MODELING OF GPS VERTICAL ERRORS

The accuracy of GPS height (or vertical or altitude) measurements is of interest to some users. Before proceeding, we need to recall that height can be measured in two ways. The ellipsoid height (h) is the height above the reference ellipsoid that approximates the earth's surface. The orthometric height (H) is the height above the geoid, which is an imaginary surface determined by the earth's gravity and approximated by mean sea level (MSL). The signed difference between the two heights, which is the difference between the ellipsoid and geoid, is the geoid height (N). The figure below shows the relationships between the different quantities.

Although GPS receivers can measure ellipsoid height, some receivers use approximations of the geoid height to estimate the orthometric height from the geoid height. As an example, Garmin receivers at the author's surveyed location give a geoid height of -34.0 m (reported on the NMEA data). Accurate surveying of the area shows -32.4 m as a more accurate value for the geoid height at that location. In order to eliminate errors caused by the GPS receiver's approximation of the geoid height, the ellipsoid height is always used below. In the case of Garmin receivers, the ellipsoid height was found by subtracting the Garmin's geoid height approximation from the orthometric height. All heights were converted to WGS-84 for comparisons.

Below is shown the vertical error histograms for an early Garmin 12XL, a Garmin III+, an Eagle Explorer and a Garmin eMap.

From the above, as well as other users' reports, it is seen that both the Garmin 12XL and Garmin III+ have significant bias in their height measurements. Both these receivers over-estimate heights by about 10 meters. From the above and other measurement sessions, the Eagle Explorer and Garmin eMap may have a small bias or no bias in their height measurements.

The relatively large height bias of the Garmin 12XL and Garmin III+ is a problem for analysis. Generally one assumes a measurement is right on average (unbiased). Additionally, it is unknown if the their height bias varies by latitude or some other parameter. For these reasons, the Garmin 12XL and Garmin III+ are dropped from the following analysis. As both the Eagle Explorer and Garmin eMap appear to have mean error near zero, they will be further considered below.

The vertical (or height) RMS (Root-Mean-Squared) error is defined as:

Modeled by a normal (Gaussian) distribution having mean 0, the plot below shows the predicted relationship between RMS error (the same as the standard deviation in this case), mean error, median error (50% error bound) and 95% error bound. Note the error is the absolute (unsigned) error.

Based on the normal distribution, the table below can be used to estimate one vertical error statistic from another. To estimate an error statistic on the top from an error statistic on the left, multiply by the corresponding number in the table.

	Vertical Median/50%	Vertical Mean/58%	Vertical RMS/68%	Vertical 95%
Vertical Median/50%	1.00	1.20	1.50	2.95
Vertical Mean/58%	0.83	1.00	1.25	2.46
Vertical RMS/68%	0.67	0.80	1.00	1.96
Vertical 95%	0.34	0.41	0.64	1.00

The plots below show the measured and predicted distributions of (absolute) vertical errors based on the respective measured RMS errors for the Eagle Explorer and Garmin eMap.

In conclusion, the Garmin 12XL and Garmin III+ exhibit significant bias in height measurements of approximately 10 meters. However, the Eagle Explorer and Garmin eMap have little or no error in their height measurements. The Garmin eMap height errors were particularly well modeled by the normal distribution; the Eagle Explorer was reasonably well modeled by the normal distribution.

Although the results presented here may be typical of GPS vertical accuracy, it should be remembered that vertical accuracy depends on latitude (errors for vertical accuracy rapidly increase with latitudes greater than 65 degrees), receiver/antenna, local geometry/multipath and satellite geometry (VDOP). A vertical error specification something like 95% within 20 meters with VDOP of perhaps 2.0 is likely.

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