METHOD OF PERTURBED METHOD OF PERTURBED OBSERVATIONS FOR BUILDING OBSERVATIONS FOR BUILDING

REGIONS OF POSSIBLE PARAMETERSREGIONS OF POSSIBLE PARAMETERSIN ORBITAL DYNAMICS IN ORBITAL DYNAMICS

INVERSE PROBLEMINVERSE PROBLEM

Avdyushev V.Avdyushev V.

Planet

ERRORS OF OBSERVATIONSERRORS OF OBSERVATIONS

SatelliteReal

We SeeSatellite

p

Op

p Error

Orbit( , )C C tp p q

Parametric Errorq

q 1q 2

Probabilistic Region

True

LEAST-SQUARES (LS) PROBLEMLEAST-SQUARES (LS) PROBLEM

System of Equations

2

( )

N(0, )

O O C

O s

P P P P q

P

Solution of System2ˆ ˆ ˆ: ( ) || ( ) || minO CS q q P P q

Linear Case1ˆ( ) ( )C T T O P q Aq q A A A P

Nonlinear Case

11

1ˆ( ), lim

2

TC C

k k k kk

S

P Pq q Q q Q q q

q q q

2(4.5 ) ( 6, 0.997)K

CONFIDENCE REGIONCONFIDENCE REGION

ˆ ˆ ˆ( ) ( ) ( ) ( )T TS S q q q q A A q q

RMS Error2 ˆ( ) /( )S N K q

Normal Distribution

11 1ˆ ˆ( ) exp ( ) ( )

2(2 ) det

T

Kf

q q q C q q

C

Covariance Matrix2 1( )T C A A

2 ( , , )K F K N K

dim ; dimN K P qVector Length:

TWO TECHNIQUES FOR BUILDING TWO TECHNIQUES FOR BUILDING REGIONS OF POSSIBLE PARAMETERS REGIONS OF POSSIBLE PARAMETERS

1 1

2

ˆ( ) ( ) ( )

N(0, )

T T O T T

q A A A P P q A A A P

P

Technique 2 (LS-scattering)

11 1ˆ ˆ( ) exp ( ) ( )

2(2 ) det

T

Kf

q q q C q q

C

1/ 2ˆ

N(0,1)

q q C η

η

Technique 11/ 2 1/ 2 1/ 2: ( )T C C C C

Cholesky Matrix

LS-SCATTERINGLS-SCATTERING

ˆ ˆ: ( ( )) minO

S P Pr T rˆ ˆ: ( ) minO

S P Pq q

ˆ ˆ( )!q T r

( )( ) ( ( ))O C O C q T rP P q P P T r

Nonlinear Problem Linear Problem

REPARAMETRIZATIONREPARAMETRIZATION

LS-EstimationsTrue Positions

Observations

GEOMETRIC INTERPRETATIONGEOMETRIC INTERPRETATION

Linear Case

P C (q )

P O

P O

P C (q ) P C (q )ˆ–

–

2ˆ ˆ: || ( ) || minO C q P P q

GEOMETRIC INTERPRETATIONGEOMETRIC INTERPRETATION

P C (q )

P O

P O

P C (q ) P C (q )ˆ–

–P C(q )

P C (q )

P O

P O

P C (q )

–

– ˆ

Planar Case Non-planar Case

MODIFICATIONMODIFICATIONˆˆ ˆ: ( ) minC

S P Pq q

NUMERICAL EXAMPLENUMERICAL EXAMPLE

PROBLEM

Object S/2003 J04 (Jupiter’s Satellite)

PO (Емельянов и др, 2006)

PC (Авдюшев, Баньщикова, 2007)

Parameters 0 0( , ) ( 6)T K q x x

( 22)N

PROBABILISTIC REGIONSPROBABILISTIC REGIONS

Small Scale

-0 .15 -0 .1 -0 .05 0 0.05x 1 (A U )

-0 .15

-0 .1

-0 .05

x 2 (

AU

)

T ru e

E stim a tion

PROBABILISTIC REGIONSPROBABILISTIC REGIONS

Vast Scale

-0 .09 -0.088 -0 .086 -0 .084x 1 (A U )

-0 .108

-0 .107

-0 .106

-0 .105

x 2 (A

U)

T ru e

Astronomy & Space GeodezyAstronomy & Space GeodezyDepartmentDepartment

Tomsk State UniversityTomsk State UniversityAvdyushev VAvdyushev V. . Thanks!Thanks!

DISTRIBUTION OF SUMS OF SQUARESDISTRIBUTION OF SUMS OF SQUARES

2

ˆ( ) ( )( )

S SU

K

q q

q

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5U

0

0.2

0.4

0.6

0.8

1

Den

sity

of

Dis

trib

uti

on F-distribution

CURVATURE OF ESTIMATE SPACECURVATURE OF ESTIMATE SPACE

ˆ ˆ( ) ( ( ) )cos(90 )

ˆ ˆ|| || || ( ) ||

O C C C

O C C C

P P P q P

P P P q P

-0 .02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2D ev iation (°)

0

0.05

0.1

0.15

0.2

0.25

Den

sity

of

Dis

trib

uti

on