discussion on the post-Newtonian inter-
pretation and neglected terms, can be found in Soffeletal.
(2003).
Equation (3) shows that the proper time interval dτ, sep-
arating two events, is less than the corresponding coordinate
time interval dT, by an amount that depends on the gravi-
tational potential V(zero at infinity, increasing towards the
attracting masses) and the velocity vrelative to the chosen
reference system. Consequently, when compared to coordi-
nate time, clocks run slower (tick slower, show less time)
when they move or are affected by gravitation; this slow-
ing of time is called time dilation and may be separated into
a (relative) velocity and a gravitational time dilation, some-
times also called special and general relativistic time dilation,
respectively, where the special relativistic time dilation is
measurable by the transverse (or second-order) Doppler shift.
On the other hand, for a stationary clock at infinity (i.e. v=0
and V=0, a “distant” observer at rest), the proper time inter-
val approaches the coordinate time interval (dτ∞=dT), and
hence offers a way, in principle at least, to directly observe
coordinate time.
2.3 The gravitational redshift effect
The gravitational time dilation is closely related to the grav-
itational redshift effect, which considers an electromagnetic
wave (light) travelling from an emitter (em—located at A)
to a receiver (rec—located at B) in conjunction with two
ideal (zero uncertainty) frequency standards (clocks) at A
and B, measuring proper time. Here the ideal clocks should
show the same time under the same conditions, e.g. they
should be atomic or nuclear clocks based on the emission of
an electromagnetic wave at a certain (natural) frequency. In
textbooks (see, for example, Cheng 2005;Lambourne 2010;
Misner et al. 1973;Moritz and Hofmann-Wellenhof 1993;
Schutz 2003,2009;Will 1993) the redshift effect is usu-
ally explained by assuming a stationary gravitational field
(with time-independent metric) as well as a static emitter
and receiver (v=0), and for better illustration, it is mostly
supposed (although not required) that both the emitter and
receiver are positioned in the same vertical one above the
other, noting that also the famous Pound and Rebka experi-
ment was carried out in this way (Pound and Rebka 1959).
Under these circumstances (stationary gravitational field,
static scenario), the trajectories (world lines) of successive
wave crests of the emitted signal are identical, and hence,
expressed in coordinate time, the interval between emission
and reception (observation) of successive wave crests is the
same (dtem =dtrec). However, the ideal clocks at the points
of emission (A) and reception (B) of the wave crests are mea-
suring proper time, i.e. based on Eqs. (2) and (3), the lower
clock runs slower than the corresponding upper clock; in
other words, an observer at the upper station will find that the
lower clock is running slow with respect to his own clock. On
the other hand, since the (proper) frequency being inversely
proportional to the proper time interval with f=1/dτ,Eq.
(2) can be used (together with dtem =dtrec) to derive
dτrecdtrec
dτemdtem =dτrec
dτem =fem
frec =√−(g00)rec
√−(g00)em
,(4)
where fem and frec are the proper frequencies of the light as
observed at points A(em) and B(rec) by the corresponding
ideal clocks, respectively. Rearranging the above relationship
and considering Eq. (3)give
f
frec =frec −fem
frec =1−fem
frec =1−1−Vrecc2
1−Vemc2
=Vrec −Vem
c2+Oc−4≈−bH
c2,(5)
where gravitational potential terms of the order V2/c4have
been neglected, while bis the gravitational acceleration, and
His the vertical distance between points A(em) and B
(rec) counted positive upward. Here it is worth mention-
ing again that the above equation holds for two arbitrary
points with corresponding gravitational potentials, while
only the rightmost part of the equation depends (to some
extent) on the assumption that both points be in the same
vertical. Hence, in the following discussion, the meaning
of “above” and “below” relates to two arbitrary points on
corresponding equipotential surfaces above or below each
other. For the case where point B(rec) is located above
point A(em), the potential difference Vrec −Vem is negative,
and hence, f=frec −fem is negative, i.e. the received
(observed) frequency frec is lower than the corresponding
emitted frequency fem, and thus, blue light becomes more
red, explaining the term “redshift effect”. On the other hand,
if point B(rec) is below point A(em), i.e. the light signal is
sent from the top to the bottom station, Vrec −Vem is positive,
and hence, f=frec −fem is also positive, leading to an
increase in frequency and thus a “blueshift”. Finally, instead
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
of using frequencies, the redshift effect can also be formu-
lated in terms of corresponding wavelengths, and Eq. (5) can
be extended for moving observers by introducing velocities
v.
It is also noteworthy that the redshift and the time dila-
tion effect can be derived solely on the basis of the (weak)
equivalence principle (physics in a frame freely falling in a
gravitational field is equivalent to physics in an inertial frame
without gravitation) and the energy conservation law; in other
words, a photon climbing in the (Earth’s) gravitational field
will lose energy and will consequently be redshifted. On the
other hand, any theory of gravitation can predict the redshift
effect if it respects the equivalence principle, i.e. gravita-
tional redshift experiments can be considered as tests of the
equivalence principle.
2.4 Time and the gravity potential
The fundamental relationship between proper time and coor-
dinate time according to Eq. (3) refers to the (non-rotating)
GCRS and therefore all quantities depend on the coordinate
time T(TCG) due to Earth rotation. However, for many
practical applications it is more convenient to work with an
Earth-fixed system (e.g. ITRS, co-rotating with the Earth),
which can be considered as static in the first instance. Then,
for an observer (clock) at rest in an Earth-fixed system, the
velocity vin Eq. (3) is simply given by v=ωp, where ω
is the angular velocity about the Earth’s rotation axis, and
pis the distance from the rotation axis. Taking all these
into account, Eq. (3) may be rearranged and expressed in
the Earth-fixed system (for an observer at rest) as
dτ
dT(TCG)=1−1
c2W(t)+O(c−4), (6)
where W(t)is the slightly time-dependent (Newtonian) grav-
ity potential related to the Earth-fixed system, as employed
in classical geodesy. W(t)may be decomposed into
W(t)=Wstatic(t0)+Wtemp(t−t0), (7)
where Wstatic(t0)is the dominant static (spatially vari-
able) part of the gravity potential at a certain reference
epoch t0, while Wtemp(t)incorporates all temporal compo-
nents of the gravity potential (inclusively tidal effects) and
indeed contains the temporal variations of all three terms
(VE,Vext,v
2/2) in Eq. (3); for a discussion of the relevant
terms and magnitudes, see, for example, Wolf and Petit
(1995), Petit et al. (2014) and Voigt et al. (2016). The accu-
racy of Eq. (6) is similar to Eq. (3), where terms below a few
parts in 1019 have been neglected. If the observer is not at
rest within the Earth-fixed system, then two more terms enter
in Eq. (6), see, for example, Nelson (2011).
The focus here is on the static part of the gravity potential,
denoted simply as Win the following, which is defined as
the sum of the gravitational potential VEand the centrifugal
potential ZEin the form (see also “Appendix 1”)
W=VE+ZE,ZE=ω2
2p2,(8)
showing that the term v2/2inEq.(3) is exactly the centrifugal
potential ZEas defined above. In general, the main advantage
of the Earth-fixed reference system is that all gravity field
quantities as well as station coordinates can be regarded as
time-independent in the first instance, assuming that (small)
temporal variations can be taken into account by appropriate
reductions or have been averaged out over sufficiently long
time periods.
The largest component in Wtemp is due to solid Earth
tide effects, which lead to predominantly vertical movements
of the Earth’s surface with a global maximum amplitude
of about 0.4 m (equivalent to about 4m2s−2in potential).
The next largest contribution is the ocean tide effect with
a magnitude of roughly 10–15% of the solid Earth tides,
but with significantly increased values towards the coast. All
other time-variable effects are a further order of magnitude
smaller and originate from atmospheric mass movements (on
a global scale, ranging from hourly to seasonal variations),
hydro-geophysical mass changes (on regional and continen-
tal scales, seasonal variations), and polar motion (pole tides).
For details regarding the computation, magnitudes, and main
time periods of all relevant time-variable components, see
Voigt et al. (2016). Based on this, the temporal and static
components can be added according to Eq. (7) to obtain the
actual gravity potential value W(t)at time t, as needed, for
example, for the evaluation of clock comparison experiments.
In this context, geodynamic effects also lead to additional
velocity contributions, and the effect of these on time and
frequency comparisons has to be considered (see Gersl et al.
2015). For instance, Earth tides (as by far the largest time-
variable component) lead to periodic variations of the Earth’s
surface with amplitudes up to about 0.4 m and time periods
of roughly 12 h, associated with maximum kinematic veloc-
ities of about 5 ×10−5ms−1; hence, they contribute well
below 10−25 to the (rightmost) second-order Doppler shift
term in Eq. (3), which depends on the square of the veloc-
ity, and therefore can safely be neglected in the foreseeable
future. Corresponding contributions to the (classical) first-
order Doppler effect may be significant, as mentioned in Mai
(2013), but fortunately first-order Doppler shifts do not play
a role in optical clock comparisons through (two-way) fibre
links, which presently is the only technique that has achieved
link performances at the level of 1 part in 1018 and below
(Droste et al. 2013).
123
H. Denker et al.
2.5 Terrestrial Time and the realization of international
timescales
The development of atomic timekeeping is nicely described
in the SI Brochure Appendix 2 (Practical realization of
the definition of the unit of time; http://www.bipm.org/
en/publications/mises-en-pratique/; BIPM—Bureau Inter-
national des Poids et Mesures) as well as in the review papers
from Guinot (2011), Arias (2005), and Guinot and Arias
(2005). The unit of time is the SI second, which is based
on the value of the caesium ground-state hyperfine transition
frequency, as adopted by the 13th Conférence Générale de
Poids et Mesures (CGPM) in 1967 (Terrien 1968). This def-
inition should be understood as the definition of the unit of
proper time (within a sufficiently small laboratory). The SI
second is realized by different caesium standards in national
metrology institutes (NMIs), and an expedient combination
of all the results (including results from a number of hydro-
gen masers) leads to an ensemble timescale, denoted as TAI.
The 14th CGPM approved TAI in 1971 (Terrien 1972) and
endorsed the definition proposed by the Comité Consultatif
pour la Définition de la Seconde (CCDS) in 1970, which
states that TAI is established from “atomic clocks operat-
ing ... in accordance with the definition of the second”. In
the framework of general relativity, this definition was com-
pleted by the CCDS in 1980, stating that “TAI is a coordinate
time scale defined in a geocentric reference frame with the
SI second as realized on the rotating geoid as the scale unit”
(Giacomo 1981). This definition was amplified in the context
of IAU Resolution A4 (1991), stating that TAI is considered
as a realized timescale whose ideal form is TT. While only a
best estimate for the constant rate between TT and TCG was
given in the IAU 1991 resolution, this rate was fixed in the
year 2000 by a further IAU Resolution B1.9 (2000), giving
dT(TT)
dT(TCG)=1−LG,LG=6.969290134 ×10−10,(9)
where LGis now a defining constant with a fixed value and
no uncertainty. The main idea behind the IAU 2000 defini-
tion of TT is to produce a timescale for the construction and
dissemination of time for all practical purposes at or near the
Earth’s surface, which is consistent with the previous defi-
nitions, but avoids explicit mention of the geoid due to the
intricacy and problems associated with the definition, real-
ization, and changes of the geoid. Of course, the numerical
value of LGwas chosen to maintain continuity with the pre-
vious definition, and as LG=W0/c2according to Eq. (6),
this implicitly defines a zero gravity potential value of
W0=62,636,856.00 m2s−2.(10)
As the speed of light cis also fixed (299,792,458 ms−1)
and has no uncertainty, the parameters LGand W0can be
considered as equivalent, both having zero uncertainty. In
this context, Pavlis and Weiss (2003) somewhat misleadingly
mention that the zero potential (W0)uncertainty “implies a
limitation on the realization of the second” and therefore “the
second can be realized to no better than ±1×10−17”.
