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