Do 1130 K10 Luo xb s240909 · ARMA modelling of GNSS residuals using different mo del...
Transcript of Do 1130 K10 Luo xb s240909 · ARMA modelling of GNSS residuals using different mo del...
ww
w.k
it.ed
u
∑∑
==
−−
+=
−p
i
qj
jt
jt
it
it
ZZ
YY
11 θ
φ
AICC
GIC
CIC
AR
MA
mo
de
lling
of G
NS
S re
sid
ua
ls u
sin
gd
iffere
nt m
od
el id
en
tifica
tion
crite
ria
Ge
od
etic
Institu
te
X. L
uo
, M. M
aye
r, B. H
eck
KIT
−T
he
Co
op
era
tion
of F
ors
ch
un
gs
ze
ntru
mK
arls
ruh
e G
mb
H a
nd
Un
ive
rsitä
tK
arls
ruh
e (T
H)
Ge
od
etic
W
ee
k 2
00
9S
ep
tem
be
r 2
2-2
4,
Ka
rlsru
he
S6
: T
he
ore
tica
l G
eo
de
sy
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
2
AR
MA
(p,q
) mo
del
Intro
du
ctio
n
AR
MA
Ap
plic
atio
ns
ge
ne
ratin
g p
rog
no
stic
mo
de
ls (e
.g. in
eco
no
mic
scie
nce
s)
an
aly
sin
g p
hysic
ally
co
rrela
ted
pro
cesses (e
.g. in
ge
oscie
nce
s)
mo
de
lling
tem
po
ral c
orre
latio
ns o
f GN
SS
ob
se
rva
tion
s (m
otiv
atio
n)
AR
MA
: Au
toR
eg
res
siv
eM
ovin
g A
vera
ge
)0(
)(
21
11
1,σ
WN
ZZ
ZZ
YY
Yt
qt
qt
tp
tp
tt
~ ,
−−
−−
++
+=
−−
−θ
θφ
φL
L
:)
,(
qp
o
rde
r pa
ram
ete
rs:)
,,
,,
,(
11
Tq
pθ
θφ
φK
K=β
mo
de
l co
effic
ien
ts
:2
σw
hite
no
ise
(WN
) va
rian
ce
Mo
del id
en
tificatio
n c
riteria
su
bje
ctiv
e m
eth
od
s (s
tatis
tica
l tests
, gra
ph
ics, e
tc.)
ob
jectiv
e m
eth
od
s (s
pe
cifie
d d
ecis
ion
crite
ria)
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
3
Crite
rion
AIC
C
Assu
mp
tion
s a
nd
a p
riori c
on
ditio
ns
:)
,,
(1
Tn
nX
XK
=X
ob
se
rvatio
ns fro
m a
Ga
ussia
nA
RM
A(p
,q) p
roce
ss
:)
,(
2σ
βψ
=tru
e m
od
el p
ara
me
ters
of A
RM
A(p
,q) p
roce
ss
:)
ˆ, ˆ
(ˆ
2σ
βψ
=m
axim
um
like
liho
od
estim
ato
ro
f ψ
ba
se
d o
n
nX
:)
,,
(1
Tn
nY
YK
=Y
ind
ep
end
en
t rea
lisa
tion
of A
RM
A(p
,q) p
roce
ss w
ith ψ
L: G
au
ssia
n lik
elih
oo
d fu
nctio
n o
f an
AR
MA
(p,q
) pro
ce
ss, e
.g.
