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 β = 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.
HISTOGRAM OF HORIZONTAL ERRORS
Garmin 12XL (Micropulse antenna)
0.200
Proportion of Measurements
Legend
0.150
Measured
hist2
Predicted
phist2
0.100
Predicted histogram is
based on the measured
RMS error of 5.0 m over
the 20 days.
0.050
0.000
0.0
20 days data
Fix every 2 seconds
5.0
10.0
15.0
Error Distance (1-meter bins)
The plot below is useful in relating the RMS error, the median (50% error bound or CEP
error), and the 95% error bound (∆HPRE95) to the Rayleigh distribution used for modeling
GPS error.
2
Probability = 1-exp(-(Distance/RMS) )
1.0
0.9
0.8
63%
0.50
1.00
1.73
50%
(CEP)
0.3
0.2
0.1
0.0
0.00
1.00 (RMS)
0.7
0.6
0.5
0.4
95%
0.83
Probability that point is less than Distance
DISTANCE/RMS VS. PROBABILITY
1.50
NOTE: Measured
position error may
be very large for a
small percentage
of the time.
2.00
2.50
3.00
Distance/RMS
(Multiply by RMS to get Distance)
CEP (Circular Error Probable) is also the median 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%
E-N
RMS/68%
E-N
95%
Horizontal
CEP/50%
Horizontal
Mean/54%
Horizontal
RMS/63%
Horizontal
95%
1.00
1.25
2.46
1.48
1.57
1.77
3.06
0.80
1.00
1.96
1.18
1.25
1.41
2.44
0.41
0.51
1.00
0.60
0.64
0.72
1.24
0.68
0.85
1.67
1.00
1.06
1.20
2.08
0.64
0.80
1.56
0.94
1.00
1.13
2.01
0.57
0.71
1.39
0.83
0.89
1.00
1.73
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.
Probability (Error < Distance)
MEASURED AND MODELED
DISTRIBUTION OF HORIZONTAL ERRORS
Garmin 12XL (Micropulse antenna)
1.00
0.90
0.80
Measured distribution
("jagged" due to 0.001
MMEA resolution)
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
20 days data
Fix every 2 seconds
Error
50% (CEP)
Mean
RMS
95%
5.0
Modeled distribution
(based on measured
5.0 m RMS error)
Meas.
3.6 m
4.1 m
5.0 m
9.0 m
10.0
Pred.
4.2 m
4.4 m
(meas.)
8.6 m
15.0
Distance [meters]
The plot below shows the similar plot obtained from 30 days of data using a Garmin eMap.
Probability(Error < Distance)
MEASURED AND MODELED
DISTRIBUTION OF HORIZONTAL ERRORS
Garmin eMap (GA-27C antenna)
1.00
0.90
0.80
Measured
distribution
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
30 days data
Fix every 2 seconds
Modeled distribution
(based on measured
4.01 m RMS error)
Error
50% (CEP)
Mean
RMS
95%
99%
Meas.
2.9 m
3.3 m
4.0 m
6.9 m
10.1 m
5.0
Pred.
3.3 m
3.6 m
(meas.)
7.0 m
8.6 m
10.0
15.0
Distance [meters]
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
RMS
Mean
CEP (50%)
95%
Measured
5.0 m (70%) [1.00]
4.1 m (58%) [0.83]
3.6 m (50%) [0.73]
9.0 m (95%) [1.81]
Predicted
5.0 m (63%) [1.00]
4.4 m (54%) [0.89]
4.2 m (50%) [0.83]
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
Mean
CEP (50%)
95%
4.0 m (71%) [1.00]
3.3 m (58%) [0.82]
2.9 m (50%) [0.72]
6.9 m (95%) [1.72]
4.0 m (63%) [1.00]
3.6 m (54%) [0.89]
3.3 m (50%) [0.83]
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.)
RMS error
Mean error
CEP (50%)
95%
Mean no. sat.
Mean HDOP
RMS HDOP
Notes
Garmin
12XL
5.5 m
5.5 m
4.6 m
4.6 m
4.1 m
4.3 m
9.8 m
10.1 m
Eagle
Explorer
3.6 m
4.0 m
3.0 m
3.5 m
2.9 m
2.9 m
7.2 m
7.1 m
Garmin
12XL
5.6 m
Garmin
III+
4.2 m
Garmin
12XL
5.6 m
Garmin
III+/ext. ant.
4.9 m
4.8 m
3.6 m
4.7 m
4.2 m
4.4 m
3.4 m
4.3 m
3.8 m
9.9 m
7.5 m
10.1 m
8.7 m
6.92
6.67
1.36
1.42
1.39
1.46
6.60
6.60
1.15
1.16
1.16
1.18
6.75
6.80
6.79
7.14
1.44
1.41
1.43
1.34
1.49
1.46
1.48
1.38
Two simultaneous
48 hour sessions
(interchanging receiver
positions)
One simultaneous
48 hour sessions
Internal
26 dB Micropulse
antenna
antenna
One simultaneous
48 hour session
RMS error
Mean error
CEP (50%)
95%
Mean no.
sat.
Mean HDOP
RMS HDOP
Notes
Garmin
12XL
5.1 m
4.4 m
4.0 m
9.0 m
Garmin
eMap/GA-27C
3.9 m
3.4 m
3.1 m
7.0 m
6.87
6.58
6.84
6.73
1.40
1.44
1.46
1.54
1.16
1.18
1.42
1.45
Internal
Garmin GA-27C
antenna
External antenna
One simultaneous
48 hour session
RMS error
Mean error
CEP (50%)
95%
Mean no. sat.
Mean HDOP
RMS HDOP
Notes
Garmin
eTrex
3.8 m
3.0 m
2.7 m
6.7 m
6.82
1.41
1.48
Lowrance
Garmin
GlobalNav 2 eMap/GA-27C
7.1 m
3.6 m
5.9 m
3.1 m
4.9 m
2.9 m
14.2 m
6.4 m
Internal
Garmin GA-27C
antenna
External antenna
One simultaneous
96 hour session
Garmin
eMap/GA-27C
3.9 m
3.2 m
2.7 m
6.9 m
6.87
1.38
1.1.44
Internal
Garmin GA-27C
antenna
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.
