GNSS Vertical Dilution of Precision Reductionusing Terrestrial Signals of Opportunity

Joshua J. Morales, Joe J. Khalife, and Zaher M. KassasUniversity of California, Riverside

BIOGRAPHIES

Joshua J. Morales is pursuing a Ph.D. from the Depart-ment of Electrical and Computer Engineering at The Uni-versity of California, Riverside. He received a B.S. in Elec-trical Engineering with High Honors from The Universityof California, Riverside. His research interests include esti-mation, navigation, computer vision, autonomous vehicles,and intelligent transportation systems.

Joe J. Khalifeh is a Ph.D. student at The University ofCalifornia, Riverside. He received a B.E. in electrical en-gineering and an M.S. in computer engineering from theLebanese American University (LAU). From 2012 to 2015,he was a research assistant at LAU. His research interestsinclude navigation, autonomous vehicles, and intelligenttransportation systems.

Zaher (Zak) M. Kassas is an assistant professor at The Uni-versity of California, Riverside. He received a B.E. in Elec-trical Engineering from The Lebanese American Univer-sity, an M.S. in Electrical and Computer Engineering fromThe Ohio State University, and an M.S.E. in AerospaceEngineering and a Ph.D. in Electrical and Computer En-gineering from The University of Texas at Austin. From2004 through 2010 he was a research and development en-gineer with the LabVIEW Control Design and DynamicalSystems Simulation Group at National Instruments Corp.His research interests include estimation, navigation, au-tonomous vehicles, and intelligent transportation systems.

ABSTRACT

Reducing the vertical dilution of precision (VDOP) of aglobal navigation satellite system (GNSS) position solu-tion by exploiting terrestrial signals of opportunity (SOPs)is considered. A receiver is assumed to make pseudorangeobservations on multiple GNSS satellite vehicles (SVs) andmultiple terrestrial SOPs and to fuse these observationsthrough an estimator. This paper studies GNSS VDOPreduction by adding a varying number of cellular SOPs,which are inherently at low elevation angles. It is demon-strated numerically and experimentally that adding SOPobservables is more effective to reduce VDOP over addingGNSS SV observables.

I. INTRODUCTION

Global navigation satellite system (GNSS) position so-lutions suffer from a high vertical dilution of precision

(VDOP) due to lack of satellite vehicle (SV) angle diver-sity. Signals of opportunity (SOPs) have been recentlyconsidered to enable navigation whenever GNSS signalsbecome inaccessible or untrustworthy [1–3]. TerrestrialSOPs are abundant and are available at varying geometricconfigurations, making them an attractive supplement toGNSS for reducing VDOP.

Common metrics used to assess the quality of the spa-tial geometry of GNSS SVs are the parameters of the ge-ometric dilution of precision (GDOP); namely, horizontaldilution of precision (HDOP), time dilution of precision(TDOP), and VDOP [4]. Several methods have been in-vestigated for selecting the best GNSS SV configuration toimprove the navigation solution by minimizing the GDOP[5–7]. While the navigation solution is always improvedwhen additional observables from GNSS SVs are used, thesolution’s VDOP is generally of worse quality than theHDOP [8]. GPS augmentation with LocataLites, whichare terrestrial transmitters that transmit GPS-like signals,have been shown to reduce VDOP [9]. However, this re-quires installation of additional proprietary infrastructure.This paper studies VDOP reduction by exploiting terres-trial SOPs, particularly cellular code division multiple ac-cess (CDMA) signals, which have inherently low elevationangles and are free to use.

In GNSS-based navigation, the states of the SVs are read-ily available. For SOPs, however, even though the positionstates may be known a priori, the clock error states are dy-namic; hence, must be continuously estimated. The statesof SOPs can be made available through one or more re-ceivers in the navigating receivers vicinity [10, 11]. In thispaper, the estimates of such SOPs are exploited and theVDOP reduction is evaluated.

The remainder of this paper is organized as follows. Sec-tion II formulates the GNSS+SOP-based navigation so-lution and corresponding GDOP parameters. Section IIIdiscusses the relationship between VDOP and observationelevation angles. Section IV presents simulation results forusing a varying number of GNSS SVs and SOPs. Section Vpresents experimental results using cellular CDMA SOPs.Concluding remarks are given in Section VI.