−
−=
∑=
−
−−
−nj
j
jj
nn
Xr
XX
rr
L
11
2
2
2/1
10
2/2
2)
ˆ(
ˆ2
1ex
p)
()
ˆ2(
)ˆ
, ˆ(
σσπ
σ
Lβ
Ku
llback-L
eib
ler
ind
ex (K
LI)
()
)(
)ˆ
()
ˆ(
ln2
)|
ˆ(
ψψ
ψψ
ψψ
YY
YL
LL
E
to
rela
tive
of
K
LI
−
=∆
(1)
(2)
()∑
=−
−−
==
nj
jj
jX
rX
Xn
Sn
1
12
12
)ˆ
(1
ˆˆ
β
σ
Ma
xim
um
like
liho
od
estim
ato
rs:
on
e-s
tep
pre
dic
tor fo
r: re
-sc
ale
d m
ea
n s
qu
are
d e
rrors
of
:ˆ
jX
1−j
r;j
Xj
X̂
(3)
Inn
ovatio
ns
alg
orith
m
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
4
Crite
rion
AIC
C
Estim
atio
n o
f KL
I
()n
SL
LY
XY
−+
−=
−2
ˆˆ
)ˆ
(ln
2)
ˆ(
ln2
σβ
ψψ
()
()
()(
)n
SE
LE
LE
YX
Y−
+−
=−
==
2ˆ
ˆ)
ˆ(
ln2
)ˆ
(ln
2)
|ˆ
(σ
∆β
ψψ
ψψ
ψψ
ψ
:K
LI
AIC
C: A
kaik
eIn
form
atio
n C
riterio
n (A
IC) w
ith s
mall s
am
ple
Co
rrectio
n
()(
))
2(
)1
(2
ˆˆ
2−
−−
++
≈q
pn
nq
pS
EY
σβ
ψ
AIC
C c
riterio
n (H
urv
ich
and
Tsa
i 19
89
):
()
2 )1
(2
)(
,ln
2:
)(
−−
−+
++
−=
qp
n
nq
pn
SL
XX
ββ
βA
ICC
(5)
(6)
Usin
g la
rge
-sa
mp
le a
pp
roxim
atio
ns:
(4)
Crite
rion
AIC
(Akaik
e1973)
AIC
C a
nd
AIC
asym
pto
tica
lly e
qu
iva
lent a
s
AIC
C p
artic
ula
rly a
dvis
ab
le fo
r sm
all s
am
ple
siz
es
()
)1
(2
)(
,ln
2:
)(
qp
nS
LX
X+
++
−=
ββ
βA
IC
∞→
n∞→
n
(7)
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
5
Crite
ria C
IC a
nd
GIC
CIC
for A
R(p
) ord
er s
ele
ctio
n (K
lees
an
d B
roers
en
2002)
−+
−−
+−
−+
++
=∏
∑=
=
p
k
p
kk
nk
n
kn
pR
ES
p
11
1 13
,1)
1/(
11
)1
/(1
1m
ax)
(ln
)(
C
IC
with
the
resid
ua
l (estim
ate
d W
N) v
aria
nce
22
1
ˆˆ
()
(1),
: Yu
le-W
alk
er a
nd
Bu
rg e
stim
ate
s
p
kk
k
RE
Sp
σϕ
ϕ=
=−
∏
CIC
: Co
mb
ined
Info
rmatio
n C
riterio
n; G
IC: G
en
era
lised
Info
rmatio
n C
riterio
n
(8)
(9)
GIC
for M
A(q
) resp
. AR
MA
(p,q
) ord
er s
ele
ctio
n
qp
mq
pq
mq
+=
=:
),
(,
:)
( A
RM
AM
A
{}
(,
)ln
()
3,
()
:G
IC
Du
rbin
's m
eth
od
sm
mR
ES
mR
ES
mn
α=
+(1
0)
Mo
del id
en
tificatio
n b
ased
on
pre
dic
tion
erro
r (PE
)
Nm
Nm
mR
ES
mP
Ek
n
kn
pR
ES
pP
EA
Rpk
AR
AR
/1
/1
)(
)(