The above zero gravity potential value W0was the best
estimate at that time, and it is also listed in the IERS con-
ventions 2010 (Petit and Luzum 2010) as the value for the
“potential of the geoid”. Although the latest definition of
TT gets along without the geoid and a numerical value for
W0, the realization of TT is linked to signal frequencies that
an ideal clock would generate on the zero level reference
surface, the latter being implicitly defined by the constant
LG. This leads to the relativistic redshift correction, anal-
ogous to Eq. (5), giving for a clock at rest on the Earth’s
surface
f
fP=fP−f0
fP=1−dτP
dτ0=WP−W0
c2+O(c−4)≈−gH
c2,
(11)
where fPand f0are the proper frequencies of an electromag-
netic wave as measured at points Pat the Earth’s surface and
P0on the zero level surface, respectively (cf. Sect. 2.3), while
WPand W0are the corresponding gravity potential values,
His the vertical distance of point Prelative to point P0,
measured positive upwards, and gis the gravity accelera-
tion. An exact relation for the potential difference is given by
W0−WP=CP=gH =γHN, where CPis the geopoten-
tial number, and Hand HNare the orthometric and normal
heights with corresponding mean gravity and normal gravity
values gand γ, respectively (for further details, see Sect. 3.1
and Torge and Müller 2012).
The above equation assumes that the two clocks at Pand
P0are earthbound and at rest within the (rotating) Earth-
fixed system, i.e. both points are affected by the Earth’s
gravity (gravitational plus centrifugal) field and relative
velocities between them are non-existent. Therefore, the term
“relativistic redshift effect” is preferred over “gravitational
redshift effect”, as not just the gravitational, but also the cen-
trifugal potential is involved; similar conclusions are drawn
by Delva and Lodewyck (2013), Pavlis and Weiss (2003)
and Petit and Wolf 1997. Furthermore, the sign considera-
tions related to Eq. (5) apply equally to the above equation
(11), as the gravity potential Wis dominated by the gravi-
tational part V. Assuming again that Pis located above P0,
WP−W0is negative and so is f, i.e. the received (observed)
frequency fPis lower than the emitted frequency f0, which
explains the term “redshift effect” (cf. Sect. 2.3). Regard-
ing international timescales, realizing TT, the frequency or
rate of an individual clock at Pabove the zero level surface
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Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
must be reduced by an amount f, given by Eq. (11), in
order to produce the desired signal corresponding to a hypo-
thetical clock at the zero level surface. In other words, the
clock frequency at Phas to be adjusted such that it will
no longer observe a redshift of the clock signal emitted at
P0. As all existing primary frequency standards are situ-
ated above the zero level surface, in practice, the relativistic
redshift correction is always negative and can become quite
significant; for example, Pavlis and Weiss (2003,2017) esti-
mated a correction f/fof about −1800 ×10−16 for the
clocks at NIST in Boulder, Colorado, USA, with an alti-
tude of about 1650 m. Furthermore, for the realization of
TT based on Eq. (11), in principle only the absolute gravity
potential (W)at the clock and its uncertainty matters, as the
parameters LGand W0are equivalent and both have zero
uncertainty.
As mentioned above, TAI is a realization of TT, a coordi-
nate time scale in the geocentric system. As UTC is directly
derived from TAI, it is by definition also a realization of TT.
Similarly, all other timescales, which aim at keeping in syn-
chrony with UTC, using coordinate synchronization, can also
be considered as realizations of TT. Such timescales include
the UTC(k), i.e. local realizations of UTC at the laboratory
k, and the reference timescales for GNSS. By construction,
these become realizations of TT without explicitly using Eq.
(11) to correct the frequency of participating clocks. At this
point, it should be noted that an ambiguity remains between
TT, now defined by IAU Resolution B1.9 (2000) without
mention of the geoid, and TAI, whose definition is still related
to “the geopotential on the geoid”. This comes from the use
of similar terms for the definition of TAI and TT before the
redefinition of TT in 2000, which meant that both TT and
TAI suffered from the uncertainty in the determination of
the geoid and the corresponding geopotential value. For this
reason, several authors (e.g. Wolf and Petit 1995) suggested
that LGshould be turned into a defining constant, a sug-
gestion eventually adopted in the IAU (2000) redefinition of
TT. However, this change was not passed into a new def-
inition of TAI. This problem is currently being addressed
by the Consultative Committee for Time and Frequency
(CCTF) and is expected to be solved with a new definition
for TAI.
With respect to the above value for W0(consistent with
the international recommendations for the definition of TT),
a further complication has emerged from the fact that the
International Association of Geodesy (IAG) has introduced
another numerical value for W0; for further details see
“Appendix 1”. Therefore, since it seems most likely that the
geodetic, time, and IAU communities will not be willing to
change their definitions in the near future, the problem of
different (conflicting) W0values must be solved by transfor-
mations and comprehensive documentation of the relevant
steps.
2.6 Orders of magnitude of relativistic terms and
chronometric geodesy
Equation (11) is the classic formula relating frequency dif-
ferences and gravity potential differences, where a fractional
frequency shift of 1 part in 1018 corresponds to about
0.1m2s−2in terms of the gravity potential difference, which
is equivalent to about 0.01 m in height. In addition, Eq.
(11) can be used to estimate the magnitude of the relativistic
redshift correction (e.g. about −1800 ×10−16 for NIST in
Boulder, see above), and it makes clear that the absolute grav-
ity potential (W) is required for contributions to international
timescales, while potential differences suffice for local and
remote clock comparisons, with the proviso that the actual
potential values and the clock frequency measurements must
refer to the same epochs. The latter requirement means that
the magnitude of time-variable effects in the gravity potential
due to solid Earth and ocean tides as well as other effects must
be taken into account for all clock measurements at a perfor-
mance level below roughly 5 parts in 1017. This is especially
important for contributions to international timescales and
remote clock comparisons over large distances in cases where
relatively short averaging times are used, since in such sit-
uations the time-variable gravity potential components may
not average out sufficiently. Moreover, the tidal peak-to-peak
signal could also prove useful for evaluating the performance
of optical clocks, by providing a detectability test.
Another important consequence of Eq. (11) is, on the one
hand, that geodetic knowledge of the Earth’s gravity poten-
tial (and corresponding gravity field related heights) can be
used to predict frequency shifts between local and remote
(optical) clocks, and vice versa, the clocks can be used to
determine gravity potential differences. To the knowledge
of the authors, the latter technique was first mentioned in
the geodetic literature by Bjerhammar (1975) within a short
section on a “new physical geodesy”. Vermeer (1983) intro-
duced the term “chronometric levelling”, while Bjerhammar
(1985) discussed the clock-based levelling approach under
the title “relativistic geodesy” and also included a defini-
tion of a relativistic geoid as the “surface closest to mean
sea level, where clocks run with the same speed”. Regard-
ing the terminology, Delva and Lodewyck (2013) consider
that “relativistic geodesy” should cover all geodetic topics
based on a relativistic framework and suggest, like Petit et al.
(2014), that the term “chronometric geodesy” should be used
for all geodetic disciplines employing (atomic) clocks. This
definition of terms is well conceived, and in this contribu-
tion, the term “chronometric levelling”—although somewhat
restrictive—is preferred, as it characterizes quite accurately
the clock-based levelling approach based on Eq. (11).
With regard to the definition of the geoid, when consid-
ering Eq. (11) and the underlying level of approximation (of
better than 10−18 in frequency or 1 cm in height), both the
123
H. Denker et al.
classical (geodetic) definition and the relativistic (chrono-
metric) definition given by Bjerhammar (1985) relate to a
selected level surface within the Earth’s gravity field, as
defined in classical physical geodesy within the Newtonian
framework. Further refinements of a relativistic geoid def-
inition are discussed, for example, in Soffel et al. (1988),
Kopeikin (1991), or Müller et al. (2008), where additional
terms at the few mm level show up, but the option for a
transformation between the different relativistic definitions
and the classical version also exists. Therefore, even if future
optical atomic clocks will (operationally) work at the level of
10−19 or below and deliver corresponding gravity potential
differences, these may still be integrated into the framework
of classical physical geodesy and gravity field modelling by
considering appropriate corrections. Consequently, as most
terrestrial geodetic applications do not require a relativistic
treatment, with only a few areas (e.g. reference systems and
time, ephemerides and satellite orbits, global geopotential
modelling in connection with satellite observations) need-
ing some relativistic background, the geodetic community is
unlikely to switch soon to a (much more complicated) fully
relativistic framework; this is because the classical Newto-
nian formulations are usually sufficient, far simpler to handle,
and only exceptionally require some relativistic corrections.
For a discussion of the classical geoid definition and differ-
ent existing numerical values for the zero level surface, see
“Appendix 1”.
3 Geodetic methods for determining the gravity
potential
This section deals with geodetic methods for determining the
gravity potential, needed for the computation of relativistic
redshift corrections for optical clock observations. The focus
is on the determination of the static (spatially variable) part
of the potential field, while temporal variations in the station
coordinates and the potential quantities are assumed to be
taken into account through appropriate reductions or by using
sufficiently long averaging times (see also Sect. 2.4). This is
common geodetic practice and leads to a quasi-static state
(e.g. by referring all quantities to a given epoch), such that the
Earth can be considered as a rigid and non-deformable body,
uniformly rotating about a body-fixed axis. Hence, all gravity
field quantities including the level surfaces are considered in
the following as static quantities, which do not change in
time.
In this context, a note on the handling of the permanent
(time-independent) parts of the tidal corrections is appropri-
ate; for details, see, for example, Mäkinen and Ihde (2009),
Ihde et al. (2008), or Denker (2013). The IAG has recom-
mended that the so-called zero-tide system should be used
(resolutions no. 9 and 16 from the year 1983; cf. Tsch-
erning 1984), where the direct (permanent) tide effects are
removed, but the indirect deformation effects associated with
the permanent tidal deformation are retained. Unfortunately,
geodesy and other disciplines do not strictly follow the IAG
resolutions for the handling of the permanent tidal effects,
and therefore, depending on the application, appropriate cor-
rections may be necessary to refer all quantities to a common
tidal system (see Sect. 4and the aforementioned references).
“Appendix 1” outlines some necessary fundamentals of
physical geodesy. This includes the introduction of the grav-
ity potential and its components according to Eq. (8)asthe
fundamental quantity, from which all other relevant gravity
field parameters can be derived, the definition of the geoid
as a selected equipotential surface and its relation to mean
sea level, as well as the choice of different (conflicting) zero
potential values (W0issue), being largely a matter of conven-
tion. In the following, two geodetic approaches for deriving
gravity potential values are discussed.
3.1 The geometric levelling approach
The classical and most direct way to obtain gravity potential
differences is based on geometric levelling and gravity obser-
vations, denoted here as the geometric levelling approach.
Based on Eq. (33) in “Appendix 1”, the gravity potential dif-
ferential can be expressed as
dW=∂W
∂xdx+∂W
∂ydy+∂W
∂zdz=gradWds =gds =−gdn,
(12)
where ds is the vectorial line element, gis the magnitude
of the gravity vector, and dnis the distance along the outer
normal of the level surface (zenith or vertical), which by
integration leads to the geopotential number Cin the form
C(i)=W(i)
0−WP=−
P
P0(i)
dW=
P
P0(i)
gdn,(13)
where Pis a point at the Earth’s surface, (i)refers to a given
height datum, and P0(i)is an arbitrary point on the selected
zero level or height reference surface (with gravity potential
W(i)
0). Thus, in addition to the raw levelling results (dn),
gravity observations (g) are needed along the path between
P0(i)and P. The spacing and uncertainty required for these
gravity points is discussed in standard geodesy textbooks,
e.g. Torge and Müller (2012). The geopotential number Cis
defined such that it is positive for points Pabove the zero
level surface, similar to heights. The zero level surface and the
corresponding potential are typically selected in an implicit
way by connecting the levelling to a fundamental national
tide gauge, but the exact numerical value of the reference
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
potential W(i)
0is usually unknown. As mean sea level deviates
from a level surface within the Earth’s gravity field due to the
dynamic ocean topography, this leads to inconsistencies of
more than 0.5 m between different national height systems
across Europe, the extreme being Belgium, which differs by
more than 2 m from all other European countries due to the
selection of low tide water as the reference (instead of mean
sea level).
