the improved WLS-AR model can improve the ERP prediction accuracy effectively , and for different prediction intervals of ERP, different weight scheme should ...
Geodesy and Geodynamics
2012,3(3) :57-64
http://www. jgg09. com Doi:l0.3724/SP.J.l246.2012.00057
Prediction of earth rotation parameters based on improved weighted least squares and autoregressive model Sun Zhangzhen 1 and Xu Tianhe2 ' 3 1
School of Geowgy Engineering and Sun;eying, Chang' an Unimersity, Xi' an 710054, China
2
State Key U.boratory of Geo-infomwtion Engineering, Xi' an 710054, Chi=
3
Xi' an Research Institute of Suroeying and Mapping, Xi' an 710054, China
Abstract:ln this paper, an improved weighted least squares(WLS), together with autoregressive(AR) model , is proposed to improve prediction accuracy of earth rotation parameters ( ERP) . Four weighting schemes are developed and the optimal power e for determination of the weight elements is studied. The results show that the improved WLS-AR model can improve the ERP prediction accuracy effectively , and for different prediction intervals of ERP, different weight scheme should he chosen.
Key words: earth rotation parameters(ERP); prediction; autoregressive(AR); weighted; least-square
as VLBI, SLR, GPS and DORIS, can provide high-
1 Introduction
precision spatial and temporal EOP resolution. However, due to the complexity of data processing, it is diffi-
The earth rotation movement characterizes the situation
cult to have real-time access to these parameters. In
of the whole earth movement, as well as the interaction
order to meet the needs of the space navigation and po-
between the earth ' s various spheres of the earth ' s
sitioning, high-precision prediction for EOP is urgent.
core , mantle , crust and atmosphere [1 l . It can be de-
Since there is more stable and accurate long-term pre-
scribed by earth orientation parameters ( EOP) , which
diction for precession and nutation parameter, there-
includes three parts : the precession and nutation pa-
fore, the current work is concentrated in the high -pre-
rameters, the polar motion ( PM) parameters and the
cision forecast of ERP. At present , there are two major
Universal Tin.e ( UTl-UTC) or the length of day
forecasting methods for ERP: linear and nonlinear
(LOD). The PM and UTl-UTC or LOD are alao called
models.
earth rotation parameters ( ERP). EOP is necessary
( LS ) extrapolation , combination of least squares ex-
parameters to achieve mutual conversion of the celestial
trapolation and autoregressive ( LS + AR) , combina-
reference frame and earth reference frame , and is im-
tion of least square extrapolation and auto-covariance
portant
and
( LS + AC) [2 - 71 and so on. The nonlinear models in-
Modem measurement techniques such
clude: sequence of threshold autoregressive ( TAR)
for
positioning.
high-precision
space
navigation
The linear models include: least squares
model, artificial neural network (ANN) [s -!OJ , Fuzzy Received;2012.Q9-11; Aocepted;2012-10-15 Corresponding author: Sun Zbangzhen, E-mail: sunzhangzben@ 126. com.
This work is supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China ( 2007B51) and Natural Science Foundatioo of Chma ( 41174008).
inference[ ttl and other methods. In the above methods , whether the independent method or the combined model , LS model should be performed based on the deterministic cycles and trends. In fact, they have the characteristic of time-varying in the observational data of the ERP["·" 1 • The
58
Vol. 3
Geodesy and Geodynamics
closer that observational data is near to the prediction
Where,
point, the greater impact on the prediction is. When u-
sing the least squares to estimate the model parameters , it regards the influence of basic data without con-
1
PM=
2
p
n
I (Z,n -
L9'iZ,_)
(3)
i=l
Pt=p+l
sidering time-varying characteristics. In other words,
The value p that minimums the FPE (p) is chosen as
the same impact of the recent data and the past data on the prediction value is adopted , which is obviously un-
the order of AR model. The model coefficients q> 1 , q> 2 , ••• ,q>P can be determined by solving the Yule-Walker
reasonable. In order to overcome this problem, Zhang
equations.
