GPS HORIZONTAL POSITION ACCURACY

To some, errors from GPS measurements seem like a mystery. With a little mathematics and simple modeling, the errors behave in a definable way. When the intentional degradation of non-military GPS accuracy (SA or Selective Availability) was turned off, GPS horizontal position errors of consumer-grade GPS receivers were reduced to 1/6 to 1/12 or so of their former values. The section that follows is concerned with SPS (Standard Positioning Service) non-differential horizontal (latitude/longitude) positioning accuracy with SA off. This analysis is thought to be somewhat typical of that obtainable with modern consumer-grade receivers. The analysis uses a precision surveyed point whose coordinates were determined by a licensed surveyor and independently repeatedly confirmed using carrier-phase post-processing with a Motorola Oncore VP GPS receiver and Waypoint GrafNav-Lite software. Many of the tests for acquiring modeling data were done with Garmin receivers as these are commonly used.

The starting point is the equation for experimentally measuring RMS (Root-Mean-Squared) error:

In simple words, one averages the squared errors of the fixes and then takes the square root. The RMS error can also be from an alternative formula, which may be easier with some software:

If the actual position is not known, the average position is often used as an approximation to the actual position. Several days are needed to obtain a reasonable good approximation of the RMS error for the studied GPS receiver/antenna/location/GPS constellation status; but there will still be a tendency to underestimate error using this approximation. Note that GPS receiver NMEA strings output horizontal position in the WGS84 datum and comparisons should be made accordingly.

The distribution of GPS fixes of a position may be approximated by a bivariate normal distribution with no correlation between the two variables. Sometimes this distribution has been inaccurately called "Gaussian"; but only a "slice" in any direction will indeed be a normal (Gaussian) distribution. For simplicity, one might assume the same variance in each direction (measurements show this is not quite actually true). With those approximating assumptions, the error distribution can be described by a very simple equation, which is known as a Weibull distribution with shape factor b = 2 or Rayleigh distribution:

It is interesting to place the horizontal errors in 1-meter bins. This yields the histogram below. Some will be surprised by the implications of this graph. For example, the true position is much more likely to be 2 to 3 meters or 3 to 4 meters away than is to be 0 to 1 meter away. The reason for this is that although the probability of a fix being within any unit area falls off with range from the true position, the circumference at that range gets larger (meaning there is more area at that range) which tends to increase the probability of the true position being at that range. These opposite effects on the probability play against each other in such a way to yield the observed effect. Even though certain size errors are more likely, since the direction of the error is not known, this cannot be used to improve the accuracy of the position.

The plot below is useful in relating the RMS error, the median (50% error bound or CEP error), and the 95% error bound (DHPRE₉₅) to the Rayleigh distribution used for modeling GPS error.

Based on the Rayleigh distribution, the table below can be used to estimate one 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. In the table, "E-N" indicates easting or northing error (the error in longitude or latitude in length units) and "Horizontal" indicates horizontal position error.

	E-N Mean/58%	E-N RMS/68%	E-N 95%	Horizontal CEP/50%	Horizontal Mean/54%	Horizontal RMS/63%	Horizontal 95%
E-N Mean/58%	1.00	1.25	2.46	1.48	1.57	1.77	3.06
E-N RMS/68%	0.80	1.00	1.96	1.18	1.25	1.41	2.44
E-N 95%	0.41	0.51	1.00	0.60	0.64	0.72	1.24
Horizontal CEP/50%	0.68	0.85	1.67	1.00	1.06	1.20	2.08
Horizontal Mean/54%	0.64	0.80	1.56	0.94	1.00	1.13	2.01
Horizontal RMS/63%	0.57	0.71	1.39	0.83	0.89	1.00	1.73
Horizontal 95%	0.33	0.41	0.81	0.48	0.50	0.58	1.00

One should note that there is some variation in terminology. In these writings, "RMS error" indicates the traditional mathematical RMS error as defined above. Some manufacturers use "RMS error" to indicated the 63% error distance; they do this believing that it may be more useful for some comparisons. These two definitions of "RMS error" exactly agree only if the Rayleigh error model is exact - which it is not. "CEP" (Circular Error Probable) in these writings indicates the median or 50% error distance. Although this is the common civilian definition, some recent military receiver specifications use "CEP" to indicate the 95% error distance. In the writings here, the 95% error distance will always be referred to as the 95% error distance, rather than as CEP or some other term. Additionally, there is some confusion over the term "2dRMS". Technically, "2dRMS" is defined as "two times the distance RMS" error. Sometimes "2dRMS" error is used interchanged with 95% error bound. Generally twice the RMS error is a pessimistic estimate of the 95% error bound.

