Die Satellitengeodsie als Tanzdisziplin?

Grace GRAVITY FIELD SOLUTIONS USING THE DIFFERENTIAL GRAVIMETRY APPROACHM. Weigelt, W. KellerGeophysical implications of the gravity field

Geodesy

Continental Hydrology

Oceano-graphy

Glacio-logy

Atmosphere

Tides

Earthcore

SolidEarthGeoid2Geophysical implications of the gravity field3

ESAObservation system

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GRACE = Gravity Recovery and Climate ExperimentInitial orbit height: ~ 485 kmInclination: ~ 89Key technologies:GPS receiverAccelerometerK-Band Ranging SystemObserved quantity:rangerange rate CSR, UTexas

Gravity field modelling 5

with gravitational constant times mass of the Earthradius of the Earthspherical coordinates of the calculation pointLegendre functiondegree, order(unknown) spherical harmonic coefficients

Spectral representations6

Spherical harmonic spectrum

Degree RMSOutlineGRACE geometrySolution strategiesVariational equationsDifferential gravimetry approachWhat about the Next-Generation-GRACE?7Geometry of the GRACE systemGeometry of the GRACE system9BA

DifferentiationIntegration

Rummel et al. 1978Solution StrategiesSolution strategies11Variational equationsIn-situ observationsClassical(Reigber 1989, Tapley 2004)Celestial mechanics approach(Beutler et al. 2010, Jggi 2007)Short-arc method(Mayer-Grr 2006)Energy Integral(Han 2003, Ramillien et al. 2010)Differential gravimetry(Liu 2010)LoS Gradiometry(Keller and Sharifi 2005)Numerical integrationAnalytical integration

Equation of motion12Basic equation: Newtons equation of motion

where are all gravitational and non-gravitational disturbing forces

In the general case: ordinary second order non-linear differential equation

Double integration yields:

Linearization13For the solution, linearization using a Taylor expansion is necessary:

Types of partial derivatives:initial positioninitial velocity

residual gravity field coefficientsadditional parameter

Homogeneous solutionInhomogeneous solutionHomogeneous solution14

Homogeneous solution needs the partial derivatives: Derivation by integration of the variational equation

Double integration !Inhomogeneous solution (one variant)15Solution of the inhomogeneous part by the method of the variation of the constant (Beutler 2006):

with being the columns of the matrix of the variational equation of the homogeneous solution at each epoch.

Estimation of by solving the equation system at each epoch:

Application to GRACE16In case of GRACE, the observables are range and range rate:

Chain rule needs to be applied:

Limitationsadditional parameters compensate errors in the initial conditionscounteract accumulation of errorsoutlier detection difficultlimited application to local areashigh computational effortdifficult estimation of corrections to the initial conditions in case of GRACE (twice the number of unknowns, relative observation)17In-situ observations:Differential gravimetry approachInstantaneous relative reference frame19

Position

VelocityIn-situ observation20Range observables:

Multiplication with unit vectors:

GRACERelative motion between two epochs21absolute motion neglected!Epoch 1Epoch 2

K-BandGPS

Limitation22

Combination of highly precise K-Band observations with comparably low accurate GPS relative velocityResidual quantitiesOrbit fitting using the homogeneous solution of the variational equation with a known a priori gravity field

Avoiding the estimation of empirical parameters by using short arcs23true orbitGPS-observationsestimated orbitResidual quantities24

ratio 1:2ratio 25:1

Approximated solution25

Next generation GRACENext generation GRACE27New type of intersatellite distance measurement based on laser interferometryNoise reduction by a factor 10 expected

M. Dehne, QuestSolution for next generation GRACE28

Variational equations for velocity termReduction to residual quantity insufficientModeling the velocity term by variational equations:

Application of the method of the variations of the constants29

Results30

Only minor improvements by incorporating the estimation of corrections to the spherical harmonic coefficients due to the velocity termLimiting factor is the orbit fit to the GPS positions

additional estimation of corrections to the initial conditions necessarySummaryThe primary observables of the GRACE system (range & range rate) are connected to gravity field quantities through variational equations (numerical integration) or through in-situ observations (analytical integration).Variational equations pose a high computational effort.In-situ observations demand the combination of K-band and GPS information.Next generation GRACE instruments pose a challenge to existing solution strategies.

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