HORIZONTAL POSITION ERROR DISTRIBUTION
Garmin eMap(GA-27C antenna) vs. Garmin 12XL
Probability(Error < Distance)
1.00
0.80
Garmin eMap has
0.0001 lat/lon
minute resolution.
Garmin 12XL has
0.001 lat/lon minute
resolution causing
"jagged" graph.
0.60
Garmin
E-map/
Error 12XL
GA-27C
--------------------RMS
5.1 m
3.9 m
Mean
4.4 m
3.4 m
50%
4.0 m
3.1 m
95%
9.0 m
7.0 m
0.40
0.20
0.00
0.0
Simultaneous 48 hr. sessions
Fix every 2 seconds.
5.0
10.0
15.0
Distance [meters]
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.
RETURNING TO A POSITION
It is often stated that when attempting to return to a previously measured location that the
error is multiplied by the square root of 2, or about 1.41. The information presented here is
more general - it includes the case where the initial measurement by GPS and returning
measurement by GPS have different error specifications due to differences in measuring
equipment or procedure. GPS data used below is non-differential SPS (Standard
Positioning Service) with SA (Selective Availability) turned off.
It follows from assuming that the GPS measured positions have approximately a bivariate
normal distribution that the error distribution in returning to a position is approximately
given by the following Rayleigh distribution:
Probability ( Returning _ Error ≤ D) = 1 − e − aD / RMS _ Returning _ Error f
2
where:
RMS _ Returning _ Error =
aRMS _ Error f + aRMS _ Error f
2
1
where RMS_Error1 is the RMS (Root-Means-Squared) error of the initial measurement and
RMS_Error2 is the RMS error of the measurement during the attempted return to the same
position. In words, the RMS error of the return is the "RSS" (Root-Sum-of-Squares) of the
individual RMS errors. If one (usually the initial fix) is averaged, then the RMS error for
that measurement should be the RMS error for the averaging period.
2
2
When both measurements are made with the same measuring procedure with the same
errors, RMS_Error1 = RMS_Error2. Letting RMS_Error denote the common RMS error in
this case, the above equations and some simple algebra give RMS_Returning_Error = 1.41
x RMS_Error, which is the commonly quoted error in returning to a position.
The following plot shows that this relationship works well when compared with actual
measurements:
Probability(Error < Distance)
ERROR IN RETURNING TO A PREVIOUS POINT
Garmin 12XL (Micropulse antenna) data
1.00
0.90
0.80
Measured
(Jaggedness due to
0.001 min resolution)
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
Predicted
(Based on 5.494 m
position RMS error)
Return Error
RMS
50% (CEP)
95%
5.0
Data is 2 days against 2 days
84168 measurements
10.0
Meas.
7.7 m
5.7 m
13.7 m
15.0
Pred.
7.8 m
6.4 m
13.4 m
20.0
Distance [m]
As a second example, we consider returning to a position that was initially measured by
simple averaging for 15 minutes or 1 hour. As would seem most likely, in the return
attempt to find the position, we assume single measurements (no averaging) as being used.
The plot below shows the measured error distribution. For comparison, the measured error
distribution where no averaging is used for either measurement is also shown for the same
set of data.
Probability (Error < Distance)
ERROR IN RETURNING TO AN AVERAGED POINT
Garmin 12XL (Micropulse antenna)
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
1 hr.
averaging
No
averaging
15 min.
averaging
0.20
0.10
0.00
0.0
Initial measurement averaged as indicated.
Return measurement not averaged.
5.0
Measured 2 days data
against 2 days data
10.0
15.0
20.0
Distance [m]
The table below summarizes the measured errors of for position averaging on the first visit
but no averaging on the return:
Measured
Error
RMS
50% (CEP)
95%
No averaging
on initial visit
7.7 m
5.7 m
13.7 m
15 min. averaging
on initial visit
6.5 m
4.9 m
11.3 m
1 hr. averaging
on initial visit
6.0 m
4.5 m
10.3 m
In conclusion, in returning to a location using a measurement of the same accuracy, the
error specifications are multiplied by 1.41 (the square root of 2). Otherwise, the RMS error
in returning is the RSS of the two separate RMS errors. The above example shows that
only a small advantage is achieved by averaging the initial visit for a period up to an hour;
however, it is wise to carefully evaluate single measurements for their validity.
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.
GARMIN 12XL
VERTICAL ERROR HISTOGRAM
7000
6000
Count
5000
4000
RMS Error
14.6 m
3000
2000
1000
0
-40
-30
-20
-10
0
10
20
30
(Signed) Vertical Error (Measured-True) [1 meter bins]
48 hours data
Fix every 2 seconds
40
GARMIN III PLUS
VERTICAL ERROR HISTOGRAM
7000
6000
Count
5000
4000
RMS Error
11.5 m
3000
2000
1000
0
-40
-30
-20
-10
0
10
20
30
(Signed) Vertical Error (Measured-True) [1 meter bins]
48 hours data
Fix every 2 seconds
40
EAGLE EXPLORER
VERTICAL ERROR HISTOGRAM
7000
6000
Count
5000
4000
RMS Error
6.1 m
3000
2000
1000
0
-40
-30
-20
-10
0
10
20
30
(Signed) Vertical Error (Measured-True) [1 meter bins]
48 hours data
Fix every 2 seconds
40
GARMIN eMAP
VERTICAL ERROR HISTOGRAM
7000
6000
Count
5000
RMS Error
6.0 m
4000
3000
2000
1000
0
-40
-30
-20
-10
0
10
20
30
40
(Signed) Vertical Error (Measured-True) [1 meter bins]
48 hours data
Fix every 2 seconds
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 overestimate 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.