Geometric levelling (also called spirit levelling) itself is
a quasi-differential technique, which provides height differ-
ences δn(backsight minus foresight reading) with respect to
a local horizontal line of sight. The uncertainty of geomet-
ric levelling is rather low over shorter distances, where it can
reach the sub-millimetre level, but it is susceptible to system-
atic errors up to the decimetre level over 1000 km distance
(see also Sect. 3.3). In addition, the non-parallelism of the
level surfaces cannot be neglected over larger distances, as
it results in a path dependence of the raw levelling results
(dn= 0), but this problem can be overcome by using
potential differences, which are path-independent because
the gravity field is conservative (dW=0). For this rea-
son, geopotential numbers are almost exclusively used as the
foundation for national and continental height reference sys-
tems (vertical datum) worldwide, but one can also work with
heights and corresponding gravity corrections to the raw lev-
elling results (cf. Torge and Müller 2012).
Although the geopotential numbers are ideal quantities
for describing the direction of water flow, they have the unit
m2s−2and are thus somewhat inconvenient in disciplines
such as civil engineering. A conversion to metric heights is
therefore desirable, which can be achieved by dividing the
Cvalues by an appropriate gravity value. Widely used are
the orthometric heights (e.g. in the USA, Canada, Austria,
and Switzerland) and normal heights (e.g. in Germany and
many other European countries). Heights also play an impor-
tant role in gravity field modelling due to the strong height
dependence of various gravity field quantities.
The orthometric height His defined as the distance
between the surface point Pand the zero level surface
(geoid), measured along the curved plumb line, which
explains the common understanding of this term as “height
above sea level” (Torge and Müller 2012). The orthometric
height can be derived from Eq. (13) by integrating along the
plumb line, giving
H(i)=C(i)
g,g=1
H(i)
H(i)
0
gdH,(14)
where gis the mean gravity along the plumb line (inside the
Earth). As gcannot be observed directly, hypotheses about
the interior gravity field are necessary, which is one of the
main drawbacks of the orthometric heights. Therefore, in
order to avoid hypotheses about the Earth’s interior gravity
field, the normal heights HNwere introduced by Moloden-
sky (e.g. Molodenskii et al. 1962) in the form
HN(i)=C(i)
γ,γ=1
HN(i)
HN(i)
0
γdHN,(15)
where γis a mean normal gravity value along the normal
plumb line (within the normal gravity field, associated with
the level ellipsoid), and γis the normal gravity acceleration
along this line. Consequently, the normal height HNis mea-
sured along the slightly curved normal plumb line (Torge and
Müller 2012).
While the orthometric and normal heights are related to
the Earth’s gravity field, the ellipsoidal heights h, as derived
from GNSS observations, are purely geometric quantities,
describing the distance (along the ellipsoid normal) of a point
Pfrom a conventional reference ellipsoid. As the geoid and
quasigeoid serve as the zero height reference surface (vertical
datum) for the orthometric and normal heights, respectively,
the following relation holds
h=H(i)+N(i)=HN(i)+ζ(i),(16)
where N(i)is the geoid undulation, and ζ(i)is the quasigeoid
height or height anomaly; for further details on the geoid and
quasigeoid (height anomalies) see, for example, Torge and
Müller (2012). Equation (16) neglects the fact that strictly the
relevant quantities are measured along slightly different lines
in space, but the maximum effect is only at the sub-millimetre
level (for further details cf. Denker 2013).
Lastly, the geometric levelling approach gives only grav-
ity potential differences, but the associated constant zero
potential W(i)
0can be determined by at least one (better sev-
eral) GNSS and levelling points in combination with the
(gravimetrically derived) disturbing potential, as described
in the next section. Rearranging the above equations gives
the desired gravity potential values in the form
WP=W(i)
0−C(i)=W(i)
0−¯gH(i)=W(i)
0−¯γHN(i),
(17)
and hence the geopotential numbers and the heights H(i)and
HN(i)are fully equivalent.
3.2 The GNSS/geoid approach
For the determination of the gravity potential W, one of the
primary goals of geodesy, gravity measurements form one
of the most important data sets. However, since gravity (rep-
resented as g=|g|= length of the gravity vector g) and
123
H. Denker et al.
other relevant observations depend in general in a nonlinear
way on the potential W, the observation equations must be
linearized by introducing an a priori known reference poten-
tial as well as a corresponding reference surface (positions).
Usually, the normal gravity field related to the level ellip-
soid is employed for this, requiring that the ellipsoid surface
is a level surface of its own gravity field. The level ellip-
soid is chosen as a conventional system, because it is easy
to compute (from just four fundamental parameters, e.g. two
geometrical parameters for the ellipsoid plus the total mass
Mand the angular velocity ω), useful for other disciplines,
and also utilized for describing station positions. However,
today spherical harmonic expansions based on satellite data
could also be employed (cf. Denker 2013).
The linearization process leads to the disturbing (or
anomalous) potential Tdefined as
TP=WP−UP,(18)
where Uis the normal gravity potential associated with the
level ellipsoid. Accordingly, the gravity vector and other
parameters are approximated by corresponding reference
quantities, leading to gravity anomalies, gravity disturbances,
vertical deflections, height anomalies, geoid undulations, etc.
(cf. Torge and Müller 2012). The main advantage of the lin-
earization process is that the residual quantities (with respect
to the known reference field) are in general four to five orders
of magnitude smaller than the original ones, and in addition
they are less position-dependent.
Accordingly, the gravity anomaly is given by
gP=gP−γQ=−∂T
∂h+1
γ
∂γ
∂hT−1
γ
∂γ
∂hW(i)
0−U0,
(19)
where gPis the gravity acceleration at the observation point
P(at the Earth’s surface or above), γQis the normal gravity
acceleration at a known linearization point Q(telluroid, Qis
located on the same ellipsoidal normal as Pat a distance HN
above the ellipsoid, or equivalently UQ=WP; for further
details, see Denker 2013), the derivatives are with respect
to the ellipsoidal height h, and δW(i)
0=W(i)
0−U0is the
potential difference between the zero level height reference
surface (W(i)
0) and the normal gravity potential U0at the
surface of the level ellipsoid. Equation (19) is also denoted as
the fundamental equation of physical geodesy; it represents
a boundary condition that has to be fulfilled by solutions of
the Laplace equation for the disturbing potential T, sought
within the framework of geodetic boundary value problems
(GBVPs). Moreover, the subscripts Pand Qare dropped on
the right side of Eq. (19), noting that it must be evaluated at
the known telluroid point (boundary surface).
In a similar way, the height anomaly is obtained by Bruns’s
formula as
ζ(i)=h−HN(i)=T
γ−W(i)
0−U0
γ=T
γ−δW(i)
0
γ=ζ+ζ(i)
0.
(20)
which also implies that ζ(i)and ζare associated with cor-
responding zero level surfaces W=W(i)
0and W=U0,
respectively. The δW(i)
0term is also denoted as height sys-
tem bias and is frequently omitted in the literature, implicitly
assuming that W(i)
0equals U0. However, when aiming at a
consistent derivation of absolute potential values, the δW(i)
0
term has to be taken into consideration.
Accordingly, the disturbing potential Ttakes over the role
of Was the new fundamental target quantity, to which all
other gravity field quantities of interest are related. As Thas
the important property of being harmonic outside the Earth’s
surface and regular at infinity, solutions of Tare developed in
the framework of potential theory and GBVPs, i.e. solutions
of the Laplace equation are sought that fulfil certain bound-
ary conditions. Now, the first option to compute Tis based on
the well-known spherical harmonic expansion, using coeffi-
cients derived from satellite data alone or in combination with
terrestrial data (e.g. EGM2008; EGM—Earth Gravitational
Model; Pavlis et al. 2012), yielding
T(θ , λ, r)=
nmax
n=0a
rn+1n
m=−n
TnmYnm (θ , λ) (21)
with
Ynm(θ , λ) =Pn|m|(cos θ) cos mλ
sin |m|λ,
Tnm =GM
aCnm
Snm for m≥0
m<0,(22)
where (θ , λ, r)are spherical coordinates, n,mare integers
denoting the degree and order, GM is the geocentric gravita-
tional constant (gravitational constant Gtimes the mass of the
Earth M), ais in the first instance an arbitrary constant, but
is typically set equal to the semimajor axis of a reference
ellipsoid, Pnm(cos θ) are the fully normalized associated
Legendre functions of the first kind, and Cnm,Snm are
the (fully normalized) spherical harmonic coefficients (also
denoted as Stokes’s constants), representing the difference
in the gravitational potential between the real Earth and the
level ellipsoid.
Regarding the uncertainty of a gravity field quantity com-
puted from a global spherical harmonic model up to some
fixed degree nmax, the coefficient uncertainties lead to the
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
so-called commission error based on the law of error propa-
gation, and the omitted coefficients above degree nmax, which
are not available in the model, lead to the corresponding omis-
sion error. With dedicated satellite gravity field missions such
as GRACE (Gravity Recovery and Climate Experiment) and
GOCE (Gravity Field and Steady-State Ocean Circulation
Explorer), the long-wavelength geoid and quasigeoid can
today be determined with low uncertainty, e.g. about 1 mm
at 200 km resolution (n=95) and 1 cm at 150 km resolution
(n=135) from GRACE (e.g. Mayer-Gürr et al. 2014), and
1.5 cm at about 110 km resolution (n=185) from GOCE
(e.g. Mayer-Gürr et al. 2015;Brockmann et al. 2014). How-
ever, the corresponding omission error at these wavelengths
is still quite significant with values at the level of several
decimetres, e.g. 0.94 m for n=90, 0.42 m for n=200, and
0.23 m for n=360. For the ultra-high-degree geopotential
model EGM2008 (Pavlis et al. 2012), which combines satel-
lite and terrestrial data and is complete up to degree and order
2159, the omission error is 0.023 m, while the commission
error is about 5–20 cm, depending on the region and the cor-
responding data quality. The above uncertainty estimates are
based on the published potential coefficient standard devi-
ations as well as a statistical model for the estimation of
corresponding omission errors, but do not include the uncer-
tainty contribution of GM (zero degree term in Eq. (21));
hence, the latter term, contributing about 3 mm in terms of
the height anomaly (corresponds to about 0.5 ppb; see Smith
et al. 2000;Ries 2014), has to be added in quadrature to the
figures given above. Further details on the uncertainty esti-
mates can be found in Denker (2013).
Based on these considerations it is clear that satellite mea-
surements alone will never be able to supply the complete
geopotential field with sufficient accuracy, which is due to
the signal attenuation with height and the required satellite
altitudes of a few 100 km. Only a combination of the highly
accurate and homogeneous (long wavelength) satellite grav-
ity fields with high-resolution terrestrial data (mainly gravity
and topography data with a resolution down to 1–2 km and
below) can cope with this task. In this respect, the satellite and
terrestrial data complement each other in an ideal way, with
the satellite data accurately providing the long-wavelength
field structures, while the terrestrial data sets, which have
potential weaknesses in large-scale accuracy and coverage,
mainly contribute to the short-wavelength features.
This directly leads to the development of regional solu-
tions for the disturbing potential and other gravity field
parameters, which typically have a higher resolution (down
to 1–2 km) than global spherical harmonic models. Based
on the developments of Molodensky (e.g. Molodenskii et al.
1962), the disturbing potential Tcan be derived from a series
of surface integrals, involving gravity anomalies and heights
over the entire Earth’s surface, which in the first instance can
be symbolically written as
T=M(g), (23)
where Mis the Molodensky operator and gare the gravity
anomalies over the entire Earth’s surface.
Further details on regional gravity field modelling are
given in “Appendix 2”, including the solution of Moloden-
sky’s problem, the remove–compute–restore (RCR) proce-
dure, the spectral combination approach, data requirements,
and uncertainty estimates for the disturbing potential and
quasigeoid heights. The investigations show that quasigeoid
heights can be obtained with an uncertainty of 1.9 cm, where
the major contributions come from the spectral band below
spherical harmonic degree 360. This uncertainty estimate
represents an optimistic scenario and is only valid for the case
that a state-of-the-art global satellite model is employed and
sufficient high-resolution and high-quality terrestrial gravity
and terrain data are available around the point of interest (e.g.
with a spacing of 2–4 km out to a distance of 50–100 km). For-
tunately, such a data situation exists for most of the metrology
institutes with optical clock laboratories—at least in Europe.
Furthermore, the perspective exists to improve the uncer-
tainty of the calculated quasigeoid heights (see “Appendix
2”).
