Reguzzoni_GOCE by mamapeirong

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									           GOCE data analysis:
      the space-wise approach and
the first space-wise gravity field model


          F. Migliaccio, M. Reguzzoni, F. Sansò
                     Politecnico di Milano


               C.C. Tscherning, M. Veicherts
                 University of Copenhagen



       ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                        The space-wise approach
 The main idea behind the space-wise approach is to estimate the
 spherical harmonic coefficients of the geo-potential model by exploiting
 the spatial correlation of the Earth gravitational field.
                        25

                        20

                        15

                        10
                 [E2]




                         5

                         0

                         -5

                        -10
                           0   1       2     3    4    5   [degrees]

                                   spatial dependent                        time dependent noise
                                   signal covariance                        covariances (spectra)




data                                                                                                model
                                                              COLLOCATION                           coeffs




               ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                     The space-wise approach
 A unique collocation solution is computationally unfeasible due to the
 huge amount of data downloaded from the GOCE satellite.
       A two-step collocation solution is implemented.

                     spherical grid with                                       spherical
                       local patches                                          harmonics
                                                                                      Ym



                                                                            1
                                                                  Tm 
                                                                          4 a    f (,  )  Ym (,  ) d
                                                                                  S




data                                         local             harmonic                              model
                                           gridding            analysis                              coeffs

                                                            space-wise solver




                 ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                            The space-wise approach
 In order to implement the local gridding:
     - a prior model is used to reduce the spatial correlation of the signal
     - a Wiener orbital filter is used to reduce the highly time correlated
           noise of the gradiometer

                 0
            10




                 -2
            10




                 -4
            10
                       -6        -4               -2     0
                      10    10               10         10
                                      [Hz]




data    Wiener                                                   local      harmonic                  model
         filter                                          -     gridding     analysis           +      coeffs

                                                       prior              space-wise solver   prior
                                                       model                                  model

                       ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                      The space-wise approach
 The procedure is iterated to:
    - recover the signal frequencies cancelled by the Wiener orbital filter
    - improve the rotation from gradiometer to local orbital reference frame



                  Wiener filter and GRF/LORF                 along track
                          corrections                        synthesis




data    Wiener                               local              harmonic                 model
         filter
                          +      -         gridding             analysis          +      coeffs

                              prior                          space-wise solver   prior
                              model                                              model

                  ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                     The space-wise approach
 Intermediate results that can be used for local applications:
     - filtered data (potential and gravity gradients) along the orbit
     - grid values at mean satellite altitude



                  Wiener filter and GRF/LORF                along track
                          corrections                       synthesis




data    Wiener                              local              harmonic                 model
         filter
                         +     -          gridding             analysis          +      coeffs

                             prior                          space-wise solver   prior
                             model                                              model

                 ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
              filtered data                       gridded data
                                                                        The processed data
                                                                                            common mode accelerations
             (from November 2009 to mid January 2010)



                                                        • from the gradiometer:             satellite attitude quaternions
                                                                                            gravity gradients

                                                                                            reduced dynamic orbits
                                                                                            (for geo-locating gravity gradients)
                                                        • from the GPS receiver:
Input data




                                                                                            kinematic orbits with their error estimates
                                                                                            (for low-degree gravity field recovery)

                                                                                            GOCE quick look as prior model
                                                        • geopotential models:
                                                                                            other geopotential models as reference
                                                                                            and to compute signal degree variances

                                                        • external information such as Sun and Moon ephemerides or ocean
                                                          tides for modelling tidal effects

Output data                                                     spherical harmonic coefficients and their error covariance matrix

                                                              ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                         The data processing

• The data analysis basically consisted of three steps:

    • Data preprocessing: outlier detection, data gap filling,
      unexpected behaviours tagging, etc.

    • SST solution: to estimate the low degrees of the field
      (that are then removed from the SGG data)

    • SST+SGG solution: to estimate the final model in terms
      of spherical harmonic expansion


• Error estimates are computed by Monte Carlo methods.
  In particular, few samples are used to control the evolution of the solution,
  while the final error covariance matrix is based on a larger set of samples.



                ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                   Preprocessing philosophy
Outliers and data gaps are replaced with values estimated by collocation.
   The idea is to preserve the stochastic characteristics of the observations




                                                                             time

                       empirical                       empirical
                      cov. function                   cov. function

                                        mean


                                       collocation




                                                                              time



                ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                      Preprocessing example
• An example of data gap filling applied to the difference between kinematic
 and reduced dynamic orbits.




  Cubic spline interpolation around               Collocation interpolation inside
  the data gap to recover the long                the gap to recover the stochastic
  period behaviour                                behaviour of the signal

                ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                     SST solution philosophy


SST data      energy                   collocation         numerical              SST model
           conservation                 gridding          integration         +


                           prior                         space-wise solver   prior
                           model                                             model



 The energy conservation approach requires to:
     • detect outliers and data gaps in the kinematic orbits;
     • derive velocities from positions by least-squares interpolation;
     • calibrate biases in the common mode accelerations;
     • correct potential estimates from non-gravitational and tidal effects.



                ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
    Estimated potential along track

Absolute differences w.r.t. EGM08
Predicted error standard deviation


                                           empirical error rms     predicted error rms
                                            (w.r.t. EGM08)             (from MC)

                                              1.704 m2/s2            1.523 m2/s2




                                                        Non-stationary noise
                                                        covariance is used
                                                        in the gridding



      ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
      Error calibration of the estimated potential

                               Error spectrum w.r.t. EGM08
                               Predicted error spectrum




                                If we do not remove spikes
                                we get this error pattern on
                                the grid



low frequency
        zoom
                                         Remarks:
                                         • some periodical behaviours are not modelled
                                           (the highest with 2 cpr period)
                2 cpr period
                                         • at very low frequency, the predicted spectrum
                                           is lower than the empirical one

                                                    Error calibration introducing
                                                    prior information (EGM2008)
                 ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                           Choice of prior model

• A degraded version of the GOCE quick-look is used as prior model
  to reduce the influence on the final solution
              Predicted (residual) degree variances
                                                               full signal degree variances
                                                               (estimated from EGM08)
                                                               quick-look predicted error
                                                               degree variances
                                                               rescaled signal degree variances
                                                               scale factor = 0.975
                                                               predicted residual degree
        DEGRADED                                               variances if a rescaled
        QUICK-LOOK
                                                               quick-look model is used



             QUICK-LOOK

                                                               Used for signal covariance
                                                                modelling in the gridding



                ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
        Estimated potential on the grid

Estimated signal [m2/s2]                          Predicted error [m2/s2]




           latitude        empirical (w.r.t. EGM08)           predicted
           interval               error rms                   error rms
       -83° <  < 83°            0.041 m2/s2                 0.026 m2/s2
       -90° <  < 90°            0.106 m2/s2                 0.135 m2/s2


         ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                   Estimated SST model
     Error degree variances
                                                  EGM08 degree variances

                                                  Error degree variances
                                                  w.r.t. ITG-GRACE
                                                  SST model predicted
                                                  error degree variances
                                                  ITG-GRACE predicted
                                                  error degree variances
SST GOCE




  GRACE




                          Above degree 60 the estimated model is the
                          (degraded) quick look model, as corrections
                          are negligible
                          Gravity gradients are needed for further improvements

           ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                         SST+SGG solution philosophy
     SST        SGG
                                                                 Data                        Final model
                                                              synthesis               test
                                                              along orbit
  Energy
conservation


                                                                                 Harmonic




                                                                                                  Space-wise solver
           -         -               FFT                                         analysis



                                 complementary               LORF/GRF              Data
               FFT                Wiener filter               correction          gridding



                                                        -1
       Wiener filter                   +          FFT             +




                Space-wise solver
                                                     Low degree model
         Data                     Harmonic
        gridding                   analysis
                         ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
Estimation error along the orbit

 Error rms of the Wiener filtered observations along the orbit
        Empirical values from differences w.r.t. EGM08
       T         TXX         TXZ         TYY         TZZ
    [m2/s2]      [mE]       [mE]        [mE]        [mE]
    0.091        2.4         4.4         4.6         6.0



 Error rms of the Wiener filtered observations along the orbit
        Predicted values from Monte Carlo simulations
       T          TXX        TXZ         TYY         TZZ
    [m2/s2]      [mE]        [mE]       [mE]        [mE]
     0.089        2.5        4.2         4.6         5.9




ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                       Signal covariance modelling
residual signal variances after removing SST-model




                          zoom




                                                      2    2m        2  (2  1) med  2m 
          variances of degree 30
                     log10 scale                      ˆ        ˆ           ˆ                    ˆ
                                                            m                               m


                                                           Approximate degree variances are used
                                                           for collocation

                                                           Single coefficient variances are used
                                                           for error modelling by Monte Carlo

                     ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                        Estimation error on the grid
            T predicted error [m2/s2]                         Trr predicted error [mE]




latitude         empirical       predicted               latitude      empirical         predicted
interval         error rms       error rms               interval      error rms         error rms
|| < 83°       0.020 m2/s2      0.016 m2/s2             || < 83°      2.64 mE          1.44 mE
|| < 90°       0.048 m2/s2      0.026 m2/s2             || < 90°      3.92 mE          1.71 mE
                              empirical error computed w.r.t. EGM08

                     ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
            Error degree variances

                                                  Estimated space-wise model

                                  GOCE vs GRACE
                                                        EGM08 degree variances

            GOCE vs EGM08                               Error degree variances w.r.t. EGM08

                                 GOCE                   Error degree variances w.r.t. ITG-GRACE

                                                        Predicted error degree variances




              Geoid error [cm]                                        Geoid error [cm]




                      ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
Differences w.r.t. EGM08 (d/o 150)  = 8.4 cm      Differences w.r.t. ITG-GRACE (d/o 150)  = 5.1 cm
 Estimated space-wise model

          Predicted error variances of
         the GOCE space-wise model




                                                   Log10
                                                   scale




ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                    Estimated space-wise model
              Geoid error [cm]                           Predicted cumulative geoid error [cm]




 Predicted         Predicted gravity                Assuming a mission length of 18 months,
geoid error         anomaly error                   (9 sets of two months + some refinement)

 10.86 cm              3.03 mgal                    one can expect an improvement of factor 3
                                                    in terms of accuracy, with the same spatial
                                                    error distribution
  for || < 83° and up to d/o 200

                  ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)
                Conclusions and future work

•   The analysis of the first two months of GOCE data shows that the
    space-wise approach is able to provide good results.

•   The main characteristic of the space-wise solution is to be a solution
    fully computed by collocation, with its pros and cons.
    Furthermore, intermediate results such as filtered data along track
    and grid values at satellite level can be used for local applications.

•   At medium-high degrees the solution is driven by GOCE data, while
    at very low degrees a dependence from prior models can be seen.
    This dependence will be removed in the next solutions.

•   A new solution will be computed for a longer data period, that implies
    to optimally combine grids at mean satellite altitude based on different
    data subsets.


               ESA Living Planet Symposium, 29 June 2010, Bergen (Norway)

								
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