Auto-covariance estimation of variable samples (ACEVS) - Springer Link

Journal of Geodesy (2001) 75: 438±447

Auto-covariance estimation of variable samples (ACEVS) and its application for monitoring random ionospheric disturbances using GPS Y. Yuan, J. Ou Laboratory of Dynamic Geodesy, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, 430077, China, e-mail: [email protected]; Tel.: +86-27-86780836; Fax: +86-27-86783841 Received: 18 August 2000 / Accepted: 12 April 2001

Abstract. Ionospheric variation may be considered as a stationary time series under quiet conditions. However, the disturbance of a stationary random process from stationarity results in the bias of corresponding samples from the stationary observations, and in the change of statistical model parameters of the process. From a general mathematical aspect, a new method is presented for monitoring ionospheric variations, based on the characteristic of time-series observation of GPS, and an investigation of the statistical properties of the estimated auto-covariance of the random ionospheric delay when changing the number of samples in the time series is carried out. A preliminary scheme for monitoring ionospheric delays is proposed.

series observation, is presented here. A study is made of the statistical properties of the estimated auto-covariance of the random ionospheric delay when changing the number of samples in the time series. A solution formula for auto-covariance estimation of variable samples (ACEVS) for an ergodic Gaussian process with zero expectation value is derived. In Sect. 3, through the determination of a stationary ionospheric observation model, the possibility of applying ACEVS when constructing an ionospheric monitoring scheme on the basis of the time-series observations of GPS is investigated. In Sect. 4, a scheme for monitoring ionospheric delay is proposed, and preliminary experimental results are presented. Finally, some conclusions and suggestions for future work are given.

Keywords: GPS ± Stationary Random Process ± Ionospheric Disturbances ± Monitoring

2 Auto-covariance estimation of variable samples Assume a realization of the ergodic Gaussian random process fxt g with zero expectation value x^i ˆ xi ‡ ei

1 Introduction GPS is an important tool for monitoring ionospheric variations. Many results show that it is very advantageous to use the time series of GPS observations to monitor and investigate ionospheric phenomena (see e.g. Beutler et al. 1988; Delikaraogou 1989; Wild et al. 1990; Georgiadiou 1994; Sardon et al. 1994; Qiu et al. 1995; Ou 1996; Pullen et al. 1996; Wang et al. 1996; Li et al. 1996; Ho et al. 1997; Schaer et al. 1997; Wang and Wilkinson 1997; Jakowski et al. 1999; Yuan and Ou 1999; Feltens and Schaer 2000). In fact, ionospheric disturbance monitoring is required by many GPS applications. A new method for monitoring ionospheric variations, based on the characteristics of GPS timeCorrespondence to: Y. Yuan

…i ˆ 1; 2; . . . ; N †

…1†

where ei is ergodic Gaussian white noise with zero expectation value (independent of x); N is the number of samples. For simplicity, the stochastic model and other relevant properties of fet g and fxt g are written as E…ei † ˆ E…xi † ˆ 0; COV…xi ; xi r † ˆ E…xi xi r † ˆ c…r†

…2a†

COV…ei ; ei r † ˆ E…ei ei r † ˆ ce …r† ˆ ce … r† ˆ Se d…r†

…2b†

COV…e; x† ˆ E…ex† ˆ 0; zce …y† ˆ 0

…2c†

…y  z  0†

where d is the Dirac delta function; COV is the covariance; ce and c are the auto-covariance function of fet g and fxt g, respectively; Se is the spectral density of fei g (positive constant); and r is the time delay. The di€erent subscripts which distinguish between a stochastic variable and its realization, and between a stochastic process and its sample sequence, are neglected here. For

439

example, x^ can express not only the sample of x but also the stochastic variable xt ‡ et , depending on the context. [Some basic mathematical properties used in the following derivation can be found in Ming et al. (1987) and Yang and Gu (1988)]. The asymptotically unbiased estimates c^N …r† and c^N ‡1 …r† for c…r† corresponding to N and N ‡ 1 samples can be calculated, respectively, as c^N …r† ˆ

N r 1X x^i x^i‡r N iˆ1

c^N ‡1 …r† ˆ

…3a†

‡1 r 1 NX N x^N ‡1 x^N ‡1 x^i x^i‡r ˆ c^N …r† ‡ N ‡ 1 iˆ1 N ‡1 N ‡1

r

…3b† When the number of samples is relatively large, the estimate c^…r† is relatively stable, and the di€erence between c^N ‡1 …r† and c^N …r†, i.e. D^ c…r† ˆ D^ cN ‡1;N …r† ˆ c^N ‡1 …r† c^N …r†, is very small, and EfD^c…r†g ˆ ˆ