Now, once the disturbing potential values Tare computed,
either from a global geopotential model by Eq. (21), or from a
regional solution by Eq. (23) based on Molodensky’s theory,
the gravity potential W, needed for the relativistic redshift
corrections, can be computed most straightforwardly as
WP=UP+TP,(24)
where the basic requirement is that the position of the given
point Pin space must be known accurately (e.g. from GNSS
observations), as the normal potential Uis strongly height-
dependent, while Tis only weakly height-dependent with
a maximum vertical gradient of a few parts in 10−3m2s−2
per metre. The above equation also makes clear that the pre-
dicted potential values WPare in the end independent of the
choice of W0and U0used for the linearization. Furthermore,
by combining Eqs. (24) with (20), and representing Uas a
function of U0and the ellipsoidal height h, the following
alternative expressions for W(at point P) can be derived as
WP=U0−γ(h−ζ) =U0−γ(h−ζ(i))+δW(i)
0,(25)
which demonstrates that ellipsoidal heights (e.g. from GNSS)
and the results from gravity field modelling in the form of
the quasigeoid heights (height anomalies) ζor the disturb-
ing potential Tare required, whereby a similar equation can
be derived for the geoid undulations N. Consequently, the
above approach (Eqs. (24) and (25)) is denoted here some-
what loosely as the GNSS/geoid approach, which is also
known in the literature as the GNSS/GBVP approach (the
123
H. Denker et al.
geodetic boundary value problem is the basis for computing
the disturbing potential T; see, for example, Rummel and
Teunissen 1988;Heck and Rummel 1990).
The GNSS/geoid approach depends strongly on precise
gravity field modelling (disturbing potential T, metric height
anomalies ζor geoid undulations N)and precise GNSS posi-
tions (ellipsoidal heights h)for the points of interest, with
the advantage that it delivers the absolute gravity potential
W, which is not directly observable and is therefore always
based on the assumption that the gravitational potential is
regular (zero) at infinity (see “Appendix 1”). In addition, the
GNSS/geoid approach allows the derivationof t he height sys-
tem bias term δW(i)
0based on Eq. (20) together with at least
one (better several) common GNSS and levelling stations in
combination with the gravimetrically determined disturbing
potential T.
3.3 Uncertainty considerations
The following uncertainty considerations are based on
heights, but corresponding potential values can easily be
obtained by multiplying the metric values with an average
gravity value (e.g. 9.81 m s−2or roughly 10 m s−2). Regard-
ing the geometric levelling and the GNSS/geoid approach,
the most direct and accurate way to derive potential dif-
ferences over short distances is the geometric levelling
technique, as standard deviations of 0.2–1.0 mm can be
attained for a 1-km double-run levelling with appropriate
technical equipment (Torge and Müller 2012). However, the
uncertainty of geometric levelling depends on many factors,
with some of the levelling errors behaving in a random man-
ner and propagating with the square root of the number of
individual set-ups or the distance, respectively, while other
errors of systematic type may propagate with distance in a
less favourable way. Consequently, it is important to keep
in mind that geometric levelling is a differential technique
and hence may be susceptible to systematic errors; exam-
ples include the differences between the second and third
geodetic levelling in Great Britain (about 0.2 m in the north–
south direction over about 1000 km distance; Kelsey 1972),
corresponding differences between an old and new levelling
in France (about 0.25 m from the Mediterranean Sea to the
North Sea, also mainly in north–south direction, distance
about 900 km; Rebischung et al. 2008), and inconsistencies
of more than ±1 m across Canada and the USA (differences
between different levellings and with respect to an accurate
geoid; Véronneau et al. 2006;Smith et al. 2010,2013). In
addition, a further complication with geometric levelling in
different countries is that the results are usually based on
different tide gauges with offsets between the correspond-
ing zero level surfaces, which, for example, reach more than
0.5 m across Europe. Furthermore, in some countries the lev-
elling observations are about 100 years old and thus may not
represent the actual situation due to possibly occurring recent
vertical crustal movements.
With respect to the GNSS/geoid approach, the uncertainty
of the GNSS positions is today more or less independent
of the interstation distance. For instance, the station coor-
dinates provided by the International GNSS Service (IGS)
or the IERS (e.g. ITRF2008) reach vertical accuracies of
about 5–10 mm (cf. Altamimi et al. 2011,2016,orSeitz
et al. 2013). The uncertainty of the quasigeoid heights (height
anomalies) is discussed mainly in “Appendix 2”, but also
mentioned in the previous subsection, showing that a stan-
dard deviation of 1.9 cm is possible in a best-case scenario
and that the values are nearly uncorrelated over longer
distances, with a correlation of less than 10% beyond a
distance of about 180 km. Aiming at the determination of
the absolute gravity potential Waccording to Eqs. (24)or
(25), which is the main advantage of the GNSS/geoid over
the geometric levelling approach, both the uncertainties of
GNSS and the quasigeoid have to be considered. Assuming
a standard deviation of 1.9 cm for the quasigeoid heights
and 1 cm for the GNSS ellipsoidal heights without cor-
relations between both quantities, a standard deviation of
2.2 cm is finally obtained (in terms of heights) for the abso-
lute potential values based on the GNSS/geoid approach.
Thus, for contributions of optical clocks to timescales, which
require the absolute potential WPrelative to a conventional
zero potential W0(see Sect. 2.5), the relativistic redshift
correction can be computed with an uncertainty of about
2×10−18. This is the case more or less everywhere in the
world where high-resolution regional gravity field models
have been developed on the basis of a state-of-the-art global
satellite model in combination with sufficient terrestrial grav-
ity field data. On the other hand, for potential differences
over larger distances of a few 100 km (i.e. typical distances
between different NMIs), the statistical correlations of the
quasigeoid values virtually vanish, which then leads to a
standard deviation for the potential difference of 3.2 cm in
terms of height, i.e. √2 times the figure given above for the
absolute potential (according to the law of error propagation),
which again has to be considered as a best-case scenario. This
would also hold for intercontinental connections between
metrology institutes, provided again that sufficient regional
high-resolution terrestrial data exist around these places. Fur-
thermore, in view of future refined satellite and terrestrial
data (see “Appendix 2”), the perspective exists to improve
the uncertainty of the relativistic redshift corrections from
the level of a few parts in 1018 to one part in 1018 or below.
According to this, over long distances across national bor-
ders, the GNSS/geoid approach should be a better approach
than geometric levelling. Finally, “Appendix 3” gives some
general recommendations for the implementation of geo-
metric levelling and GNSS observations at typical clock
sites.
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Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
4 Practical results for optical clock sites in
Germany and France
4.1 Geometric levelling and GNSS observations
In order to demonstrate the performance of geodetic meth-
ods for determining the gravity potential and corresponding
differences at national and intercontinental scales, this sec-
tion discusses some practical results for three optical clock
sites in Germany and one in France. Following the recom-
mendations for geometric levelling and GNSS observations
outlined in “Appendix 3”, corresponding surveys were car-
ried out at the Physikalisch-Technische Bundesanstalt (PTB)
in Braunschweig, Germany, the Leibniz Universität Han-
nover (LUH) in Hannover, Germany, the Max-Planck-Institut
für Quantenoptik (MPQ) in Garching (near Munich), Ger-
many, and the Paris Observatory (l’Observatoire de Paris,
OBSPARIS), Paris, France. The locations of the four selected
clock sites are shown in Fig. 1; the linear distances between
the German sites range from 52 km for PTB–LUH to 457
km for PTB–MPQ and 480 km for LUH–MPQ, while the
corresponding distances between OBSPARIS and the Ger-
man sites are 690 km (PTB), 653 km (LUH), and 690 km
(MPQ).
The coordinates of all GNSS stations were referred to
the ITRF2008 at its associated standard reference epoch
2005.0. The geometric levelling results were based in the
first instance on the corresponding national vertical reference
networks and then converted to the EVRS (European Verti-
cal Reference System) using its latest realization EVRF2007
(European Vertical Reference Frame), which is based on
Fig. 1 Map showing the locations of the PTB, LUH, MPQ, and
OBSPARIS sites
a common adjustment of all available European levelling
observations (UELN, United European Levelling Network).
The measurements within the UELN originate from very dif-
ferent epochs, but reductions for vertical crustal movements
were only applied for the (still ongoing) post-glacial iso-
static adjustment (GIA) in northern Europe, using the Nordic
Geodetic Commission (NKG) model NKG2005LU (Ågren
and Svensson 2007) with the epoch 2000.0; for further details
on EVRF2007, see Sacher et al. (2008). However, as GIA
hardly affects the aforementioned clock sites, while other
sources of vertical crustal movements are not known, the
EVRF2007 heights are considered as stable in time in the
following.
Further details on the local levelling results and the con-
version to EVRF2007 as well as the corresponding GNSS
observations and results are given in “Appendix 4”. In gen-
eral, the uncertainty of geometric levelling is at the few mm
level, and the uncertainty of the GNSS ellipsoidal heights is
estimated to be better than 10 mm.
Before checking the consistency between the GNSS and
levelling heights at each site, it is important to handle the
permanent parts of the tidal corrections in a consistent man-
ner. While the European height reference frame EVRF2007
and the European gravity field modelling performed at LUH
follow the IAG resolutions to use the zero-tide system, most
GNSS coordinates (including the ITRF and IGS results) refer
to the “non-tidal (or tide-free) system”. Therefore, for consis-
tency with the IAG recommendations and the other quantities
involved (EVRF2007 heights, quasigeoid), the ellipsoidal
heights from GNSS were converted from the non-tidal to
the zero-tide system based on the following formula from
Ihde et al. (2008) with
hzt =hnt +60.34 −179.01 sin2ϕ−1.82 sin4ϕ(mm),
(26)
where ϕis the ellipsoidal latitude, and hnt and hzt are the non-
tidal and zero-tide ellipsoidal heights, respectively. Hence,
the zero-tide heights over Europe are about 3–5 cm smaller
than the corresponding non-tidal heights.
All results for the PTB, LUH, MPQ, and OBSPARIS sites
are documented in Table 1, which contains the ellipsoidal
heights, the normal heights referring to the national and the
EVRF2007 height networks, and the gravimetric quasigeoid
heights based on the European Gravimetric (quasi)Geoid
EGG2015. Here, it should be noted that the zero level of
the French levelling network based on the tide gauge in Mar-
seille at the Mediterranean Sea is almost half a metre below
the zero level surface of the German and EVRF2007 height
networks related to the Amsterdam tide gauge at the North
Sea. Furthermore, ellipsoidal coordinates are given in Table 2
123
H. Denker et al.
at the epoch 2005.0, the standard reference epoch associated
with the ITRF2008.
4.2 The European gravimetric quasigeoid model
EGG2015
In this contribution, the latest European gravimetric quasi-
geoid model EGG2015 (Denker 2015) is employed. The
major differences between EGG2015 and the previous
EGG2008 model (Denker 2013) are the inclusion of addi-
tional gravity measurements carried out recently around all
major European optical clock sites within the ITOC project
(Margolis et al. 2013;Margolis 2014; see also http://projects.
npl.co.uk/itoc/) and the use of a newer geopotential model
based on the GOCE satellite mission instead of EGM2008.
EGG2015 was computed from surface gravity data in com-
bination with topographic information and the geopotential
model GOCO05S (Mayer-Gürr et al. 2015) based on the RCR
technique, as outlined in “Appendix 2”. The estimated uncer-
tainty (standard deviation) of the absolute quasigeoid values
is 1.9 cm; further details including correlation information
can be found in “Appendix 2” as well as Denker (2013).
In order to make EGG2015 consistent with GNSS and the
EVRF2007, the final model values were computed according
to Eq. (20)as
ζ(EGG2015)=T
γ−δW(EGG2015)
0
γ=ζ+ζ(EGG2015)
0,with
ζ(EGG2015)
0=+0.300 m,(27)
where a slightly rounded value for ζ(EGG2015)
0was imple-
mented; the original value of +0.305 m with a formal
uncertainty of 0.002 m resulted from the comparison with
1139 stations from the EUVN_DA GNSS/levelling data set
(EUVN_DA: European Vertical Reference Network – Den-
sification Action; Kenyeres et al. 2010). The rounded value
of +0.300 m for ζ(EGG2015)
0implies a zero potential for the
EGG2015 model of W(EGG2015)
0=62,636,857.91 m2s−2,
while the original value of + 0.305 m leads to a corresponding
zero potential of W(EVRF2007)
0=62,636,857.86 m2s−2for
EVRF2007, both having a formal uncertainty of 0.02 m2s−2.