1.0
0.9
0.8
50%
0.3
0.2
0.1
0.00
0.50
1.96
58%
1.00
(RMS or standard deviation)
68%
0.80 (mean error)
0.7
0.6
0.5
0.4
95%
0.67
Probability that Height Error
is Less Than Error
HEIGHT ERROR/RMS VS. PROBABILITY
(Based on Normal Distribution)
1.00
1.50
2.00
2.50
Error/RMS
(Multiply by RMS to get Error)
3.00
Error is absolute value of difference between measured and true altitude.
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%
Vertical
Mean/58%
Vertical
RMS/68%
Vertical
95%
1.00
1.20
1.50
2.95
0.83
1.00
1.25
2.46
0.67
0.80
1.00
1.96
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.
EAGLE EXPLORER
VERTICAL ERROR DISTRIBUTION
Probability (Error < Distance)
1.00
0.80
0.60
The Eagle Explorer has a 1 m
altitude resolution which
causes the "jagged" effect
in the measured error
distribution plot.
Blue is measured distribution.
Pink is the predicted distribution
from the measured RMS error.
0.40
0.20
0.00
0.0
48 hours data
Fix every 2 seconds
Error
RMS
Mean
50%
95%
5.0
Measured
6.1 m
4.7 m
3.5 m
12.5 m
10.0
Distance [meters]
Predicted
(meas.)
4.8 m
4.1 m
11.8 m
15.0
GARMIN eMAP (GA-27C ANTENNA)
VERTICAL ERROR DISTRIBUTION
1.0
Probability (Error < Distance)
Measured
0.8
Predicted using
measured vertical
RMS error
0.6
Error
50% (CEP)
Mean
RMS
95%
99%
0.4
0.2
0.0
0.0
30 days data
Fix every 2 seconds
5.0
Meas.
4.1 m
5.2 m
7.3 m
13.7 m
19.6 m
10.0
Pred.
4.9 m
5.8 m
(meas.)
14.3 m
18.8 m
15.0
20.0
Distance [meters]
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
GPS ERROR WHEN AVERAGING POSITION
A way to improve GPS measurement of position accuracy without additional equipment is
to simply average the coordinates. In this section, we consider only simple averaging (no
weighting by DOP) of receiver NMEA position data. As with all pages at this site, the
study is for SA off unless explicitly stated otherwise. Not only does simple averaging
decrease random errors in the measurement, it also allows interpolation beyond the
resolution of the measurement; thus averaging may yield accuracy better than the 0.001
seconds of latitude/longitude resolution or 1 meter of height resolution reported in the
NMEA data from some GPS receivers.
As is well known, if the horizontal (latitude and longitude) errors were not correlated, the
RMS error would be inversely proportional to the square root of the number of
measurements. However, the errors are correlated and this causes the error from averaging
to decrease at a slower rate than if the errors were not correlated.
As a first look at the effect of position averaging on horizontal error, consider the plot
below of twelve 24-hour averaging sessions - all starting at midnight local times.
12 RUNS AVERAGING FOR 24 HOURS
Garmin 12XL (Micropulse antenna)
8
Runs started at midnight local time.
7
Error [meters]
6
5
Note error peaks sometimes at
almost same time on different days.
4
3
2
1
0
0
Fix every 2 sec.
6 hr.
12 hr.
Averaging Period
18 hr.
24 hr.
From the plot, we see that, as expected, position-averaging tends to decrease the error.
However, note the occasional error peaks on about the same time on different days. This
may be due to similar larger (poor) HDOP, particular satellites yielding poorer accuracy,
multipath at near the same time daily, or some other reason. Clearly, if one is collecting
short sessions at the same point on different days, it is best to use different times (as
different as possible) on the different days. If not continuously collecting data but doing
short sessions on a single day, one should separate the sessions as much as possible in times
to attempt to avoid correlated errors.
The points in the plot below are measured RMS errors from position-averaging over the
corresponding different periods of every possible interval of each period in the 12 days of
data. Only periods up to 8 hours are plotted, as longer periods would mean a "small"
number of non-overlapping averaging periods. The separate results from the 6 days in May
and 6 days in June as well as the combined 12 days are shown to indicate the robustness or
variability of the measurements. The curve fits are explained in what follows.
AVERAGING POSITION HORIZONTAL RMS ERROR
Garmin 12XL (Micropulse antenna)
5
4
RED
12 days May/June
LIGHT GREEN
6 days in May
LIGHT BLUE
6 days in June
3
Avg. RMS Error
Averaging RMS Error [meters]
6
FIRST HOUR
5
4
3
2
1
Averaging Period [minutes]
10
20
30
40 50
2
1
Prediction for beyond 8 hours
0
0.0
5.0
10.0
15.0
20.0
Averaging Period [hours]
When SA (Selective Availability) was on, it was noted that if one position-averaged, the
error when position-averaging roughly fell by the reciprocal of the square root of the
number of fixes divided by a constant. This constant was twice the correlation time of the
fixes; this allowed the previous effective fix to "decay" and the next effective fix to "build".
(Whether one calls L or 2L the correlation time depends on one's choice of terminology.)
One took into account the first fix (with averaging time of zero) acting as the first "true"
measurement by adding a "1" under the radical. Thus we had the following formula for
calculating the error from position-averaging from the error of single measurements and the
period over which the averaging was done:
Averaging _ RMS _ Error =
RMS _ Error
Averaging _ Period
1+
2L
This equation to model error will be called "Model 1". When SA was turned off, smaller
errors became significant and Model 1 did not work as well for modeling RMS when
position-averaging.
Modeling the error as two errors as above added in quadrature seemed to work better when
SA was turned off:
Averaging _ RMS _ Error
2
2
E1
E2
=
+
Averaging _ Period
Averaging _ Period
1+
1+
2L1
2L2
This equation to model RMS error when averaging will be called "Model 2". At present, it
has not been possible to associate either error component with known error sources such as
receiver hardware or algorithm, multipath, satellite geometry or GPS satellite constellation
status.