N r X 1 E‰^ xN ‡1 x^N ‡1 r Š …4a† E‰^ xi x^i‡r Š ‡ N …N ‡ 1† iˆ1 N ‡1

r‰c…r† ‡ ce …r†Š rc…r† ˆ 0 N …N ‡ 1† N …N ‡ 1†

…4b†

Similarly r‡v c…r ‡ v†  0 EfD^ c…r ‡ v†g ˆ N …N ‡ 1† EfD^ c…r†gEfD^ c…r ‡ v†g ˆ

r…r ‡ v† N 2 …N

‡ 1†2

…v  0†

…5a†

c…r†c…r ‡ v†  0

E…D^c…r ‡ v††Šg

E‰D^c…r†ŠE‰D^c…r ‡ v†Š

…6†

To further develop the above expression, it is necessary to determine E‰D^c…r†D^c…r ‡ v†Š. Then D^c…r†D^c…r ‡ v† ( ) N r X 1 x^N ‡1 x^N ‡1 r x^t x^t‡r ‡ ˆ N …N ‡ 1† tˆ1 N ‡1

r v

r v 1 NX x^s x^s‡r‡v x^N ‡1 x^N ‡1 r N sˆ1 )

‡ x^N ‡1 x^N ‡1 r x^N ‡1 x^N ‡1

…7†

r v

where both fxt g and fet g are stationary Gaussian processes with zero expectation, which are independent of each other. Hence E‰^ xt x^t‡r x^s x^s‡r‡v Š ˆ E‰xt xt‡r xs xs‡r‡v Š ‡ E‰et et‡r es es‡r‡v Š ˆ E‰xt xt‡r ŠE‰xs xs‡r‡v Š ‡ E‰xt xs ŠE‰xt‡r xs‡r‡v Š ‡ E‰xt xs‡r‡v ŠE‰xt‡r xs Š ‡ E‰et xt‡r ŠE‰es es‡r‡v Š ‡ E‰et es ŠE‰et‡r es‡r‡v Š ‡ E‰et es‡r‡v ŠE‰et‡r es Š ˆ c…r†c…r ‡ v† ‡ c…s

t†c…s ‡ v

t†

‡ c…s ‡ r ‡ v t†c…s t r† ‡ ce …r†ce …r ‡ v† ‡ ce …s t†ce …s ‡ v t†ce …s

t

t†

r†

…8a†

Similarly

ˆ c…r†c…r ‡ v† ‡ c…N ‡ 1

COV‰D^c…r†; D^c…r ‡ v†Š E…D^ c…r††Š‰D^c…r ‡ v†

N r 1X x^t x^t‡r x^N ‡1 x^N ‡1 N tˆ1

) r v

E‰^ xt x^t‡r x^N ‡1 x^N ‡1 r v Š ˆ E‰^ xs x^s‡r‡v x^N ‡1 x^N ‡1 r Š

However, the di€erence between c^N ‡1 …r† and ^cN …r† will obviously be large when a sudden disturbance of fxt g results in the bias of the N ‡ 1th sample from all of its previous epochs (possible gross errors in the observations are not considered here). By investigating the change in the stochastic property of D^ c…r†, it is possible to identify the change of D^ c…r†, and further improve the monitoring of the state of fxt g. According to the de®nition of covariance

ˆ E‰D^c…r†D^c…r ‡ v†Š

NX r v 1 x^N ‡1 x^N ‡1  x^s x^s‡r‡v ‡ N …N ‡ 1† sˆ1 N ‡1 ( N r NX r v 1 1 X x^t x^t‡r x^s x^s‡r‡v ˆ …N ‡ 1†2 N 2 tˆ1 sˆ1

