An M-estimator, one kind of a robust estimator, has been used in satellite laser ranging (SLR) data preprocessing. It has been shown that the M-estimator has a 50 percent or more breakdown point

Jia, Peiz-Hang

Tan, De-Tong

Xu, Hua-Guan

Zhang, Zhong-Ping

English

NASA Technical Reports Server

N94-15597
APPLICATIONOF THE ROBUST ESTIMATE
IN SLR DATA PREPROCESSING
T. Detong, Z. Zhongping. X. Huaguan
Shanghai Observatory
Academia Sinica
J. Peizhang
Institute of Systems Science
Academia Sinica
ABSTRACT
M-estimator. one kind of robust estimator, has been used
in SLR data preprocessing. It has been shown that, the
M-estimator has 50% or more breakdown point.
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https://ntrs.nasa.gov/search.jsp?R=19940011124 2020-06-16T19:13:19+00:00Z
APPLICATION OF THE ROBUST ESTIMATE
IN SLR DATA PREPROCESSING
Tan Detong, Zhang Zhongping, Xu Huaguan
Shanghai Observatory
Academia Sinica
Jia Peizhang
Institute of Systems Science
Academia Sinica
INTRODUCTION
There are three purposes in preprocessing from a. pass of
raw satellite range measurements:
a)Correcting system errors for raw SLR data and forming
observational files;
b)Fitting a smoothing function to the range residuals
from the predicted orbit, rejecting noises and outliers
then estimating accurate for this pass;
c_Forming QL, FL and NP data files.
The second term is very impotant for data preprocess, because
the smoothing function will have effect on quality of NP
data.
The smoothing function we used is simply a polynomial in
time. Generaly, the least squares(LS) estimation is:used to
solve the parameters of polynominal. But, the LS estimation
is not a robust estimetion. Sometime, there are a large
number of noises in raw SLR data, especialy those passes are
in daylight, the solution of the LS estimation will converge
to false values.
In this paper, M-estimator, one kind of robust estimator
has been used in SLR data preprocessing. It .has been shown
that the M-estimators has 50% or more breacdown point e .
The breakdown point means that, when the probability of
noises e ,increases to e ,this method will fail.
M-ESTIMATOR
The linear equation is writen:
Yi=X: 0 +e i
where
Yi are observations
9-58
(I)
e is the vector of paramaters to be estimated
X i is the vector of coefficients
e i are random errors.
The M-estimator, called Maximum Likelihood Type Estimator, is
such an estimator which makes the following objective
function minimum:
N
ZF{( yi-X:e )/a}=min. (2)
are values estimated for O
2
a is variance
F{'} is an even function
Different objective functions have different M-estimator.
this paper we used Hampel estimator, here ....
In
___ 2
;toO(Iril - ;toO)
;tO (;t air i____r:i
F(ri)= l ;t2-;tl Xo _. 1_2 2
Xl _r- a ---2Xoe
;t2-11
;to (X2+;tt-Xo) a2/2
;ti_< Jr i J-<;t2cr
J r i J );t2(_
(3)
r.=y.-X.XO
1 1 1
;to=3, ;t1=4, t2=6.
Equation (2) can be rewriten as"
N
_, x++{ (yi-x:_)/0} =0
i=l
where
+ {-}=F' {-}.
Then
:]'E, ]O= [i._XiWi X' _- XiWiYi
and
Wi=4 (ri/a) / (ri/a) •
From (3) we have
(4)
(5)
(6)
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and
W(ri/a)=_
"1 [r I I_Xoa
Xo_/I r_ I Xoa< I r_ I-_x,
0
(7)
N
_2 1 2
- _, r i (8)N--p
i=l
p is number of the paramaters estimated.
When given the starting values 0 ° and a o ,we can solve 0 by
(5), (6) and (7). The solution is then iterated betw-een (5)
and (8) until convergence of the object function.
IUJ+I-UJI / uJ <10 -3
here
N
u= _ F {(Yi-XT6,)/:}
i=l
is the times of the iteration.
PROCEDURES
The predicted and observed ranges" is R and R at each
C 0
instant of observation T. After atmospheric correction,
center of mass correction and delay calibration, we have
the following range residual equation:
y.=AR.
1 1
=a+bp i +e i •
Where a, b are range bias and time bias.
is the variability of range.
(9)!-
Reference show a mathod of caculation which have 50%
breakdown point.
a) If totel observation data points are N ,which are
divided into n subgroups equaly and every subgroup inclodes
four data points, as:
Yl Yn+l Y2n+l Y3n+l