et al[l3] proposed weighted least squares combined with AR model to improve the prediction accuracy for polar
2. 2
WLS model
motion. However, the weighted method mentioned a-
For the prediction of ERP, the first step is to remove
bove is relatively simple , and there is no discussion a-
the leap second from UT1-UTC to get UT1-TAl based
bout the selection of the optimal weight. In this paper,
on IERS Convention 2003 , and then remove the earth ' s
an improved method of weighted least squares and AR
zonal harmonic tidal from UTl-TAl and LOD to gener-
model ( WLS + AR) for the ERP prediction are devel-
ate the series of UT1R-TAI and LODR. The equation
oped and the optimal weighting scheme is studied.
of removing tidal can be expressed as :
WLS + AR model
2
62
L B,sin£, + C,co~
lJUTI =
(4)
i=l
2.1
62
AR model
g.:1
=
L B(sin£ + C(co~, j
=1
AR model is a random series z, ( t = 1 , 2 , · · · , N) related with the regular changes before time t and the white noise of time t. The expression can be written as:
Where the meanings of various symbols in g, =
L' a 0ai
j=l
can refer to IERS Convention 2003. p
L cJ>;.ZH
After that, the LS is used to fit the ERP series con-
+ a,
(1)
(/Jp
are the autoregressive coeffi-
are included. For LODRIUT1R-TAI the long-term of
cients, and a, is the white noise with zero mean, p is
18.6 years and 9. 3 years, annual, semi-annual and
the model order. The above equation denoted by AR
1/3 annual periodic terms should be considered. The
(p) is known as the autoregressive model of the order
fitting equation of LS model can be expressed as :
z, =
i=l
sidering the linear and periodic terms. For PM Chandler wobble , annual and semi -annual periodic terms
Where q; 1 , tp 2 ,
•· · ,
p. The AR model requires the time series to be stationary and random, which means steady, normal and zero mean conditions. The key of using the AR model is the determination the order p. There are three methods to obtain order p , the final prediction error criterion, the information criterion and the delivery function criteri-
21Tt
. ( 21Tt
C1 cos ( R+C 2 sm
R)
2
2
+
. (21Tt Dcos( 21Tt) +Dsm-+·" 1
R,
2
R,
(5)
on. In practice, these three methods are almost equivalent. In this paper, the final prediction error criterion
Where a 0 is the constant term, a 1 is the linear term,
is adopted. The final prediction error criterion can be
Bj, Ci and Dj are the coefficients for periodic terms, Ri
expressed as :
is the corresponding periodic , t is the time of UTC. The estimator by using the WLS for parameter solution
FPE(p) =PM(n+p+1)/(n-p-1)
(2)
can be written as :
Sun Zhangzhen, et al. Prediction of earth rotation parameters based on improved
59
weighted least squares and autoregressive model
No.3
X= (BrPB) -lBTPL
(6)
to 10 equal parts , that is :
pr J
Where
X=[a0
a 1 B1 B,
C1
C2
D1
P,
···](7)
D,
(10)
••
X is the estimated parameter vector, B is the coefficient matrix with the expression as :
B=
Where , P 1 , P2 ,
1
t,
(21Tt,) cos--
. (21Tt,) sm --
. (21Tt,) sm --
1
t,
(21Tt,) cos--
. (21Tt,) sm --
. (21Tt,) sm --
R, R,
R,
R,
R,
•• •
and P 10 are the diagonal matrix
with diagonal elements as
1~,
~
, ··· and 1, respec-
tively . Scheme 3 : chooses different weight for different ob-
R,
servation sequence with weight element as :
t.
(21Tt.) cos--
(21Tt,) cos--
(21Tt,) cos--
. (21Tt,) sm --
(21Tt,) cos--
(21Tt,) cos--
. (21Tt,) sm --
1
R, R,
(21Tt.) cos--
R,
R,
R, R,
(21Tt.) cos--
R,
. (21Tt.) sm --
R,
. (21Tt") sm --
R,
R, R,
P(i,i)=(U Where
ti
( 11)
is time interval from the prediction origin .
Scheme 4: same as the scheme 3 with different
(8)
choice for the weight element expressed as following:
. (21Tt") sm --
P(i,i) =
R,
(-tr T
(12)
L is observation vector of ERP series, P is the weight
Where T is total time interval of the basic observation
matrix. A diagonal matrix is considered for P in this
sequence.
paper.