For time in minutes and RMS error in meters, the constants obtained by non-linear leastsquares regression are shown in the following table for the above Garmin 12XL test data:
6 days (May)
6 days (June)
12 days (May+June)
E1
4.33
3.27
3.85
L1
1.54
0.48
1.06
E2
3.38
3.50
3.43
L2
69.13
32.82
48.85
The figures below show three pairs of measured position-averaging RMS errors. In each
case, the two receivers antennas were separated by 1.23 meters to avoid interference but
attempt similar reception condition geometry.
Horizontal Position RMS Error [meters]
GARMIN 12XL AND EAGLE EXPLORER
SIMULTANEOUS AVERAGING SESSIONS
6
Garmin 12XL
5
4
3
Eagle Explorer
2
1
0
0
10
20
48-hour sessions
Fix every 2 seconds
30
40
50
60
70
80
90
Averaging Period [minutes]
Horizontal Position RMS Error [meters]
GARMIN 12XL AND GARMIN III PLUS
SIMULTANEOUS AVERAGING SESSIONS
6
5
Garmin 12XL
4
3
2
Garmin III Plus
1
0
0
10
20
48-hour sessions
Fix every 2 seconds
30
40
50
60
Averaging Period [minutes]
70
80
90
Horizontal Position RMS Error [meters]
GARMIN 12XL AND GARMIN eMAP
SIMULTANEOUS AVERAGING SESSIONS
6
5
Garmin eMap
4
3
2
Garmin 12XL
1
0
0
10
20
48-hour sessions
Fix every 2 seconds
30
40
50
60
70
80
90
Averaging Period [minutes]
In each of the above three figures, an early production Garmin 12XL was used as the
common comparison receiver. Note how both the Garmin 12XL and Garmin III+ have a
significant error component that rapidly falls off in the first few minutes (that is, that error
component has a short correlation length). The Eagle Explorer and Garmin eMap also
seem to have a similar component but much smaller in size, as is indicated by the relative
smallness of E1 matched with the short L1 in the table below for these two receivers:
E1
L1
E2
L2
Simultaneous session
Garmin III+
Garmin 12XL
3.09
4.50
1.10
1.25
2.92
3.51
55.39
106.52
Simultaneous session
Eagle Explorer
Garmin 12XL
0.49
4.80
0.07
1.23
3.61
2.86
36.45
91.57
Simultaneous session
Garmin eMap
Garmin 12XL
1.90
4.11
3.17
1.09
3.45
3.19
117.82
69.33
As can be seen in the table, the Garmin 12XL values varied in the three 48-hours sessions.
The figure below shows the results of measuring position-averaging RMS error for the
Garmin 12XL in the separate sessions.
Horizontal Position RMS Error [meters]
GARMIN 12XL
RMS ERROR FROM AVERAGING
MEASURED IN THREE 48-HOURS SESSIONS
6
5
4
3
These curves show that the RMS errors
measured for the same Garmin 12XL
varied a little in the separate 48-hour
sessions.
2
1
0
0
10
20
48-hour sessions
Fix every 2 seconds
30
40
50
60
70
80
90
Averaging Period [minutes]
It is clear from the above that longer sessions would be required to obtain model parameters
accurate for long periods of time. The above plots and formulas would seem to imply that
1 to 2 days might get a position-average RMS error down to the 1-meter level. Note well:
this is RMS error--not every measurement error.
The plot below shows 20 1-day (24 hour) horizontal position-averages using the Garmin
12XL (and Micropulse antenna).
Garmin 12XL Daily Average Position
(Micropulse antenna)
2.0
N/S Error [meters]
1.5
1 meter
1.0
0.5
0.0
-0.5
-1.0
16 out of 20 (80%)
within 1meter
-1.5
-2.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
E/W Error [meters]
20 Days
For this small sample, the RMS error for 24-hour position-averaging was 0.78 m while the
initial Model 2 parameters (based on 12-days) for the Garmin 12XL data predict 0.87 m -this is fairly good agreement considering the approximation of values due to the relatively
short periods of the measurements involved and that the results probably vary with time. In
each of the 20 1-day averages, the error of the position-average was less than 2 meters.
The plot below shows averaging results for 30 days using a Garmin eMap. As the plot
indicates, the 30-day average is displaced about a meter and a half from the true position.
This possible bias indicates that there is a limit in the accuracy that may be obtained from
averaging position.
N/S (Latitude) Error [meters]
GARMIN eMAP (GA-27C antenna)
ERRORS FOR 30 24-HOUR AVERAGE POSITIONS
3.00
2.00
Measured 30 day
mean horizontal
position
1.00
True WGS-84
Position
0.00
-1.00
-2
-1
30 days data
Fix every 2 seconds
0
1
2
3
4
E/W (Longitude) Error [meters]
Finally, we take a look at averaging height data with a Garmin eMap and an Eagle Explorer
to improve vertical accuracy.
POSITION-AVERAGING VERTICAL RMS ERROR
GARMIN eMAP (GA-27C ANTENNA)
EAGLE EXPLORER
Vertical RMS Error from Averaging
7
Eagle Explorer
6
5
4
Garmin eMap
3
2
1
0
0
10
20
48 hour sessions
(Non-simultaneous)
Fix every 2 seconds
30
40
50
60
70
80
90
Period of Averaging [minutes]
Again we see that averaging improves the accuracy of the vertical measurement. In this
case, Model 2 formulas were again used to model the measured vertical error. The values
for the constants to model the height measurements from the (non-simultaneous) sessions
with these two receivers are:
Garmin eMap
Eagle Explorer
E1
3.30
2.16
L1
6.65
11.07
E2
5.07
5.65
L2
306.51
107.61
In summary, different GPS receivers perform differently when position-averaging. Several
days of position-averaging appear to be needed to obtain 1-meter level horizontal accuracy.
High-end (survey-grade) units will do significantly better. Finally, the statistics vary
somewhat and extensive measurements may be required to obtain accurate model values.
For this reason, predictions should be taken only as approximations. Remember that the
analysis done here were for simple position-averaging done on the NMEA data. Any
"tricks" or re-configuring of the receiver algorithm for firmware position-averaging have
not been analyzed.