Y2 Yn+2 Y2n+2 Y3n+2
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Yk Yn+k Y2n+k Y3n+k
• . o o • •
Yn Yn+n Y2n+n Y3n+n
n=N/4.
,
When noise numbers in raw obsrevation data are less then
N/2, there must be a subgroup in which contains one noise
point at most.
b) For the linear model as (9), we can find the linear
estimated value of b for any subgroup k:
k
4
bk = Z BIY 1 _10)
I=I
If b is no-bias, we have"
k
4
_ BI=O
I=l
4
B,_=I (11/
1=1
and
where
412
-"g--B 1 =m i n
I=I 1
5 =5 =1
2 3
T
6- 2 C
2T 1 +T 2
T
26 - C
4 2T 3 +T 2
Tl=ol÷l-P 1
By solving" equations (11), we get_."
where
Bi=611T i
4
2
T
1=I
TI=6 1 --bO
(C is a constant to be selected)
(I=1, 2, 3)
(1=1, 2, 3, 4)
¢ 4
[ T2 +T 3 T i +T 2
Thus, the residuats of k-subgroup are
9-61
rkl"Ykl--6kPk i (1=1, 2, 3, 4)
c) For each subgroup, the largest and smallest values of
r are rejeated. And we can get the initial values a b
kl kO' kO
from remained two data points through follows:
Ykl=ako+bkoPk j (j=l, 2)
d) Then calcuiation the object function of M-estimate
used all observations for every subgroups:
N
Uk= _ F{yi-ako-bkoPi} (k=l, 2 .... n)
i=l
where F(-} can be taken from (3),and the initial value of g
can be arbitrarily given, for example 0.5'meters.
e) Select the
Suppose k=m, that is
minimum value from U
k
(k=l ,-2 .... n).
U =min.
m
Then amo and bmo, those are taken from m-subgrop, can be used
as the initial values ao, bo. It is sure that the a ° and b °
are taken from 'good' observation points.
f) Then we can get
ri =Yt-ao-boP i ( i = 1, 2 .... N )
N
a2_ 1
0 N-2 _ rl"
I=I
Because a o, b ° are obtaied by two data points, they have just
lower accuracy. From (5) to (8) and iterated until
covergence, the accurate results a,b can be get as above.
g) A_'ter correcting range bias and time bias, we can get
a polynomial in time as following:
Ap_ =yi-a-bp,
t2+a tJ+ .....
=ao+alti+a2 1 3 i "
Useing M-estimator, the
ao,al,a2,a 3 ...... can be solved.
parameters of polynomial
CONCLUSION AND DISCUSSION
Comparing with the LS estimator, M-estimator has its
advantage as follows:
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a) It can" be preprocess observation data that contain a
large amount of noises, for example , a pass for LAGEOS
in daytime are shown in fig I, (12/20/1991 8:45 UT). In this
pass rate of noise is up to 70%.
i
!
b) At same accuracy, the order of polynomial fitting is
only 4 using M-estimator, while the order is up to 6-8 or
more with LS estimator. Seeing table I.
c) Noise mixed at the parts near the both ends of the
curve can be detected and deleted.
Besides, comparison with the method of screen-processin
and LS estimator, one third time is saved with M-estimator.
REFERENCE
J. peizhang On ,Joint Robust Estimation of The
Parameters and The Variance. Acta Astronomica
Sinica. Vol.33.No 1. 1992.
9-63
Table 1.
Comparison for Two Estimators(Lageos)
Passes
Y M D H
92011011
92011110
92011120
92011317
92011321
92011416
92011419
92011512
92022216
92031116
92041514
92041616
92042011
92042018
92042613
92043015
92050815
92052114
92052312
92060216
M-estimator
Order RMS(cm) Points
4 5.8 28
4 5.9 778
4 5.1 452
4 6.0 216
4 4.8 170
4 4.6 169
4 5.9 94
4 5.4 457
4 6.2 187
4 5.6 326
4 5.3 425
4 4.9 419
6 5.6 60
4 5.3 41
4 5.1 583
4 4.1 77
4 3.1 91
4 2.7 212
4 2.6 170
4 3.2 503
LS-estimator
Order RMS(cm) Points
8 5.9 29
8 5.7 752
8 5.4 419
4 6.6 212
4 5.0 169
8 5.1 169
8 5.9 86
8 5.8 453
8 6.1 183
8 6.1 326
8 5.9 422
8 7.0 417
8 5.3 56
4 5.8 41
4 6.4 585
8 5.3 83
8 2.9 91
16 2.7 210
12 2.5 170
8 2.9 471
9-64
Fig. l.
Residual for A Lageos Pass in Daytime
A
uJ
r_9
Z
<[
U_
O
O9
_J
cn
b-q
O9
UJ
CC
STATION ID • 7837 SATELLITE: LAGEOS N= 1505
DATA: 12/20/91 TIME (UTC) " 8:45 RMS=6CM
__--_ ....__......:.-.:.-:..........................._-.. .. ................................................
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ELAPSE TIME (SECOND)
9-65

Lunar Laser Ranging



Application of the robust estimate in SLR data preprocessing

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