Acta Astronautica 65 (2009) 1149–1157www.elsevier.com/locate/actaastro

An algorithm for estimation and separation of ephemeris and clockerrors in SBAS

S. Mishra∗, R. Gupta, A.S. GaneshanISRO Satellite Center, Bangalore, India

Received 5 December 2007; accepted 9 March 2009Available online 18 April 2009

Abstract

The estimation and separation of ephemeris and clock errors is an integral part of a SBAS (Space Based Augmentation System).Generally, the global solution is based on the full state approach for satellite errors (ephemeris and clock) and station errors,using a large least square estimator; or the other way is to sequentially estimate the ephemeris and clock through a Kalmanfilter, using a complex model of the satellite dynamics. In this paper, the estimation and separation of ephemeris and clockerrors is addressed through a unique approach of combining both the methods. The algorithm employs measurements, which arepre-processed for various errors and known biases. A single difference technique is used to separately estimate the ephemerisand clock components. The ephemeris Kalman filter uses a priori information of ephemeris errors along with measurementsthrough a minimum variance estimator to provide ephemeris error estimate. A similar approach is adopted in the clock errorestimation process, to provide clock and clock rate estimates. The algorithm results are presented using simulated data for knownerrors in ephemeris/clock and subsequent retrieval. This algorithm estimates these errors as corrections to the broadcast GlobalPositioning System (GPS) navigation data, required by a SBAS user for accuracy improvement.© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The SBAS (Space Based Augmentation System) isdesigned to improve the user position accuracy and in-tegrity for the approach and landing phase of an air-craft flight. The SBAS system is conceptualized aroundGPS (Global Positioning System) and improves the ba-sic accuracy provided by the GPS. The primary ob-jective of SBAS is to generate real time correctionsand integrity for the GPS ephemeris, GPS clock andthe vertical ionosphere delay, apart from various other

∗Corresponding author.E-mail addresses: saumyakm@isac.gov.in (S. Mishra),

rgupta_india@yahoo.com (R. Gupta), asganesh@isac.gov.in(A.S. Ganeshan).

0094-5765/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.actaastro.2009.03.044

parameters, defined by RTCA MOPS [1]. The primarychallenge in providing the ephemeris and clock error isto separate these, from the line-of-sight errors [2]. Thereare various approaches and algorithms to estimate theGPS ephemeris and clock errors in real time [3–6].The real time estimation of GPS ephemeris and clock

corrections in SBAS requires a certain processing onraw measurements, for removing known errors and bi-ases. The algorithm accesses the processed measure-ments and involves the following steps for the solution:

• A single difference technique is used on the pro-cessed measurements, considering various line-of-sight measurements from all stations to the visibleGPS.

• A least square algorithm is used to produce an initialestimate and variance of the ephemeris errors.

1150 S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157

Fig. 1. Geometry of GPS to ground stations.

• The least square estimate and its variance are pro-vided as a priori guess to a Kalman filter to estimatethe ephemeris errors and their rate in real time. Thepropagation model in Kalman filter is a simple kine-matics model.

The next step is to prepare the measurements for clockerror estimates. Here, the estimated ephemeris error isremoved from the processed measurements and an ap-proach (similar to ephemeris) is adopted to estimate theclock errors, described subsequently in detail.The paper addresses the mathematical description

related to the measurement modeling, followed by thedescription of ephemeris and clock error estimationusing epoch least square and Kalman filter methods.In the end, the data simulation procedure and algo-rithm results are presented, including the concludingremarks.

2. Measurement modeling

The mathematical description involves the procedureof measurements generation for the algorithm, begin-ning with the raw GPS measurements and subsequentremoval of known errors from the measurements.In SBAS, the reference receiver is a specially de-

signed dual frequency GPS receiver. This receiver pro-vides pseudo range and carrier phase measurements inL1 and L2 frequency at every 1-s interval. It is im-portant to note that the user segment in SBAS is cen-tered on the L1 pseudo range. However, in SBAS, thesmoothed L1 range (ionosphere-free) is derived fromthe dual frequency smoothing of code with carrier phasemeasurements. This range needs to be corrected for var-ious other known errors and biases. The typical geom-etry of a GPS satellite to the number of SBAS stationsis shown in Fig. 1.The GPS observation in L1 frequency can be ex-

pressed (from a reference station “m” to the visible

GPS satellite “ j”) in the form of a pseudo rangemeasurement

� jm,L1 = D j

m + I jm + T jm + bm − B j + � j

m,L1 (1)

where � jm,L1is the L1 pseudo range, D

jm is the geometric

range, I jm is the ionosphere delay, T jm is the troposphere

delay, bm is the receiver clock error, B j is the satelliteclock error and � j

m,L1 is the measurement noise.The next step is to model and remove the above errors

as follows:

• The geometric range from the station to the satelliteis computed using satellite broadcast ephemeris andthe precisely known station positions, wherein thesatellite ephemeris errors are manifested in geometricrange due to the errors in broadcast GPS ephemeris.