4.3 Consistency check of GNSS and levelling heights
In order to check the consistency between the GNSS and
levelling heights, corresponding quasigeoid heights were
computed as ζGNSS =hzt −HN(EVRF2007), where hzt is
the ellipsoidal height from GNSS (ITRF2008, epoch 2005.0,
zero-tide system) and HN(EVRF2007)is the normal height
based on the EVRF2007, see Table 1. As the maximum
distance between the GNSS stations at each NMI is only
about 600 m for the PTB site, and less for the three other
sites, the quasigeoid at each site can be approximated in the
first instance as a horizontal plane, i.e. a constant value;
the remaining residuals (#1) are listed in Table 1in col-
umn (8), attaining maximum values of 11 mm (RMS 7 mm;
RMS: root mean square) for PTB, 9 mm (RMS 5 mm) for
LUH,2mm(RMS2mm)forMPQ,and5mm(RMS4
mm) for OBSPARIS. Besides this simple internal evaluation,
a comparison with the independent gravimetric quasigeoid
model EGG2015 is also performed, which is considered as an
external evaluation. Table 1shows the EGG2015 quasigeoid
heights ζEGG2015, the (raw) differences ζGNSS −ζEGG2015
and the residuals about the mean difference in columns 9–
11, respectively. Of most interest are the residuals (#2) about
the mean difference (in column 11), which attain a maximum
valueofonly6mm(RMS4mm)forPTB,1mm(RMS1
mm) for LUH, 2 mm (RMS 1 mm) for MPQ, and 5 mm (RMS
4 mm) for OBSPARIS, proving the excellent consistency
of the GNSS and levelling results. Although initial results
were worse for the PTB and OBSPARIS sites, the problem
was traced to an incorrect identification of the corresponding
antenna reference points (ARPs); at the PTB site, an error of
16 mm was found for station PTBB, and at OBSPARIS, there
was a difference of 29 mm between the ARP and the level-
ling benchmark and an additional error in the ARP height of
8 mm at station OPMT. It should be noted that, due to the
high consistency of the GNSS and levelling data at all four
sites, even quite small problems in the ARP heights (below
1 cm) could be detected and corrected after on-site inspec-
tions and additional verification measurements. This strongly
supports the recommendation of “Appendix 3” to have suf-
ficient redundancy in the GNSS and levelling stations. The
mean differences are 23 mm or below for the three German
sites, whereas 106 mm is obtained for the OBSPARIS site.
The magnitude of the mean values for the German sites is
excellent and proves the low uncertainty of both the geomet-
ric levelling and the GNSS/quasigeoid results in Germany
as well as the correct implementation of the corresponding
height system bias terms (δW(i)
0). The somewhat larger value
for OBSPARIS is believed to be mainly related to accumu-
lated systematic levelling errors (see Sect. 5for a detailed
discussion). Nevertheless, the results show that the different
zero levels of the German and French height reference net-
works have been properly taken into account, recalling that
the difference between the French and German zero level
surfaces is about half a metre.
4.4 Gravity potential determination
In order to apply the GNSS/geoid approach according to Eq.
(24)or(25), ellipsoidal heights are required for all stations of
interest. However, initially GNSS coordinates are only avail-
able for a few selected points at each NMI site, while for most
of the other laboratory points near the clocks, only levelled
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
Tabl e 1 Evaluation of GNSS and levelling data for the PTB, LUH, MPQ, and OBSPARIS sites by considering the (quasi)geoid as a horizontal plane and by utilizing the European Gravimetric
(Quasi)Geoid model EGG2015 with corresponding residuals #1 and #2, respectively
Site Station hzt (ITRF2008) HN(i)(national) HN(i)HN(EVRF2007)ζGNSS Residual #1 ζEGG2015 ζGNSS −ζEGG2015 Residual #2
(4) + (5) (3)−(6)Mean −(7)Mean −(10)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
PTB, Braunschweig, Germany
PTB PTBB (IGS) 130.196 87.436 +0.006 87.442 42.754 +0.007 42.772 −0.018 +0.005
PTB LB03 143.514 100.752 +0.006 100.758 42.756 +0.006 42.769 −0.013 +0.001
PTB AF02 123.717 80.939 +0.006 80.945 42.772 −0.010 42.784 −0.012 −0.001
PTB MB02 144.937 102.167 +0.006 102.173 42.764 −0.003 42.772 −0.008 −0.005
Mean, RMS: 42.762 0.007 Mean, RMS: −0.013 0.004
LUH, Hannover, Germany
LUH 362400200 98.061 55.359 +0.006 55.365 42.696 −0.009 42.719 −0.023 0.000
LUH MSD1 113.303 70.612 +0.006 70.618 42.685 +0.003 42.708 −0.023 0.000
LUH MSD8 113.290 70.600 +0.006 70.606 42.684 +0.004 42.708 −0.024 +0.001
LUH MSD9 117.744 75.052 +0.006 75.058 42.686 +0.002 42.708 −0.022 −0.001
Mean, RMS: 42.688 0.005 Mean, RMS: −0.023 0.001
MPQ, Garching, Germany
MPQ MPQ-4001 521.635 476.181 +0.031 476.212 45.423 −0.003 45.402 +0.021 −0.002
MPQ MPQ-4002 528.247 482.796 +0.031 482.827 45.420 +0.001 45.402 +0.018 +0.001
MPQ MPQ-4003 535.946 490.496 +0.031 490.527 45.419 +0.002 45.401 +0.018 +0.001
Mean, RMS: 45.421 0.002 Mean, RMS: 0.019 0.001
OBSPARIS, Paris, France
OBSPARIS 100 105.655 61.873 −0.479 61.394 44.261 −0.003 44.364 −0.103 −0.003
OBSPARIS A 130.958 87.185 −0.479 86.706 44.252 +0.005 44.364 −0.112 +0.005
OBSPARIS OPMT (IGS) 122.548 78.767 −0.479 78.288 44.260 −0.002 44.364 −0.104 −0.002
Mean, RMS: 44.258 0.004 Mean, RMS: −0.106 0.004
The GNSS and the European levelling heights refer to ITRF2008 (Epoch 2005.0; GRS80 ellipsoid; zero-tide system) and EVRF2007, respectively; for further details, see Sects. 4and 5. All units
are in m
123
H. Denker et al.
heights exist. Therefore, based on Eq. (20), a quantity δζ is
defined as δζ =(h−HN(i))−ζ(i). This should be zero in
theory, but is not in practice due to the uncertainties in the
quantities involved (GNSS, levelling, quasigeoid). However,
if a high-resolution quasigeoid model is employed (such as
EGG2015), the term δζ should be small and represent only
long-wavelength features, mainly due to systematic levelling
errors over large distances as well as long-wavelength quasi-
geoid errors. In this case an average (constant) value δζ can
be used at each NMI site to convert all levelled heights into
ellipsoidal heights by using
h(adj.) =HN(i)+ζ(i)+δζ =HN(i)+(ζ +ζ(i)
0)+δζ, (28)
which is based on Eq. (16). This has the advantage that locally
(at each NMI) the consistency is kept between the levelling
results on the one hand and the GNSS/quasigeoid results
on the other hand and consequently that the final potential
differences between stations at each NMI are identical for
the GNSS/geoid and geometric levelling approach, which is
reasonable, as locally the uncertainty of levelling is usually
lower than that of the GNSS/quasigeoid results. The quantity
δζ is in fact the mean value given in column (10) of Table 1
for each of the four sites. Furthermore, regarding the com-
mon GNSS and levelling stations, their (adjusted) ellipsoidal
heights according to Eq. (28) (see Table 2) are identical with
the observed ellipsoidal heights (column 3 in Table 1)plus
the corresponding residual #2 (column 11 in Table 1).
Based on the GNSS and levelling results at the selected
four sites, the gravity potential values can finally be derived
for all relevant stations. The gravity potential values obtained
are documented in Table 2for all GNSS and levelling points
given in Table 1plus two additional points on the PTB cam-
pus as an example for stations that have only levelled heights
(KB01, KB02; points at the Kopfermann building, hosting the
caesium fountains). The gravity potential values are given in
Table 2in the form of geopotential numbers according to Eq.
(13) with
C=W0(IERS2010)−WP,(29)
where the conventional value W0(IERS2010) according to
Eq. (10) is used, following the IERS2010 conventions and
the IAU resolutions for the definition of TT. The geopo-
tential numbers Care more convenient than the absolute
potential values WPdue to their smaller numerical values
and direct usability for the derivation of the relativistic red-
shift corrections according to Eq. (11). Table 2gives the
geopotential numbers Caccording to Eq. (29) in the geopo-
tential unit (gpu; 1 gpu =10 m2s−2), resulting in numerical
values of Cthat are about 2% smaller than the numeri-
cal height values, for both the geometric levelling approach
(C(lev))and the GNSS/geoid approach (C(GNSS/geoid))based
on Eqs. (17)or(24) and (25), respectively. Regarding the
geometric levelling approach, the above-mentioned value
W(EVRF2007)
0=62,636,857.86 m2s−2based on the Euro-
pean EUVN_DA GNSS/levelling data set from Kenyeres
et al. (2010) is utilized in Eq. (17). The GNSS/geoid approach
according to Eqs. (24) and (25) is based on the disturbing
potential Tor the corresponding height anomaly values ζ
from the EGG2015 model (see above and Eq. (27)), as well
as the normal potential U0=62,636,860.850 m2s−2, asso-
ciated with the surface of the underlying GRS80 (Geodetic
Reference System 1980; see Moritz 2000) level ellipsoid,
and furthermore, the mean normal gravity values γare also
based on the GRS80 level ellipsoid; for further details, see
Torge and Müller (2012). Besides the geopotential numbers
derived by the two independent approaches, Table 2also
shows the differences between the two approaches as well
as corresponding ITRF2008 coordinates and normal heights
referring to EVRF2007.
5 Discussion and conclusions
The differences between the geopotential numbers from the
two approaches (geometric levelling and GNSS/geoid) range
from +0.014 to −0.109 gpu, which is equivalent to +0.014 to
−0.111 m in terms of height, and again the metric (height)
values are preferred in the following, as earlier. Regarding
gravity potential differences between two stations, for the
nearby sites PTB and LUH (52 km linear distance) the dif-
ference between both approaches is only 0.010 m, while the
corresponding figures for the connection PTB–MPQ (457 km
distance) and LUH–MPQ (480 km distance) are 0.032 m and
0.042 m, respectively. On the other hand, the (international)
potential differences between OBSPARIS (France) and the
German sites show larger differences between the geometric
levelling and the GNSS/geoid approach of the order of 0.10
m (0.094 m for the difference to PTB, 0.084 m for LUH, and
0.125 m for MPQ) over distances between about 650 and 700
km.
Regarding the significance of the 1 dm discrepancies in the
potential differences between the two geodetic approaches
for the German stations and OBSPARIS, these have to be
discussed in relation to the corresponding uncertainties of
levelling, GNSS, and the quasigeoid model (EGG2015).
