A Mission to Planet Mars Gravity Field Determination

Transcription

1 A Mission to Planet Mars Gravity Field Determination Department for Theoretical Geodesy Graz University of Technology and Space Research Institute Austrian Academy of Sciences

2 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

3 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

4 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

5 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

6 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

7 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 )

8 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 ( )

9 Turning inside out mass mass Gravitational potential shape shape V = G l 1 Earth ρ dv Geoid Geoid

10 Turning inside out mass mass Gravitational potential shape shape V = G l 1 Earth ρ dv Gravity Gravity

11 The dual role of the gravity field yr Time Scales 1-1 yr Post-glacial rebound Volcanic activities Ice Sheet Melting Ocean circulation & heat transport Sea level change

12 POD for gravity field recovery The idea: Satellite orbit orbit & r& V = Mass Mass distribution Gravitational field field

13 POD for gravity field recovery The idea: Satellite orbit orbit & r& V = Mass Mass distribution Gravitational field field

14 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 ]

15 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)

16 The geoid

17 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{ }

18 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

19 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.

20 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

21 POD for gravity field recovery Current knowledge Earth: Harmonic coeff. error pattern basedonterrestrial satellite tracking

22 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

23 POD for gravity field recovery Current knowledge Mars Global Surveyor: Launch: 1997 Transit time: 1 months Signal travel time: 14 min. Orbit altitude: 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

24 POD for gravity field recovery Current knowledge Mars Global Surveyor: Mars Orbital Laser Altimeter

25 POD for gravity field recovery Current knowledge Mars Global Surveyor: Mars Orbital Laser Altimeter Surface topography

26 POD for gravity field recovery Current knowledge Mars Global Surveyor: Gravity Mapping by Doppler Tracking, supported by Orbit Laser Altimetry

27 POD for gravity field recovery Current knowledge Mars Global Surveyor: Gravity Mapping by Doppler Tracking (X-band), supported by Orbit Laser Altimetry

28 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

29 Love affairs with body Earth (... put your body close to mine ) CHAMP () GRACE () GOCE (6)

30 High-low Satellite-to-Satellite Tracking GPS - satellites SST - hl 3-D accelerometer mass anomaly Earth

31 High-low Satellite-to-Satellite Tracking GPS - satellites SST - ll SST - hl mass anomaly Earth

32 High-low Satellite-to-Satellite Tracking GPS - satellites SST - hl SGG mass anomaly Earth

33 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)

34 POD for gravity field recovery GOCE / hi-lo SST Harmonic coeff. error pattern: h = 5 km 1 days GPS-SST

35 Observation sensitivities ( ) ( ) ( ) ( ) ( ) ( ) 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 ( ) ( ) ( )( ) 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

36 The GOCE challenge Vxx V xy Vxz 1 observations V yy V yz V zz 1 parameters

37 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 km km < 1 mm < 1 mgal 1 km km 3 ~ 45 ~

38 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