2. 3
2. 4 Determination of the weight elements
Error analysis
In order to evaluate the prediction accuracy , we use
The main principle for the choice of weight matrix is
the mean absolute error ( MAE) standards. It can be
that the closer the data is near to the predicted value,
expressed as following:
the greater the weight is. In this paper, four schemes
( 13)
are designed to choose weight according to this principle. Scheme 1 : divide the basic observation sequence to three equal parts and each part has different weight
(14)
matrix, that is :
Where P, is the predicted value of the i-th prediction, Oj is the corresponding observation value, Ei is real er-
(9)
ror ( assumed the observation value as true value) , n is the total prediction number.
Where P 1 , P 2 and P 3 are the diagonal matrix with diagonal elements as
!,~
3
Calculation and analysis
and 1 , respectively, and e is
the power. Scheme 2 : divide the basic of observation sequence
3. 1
Data description
In this paper, the ERP time series EOP 05 C04 file
60
Vol. 3
Geodesy and Geodynamics
provided by the IERS is used for calculation and analy-
with the span of 1 to 360 days of the forecast are calcu-
sis ( http :ffhpiers. obspm. fr/ eoppc/eop/) . The sam-
lated and compared. Firstly, the optimal power value
pling interval of these data is one day. The time-series
of each case is determined (Fig. 1 (a) to 1 (e) ) . For
is collected from January 1962 ( 37665 MID) to Janu-
the length limitation, only the forecast accuracy of near
ary 2012 after removing the earth's zonal harmonic tid-
real-time forecasting ERP forecast of 5 days, short-term
al from UT1-TAI and LOD according to the formula
forecast of 30 days and 60 days, 150 days of the medi-
( 4) . Then the WLS is used to estimate the model pa-
um-term forecasts and long-term forecast of 300 days
rameters according to the formula ( 5 ) and ( 6 ) , and
with different power e are shown here. Figure 6 shows
the AR model is used to predict the residuals. Differ-
the prediction accuracy comparison between the WLS
ent weighted schemes are adopted and the optimal
+ AR model and LS + AR model. Table 1 shows the
weight setting is found.
statistical results of prediction accuracy.
3. 2
It can be seen that the optimal power value e is dif-
Data processing and analysis
ferent for different weighting scheme , and also different
In order to find a relatively better weighted scheme,
with different prediction span. Figure 1 shows that for
the above-mentioned four schemes are used and their
near real-time ERP, the prediction accuracy of four ca-
optimal power e is determined respectively. So four ca-
ses with the optimal power e by WLS + AR prediction
ses are conducted as following :
model are all higher than the those of LS + AR predic-
Case 1 : application of scheme 1 , change the power
e , to determine an optimal weights ;
tion model. For the polar motion component X, the best prediction accuracy is the case 4 with the optimal
Case 2 : application of scheme 2, change the power
value e = 8. 5. For the component Y, the best prediction accuracy is in the case 4 with e = -0. 5. For
e , to determine an optimal weights ;
Case 3 : application of scheme 3 , change the power
LOD, the best prediction accuracy is in the case 1 with
e = - 1. For UT1-UTC , the best prediction accuracy is
e , to determine an optimal weights ; Case 4 : application of scheme 4, change the power
in the case 1 with e =6. 5.
e , to determine an optimal weights ;
Figures 2 and 3 show that for short-term ERP, the
The basic sequence length of the polar motion com-
forecast accuracy of the 4 cases with the optimal power
ponents X and Y is chosen as 10 years , UT1-UTC as
e in WLS + AR prediction model are also all higher
12 years and LOD is as 18 years. Forecast accuracy
than those of the LS + AR prediction model. For the po-
statistics from January 1, 2004 to January 15, 2012,
lar motion component X, the prediction accuracy in case
1.68 -,--- -·-r·-·-r·-·- -·-- -·---,-----}!____ ,_____ ,-- --·-- -·-- -- -·-·- --·
---+- case1
1.67 ···- ···- ········-· - ...- ........____..,_ .. ---+--- case 2 --+- case3 ~- case4
I
~
1.11 1.1 1.09'--'---~~~~~~~~~~~___J
1.621-o....~~~~~~~~~~-
-1
0 1 2
3 4 5 6
7 8 9
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 e
e LOD
UI'l-UI'C
0.44
!~
0.43 0.42 0.41 0~4
-3
-2
-1
2
0 e
3
...I,.,.,,.,.J,,,...,.,.,J,...,.,,.,..,I,,,.,.....I..-,.,.,.&.,,.,.,_J.,.,..,,...I,-.,-1...,-J.,...-,J ......,. ~
0 1 2 3 4 5 6 7 8 9 e
Figure 1
Prediction interval of 5 days
Sun Zhangzhen, et al. Prediction of earth rotation parameters based on improved
61
weighted least squares and autoregressive model
No.3
1 (the optimal value e = 4 ) is almost the same as in case 4 ( the optimal value e
=8. 5 ) .
