Theoretical fundamentals of gravity field modelling using airborne, satellite and surface data
Rene Forsberg, DTU-Space, Denmark Gravity field - an old science with new applications -
Geodesy – corrections for levelling, geoid, deflections of the vertical ..
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 ..
Typical geoid applications: RTK-GPS, lidar, hydrography, marine vertical datum ..
Gravity field
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
NORTH POLE
GREENLAND
Greenland examples (Nunaoil / UNCLOS): Top: seismic + gravity .. saltdomes detected! Right: integrated modelling of East Greenland ridge Basic physical geodesy
Anomalous potential (non-ellipsoidal potential):
= − λϕλϕλϕ rUrWrT ),,(),,(),,(
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:
hT = )0( Geoid: TLN )( == N γ
= hhT terrain )( 1 ∂T Quasi-geoid: ς ς TL )( == ξ TL )( −== γ ξ r ∂ϕγ
∂ 1 ∂T Gravity anomaly: T T Deflections: η −== ∆ TLg −==∆ − 2)( η TL )( g ∂r r r cos( ) ∂λϕγ Geoid and heights
H = Orthometric Height Geoid = Actual Equipotential Surface
Ellipsoid = Reference Model Equipotential Q N = Geoid Height Surface gQ
P Unmodeled Mass γp
• GRAVITY ANOMALY: ∆ g = | g Q| - | γ P|
• H (Orthometric Height) = h (Ellipsoid Height) – N (Geoid Height) 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):
n n n GM max R ref ()rT ,, λφ = ∑∑ [ nm cos + sin ] PmSmC nm ()sinφλλ r n=2m = 0 r
• 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).
Linear interpolation in height Level 1: 3 km
Level 1: 0 km Gravity anomaly definition
Normal gravity – the gravity from the “normal” field with constant potential on the WGS84 ellipsoid - “GRS80 formula”:
2 2 a a cos + b b sin φγφγ 0 ϕγ )( = a cos 22 φ + b sin 22 φ
Free-air anomaly – removes field due to reference earth ellipsoid interior mass (note quadratic term – important for airborne gravity:
− 28 g γ )( gHg γ 0 +−≈−=∆ H − mmgalH ]/[10*2.73086.0
H above is orthometric height above geoid – not GPS ellipsoid height Gravity disturbance is obtained if GPS ellipsoidal heights are used
− 28 δg )( ghg γγ 0 h −+−≈−= mmgalh ]/[10*2.73086.0
Important – large difference: 2T g - g - δg =−=∆ 0.3086 N (e.g. case for IceBridge data) R
Bouguer anomalies
Bouguer anomalies – removing the terrain density effect above the geoid
Simple Bouguer: gBA 2 gHGg γρπ 0 +−≈−∆=∆ mmgalH ]/[1967.0
Complete Bouguer: (c terrain correction) gBA 2 ρπ +−∆=∆ cHGg
For airborne gravity Bouguer anomalies must be computed by 3D mass integration (and filtered appropriately)
Classical terrain correction integral – can be computed by prisms or FFT
= () ∞ , yxHz − Hz () = GPc ρ P dzdydx ∫∫ ∫ 2 2 2 2/3 QQQ ∞− =Hz P [()()()PQ PQ −+−+− Hzyyxx PQ ] Anomalies and terrain
Correlation with height: South Greenland fjord region
Correlation of free-air anomalies, terrain corrections (c) and Bouguer anomalies with height in a 100 x 100 km local area 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)
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.
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, ...) ArcGP core data
Airborne gravity surveys: 1992-2003 US NRL, DNSC-Norway, Canada, AWI Germany, Russia ..
