Thursday, November 7th, 2013
HDOP AND GPS HORIZONTAL POSITION ERRORS
The horizontal dilution of precision (HDOP) allows one to more precisely estimate the accuracy of GPS horizontal (latitude/longitude) position fixes by adjusting the error estimates according to the geometry of the satellites used. Theoretically, given the HDOP, one can obtain error estimates that are good for all fixes with that HDOP, rather than the more general error estimates for all position fixes. In probability terminology, HDOP is an additional variable that allows one to replace the overall accuracy estimates with conditional accuracy ones for the given 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 is less than 4″ then the (conditional) probability of getting a “2” is 1/3. Knowing this is somewhat similar to knowing “the number is less than 4″ in the analogy.
The notation “RMS_Error” is used here to indicate the RMS error of all fixes with a given HDOP value; for example, RMS_Error(HDOP = 1.2) would indicate the RMS error of all fixes with HDOP = 1.2. The value of RMS_Error increases as HDOP increases, as higher values indicate a satellite geometry that will tend to give less accurate fixes.
When a set of position measurements is analyzed, just as the RMS error is used to represent the error of the set of measurements, the RMS of the HDOP, denoted here as RMS_HDOP can be used to represent the HDOP of the set. The RMS of the HDOP is defined in the usual manner:
As can any RMS or “quadratic mean”, RMS_HDOP can instead be found from the mean and standard deviation:
Below is plotted HDOP versus RMS_Error(HDOP) for a 20-day session using a Garmin 12XL. Actually, because of the need for sufficient sample sizes, the data is binned according to HDOP with bins of width 0.2, and then using the data in each bin, the RMS that was plotted against the RMS error. These measured data points are indicated in red in the following plot:
In theory, if satellite geometry were the only component of the horizontal error of position, the RMS error would be directly proportional to HDOP; thus the points in the plot would lie on a straight line:
The solid green line indicates the prediction by this linear model if one uses the sometimes quoted RMS_Error(HDOP=1) = 4.0 meters. Linear regression actually gives RMS_Error(HDOP = 1) = 3.71 meters, or 3.98 meters if the point for HDOP > 2 is excluded. The difference between this and 4.0 meters is marginal when the scatter of the points is considered.
The broken blue curve indicates a curve-fit that was obtained from weighted (by frequency of occurrence) non-linear least-squares regression:
Using A=3.04 m and B=3.57 m, this curve seems to fit the data better. This curve-fit form was to allow a fixed RMS error component (3.57 meters) added in quadrature to a component directly proportional to the HDOP (that is, 3.04 x HDOP).
The plot below shows the corresponding plot from a later 31-day collection using a Garmin eMap and external GA-27C antenna. As there was more data, it was grouped by each individual HDOP value rather than by binning values. In this case, the values obtained from weighted non-linear regression using the previous curve fit family were A=2.77 m and B=3.70 m. The plot of this regression/prediction is again the broken blue curve. The fit for values between 0.9 and 2.3 is excellent. Outside that range of values, there were significantly fewer data points and the measurement of RMS errors for those HDOP values is thus less accurate.
One can approximate the GPS position distribution by a bivariate normal distribution having equal variance in both variables (directions) and correlation of zero between the two variables. When this is done, for our RMS_Error, we obtain a (conditional) Rayleigh error probability distribution given the HDOP:
As the number of satellite in view will influence HDOP and possible other error causes, one is tempted to try using the number of satellites in view to predict the HDOP as a function of the number of satellites in view. Of course, regardless of the number of satellites, there will be times when the HDOP will be very large or even times when no fix is possible. The next plot shows HDOP, or rather actually RMS of HDOP, as a function of the number of satellites in view.
The curve-fit is that given by:
where values of C=30.0 and D=0.66 were obtained for the Garmin 12XL data.
The plot below is the corresponding plot of the number of satellites versus RMS_HDOP for data obtained from the 31-day session with a Garmin eMap and GA-27C antenna. In this case, weighted non-linear regression gave C=32.38 and D=0.71 in the previous fitting equation. As the Garmin eMap C and D values are quite close to that for the Garmin 12XL, it is reasonable to conclude the GPS satellite constellation is basically the same during the two long observing periods and that both receivers compute HDOP the same way.
In summary, given the HDOP, one can refine the horizontal RMS error to reflect the measured HDOP and more precisely estimate the distribution of the horizontal errors. This requires measuring the HDOP (or RMS_HDOP in the case of a set of more than one measurement and assuming the linear model relating HDOP and RMS error to be valid) when estimating the RMS error of the GPS receiver/antenna and satellite constellation status. This conditional RMS error can be used in the Rayleigh distribution formula to predicted error probabilities for the particular HDOP (or RMS HDOP of a set of fixes). Note that Eagle-Lowrance receivers and probably other manufactures appear to be using a different algorithm than Garmin to calculate HDOP. Users should verify the applicability of these tentative results (based on Garmin values) to that reported by their GPS receiver.
Finally, histograms are shown below for HDOP and the number of satellites in view. Note that lower values and higher number of satellites in view values have at times been observed in the past at times with other receivers and antennas.
Figures and report courtesy of dlwilson. To learn more about Dilution of Precision you can read here.