• The ionosphere error is estimated from dual fre-quency measurements of code and carrier.

• The troposphere delay is computed using a simpleSBAS model, requiring no meteorological measure-ments.

• The satellite clock bias is computed from theGPS clock coefficients broadcast in the navigationmessage.

• The receiver clock error is estimated using variousGPS line-of-sight measurements from a station.

After correcting for all the above errors, the expressionfor pseudo range residual measurements for the k thGPS satellite from the m th ground station is

��km = �Rk .1km − �Bk + �km (2)

where �Rk is the ephemeris error vector, 1km is line-of-sight unit vector, �Bk is satellite clock error and �km isthe measurement noise.

3. Ephemeris error estimation

The ephemeris error can be estimated by using atleast two line-of-sight measurements to a GPS satellite.This requires the clock error to be decoupled, from themeasurements. The concept of single difference is usedto remove �Bk from the range residuals in Eq. (2).

First, a pivotal station “p” (smallest variance) is se-lected such that all other station residuals are differ-enced with the residual of station “p”

��km − ��kp = (1km)�Rk + �km (3)

S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157 1151

3.1. Minimum variance estimate

Eq. (3) can be represented in the matrix form asfollows:

Z = H �Rk + �km (4)

where

Z =

⎡⎢⎣

��k1 − ��kp...

��km − ��kp

⎤⎥⎦ , H =

⎡⎢⎣1k1 − 1kp

...

1km − 1kp

⎤⎥⎦

E[(�km)2] = E[(�km)

2] + E[(�kp)2] = (�km)

2 + (�kp)2 (5)

The matrix Z has order (M − 1) X 1, wherein M is thetotal number of pseudo range residual measurementsand (M − 1) measurements are used in the above equa-tion. The method of minimum variance estimation isfirst used for obtaining an initial value of ephemeris er-ror vector, by solving the matrix Eq. (4).The notation X is used for the ephemeris error vector,

which needs to be estimated. Using minimum variance,for

Z = HX + � (6)

with the following assumptions:

E[x] = x

E[xxT ] = � (7)

(a) First compute HX .(b) Compute P= (�−1+HT V−1H )−1 where � is co-

variance matrix for X and V is measurement noisematrix.

(c) Compute GR = �−1(x − x) − HT V−1[z − h(x)].(d) If |GR| < �, set X∧ = X and stop.(e) Replace X by (X − P.GR) and go to (a).

The minimum variance estimate is used as a measure-ment and the initial guess in the Kalman Filter, whereinthe kinematic orbit model for state propagation is used.

3.2. Kinematic orbit model

The kinematic orbit model is used for the estimationof ephemeris error, where X is containing ephemeriserror and ephemeris error rate

X (k + 1) = X (k) + X (k) ∗ T

X (k + 1) = X (k) + wd (k) (8)

where T is the sampling interval and wd is the discreteprocess noise. Above kinematic model for the ephemeriserrors can be converted to the following discrete-timesystem:

X0(k + 1) = �0(k)X0(k) + 0(k)w0(k) (9)

And, the measurement vector, state transition matrix,measurement sensitivity matrix and initial state are

Z0(k) = H0(k)X0(k) + �0(k)

0(k) =

⎡⎢⎢⎢⎢⎢⎣

1 0 0 T 0 00 1 0 0 T 00 0 1 0 0 T0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎦

H0(k) =[1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

]

X0=[�Rk �Rk]T=[�Xk, �Y k, �Zk, �X k, �Y k, �Z k]

(10)

Here measurement vector Z0 is available from the snap-shot minimum variance estimate. As the linear systemdescribed above is time invariant, �0, H0 and 0 areconstant matrices. 0(k) is an identity matrix here.

