Height and time systems in geodesy
and the impact of clock networks
Jürgen Müller
Institut für Erdmessung
Leibniz Universität Hannover
and
SFB 1128 Relativistic geodesy and
gravimetry with quantum sensors (geo-Q)
Goal: Determination of physical heights and/or geoid
Gravimetric quasi-geoid (like EGG2016, GCG2016) from
various data
Differences between two
realisations of the national
height system (over 25 years)
Discrepancies in height systems
Feldmann-Westendorff et al. 2016
Inconsistencies in
classical height
systems
Gruber et al. 2014
Pp
CH
g
Heights from geometric levelling
Ocean surface
geoid = equipotential surface
Earth surface
Classically establishing a (physical) height system by
levelling + terrestrial gravimetry
geopotential numbers
resp.
related to a vertical datum
• (levelling) errors increase with distance
• systematic errors (measurement periods)
• time consuming, repeated measurements
Pp
CH
g
00 0
P P
P PC W W dW g dn
Geopotential numbers
2 2
1,2
1( ) 1( )
i i i
i n i n
C g n W
Differences of
geopotential numbers
00 0
P P
P PC W W dW g dn
obtained via
gravimetry and levelling
Mean sea level
Physical heights
Orthometric heights
(Switzerland …)
Pp
CH
g
Normal heights
(Germany, …)
N PP
CH
…, Dynamic heights, Normal-orthometric heights normal plumbline
plumbline
Q PU W
Normal heights and geometric levelling
Conversion of levelled height differences into normal height differences
requires the normal correction EN(1,2)
g - mean gravity value between the two points at the surface
γ - mean normal gravity value along the normal plumb line
𝛾045 - normal gravity on the ellipsoid at the latitude of 45o
2,12,1122,1 NNNN EnHHH
245
0
45
02145
0
45
012
1 45
0
45
02,1 NNN HHdn
gE
Evaluation of different height types
Height type
unit
depending
on the path
correction
to
levelling
points with
same height
on same W?
geopotential number
m² / s²
no
no yes
raw levelling
m
yes -
no
dynamic height
m
no
large yes
orthometric height
m
no
small no
normal height
m
no
small no
normal-orthometric
height
m
yes
small no
Effect of permanent tide
real sea surface
mean-tide system (ITRS, geoid?)
zero-tide system (gravimetry)
tide-free system (IERS eqs. ?!)
Maximum difference up to several decimeters
Different use for geometric and gravimetric quantities
Possible reference surfaces
Relativistic geoid definition
C. F. Gauss: Bestimmung des Breitenunterschiedes zwischen den Sternwarten von
Göttingen und Altona, Göttingen 1828. C. F. Gauss Werke, Band IX, Leipzig 1903, p. 49
Gauss “Was wir im geometrischen Sinn Oberfläche der Erde nennen, ist nichts anderes als
diejenige Fläche, welche überall die Richtung der Schwere senkrecht schneidet, und von der die
Oberfläche des Weltmeers einen Theil ausmacht…”
What we use/realize is the gravity potential W = V + Φ = const., i.e.
only Newtonian (gravitational V and centrifugal Φ) potentials
A relativistic geoid can be defined
through surfaces of the same clock
redshift (isochronometric surface):
Wrel ~ V + Φ + 1/2 V2/c2 … = const.,
which is where clocks are sensitive to –
and it is closer to the definition of Gauss
Hackmann, Phillip, ZARM, 2017
2
relWf
f c
Geodetic reference system (GRS)
• recommended by IUGG/IAG with 4 defining constants:
o semi-major axis a
o geocentric gravitational constant GM (including the
atmosphere)
o dynamic flattening of the Earth J2 (without the permanent tide)
o angular velocity of the Earth ω
• for geodetic work and calculations in the exterior of the Earth’s
• defines the geodetic Earth’s model, incl. the level ellipsoid
• today: GRS 1980
a = 6378137 m
J2 = 1082.63 • 10-6
GM = 398 600.5 • 109 m³ / s²
ω = 7.292115 • 10-5 rad / s
Current values for GRS parameters and their temporal variations
are determined from space geodetic observations (VLBI, GNSS,
SLR/LLR, DORIS, altimetry) and gravimeter measurements:
large secular temporal change of the angular velocity w due to tidal
friction and post-glacial rebound
However, the defined values of the GRS80 are kept fixed to have
(and to maintain) a consistent basis.