The case 1 has
the polar motion component X of medium-term prediction , the case 1 with the optimal value e
=4. 5
is the
advantage with the increase of prediction span. For the
best one and can reach about 1mas accuracy level com-
component Y, the best prediction accuracy is in the
pared to 5 mas level by the LS + AR model. For the
case 4 with the optimal value e = - 1. For LOD, the
long-term forecasting, the case 3 with e =2 performs
best prediction accuracy is in the case 1 with e = - 1.
the best. For the component Y of medium-term predic-
For UT1-UTC, the best prediction accuracy is in the
tion , the best prediction accuracy is the case 1 with
case 1 withe= 6. 5.
e
=4,
and for long-term prediction the case 4 with e =
Figures 4 and 5 show that for medium- and long-term
4 is the best. For LOD, the best prediction accuracy is
ERP , the forecast accuracy of four cases in the optimal
in the case 3 with e = 0. 5 . For UT1-UTC , the best
power e in WL'5 + AR prediction model are obviously
prediction accuracy is in the case 3 with e = 1. 5.
higher than those of the LS + AR prediction model. For 9.4 .....-~~.......~~::.x_~..,----------,--, ----+- • ..., 1
9.2
-------------------- ----+--- cue2
1
9
------- ------------- ~- Cllle4
~
8.8
---+---- caae3
'-"
6.4
16.2 .._, 6
~ 5.8 5.6 -
8.6
5.4
8.4 -1
0
3 4 5 6 e
1 2
7
0 1 2 3 4 5 6 7 8 9
~ ~ ~
8 9
e
LOD
0.19
UTl-UTC
0.185
]:
~
0.18 0.175 0.17 -3
-2
-1
0
2
~
3
0 I 2 3 4 5 6 7 8 9
e
e
Figure 2
Prediction interval of 30 days y
X
18 -----·· ---- --- ----------- -- ------- - ----+- easel ';;' 17.5 --------------------------------- ----+--- caae2 ~ 17 --+- cue3
15
~case4
,!:.
~ 16.5 . 16 11~~~~~~~~~~~~~
15
0.2 ~
!
~ ~
~ ~ ~
-I 0 1 2 3 4 5 6 7 8 9 e LOD
14
-~--4 ·-- ·
0 1 2 3 4 5 6 7 8 9 e
UTl-UTC
-;
-- -- -
-,
-
-·
---,------·~--
· --
-- T-- ·· -- -, -- -----,---- ·· -- -; -----
·r·· -·-· ·r· ----
13
0.195
';;' 12
a
;;r
0.19
~
0.185
11
10 9 s~~~~~~~~~~~~
0.18 -3
-2
-1
2
0 e
Figure 3
3
-10123456789 e
Prediction interval of 60 days
62
Vol. 3
Geodesy and Geodynamics
30 29
!2s
~ 27 26 25 -3
.2,
-1 0 1 2 3 4 5 6 7 8 9
LOD
0.26 -- ------· ----
UTI·UI'C
0.25
jo.24 ......., ~ 0.23 :::::: ::::: _:::::: _-_ -------- ------ --- --- - --
~ 0.22 0.21
---- --- -·-- ----·---------- ------------------------·----·---- -----·----
0.2 -3
-2
-1
2
0
i._, 35
~
30
~ ~
3
0 1 2 3 4 5 6 7 8 9 e
Prediction interval of 150 days
Figure 4
y
X
-,· ·-·-· ·-·--·T·-·-·-·-,·-·--·r···-···r·--·-·r·-·-·-·r··-···r···-·-,--·-·-,-·-·-·,-·-··
----+- caacl
I
34 ---- --------------------------32 _________
--------------------
----4---- .... 2 --+- cue3 .... 4
~
~
32 31
130
~ 30 --------
~ 29 28
26~~~~~~~~~~~~
~
27
0 1 2 3 4 5 6 7 8 9
-3
.2,
-1 0 1 2 3 4 5 6 7 8 9 e
e
LOD
0.34 --· ------· -----
UTI·UI'C
100 90
!
