A Mission to Planet Mars Gravity Field Determination Department for Theoretical Geodesy Graz University of Technology and Space Research Institute Austrian Academy of Sciences
Gravity field CHAMP GRACE GOCE Satellite Altimetry ERS-1/ TOPEX/Poseidon Envisat Jason Topography (DTM) SRTM Earth Observation Seismic Tomography Magnetic field OERSTED CHAMP Global Positioning GPS GLONASS GALILEO Remote Sensing ERS-1/ Envisat TOPEX/Poseidon
Precise Orbit Determination (POD) Dual role of POD: 1. POD for geolocating satellite sensors in an Earth-fixed reference frame: geo-referencing. POD for the determination of the gravitational field of the Earth: gravity field recovery
POD for geo-referencing (1/) (W W W: what - where - when) f ( P, t) Function f of position P and time t J f ˆ ( P, t) = c j ( t) ϕ j ( P) Model function f ˆ ( P, t) j= 1 ϕ (P) (t) L f P, t ) + n i i i i (... base functions,... parameters) j ( Observation i (functional of f ),... noise c j n { L f ( P, t n } l = ) + i Observations i i i Linear estimation (LSA, LSC) cˆ = { ˆ } c j parameters ˆ) Σ(c covariance matrix
POD for geo-referencing (1/) LS estimation equations: cˆ = ( A T Σ 1 ll Σ(ˆ) c = ( A T A) Σ 1 1 ll A A) T 1 Σ 1 ll l Estimated parameters: Estimated parameter statistics: Σ (cˆ ) ĉ j fˆ( P) σ ( fˆ P J = j= 1 cˆ j ) = A P ϕ ( P) = A j Σ(ˆ) c A T P P cˆ Estimated function Estimated error: f ˆ( P ) σ ( ˆ f P ) Geolocation of satellite sensors in an Earth-fixed reference frame
POD for gravity field recovery Example 1 Vertical free fall: Equation of motion: z= & r& V = && z = g z Initial state vector at : z( ) = z&( ) = Position at Velocity at t = t t Trajectory model: z( t) = z( t ) + z& ( t z ( t) = g t ) ( t t ) ( t t + g ) Gravity from observation of space and time
POD for gravity field recovery Example [, ] V T =, g Bullet trajectory in a flat gravity field: z Equation of motion: & r& V = Observation-based trajectory: r ( t ) =, r& ( t 5 ) =, 5 V x = 9.81 Trajectory model: r( t) = r( t ) + r& ( t ) ( t t Initial state vector: r( t ) r& ( t ) ) Position at Velocity at + V t t ( t t )
POD for gravity field recovery Example 3 Elliptic orbit around a masspoint: Keplerian 3rd law: r a GM = 4π a T 3 Mass ( M) determination from observation of space ( ) and time ( T) a V GM r a 3 = = 4π Gravitational potential ( V ) determination rt a, r, T from orbit observation ( )
Turning inside out mass mass Gravitational potential shape shape V = G l 1 Earth ρ dv Geoid Geoid
Turning inside out mass mass Gravitational potential shape shape V = G l 1 Earth ρ dv Gravity Gravity
The dual role of the gravity field 1 8-1 6 yr Time Scales 1-1 yr Post-glacial rebound Volcanic activities Ice Sheet Melting Ocean circulation & heat transport Sea level change
POD for gravity field recovery The idea: Satellite orbit orbit & r& V = Mass Mass distribution Gravitational field field
POD for gravity field recovery The idea: Satellite orbit orbit & r& V = Mass Mass distribution Gravitational field field
The gravitational potential Properties of the gravitational potential: 1. V is harmonic outside the body:. V decreases to zero towards infinity T V 3. V belongs to an infinite dimensional space V Consequence: is represented by a linear combination of T harmonic functions (= solutions of ) = V = V GM R = l= l m= R r l+ 1 P (cosϑ) [ C cos mλ + S sin mλ ]
The gravity potential V(P) Φ(P) W(P) Gravitational potential ( ) Rotational potential ( Φ( P) = ϖ h P / ) Gravity potential ( W) V W ( P) = const. = W W(P) Unique global horizontal surface of constant gravity potential ( W ) at mean sea level: geoid Global reference surface for orthometric height Unique local vertical Reference direction for local-horizontal reference system (see lecture by R. Rummel)
The geoid
POD for gravity field recovery & r V = F Equation of motion, defined in a space-fixed geocentric reference system free fall (around a body) Satellite motion due to surface forces F r = r ( r, r & ; t ; C, S ) Satellite orbit as a function of gravitational C, field parameters { } S V = V ( C, S ) Reference gravitational field controlled by C S r parameters { }, = r ( r, r & ; t ; C, S ) Reference satellite orbit as a function of C S, gravitational field parameters{ }
POD for gravity field recovery Principle: r = r ( C, S ) Real orbit from satellite tracking r = r ( C, S ) Reference orbit based on a priori gravitational field C S = C = S C S Residual harmonic coefficients unknowns r = r r = r( C, S ) Functional relation
POD for gravity field recovery T i [ x y, z ] = f ( C S ) r =, i, Pseudo-observations i i A = ( x ) i, yi, zi ( C, S ) Design matrix from partials { x, y, z } i i Observation residuals i LSA { Cˆ, ˆ } S Res. harmonic coeff.
