D. Coulot (1), Ph. Berio (2) & A. Pollet (1)

Least square mean effect.Application to the analysis

of Satellite Laser Ranging time series

(1) IGN/LAboratoire de REcherche en Géodésie – Marne la Vallée - France(2) OCA/Département GEMINI - Grasse - France

1

15th International Laser Ranging Workshop

2Summary

1- Least square mean effect

2- Alternative models

3- New model for SLR data processing

● Theoretical considerations● Numerical examples

● Periodic series● Wavelets

● General considerations● First results● Towards global estimations over a long period

4- Prospects

3Least square mean effect Theoretical considerations

Quality of space-geodetic measurements

Representation of studied physical parameters as time series

Example : terrestrial observing station position time series

Modeling currently used

« Well-known » physical effects=

Modeled+ Other physical effects

=Constant estimations

!! We need to get exact and judicious representations !!

4Least square mean effect Theoretical considerations

Xr

Vector of physical parametersTime-varying parameters

)()()( 0 tXtXtXrrr

δ+=Modeled effects Studied effects

)(tXr

δ [ ]mtt ,1BUT they are supposed to be constant over Xr

δ

),()( Xtftmr

≅Measurements modeled as

XtXtXf

tXtftmrr

rr

δ)).(,())(,()( 00 ∂∂+≅Linearization of the model

))(,( 0 tXtXf rr

∂∂ ))(,( 0 tXt

r= f partial derivative matrix at the point

Physical measurement )()).(,())(,()( 00 tXtXtXftXtftm

rrr

rδ

∂∂+≅

-> averages of physical signals over the interval

5Least square mean effect Theoretical considerations

Estimation model

XtXtXf

tXtftmrr

rr

δ)).(,())(,()( 00 ∂∂+≅

Linearization of physical measurements

)()).(,())(,()( 00 tXtXtXftXtftm

rrr

rδ

∂∂+≅

Observation equation)()).(,())(,()).(,())(,( 0000 tXtXt

XftXtfXtXt

XftXtf

rrr

rrrr

rδδ

∂∂+≅

∂∂+

∂∂

∂∂∂∂

=

))(,(

))(,(

))(,(

0

202

101

mm tXtXf

tXtXf

tXtXf

A

rr

M

rr

rr

∂∂

∂∂

∂∂

=

))(,(00

0))(,(0

00))(,(

~

0

202

101

mm tXtXf

tXtXf

tXtXf

A

rrL

MOMM

Lr

r

Lr

r

=

)(

)()(

~2

1

mtX

tXtX

XrM

rr

r

δ

δδ

δ

Matricial relation

XAXA~

.~

.rr

δδ ≅

Least square estimation

( ) )~~

.(~ˆ 1

averageTT

average XXAPAPAAXXrrrr

δδδδ −+≅ −

=

X

XX

X

average

average

average

average

r

r

r

rM

δ

δδ

δ~

6Least square mean effect Numerical examples: method of simulation

GINS

MeasurementsPhysical models

7-dayLAGEOS orbits

7-dayLAGEOS-2 orbits

MATLO POSGLOB

Estimatedstation positions

!! Real orbits and real SLR measurement times are used in simulations !!

Atmospheric loading effects are derived from the ECMWF pressure fieldshttp://www.ecmwf.int .

!! Estimated station position time series contain atmospheric loading signals !!

Simulated measurements

ITRF2000

Partial derivatives

Atmospheric loadingeffects

AVERAGE

Signal weeklytemporal means

7Least square mean effect Numerical examples: results of simulations

0.130.151.262.37 10-4East

0.320.342.002.29 10-5Up0.120.130.951.86 10-5North

RMSAverageMaximumMinimumValues (mm)

0.210.192.281.57 10-4East

0.510.424.493.14 10-5Up0.220.191.963.87 10-4North

RMSAverageMaximumMinimumValues (mm)

8Alternative models Periodic series

∑=

+≅n

i ii

ii t

Tbt

Tat

1

)2sin()2cos()( ππϕModel used

niiT ,1)( = = characteristic periods of studied signal

niia ,1)( = niib ,1)( =New parameters = sets of coefficientsfor each positioning component

Advantage : no sampling a priori imposedBUT

– minimal period allowed imposed by measurements- knowledge of characteristic periods ?!- « discontinuities » of physical signals

(earthquakes, seasonality, etc.)