~
80
70 60
0.26 -------------- ---------------------------------3
-2
-1
2
0
3
-10123456789 f!
Figure 5
Prediction interval of 300 days
From the above calculations and analysis, the fol-
tion, case 1 for the medium-term prediction and case 2
lowing remarks can be drawn. ERP prediction accuracy
or case 4 for the long-term prediction. For UT1-UTC
using the proposed Wl.S + AR model is higher than that
and LOD, it is appropriate to select case 1 for the near
of l.S + AR model in different prediction span. For the
real-time and the short-term prediction, case 3 for the
polar motion component X, we can select case 4 for the
medium-term and long-term predictions.
near real-time prediction, case 1 for short- and medi-
shows the application of the WIB + AR model and l.S
um-term prediction, and case 3 for long-term forecast. For the component Y , we can select case 4 for the near
+ AR model prediction accuracy comparison. Table 1 shows the ERP of each span of LS + AR model and
real-time prediction, case 2 for the short-term predic-
the Wl.S + AR model prediction accuracy.
Figure 6
Sun Zhangzhen, et al. Prediction of earth rotation parameters based on improved weighted least squares and autoregressive model
No.3
63
10 - -- ·----· ----·- ·--·-· --- -·- --- ·--- -·--- --- ·-- · o ~~--~--~~--~--~_w
0
50
100 150 200 250 300 350
50
100 150 200 250 300 350
Prediction interval (day)
Prediction interval (day)
120
0.4 ·---·-·,·---·· --, -··-·-·-,-·WQ,._ --·- ·-,---·· --, --·- ·-·- •
!
0.3
~
0.2
-----,-·----,------~H/1'{; ___ ,_______,_____ _
! ~~~ ::::::::::::::::::::::::::::::::::--:::: ______ _
oL_~--~--~~--~--~_w
0
50
~ : ===~~~~-~:~~~==:· 0
100 150 200 250 300 350
0
50
100 150 200 250 300 350
Prediction interval (day)
Prediction interval (day)
Figure 6 MAE for different prediction intervals for ERP Table 1
Forecast span
MAE different predictions for ERP
Y{mas)
X (mas)
LOD (ms)
UT1-UI'C (msL)
S+AR
WI.S+AR
S+AR
WLS+AR
S+AR
WLS+AR
S+AR
WLS+AR
0.2449
0.243
0.2003
0.1997
0.0268
0. 0268
0.0284
0.0281
5
1.6643
1.6302
1.1156
1.1007
0. 1047
0.1045
0.4266
0. 4075
6
2.0058
1.9604
1. 3156
1. 2971
0.1156
0.1155
0.5717
0.5446
10
3.2041
3.0860
2.0443
2.0039
0.1418
0.1419
1.2232
1.1474
20
6.2102
5.8551
3.7921
3.6594
0.1699
0. 1688
3.1421
2. 8026
30
9.1866
8.5177
5.6485
5.4120
0. 1729
0.1720
4.8979
4.2200
60
16.9305
15.4135
11.7197
11.4248
0.1842
0.1833
10.4493
9.0568
90
23.8394
21.1173
17.9813
17.6493
0.1993
0.1961
17.4364
15.0892
120
28.2801
24.2281
23.2525
22.3614
0. 2085
0.2041
25.9344
22. 0216
180
30.0425
25.056
27.2737
25.7173
0. 2332
0.2244
46.0779
31.8683
240
30.8296
25.9977
25.8545
25.0952
0.2545
0.2426
69.3185
41.5955
300
33.9825
27.5630
28.8677
27.3027
0.2763
0. 2647
94.2463
53.3354
360
35.7399
29.0960
32.9789
30.3491
0. 2942
0.2774
121.248
62. 615
model, especially for the medium-term and long-term
4 Conclusions
forecast accuracy.
Due to the influence of the periodic term and linear
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