theoretical fundamentals of gravity field modelling using … · 2017. 5. 16. · linear...
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Theoretical fundamentals of gravity field modelling using airborne, satellite and surface data
Rene Forsberg, DTU-Space, Denmark
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Geodesy – corrections for levelling, geoid, deflections of the vertical .. Typical geoid applications: RTK-GPS, lidar, hydrography, marine vertical datum ..
Heights from GPS: H = hellipsoidal – N The 1 cm-geoid is within reach in countries with good gravity coverage or for special projects like large bridges ..
Gravity field - an old science with new applications -
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Geophysics – gravity integrated part of geophysical studies with seismic and magnetics Regional geology .. oil & gas exploration, mining .. UNCLOS .. bathymetry in ice-covered regions, ocean ridges, continental shelf limits
GREENLAND
NORTH POLE
Greenland examples (Nunaoil / UNCLOS): Top: seismic + gravity .. saltdomes detected! Right: integrated modelling of East Greenland ridge
Gravity field
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Basic physical geodesy
Anomalous potential (non-ellipsoidal potential): Full-fill Lapace equation ∇2T = 0 => classical potential field theory can be used .. - spherical harmonic expansions, boundary value problems Gravity field quantities become functionals of T: Geoid: Quasi-geoid: Gravity anomaly: Deflections:
),,(),,(),,( rUrWrT λϕλϕλϕ −=
γ)0()( === hTTLN N
γς ς
)()( terrainhhTTL ===
rT
rTTLg g 2)( −∂∂
−==∆ ∆ λϕγη
ϕγξ
η
ξ
∂∂
−==
∂∂
−==
Tr
TL
Tr
TL
)cos(1)(
1)(
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Geoid and heights
Geoid = Actual Equipotential Surface
Unmodeled Mass
gQ
• GRAVITY ANOMALY: ∆ g = | g Q| - | γ P|
γp
Q
P
Ellipsoid = Reference Model Equipotential Surface
H = Orthometric Height
N = Geoid Height
• H (Orthometric Height) = h (Ellipsoid Height) – N (Geoid Height)
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Spatial gravity field
Challenge in gravity field modelling: handling spatial data in full 3D (r,ϕ,λ) Satellite data: Easy … comes as spherical harmonics (e.g. EGM2008 nmax = 2190, GOCE R5 ”direct” nmax = 300):
( ) [ ] ( )∑∑= =
+
=
max
2 0sinsincos,,
n
nnm
n
mnm
n
ref PmSmCrR
rGMrT φλλλφ
Level 1: 0 km
Level 1: 3 km
Linear interpolation in height
• Function in space, and reference gravity at a point P should be evaluated at the correct elevation r = R + hP • 3-dimensional interpolation between reference grids (“sandwich grid” interpolation).
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Free-air anomaly – removes field due to reference earth ellipsoid interior mass (note quadratic term – important for airborne gravity: H above is orthometric height above geoid – not GPS ellipsoid height Gravity disturbance is obtained if GPS ellipsoidal heights are used Important – large difference: (e.g. case for IceBridge data)
Gravity anomaly definition
N 0.3086 2g - g =−=∆RTδ
φφ
φγφγϕγ
2222
22
0sincos
sincos)(ba
ba ba+
+=
Normal gravity – the gravity from the “normal” field with constant potential on the WGS84 ellipsoid - “GRS80 formula”:
]/[10*2.73086.0)(g 280 mmgalHHgHg−−+−≈−=∆ γγ
]/[10*2.73086.0)(g 280 mmgalhhghg−−+−≈−= γγδ
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For airborne gravity Bouguer anomalies must be computed by 3D mass integration (and filtered appropriately)
Bouguer anomalies
cHGg +−∆=∆ ρπ2gBA
Bouguer anomalies – removing the terrain density effect above the geoid
]/[1967.02g 0BA mmgalHgHGg +−≈−∆=∆ γρπ
Simple Bouguer: Complete Bouguer: (c terrain correction)
( )( ) ( ) ( )[ ]
( )
∫ ∫ ∫∞
∞−
=
= −+−+−
−=
yxHz
HzQQQ
PQPQPQ
P
P
dzdydxHzyyxx
HzGPc,
2/3222ρ
Classical terrain correction integral – can be computed by prisms or FFT
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Correlation of free-air anomalies, terrain corrections (c) and Bouguer anomalies with height in a 100 x 100 km local area
Correlation with height: South Greenland fjord region
Anomalies and terrain
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Airborne gravity principle
Operational since the 1990’s … large scale surveys pioneered by US NRL Basic principle: ∆g = y - h´´ - δgeotvos - δgtilt - y0 + g 0 - γ0 + 0.3086 (h - N) + 2nd order terms y: measured acceleration h´´: acceleration from GPS y0: airport base reading g0: airport reference gravity h : GPS ellipsoidal height δgeotvos: Eotvos correction δgtilt: Gravimeter tilt correction Current accuracy approx 1-2 mGal @ 5 km along-track filtering (platform systems)
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NRL Greenland survey
Airborne gravity .. Greenland Aerogeo- physical project 1991-92 Cooperation: US Naval Research Lab (J. Brozena) NOAA (G. Mader) Danish National Survey (now DTU Space) NIMA (now NGA) First continental-scale airborne survey Lots of problems .. GPS in its infancy Processing not refined, accuracy ~ 4-5 mGal Nominal flight elevation 4100 m.
