Dilution of Precision

centered on the transmitters (see Figure 1).There is some uncertainty, however, in thereceivers measurements, and so the locationof the range circles will be inexact and resultin an error in the computed position. Thiserror depends on the geometry relating thereceiver and the transmitters.

In Figure 1a, the transmitters are far apart,giving a relatively small region in which thereceiver must lie with some degree of cer-tainty. Transmitter 1 lies in a directionorthogonal to that of transmitter 2, so thereceivers X and Y coordinates are deter-

52 GPS WORLD May 1999 www.gpsworld.com

How accurate is GPS? This is a question thatalmost every newcomer to GPS asks. And theanswer? It depends. It depends on whether weare talking about standalone (single receiver)or differential positioning, single- or dual-fre-quency receivers, real-time or postprocessedoperation, and so on. Even if we confine our-selves to the Standard Positioning Service(SPS), the official, standalone service theUnited States government provides to all usersworldwide, the answer is still it depends.

The specified SPS accuracy is given interms of minimum performance levels; thatis, accuracy will be no worse than a certainlevel for a certain percentage of time. For anypoint on the globe, the horizontal accuracy isequal to or better than 100 meters based onthe twice-distanceroot-mean-square errormeasure. This means that over a 24-hourperiod, the horizontal coordinates of a posi-tion determined by GPS will be within 100meters of the true position about 95 percentof the time. The corresponding specifiedaccuracy for heights is 156 meters and 340nanoseconds for time transfer.

These predicted accuracies are predicatedon a 24-satellite constellation (additionalsatellites are a bonus), a 5-degree satellite ele-vation mask angle with no obstructions, andat least four satellites in view with a positiondilution of precision (PDOP) of six or lower.So, even the basic SPS accuracy is qualified.This means that depending on where we areand the time of day, actual SPS accuracy willvary. In urban canyons, we may in fact noteven have four satellites in view, and if we do,the PDOP may be greater than six.

The variability of actual SPS accuracyfrom place to place and time to time is domi-nated by the effects of dilution of precision, ageometric factor that, when multiplied bymeasurement and other input errors, gives theerror in position, some component of posi-tion, or time. Before we examine how obser-vation geometry affects GPS, lets look at asimple, non-GPS example.

GEOMETRY: A SIMPLE EXAMPLEImagine a radio positioning system in whicha receiver measures the ranges to two terres-trial transmitters to determine its horizontalcoordinates. The receiver lies at the intersec-tion of the circular lines of position that are

Dilution of Precision Richard B. Langley

University of New Brunswick

Dilution of precision, or DOP: weve all seen the term, and most of us know that smaller DOP values are better than larger ones.Many of us also know thatDOP comes in variousflavors, including geometrical(GDOP), positional (PDOP),horizontal (HDOP), vertical(VDOP), and time (TDOP).But just what are theseDOPs? In this monthscolumn, we examine GPSdilution of precision and how it affects the accuracywith which our receivers can determine position and time.

Innovation is a regularcolumn featuring discussionsabout recent advances in GPStechnology and its applicationsas well as the fundamentals of GPS positioning. The column is coordinated byRichard Langley of the Department of Geodesy andGeomatics Engineering at theUniversity of New Brunswick,who appreciates receiving your comments as well as topic suggestions for futurecolumns. To contact him, see the Columnists section on page 4 of this issue.

I N N O V A T I O N

Figure 1. In any ranging system,receivertransmitter geometry influ-ences position precision. In this figure,the uncertainty in the receivers positionis indicated by the patterned areas. In(a), the position uncertainty is small (lowdilution of precision). In (b), transmitter 2is moved closer to transmitter 1, and,although the measurement uncertaintyis the same, the position uncertainty isconsiderably larger (high dilution ofprecision).

yy

yy

yy

y

Transmitter 1

Transmitter 2

Transmitter 1

Transmitter 2

X

X

Y

(a)

(b)

Y


[5]in which C

D x is the covariance matrix of theparameter estimates.

Equation 5 represents a fundamental rela-tionship widely used in science and engineer-ing not only for actual measurement analysisbut also for experiment and system designstudies. It allows a scientist or engineer toexamine the effect a particular design or measurement capability will have on speci-fied parameters without actually making anymeasurements.

In GPS-related studies, for example, wemight use the equation to answer a variety ofquestions: What is the behavior of the esti-mated parameter covariance matrix as a func-tion of particular satellite configurations?How do various model errors propagate intothe receiver coordinates as a function of satel-lite configurations? What is the tolerancevalue that a particular model error should notexceed to achieve a desired positioning accu-racy? Such questions are not limited to theanalysis of pseudoranges; they can also beasked about more precise carrier-phase mea-surements and differenced observables.