The residual gravitational potential and derived quantities V = GM R L l= l m= R r l+ 1 P (cosϑ) [ Cˆ cos mλ + Sˆ sin mλ ] Earth: 1 km resolution requires L = km / 1 km = N L = R l l= m= P (cosϑ) [ Cˆ cosmλ + Sˆ sin mλ ] L g = γ ( ) [ l 1 P (cosϑ) Cˆ cosmλ + Sˆ sin mλ ] l= m= l
POD for gravity field recovery Current knowledge Earth: Harmonic coeff. error pattern basedonterrestrial satellite tracking
POD for gravity field recovery Current knowledge Mars: Signal and noise degree variances of gravitational potential c l = l m= C + GMM: Goddard Mars Model S GMM-1... Viking GMM-... Mars Global Surveyor 5 Kaula s rule: 13x1 l c l
POD for gravity field recovery Current knowledge Mars Global Surveyor: Launch: 1997 Transit time: 1 months Signal travel time: 14 min. Orbit altitude: 17-378 km Sun-synchronous orbit 6 scientific investigations: Mars Orbital Camera Thermal Emission Spectrometer Mars Orbital Laser Altimeter Radio Science Investigation Magnetic Field Investigation Mars Relay
POD for gravity field recovery Current knowledge Mars Global Surveyor: Mars Orbital Laser Altimeter
POD for gravity field recovery Current knowledge Mars Global Surveyor: Mars Orbital Laser Altimeter Surface topography
POD for gravity field recovery Current knowledge Mars Global Surveyor: Gravity Mapping by Doppler Tracking, supported by Orbit Laser Altimetry
POD for gravity field recovery Current knowledge Mars Global Surveyor: Gravity Mapping by Doppler Tracking (X-band), supported by Orbit Laser Altimetry
Gravity Field Recovery Iterative Improvement xˆ k = xˆ k Modelling, Analysis, Interpretation + Curiosity Necessity K k (l k A Σ k = (I K k A k ) Σ k xˆ k ) Technological Development k Observation
Love affairs with body Earth (... put your body close to mine ) CHAMP () GRACE () GOCE (6)
High-low Satellite-to-Satellite Tracking GPS - satellites SST - hl 3-D accelerometer mass anomaly Earth
High-low Satellite-to-Satellite Tracking GPS - satellites SST - ll SST - hl mass anomaly Earth
High-low Satellite-to-Satellite Tracking GPS - satellites SST - hl SGG mass anomaly Earth
POD for gravity field recovery GOCE / hi-lo SST GPS performance: Measurement noise for ionospheric-free combinations of carrier phase observations: 9 mm Measurement rate: 1Hz Error contribution X [mm] Y [mm] Z [mm] GPS measurement noise 9 / 5 8 / 5 19 / 6 GPS station coordinates (1 cm) 4 / 4 / 6 / GPS ephemeris error (5 cm) 7 / 5 6 / 5 14 / 1 Tropospheric corr. Error (.5 %) 3 / 3 3 / 3 6 / 3 Phase center location error 5 / 5 5 / 5 5 / 5 COM location error 3 / 3 1 / 1 / Remaining dynamical model errors / 5 / 1 / 8 Total error for single position determination 14 / 1 1 / 15 6 / 1 Kinematic POD error budet (beginning of data processing) Dynamic POD error budget (more accurate gravity field available)
POD for gravity field recovery GOCE / hi-lo SST Harmonic coeff. error pattern: h = 5 km 1 days GPS-SST
Observation sensitivities ( ) ( ) ( ) ( ) ( ) ( ) + + + + 1 1 1 1 1 1 3 1 n l k n n l ik R r R z y x l β β β β β β β β Orbit smoother SST amplifier ( ) ( ) ( )( ) + + + + + + 1 1 1 1 3 l l l k l k R r R V V V l zz yy xx Gradiometer data SGG: Orbit perturbations SST (hi-lo): Orbit smoother SGG amplifier
The GOCE challenge Vxx V xy Vxz 1 observations V yy V yz V zz 1 parameters
GOCE performance ( cumulative error ) simulated GOCE performance Goal: spatial resolution D (half wavelength) maximum degree L (corresponds to D ) geoid height [m m ] gravity anomaly [m Gal] 1 km.4.6 4 km 5.5.1 km 1.6.3 < 1 mm < 1 mgal 1 km.5.8 65 km 3 ~ 45 ~
Benefits Oceanography: Absolute ocean circulation Sea level changes Ice mass balance Solid Earth Physics: Geotomography Processes in the deep Earth s interior... Earthquake prediction Geodesy: Unified height datum GPS levelling Orbit prediction Inertial navigation