,)1
/(1
1
)1
/(1
1)
ˆ(
)ˆ
(
ˆ
1− +
⋅=
−+
−−
++
=∏
=
qp
mq
pq
mq
MA
ˆˆ
:)
,(
,ˆ
:)
(+
==
AR
MA
MA
(11
)
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
6
Ca
se s
tud
y: d
ata
ba
se
HE
DA
54.1
km
TA
AF
53.7
km
RA
TA
203.7
km
SIB
I
42.5
km
AF
LO
32.4
km
Mu
ltipath
(MP
): stro
ng
MP
: weak
Fig
. 1: S
AP
OS
®n
etw
ork
in th
e a
rea o
f the s
tate
o
f Bad
en
-Wü
rttem
berg
(So
uth
west G
erm
an
y)
Ind
ivid
ual a
bso
lute
calib
ratio
nA
nt. c
orre
ctio
n
Based
on
sig
nal q
uality
measu
res (S
NR
*) O
bs. w
eig
htin
g
1-H
z G
PS
ph
ase d
ou
ble
diffe
ren
ces
Ob
serv
atio
ns
Nie
lldry
(a p
riori), M
F**: N
iellw
et(1
996)
Atm
os. m
od
ellin
g
Calc
ula
ting
ep
och
-wis
e m
ean
valu
es o
fh
igh
ly c
orre
late
d S
DD
R tim
e s
erie
sT
ren
d m
od
ellin
g
Stu
den
tised
ph
ase d
ou
ble
diffe
ren
ce
resid
uals
(SD
DR
) in s
idere
al
time
Tim
e s
erie
s d
ata
(No
. 210, n
=3600)
Stro
ng
(HE
DA
), weak (o
ther b
aselin
es)
Mu
ltipath
imp
act
Tab
. 1: G
PS
pro
cessin
g s
trate
gie
s a
nd
data
ch
ara
cte
ristic
s
AR
MA
sel
(Bro
ers
en
2000)
Min
imum
of
CIC
, GIC
and
pre
dic
tion e
rrors
Burg
’s a
lgorith
m
Durb
in’s
meth
ods
(least-s
quare
s)
CIC
, GIC
ITS
M2000
(Bro
ckw
ell
and D
avis
2002)
Min
imum
of
AIC
C v
alu
es
Hannen-R
issannen
Innovatio
ns a
lgorith
m
(maxim
um
likelih
ood)
AIC
C
So
ftware
packag
e
AR
MA
iden
tificatio
n
Para
mete
r
estim
atio
nC
riterio
n
Tab
. 2: A
RM
A m
od
ellin
g u
sin
g d
iffere
nt id
en
tificatio
n c
riteria
*SN
R: S
igna
l-to-N
ois
e R
atio
, **MF
: mappin
g fu
nctio
n
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
7
searc
hare
a:
searc
hare
a:
p=
q+
1p
=q
AIC
CC
IC, G
IC
}1
0,
,1{
}1
0,
,1{
K K
∈ ∈q p
1
}1
0,
,1{
−= ∈
pq p
K
Ord
er s
ele
ctio
n
Fig
. 2: C
om
paris
on
of o
rder s
ele
ctio
n u
sin
g d
iffere
nt m
od
el id
en
tificatio
n c
riteria
10
max
max
==
qp
MP
stro
ng
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
8
∑≈
→=
03
.
0|
|5.