3.3. Kalman filter estimation

The estimate for the state vector X0 at tk is

X0(k) = X0(k) + K0(k)(Z0(k) − H0 X0(k)) (11)

where Kalman gain matrix

K0(k) = P0(k)HT0 (H0 P0(k)H

T0 + R(k))−1 (12)

and X0, P0 are the a priori information.The updated error covariance matrix for X0hat (k) is

P0(k) = (I − K0(k)H0)P0(k) (13)

The time update for the Kalman filter predicts the statevector and its error covariance matrix for the next timestep

X0(k + 1) = �0 X0(k)

P0(k + 1) = �0 P0(k)�T0 + Q0 (14)

After estimating ephemeris error, the clock errors areestimated next.

1152 S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157

4. Clock error estimation

The clock error measurement for the k th satellitefrom the m th reference station is computed as follows:

zkc,m = �Rk .1km − ��km = �Bkm + nkm (15)

and, its measurement variance can be computed asfollows:

�kc,m2 = 1km .P3.1

km + �2��km

(16)

where P3hat is the upper 3 × 3 diagonal block matrixfrom the ephemeris algorithm. The subscript c is usedto denote clock. This variance combines the covariancematrix of the ephemeris error estimate and the vari-ance of the original measurement, i.e., the pseudo rangeresidual.In Matrix form, the above equation can be re-

written as

zc = Hc�Bk + nc (17)

where Hc is the column vector with all 1 and nc is themeasurement noise with covariance matrix Wc whosediagonal elements are given by previous equation.The least square estimation for the abovematrix equa-

tion is obtained from

�Bksnap = (HT

c W−1c Hc)

−1HTc W−1

c zc

P∧kc,snap = (HT

c W−1c Hc)

−1 (18)

where Wc is a diagonal matrix whose elements arecomputed. Pk

c,snap is the variance of the weighted leastsquare estimate.

4.1. Model equations for clock

With the measurement equation

�Bksnap = Hc(k)Xc(k) + �c(k) (19)

where

Xc = [�Bk �Bk]T

Hc = [1, 0] (20)

The measurement noise vc(k) has the variance Pkc,snap hat

from least square estimate. The clock propagationmodel and the state transition matrices are

Xc(k + 1) = c(k)Xc(k) + wc(k)

� =[1 T0 1

](21)

4.2. Kalman filter estimation

The Kalman estimate for the state vector, which in-cludes clock error and clock error rate Xc at time tk is

Xc(k) = Xc(k) + Kc(k)(�Bksnap − Hc Xc(k))

Kc(k) = Pc(k)HTc (Hc Pc(k)H

Tc + Rc(k))

−1

Pc(k) = (I − Kc(k)Hc)Pc(k) (22)

where Xc(k) and Pc(k) are the a priori information (timeupdate estimate to be described below) and Kc is theKalman gain matrix.The time update for Kalman filter predicts the state

vector and its error covariance matrix for the next timestep. The superscript bar is used to denote the predictedestimate and covariance matrix.

Xc(k + 1) = �c Xc(k)

Pc(k + 1) = �c Pc(k)�Tc + Qc (23)

The above results are used as the a priori informationfor the Kalman filter measurement update in next timestep tk+1.

5. Results and discussions

The algorithm results are based on the simulation ofknown ephemeris and clock errors in the measurementvector for a selected GPS PRN for a set of eight stations.The retrieval of the parameters of ephemeris and clockis achieved through the estimation algorithm.

5.1. Data simulation

The simulated measurements were generated using

• precise GPS ephemeris and clock,• broadcast GPS ephemeris and clock,• known locations of stations.

Further, the measurement noise (0.1m sigma) wasadded as a Gaussian noise in the measurements. Thedata duration for simulation was taken to be 2h.

5.2. Ephemeris parameters

The estimated ephemeris, clock and rate componentsfrom the algorithm are shown in Figs. 2–21.The true values of ephemeris errors along the three

axes are provided in Figs. 2–4.Fig. 5 displays the estimated value of ephemeris

“dx”, which follows the same pattern as the true value.The estimated error covariance bounds the estimated

S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157 1153

1.1

1.05

1

0.95

0.9

0.850 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

true

dx (m

)

Ephemeris error : true values

Fig. 2.

1.2

1

0.8

0.6

0.40 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

true

dy (m

)

Fig. 3.

0

-0.05

-0.1

-0.15

-0.2

-0.250 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

true

dz (m

)

Fig. 4.

error in “dx” (Fig. 6). Likewise, the ephemeris es-timates of “dy” and “dz” have been provided inFigs. 7–10, respectively.

1.1

1.05

1

0.95

0.9

0.850 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ated

dx

(m)

Ephemeris estimate x

Fig. 5.