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Gravity – Arctic Ocean example
• Prime example of gravity signatures – submarine, surface, airborne data • Used for bathymetry inversion and sediment structures in UNCLOS projects (Denmark, Canada, US, Russia (VNIIOkeangeologia)
Arctic gravity project gravity compilation (DTU, SK, NGA, VNIIO, Tsniigaik, NRCan, ICESat, ...)
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Airborne gravity surveys: 1992-2003 US NRL, DNSC-Norway, Canada, AWI Germany, Russia ..
US Naval Research Lab (Brozena)
ArcGP core data
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Lomonosov Ridge airborne gravity and magnetic survey
DC3 used for airborne survey 2009 Russian icebreaker 50 Let Pobedy (LOMROG07 Denmark-Sweden-Russia)
• Airborne gravity and magnetics (LOMGRAV09 – DTU+NRCan) ~ 1.5 mGal r.m.s. error • Russian airborne surveys from Tiksi and Murmansk 2003-2006 • Icebreaker cruises with marine and helicopter gravimetry Grav + Mag -> structure and sediment thickness Fills GOCE polar gap Geoid useful for sea-ice altimetry
Arctic Ocean 2009 survey
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Example: Malaysia 2002-3
Fig. 2a. Flight lines in East Malaysia. Colour coding represents flight elevation.
East Malaysia 2002 flight tracks First national large-scale survey dedicated for national geoid (GPS-RTK support) Carried out for JUPEM
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West Malaysia flight tracks + existing data
Fig. 2b. Flight lines in East Malaysia. High elevation mainly due to airspace restrictions.
Fig. 3. Surface gravity coverage in East Malaysia (colours indicate anomalies)
Example: Malaysia 2002-3
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Example: Malaysia 2002-3
Malaysia airborne gravity results – statistics Original data or Bouguer anomaly continued to constant elevation
Unit: mgal
Mean x-over
R.m.s. x-over
R.m.s. error
Original free-air data at altitude
0.18
3.16
2.23
Bouguer anomalies at 2700 m
0.12
2.78
1.96
Do, after bias adjustment
-0.05
2.26
1.60
Unit: mgal
Mean x-over
R.m.s. x-over
R.m.s. error
Original free-air data at altitude
-.09
2.37
1.68
Bouguer anomalies at 3400 m
-.06
2.36
1.67
Do, after bias adjustment
-.06
1.81
1.28
East
West Malaysia
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Malaysia airborne gravity results – Bouguer gravity maps
Example: Malaysia 2002-3
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Final geoid models – revised 2008 due to Sumatra earthquake (fit in KL area after new re-levelling ~ 2 cm)
Example: Malaysia 2002-3
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New developments: IMU data
IMU + LCR meter ideal combination: high linearity combined with bias stability First DTU test: Chile 2013 (with IGM/NGA and TU Darmstadt) – demonstrate 1-2 mGal rms
Chile 2013 gravity flt elevations
iMAR IMU Green, LCR blue (D. Becker, TU Darmstadt)
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Satellite gravity: GOCE
Global gravity field from CHAMP, GRACE and GOCE … long-wavelength help aerogravity
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Satellite gravity: GOCE
GOCE gradiometer GPS Tracking Drag-free satellite measurements Orbit inclination 83° => Polar gap!