II. PROBLEM FORMULATION

Consider an environment comprising a receiver, M GNSSSVs, and N terrestrial SOPs. Each SOP will be assumed

Copyright c© 2016 by J.J. Morales, J.J. Khalife, and Z.M.Kassas

Preprint of the 2016 ION ITM ConferenceMonterey, CA, January 25–28, 2016

to emanate from a spatially-stationary transmitter, andits state vector will consist of its position states rsop

n,[

xsopn, ysop

n, zsop

n

]Tand clock error states cxclk,sop

n,

c[δtsop

n, δtsop

n

], where c is the speed of light, δtsop

nis

the clock bias, and δtsopn

is the clock drift [12], wheren = 1, . . . , N .

The receiver draws pseudorange observations from theGNSS SVs, denoted {zsvm

}M

m=1, and from the SOPs, de-

noted{zsop

n

}N

n=1. These observations are fused through

an estimator whose role is to estimate the state vector ofthe receiver xr =

[rTr , cδtr

]T, where rr , [xr, yr, zr]

Tand

δtr are the position and clock bias of the receiver, respec-tively. The pseudorange observation made by the receiveron the mth GNSS SV, after compensating for ionosphericand tropospheric delays, is related to the receiver statesby

z′svm

= ‖rr − rsvm‖2 + c · [δtr − δtsvm

] + vsvm,

where, z′svm

, zsvm− δtiono − δttropo; rsvm

and δtsvmare

the position and clock bias states of the mth GNSS SV,respectively; δtiono and δttropo are the ionospheric and tro-pospheric delays, respectively; and vsvm

is the observationnoise, which is modeled as a zero-mean Gaussian randomvariable with variance σ2

svm. The pseudorange observation

made by the receiver on the nth SOP, after mild approxi-mations discussed in [12], is related to the receiver statesby

zsopn= ‖rr − rsop

n‖2 + c ·

[δtr − δtsop

n

]+ vsop

n,

where vsopnis the observation noise, which is modeled as a

zero-mean Gaussian random variable with variance σ2sop

n.

The measurement residual computed by the estimator hasa first-order approximation of its Taylor series expansionabout an estimate of the receiver’s state vector xr givenby

∆z = H∆xr + v,

where ∆z , z − z, i.e., the difference between the obser-

vation vector z ,[z′sv1

, . . . , z′svM, zsop

1, . . . , zsop

N

]Tand

its estimate z; ∆x , xr − xr, i.e., the difference be-tween the receivers’s state vector xr and its estimate xr;

v ,[vsv1

, . . . , vsvM, vsop

1, . . . , vsop

N

]T; and H is the Ja-

cobain matrix evaluated at the estimate xr. Without lossof generality, assume an East, North, UP (ENU) coordi-nate frame to be centered at xr. Then, the Jacobian inthis ENU frame can be expressed as

H =[HT

sv, HT

sop

]T,

where

Hsv=

c(elsv1

)s(azsv1) c(elsv1

)c(azsv1) s(elsv1

) 1...

......

...c(elsvM)s(azsvM) c(elsvM)c(azsvM) s(elsvM) 1

and

Hsop=

c(elsop

1)s(azsop

1) c(elsop

1)c(azsop

1) s(elsop

1) 1

......

......

c(elsopN)s(azsop

N) c(elsop

N)c(azsop

N) s(elsop

N) 1

,

where c(·) and s(·) are the cosine and sine functions,respectively, elsvm

and azsvmare the elevation and az-

imuth angles, respectively, of the mth GNSS SV, and elsopn

and azsopnare the elevation and azimuth angles, respec-

tively, of the nth terrestrial SOP as observed from the re-ceiver. To simplify the discussion, assume that the ob-servation noise is independent and identically distributed,i.e., cov(v) = σ2I, then, the weighted least-squares esti-mate xr and associated estimation error covariance Pxrxr

are given by

xr =(HTH

)−1

HTz, Pxrxr

= σ2

(HTH

)−1

.

The matrix G ,

(HTH

)−1

is completely determined by

the receiver-to-SV and receiver-to-SOP geometry. Hence,the quality of the estimate depends on this geometry andthe pseudorange observation noise variance. The diago-nal elements of G, denoted gii, are the parameters of thedilution of precision (DOP) factors:

GDOP ,

√tr[G]

HDOP ,√g11 + g22

VDOP ,√g33.

Therefore, the DOP values are directly related to the es-timation error covariance; hence, the more favorable thegeometry, the lower the DOP values [13]. If the observa-tion noise was not independent and identically distributed,the weighted DOP factors must be used [14].

The following sections illustrate the VDOP reduction byincorporating additional GNSS SV observations versus ad-ditional SOP observations.