9Alternative models Periodic series

10Alternative models Wavelets

∑∑−= =

=2

1

max

0,, )()(

j

jj

n

nnjnj tat ψϕModel used

where <−

=not if 0

0 if 12max

jn

j

and

−= j

j

jnj

ntt

2

2

21)(, ψψ

with Haar waveletψ

<≤−

<≤

=

not if 0

12

1 if 1

2

10 if 1

)( t

t

tψ

Advantages : no sampling a priori imposeddiscontinuities can be taken into accountrepresentation in time and in frequency

nja ,New parameters = sets of coefficients

11Alternative models Wavelets

Whatever the model used :

For a computation over the global network, we need to guaranteethe homogeneity of the involved Terrestrial Reference Frame.

We can take the opportunity of this global computation to derive geodynamical signals from global parameters.

12New model for SLR data processing General considerations

δX=T+DX0+RX0

T=(TX,TY,TZ)T

R=

Station positions Earth Orientation Parameters

withand

with f=1.002737909350795

Classical approach

This model must allow us to compute together EOPs, station positions in a homogeneous reference frame and weekly Helmert’s transformations

between weekly TRFs and this reference frame

Observation system : Y=A.δX

Y: pseudo measurements or a priori residualsA: design matrix (partial derivatives)δX: updates of parameters (mainly EOPs and station positions)

Weak or minimum constraints

Weekly estimated solution : δX

Typically,Daily EOPs (xp, yp, UT1 and their derivatives)Weekly station positions for the global network

Helmert’s transformation

−

−−0

00

RRRRRR

XY

XZ

YZ

−=

=

=

RUT

RyRx

Z

XP

YP

f

11δ

δδ

Solutions :- Station positions in the a priori reference frame (ITRF2000)- Coherent EOPs- Transformation parameters between the weekly TRF and the a priori reference frame

Goal of the new model = to obtain all these parameters in a unique processand directly at the measurement level

13New model for SLR data processing General considerations

New approach

Observation system : Y=A.δX

δX= δXC+T+DX0+RX0

δEOP= δEOPC+εR{X,Y,Z}New parameters to be estimated

Theoretical considerations and numerical tests for SLR techniqueWe do not need rotations

XX

Rank deficiency of weekly normal matrices so obtained = 77 = 3 (physical orientation not defined)

+ 4 (estimation of the parameters T and D)= definition of the TRF underlying the estimated δXC

New Observation system : Y=A’.δX’with δX’=(δEOPC,δXC,TX,TY,TZ,D)T

The weekly TRF underlying the δXC is defined by minimum constraintswith respect to ITRF2000 with a minimum network

14New model for SLR data processing First results : transformation parameters

Minimum network used for minimum constraints

15New model for SLR data processing First results: EOP and station position time series

5 +/- 280 µas

23 +/- 280 µas

Mount Stromlo (7849)Yarragadee (7090)

16New model for SLR data processing Towards global estimations over a long period

Observation system : Y=A.δX’ δX= δXC+T+DX0

δEOP= δEOPC

How to use this new model to reduce least square mean effect ?

For each parameter δZ, we can use the model

δZ(t)=δZ0+Σi [azi (t)cos(2πt/Ti )+bzi (t)cos(2πt/Ti )]

But

Each harmonic estimated for station positions creates an additionalRank deficiency Generalization of minimum constraints

The number of parameters involved is large (several tens of thousands) Manipulation of large normal systems

17New model for SLR data processing Towards global estimations over a long period

A first experiment …

Computation of the amplitudes of annual signals for the three translations and the scale factor

TX : 2.1 mmTY : 3.6 mmTZ : 1.1 mmD : 0.9 mm

Furthermore, frequency analyses show the disappearance of the annual frequency in the weekly parameters estimated

with respect to the annual harmonics.

18Prospects

Generalization of the « periodic » modelGlobal parameters + station positionsHarmonics linked to the oceanic tides ?Diurnal and semi-diurnal signals on EOPs ?

Coupling of periodic series and wavelets to get a more robust model

Stochastic approaches ?