In Equation 5, if we assume that the measurement and model errors are the samefor all observations with a particular stand-ard deviation (s ) and that they are uncorre-lated, then C

D PC is Is2 (in which I is the iden-

tity matrix). The expression for thecovariance of D x thus simplifies to

[6](A more accurate error analysis can be car-ried out using slightly more realistic non-homogeneous variances and nonzerocorrelations, but for pedagogical as well asplanning and assessment purposes, the equal-variance, zero-correlation assumption is usu-ally more than adequate.)

Because the least-squares estimates of theparameter offsets are simply added to the lin-earization point values a linear operation the parameter estimates and the correc-tions have the same covariance. The diagonalelements of C

D x are the estimated receiver-coordinate and clock-offset variances, andthe off-diagonal elements (the covariances)indicate the degree to which these estimatesare correlated.

UERE. As we mentioned, s represents thestandard deviation of the pseudorange mea-surement error plus the residual model error,which weve assumed to be equal for allsimultaneous observations. If we furtherassume that the measurement error and the

Cx = ATA1

2 = D2

Cx = ATWA1

ATW CPc ATWA

1ATW

T

= ATCPc1 A

1

availability[SA]). There are n such equationsthat a receiver must solve using the n-simul-taneous measurements.

The parameter r is a nonlinear function ofthe receiver and satellite coordinates. Todetermine the receiver coordinates, we canlinearize the pseudorange equations usingsome initial estimates or guesses for thereceivers position (the linearization point).We can then determine corrections to theseinitial estimates to obtain the receiversactual coordinates and clock offset. Groupingour equations together and representing themin matrix form, our model is now

[3]in which D Pc is the n-length vector of differ-ences between the corrected pseudorangemeasurements and modeled pseudorangevalues based on the linearization point coordinates; D x designates the four-elementvector of unknowns the receiver positionand clock offset (in distance units) fromthe linearization point; A is the n 2 4 matrix of the partial derivatives of thepseudoranges with respect to the unknowns;and ec is the n-length vector of measurementand other errors. The first three columns ofthe A matrix are simply the components ofthe unit vectors pointing from the lineariza-tion point to the satellites; the fourth columnis all ones.

The receiver (or postprocessing software)solves the matrix equation using leastsquares. (The receiver might use a Kalmanfilter, which is a more general form of con-ventional least squares.) Equation 4 gives thissolution:

[4]A weight matrix (W) characterizes the dif-

ferences in the errors of the simultaneousmeasurements as well as any correlations thatmay exist among them. This weight matrix is also equal to s 02C D PC

-1, in which C

D PC is the covariance matrix of the pseudorangeerrors and s 02 is a scale factor known as the apriori variance of unit weight. In general, thesolution of a nonlinear problem must be iter-ated to obtain the result. However, if the lin-earization point is sufficiently close to thetrue solution, then only one iteration isrequired.

The Covariance Matrix. So how accurate are thereceivers coordinates and clock offset fromsuch a solution? What we are actually askingis how do the pseudorange measurement andmodel errors affect the estimated parametersobtained from Equation 4? This is given bythe law of propagation of error also knownas the covariance law:

x = ATWA1

ATWPc

Pc = A x+ ec

mined with equal precision. In panel (b), thetransmitters are closer together resulting in aconsiderably larger uncertainty region, withthe confidence in the Y coordinate beingsmaller than the X coordinate. We say thatthe precision in case (b) is diluted in compar-ison to that of (a).

Although fictitious, this simple example isnot too far removed from the case of Loran-Cradionavigation (although in Loran-C,because we typically measure range differ-ences, the lines of position are usually hyper-bolas, not circles). In fact, the concept ofdilution of precision originated with Loran-Cusers.

With this simple analogy under our collec-tive belt, we can now examine the effect ofgeometry on GPS accuracy. First, though,lets quickly review the basics of GPS posi-tioning using pseudoranges.

PSEUDORANGE MEASUREMENTSA GPS receiver computes its three-dimen-sional coordinates and its clock offset fromfour or more simultaneous pseudorange mea-surements. These are measurements of thebiased range (hence the term pseudorange)between the receivers antenna and the anten-nas of each of the satellites being tracked.This is derived by cross-correlating thepseudorandom noise code received from asatellite with a replica generated in thereceiver. The accuracy of the measuredpseudoranges and the fidelity of the modelused to process those measurements deter-mine, in part, the overall accuracy of thereceiver-derived coordinates.