As the Rayleigh distribution would approximately model the horizontal position-averaging
error distribution, the 95% error bound would be predicted to be approximately 1.73 times
the horizontal position-averaging RMS error and other error parameters could be similarly
predicted by multiplying the position-averaging RMS by the appropriate factor due to the
Rayleigh distribution. The normal distribution should be used to model the height errors
when position-averaging to improve vertical accuracy. Thus the 95% error bound would be
predicted to be approximately 1.96 times the vertical position-averaging RMS error.
Extreme caution is recommended in applying these results here to other GPS receivers at
other times and places, especially considering that there is some variation with latitude,
GPS satellite constellation status and local reception of signal effects. See the section on
correlation of errors for related material.
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 (regardless of HDOP). In probability
terminology, HDOP is an additional variable that allows one to replace the overall accuracy
estimates with conditional 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 is less than 4" then the (conditional)
probability of getting a "2" is 1/3. Knowing HDOP is somewhat similar to knowing "the
number is less than 4" in the analogy.
The notation "RMS_Error(HDOP)" 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(HDOP) increases as HDOP
increases, as higher values of HDOP 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 of
the HDOP was plotted against the RMS error. These measured data points are indicated in
red in the following plot:
HORIZONTAL RMS ERROR AS A FUNCTION OF HDOP
Garmin 12XL (Micropulse antenna)
RMS_ERROR (HDOP) [meters]
10
9
8
7
6
5
4
3
2
1.00
1.50
2.00
2.50
3.00
Horizontal Dilution of Precision (HDOP)
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:
a
f
a
f
RMS _ Error HDOP = HDOP ⋅ RMS _ Error HDOP = 1
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:
RMS _ Error ( HDOP ) =
( AHDOP )
2
+ B2
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 HDOP 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 HDOP values between 0.9 and 2.3 is excellent. Outside that range of
HDOP values, there were significantly fewer data points and the measurement of RMS
errors for those HDOP values is thus less accurate.
HORIZONTAL RMS ERROR AS A FUNCTION OF HDOP
Garmin eMap (GA-27C antenna)
RMS_Error (HDOP) [meters]
10
9
8
7
6
5
4
Fewer than 10000 samples
with these HDOP values.
3
2
1.00
1.50
2.00
2.50
3.00
HDOP
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(HDOP), we obtain a (conditional)
Rayleigh error probability distribution given the HDOP:
Probability ( Error ≤ Distance | HDOP )
=1− e
−( Distance / RMS _ Error ( HDOP ) )
2
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.
NUMBER OF SATELLITES VS. RMS_HDOP
Garmin 12XL (Micropulse antenna)
4.00
Legend
3.00
RMS_HDOP
pred.
PP
Small sample size
meas.
hh
2.00
1.00
0.00
4
5
6
6 days data
Fix every 2 seconds
7
8
9
10
11
12
Number of Satellites
The curve-fit is that given by:
RMS _ HDOP =
C
( Number _ of _ satellites )
2
+D
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.
Horizontal Dilution of Precision (HDOP)
NUMBER OF SATELLITES VS. RMS_HDOP
Garmin eMap (GA-27C antenna)
5
Note: The fix from 3 satellites
uses a different algorithm than
that for more than 4 satellites.
4
Legend
Y
Pred.
rmshdop
Meas.
3
2
1
0
3
4
5
31 days data
Fix every 2 seconds
6
7
8
9
10
11
12
Number of Satellites
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 HDOP values) 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 higher number of satellites in view values have at times been
observed in the past at times with other receivers and antennas.
HISTOGRAM OF HDOP
Observed with Garmin eMap (GA-27C antenna)
14.0
12.0
Mean HDOP 1.50
RMS HDOP 1.65
Percent of Fixes
10.0
8.0
6.0
About 1.1% of the data
had HDOP > 2.9
4.0
2.0
0.0
1.00
1.50
2.00
2.50
Horizontal Dilution of Precision (HDOP)
31 days of data
Fix every 2 seconds
1338681 fixes
3.00
HISTOGRAM OF NUMBER OF SATELLITES IN VIEW
Observed with a Garmin eMap (GA-27C antenna)
35.0
Percent of Fixes
30.0
Mean Number of Satellites 6.46
25.0
20.0
15.0
10.0
5.0
0.0
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0
31 days of data
Fix every 2 seconds
1338682 fixes
Number of Satellites
GPS WAAS ACCURACY
The Wide Area Augmentation System (WAAS) is a form of differential GPS (DGPS)
giving enhanced position accuracy developed primarily for aeronautical navigation but
usable by other users. Each Wide Area Reference Station (WRS) provides correction data
to a Wide Area Master Station (WMS), which computes a grid of correction data to be
uplinked to a geostationary satellite (GEO) via a Ground Earth Station (GES) in the Ground
Uplink System (GUS). The geostationary satellite transmits the correction data (and also
navigation data) to the user on the L1 GPS navigation frequency (1575.42 MHz). The user
GPS receiver uses the downlink WAAS data to correct received navigation data. The goal
of WAAS is to obtain at least a 7-meter horizontal and vertical accuracy.
In the analysis reported here, a Garmin GPSMAP 76 receiver was used with a Garmin GA
29 pole mount GPS antenna. The WAAS corrections were received from the INMARSAT
3F4 satellite at 54 degrees west, which is known as AOR-W. This satellite has GPS PRN
number 122. Although some users have reported difficulty receiving the WAAS signals,
they were copied 100% of the time during these tests.
WAAS corrections are WGS84 rather than USCG DGPS, which is NAD83/NAVD88. As
the accuracy of the system is very good, this distinction is significant. In the analysis
presented here, the surveyed NAD83/NAVD88 position was converted to a WGS84
position using the NGS program HTDP (see the FAQ page for a link to obtain the
software).
The plot and embedded table below show the distribution of horizontal and vertical errors
that were obtained during the test session.