III. VDOP REDUCTION VIA SOPs

With the exception of GNSS receivers mounted on high-flying and space vehicles, all GNSS SVs are typically abovethe receiver [13], i.e., the elevation angles in Hsv are the-oretically limited between 0◦ ≤ elsvm ≤ 90◦. GNSS re-ceivers typically restrict the lowest elevation angle to someelevation mask, elsv,min, so to ignore GNSS SV signals thatare heavily degraded due to the ionosphere, the tropo-sphere, and multipath. As a consequence, GNSS SV ob-servables lack elevation angle diversity and the VDOP of aGNSS-based navigation solution is degraded. For groundvehicles, elsv,min is typically between 10◦ and 20◦. Theseelevation angle masks also apply to low flying aircrafts,

2

such as small unmanned aircraft systems (UASs), whoseflight altitudes are limited to 500ft (approximately 152m)by the Federal Aviation Administration (FAA) [15].

In GNSS + SOP-based navigation, the elevation anglespan may effectively double, specifically −90◦ ≤ elsop

n≤

90◦. For ground vehicles, useful observations can bemade on terrestrial SOPs that reside at elevation anglesof elsop

n= 0◦. For aerial vehicles, terrestrial SOPs can

reside at elevation angle as low as elsopn= −90◦, e.g., if

the vehicle is flying directly above the SOP transmitter.

To illustrate the VDOP reduction by incorporating addi-tional GNSS SV observations versus additional SOP obser-vations, an additional observation at elnew is introduced,and the resulting VDOP(elnew) is evaluated. To this end,M SV azimuth and elevation angles were computed usingGPS ephemeris files accessed from the Yucaipa, Californiastation via Garner GPS Archive [16], which are tabulatedin Table I. For each set of GPS SVs, the azimuth an-gle of an additional observation was chosen according toAnew ∼ U(0◦, 359◦). The corresponding VDOP for intro-ducing an additional measurement at a sweeping elevationangle −90◦ ≤ elnew ≤ 90◦ are plotted in Fig. 1 (a)-(d) forM = 4, . . . , 7, respectively.

The following can be concluded from these plots. First,while the VDOP is always improved by introducing an ad-ditional measurement, the improvement of adding an SOPmeasurement is much more significant than adding an ad-ditional GPS SV measurement. Second, for elevation an-gles inherent only to terrestrial SOPs, i.e., −90◦ ≤ elsop

n≤

0◦, the VDOP is monotonically decreasing for decreasingelevation angles.

TABLE I

SV AZIMUTH AND ELEVATION ANGLES (DEGREES)

M = 4 M = 5 M = 6 M = 7

(m) azsvm elsvm azsvm elsvm azsvm elsvm azsvm elsvm

1 185 79 189 66 46 40 61 21

2 52 60 73 69 101 58 57 49

3 326 52 320 41 173 59 174 30

4 242 47 56 27 185 38 179 66

5 - - 261 51 278 67 269 31

6 - - - - 314 41 218 56

7 - - - - - - 339 62

IV. SIMULATION RESULTS

This section presents simulation results demonstrating thepotential of exploiting cellular CDMA SOPs for VDOPreduction. To compare the VDOP of a GNSS only nav-igation solution with a GNSS + SOP navigation solu-

VDOP(el new

)(c)

(d)

elnew [degrees]

elnew [degrees]

elnew [degrees]

VDOP(el new

)VDOP(el new

)

elnew [degrees]

VDOP(el new

)

(b)

(a)

Fig. 1. A receiver has access to M GPS SVs from Table I.Plots (a)-(d) show the VDOP for each GPS SV configuration be-fore adding an additional measurement (red dotted line) and theresulting VDOP(elnew) for adding an additional measurement (bluecurve) at an elevation angle −90 ≤ elnew ≤ 90 for M = 4, . . . , 7,respectively.

tion, a receiver position expressed in an Earth-Centered-Earth-Fixed (ECEF) coordinate frame was set to rr ≡

(106) · [−2.431171,−4.696750, 3.553778]T. The elevationand azimuth angles of the GPS SV constellation above thereceiver over a twenty-four hour-period was computed us-ing GPS SV ephemeris files from the Garner GPS Archive.

3

The elevation mask was set to elsv,min ≡ 20◦. The az-imuth and elevation angles of three SOPs, which werecalculated from surveyed terrestrial cellular CDMA towerpositions in the receivers vicinity, were set to azsop ≡

[42.4◦, 113.4◦, 230.3◦]T and elsop ≡ [3.53◦, 1.98◦, 0.95◦]T.The resulting VDOP, HDOP, GDOP, and associated num-ber of available GPS SVs for a twenty-four hour periodstarting from midnight, September 1st, 2015, are plotted inFig. 5. These results were consistent for different receiverlocations and corresponding GPS SV configurations.