The basic pseudorange model is given by

[1]in which P denotes the pseudorange mea-surement; r is the geometric range betweenthe receivers antenna at signal receptiontime and the satellites antenna at signaltransmission time; dT and dt representreceiver and satellite clock offsets from GPSTime, respectively; dion and dtrop are theionospheric and tropospheric propagationdelays; e accounts for measurement noise aswell as unmodeled effects such as multipath;and c stands for the vacuum speed of light.

Assuming the receiver accounts for thesatellite clock offset (using the navigationmessage) and atmospheric delays (from mod-els programmed into its firmware), we cansimplify the pseudorange model as follows:

[2]In this equation, ec represents the original

measurement noise plus model errors andany unmodeled effects (such as selective

Pc = + c dT + ec

P = + c(dT dt) + dion + dtrop + e

I N N O V A T I O N


model error components are all independent,then we can simply root-sum-square theseerrors to obtain a value for s . When we com-bine receiver noise, satellite clock andephemeris error, atmospheric error, multi-path, and SA all expressed in units of dis-tance we obtain a quantity known as thetotal user equivalent range error (UERE),which we can use for s .

For SPS, the total UERE is typically in theneighborhood of 25 meters. When SA isturned off, total UERE could be less than 5meters, with the actual value dominated byionospheric and multipath effects. Dual-fre-quency Precise Positioning Service users,with the capability to remove almost all ofthe ionospheric delay from the pseudorangeobservations, can experience even smallerUEREs. Future users of the proposed newcivilian GPS signals will likewise be able tocompensate for ionospheric effects andachieve superior UEREs.

THE DOPSWith a value for s , we can compute the com-ponents of C

D x using Equation 6. We thencan get a measure of the overall quality of theleast-squares solution by taking the squareroot of the sum of the parameter estimatevariances:

[7]in which s E2, s N2, and s U2 are the variances ofthe east, north, and up components of thereceiver position estimate, and s T2 is the vari-ance of the receiver clock offset estimate. Ifthe solution algorithm is parameterized interms of geocentric Cartesian coordinates, itis a straightforward procedure to transformthe solution covariance matrix to the localcoordinate frame. This estimate of solutionaccuracy the square root of the trace of thesolution covariance matrix is equal to thepseudorange measurement and modelingerror standard deviation (s ) multiplied by ascaling factor equal to the square root of thetrace of matrix D. The elements of matrix Dare a function of the receiversatellite geom-etry only. And because the scaling factor istypically greater than one, it amplifies thepseudorange error, or dilutes the precision, ofthe position determination. This scaling fac-tor is therefore usually called the geometricdilution of precision (GDOP).

Rather than examining the quality of theoverall solution, we may prefer to look atspecific components such as the three-dimen-sional receiver position coordinates, the hori-zontal coordinates, the vertical coordinate, or

G = E2 + N

2 + U2 + T

2

= D11 + D22+ D33+ D44

the zenith and three satellites are below theearths horizon at an elevation angle of19.47 degrees and equally spaced inazimuth: GDOP works out to be 1.581. Ofcourse, a GPS receiver on or near the earthssurface cannot see the three below-horizonsatellites, so in this case, the lowest possibleGDOP (1.732) is obtained with one satelliteat the zenith and three satellites equally-spaced on the horizon. Some early GPSreceivers could only track a maximum offour satellites simultaneously. Such receiversmade use of a satellite selection algorithm tochoose the best four satellites of those visible the four that would produce the lowestDOP values.

HDOP versus VDOP. Generally, the more satel-lites used in the solution, the smaller the DOPvalues and hence the smaller the solutionerror. Figure 3 shows the DOP values com-puted for the current GPS constellationviewed from Fredericton, New Brunswick,Canada, with an elevation mask angle of 15degrees. HDOP values are typically betweenone and two. VDOP values are larger thanthe HDOP values indicating that verticalposition errors are larger than horizontalerrors. We suffer this effect because all of thesatellites from which we obtain signals areabove the receiver. The horizontal coordi-nates do not suffer a similar fate as we usu-ally receive signals from all sides.

the clock offset. To do this, we simply take orcombine the appropriate C

D x variances:

[8]For each of these error measures, we can

determine the corresponding position, hori-zontal, vertical, and time DOPs:

[9]

Note that PDOP2 = HDOP2 + VDOP2, andGDOP2 = PDOP2 + TDOP2. These relation-ships may be useful for interrelating the vari-ous DOP values. Because the various DOPsare functions only of receiver and satellitecoordinates, they may be predicted ahead oftime for any given set of satellites in viewfrom a specified location using a satellitealmanac.