The geometric levelling results are based on the EVRF2007
adjustment, where a posteriori standard deviations of about
1mm ×√d(dis the distance in km) are reported for the
German levelling data and 2 mm ×√dfor the French data
(Sacher et al. 2008). Starting with a rough uncertainty esti-
mation for the levelling connections between the German
stations and OBSPARIS by assuming that the levelling is
half in Germany and half in France, and considering that
the levelling lines will be longer than the linear distances
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
Tabl e 2 Ellipsoidal coordinates (latitude, longitude, height; ϕ,λ,h(adj.) )referring to ITRF2008 reference frame (epoch 2005.0; GRS80 ellipsoid; zero-tide system), normal heights HN(EVRF2007)
based on EVRF2007, geopotential numbers based on the geometric levelling (C(lev))and GNSS/geoid approach (C(GNSS/geoid))relative to the IERS2010 conventional reference potential
W0=62,636,856.00 m2s−2and differences Cthereof, as well as the relativistic redshift correction based on the GNSS/geoid approach; for further details, see Sects. 4and 5
Site Station ϕλ h(adj.) HN(EVRF2007)C(lev)C(GNSS/geoid)CRedshift
(◦)(
)(
)(
◦)(
)(
) (m) (m) (10 m2s−2)(10m
2s−2)(10m
2s−2)(10
−16)
PTB, Braunschweig, Germany
PTB PTBB (IGS) 52 17 46.28177 10 27 35.08676 130.201 87.442 85.617 85.600 −0.017 −95.243
PTB LB03 52 17 49.94834 10 27 37.63590 143.514 100.758 98.684 98.667 −0.017 −109.782
PTB AF02 52 17 30.90851 10 27 28.21874 123.716 80.945 79.242 79.225 −0.017 −88.150
PTB MB02 52 17 47.22270 10 27 50.49262 144.932 102.173 100.072 100.055 −0.017 −111.326
PTB KB01 52 17 45.2102733.1 119.627 76.867 75.241 75.224 −0.017 −83.698
PTB KB02 52 17 46.3102735.1 119.708 76.949 75.321 75.304 −0.017 −83.787
LUH, Hannover, Germany
LUH 362400200 52 22 54.38687 9 43 3.64026 98.061 55.365 54.142 54.115 −0.027 −60.211
LUH MSD1 52 23 7.24990 9 42 44.90267 113.303 70.618 69.109 69.082 −0.027 −76.864
LUH MSD8 52 23 8.30803 9 42 43.62979 113.291 70.606 69.098 69.070 −0.027 −76.859
LUH MSD9 52 23 7.58097 9 42 45.17101 117.743 75.058 73.466 73.439 −0.027 −76.856
MPQ, Garching, Germany
MPQ MPQ-4001 48 15 33.82899 11 39 57.67654 521.633 476.212 466.902 466.916 0.014 −76.869
MPQ MPQ-4002 48 15 37.08423 11 39 57.95797 528.248 482.827 473.390 473.404 0.014 −76.863
MPQ MPQ-4003 48 15 35.02523 11 39 59.74055 535.947 490.527 480.942 480.956 0.014 −76.853
OBSPARIS, Paris, France
OBSPARIS 100 48 50 7.99682 2 20 8.38896 105.652 61.394 60.039 59.930 −0.109 −76.845
OBSPARIS A 48 50 10.90277 2 20 10.55555 130.964 86.706 84.868 84.759 −0.109 −76.851
OBSPARIS OPMT (IGS) 48 50 9.31198 2 20 5.77891 122.546 78.288 76.611 76.502 −0.109 −81.712
123
H. Denker et al.
of about 700 km, leads to a standard deviation of about 50
mm. However, a more thorough uncertainty estimate has
to consider that the EVRF2007 heights are not the result
of single-line connections but rest upon an adjustment of
a whole levelling network, which leads to a corresponding
standard deviation of about 20 mm (M. Sacher, Bundesamt
für Kartographie und Geodäsie, BKG, Leipzig, Germany,
personal communication, 10 May 2017), indicating a factor
2.5 improvement due to the network; nevertheless, neither
uncertainty estimate considers any systematic levelling error
contributions. Furthermore, the GNSS ellipsoidal heights
have uncertainties below 10 mm, as they are directly based
on permanent reference stations or connected to such sta-
tions located nearby. The uncertainty of EGG2015 has been
discussed in “Appendix 2” and above, yielding a standard
deviation of 19 mm for the absolute values and about 27
mm for corresponding differences between the German and
French sites over about 700 km distance, presuming that the
correlations are insignificant over these distances. Finally,
accepting that all three quantities involved in the comparison
(levelling, GNSS, quasigeoid) are uncorrelated, the corre-
sponding uncertainty components add in quadrature, giving
a standard deviation of 59 and 36 mm based on the single-line
levelling and the corresponding network uncertainty esti-
mates, respectively. Hence, the discrepancies of about 100
mm between the two geodetic approaches have to be consid-
ered as statistically significant at a confidence level of 95%
for the network-based levelling uncertainty estimates, but not
for the simple single-line estimates.
Nevertheless, as the quasigeoid model EGG2015 is based
on an up-to-date, consistent and quite homogeneous GOCE
satellite model and also includes high-quality and high-
resolution terrestrial gravity field data around the investigated
clock sites, the corresponding uncertainty estimates are con-
sidered as realistic, while the GNSS component plays only a
minor role. It is therefore hypothesized that the largest uncer-
tainty contribution comes from geometric levelling, which is
very accurate over short distances, but susceptible to sys-
tematic errors at the decimetre level over larger distances in
the order of 1000 km. In France, differences between an old
and a new levelling exist, mainly in the north–south direc-
tion (about 0.25 m from the Mediterranean Sea to the North
Sea over a distance of about 900 km, i.e. 28 mm/100 km),
but to a lesser extent also in the east–west direction (about
0.04 m from Strasbourg to Brest, i.e. 5 mm/100 km), where
an evaluation by independent GNSS/quasigeoid data clearly
shows a much better agreement with the new levelling results
(Rebischung et al. 2008;Denker 2013). Since the old level-
ling (NGF -IGN69: Nivellement Général France–L’Institut
National de l’Information Géographique et Forestière, IGN,
1969; see “Appendix 4”) is the basis for the results in Tables 1
and 2, using the new French levelling could significantly
reduce the existing discrepancies of about 1 dm, and in fact a
preliminary EVRF adjustment from 2017 with new levelling
data for France and other countries (M. Sacher, BKG, per-
sonal communication, 10 May 2017) is indicating a reduction
in the current differences between the German stations and
OBSPARIS by 68 mm.
Based on the preceding discussion, the geometric levelling
approach, which gives in the first instance only height and
potential differences, is recommended over shorter distances
of up to several ten kilometres, where it can give millimetre
uncertainties. However, it is problematic over long distances,
the data may not be up-to-date due to recent vertical crustal
movements, and it is further complicated across national bor-
ders due to different zero levels. Consequently, over longer
distances (of more than about 100 km) across national bor-
ders, the GNSS/geoid approach should be better. Moreover,
this approach has the advantage that it gives absolute grav-
ity potential values, presently with an uncertainty of about
two centimetres in terms of heights (best-case scenario, see
above). Hence, for contributions of optical clocks to inter-
national timescales, which require absolute potential values
WPrelative to a conventional zero potential W0, the relativis-
tic redshift corrections can be derived from the GNSS/geoid
approach with a present uncertainty of about 2 ×10−18.
This is the case more or less everywhere in the world, where
sufficient terrestrial gravity field data sets exist, and there
is still potential for further improvements (see Sect 3.3 and
“Appendix 2”). Based on this reasoning, Table 2gives the rel-
ativistic redshift corrections according to Eq. (11) only for
the GNSS/geoid approach, which can be considered as the
recommended values. On the other hand, for optical clock
(frequency) comparisons over shorter distances, requiring
only potential differences, the geometric levelling approach
can also be employed, giving the differential redshift correc-
tions between the clock sites with uncertainties down to a
few parts in 1019.
Finally, the results given in Tables 1and 2show that
the geodetic heights and corresponding gravity potential
values derived from the geometric levelling approach and
the GNSS/geoid approach are presently inconsistent at the
decimetre level across Europe. For this reason, the more
or less direct observation of gravity potential differences
through optical clock comparisons (with targeted fractional
accuracies of 10−18, corresponding to 1 cm in height) is
eagerly awaited as a means for resolving the existing discrep-
ancies between different geodetic techniques and remedying
the geodetic height determination problem over large dis-
tances. A first attempt in this direction was the comparison
of two strontium optical clocks between PTB and OBSPARIS
via a fibre link, showing an uncertainty and agreement with
the geodetic results of about 5 ×10−17 (Lisdat et al. 2016).
This was mainly limited by the uncertainty and instability
of the participating clocks, which is likely to improve in the
near future. For clocks with performance at the 10−17 level
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
and below, time-variable effects in the gravity potential, espe-
cially solid Earth and ocean tides, have to be considered and
can also serve as a method of evaluating the performance
of the optical clocks (i.e. a detectability test). Then, after
further improvements in the optical clock performance, con-
clusive geodetic results can be anticipated in the future, and
clock networks may also contribute to the establishment of
the International Height Reference System (IHRS).
Acknowledgements The authors would like to thank Thomas Udem,
Ronald Holzwarth, and Arthur Matveev (Max-Planck-Institut für Quan-
tenoptik, MPQ, Garching, Germany) for their support and facilitating
the access to the MPQ site, Christof Völksen and Torsten Spohnholtz
(Bayerische Akademie der Wissenschaften, Kommission für Erdmes-
sung und Glaziologie, Munich, Germany) for carrying out the GNSS
observations and the data processing for the MPQ site, Nico Linden-
thal and Tobias Kersten (Leibniz Universität Hannover, LUH, Institut
für Erdmessung, Hannover, Germany), Cord-Hinrich Jahn and Peter
Lembrecht (Landesamt für Geoinformation und Landesvermessung
Niedersachsen, LGLN, Landesvermessung und Geobasisinformation,
Hannover, Germany) for corresponding GNSS work at LUH and PTB,
and Martina Sacher (Bundesamt für Kartographie und Geodäsie, BKG,
Leipzig, Germany) for providing information on the EVRF2007 heights
and uncertainties, the associated height transformations, and a new
adjustment of the UELN from 2017. This research was supported
by the European Metrology Research Programme (EMRP) within
the framework of a Researcher Excellence Grant associated with the
Joint Research Project “International Timescales with Optical Clocks”
(SIB55 ITOC), as well as the Deutsche Forschungsgemeinschaft (DFG)
within the Collaborative Research Centre 1128 “Relativistic Geodesy
and Gravimetry with Quantum Sensors (geo-Q)”, project C04. The
EMRP is jointly funded by the EMRP participating countries within
EURAMET and the European Union. We also thank the reviewers for
their valuable comments, which helped to improve the manuscript sig-
nificantly.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Appendix 1: Some fundamentals of physical geodesy
Classical physical geodesy is largely based on the Newto-
nian theory with Newton’s law of gravitation, giving the
gravitational force between two point masses, to which a
gravitational acceleration (also termed gravitation) can be
ascribed by setting the mass at the attracted point Pto unity.
Then, by the law of superposition, the gravitational acceler-
ation of an extended body like the Earth can be computed as
the vector sum of the accelerations generated by the individ-
ual point masses (or mass elements), yielding
b=b(r)=−G
Earth
r−r
|r−r|3dm,dm=ρdv, ρ =ρ(r),
(30)
where rand rare the position vectors of the attracted point
Pand the source point Q, respectively, dmis the differen-
tial mass element, ρis the volume density, dvis the volume
element, and Gis the gravitational constant. The SI unit of
acceleration is m s−2, but the non-SI unit Gal is still used
frequently in geodesy and geophysics (1 Gal =0.01 m s−2,
1 mGal =10−5ms
−2), see also BIPM (2006). While an
artificial satellite is only affected by gravitation, a body rotat-
ing with the Earth also experiences a centrifugal force and
a corresponding centrifugal acceleration z, which is directed
outwards and perpendicular to the rotation axis:
z=z(p)=ω2p.(31)
In the above equation, ωis the angular velocity, and pis
the distance vector from the rotation axis. Finally, the grav-
ity acceleration (or gravity) vector gis the resultant of the
gravitation band the centrifugal acceleration z:
g=b+z.(32)
As the gravitational and centrifugal acceleration vectors b
and zboth form conservative vector fields or potential fields,
these can be represented as the gradient of corresponding
potential functions by
g=gradW=b+z=gradVE+gradZE=grad(VE+ZE),
(33)
where Wis the gravity potential, consisting of the gravita-
tional potential VEand the centrifugal potential ZE. Based
on Eqs. (30)–(33), the gravity potential Wcan be expressed
as[seealsoEq.(8)]
W=W(r)=VE+ZE=G
Earth
ρdv
l+ω2
2p2,(34)
where land pare the lengths of the vectors r−rand p,
respectively. All potentials are defined with a positive sign,
which is common geodetic practice (cf. Sect. 2.2). The grav-
itational potential VEis assumed to be regular (i.e. zero)
at infinity and has the important property that it fulfils the
Laplace equation outside the masses; hence, it can be repre-
sented by harmonic functions in free space, with the spherical
harmonic expansion playing a very important role. Further
details on potential theory and properties of the potential
functions can be found, for example, in Heiskanen and Moritz
(1967), Torge and Müller (2012), or Denker (2013).