The following can be concluded from these plots. First, theresulting VDOP using GPS + N SOPs, for N ≥ 1, is al-ways less than the resulting VDOP using GPS alone. Sec-ond, using GPS + N SOPs, forN ≥ 1 prevents large spikesin VDOP when the number of GPS SVs drops. Third, us-ing GPS + N SOPs, for N ≥ 1 also reduces both HDOPand GDOP.

V. EXPERIMENTAL RESULTS

A field experiment was conducted using software definedreceivers (SDRs) to demonstrate the reduction of VDOPobtained from including SOP pseudoranges alongside GPSpseudoranges for estimating the states of a receiver. Tothis end, two antennas were mounted on a vehicle to ac-quire and track: (i) multiple GPS signals and (ii) three cel-lular base transceiver stations (BTSs) whose signals weremodulated through CDMA. The GPS and cellular signalswere simultaneously downmixed and synchronously sam-pled via two National Instruments R© universal softwareradio peripherals (USRPs). These front-ends fed theirdata to a Generalized Radionavigation Interfusion Device(GRID) software receiver [17], which produced pseudor-ange observables from five GPS L1 C/A signals in view,and the three cellular BTSs. Fig. 2 depicts the experi-mental hardware setup.

The pseudoranges were drawn from a receiver located atrr = (106) · [−2.430701,−4.697498, 3.553099]T, expressedin an ECEF frame, which was surveyed using a Trimble5700 carrier-phase differential GPS receiver. The corre-sponding SOP state estimates {xsop

n}Nn=1, were collabora-

tively estimated by receivers in the navigating receiver’svicinity. The pseudoranges and SOP estimates were fedto a least-squares estimator, producing xr and associatedPxrxr

, from which the VDOP, HDOP, and GDOP werecalculated and tabulated in Table II for M GPS SVs andN cellular CDMA SOPs. A sky plot of the GPS SVs usedis shown in Fig. 4. The tower locations, receiver location,and a comparison of the resulting 95th-percentile estima-tion uncertainty ellipsoids of xr for {M,N} = {5, 0} and{5, 3} are illustrated in Fig. 3. The corresponding verticalerror was 1.82m and 0.65m respectively. Hence, addingthree SOPs to the navigation solution that used five GPSSVs reduced the vertical error by 64.5%. Although thisis a significant improvement over using GPS observables

alone, improvements for aerial vehicles are expected to beeven more significant, since they can exploit a full span ofobservable elevation angles.

NI USRPs

Storage

GPS and cellular anetnnasSoftware-defined radios (SDRs)

GRID SDR

MATLAB-Based

Filter

Fig. 2. Experiment hardware setup

Five SVs Five SVs + three SOPs

Tower locations Receiver location

Fig. 3. Top: Cellular CDMA SOP tower locations and receiver lo-cation. Bottom: uncertainty ellipsoid (yellow) of navigation solutionfrom using pseudoranges from five GPS SVs and uncertainty ellip-soid (blue) of navigation solution from using pseudoranges from fiveGPS SVs and three cellular CDMA SOPs.

0o

90o270

o

180o

27

22

14

21

0o

90o270

o

180o

27

22

14

21

18

Fig. 4. Left: Sky plot of GPS SVs: 14, 21, 22, and 27 used for thefour SV scenarios. Right: Sky plot of GPS SVs: 14, 18, 21, 22, and27 used for the five SV scenarios. The elevation mask, elsv,min, wasset to 20◦ (dashed red circle).

4

Time [Hours]

HDOP

GDOP

Number

ofSatellites

(a)

(d)

(c)

VDOP

(b)

GPS Only GPS+1 SOPs GPS+2 SOPs GPS+3 SOPs

Fig. 5. Fig. (a) represents the number of SVs with an elevation angle > 20◦ as a function of time. Fig. (b)–(d) correspond to the resultingVDOP, HDOP, and GDOP, respectively, of the navigation solution using GPS only, GPS + 1 SOP, GPS + 2 SOPs, and GPS + 3 SOPs.