Global geoid models available as spherical harmonics data to degree 260 from European GOCE Consortium Latest model R5 Complete GOCE Mission data GOCE dived into low
Earth orbit in final months (240 -> 230 -> 225 km)
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Satellite gravity: GOCE
2160
Airborne gravity
GOCE observations: Gravity gradients … (SGG: Tzz, Txx etc) GOCE Level-2 data in spherical harmonics:
[ ] )(sinPsincos)(),,( ϕλλλφ nmnmnmn
=0m2=n
m D + m Cn
Rr
rGM = rT −∑∑
∞
GRACE limitation ~ degree 90 GOCE to degree ~ 220-240 Airborne ~ degree 2160 (5’) Surface data ~ degree 10000 (1’)
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Gravity validation: GOCE
SE-Asia Comparison to GOCE
as a function of max degree N (R5 direct)
Unit: mGal
N Mean Stddev
Data 45.3 31.2
180 1.1 39.7
200 0.6 36.9
220 0.2 34.3
240 0.0 32.6
260 -0.1 30.7
280 -0.0 30.7 DTU Surveys - r.m.s. error 1.8-2.5 mGal
Philippines 2012-13
Malaysia 2002-3
Indonesia 2008-11
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Gravity validation: GOCE
-21
-19
-17
-15
0 300 600 900 1200 1500 1800 2100
Degree
Log
[deg
ree
varia
nce]
Surface data
GOCE data
Surface – GOCE data GOCE
Zoom-In
Spectral analysis for GOCE validation Malaysia/Indonesia/Phillipines DTU surveys Method: - Fill-in by EGM08 (mainly marine) - 2D PSD estimation with FFT - Isotropic averaging of PSD - Conversion to degree variance σ
Surface – GOCE
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Back to theory ..
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Stokes formula
σψπγ
σ
d ) S(g 4
R = N ∫∫ ∆
)2
+2
( 3 - 5 - 1 + 2
6 - )
2(
1 = )S( 2ψψψψψψ
ψ sinsinlogcoscossinsin
Stokes formula – the geoid N can be determined by a global integral … ∆g assumed on geoid
Stokes formula for geoid In point P can be integrated either in (lat,lon cells) or (ψ, A)
The classical geoid determination … requires downward continuation of airborne gravity
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σψπγ
ζσ
d ) S()g+g( 4
R = 1∆∫∫
Non-level surface => Molodenskys formula: ζ is quasi-geoid
Definition of gravity anomaly: Refers to surface of topography!
H h
+ -g -g = g oobservedPP
observedPP ∂
∂≈∆ ′
γγγ
Geoid and quasigeoid
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Remove-restore methods
General remove-restore terrain reductions
“Remove” “Restore” Case of geoid computation from gravity Gravity and geoid functionals Case of downward/upward continuation Make RTM- or Bouguer anomalies -> downward continue -> restore terrain
( ) ( ) ( )mobsobscobs TLTLTL −=
( ) ( ) ( )mpredcpredpred TLTLTL += predobs LL →
( ) ( ) Trr
TTLTL gobs2
−∂∂
−−= ∆
( ) ( )γζTTLTLpred ==
Remove long wavelengths: EGM2008/GOCE combination Remove shorter wavelengths: terrain effects (e.g. from SRTM)
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Terrain effect types
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RTM gravity anomalies Correspond to the use of spherical harmonics reference field T’=T-Tref Smooth mean elevation surface href(φ,λ)
( ) ( ) ( )[ ]( )( )
∫ ∫ ∫∞
∞−
=
= −+−+−
−=∆
yxhz
yxhzQQQ
PQPQPQ
PRTM
ref
dzdydxhzyyxx
HzGg,
,2/3222
ρ
( ) chhGg refRTM −−≈∆ ρπ2
when the mean elevation surface is a sufficiently long-wavelength surface
RTM effects on gravity
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Terrain effects: prisms
• The rectangular prism of constant density is a useful "building block" for numerical integrations of the basic effects – gravity and geoid formulas:
z+y+x =r
,|||zrxyz -r)+(xy + r)+(y x|||G = g
222
zz
yy
xxm
12
12
12
arctanloglogρδ
||| zrxy
2z -
yrxz
2y -
xryz
2x -
r)+(xyz +r)+(yxz + r)+(zxy |||G = T
zz
yy
xx
222
m
12
12
12
arctanarctanarctan
logloglogρ
• Bouguer anomalies .. require DEM and terrain corrections .. 3’ SRTM data perfect data source.