Probability (Error < Distance)
GPS ACCURACY WITH WAAS ENABLED
Garmin GPSMAP 76 with GA 29 antenna
1.00
0.90
0.80
Horizontal
0.20
0.10
0.00
Horiz. Vert.
2.2 m
1.4 m
2.6 m
1.5 m
3.2 m
1.8 m
6.0 m
3.2 m
8.0 m 15.7 m
Error
50%
Mean
RMS
95%
Max.
0.70
0.60
0.50
0.40
0.30
Vertical
0
1
2
3
Note: Max. error depends greatly on the
length of the observation period and is
generally not a robust statistic.
4
5
6
7
8
9
Distance [meters]
172815 obervations (4 days)
Sample every 2 seconds
The plot below shows a comparison of WAAS with non-WAAS using the same receiver.
As only one such receiver was available, the WAAS and non-WAAS session were nonsimultaneous; however, looking at sub-sessions, the accuracy of each mode seemed fairly
stable during these observations. The mean number of satellites received during the WAAS
session was 8.43, the mean HDOP was 1.09, and the RMS of the HDOP was 1.11. The
mean number of satellites received during the non-WAAS session was 7.44, the mean
HDOP was 1.27, and the RMS of the HDOP was 1.31.
Probability (Error < Distance)
WAAS VS. NON-WAAS COMPARISON
Garmin GPSMAP 76 with GA29 antenna
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
Horiz.
WAAS
W/O = without
Vert.
WAAS
Horiz.
W/O-WAAS
Vert.
W/O-WAAS
Error
50%
Mean
RMS
95%
Max.
5.0
Horizontal
WAAS W/O
1.4 2.4
1.5 2.6
1.8 3.0
3.2 5.3
8.0 15.5
Vertical
WAAS W/O
2.2 3.0
2.6 3.6
3.2 4.5
6.0 8.9
15.7 19.0
10.0
Distance [meters]
172815 samples (4 days) each session
Note: Max. error depends greatly on the
WAAS & Non-WAAS non-simultaneous length of the observation period and is
generally not a robust statistic.
Samples every 2 seconds
The above plot and table show the improvement in both horizontal and vertical inaccuracy
due to WAAS.
WAAS accuracy performance using this type of GPS equipment is comparable to the
accuracy obtained by using DGPS beacon stations as the plot below idicates.
Probability (Error < Distance)
WAAS VS. USCG BEACON DGPS COMPARISON
Garmin GPSMAP 76 with GA29 antenna
1.00
0.90
0.80
Horiz. WAAS
& DGPS
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
Vert.
WAAS
Vert.
DGPS
Error
50%
Mean
RMS
95%
1.0
WAAS and DGPS
performance were
essentially the same.
2.0
3.0
4.0
Horizontal
WAAS DGPS
1.4
1.4
1.5
1.5
1.8
1.9
3.2
3.2
5.0
6.0
Distance [meters]
4 days WAAS &
3 days Non-WAAS (non-simultaneous)
Samples every 2 seconds
Vertical
WAAS DGPS
2.2
1.7
2.6
2.8
3.2
3.4
6.0
5.9
7.0
8.0
9.0 10.0
USCG DGPS beacon
at 169 km (105 mi)
Finally, the plot below shows the improvements in horizontal and vertical accuracy
obtained by averaging a position over time. Note that the NMEA reported position was
averaged rather than using the receiver’s own waypoint averaging; this was done in order to
obtain a sufficient sample size. Also note that the plot is for RMS-errors and that the 95%
error distances will generally be something less than twice the RMS-errors.
ERROR WHEN AVERAGING POSITION WITH WAAS
Garmin GPSMAP 76 with GA 29 antenna
3.50
RMS Error [meters]
3.00
2.50
Vertical error
2.00
1.50
Horizontal error
1.00
0.50
0.00
Predicted
0
60
120
180
240
300
360
420
480
Period [minutes]
4 days data (every 2 seconds)
All possible sessions of each period used
Although four days of data were collected, that is only perhaps sufficient to give reliable
statistics for averaging up to 180 minutes (3 hours), as longer periods may yield too few
disjoint periods. The curve-fit extrapolation beyond 3 hours is only a model prediction for
periods up to 480 minutes (8 hours). This curve-fit is the one given by LevenburgMarquadt non-linear regression on the measured data using the family:
Averaging _ RMS _ Error
2
2
E1
E2
=
+
Averaging _ Period
Averaging _ Period
1+
1+
2L1
2L2
The measured values of the constants were:
Horizontal
Vertical
E1
1.43
2.37
L1
3.77
5.24
E2
1.09
2.11
L2
251.07
210.07
The applicability of this fit to other receivers, times or locations is not know so the reader is
cautioned that these numbers should only be considered as representative.
The above analysis shows that WAAS can improve the accuracy of position measurement.
WAAS gave 95% of the time horizontal position within 3.2 meters and vertical position
within 6.0 meters in these tests. Averaging for2 or 3 hours reduced this to 95% of the time
horizontal position within 2 meters and vertical position within 4 meters.
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 and the WAAS system, the surroundings of the antenna
and ionosphere conditions/modeling. At the same location with the same receiver and
antenna, daily RMS error of horizontal and vertical positions have been seen to vary.
CORRELATION OF ERRORS
This section shows two interesting results concerning the relationship between the errors in
measuring position and the time differences (lag times) between the measurements. The
first result concerns how errors are related in the long-term; the second concerns
relationships in the short-term.
A common question asked is whether there is a diurnal variation of GPS errors in
measuring position. By this, we mean whether there is a tendency for errors at certain
times of day to follow some pattern. The plot below shows the horizontal position RMS
errors obtained from 18 24-hour periods. The error of each plot was truncated at 15 meters
to prevent the plots from overlapping. The plots in red were taken on 6 consecutive days in
June while those in blue were taken on 12 consecutive days in July. The lower plots were
taken on earlier days than those higher in the figure.