TABLE II

DOP values for M Svs + N SOPs

(M) SVs, (N) SOPs: {M,N} {4, 0} {4, 1} {4, 2} {4, 3} {5, 0} {5, 1} {5, 2} {5, 3}

VDOP 3.773 1.561 1.261 1.080 3.330 1.495 1.241 1.013

HDOP 2.246 1.823 1.120 1.073 1.702 1.381 1.135 1.007

GDOP 5.393 2.696 1.933 1.654 4.565 2.294 1.880 1.566

5

VI. CONCLUSIONS

This paper studied the reduction of VDOP of a GNSS-based navigation solution by exploiting terrestrial SOPs.It was demonstrated that the VDOP of GNSS solution canbe reduced by exploiting the inherently small elevation an-gles of terrestrial SOPs. Experimental results using groundvehicles equipped with SDRs demonstrated VDOP reduc-tion of a GNSS navigation solution by exploiting a vary-ing number of cellular CDMA SOPs. Incorporating ter-restrial SOP observables alongside GNSS SV observablesfor VDOP reduction is particularly attractive for aerialsystems, since a full span of observable elevation anglesbecome available.

References

[1] J. Raquet and R. Martin, “Non-GNSS radio frequency nav-igation,” in Proceedings of IEEE International Conference onAcoustics, Speech and Signal Processing, March 2008, pp. 5308–5311.

[2] K. Pesyna, Z. Kassas, J. Bhatti, and T. Humphreys, “Tightly-coupled opportunistic navigation for deep urban and indoor po-sitioning,” in Proceedings of ION GNSS Conference, September2011, pp. 3605–3617.

[3] Z. Kassas, “Collaborative opportunistic navigation,” IEEEAerospace and Electronic Systems Magazine, vol. 28, no. 6, pp.38–41, 2013.

[4] P. Massat and K. Rudnick, “Geometric formulas for dilution ofprecision calculations,” NAVIGATION, Journal of the Instituteof Navigation, vol. 37, no. 4, pp. 379–391, 1990.

[5] N. Levanon, “Lowest GDOP in 2-D scenarios,” IEE ProceedingsRadar, Sonar and Navigation, vol. 147, no. 3, pp. 149–155, 2000.

[6] I. Sharp, K. Yu, and Y. Guo, “GDOP analysis for positioningsystem design,” IEEE Transactions on Vehicular Technology,vol. 58, no. 7, pp. 3371–3382, 2009.

[7] N. Blanco-Delgado and F. Nunes, “Satellite selection method formulti-constellation GNSS using convex geometry,” IEEE Trans-actions on Vehicular Technology, vol. 59, no. 9, pp. 4289–4297,November 2010.

[8] P. Misra and P. Enge, Global Positioning System: Signals, Mea-surements, and Performance, 2nd ed. Ganga-Jamuna Press,2010.

[9] M. K. J. Barnes, C. Rizos and A. Pahwa, “A soultion to toughGNSS land applications using terrestrial-based transciers (Lo-cataLites),” in Proceedings of ION GNSS Conference, Septem-ber 2006, pp. 1487–1493.

[10] Z. Kassas, V. Ghadiok, and T. Humphreys, “Adaptive estima-tion of signals of opportunity,” in Proceedings of ION GNSSConference, September 2014, pp. 1679–1689.

[11] J. Morales and Z. Kassas, “Optimal receiver placement for col-laborative mapping of signals of opportunity,” in Proceedings ofION GNSS Conference, September 2015, pp. 2362–2368.

[12] Z. Kassas and T. Humphreys, “Observability analysis of col-laborative opportunistic navigation with pseudorange measure-ments,” IEEE Transactions on Intelligent Transportation Sys-tems, vol. 15, no. 1, pp. 260–273, February 2014.

[13] J. Spilker, Jr., Global Positioning System: Theory and Appli-cations. Washington, D.C.: American Institute of Aeronauticsand Astronautics, 1996, ch. 5: Satellite Constellation and Geo-metric Dilution of Precision, pp. 177–208.

[14] D. H. Won, J. Ahn, S. Lee, J. Lee, S. Sung, H. Park, J. Park,and Y. J. Lee, “Weighted DOP with consideration on elevation-dependent range errors of GNSS satellites,” IEEE Transactionson Instrumentation and Measurement, vol. 61, no. 12, pp. 3241–3250, December 2012.

[15] Federal Communications Commission, “Overviewof small UAS notice of proposed rulemaking,”https://www.faa.gov/uas/nprm/, February 2015, accessedNovember 25, 2015.

[16] University of California, San Diego, “Garner GPS archive,”http://garner.ucsd.edu/, accessed November 23, 2015.

[17] T. Humphreys, J. Bhatti, T. Pany, B. Ledvina, andB. O’Hanlon, “Exploiting multicore technology in software-defined GNSS receivers,” in Proceedings of ION GNSS Con-ference, September 2009, pp. 326–338.

6