02
vv
Au
toc
orre
latio
n fu
nc
tion
(AC
F)
SD
DR
: TA
AF
18261
68
mu
ltipath
: weak
SD
DR
: HE
DA
1826168
mu
ltipath
: stro
ng
Fig
. 3: C
om
paris
on
of a
uto
co
rrela
tion
fun
ctio
n u
sin
g d
iffere
nt m
od
el id
en
tificatio
n c
riteria
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
9
∑≈
→=
dB
/Hz
6.0
||
10
02
vv
Po
wer s
pe
ctra
l de
nsity
(PS
D)
SD
DR
: TA
AF
18261
68
mu
ltipath
: weak
SD
DR
: HE
DA
1826168
mu
ltipath
: stro
ng
Fig
. 4: C
om
paris
on
of p
ow
er s
pectra
l den
sity
usin
g d
iffere
nt m
od
el id
en
tificatio
n c
riteria
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
10
Wh
ite n
ois
eσ̂
Sate
llite p
air: P
RN
0917
Site
: Ta
ub
erb
isch
ofs
heim
AR
MA
sim
ula
tion
SD
DR
data
: TA
AF
18
26168 (M
P: w
eak)
Fig
. 5: C
om
paris
on
of A
RM
A s
imu
latio
n b
ase
d o
n d
iffere
nt m
od
el id
en
tificatio
n c
riteria
Sim
ula
tion
(AIC
C): A
RM
A(3
, 3)
Sim
ula
tion
(CIC
, GIC
): AR
MA
(2, 1
)
SD
DR
data
: HE
DA
18
26168 (M
P: s
tron
g)
Sim
ula
tion
(AIC
C): A
RM
A(5
, 10)
Sim
ula
tion
(CIC
, GIC
): AR
MA
(3, 2
)
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
11
Facto
rs im
pactin
g th
e id
en
tificatio
n p
erfo
rman
ce
GN
SS
ob
se
rva
tion
al d
ata
(e.g
. da
ta q
ua
lity, s
am
ple
siz
e)
ap
plie
d a
lgo
rithm
s fo
r pa
ram
ete
r estim
atio
n
the
hig
he
st c
an
did
ate
ord
er fo
r se
lectio
n
Co
mp
aris
on
of th
e u
sed
iden
tificatio
n c
riteria
Crite
rion
AIC
C
hig
he
r sele
cte
d o
rde
rs w
ith s
trong
va
riability
be
tter p
erfo
rman
ce
in th
e c
ase o
f low
-qua
lity d
ata
larg
e o
rde
r searc
h a
rea
→tim
e-c
on
su
min
g c
om
pu
tatio
n
Crite
ria C
IC, G
IC
low
er s
ele
cte
d o
rde
rs w
ith c
om
pa
rable
mo
dellin
g re
su
lts
rea
so
na
ble
redu
ctio
n o
f ord
er s
ea
rch
are
a →
less c
om
pu
tatio
nal c
ost
No
t the v
ery
best, b
ut th
e m
ost re
liab
le a
nd
effic
ien
t crite
rion
!
Co
nclu
sio
ns
iden
tical w
ithin
th
is c
ase s
tud
y
Ge
od
etic
Ins
titute
: X. L
uo
, M. M
aye
r, B. H
ec
k –
luo
@g
ik.u
ni-k
arls
ruh
e.d
eA
RM
A m
od
ellin
g o
f GN
SS
res
idu
als
us
ing
diffe
ren
t mo
de
l ide
ntific
atio
n c
riteria
12
Ge
od
etic
Ins
titute
−U
niv
ers
ität
Ka
rlsru
he
(TH
) −E
ng
lers
traß
e7
, 76
13
1 K
arls
ruh
e, G
erm
an
yT
el.: +
49
(0)7
21
60
8 3
668
, Fa
x: +
49
(0)7
21
60
8 6
808
, ho
me
pa
ge
: ww
w.g
ik.u
ni-k
arls
ruh
e.d
e
GP
S s
yste
m m
od
ern
isatio
ns +
ad
van
ce
d m
ath
em
atic
al m
od
ellin
g =
ac
cu
rate
an
d re
liab
le p
ositio
nin
g re
su
lts
Th
an
k y
ou
very
mu
ch
for y
ou
r atte
ntio
n!
Qu
estio
ns
& c
om
men
ts
Th
e p
roje
ct “Im
pro
vin
g th
e s
toch
astic
mo
de
l of G
PS
ob
se
rva
tion
s b
y
mo
de
lling
p
hysic
al
co
rrela
tion
s”
(HE
14
33
/16
-1/2
) is
su
ppo
rted
by
the
De
uts
ch
e F
ors
ch
ung
sge
me
insch
aft
(DF
G).