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Prism approximations
Approximation in spherical harmonics (larger distances)
Efficient implementations (GRAVSOFT TC):
• Coarse/detailed grid
• Splines in inner zone
• Use of station heights (inner zone modification)
( ) ( )[
( ) ] [ ]
+++∆+∆−∆−+
∆−∆+∆−+∆−∆−∆+∆∆∆=
4492222
222222225
28812
2224
11
yxr
zzyx
yzyxxzyxrr
zyxGTm
βα
ρ
121212 ,, zzzyyyxxx −=∆−=∆−=∆
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• Basic definition of 2-D Fourier transform
• kx and ky are called wavenumbers (like frequency in 1-D time domain) … defined on infinite x-y plane
yxykxki
yx
ykxkiyx
dkdkekkGyxgGF
dxdyeyxgkkFgF
yx
yx
∫∫∫∫
+−
+−
==
==
)(2
1
)(
),(4
1),()(
),(),()(
π
Advantage of Fourier transforms: convolution theorem
• Convolutions must faster in frequency domain than space domain …many geodetic integrals can be expressed as convolutions
)()()*(
'')','()','(),(*
gFfFgfF
dydxyxgyyxxfyxgf
⋅=⇒
−−= ∫∫
g F(g)
2-D Fourier transforms
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• Derivates of Fourier transform:
• Vertical derivates from upward continuation formula:
• Anomalous potential relationships follows from these (allow the direct determination of geoid: transform + filter + inverse transform!)
)()(
)()()()/2()(
TFkF
TFkFTFrkgF
x
y
γη
γξ
−=
−=+=∆
)()( gFikxgF x δ=∂∂
22
))0,,(()),,((
yx
kz
kkk
eyxgFzyxgF
+=
= −δδ
FFT and gravity field quantities
Basic equations for geoid determination, deflections, upw.continuation – in planar approximation (on sphere: spherical FFT ….)
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The terrain correction as convolutions
[ ]dxdy
syxhyxhGyxc
E
PPPP ∫∫
−= 3
0
2),(),(21),( ρ ),(),(),,(),(,2
1 30
2 yxsyxryxhyxnGK −=== ρ
( ) ( ) ( )[ ]PpPpPpPP yxtyxtyxtKyxc ,,,),( 321 ++=
( ) ∫∫ −−=E
PPPP dxdyyyxxryxnyxt ),(),(,1
( ) ( )∫∫ −−−=E
PPPPPP dxdyyyxxryxhyxhyxt ),(),(,2,2
( ) ∫∫ −−=E
PPPPPP dxdyyyxxryxnyxt ),(),(,3
( ) ( )[ ])0,0(2 22 RhrhhrhKc PPP +∗−∗=Final formula – c as convolutions in h and h2: Convolutions very fast evalutated by FFT: Much more in IAG geoid schools …
Terrain corrections by FFT
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Airborne terrain effects by FFT
( ) ( ) ( ) ( )[ ]yxkyxhyxkyxhGzGzyxc avPP ,,,,2),,( 22110 ∗+∗+−= ρρπ
• Integration with respect to z
• Applying some analytical evaluations
• Introducing into the derivations Zav (mean height of elevations)
avzyxhyxh −= ),(),(1
[ ]22 ),(),( avzyxhyxh −=
( ) ( )( )[ ] 2/320220
1 ,av
av
zzyx
zzyxk
−++
−−= ( )
( )[ ]( )
( )[ ] 2/320220
2/320
222
2
3
2
1,av
av
av zzyx
zz
zzyxyxk
−++
−−
−++=
Analytical derivation in Tziavos et al. (1988)
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Downward continuation
Downward continuation of airborne gravity important application of spatial least-squares collocation (planar or spherical self-consistent models)
s is ”signal” (prediction quantity), x ”observation”, Csx = cov(Ls(T), Lx(T)) from model
- The covariance model must be harmonic: ∆ {cov( · ,T)} = 0
- Use of remove-restore terrain reductions stabilizes solution
- May be done blockwise .. e.g. 1º x 1 º blocks with overlap … makes solution fast, avoiding very large sets of linear equations
Alternative for constant-elevation surveys: Fourier transformation
F [∆g(x,y,h)] = e-kh F [∆g(x,y,0)]
k = √(kx2+ky2)
1][ −+= DCCs xxsx
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Downward continuation (2)
Self-consistent covariance model for aerogravity
iTDD
hhDsDCggC
i
iiii
hh
−=
++++−=∆∆ ∑=
4
1
221
20 )(log(),( 21 α
• Planar domain ok for downward continuation of airborne data • Requirement: spatial analytical covariance function model - e.g. Tscherning-Rapp model on sphere - globall - e.g. planar logaritmic model - planar