One does see some tendency for errors at certain times of day to be greater. The cause for
this might be a tendency for HDOP to be larger at those times, certain satellites to perhaps
yield larger error results or more likely multipath (reflected signals from objects in the
antenna’s vicinity). No day versus night pattern is apparent.
On the other hand, in the short term, GPS errors are strongly correlated. The section on
averaging to improve the accuracy of GPS horizontal position measurements presented an
equation (Model 2) with two error components of different time constants (correlation
lengths) to model the RMS error when position-averaging. The question arises as to
whether these time constants and error components are just curve-fit artifacts or do they
represent some underlying correlation of errors effect. The natural way to attempt to
answer this is to plot the autocorrelation of the horizontal errors. For a given lag time, an
autocorrelation of 1 would indicate the error after the lag time is simply a multiple of the
original error. An autocorrelation of 0 would indicate that there is no (linear) relationship
between the errors, or that the errors are not correlated. Generally, for “small” lag times,
we might expect an autocorrelation between 0 and 1 indicating some degree of correlation.
With a Garmin 12XL and a Garmin III+, the shorter time constant was of the order of 1
minute, while the other was much longer--as large as roughly 1 or 2 hours depending on the
data session. Both error components contributed significantly to the position-averaging
RMS error as indicated by the magnitudes of E1 and E2.
The plot below show the measured autocorrelation for 70 minutes of horizontal errors
obtained for the Garmin 12XL twenty-day test.
AUTOCORRELATION OF ERRORS
GARMIN 12XL (Micropulse antenna)
1.0
Legend
Autocorrelation
0.8
Longitude
rmsloncor
Latitude
rmslatcor
0.6
Horizontal
rmshorcor
0.4
0.2
0.0
0
10
20 days data
Fix every 2 seconds
20
30
40
50
Time Lag [minutes]
The plot below magnifies the first 10 minutes of the above figure.
60
70
AUTOCORRELATION OF ERRORS
GARMIN 12XL (Micropulse antenna)
1.0
Legend
Autocorrelation
0.8
rmsloncor
Longitude
Latitude
rmslatcor
0.6
rmshorcor
Horizontal
0.4
0.2
0.0
0.0
1.0
20 days data
Fix every 2 seconds
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0 10.0
Time Lag [minutes]
The autocorrelation of errors for the Garmin 12XL makes a dramatic change (bend) in its
general direction around 1 minute. This is roughly the same as the smaller correlation
length L1 in the modeling of position-averaging horizontal error. This equation to model
RMS error when averaging will be called "Model 2". At present, it has not been possible to
this error component with known error sources such as receiver hardware or algorithm,
multipath, satellite geometry or GPS satellite constellation status.
The figure below shows that the Eagle Explorer autocorrelation of errors behaves
differently.
AUTOCORRELATION OF ERRORS
EAGLE EXPLORER
1.00
Legend
0.80
Autocorrelation
Longitude
b3
Latitude
a3
0.60
Horizontal
c3
0.40
0.20
0.00
0
10
4 days data
Fix every 2 seconds
20
30
40
50
60
70
80
Lag [minutes]
The Eagle Explorer autocorrelation decreases with slight upward concavity until long-term
errors dominate causing it to go nearly flat near zero or negative due to random error in the
measurement. This would seem to agree with the very small correlation length L1 and
small error coefficient E1 in the modeling of the position-averaging horizontal error of this
receiver having only a very small effect; thus the other component (with much longer
correlation length L2 and much larger magnitude E2) appears to dominate the
autocorrelation throughout and no "bend" in the autocorrelation curve is perceived.
Finally, the figure below shows the autocorrelation of vertical errors for the Garmin eMap
and Eagle Explorer.
AUTOCORRELATION OF VERTICAL ERRORS
OF TWO GPS RECEIVERS
Autocorrelation of Vertical Error
1.0
Legend
0.8
eecorrExplorer
Eagle
empcorreMap
Garmin
0.6
0.4
0.2
0.0
0
10
20
30
40
50
60
70
48 hour session
Lag [minutes]
Fix every 2 seconds
Garmin eMap using GA-27C antenna
Note that the "bend" in the Garmin eMap autocorrelation plot around 5 minutes roughly
corresponds to the value of L1 of about 6.6 minutes in the vertical averaging portion of the
section on position-averaging. The "bend" in the Eagle Explorer autocorrelation plot
corresponding to the value of L1 of about 11.1 minutes for it in that section is harder to see.
In conclusion, the modeling in the position-averaging section and the approximate time
constants used in them appear to be confirmed by autocorrelation of the errors.
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 (regardless of HDOP). In probability
terminology, HDOP is an additional variable that allows one to replace the overall accuracy
estimates with conditional 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 is less than 4" then the (conditional)
probability of getting a "2" is 1/3. Knowing HDOP is somewhat similar to knowing "the
number is less than 4" in the analogy.
The notation "RMS_Error(HDOP)" 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(HDOP) increases as HDOP
increases, as higher values of HDOP 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 of
the HDOP was plotted against the RMS error. These measured data points are indicated in
red in the following plot:
HORIZONTAL RMS ERROR AS A FUNCTION OF HDOP
Garmin 12XL (Micropulse antenna)
RMS_ERROR (HDOP) [meters]
10
9
8
7
6
5
4
3
2
1.00
1.50
2.00
2.50
3.00
Horizontal Dilution of Precision (HDOP)
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:
a
f
a
f
RMS _ Error HDOP = HDOP ⋅ RMS _ Error HDOP = 1
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:
RMS _ Error ( HDOP ) =
( AHDOP )
2
+ B2
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 HDOP 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 HDOP values between 0.9 and 2.3 is excellent. Outside that range of
HDOP values, there were significantly fewer data points and the measurement of RMS
errors for those HDOP values is thus less accurate.
HORIZONTAL RMS ERROR AS A FUNCTION OF HDOP
Garmin eMap (GA-27C antenna)
RMS_Error (HDOP) [meters]
10
9
8
7
6
5
4
Fewer than 10000 samples
with these HDOP values.