If the tips of the receiversatellite unitvectors lie in a plane, the DOP factors areinfinitely large. In fact, no position solutionis possible with this receiversatellite geom-etry, as the matrix ATA (see Equation 6) issingular: The solution cannot distinguishbetween an error in the receiver clock and anerror in the position of the receiver. DOP val-ues are smaller and hence solution errors aresmaller when the satellites used in computingthe solution are spread out in the sky.

We can most easily visualize the depen-dence of solution error on receiversatellitegeometry if we assume a receiver is observ-ing only four satellites. This scenario has nomeasurement redundancy and makes possi-ble a direct solution of the linearized obser-vation equations (as long as A is notsingular). However, the covariance of thesolution, again assuming equal uncorrelatederrors, has the same form as that of the least-squares solution given in Equation 6.

The Tetrahedron. The tips of the fourreceiversatellite unit vectors form a tetra-hedron (see Figure 2). The volume of thisgeometrical figure is related to the DOP values. The larger the tetrahedrons volume,the smaller the DOPs. The largest possibletetrahedron is one for which one satellite is at

TDOP = T = D44

VDOP= U = D33

HDOP= E2 + N

2

= D11+ D22

PDOP= E2 + N

2 + U2

= D11+ D22+ D33

T = T2

U = U2

H = E2 + N

2

P = E2 + N

2 + U2

I N N O V A T I O N

Figure 2. If only four GPS satellites are observed, the tips of the receiver-satellite unit vectors form a tetrahedroncircumscribed by a unit sphere. Twofaces of the tetrahedron formed byone satellite at the zenith and three at a10-degree elevation angle, equallyspaced in azimuth are shaded in thisfigure. The tetrahedrons volume ishighly correlated with GDOP. Maximiz-ing the volume tends to minimize GDOP.


available and the HDOP value drops to aboutone, with a correspondingly small GDOP(see Figure 5). In fact, the HDOP value staysclose to one for the whole day except for shortperiods when it grows to about 1.5 or so.

Latitude. The disparity between HDOP andVDOP values is larger for higher (north orsouth) latitudes because there are fewer satel-lites high in the sky. This limitation comesfrom the fact that the inclination of the GPSsatellite orbits is about 55 degrees, whichmeans that you can never have a satellitedirectly overhead at a latitude north of 55degrees north (or south of 55 degrees south).At the poles, the highest elevation angle pos-sible is about 45 degrees.

If we use an elevation mask angle of 15degrees and track only four satellites thefour that produce the lowest DOP values we find that while HDOP values are alwaysbetween one and two, VDOP values arealmost always above three and sometimes aslarge as seven. This isnt too surprisingbecause we are only using satellites in an ele-vation-angle band of about 30 degrees aroundthe sky. How bad is a VDOP of seven? If theUERE is 25 meters, the root-mean-squarevertical error would be about 175 meters, andat the 95 percent uncertainty level, this errorwould increase to 350 meters. Dropping theelevation mask angle to 5 degrees improvesthe VDOP values to between two and threewith an occasional excursion to four.

More Satellites. High DOP values can some-times occur even for all-in-view receiversoperating at midlatitudes. In some environ-ments, such as heavily forested areas orurban canyons, a GPS receivers antennamay not have a clear view of the whole skybecause of obstructions. If it can only receiveGPS signals from a small region of the sky,the DOPs will be large, and position accuracywill suffer. Being able to track more satellites

If the Earth weretransparent to radiowaves, we would beable to determine vertical coordinateswith about the sameaccuracy as horizon-tal coordinates. Morerealistically, we canalso get improvedvertical coordinates if we have an accu-rate receiver clock or one whose offsetfrom GPS Time canbe accurately deter-mined so that thereceiver only needs to estimate its posi-tion. For example,lets assume we are

observing one satellite at the zenith and threeequally spaced in azimuth at a 15-degree ele-vation angle. If we estimate the receiverclock offset along with the receiver coordi-nates, HDOP is 1.195, and VDOP is 1.558.If, however, we assume the clock error to bezero, and we only estimate the receiver coor-dinates, then HDOP is still 1.195, but VDOPis 0.913 actually better than HDOP.