‡ ce …s ‡ r ‡ v

…5b†

ˆ Ef‰D^c…r†

(

t†c…N ‡ 1

2r

‡ c…N ‡ 1 r v t†c…N ‡ 1 t r† ‡ ce …r†ce …r ‡ v† ‡ ce …N ‡ 1 t†ce …N ‡ 1 ‡ ce …N ‡ 1

r

v

E‰^ xN ‡1 x^N ‡1 r x^N ‡1 x^N ‡1

t†ce …N ‡ 1

t

v 2r

t† v

t†

r†

r vŠ

ˆ c…0†c…v† ‡ 2c…r†c…r ‡ v† ‡ ce …0†ce …v† ‡ 2ce …r†ce …r ‡ v† Considering Eqs. (7) and (8), then E‰D^c…r†D^c…r ‡ v†Š ( N r NX r v 1 1 X E‰xt xt‡r xs xs‡r‡v Š ˆ …N ‡ 1†2 N 2 tˆ1 sˆ1 N r 1X E‰xt xt‡r xN ‡1 xN ‡1 N tˆ1

r vŠ

r v 1 NX E‰xs xs‡r‡v xN ‡1 xN ‡1 r Š N sˆ1

‡ E‰xN ‡1 xN ‡1 r xN ‡1 xN ‡1

) r vŠ

…8b†

440

‡

(

1 …N ‡ 1†2

N r NX r v 1 X E‰et et‡r es es‡r‡v Š N 2 tˆ1 sˆ1

N r 1X E‰et et‡r eN ‡1 eN ‡1 N tˆ1

r

‡ ‡

vŠ

r

…11†

N 2 …N ‡ 1†2

r…r ‡ v†

EfD^c…r†gEfD^c…r ‡ v†g ˆ

r vŠ

…9†

N 2 …N

‡ 1†2

c…r†c…r ‡ v†

into Eq. (6), and after rearrangement COV‰Dc…r†; Dc…r ‡ v†Š

Let 8 0 vm0 1 …N r†  m 

‡

v

t†c…s ‡ v

t†

‡ c…m ‡ r ‡ v†c…m

1 N

r†Š

COV‰D^c…r†D^c…r ‡ v†Š ( r v 1  1 1 NX 1 ˆ …N ‡ 1†2 N mˆ …N r†‡1 …10a†

m†c…N ‡ 1 r

‰c…N ‡ 1

v

2r

v

m†

m†c…N ‡ 1

r

m†Š

m†c…N ‡ 1

2r

v

r

v

m†c…N ‡ 1

r

‡ c…N ‡ 1

m†

…10b† De1 and De2 are de®ned in the same manner as D1 and D2 above, except that ce is substituted for c. Then, using the property of ce that zce …y† ˆ 0 …y  z  0†, we can ®nd that D ˆ D1 ‡ D2 De ˆ De1 ‡ De2 …N r v†‰ce …0†ce …v† ‡ ce …r†ce …r ‡ v†Š ˆ N2 D ‡ De E‰Dc…r†Dc…r ‡ v†Š ˆ …N ‡ 1†2 …N 2 ‡ r2 ‡ vr†‰c…r†c…r ‡ v† ‡ ce …r†ce …r ‡ v†Š ‡ N 2 …N ‡ 1†2 ‰c…0†c…v† ‡ ce …r†ce …r ‡ v†Š ‡ N …N ‡ 1†2 D …N 2 ‡ r2 ‡ vr†c…r†c…r ‡ v†† ˆ ‡ 2 …N ‡ 1† N 2 …N ‡ 1†2

‡ ‡

m†c…N ‡ 1

r

r†Š

2r

v

m†

v

m†c…N ‡ 1

r

m†Š

r v 1 NX ‰c…N ‡ 1 N mˆ1

m†c…N ‡ 1

2r

v

‡ c…N ‡ 1

m†Š

 #…m† ‡ r ‡ v N

 ‰c…m†c…m ‡ v† ‡ c…m ‡ r ‡ v†c…m 1 ‰c…N ‡ 1 N mˆ1

mˆ1

‡ c…N ‡ 1

N 2 …N ‡ 1†2

N r X

N r 1X ‰c…N ‡ 1 N mˆ1 NX r v

…N ‡ 1†2 …N 2 ‡ N r v†‰ce …0†ce …v† ‡ ce …r†ce …r ‡ v†Š

Using D1 and D2 , the above formula becomes

‡ c…s ‡ r ‡ v t†c…s t r†Š  r v 1  1 NX #…m† ‡ r ‡ v ‰c…m†c…m ‡ v† 1 ˆ N mˆ1 …N r† N

‡ c…N ‡ 1

D ‡ c…0†c…v† ‡ c…r†c…r ‡ v†

…12†

N r NX r v 1 X D1 ˆ 2 ‰c…s N tˆ1 sˆ1

D2 ˆ

…N

…N ‡ 1†2 v†‰ce …0†ce …v† ‡ ce …r†ce …r ‡ v†Š

Inserting the above and

r v 1 NX E‰es es‡r‡v eN ‡1 eN ‡1 r Š N sˆ1 )