3

2

1

0

-1

-2

-30 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

dx e

stim

ate

(m)

Fig. 6.

Fig. 11 represents the ephemeris rate estimate“dvx”, and the estimated covariance bounds the es-timation error, as shown in Fig. 12. The other rate

1154 S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.50 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ated

dy

(m)

ephemeris estimate y

Fig. 7.

3

2

1

0

-1

-2

-30 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

dy e

stim

ate

(m)

Fig. 8.

0

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

-0.350 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ate

dz (m

)

Ephemeris estimate z

Fig. 9.

3

2

1

0

-1

-2

-30 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

dz e

stim

ate

(m)

Fig. 10.

S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157 1155

6

5

4

3

2

1

0

-10 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ated

dvx

(m/s

)

Ephemeris rate estimate dvxx 10-3

Fig. 11.

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-10 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

dvx

estim

ate

(m/s

)

Fig. 12.

1

0.5

0

-0.5

-1

-1.5

-20 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ated

dvy

(m/s

)

Ephemeris rate estimate dvyx 10-4

Fig. 13.

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-10 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

dvy

estim

ate

(m/s

)

Fig. 14.

1156 S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157

3.5

3

2.5

2

1.5

1

0.5

0

-0.50 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ated

dvz

(m/s

)

Ephermeris rate estimate dvzx 10-3

Fig. 15.

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-10 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

dvz

estim

ate

(m/s

)

Fig. 16.

9

8.5

8

7.5

7

6.5

6

5.5

50 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

true

cloc

k (m

)

Clock error : true value

Fig. 17.

9

8.5

8

7.5

7

6.5

6

5.5

5

cloc

k es

timat

e (m

)

0 1000 2000 3000 4000 5000 6000 7000 8000time (sec)

Clock error estimate

Fig. 18.

estimates of “dvy” and “dvz” have been displayed inFigs. 13–16.

5.3. Clock parameters

The true clock error is as shown in Fig. 17, whereasthe estimated error is provided in Fig. 18. The estimatederror covariance in Fig. 19 bounds the error in clockestimate.Fig. 20 provides the delta clock rate estimate. The

associated error covariance contains well the error inestimate, as shown in Fig. 21.

S. Mishra et al. / Acta Astronautica 65 (2009) 1149–1157 1157

3

2

1

0

-1

-2

-30 1000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

cloc

k es

timat

e (m

)

Fig. 19.

12

10

8

6

4

2

0

-2

-41000 2000 3000 4000 5000 6000 7000 8000

time (sec)

estim

ated

clo

ck ra

te (m

/s)

x 10-4 Clock rate estimate

Fig. 20.

6. Conclusions

The paper has presented an algorithm for the sep-aration of ephemeris and clock errors in SBAS. TheKalman filters design for the estimation has been ex-plained, providing details of the measurement and state

2

1.5

1

0.5

0

-0.5

-1

-1.5

-21000 2000 3000 4000 5000 6000 7000 8000

time (sec)

erro

r in

cloc

k ra

te e

stim

ate

(m/s

)

x 10-3

Fig. 21.

propagation models. A priori values for Kalman fil-ter initialization are generated through least square andminimum variance approaches. The algorithm resultsare presented for ephemeris, clock and their rate param-eters. The estimation has been verified with the corre-sponding truth values and estimation error is validatedwith the error covariance.

References

[1] Minimum operational performance standards for globalpositioning system/wide area augmentation system airborneequipment, RTCA Document no. RTCA/DO-229A, June 8,1998.

[2] J.F. Zumberge, W.I. Bertiger, Global positioning system: theoryand applications, in: B.W. Parkinson, J.J. Spilker (Eds.),Ephemeris and Clock Navigation Message Accuracy, AIAAPublications, New York, vol. I, 1996 (Chapter 16).

[3] M.S. Grewal, L.R. Weill, A.P. Andrews, Global PositioningSystems, Inertial Navigation and Integration, Wiley, New York,2001.

[4] E.D. Kaplan, Understanding GPS: Principles and Applications,Artech House Inc., Norwood, MA, USA, 1996.

[5] B.W. Parkinson, Global Positioning System: Theory andApplications (volumes I and II), Progress in Astronauticsand Aeronautics, American Institute of Aeronautics andAstronautics (AIAA), vol. 164, 1996.

[6] R.G. Brown, Introduction to Random Signal Analysis andKalman Filtering, Wiley, New York, 1983.