m1.066378136 .a
6
2 100001.06359.1082 J 11
2 2.6 0.3 10 1/yr J
m³/s²100008.04418.398600 9GM
rad/s10292115.7 5w
13/ 1 10 1/yr G G
Geodetic reference systems (GRS)
22 24.5 0.1 10 rad/sw
1mm/yra
… if mass and potential of the reference ellipsoid do
not agree with those of the geoid
mass
potential
using
Corrections: GRS80 ellipsoid - geoid
U0 = 62 636 860.85 m2 / s² (GRS80)
W0 = 62 636 853.4 m2 / s² (IHRS, IAG 2015)
0 02 76cm
U WN
1 93cm G m
NR
Heiskanen/Moritz 1967
German height system DHHN2016
Levelling point in Wallenhorst
at Neue St.-Alexander church,
88 m above NHN,
related to Amsterdam
vertical datum Normal heights
Related systems
Geodetic Reference System 1980:
GRS80 (constants a, b(J2), M, ω)
European Terrestrial Reference
System 1989 and German System:
ETRS89/DREF91-Realisation 2016
(coordinates based on GNSS, etc.)
German Combined Quasigeoid 2016:
GCG2016 (vertical reference surface)
Time systems in geodesy
stellar time scales
solar time scales
relativistic time scales
atomic time
1 – d(TT)/d(TCG) = LG WGeoid / c2 0.6969…*10-9
Transformation between time systems
TDB Temps Dynamique Barycentric
T(D)T Temps (Dynamique) Terrestre
TAI Temps Atomic International
T(GPS) GPS Time
JD Julian Date
UT1 Universal Time 1
GMST Greenwich Mean Sideral Time
GAST Greenwich Apparent Sideral Time
MEZ Central European Time
relativity theory
-32.s184
-19s
UTC Coordinated Universal Time
-37s (currently)
1h TJD
tables with corrections
mo
de
l a
tom
ic tim
e
na
tura
l
Using clocks to determine differences of the gravity
potential („relativistic geodesy“)
Highly precise optical clocks,
e.g. Sr lattice clock, PTB Linked via fibres or satellite
2
2 2
proper time, speed of light,11 ,
velocity, ravitational potential2
cd v V
v V gdt c c
1 2
2
2 1
1d df W
d df c
1
2
Relativistic frequency change
Normal height
clock@geoid
N PP
CH
Clocks for determining physical heights
2 2
PCf W
f c c2,
N PP P
C fH C c
f
Ocean surface
quasi-geoid
(no equipotential surface)
Earth surface
0PW W W
0W
1 2
2
2 1
1d df W
d df c
Use the relation
height anomaly (h ellipsoidal height from GNSS)
Normal height difference, if none of the clocks is at the geoid
22
2
N CH
Clocks for determining physical heights
Ocean surface
quasi-geoid
Earth surface
2 1 21 f f f1f
11
1
N CH
2
fW C c
f
„chronometric levelling“ as new geodetic tool
Nh Hellipsoid
2h
2
𝐻2𝑁 = 𝐻1
𝑁1
2−
𝑐2
2
∆𝑓21
𝑓1
ellipsoid
quasi-geoid
surface
GNSS heights h
levelled heights ocean
surface
quasi-geoid
heights
Difference
few cm (Germany)
up to decimeter (Europe, USA)
Recent quasi-geoid accuracy: difference of
GNSS/levelling vs. gravimetric methods
/
N
G L Ph H
/gravimetric G L
N
PH
GNSS, levelling and gravimetric quasigeoid EGG2015 agree at the
level of 1 – 4 mm RMS at each of the three sites
Height difference results
GNSS/levelling results
Physical & geometrical heights
Nh H (HN: normal height, h: ellipsoidal height, : quasigeoid height)
Station 1 Station 2 ΔHN
Levelling
ΔHN
GNSS/EGG2015
Difference Distance
PTB LUH -32.077 m -32.082 m -0.005 m 52 km
PTB MPQ 388.770 m 388.809 m +0.039 m 457 km
LUH MPQ 420.847 m 420.891 m +0.044 m 480 km
PTB Braunschweig, LUH Hannover, MPQ Garching/Munich
Denker et al. 2016
Clock comparison between Paris and Braunschweig
Distance: 700 km
Height difference: 24.7 m
Lisdat, PTB, 2016/2017
uclocks
=30 cm (∼3m2 s2 ∼3´10-17)
ugeodesy
= 4 cm (∼0.4m2 s2)
~ ~
~
Clocks for unifying height systems
Simulation study
4 European regions where height systems differ due to
biases, different tilts and noise
Number of clocks: 2 resp. 3 per region
Clock accuracy: 10-18 (1 cm)
Hu Wu, IfE, 2017
Pros
• clocks can connect distant areas
• no (good) ground gravity data are needed, e.g. in
underdeveloped countries, or when rough environment
• discrepancies in classical realisations of height systems and
geoid solutions (e.g. using GNSS, levelling and gravimetric
data) can be resolved,
today, we still have decimeter differences when comparing,
e.g., Syrte, Paris and PTB, Braunschweig, i.e. h ≠ HN + ς
• comparison of different national height systems with
different datum (i.e. reference levels such as in South
America have decimeter discrepancies)
Advantage of using clocks for height systems
Different height systems in South America
Sanchez et al. 2015
Difference of regional vertical reference levels to a global one
(related to W0) derived from geodetic measurements (gravity, GPS,
levelling, altimetry), unit: cm
ΔWi=Wi-W0
Temporal variations of the gravitational potential
Sources
• solid earth tides
• ocean tides
• non-tidal effects (atmosphere, hydrology ….)