The determination of the gravity potential Was a function
of position is one of the primary goals of physical geodesy; if
W(r) were known, then all other parameters of interest could
be derived from it, including the gravity vector gaccording
to Eq. (33) as well as the form of the equipotential surfaces
123
H. Denker et al.
[by solving the equation W(r) = const.]. Furthermore, the
gravity potential is also the ideal quantity for describing the
direction of water flow, i.e. water flows from points with
lower gravity potential to points with higher values. How-
ever, although above Eq. (34) is fundamental in geodesy, it
cannot be used directly to compute the gravity potential W
due to insufficient knowledge about the density structure of
the entire Earth; this is evident from the fact that densities
are at best known with two to three significant digits, while
geodesy generally strives for a relative uncertainty of at least
10−9for all relevant quantities (including the potential W).
Therefore, the determination of the exterior potential field
must be solved indirectly based on measurements taken at or
above the Earth’s surface, which leads to the area of geodetic
boundary value problems (see Sect. 3.2 and “Appendix 2”).
The gravity potential is closely related to the question of
heights as well as level or equipotential surfaces and the
geoid, where the geoid is classically defined as a selected
level surface with constant gravity potential W0, conceptu-
ally chosen to approximate (in a mathematical sense) the
mean ocean surface or mean sea level (MSL). However, MSL
does not coincide with a level surface due to the forcing of
the oceans by winds, atmospheric pressure, and buoyancy in
combination with gravity and the Earth’s rotation. The devi-
ation of MSL from a best fitting equipotential surface (geoid)
is denoted as the (mean) dynamic ocean topography (DOT);
it reaches maximum values of about ±2m(Bosch and Sav-
cenko 2010) and is of vital importance for oceanographers
for deriving ocean circulation models (Condi and Wunsch
2004).
On the other hand, a substantially different approach was
chosen by the IAG during its General Assembly in Prague,
2015, within “IAG Resolution (No. 1) for the definition
and realization of an International Height Reference Sys-
tem (IHRS)” (Drewes et al. 2016), where a numerical value
W0=62,636,853.4m
2s−2(based on observations and data
related to the mean tidal system) is defined for the realization
of the IHRS vertical reference level surface, with a corre-
sponding note, stating that W0is related to “the equipotential
surface that coincides (in the least-squares sense) with the
worldwide mean ocean surface, the most accepted defini-
tion of the geoid” (Ihde et al. 2015). Although the classical
geodetic geoid definition and the IAG 2015 resolution both
refer to the worldwide mean ocean surface, so far no adopted
standards exist for the definition of MSL, the handling of
time-dependent terms (e.g. due to global sea level rise), and
the derivation of W0, where the latter value can be determined
in principle from satellite altimetry and a global geopoten-
tial model (see Burša et al. 1999 or Sánchez et al. 2016).
Furthermore, the IHRS value for the reference potential is
inconsistent with the corresponding value W0used for the
definition of TT (see also Sect. 2.5); Petit et al. (2014) denoted
these two definitions as “classical geoid” and “chronometric
geoid”.
In this context, it is somewhat unfortunate that the
same notation (W0)is used to represent different estimates
for a quantity that is connected with the (time-variable)
mean ocean surface, but this issue can be resolved only
through future international cooperation, even though it
seems unlikely that the different communities are willing to
change their definitions. In the meantime, this problem has to
be handled by a simple constant shift transformation between
the different level surfaces, associated with a thorough doc-
umentation of the procedures and conventions involved. It is
clear that the definition of the zero level surface (W0issue) is
largely a matter of convention, where a good option is proba-
bly to select a conventional value of W0(referring to a certain
epoch, without a strict relationship to MSL), accompanied by
static modelling of the corresponding zero level surface, and
to describe then the potential of the time-variable mean ocean
surface for any given point in time as the deviation from this
reference value.
Appendix 2: Regional gravity field modelling
Molodensky’s boundary value problem can be solved in var-
ious ways, and detailed derivations can be found in Moritz
(1980). An efficient solution, avoiding integral equations, is
provided by the method of analytical continuation (Moritz
1980;Sideris 1987), which finally leads to
T=S(g)+∞
i=1
S(gi)=∞
i=0
Ti,(35)
where Sis Stokes’s integral operator, T0=S(g)is the
main contribution and denoted as the Stokes term (i=0),
while the further giand Titerms (for i>0) are the so-called
Molodensky correction terms, which consider that the data
are referring to the Earth’s surface and not to a level surface.
The Stokes term can be written in full as
T0=R
4π
σ
gS(ψ)dσ, S(ψ ) =∞
n=2
2n+1
n−1Pn(cos ψ),
(36)
where ψis the spherical distance between the computation
and data points, nis the spherical harmonic degree, Ris a
mean Earth radius, and σis the unit sphere.
The above integral formulae require global gravity ano-
maly data sets, but in practice, only local and regional discrete
gravity field data sets are typically available for the area of
interest and the immediate surroundings. The very short-
wavelength gravity field information is therefore not properly
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
represented by the discrete observations (aliasing) and the
long-wavelength (global) gravity field information is also
incomplete. This problem is remedied by the RCR tech-
nique, where the short- and long-wavelength gravity field
structures are obtained from digital elevation models and a
global geopotential model, respectively, while the medium
wavelength field structures are derived from the regional dis-
crete gravity field observations. The general scheme of the
RCR technique is based on a residual disturbing potential
given by Tres =T−TM−TT, where TMand TTare
the contributions from the global geopotential model and
the topographic information (or more generally the anoma-
lous masses), and accordingly all observations (e.g. g)are
converted to corresponding residual quantities. Then, after
applying the gravity field modelling techniques to the resid-
ual quantities, the effects of the global geopotential model
and the topography are restored as the final stage.
The removal of the short- and long-wavelength gravity
field information is equivalent to a spectral (low-pass and
high-pass) filtering; this leads to residual quantities which
are typically much smaller and smoother (as well as statis-
tically more homogeneous and isotropic) than the original
ones, facilitating, for example, the tasks of interpolation and
gridding, as well as field transformations by integral formu-
lae or least-squares collocation (LSC), with the additional
side effect that the collection of observational data can be
restricted to the region of interest plus a narrow edge zone
around it.
Within the RCR combination strategy, the very long-
wavelength gravity field structures can be determined much
more accurately from satellite data (GRACE and GOCE mis-
sions) than from terrestrial data, as the latter data set still
includes significant data gaps and possibly also (small) sys-
tematic long-wavelength errors due to the underlying largely
historic gravity observations and networks. Hence, within the
combination process, the very accurate long-wavelength fea-
tures of state-of-the-art global geopotential models should be
retained, while the terrestrial data should mainly contribute
to the shorter-wavelength components. This can be reached
by a spectral combination, which can also take into account
the uncertainties of the satellite data and the terrestrial data.
Furthermore, the spectral combination approach is easy to
implement for the integration techniques, as it results only in
a small change in the integration kernel. Thus, instead of the
Stokes kernel S(ψ), a modified integration kernel has to be
implemented in the form
W(ψ) =∞
n=2
wn
2n+1
n−1Pn(cos ψ), (37)
where the so-called spectral weights wnform the only differ-
ence compared to the Stokes kernel in Eq. (36). Therefore,
the Stokes kernel can also be considered as a special case of
the above modified integration kernel W(ψ) with wn=1.0
for degrees nequal 2 to ∞, which also makes clear that the
Stokes formula always extracts all degrees from 2 to ∞from
the (terrestrial) input gravity data, while the spectral weights
allow control of which degrees are taken from the terrestrial
gravity data. The spectral weights can either be determined
empirically, e.g. as filter coefficients, or within the framework
of a least-squares adjustment or a least-squares collocation
solution. The latter two approaches allow the uncertainties
of the satellite and terrestrial data to be taken into account
through corresponding (so-called) error degree variances.
Concerning the regional calculation of the disturbing
potential Tor corresponding height anomalies ζbased on
the Molodensky theory, uncertainty estimates can also be
obtained based on the law of error propagation, which is
most easily implemented on the basis of the degree vari-
ance approach with corresponding error degree variances.
Taking into account the given uncertainty estimates for the
global geopotential model coefficients (including the uncer-
tainty contribution of GM, see above) and the terrestrial data
(assuming a standard deviation of 1 mGal with correlated
noise), and supposing that a least-squares spectral combina-
tion of the satellite and terrestrial data is performed, results
in a standard deviation of 3.1 cm for the combined (abso-
lute) height anomalies based on the global model EGM2008,
while the corresponding standard deviations based on the
latest GRACE and GOCE (e.g. 5th generation; Brockmann
et al. 2014) global geopotential models are 2.5 cm and 1.9
cm, respectively. The major uncertainty contributions come
from the spectral band 90 <n≤360, while today the
very long wavelengths (n≤90) are accurately known from
the GRACE mission, and the short wavelengths (n >360)
can be obtained from high-quality terrestrial data. Regard-
ing the GOCE-based solution, the uncertainty contributions
from the different spectral bands are <0.1 cm for n≤90,
1.2 cm for 90 <n≤180, 1.1 cm for 180 <n≤270, 0.7
cm for 270 <n≤360, and 0.7 cm for n>360, respec-
tively, which sum in quadrature to the aforementioned total
uncertainty of 1.9 cm. These uncertainty estimates represent
an optimistic scenario, and they are only valid for the case
that a state-of-the-art global satellite model is employed and
sufficient high-resolution and high-quality terrestrial grav-
ity and terrain data are available around the point of interest
(e.g. within a distance of 50–100 km). The terrestrial grav-
ity data should have a spacing of roughly 2–4 km, while
the uncertainty demands are only at the level of about 1
mGal, presupposing that this is purely random noise with-
out systematic errors over large regions, with similar data
requirements being reported by Forsberg (1993) and Jekeli
(2012). Fortunately, such a data situation exists for most of
the metrology institutes with optical clock laboratories—at
least in Europe. In addition, a covariance function has been
123
H. Denker et al.
derived for the computed height anomalies, which has a half-
length of about 40 km and zeros at about 80, 220, 370 km, and
so on. Hence, over longer distances, e.g. beyond the second
and third zero of the covariance function, the height anoma-
lies are nearly uncorrelated; for further details cf. Denker
(2013). In this context, further improvements would be pos-
sible by using better terrestrial gravity data over a sufficiently
large area and enhanced satellite gravity field missions up to
about degree and order 300, such that standard deviations of
1 cm or below may become feasible; this would foster further
applications, including optical clocks.
Appendix 3: General recommendations for geomet-
ric levelling and GNSS observations at clock sites
Optical clocks are presently developed at different NMIs,
universities, and other research institutes, where typically
several different optical and microwave clocks are operated in
different laboratories and buildings on a local site. In this sit-
uation, gravity potential differences are required for local and
remote clock comparisons, and absolute potential values are
needed for contributions to international timescales. Based
on the uncertainty considerations in Sect. 3.3, it is recom-
mended that all local clock laboratories should be connected
by geometric levelling with millimetre uncertainty to support
local clock comparisons at the highest level. For remote clock
comparisons and contributions to international timescales,
both the geometric levelling and the GNSS approach ought
to be employed to obtain not only the potential differences
but also the absolute potential values, while at the same time
improving the redundancy and allowing a mutual control of
GNSS, levelling, and (quasi)geoid data. For this purpose, it is
first of all useful to have permanent survey markers for level-
ling benchmarks and GNSS stations. The levelling markers
should be installed inside the buildings as close as possi-
ble to the clock tables, with one additional marker placed
outside the building (e.g. near the entrance, however, out-
side markers are not mandatory). The main purpose of the
interior markers is to allow an easy height transfer to the opti-
cal clocks (e.g. with a simple spirit level used for building
construction), while the outside point can serve as a securing
point and for separating the inside and outside levelling oper-
ations. After installing the markers and allowing some time
for them to settle, all markers should be observed by geo-
metric levelling connected to at least two (preferably three)
existing national levelling benchmarks in order to validate
the given heights and to exclude local vertical movements
of the selected benchmarks. If it turns out that significant
discrepancies exist between the performed levelling and the
existing heights, further benchmarks have to be employed
until the necessary tolerances are achieved. The national
levelling benchmarks should be located close to the clock
sites in order to minimize the effect of levelling uncertain-
ties, which depend strongly on the levelled distances. The
levelling observations should be carried out in two direc-
tions (double-run levelling) to allow a consistency check.