‡ E‰eN ‡1 eN ‡1 r eN ‡1 eN ‡1

‰c…0†c…v† ‡ ce …0†e c…v† ‡ ce …r†ce …r ‡ v†Š

r

m† )

v

m†c…N ‡ 1

r

m†Š

c…0†c…v† ‡ c…r†c…r ‡ v† …N ‡ 1†2 …N 2 ‡ N r v†‰ce …0†ce …v† ‡ ce …r†ce …r ‡ v†Š N …N ‡ 1†2

…13a† When v=0 VAR‰D^ c…r†Š ˆ

(

1 N …N ‡ 1†2



NX r 1

1

mˆ1 …N r†

 ‰c2 …m† ‡ c…m ‡ r†c…m 2

N r X

‰c…N ‡ 1

)

‡ c …N ‡ 1 ‡

…N 2 ‡ N

r

m†Š

‡



2r

m†

c2 …0† ‡ c2 …r† …N ‡ 1†2

r†‰c2e …0† ‡ c2e …r†Š

N 2 …N ‡ 1†2

r

r†Š

m†c…N ‡ 1

mˆ1 2

#…m† N

…13b†

441

It can be shown that c^ asymptotically converges to a Gaussian distribution (Yang and Gu 1988). Based on this and the discussion above, when N is relatively large, D^ cN ‡1;N …r† ˆ c^N ‡1 …r† c^N …r† will converge to a Gaussian distribution with zero expectation, i.e.

When N is very large COV‰D^ c…r†; D^ c…r ‡ v†Š ( 1 X 1 ‰c…m†c…m ‡ v†  2 N …N ‡ 1† mˆ 1 ‡ c…m ‡ r ‡ v†c…m r†Š 1 X ‰c…N ‡ 1 m†c…N ‡ 1 2

2r

cN ‡1;N …r†Š† D^ cN ‡1;N …r†  N …0; VAR‰D^ v

mˆ1

)

‡ c…N ‡ 1 ‡

m†

r

v

m†c…N ‡ 1

r

m†Š



(

1

…14a†

1 X

‰c2 …m† ‡ c…m ‡ r†c…m

N …N ‡ 1† mˆ 1 1 X 2 ‰c…N ‡ 1 m†c…N ‡ 1 mˆ1

2r

k f^ qr …r†  N …0 ; 1†gNrˆM

r†Š

2

‡ c …N ‡ 1 c2 …0† ‡ c2 …r†

r

m†

QN ‡1 …k† ˆ QN ‡1 …Nk

m†Š

c2 …0† ‡ c2e …r† ‡ e N …N ‡ 1†

c2 …0† ‡ c2e …0† N …N ‡ 1†

…14b†

…15a†

COV‰D^ c…r†; D^ c…r ‡ v†Š ( 1 X 1  …‰c…m†c…m ‡ v†Š N …N ‡ 1†2 mˆ 1 ) 1 X c…0†c…v† ce …0†ce …v† 2 ‰c…m†c…m ‡ v†Š ‡ ‡ …N ‡ 1†2 N …N ‡ 1† mˆ1 c…0†c…v† ‡ ce …0†ce …v† N …N ‡ 1†

Nk X

q^2r …r†

…17†

According to QN ‡1 …k†, the state of a signal can be tested.

and

ˆ

M ‡ 1† ˆ

rˆM

In fact, when r is large enough, the terms dependent on r are almost zero, i.e. c…s…r††  0. Thus, when r > M (one time delay) VAR‰D^ c…r†Š 

(16b)

Based on Eq. (16b), a v2a …k† statistic can be constructed and denoted as QN ‡1 …k†

)

…N ‡ 1†2

D^ cN ‡1;N …r† q^r …r† ˆ p c^2 …0†=N …N ‡ 1† then an asymptotically independent normal Gaussian sequence can be obtained from Eq. (16a)

2

‡

c^2 …0† N …N ‡ 1†

and if

…N ‡ 1†2 c …0†ce …v† ‡ ce …r†ce …r ‡ v† ‡ e N …N ‡ 1† VAR‰D^ c…r†Š

So, fD^ cN ‡1;N …r†g can be considered to be an asymptotically independent Gaussian sequence. Because, when r  M, VAR‰D^ cN ‡1;N …r†Š 

c…0†c…v† ‡ c…r†c…r ‡ v†

(16a)

…15b†

Because c and ce are bounded functions, the expression in Eq. (15b) approaches zero for large N . This shows that fD^ cN ‡1;N …r†g is an asymptotically independent sequence. Usually, in practice, only c^ can be obtained. By using c^ instead of c and neglecting the ce terms, then Eqs. (13) and (15) will be changed into equations which use the estimated quantities c^. Thus, Eqs. (4), (13) and (15) can be referred to as the solution formulae of the stochastic model parameters of the ACEVS.

