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Theoretical Fundamentals of Gravity Field Modelling Using Airborne, Satellite and Surface Data

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Theoretical Fundamentals of Gravity Field Modelling Using Airborne, Satellite and Surface Data 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): T(,,)(,,)(,,)ϕ λ r= W ϕ λ r− U ϕ λ r 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: T( h = 0) Geoid: NLT=() = N γ T() h= hterrain 1 ∂T Quasi-geoid: ς =LTς () = ξ =LT() = − γ ξ γr ∂ ϕ ∂ 1 ∂T Gravity anomaly: T T Deflections: η = = − ∆g = L∆ ( T ) =− − 2 LTη () g ∂r r γr cos( ϕ) ∂ λ 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 nmax n GM R Tref () r,,φ λ = ∑∑ [Cnm cos mλ+ Ssin m λ] Pnm ()sin φ 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 aγa cos φ+ b γb sin φ γ0 () ϕ = a 2cos 2 φ + b 2sin 2 φ Free-air anomaly – removes field due to reference earth ellipsoid interior mass (note quadratic term – important for airborne gravity: −8 2 ∆g =g −γ () H ≈ g −γ 0 + 0.3086H − 7.2*10H [ mgal / m ] H above is orthometric height above geoid – not GPS ellipsoid height Gravity disturbance is obtained if GPS ellipsoidal heights are used −8 2 δg =g −γ() h ≈ g − γ 0 +0.3086h − 7.2*10h [ mgal / m ] Important – large difference: 2T ∆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 = ∆g −2π G ρ H ≈ g − γ 0 + 0.1967H [ mgal / m ] Complete Bouguer: (c terrain correction) ∆gBA = ∆g −2π G ρ H + c 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 = () ∞ z H x, y z− H c() P= Gρ P dx dy dz ∫∫ ∫ 2 2 2 3/ 2 QQQ −∞ z= H P [()()()xQP− x + yQP − y + zQP − H ] 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 − T(,,) rφ λ = () n[ cos mλ + sin mλ] (sin ϕ) spherical harmonics: ∑∑ C nm Dnm Pnm r n=2 m=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 = ∆g S(ψ ) d σ 4πγ ∫∫ σ 1 ψ ψ ψ S(ψ ) = - 6 sin + 1 - 5 ψ - 3 coscos ψ 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 ∆ ψ σ = = ∫∫ ( g + g1 ) S( ) d 4πγ σ ∂γ Definition of gravity anomaly: observed observed ∆ g PP = g - 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: LTLTLTobs ()()()=obs − obs m EGM2008/GOCE combination LLobs → pred Remove shorter wavelengths: terrain effects (e.g. from SRTM) LTLTLT () = c + “Restore”() pred pred () pred m Case of geoid computation from gravity ∂T 2 LTLTobs ()()=∆g − − − T Gravity and geoid functionals ∂r r T LTLT()()= = pred ζ γ Case of downward/upward continuation Make RTM- or Bouguer anomalies -> downward continue -> restore terrain Terrain effect types RTM
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