Model fitted to empirical data by three parameters: C0, D, T - D corresponding to Bjerhammar sphere depth - T is a long-wavelength attenuation ”compensation depth” - Complete formulas for gravity, geoid, 2nd order gradients in Forsberg (1987) … useful for downward continuation, deflections and gradiometry
Spherical Tscherning-Rapp model
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Deflections of the vertical
Vector gravimetry – PEI, Canada (S. Ferguson/SGL & Forsberg, AGU 2012)
Data: SRTM DEM (Quite benign region) NRCan /GSDgravimetry (M. Veronneau)
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Vector gravimetry data – SGL survey
SGL AIRGRAV survey tied to g-value at airport; converted to gravity anomaly using EGM08 geoid. Deflections of the vertical fitted by survey-wide bias and slope to EGM08. Comparison SGL ∆g minus GSD ∆g: mean = 0.3 mgal, r.m.s. 1.5 mgal (points within 500 m; no downward continuation) => excellent quality!
Flight elevation ~ 1000 ft
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PEI geoid comparison
Geoid predictions by FFT compared to GSD GPS-levelling on PEI (18 1st order points) Blue rows: What would happen if only GOCE and SGL data available?
GPS levelling – geoid from NRCan gravity Comparison of geoid solutions
Geoid model
Mean (m)
Std.dev. (m)
Geoid from NRCan data only + RTM
-0.056 0.031
Geoid from SGL gravity + RTM -0.069 0.032
CGG10 NRCan geoid model -0.013 0.039
Geoid from GOCE only -0.122 0.262
Geoid from GOCE and SGL gravity -0.124 0.216
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DoV comparison
Line 25 – E-W line through center of island (deflections fitted by bias and slope). Black: observed data; green: ”Best” FFT solution; pink: CGG10 deflections Results show horizontal gravity accurate @ 1-2 mgal accuracy (1” ~ 4.8 mGal)
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Repeated DoV line example
Airborne deflections of vertical – East component – repeat flights (Alexandria line near Ottawa; Ferguson, SGL – 2006)
0.784 0.786 0.788 0.79 0.792 0.794 0.796 0.798 0.8-8
-6
-4
-2
0
2
4
6Alexandria repeat lines, uncorrected
latitude (radians)
Eas
t def
lect
ion,
arc
sec
0.784 0.786 0.788 0.79 0.792 0.794 0.796 0.798 0.8-4
-3
-2
-1
0
1
2
3Alexandria repeat lines, linear trend removed
latitude (radians)
Eas
t def
lect
ion,
arc
sec
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Summary
• Airborne gravity complements surface and satellite gravity … GOCE R5 models agree with airborne data to degree 200-220 (except polar gaps >83°)
• Important to understand processing parameters for merging data from separate surveys (e.g., disturbance or free-air anomalies, atmospheric corrections applied, filtering parameters etc.)
• Gravity field modelling in 3D allows optimal combination of all available data - least-squares collocation combined by Fourier methods efficient
• Terrain effects important – Bouguer anomaly computations for geophysics, stabilization of downward continuation for merged free-air anomaly grid for geoid determination or prediction of deflections of the vertical
• Deflections of the vertical can be handled analogous to (vertical) gravity – full vector gravimetry systems operational ..
More on geoid determination and GRAVSOFT in Wednesday talk/demo
Theoretical fundamentals of gravity field modelling using airborne, satellite and surface data��Rene Forsberg, DTU-Space, DenmarkSlide Number 2Slide Number 3Basic physical geodesy Geoid and heightsSpatial gravity fieldGravity anomaly definitionBouguer anomaliesSlide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45