The plot below shows the measured error distribution of a test configuration at the author’s test point collected over 20 days after selective availability was turned off. The test configuration was an early Garmin 12XL with a 26 dB gain external Micropulse antenna. Note that later manufactured Garmin 12XL receivers may perform differently. The test location does show perhaps brief multipath; this may not be uncommon with continuous observations at most locations. The "jaggedness" is due to the fact that the receiver NMEA data, like that of some other models, outputs latitude and longitude in steps of 0.001 minutes. This gives rise to a lattice of possible fix locations with N/S spacing of about 1.8 meters and E/W spacing of about 1.5 meters at the test location. However, this effect has a contribution at only the centimeters level in the RMS error and other error statistics. Also shown in the plot is the predicted Rayleigh distribution based on the measured RMS error.

The plot below shows the similar plot obtained from 30 days of data using a Garmin eMap.

Note in the above two plots that the agreement between measured and predicted error statistics is only approximate due primarily to the Rayleigh distribution approximation (assuming the error distribution is the same in all horizontal directions); unfortunately, to do better than this is an intractable mathematical problem.

In the table below, the predicted (from the Rayleigh distribution and measured RMS error) and measured errors from the Garmin 12XL test configuration 20-day data are compared. The numbers in parenthesis "( )" are the percentage of fixes closer than the stated error distance. The numbers within brackets "[ ]" are the ratios of that error distance to the RMS error distance. All distances are in meters. Entries in bold have their values defined by the type of error so they will always be exact in any set of data.

Error	Measured	Predicted
RMS	5.0 m (70%) [1.00]	5.0 m (63%) [1.00]
Mean	4.1 m (58%) [0.83]	4.4 m (54%) [0.89]
CEP (50%)	3.6 m (50%) [0.73]	4.2 m (50%) [0.83]
95%	9.0 m (95%) [1.81]	8.6 m (95%) [1.73]

The table below is the corresponding table for the 30-day Garmin eMap data. Note the close agreement of the measured ratios to RMS errors in the two tables.

Error	Measured	Predicted
RMS	4.0 m (71%) [1.00]	4.0 m (63%) [1.00]
Mean	3.3 m (58%) [0.82]	3.6 m (54%) [0.89]
CEP (50%)	2.9 m (50%) [0.72]	3.3 m (50%) [0.83]
95%	6.9 m (95%) [1.72]	7.0 m (95%) [1.73]

The percentages are those within the stated error. The differences between predictions and measurements are probably a combination of the assumptions made, biases in the receiver measurement and the NMEA latitude/longitude resolution. Note that the measured distances, although perhaps somewhat typical, are for a particular receiver/antenna, surroundings, ionosphere conditions and constellation status. Maximum errors generally cannot be modeled as they represent rare events (such as multipath due to surrounding a particular satellite geometry); thus reporting of maximum errors is of little value.

The tables below show error measurements six sets of simultaneous tests using two GPS receiver antennas separated by 1.23 meters to avoid interference between the receivers but close enough together to attempt similar receiving conditions. The earlier Garmin 12XL test gave smaller horizontal errors with an external antenna than the above tests with the same Garmin 12XL using the internal antenna. As might be expected, the Eagle Explorer, Garmin eMap and Garmin III+ gave smaller errors than the early production Garmin 12XL that was tested. The tests suggest that the Garmin III+ does perhaps better with its supplied helix antenna than with the Micropulse external antenna; however, more tests would be suggested to confirm this. (Text continues after the tables.)

	Garmin 12XL	Eagle Explorer	Garmin 12XL	Garmin III+	Garmin 12XL	Garmin III+/ext. ant.
RMS error	5.5 m 5.5 m	3.6 m 4.0 m	5.6 m	4.2 m	5.6 m	4.9 m
Mean error	4.6 m 4.6 m	3.0 m 3.5 m	4.8 m	3.6 m	4.7 m	4.2 m
CEP (50%)	4.1 m 4.3 m	2.9 m 2.9 m	4.4 m	3.4 m	4.3 m	3.8 m
95%	9.8 m 10.1 m	7.2 m 7.1 m	9.9 m	7.5 m	10.1 m	8.7 m
Mean no. sat.	6.92 6.67	6.60 6.60	6.75	6.80	6.79	7.14
Mean HDOP	1.36 1.42	1.15 1.16	1.44	1.41	1.43	1.34
RMS HDOP	1.39 1.46	1.16 1.18	1.49	1.46	1.48	1.38
Notes	Two simultaneous 48 hour sessions (interchanging receiver positions)		One simultaneous 48 hour sessions		Internal antenna	26 dB Micropulse antenna
					One simultaneous 48 hour session