(1) The signal is in the stable state if QN ‡1 …k†  v2a …k†. (2) The signal is in the disturbed state if QN ‡1 …k† > v2a …k†. From the estimation formulae above [i.e. Eqs. (4), (13) and (15)], it can be seen that for a stable signal the cN ‡1;N …r†Š) as estimate of E‰r^ cN ‡1;N …r†Š (and VAR‰r^ calculated when c^N ‡1 …r† or c^N …r† represents c^ will be very close. In contrast, for the disturbed signal these estimates are obviously di€erent. Because this paper assumes that the previous N samples of the signal are stable, the normal estimation of VAR‰r^ cN ‡1;N …r†Š in Eq. (16) will be calculated using c^N …r† only. In real-time applications, especially for kinematic applications, the selected values of r, M and N cannot be very large; however, for post-processing, the selection conditions may be broadened whenever necessary and possible. Therefore, for the two cases above, the estimation formulae, Eqs. (15) and (13) respectively, may be selected. The ecient and precise selection of r, M and N depends on the results of extensive experiments. In the later part of this paper some preliminary discussions will be presented. 3 ACEVS and ionospheric monitoring using GPS GPS observations include the deterministic I (such as a trend and a period) and stochastic (dI) delays due to the ionosphere. Usually, for short time scales, for deterministic e€ects, only the trend variation can be considered,

442

P i and I can be written as a polynomial It ˆ m iˆ0 ai t . The stochastic e€ects dIt may be considered as a Gaussian random process with zero expectation value. When random disturbances of the ionosphere happen, their e€ects on GPS signals will usually destroy the steady state of dIt . Therefore, it is possible to test the change of state of dIt using ACEVS.

ˆ

kˆ0 jˆ0

‡ ˆ

ai ti ‡ dIt ‡ e

…18†

De®ne the di€erence operator as I^t ; rk I^t ˆ r…rk 1 It † ˆ

k X iˆ0

… 1†i Cki I^t‡k

i

…19† where Cki is the combination operator. To reduce the trend term It , a q ˆ m ‡ 1-order difference operation can be used for Eq. (18) rq I^t ˆ rq dIt ‡ rq et ˆ rq …dIt ‡ et †

(20a)

E…rq I^t † ˆ rq E…dIt ‡ et † ˆ rq E…dIt † ‡ rq E…et † ˆ 0 (20b) Similarly rq I^t‡h ˆ rq dIt‡h ‡ rq et ; E…rq I^t‡h † ˆ 0

(20c)

rq I^t

is a linear From the above, it can be seen that combination of dIt‡q i ‡ et‡q i …i ˆ 0; 1; 2; . . . ; q†, while dIt‡q i ‡ et‡q i is a Gaussian random variable with zero expectation value. So, according to the invariance property of linear transformations of Gaussian distributions, rq I^t is a Gaussian random variable with zero expectation value as well. Based on the above discussion, the following can be obtained: COV…rq I^t‡h ; rq I^t † ˆ ˆ ˆ

ˆ crq …h† q^ Ef‰r It‡h E…rq I^t‡h †Š‰rq I^t‡h E…rq I^t †Šg Ef‰rq dIt‡h ‡ rq et‡h †Š‰rq dIt ‡ rq et †Šg E…rq dIt‡h rq dIt † ‡ E…rq et‡h rq et † "