The uncertainty of levelling is generally characterized by the
standard deviation for the height difference of 1 km double-
run levelling, which should not exceed 2–3 mm in the case
of an engineer’s levelling. Today, digital self-levelling levels
are generally used to achieve these uncertainty specifica-
tions. Further details on the procedure and error theory of
geometric levelling can be found in standard surveying text-
books, for instance Kavanagh and Mastin (2014), Anderson
and Mikhail (2012), Kahmen and Faig (1988), or Torge and
Müller (2012).
In addition, at least two (preferably three) GNSS stations
should be utilized at each clock site, and these should also
be connected to the levelling network in order to improve
the redundancy and uncertainty and to allow the detection of
any gross errors such as wrong antenna heights or reference
points. The GNSS stations should be observed by traditional
static measurements in connection with reference stations or
the PPP (precise point positioning) approach. Based on the
IGS products, both these approaches give positions at the
mm–cm uncertainty level, directly in the ITRF; for further
details, see Kouba (2009). As station-dependent effects are
among the main limiting factors for precise height determi-
nation, the stations should be carefully selected, considering
shadowing and reflection effects, and every effort should be
made to reach the best possible uncertainty. In order to aver-
age out random effects, and also to some extent systematic
effects, the station occupation should preferably be several
days, but at least one full day (with 24-h observations). Post-
processing with state-of-the-art software should target an
uncertainty of about 5 mm for the ellipsoidal heights, with
coordinates referring to the ITRF. Furthermore, the con-
nection of the GNSS points to the levelling network may
necessitate some effort for typical permanent stations on
rooftops (e.g. requiring geometric levelling and tape mea-
surements inside and outside buildings, trigonometric height
determination). Additional information on GNSS surveying
procedures, error sources, processing strategies, etc., can be
found in the textbooks from Kavanagh and Mastin (2014)or
Anderson and Mikhail (2012).
Appendix 4: Practical implementation of geomet-
ric levelling and GNSS observations at clock sites in
Germany and France
The German national levelling network is denoted as
DHHN92 (Deutsches Haupthöhennetz 1992; internal des-
ignation “height status 160”) and is based on normal heights,
referring to the level of the “Normaal Amsterdams Peil”
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
Fig. 2 Levelling benchmarks and lines, GNSS stations, and an absolute gravity point at PTB
(NAP) tide gauge. The French national levelling network is
denoted as NGF-IGN69 (see Sect. 5) and is similarly based
on normal heights, but referring to the French fundamental
tide gauge in Marseille. For the conversion of the national
heights into the vertical reference frame EVRF2007, nearby
common points with heights in both systems were utilized;
this information was kindly provided by BKG in Germany
(M. Sacher, personal communication, 9 October 2015). If
such information is not available, the CRS-EU webpage
(Coordinate Reference Systems in Europe; http://www.crs-
geo.eu, also operated by BKG) can be used, which gives,
besides a description of all the national and international
coordinate and height reference systems for the participating
European countries, up to three transformation parameters
(height bias and two tilt parameters) for the transformation
of the national heights into EVRF2007 and a statement on the
quality of this transformation; the uncertainty of the trans-
formation depends on the uncertainty of the input heights as
well as the number of identical points, giving RMS residuals
of the transformation between 2 mm (Germany) and 5 mm
(France).
Regarding the individual GNSS and levelling surveys at
the four selected clock sites, very similar procedures were
employed in all cases and so the focus here is on details
for the PTB case, which serves as a typical example. On
the PTB campus, several clock laboratories exist in differ-
ent buildings, in which height markers were installed and
observed by double-run geometric levelling in connection to
three existing national levelling benchmarks. For the level-
ling observations, a digital automatic level Leica NA3003
(includes an automatic data control and storage system) with
bar-code fibreglass staffs was employed; the uncertainty of
the entire system, as specified by the manufacturer, is 1.2
mm for 1 km double-run levelling with fibreglass staffs (0.4
mm with invar staffs). All levelling discrepancies (loops and
double-run results) were found to be less than 1 mm, and
the official heights of the national benchmarks and the lev-
elling observations showed differences between 0.5 and 2
mm, which is fully compatible with the corresponding uncer-
123
H. Denker et al.
Fig. 3 PTB, Kopfermann-Bau: entrance and wall marker (bolt) (a), floor marker (dome) with bar-code staff in clock hall (b), height transfer to a
clock table (c), and IGS station PTBB with GNSS antenna (arrow points to ARP) and levelling on rooftop (d)
tainty budgets. In addition to this, four GNSS points were
connected to the levelling network. All observed levelling
lines and the four GNSS stations are depicted in Fig. 2.As
only one GNSS station is at ground level, while the other
three stations are located on rooftops, different height deter-
mination techniques had to be used, including a combination
of levelling outside and inside buildings (e.g. in the stair-
cases), vertical steel tape and laser distance measurements,
and trigonometric height determination, which can be quite
challenging (see Fig. 3). Nevertheless, even under these dif-
ficult circumstances, all levelling lines closed within 1 mm,
and only one station on the rooftop showed a discrepancy of 4
mm between levelling, vertical steel tape measurements, and
trigonometric height determination. Based on these findings,
the overall (relative) levelling height uncertainty within the
internal network on the PTB campus is estimated to be bet-
ter than 2–3 mm. Figure 3shows as an example of some
of the markers and measurements taken in the Kopfermann
building, which hosts PTB’s caesium fountains as well as an
ytterbium ion optical clock; the building also has a GNSS
antenna on the rooftop, which is part of the IGS network and
is denoted as PTBB.
The levelling observations at PTB were for the most part
performed during three different days in 2011, with supple-
mentary information and data being collected until mid-2014,
while the GNSS observations were carried out in a campaign
over four full days from 15 to 19 September 2011 (i.e. four
full 24-h sessions). The GNSS observations were processed
with the Bernese 5.0 software using the IGS station PTBB
as a reference station (with fixed coordinates). Furthermore,
IGS precise satellite orbits and clock information as well as
individual antenna phase centre corrections (phase centre off-
set PCO plus phase centre variations PCV) were considered.
Due to the short baselines, the station coordinates were com-
puted as L1 solutions with an elevation cut-off angle of 15◦
(chosen due to obstructions limiting satellite visibility) and
123
Geodetic methods to determine the relativistic redshift at the level of 10−18 in the...
tropospheric parameter estimation; the corresponding phase
ambiguities (about 100 per 24 h session) could be resolved
in 99% of all cases. In addition to the Bernese solution,
PPP solutions were also computed; the two results showed
excellent agreement, with maximum differences of only 4
mm in the ellipsoidal heights. The uncertainty of the GNSS
ellipsoidal heights is estimated to be better than 10 mm.
The GNSS processing resulted in station coordinates at the
epoch of the observations (i.e. 2011.71) in the ITRS using the
frame ITRF2008 (as recommended in Boucher and Altamimi
2011). The ITRF2008 Cartesian coordinates and the given
velocities were then transformed to ellipsoidal quantities
based on the GRS80 reference ellipsoid. In principle, all
quantities (GNSS coordinates, geoid, levelling) should be
estimated at the actual epoch of a clock experiment, i.e. for
the ellipsoidal heights by using the corresponding coordinate
velocities; however, this is not considered further here, as all
ellipsoidal height velocities are below 0.5 mm/year in the
present case. Ellipsoidal coordinates are given in Table 2at
the epoch 2005.0, the standard reference epoch associated
with the ITRF2008 (see Altamimi et al. 2011).
Similar GNSS and levelling surveys were also carried
out at the LUH, Hannover, and at the MPQ, Garching (near
Munich), both located in Germany, as well as at OBSPARIS
in Paris, France. All measurements at the German sites were
taken under the responsibility of LUH, while OBSPARIS
commissioned IGN to do corresponding observations on their
campus. IGN used a Leica DNA03 instrument with a veri-
fied uncertainty below 1 mm for the levelling measurements
and carried out GNSS observations in connection to the IGS
station OPMT on the OBSPARIS campus. In total, 4 GNSS
and levelling stations are available on the PTB campus, while
the corresponding figures for the other sites are 4 (LUH), 3
(MPQ), and 3 (OBSPARIS), all having similar uncertainties
as outlined for the PTB case. In addition, further benchmarks
were installed in different physical (optical clock) laborato-
ries and connected to the local GNSS and levelling network
established at each site.
Appendix 5: List of abbreviations
ARP Antenna reference point
BCRS Barycentric Celestial Reference System
BIH Bureau International de l’Heure
BIPM Bureau International des Poids et Mesures,
Sèvres, France
BKG Bundesamt für Kartographie und Geodäsie,
Frankfurt am Main and Leipzig, Germany
CCDS Comité Consultatif pour la Définition de la Sec-
onde
CCTF Consultative Committee for Time and Fre-
quency
CGPM Conférence Générale de Poids et Mesures
CIP Celestial Intermediate Pole
CNRS Centre National de la Recherche Scientifique,
Paris, France
CRS-EU Coordinate Reference Systems in Europe
DFG Deutsche Forschungsgemeinschaft
DHHN Deutsches Haupthöhennetz
DOT Dynamic ocean topography
EGG European Gravimetric (quasi)Geoid
EGM Earth Gravitational Model
EMRP European Metrology Research Programme
EOP Earth orientation parameter
ERA Earth Rotation Angle
ETRF European Terrestrial Reference Frame
ETRS European Terrestrial Reference System
EUVN_DA European Vertical Reference Network—
Densification Action
EVRF European Vertical Reference Frame
EVRS European Vertical Reference System
GBVP Geodetic boundary value problem
GCRS Geocentric Celestial Reference System
GFZ Deutsches GeoForschungsZentrum, Helmholtz-
Zentrum Potsdam
GIA Glacial isostatic adjustment
GNSS Global Navigation Satellite Systems
GOCE Gravity Field and Steady-State Ocean Circula-
tion Explorer
GPS Global Positioning System
GRACE Gravity Recovery and Climate Experiment
GRS80 Geodetic Reference System 1980
GRT General relativity theory
IAG International Association of Geodesy
IAU International Astronomical Union
ICRF International Celestial Reference Frame
ICRS International Celestial Reference System
IERS International Earth Rotation Service
IGN L’Institut National de l’Information
Géographique et Forestière, Paris, France
IGS International GNSS Service
IHRS International Height Reference System
INRIM Istituto Nazionale di Ricerca Metrologica,
Torino, Italy
ITOC International Timescales with Optical Clocks,
EMRP project
ITRF International Terrestrial Reference Frame
ITRS International Terrestrial Reference System
IUGG International Union of Geodesy and Geophysics
LGLN Landesamt für Geoinformation und Landesver-
messung Niedersachsen
LNE Laboratoire National de Métrologie et d’Essais,
Paris, France
LSC Least-squares collocation
123
H. Denker et al.
LUH Leibniz Universität Hannover, Hannover, Ger-
many
MPQ Max-Planck-Institut für Quantenoptik, Garch-
ing, Germany
MSL Mean sea level
NAP Normaal Amsterdams Peil (tide gauge)
NGA National Geospatial-Intelligence Agency, USA
NGF Nivellement Général France
NIST National Institute of Standards and Technology,
Boulder, Colorado, USA
NKG Nordic Geodetic Commission
NMI National metrology institute
NPL National Physical Laboratory, UK
OBSPARIS l’Observatoire de Paris, Paris, France
PCO Phase centre offset
PCV Phase centre variations
PPP Precise point positioning
PSL Research University Paris, Paris, France
PTB Physikalisch-Technische Bundesanstalt, Braun-
schweig, Germany
RCR Remove–compute–restore (technique)
RMS Root mean square
SI International System of Units
SYRTE Système de Références Temps-Espace, located
at OBSPARIS, Paris, France
TAI International Atomic Time
TCB Barycentric Coordinate Time
TCG Geocentric Coordinate Time
TT Terrestrial Time
UELN United European Levelling Network
UK United Kingdom
UPMC Université Pierre et Marie Curie—Paris 6,
Paris, France
USA United States of America
UTC Coordinated Universal Time
VLBI Very Long Baseline Interferometry
WGS84 World Geodetic System 1984
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