	Garmin 12XL	Garmin eMap/GA-27C		Lowrance GlobalNav 2	Garmin eMap/GA-27C
RMS error	5.1 m	3.9 m		7.1 m	3.6 m
Mean error	4.4 m	3.4 m		5.9 m	3.1 m
CEP (50%)	4.0 m	3.1 m		4.9 m	2.9 m
95%	9.0 m	7.0 m		14.2 m	6.4 m
Mean no. sat.	6.87	6.58		6.84	6.73
Mean HDOP	1.40	1.46		1.16	1.42
RMS HDOP	1.44	1.54		1.18	1.45
Notes	Internal antenna	Garmin GA-27C External antenna	Internal antenna		Garmin GA-27C External antenna
	One simultaneous 48 hour session		One simultaneous 96 hour session

	Garmin eTrex	Garmin eMap/GA-27C
RMS error	3.8 m	3.9 m
Mean error	3.0 m	3.2 m
CEP (50%)	2.7 m	2.7 m
95%	6.7 m	6.9 m
Mean no. sat.	6.82	6.87
Mean HDOP	1.41	1.38
RMS HDOP	1.48	1.1.44
Notes	Internal antenna		Garmin GA-27C External antenna
	One simultaneous 48 hour session

As interchanging the receiver positions made little difference in the first comparison, it was judged unnecessary to do so in the subsequent tests. Note that the period lengths in this last table are too short to give robust error statistics; the table is useful though in that simultaneous comparisons were made. It is clear that Garmin and Eagle-Lowrance use different algorithms to calculate HDOP. No theory is presented for why the Lowrance GlobalNav 2 had such a larger error compared to the Garmin eMap tested at the same time; the difference is too large to be due to antenna and the difference appeared consistently on each of the four days. Comparing across sessions, even the similar Eagle Explorer appeared to do better than the Lowrance GobalNav 2. Also note that the Garmin eTrex and Garmin eMap/GA-27C showed essentially the same accuracy during their simultaneous test.

The figure below compares the distributions of horizontal errors for the Garmin 12XL and Garmin eMap simultaneous 48-hour session. Not only was the Garmin eMap more accurate, but also the effect of the Garmin eMap .0001- minute latitude/longitude resolution compared to the Garmin 12XL .001-minute resolution can be seen. Although not discussed further in this section, the eMap also does not have the roughly 10 meter altitude bias that is present in the Garmin 12XL.

In summary, not all GPS receivers, even from the same manufacturer, have the same horizontal accuracy. If one wishes to study the accuracy of a given receiver/antenna, one should start by measuring the RMS error of the receiver/antenna and satellite constellation status. For reasons given in other sections, HDOP (Horizontal Dilution Of Precision) should also be recorded. These measurements should occur over at least a couple days. There is a variation of error with latitude, which is why those in the northern United States report smaller errors (due to, on average, seeing more satellites). The latitude variation of error gives the greatest horizontal position error at about 43 degrees, which is near author’s latitude of 38 degrees. The error distribution is modeled as a Rayleigh distribution and allows us to estimate the mean error, median error (CEP), 95% error bound and other errors from the measured RMS error.

The numbers presented here are only presented as being somewhat typical. Position accuracy is a function not only of the GPS receiver and antenna, but also a function of the geometry and status of the satellites, the surroundings of the antenna and ionosphere conditions/modeling. At the same location with the same receiver and antenna, daily RMS error of horizontal position has been seen to vary by a meter or more. Because of this, one should never depend on a belief that the RMS error or any other error statistic is known more accurately than within a couple meters.

Although some consumer-grade receiver/antenna configurations are seeing horizontal RMS errors closer to 4 meters and 95% errors around 7 to 8 meters, some sources, including some receiver specifications, are now stating a possible horizontal specification of a CEP (50%) of 8 meters and 95% within 15 meters (implying an RMS error of about 9 meters), when HDOP is perhaps 1.5. Remember that the horizontal error is a "random variable". Some observations may yield errors near zero or very large ones, but neither case is of any particular significance.

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