• …
Left: full tidal signal Vel=Vt (1+kl-hl) at PTB ~ ± 25 cm
Right: difference between PTB, Braunschweig,
and NPL, London ~ ± 8 cm
always direct effect (1+kl) and deformation (hl) combined
Potential difference due to solid Earth tides
Voigt, Timmen 2015
Gravity potential due to non-tidal mass variations
• Potential variations from
• coastDat2
• GLDAS
• Extreme example
Helgoland
Station Epochs > |0.1| m²/s²
Braunschweig 13 %
Paris 2 %
Helgoland 28 %
Lion et al. (2017): study using clocks for gravity field recovery;
input gravity anomalies Δg and clock potential values Tc, output
gravity potential T
Further applications/challenges
Massive Centrale, France
Clocks in motion
• at ships on lakes and ocean:
Wi ≈ const., ΔWi ≈ const.
• on (land) vehicles:
ΔWi ~ integral of relativistic
time/frequency equation
Further applications/challenges
zazzle.de
Relativistic effects on a transported
and continuously operating (optical) clock
Nelson 2011, Metrologia
Changes due to
1. varying gravity potential ΔW along the path
2. velocity v of the transported clock (in the Earth-fixed reference system)
3. Sagnac effect
Test case
• virtual transport of an optical clock from Braunschweig, Germany to Paris, France
• height variations about 200 m
• mean velocity 80 km/h
Result
• relativistic effect of about 2 ns
• uncertainty 0.1-1 ps, depending on the navigation quality
• [and –1.5 ns when moving in opposite direction due to Sagnac effect]
2
0 2 2 2
1 11
2
B B
A A
W vd d
c c c
ω r v
Denker et al., personal communication, 2016, results from the EMRP(REG) „Gravity
Potential for Optical Clock Comparisons“ within the ITOC project (International
Timescales with Optical Clocks)
Time interval of a clock at rest on the geoid related to a clock (transported
with velocity v) on the Earth rotating with angular velocity ω
Clocks in motion
• at ships on lakes and ocean:
Wi ≈ const., ΔWi ≈ const.
• on (land) vehicles:
ΔWi ~ integral of relativistic
time/frequency equation
• in space (reference/master clock):
averaging time
link requirements
geo-referencing of measured values
…?
Further applications/challenges
zazzle.de
2
2 2
2
2 2
( )11
2
( )11
2
/
E E A ESE
E
S s A SSSE
SE SE SE
V
f c c c
Vf
c c c
r r re
r r re
e r r
(Relativistic) Doppler effect
Velocity and potential-dependent parts
SEr
SrEr
SEr
• Low satellite (fE = 400 MHz)
Δf = fE-fS = 2250 Hz
• GPS (fE = 1227 MHz), relativistic
Δfrel = fE-fS = 0.546 Hz
Dirkx et al. (2016): clock measurements along satellite orbit,
determination of gravity potential based on redshift equation
Further applications/challenges
What we hope to get for geodetic applications
• transportable clocks with 1 cm height
accuracy
• mobile clocks (operating during transport)
• clocks in space & optical satellite links
• application of relativistic geodesy for height
unification and regional gravity field recovery
establish chronometric levelling as
standard technique in geodesy
ocean surface
geoid,
W0
Earth
surface
0f f f
0f
ellipsoid
h
N h H
2f cH
f g
Quantum metrology and relativistic geodesy provide novel
methods for geodesy and Earth observation
Relativistic geodesy
• Clocks support physical height and geoid determination by
providing independent gravity information
• Chronometric levelling will develop as a new geodetic tool
• Clocks may support gravity field recovery
Conclusions
Novel Concepts for Gravimetric Earth Observation
… are studied in SFB 1128 “Relativistic geodesy and gravimetry with
quantum sensors (geo-Q)” at the Leibniz Universität Hannover