US Naval Research Lab (Brozena) Arctic Ocean 2009 survey
• Airborne gravity and magnetics Lomonosov Ridge airborne gravity and magnetic survey (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
Russian icebreaker 50 Let Pobedy DC3 used for airborne survey 2009 (LOMROG07 Denmark-Sweden-Russia) Example: Malaysia 2002-3
East Malaysia 2002 flight tracks First national large-scale survey dedicated for national geoid (GPS-RTK support) Carried out for JUPEM
Fig. 2a. Flight lines in East Malaysia. Colour coding represents flight elevation. Example: Malaysia 2002-3
West Malaysia flight tracks + existing data
Fig. 2b. Flight lines in East Malaysia. High elevation mainly due to Fig. 3. Surface gravity coverage in East Malaysia (colours airspace restrictions. indicate anomalies) Example: Malaysia 2002-3
Malaysia airborne gravity results – statistics Original data or Bouguer anomaly continued to constant elevation
Unit: mgal Mean x- R.m.s. x- R.m.s. over over error Original free-air data at 0.18 3.16 2.23 East altitude Bouguer anomalies at 0.12 2.78 1.96 2700 m Do, after bias -0.05 2.26 1.60 adjustment
Unit: mgal Mean x- R.m.s. x- R.m.s. over over error Original free-air data at -.09 2.37 1.68 West altitude Malaysia Bouguer anomalies at -.06 2.36 1.67 3400 m Do, after bias -.06 1.81 1.28 adjustment
Example: Malaysia 2002-3
Malaysia airborne gravity results – Bouguer gravity maps
Example: Malaysia 2002-3
Final geoid models – revised 2008 due to Sumatra earthquake (fit in KL area after new re-levelling ~ 2 cm)
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)
Satellite gravity: GOCE
Global gravity field from CHAMP, GRACE and GOCE … long-wavelength help aerogravity 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) Satellite gravity: GOCE
GOCE observations: Gravity gradients … (SGG: Tzz, Txx etc)
GOCE Level-2 data in GM ∞ n r − = rT λφ ),,( = )( n[ cos m λ + sin m ] (sinϕλ ) spherical harmonics: ∑∑ C nm Dnm Pnm r m2=n =0 R
GRACE limitation ~ degree 90 GOCE to degree ~ 220-240 Airborne ~ degree 2160 (5’) Surface data ~ degree 10000 (1’)
Airborne gravity
2160 Gravity validation: GOCE
SE-Asia Comparison to GOCE as a function of max degree N (R5 direct) Philippines 2012-13 Unit: mGal
N Mean Stddev Malaysia 2002-3 Data 45.3 31.2
180 1.1 39.7
200 0.6 36.9
Indonesia 2008-11 220 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 Gravity validation: GOCE
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
Zoom-In
-15
-17 Surface data
-19 Surface – Surface –
Log [degree variance] [degree Log GOCE data -21 GOCE data GOCE GOCE
0 300 600 900 1200 1500 1800 2100 Degree Back to theory .. Stokes formula
Stokes formula – the geoid N can be determined by a global integral … ∆g assumed on geoid
R = N = ∆ S(g ) d σψ 4πγ ∫∫ σ 1 ψ ψψ ψ )S( = - 6 sin + 1 - 5 - 3 coscos ψψ log( sin + 2 ) ψ sin sin( ) 2 2 2 2
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 Geoid and quasigeoid
Non-level surface => Molodenskys formula: ζ is quasi-geoid
R ζ ∆ σψ = = ∫∫ +g( g1 S() ) d 4πγ σ
∂γ Definition of gravity anomaly: observed observed ∆ g = g PP - γ P′ ≈ g P - γ o + H Refers to surface of topography! ∂h Remove-restore methods
General remove-restore terrain reductions
“Remove” c Remove long wavelengths: obs ()()()obs −= obs TLTLTL m EGM2008/GOCE combination obs → LL pred Remove shorter wavelengths: terrain effects (e.g. from SRTM) () = c + “Restore”()TLTLTL pred pred () pred m
Case of geoid computation from gravity ∂ T 2 obs ()()∆g TLTL −−= − T Gravity and geoid functionals ∂ rr T ()()TLTL == pred ζ γ
Case of downward/upward continuation
Make RTM- or Bouguer anomalies -> downward continue -> restore terrain Terrain effect types RTM effects on gravity
RTM gravity anomalies Correspond to the use of spherical harmonics reference field T’=T-Tref Smooth mean elevation surface href(φ,λ)
∞ = () ,yxhz − Hz =∆ Gg ρ P dzdydx RTM ∫ ∫ ∫ 2/3 QQQ ∞− = (),yxhz 2 2 2 ref [()()()PQ −+−+− hzyyxx PQPQ ]
when the mean elevation surface is a sufficiently long-wavelength surface
gRTM 2 ρπ ()ref −−≈∆ chhG 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:
xy x1 y1 z1 gm = ρδ x|||G log(y +r) + y log(x+r)- z arctan ||| , zr x2 y2 z2
2 r = x2 + y + z2
T m = ρ |||G xy log(z +r) + xz log(y +r)+ yz log(x+r) 2 2 2 x yz y xz z xy y - - arctan - arctan - arctan x1 1 ||| z1 2 xr 2 yr 2 zr x2 y2 z2
• Bouguer anomalies .. require DEM and terrain corrections .. 3’ SRTM data perfect data source.
