HDOP AND GPS HORIZONTAL POSITION ERRORS

 

转自   http://users.erols.com/dlwilson/gpshdop.htm

 

Thehorizontal dilution of precision (HDOP) allows one to more precisely estimatethe accuracy of GPS horizontal (latitude/longitude) position fixes by adjustingthe error estimates according to the geometry of the satellites used.  Theoretically, given the HDOP, one canobtain error estimates that are good for all fixes with that HDOP, rather thanthe more general error estimates for all position fixes (regardless of HDOP).  In probability terminology, HDOP is an additionalvariable that allows one to replace the overall accuracy estimates withconditional accuracy ones for the given HDOP value.  As an analogy, consider the probability of getting a"2" when rolling a fair die. The probability of getting a "2" is 1/6.  But if you already know "the number isless than 4" then the (conditional) probability of getting a "2"is 1/3.  Knowing HDOP is somewhatsimilar to knowing "the number is less than 4" in the analogy.

 

Thenotation "RMS_Error(HDOP)" is used here to indicate the RMS error ofall fixes with a given HDOP value; for example, RMS_Error(HDOP = 1.2) wouldindicate the RMS error of all fixes with HDOP = 1.2.  The value of RMS_Error(HDOP)increases as HDOP increases, as higher values of HDOP indicate a satellitegeometry that will tend to give less accurate fixes.

 

Whena set of position measurements is analyzed, just as the RMS error is used torepresent the error of the set of measurements, the RMS of the HDOP, denotedhere as RMS_HDOP can be used to represent the HDOP of the set.  The RMS of the HDOP is defined in the usualmanner:

 

 

Ascan any RMS or "quadratic mean", RMS_HDOP can instead be found fromthe mean and standard deviation:

 

 

Belowis plotted HDOP versus RMS_Error(HDOP) for a 20-day session using a Garmin12XL.  Actually, because of the need forsufficient sample sizes, the data is binned according to HDOP with bins ofwidth 0.2, and then using the data in each bin, the RMS of the HDOP was plottedagainst the RMS error.  These measureddata points are indicated in red in the following plot:

 

 

In theory, if satellite geometry were the only componentof the horizontal error of position, the RMS error would be directlyproportional to HDOP; thus the points in the plot would lie on a straight line:

 

 

 

Thesolid green line indicates the prediction by this linear model if one uses thesometimes quoted RMS_Error(HDOP=1) = 4.0 meters.   Linear regression actually gives RMS_Error(HDOP = 1) = 3.71meters, or 3.98 meters if the point for HDOP > 2  is excluded.  The difference between this and 4.0 metersis marginal when the scatter of the points is considered.

 

Thebroken blue curve indicates a curve-fit that was obtained from weighted (byfrequency of occurrence) non-linear least-squares regression:

 

 

UsingA=3.04 m and B=3.57 m, this curve seems to fit the databetter.  This curve-fit form was toallow a fixed RMS error component (3.57 meters) added in quadrature to acomponent directly proportional to the HDOP (that is, 3.04 x HDOP). 

 

Theplot below shows the corresponding plot from a later 31-day collection using aGarmin eMap and external GA-27C antenna. As there was more data, it was grouped by each individual HDOP valuerather than by binning HDOP values.  Inthis case, the values obtained from weighted non-linear regression using theprevious curve fit family were A=2.77 m and B=3.70 m.  The plot of this regression/prediction isagain the broken blue curve.  The fitfor HDOP values between 0.9 and 2.3 is excellent.  Outside that range of HDOP values, there were significantly fewerdata points and the measurement of RMS errors for those HDOP values is thusless accurate.

 

 

Onecan approximate the GPS position distribution by a bivariate normaldistribution having equal variance in both variables (directions) andcorrelation of zero between the two variables. When this is done, for our RMS_Error(HDOP), we obtain a (conditional)Rayleigh error probability distribution given the HDOP:

 

 

Asthe number of satellite in view will influence HDOP and possible other errorcauses, one is tempted to try using the number of satellites in view to predictthe HDOP as a function of the number of satellites in view.  Of course, regardless of the number ofsatellites, there will be times when the HDOP will be very large or even timeswhen no fix is possible.  The next plotshows HDOP, or rather actually RMS of HDOP, as a function of the number ofsatellites in view.

 

 

Thecurve-fit is that given by:

 

 

wherevalues of C=30.0 and D=0.66 were obtained for the Garmin 12XLdata.

 

Theplot below is the corresponding plot of the number of satellites versusRMS_HDOP for data obtained from the 31-day session with a Garmin eMap andGA-27C antenna.  In this case, weightednon-linear regression gave C=32.38 and D=0.71 in the previousfitting equation.  As the Garmin eMap Cand D values are quite close to that for the Garmin 12XL, it isreasonable to conclude the GPS satellite constellation is basically the sameduring the two long observing periods and that both receivers compute HDOP thesame way.

 

 

Insummary, given the HDOP, one can refine the horizontal RMS error to reflect themeasured HDOP and more precisely estimate the distribution of the horizontalerrors.  This requires measuring theHDOP (or RMS_HDOP in the case of a set of more than one measurement andassuming the linear model relating HDOP and RMS error to be valid) whenestimating the RMS error of the GPS receiver/antenna and satelliteconstellation status.   This conditionalRMS error can be used in the Rayleigh distribution formula to predicted errorprobabilities for the particular HDOP (or RMS HDOP of a set of fixes).  Note that Eagle-Lowrance receivers andprobably other manufactures appear to be using a different algorithm thanGarmin to calculate HDOP.  Users shouldverify the applicability of these tentative results (based on Garmin HDOPvalues) to the HDOP reported by their GPS receiver.

 

Finally,histograms are shown below for HDOP and the number of satellites in view.  Note that lower HDOP values and highernumber of satellites in view values have at times been observed in the past attimes with other receivers and antennas.

 

 

 

( Return to http://www.erols.com/dlwilson/gps.htm)