ˆE

q X … 1†k Cqk dIt‡h‡q

q X k

kˆ0

jˆ0

"

q X … 1†k Cqk et‡h‡q ‡E kˆ0

#

… 1†j Cqj dIt‡q

q X k jˆ0

j

# … 1†

j

q X q X

… 1†k‡j Cqk Cqj E…et‡h‡q k et‡q j †

… 1†k‡j Cqk Cqj ‰cdI …h ‡ j

k†‡ce …h ‡ j

k†Š …21†

iˆ0

rI^t ˆ I^t‡1

q X q X

kˆ0 jˆ0

Assume that I^t is the ionospheric delay in GPS L1 phase observation at an arbitrary epoch t and et is its Gaussian white noise [E…et † ˆ 0Š, independent of dI [i.e. E…dIt et‡i † ˆ 0]. Further, fdI ‡ eg is a Gaussian process with zero expectation value. Thus, the ionospheric observation model can be written as m X

… 1†k‡j Cqk Cqj E…dIt‡h‡q k dIt‡q j †

kˆ0 jˆ0

3.1 Determining a stationary ionospheric model for GPS and the corresponding ACEVS

I^t ˆ It ‡ dIt ‡ et ˆ

q X q X

Cqj et‡q j

Equation (21) shows that for a certain value of q, COV…rq I^t‡h ; rq I^t † depends only on h and is independent of t. E…rq I^t‡h;t † ˆ 0 and, therefore, frq I^t‡h;t g is stationary and obviously is an ergodic process as well. Because frq I^t‡h;t g is an ergodic Gaussian process with zero expectation value, and if x^t ˆ rq I^t‡h;t and xt ˆ rq It‡h;t , then, under normal observation conditions, the ACEVS of f^ xt ˆ rq I^t‡h;t g can be calculated according to the method discussed in Sect. 2 and it may be considered as the approximate value of ACEVS of fxt ˆ rq It‡h;t g. In time-series observations of GPS, the change of the statistical properties of the random ionospheric delay fxt ˆ rq It‡h;t g from a situation of stability to one of disturbance can be distinguished by the corresponding change of its ACEVS. So, it is possible to test by using GPS time series and monitoring the corresponding ACEVS of f^ xt ˆ rq I^t‡h;t g for any abnormal ionosphere variation. This is the main result of this paper. 3.2 Analyses using ACEVS on random ionospheric delay 3.2.1 GPS data and ionospheric observables Some preliminary results are obtained using dualfrequency GPS phase data (the sample period is 30 s) collected by an IGS station (SHAO). Only some of them are displayed here (the date is 31/10/1997, the time period is 12:00 to 18:00 h, local time). The others are very similar. During this data interval, the ionosphere activity is quiet. In this paper, all total electron content (TEC) values are transferred to ionospheric delays in the carrier L1 observation. The ionospheric observables I^t are formed as (see more detail in Yuan and Ou 1999) I^t ˆ L/ ‡ D ‡ e/ where D ˆ F ‰…N2 F ˆ

f22 =… f12

N1 † ‡ …S/2 f22 †;

S/1 † ‡ …R/2

R/1 †Š;

fi …i ˆ 1; 2†

is the frequency of the carrier Li …i ˆ 1; 2†; L/ ˆ F …/1 /2 †; S and R are the instrumental biases (constant) of satellite and receiver in the corresponding GPS data, respectively; /i …i ˆ 1; 2† is Li …i ˆ 1; 2† carrier phase observation; Ni …i ˆ 1; 2† is the integer ambiguity in carrier Li …i ˆ 1; 2† phase observation; e is the combined term of observation noise and other random errors.

443

Fig. 1. Imitated result of auto-covariance of ideal model

is much reduced and the estimation of auto-covariance is very close to the ideal result [hence let q ˆ 2 for the later discussion, though q may have other values such as 3 or 4, etc. for other GPS data (Delikaraogou 1989)]. The above results show that the statistical model and observation model of the random ionospheric delay for the GPS data considered here is reasonable. Here, and later, the number of experimental samples is always N ‡ 1 ˆ 331, unless otherwise stated. Figure 4 illustrates the calculated auto-covariance of fD^ crq …r†N ‡1;N gNr r from the di€erence value rq I^t of N ‡ 1 random ionospheric delay changes with time delay v. From Fig. 4, it can be seen that fCOV‰D^ crq …r†; D^ crq …r ‡ v†ŠgNr r  0 …v 6ˆ 0† and this indicates that fD^ crq …r†N ‡1;N gNr r is an asymptotically independent sequence when N is relatively large. Figure 5a suggests that the estimated expectation value of fD^ crq …r†N ‡1;N gNr r is asymptotically zero, i.e. fEfD^ crq …r†N ‡1;N g  0gNr r . This is similar to what was discussed before. Figure 5b shows that the estimated variance value of the sequence fD^ crq …r†N ‡1;N gNr r is asymptotically a constant, i.e. r . Based on this, and fVAR‰D^ crq …r†N ‡1;N Š  constgNrˆ1 the fact that D^ crq …r†N ‡1;N converges to an asymptotically Gaussian distribution, it can be assumed that fD^ crq …r†N ‡1;N gNr r is a `near-Gaussian white-noise process'. Therefore, in theory, M equals 1. The previous