Prism approximations
Approximation in spherical harmonics (larger distances)
11 2222 2222 m ρ zyxGT +∆∆∆= [(2 )( 2 ∆−∆+∆−+∆−∆−∆ )yzyxxzyx r 24r 5 1 2 zzyx 2222 +∆+∆−∆−+ βα yx 44 ++ ( ) ] 9 [ ] 288 r
12 , 12 , −=∆−=∆−=∆ zzzyyyxxx 12
Efficient implementations (GRAVSOFT TC):
• Coarse/detailed grid
• Splines in inner zone
• Use of station heights (inner zone modification) 2-D Fourier transforms
• Basic definition of 2-D Fourier transform g F(g)
+− = = yx ykxki )( yx ∫∫ ),(),()( eyxgkkFgF dxdy
−1 1 + yx ykxki )( yxgGF ),()( == yx ),( dkdkekkG yx 4π 2 ∫∫
• kx and ky are called wavenumbers (like frequency in 1-D time domain) … defined on infinite x-y plane
Advantage of Fourier transforms: convolution theorem
−−= dydxyxgyyxxfyxgf '')','()','(),(* ∫∫ ⇒ ⋅= gFfFgfF )()()*(
• Convolutions must faster in frequency domain than space domain …many geodetic integrals can be expressed as convolutions FFT and gravity field quantities
• Derivates of Fourier transform:
∂g F =ikx δgF )()( ∂x • Vertical derivates from upward continuation formula:
δ = δ ))0,,(()),,(( eyxgFzyxgF −kz
22 += kkk yx
• Anomalous potential relationships follows from these (allow the direct determination of geoid: transform + filter + inverse transform!)
+=∆ TFrkgF )()/2()(
−= yγξ TFkF )()(
−= xγη TFkF )()(
Basic equations for geoid determination, deflections, upw.continuation – in planar approximation (on sphere: spherical FFT ….) Terrain corrections by FFT
The terrain correction as convolutions
2 1 2 −3 1 [ − yxhyxh PP ),(),( ] = ρ = = ),( = Gyxc ρ dxdy 0 yxsyxryxhyxnGK ),(),(),,(),(, PP ∫∫ 3 2 2 E s0
PP = [ 1()()(),),( Pp + 2 , Pp + 3 ,yxtyxtyxtKyxc Pp ]
= −− 1()PP ∫∫ P yyxxryxnyxt P ),(),(, dxdy E −= −− 2 ()()PP PP ∫∫ P yyxxryxhyxhyxt P ),(),(,2, dxdy E = −− 3 ()PP PP ∫∫ P yyxxryxnyxt P ),(),(, dxdy E
Final formula – c as convolutions in h and h2: 2 +∗−∗= 2 Convolutions very fast evalutated by FFT: P [() 2 P () P RhrhhrhKc )0,0( ] Much more in IAG geoid schools … Airborne terrain effects by FFT
Analytical derivation in Tziavos et al. (1988)
• Integration with respect to z
• Applying some analytical evaluations
• Introducing into the derivations Zav (mean height of elevations)
1 ),(),( −= zyxhyxh av PP 0 2),,( av +−= ρρπ [ 1()()()()1 2 ∗+∗ 2 ,,,, yxkyxhyxkyxhGzGzyxc ] 2 2 [ ),(),( −= zyxhyxh av ]
()−− zz 1 3()− zz = 0 av = − 0 av 1(),yxk 2/3 2 (),yxk 2/3 2/3 22 2 22 −++ 2 22 −++ 2 [ ()0 −++ zzyx av ] 2[ ()0 zzyx av ] 2[ ()0 zzyx av ] Downward continuation
Downward continuation of airborne gravity important application of spatial least-squares collocation (planar or spherical self-consistent models)
−1 sx xx += DCCs ][ 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)]
2 2 k = √(kx +ky )
Downward continuation (2)
Self-consistent covariance model for aerogravity
• Planar domain ok for downward continuation of airborne data Spherical Tscherning-Rapp model • Requirement: spatial analytical covariance function model - e.g. Tscherning-Rapp model on sphere - globall - e.g. planar logaritmic model - planar
4 1 hh 2 −=∆∆ α 2 ++++ 2 ),( CggC 0 ∑ log( ii ( i hhDsD 21 ) i=1
i −= iTDD
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
Deflections of the vertical
Vector gravimetry – PEI, Canada (S. Ferguson/SGL & Forsberg, AGU 2012)
Data: SRTM DEM (Quite benign region) NRCan /GSDgravimetry (M. Veronneau)
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 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
Mean Std.dev. Geoid model (m) (m) Geoid from NRCan -0.056 0.031 data only + RTM Geoid from SGL -0.069 0.032 gravity + RTM CGG10 NRCan geoid -0.013 0.039 model Geoid from GOCE only -0.122 0.262 Geoid from GOCE -0.124 0.216 and SGL gravity 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)
Repeated DoV line example
Airborne deflections of vertical – East component – repeat flights (Alexandria line near Ottawa; Ferguson, SGL – 2006)
Alexandria repeat lines, uncorrected Alexandria repeat lines, linear trend removed 6 3
4 2
2 1
0 0
-2 -1 East deflection, arcsec deflection, East East deflection, arcsec deflection, East -4 -2
-6 -3
-8 -4 0.784 0.786 0.788 0.79 0.792 0.794 0.796 0.798 0.8 0.784 0.786 0.788 0.79 0.792 0.794 0.796 0.798 0.8 latitude (radians) latitude (radians) 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