Grace gravity field solutions using the differential gravimetry approach
This presentation is the property of its rightful owner.
Sponsored Links
1 / 31

Grace GRAVITY FIELD SOLUTIONS USING THE DIFFERENTIAL GRAVIMETRY APPROACH PowerPoint PPT Presentation


  • 118 Views
  • Uploaded on
  • Presentation posted in: General

M. Weigelt, W. Keller. Grace GRAVITY FIELD SOLUTIONS USING THE DIFFERENTIAL GRAVIMETRY APPROACH. Continental Hydrology. Geodesy. Oceano - graphy. Solid Earth. Glacio - logy. Earth core. Tides. Atmosphere. Geophysical implications of the gravity field. Geoid.

Download Presentation

Grace GRAVITY FIELD SOLUTIONS USING THE DIFFERENTIAL GRAVIMETRY APPROACH

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Grace gravity field solutions using the differential gravimetry approach

M. Weigelt, W. Keller

Grace GRAVITY FIELD SOLUTIONS USING THE DIFFERENTIAL GRAVIMETRY APPROACH


Geophysical implications of the gravity field

Continental Hydrology

Geodesy

Oceano-

graphy

Solid

Earth

Glacio-

logy

Earth

core

Tides

Atmosphere

Geophysicalimplicationsofthegravityfield

Geoid


Geophysical implications of the gravity field1

Geophysical implications of the gravity field

ESA


Observation system

Observation system

GRACE = Gravity Recovery and Climate Experiment

  • Initial orbit height: ~ 485 km

  • Inclination: ~ 89°

  • Key technologies:

    • GPS receiver

    • Accelerometer

    • K-Band Ranging System

  • Observed quantity:

    • range

    • range rate

CSR, UTexas


Gravity field modelling

Gravity field modelling

with gravitational constant times mass of the Earth

radius of the Earth

spherical coordinates of the calculation point

Legendre function

degree, order

(unknown) spherical harmonic coefficients


Spectral representations

Spectral representations

Degree RMS

Spherical harmonic spectrum


Outline

Outline

  • GRACE geometry

  • Solution strategies

    • Variational equations

    • Differential gravimetry approach

  • What about the Next-Generation-GRACE?


Grace gravity field solutions using the differential gravimetry approach

Geometry of the GRACE system


Geometry of the grace system

Geometry of the GRACE system

B

A

Differentiation

Rummel et al. 1978

Integration


Grace gravity field solutions using the differential gravimetry approach

Solution Strategies


Solution strategies

Solution strategies

Variational equations

In-situ observations

Energy Integral

(Han 2003, Ramillien et al. 2010)

Classical

(Reigber 1989, Tapley 2004)

Celestial mechanics approach

  • (Beutler et al. 2010, Jäggi 2007)

Differential gravimetry

(Liu 2010)

LoS Gradiometry

(Keller and Sharifi 2005)

Short-arc method

(Mayer-Gürr 2006)

Numericalintegration

Analyticalintegration


Equation of motion

Equation of motion

Basic equation: Newton’s 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:


Linearization

Linearization

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

  • Types of partial derivatives:

  • initial position

  • initial velocity

  • residual gravity field coefficients

  • additional parameter

Homogeneous solution

Inhomogeneous solution


Homogeneous solution

Homogeneous solution

Homogeneous solution needs the partial derivatives:

Derivation byintegrationofthevariationalequation

Double integration !


Inhomogeneous solution one variant

Inhomogeneous solution (one variant)

Solution 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 grace

Application to GRACE

In case of GRACE, the observables are range and range rate:

Chain rule needs to be applied:


Limitations

Limitations

  • additional parameters

    • compensate errors in the initial conditions

    • counteract accumulation of errors

  • outlier detection difficult

  • limited application to local areas

  • high computational effort

  • difficult estimation of corrections to the initial conditions in case of GRACE (twice the number of unknowns, relative observation)


Grace gravity field solutions using the differential gravimetry approach

In-situ observations:

Differential gravimetry approach


Instantaneous relative reference frame

Instantaneous relative reference frame

Position

Velocity


In situ observation

In-situ observation

Range observables:

Multiplication with unit vectors:

GRACE


Relative motion between two epochs

Relative motion between two epochs

absolute motion neglected!

GPS

Epoch 1

K-Band

Epoch 2


Limitation

Limitation

Combination of highly precise K-Band observations with comparably low accurate GPS relative velocity


Residual quantities

Residual quantities

estimated orbit

GPS-observations

true orbit

Orbit 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 arcs


Residual quantities1

Residual quantities

ratio≈ 1:2

ratio≈ 25:1


Approximated solution

Approximated solution


Grace gravity field solutions using the differential gravimetry approach

Next generation GRACE


Next generation grace

Next generation GRACE

M. Dehne, Quest

New type of intersatellite distance measurement based on laser interferometry

Noise reduction by a factor 10 expected


Solution for next generation grace

Solution for next generation GRACE


Variational equations for velocity term

Variational equations for velocity term

Reduction to residual quantity insufficient

Modeling the velocity term by variational equations:

Application of the method of the variations of the constants


Results

Results

  • Only minor improvements by incorporating the estimation of corrections to the spherical harmonic coefficients due to the velocity term

  • Limiting factor is the orbit fit to the GPS positions

  • additional estimation of corrections to the initial conditions necessary


Summary

Summary

The 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.


  • Login