Fig. 2. a First-order di€erence ionospheric variations (FDIV) and b second-order di€erence ionospheric variations (SDIV)

Fig. 3. Estimated auto-covariance of FDIV (a) and SDIV (b)

3.2.2 Preliminary analyses Figure 1 shows the result for an auto-covariance of an ideal statistical model of the random ionosphere delay. Figures 2 and 3 show the variation, and its corresponding auto-covariance COV…rq I^t‡h ; rq I^t † ˆ crq …h†, of the ®rst- and second-order di€erence of the real ionospheric delay I^t , respectively. From the comparison of Figs. 1, 2 and 3, it can be seen that the trend variation of the ionospheric delay in the second-order di€erence process

444

Fig. 4. Auto-covariance of ACEVS of SDIV for r ˆ 0 (a) and r ˆ 10 (b)

Fig. 5. Variation of the estimated expectation (a) and the estimated variance of ACEVS of SDVID (b)

^ of M is results also show that the estimated value M almost equal to 1. That is, when r 1, c^r^c …r† ˆ ^ VAR…r^ c…r††  const. According to the value of M, some other values of r can be selected, and for every r, di€erent variation curves of VAR…r^ c…r†† for increasing Ni ˆ r ‡ 1; . . . ; N ‡ 1, the number of samples, can be found. Through the comparison of di€erent curves, it is possible to determine a number NLM , which is the value of N when VAR…r^ c…r†† is almost stable. Figure 6 displays the results corresponding to r ˆ 0 and r ˆ 10, respectively. The preliminary results show that NLM may be about 50. Assume that (1) case I expresses the case that all of the N ‡ 1 samples are free from ionospheric disturbance; and (2) case II is the case that the previous N samples are calm like case I while only the N ‡ 1th sample is in the disturbed period of the ionosphere. Then, some r (r  1) values are selected. For case I, when Ni  Nmin ˆ r ‡ cNi ‡1;Ni …r†  0] can be seen NLM , c^Ni ‡1 …r†  c^Ni …r† [i.e. D^ from Fig. 5a (refer to the earlier discussion). Figure 5b shows that the variation of VAR…r^ cN ‡1;N …r†† is also steady for this case. For case II, Fig. 7 suggests that the results are similar to that of case I when N  Ni  Nmin , and the di€erence between c^N ‡1 …r† and c^N …r† is very large when Ni ˆ N ‡ 1. This results in the obvious biases between the calculated D^ cN ‡1;N …r† and the estimated value of fE…D^ cNi ;Ni 1 …r††g [the bias values asymptotically equal D^ cN ‡1;N …r† fE…D†^ cNi ;Ni 1 …r†gNNi ˆNmin  D^ cN ‡1;N …r†Š

when Ni ˆ N ‡ 1. Figure 8 also shows that the variations of VAR…r^ cNi ;Ni 1 …r†† are like case I if Ni ˆ r ‡ 1; . . . ; N , while the estimated values of VAR…r^ cN ‡1;N …r†† have the obvious biases from the other estimated values of VAR…r^ cNi ;Ni 1 …r††. From the preliminary results above, it can be seen r r and fVAR‰r^ cN ‡1;N …r†ŠgNrˆ1 that both fr^ cN ‡1;N …r†gNrˆ1 of case II obviously have large biases compared to those of case I. This shows that the statistical property of ACEVS is sensitive to the change of the state of the signal. Based on this, a new way to monitor ionospheric actions using GPS time series observations is proposed. 4 A framework scheme for monitoring the disturbance of the random ionospheric delay and its preliminary experimental results Following the above discussion and the characteristic of the time-series observations of GPS, a preliminary framework scheme for monitoring the disturbance of the ionospheric delay is suggested: (1) use the previous Ni and Ni‡1 (the present epoch) samples to calculate the estimated values c^Ni …r† and c^Ni ‡1 …r† of c…r†, respectively; (2) according to Eq. (15) [or Eq. (13)], determine the auto-covariance estimate (v ˆ 0) fVAR and