Getting back to Figure 3, we notice a largespike in the HDOP values (and hence in thePDOP and GDOP values as well) just before0000 hours. Whats going on here? At thistime, the number of visible satellites abovethe mask angle has dropped to five. Whilethis is not particularly unusual, the arrange-ment of these five satellites in the sky is (seeFigure 4). The satellite positions, projectedonto the users horizon plane, are almost col-inear, which makes the elements of matrix Dlarge. If we accept a lower elevation maskangle of 5 degrees, several more satellites are

Figure 3. DOP values at Fredericton, New Brunswick, Canada, computed from broadcast satellite ephemerides using a 15-degree elevation mask angle are sufficiently small except just before 0000 hours when HDOP, PDOP, and GDOP increase to more than 12 (#SV represents the numberof visible satellites [space vehicles]).

12 18 0 6 120

5

10

15#S

V

12 18 0 6 120

5

10

15

GDO

P

12 18 0 6 120

5

10

15

PDO

P

12 18 0 6 120

5

10

15

HD

OP

12 18 0 6 120

5

10

15

VDO

P

12 18 0 6 120

5

10

15

TDO

P

Hours (Coordinated Universal Time)

Figure 4. The spike in the DOP valuesin Figure 3 is caused by the almostperfect alignment of the five satellitesabove the elevation mask angle.

E

N

0

30

60

Figure 5. Lowering the elevation mask angle to 5 degreesremoves the spikes just before 0000 hours in the DOPs ofFigure 3. Now, the GDOP has an average value of about twoand never exceeds 3.5 (#SV represents the number of visiblesatellites [space vehicles]).

12 18 0 6 120

5

10

15

#SV

12 18 0 6 12012345

GDO

P

Hours (Coordinated Universal Time)

I N N O V A T I O N

www.gpsworld.com GPS WORLD May 1999 59

can help in such situations, and a combinedGPS/GLONASS receiver may provideacceptable accuracies. New receiver technol-ogy permitting use of weaker GPS signals,even those present inside buildings, will alsobe beneficial.

CONCLUSIONIn this brief article we have introduced theconcept of dilution of precision and exam-ined the important role that receiversatellite

geometry plays in determining GPS positionaccuracy. While this geometry will always beof some concern in GPS positioning, the newsatellite signals, improvements in receiverdesign, and use of additional signals fromGLONASS or the proposed EuropeanGalileo constellations of satellites will helpto minimize its impact. And in the not toodistant future, we might expect real-time,

FURTHER READINGFor a more in-depth analysis of GPS dilutionof precision (DOP), see

n Satellite Constellation and GeometricDilution of Precision, by J.J. Spilker Jr. andGPS Error Analysis, by B.W. Parkinson, inGlobal Positioning System: Theory andApplications, Vol. 1, edited by B.W. Parkin-son and J.J. Spilker Jr., Progress in Astro-nautics and Aeronautics, Vol. 163. AmericanInstitute of Aeronautics and Astronautics,Washington, D.C., 1996, pp.177208 and469483.

n Performance of Standalone GPS, byJ.L. Leva, M.U. de Haag, and K. Van Dyke,in Understanding GPS: Principles andApplications, edited by E.D. Kaplan, ArtechHouse Publishers, Norwood, Massachu-setts, 1996, pp. 237320.

For a discussion about how DOP andvarious error sources affect positioningaccuracy, see

n GPS Performance in Navigation, by P.Misra, B.P. Burke, and M.M. Pratt, Proceed-ings of the IEEE (Special Issue on GPS), Vol.187, No. 1, January 1999, pp. 6585.

Several Internet sites provide onlinecomputation of DOPs for a specifiedlocation and time interval, including the U.S.Naval Air Warfare Center Weapons Divisionat China Lake, California:

n .For a Java-based DOP demonstration, seen .For a detailed examination of the role

geometry plays in high-precision relativeGPS positioning, see

n Impact of GPS Satellite Sky Distribu-tion, by R. Santerre, in ManuscriptaGeodaetica, Vol. 16, 1991, pp. 2853.

For a discussion of GDOP in terrestrialhyperbolic ranging systems, see

n Geometric Dilution of Precision, byE.R. Swanson, in Navigation: Journal of TheInstitution of Navigation, Vol. 25, No. 4,197879, pp. 425 429.

Circle 38

Odetics Telecom1/2 Page IslandAd Goes Here

Keyline does not printpage 59

I N N O V A T I O N

standalone GPS position accuracies even inurban areas of a few meters or perhaps better.Stay tuned. n

ACKNOWLEDGMENTThanks to University of New BrunswicksPaul Collins for generating the DOP plots forFigures 3 and 5.

Title: CORRECTIONTest: The time scales of Figs. 3 and 5 in the original article were incorrect. They should run from 1200 to 1200 UTC and have been corrected in this version. R.B.L.blank:

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Dilution of Precision