445

Fig. 6. Relationship between the estimated variance and number of samples for r ˆ 0 (a) and r ˆ 10 (b) r r ‰r^ cN ‡1;N …r†ŠgNrˆm for fD^ cNi ‡1;Ni …r†gNrˆm , Nr;m the selected maximum and minimum number of r, respectively (3) calculate Nr  2 Ni …Ni ‡ 1† X QNi ‡1 ˆ D^ cN i ‡1;Ni …r† c^Ni …0† rˆm or Nr n X  o QNi ‡1 ˆ ‰D^ cN i ‡1;Ni …r†Š2 =VAR D^ cN i ‡1;Ni …r†

rˆm

where Nr ˆ 15; m ˆ 1; K ˆ Nr m ‡ 1 ˆ 15, in this paper (the k value may be selected di€erently for di€erent conditions). (4) if QNi ‡1  v2a …k†, then the ionosphere is in a normal condition. Use epoch Ni +1 instead of epoch Ni and epoch Ni +2 (the present epoch) instead of epoch Ni +1, and reiterate. Here a is the disturbance test factor and equals 0.005 in this paper. (5) if QNi ‡1 > v2a …k†, then the ionosphere is in the disturbed state. Let ^INi ‡1 ˆ ^INi then Ni +1 instead of Ni and Ni +2 (the present epoch) instead of epoch Ni +1, and recalculate. Based on this reference method, a static real-time monitoring scheme is designed to test a simulated ionosphere disturbance in the next epochs (N  331). In fact, the real ionosphere is quiet; the simulation technique is only for testing the e€ectiveness of the scheme easily) of the previous GPS data (N < 331). In total, there are 14

Fig. 7. Variations of the estimated expectation with number of samples for r ˆ 0 (a) and r ˆ 10 (b)

epochs, corresponding to 7 minutes when the ionosphere is simulated as a disturbed state (equals real quiet ionosphere plus simulated random ionospheric disturbance). Figure 9a shows the experimental samples of GPS and the time interval during epoch 331 to 344 is the simulated ionospheric disturbance period. From Fig. 9b it can be seen that the tested disturbed period is similar to the simulated disturbed interval. Although the above method is proposed for real-time monitoring, it can be easily applied to post-processing of GPS data. Note that, in the design of di€erent concrete schemes for di€erent real cases, there will be di€erences in the selection of N and r. It is necessary to devise practical schemes based on the di€erent demands of Nmin and rmin and other aspects from di€erent applications. For post-processing, Nmin and rmin are not very important and the number of N and r may be selected to be as large as possible. For static real-time processing, one should select the values as large as possible on the basis of the requirements of accuracy and storage and calculation speed. For a kinematic application, in order to meet the demands of accuracy, storage and calculation eciency, N and r should be selected to be as small as possible based on Nmin and rmin . Test schemes of ionospheric disturbance of GPS mainly include post- and real-time methods while static and real-time scheme designs are basically similar and only have di€erent hardware requirements. A more

446

Fig. 8. Relationship between the estimated variance and number of samples for r ˆ 0 (a) and r ˆ 10 (b)

detailed discussion concerning these will be given in another paper. 5 Conclusions and suggestions Preliminary experimental results indicate that it is possible to monitor the random ionosphere disturbance in GPS observations using the ACEVS method proposed in this paper. The framework scheme based on ACEVS can be used to design many concrete practical schemes for monitoring ionosphere variation using a single (static or kinematic) dual-frequency GPS reciever. However, these are only preliminary results. All procedures presented here should be further veri®ed and improved using more GPS experimental data. In particular, the relationship between the number N and r and the storage requirements for di€erent schemes needs to be investigated. More ecient statistics for the determination of di€erent ionospheric situations, as well as calculation methods, should be selected to meet the demands of ionospheric monitoring for kinematic GPS. Acknowledgements. We would like to thank the Chinese Academy of Sciences (Grant N0. KZCX2-106) and the China Defence Committee of Science and Technology for their ®nancial support and for providing us with GPS data. Special thanks should also be

Fig. 9. a Calculated ionospheric delay based on dual-frequency GPS phase observations; b test results for ionospheric disturbance using ACEVS and dual-frequency GPS phase observation

given to Prof. C.C. Tscherning and the other reviewers for their many valuable suggestions and critical comments, and to Dr. Per Knudsen, Ms. Anna O. Jensen and Mr. Abbas Khan for their help.

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