3
2
1.00
1.50
2.00
2.50
3.00
HDOP
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(HDOP), we obtain a (conditional)
Rayleigh error probability distribution given the HDOP:
Probability ( Error ≤ Distance | HDOP )
2
− Distance / RMS _ Error ( HDOP ) )
=1− e (
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.
NUMBER OF SATELLITES VS. RMS_HDOP
Garmin 12XL (Micropulse antenna)
4.00
Legend
3.00
RMS_HDOP
pred.
PP
Small sample size
meas.
hh
2.00
1.00
0.00
4
5
6
6 days data
Fix every 2 seconds
7
8
9
10
11
12
Number of Satellites
The curve-fit is that given by:
RMS _ HDOP =
C
( Number _ of _ satellites )
2
+D
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.
Horizontal Dilution of Precision (HDOP)
NUMBER OF SATELLITES VS. RMS_HDOP
Garmin eMap (GA-27C antenna)
5
Note: The fix from 3 satellites
uses a different algorithm than
that for more than 4 satellites.
4
Legend
Y
Pred.
rmshdop
Meas.
3
2
1
0
3
4
5
31 days data
Fix every 2 seconds
6
7
8
9
10
11
12
Number of Satellites
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 HDOP values) 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 higher number of satellites in view values have at times been
observed in the past at times with other receivers and antennas.
HISTOGRAM OF HDOP
Observed with Garmin eMap (GA-27C antenna)
14.0
12.0
Mean HDOP 1.50
RMS HDOP 1.65
Percent of Fixes
10.0
8.0
6.0
About 1.1% of the data
had HDOP > 2.9
4.0
2.0
0.0
1.00
1.50
2.00
2.50
Horizontal Dilution of Precision (HDOP)
31 days of data
Fix every 2 seconds
1338681 fixes
3.00
HISTOGRAM OF NUMBER OF SATELLITES IN VIEW
Observed with a Garmin eMap (GA-27C antenna)
35.0
Percent of Fixes
30.0
Mean Number of Satellites 6.46
25.0
20.0
15.0
10.0
5.0
0.0
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0
31 days of data
Fix every 2 seconds
1338682 fixes
Number of Satellites
AVERAGING HORIZONTAL POSITION
WITH WEIGHTING BY HDOP
Rather than simple averaging to improve the accuracy of a position measurement, one
might consider weighting each measurement by the dilution of precision (DOP) to even
better improve the accuracy. This section explores averaging with weighting using the
horizontal dilution of precision (HDOP) to attempt to improve the horizontal (latitude and
longitude) accuracy when measuring a position. (If one finds the below too technical to
read, I suggest going to the conclusion at the end.)
We start by assuming the distribution of latitude and longitude measurements are normally
(Gaussian) distributed and unbiased (with mean being the true value). Although
measurements close in time are correlated, as equal weighting would be applied to these in
what follows, the results from assuming the measurements are independent generally can
be applied.
It is easily derived that the “maximum likelihood estimator” for measurements of the mean
from unbiased Gaussian distributions having the same mean but possible different standard
deviations σi is given by:
n
xˆ = ∑ λi xi
i =1
where:
1
λi =
σ i2
n
1
∑σ
j =1
2
j
(This is different from the simple average in which all the λi are 1/n.) The above formulas
would both be applied to separately calculate weighted averages for the latitude and
longitude.
If we assume that σi is proportional to HDOPi (that is, σi = HDOP⋅σ1), a little simple
algebra will give:
1
HDOPi 2
λi = n
1
∑
2
j =1 HDOPj
The predicted ratio of the RMS of horizontal error from this weighted average to the RMS
error of horizontal error from simple averaging is then:
1
n
RMS ( xˆ )
RMS ( x )
=
1
∑
2
i =1 HDOPi
n
∑ HDOP
2
j
j =1
n
When the HDOP distribution obtained from 6 days using a Garmin 12XL were inserted
into the above formula, the result obtained was 0.89. In other words, under the
assumptions, one should obtain an 11% reduction in RMS error of latitude and longitude
from averaging if one uses the above HDOP weighted average rather than the simple
average. However, in looking at actual data, the author has failed to see evidence of this
predicted small improvement by using the weighting described above.
As the relationship between HDOP and error is only approximate, inaccuracies due to that
modeling might be the cause of the failure to observe the expected improvement in
accuracy by weighting using HDOP in the above effort. One is then tempted to try using
the apparently better model, described elsewhere in this work, relating HDOP and error
based on measured Garmin 12XL data:
In this case, the desired weighting of coordinates would be given by:
and the predicted ratio of RMS of horizontal error from this new weighted average to the
RMS error of horizontal error from simple averaging is given by:
Using the 6 days of Garmin12XL data again, one obtains a result of 0.96. In other words,
under the assumptions, one should obtain a 4% reduction in RMS error of latitude and
longitude from averaging if one uses the new HDOP weighted average rather than the
simple average. Although this is smaller than the earlier 11%, it is believed to more
accurately reflect the possible improvement in horizontal accuracy by weighting the
average. It is a very small improvement and probably too small to detect.
In conclusion, evidence suggests that in general, weighting horizontal position
measurements using HDOP in averaging to improve accuracy is of minimal value. The
exception is probably in the case where a few values are being averaged with very different
HDOP, rather that the distribution of HDOP usually seen in continuously recording GPS
data. Finally, some people eliminate fixes with large HDOP when simple averaging. A
better approach in the same vein for large samples would be to eliminate fixes with any
coordinate being an outlier (say perhaps more than 3 standard deviations from the sample
mean).
AVERAGING HORIZONTAL POSITION
WITH WEIGHTING BY HDOP
Rather than simple averaging to improve the accuracy of a position measurement, one
might consider weighting each measurement by the dilution of precision (DOP) to even
better improve the accuracy. This section explores averaging with weighting using the
horizontal dilution of precision (HDOP) to attempt to improve the horizontal (latitude and
longitude) accuracy when measuring a position. (If one finds the below too technical to
read, I suggest going to the conclusion at the end.)
We start by assuming the distribution of latitude and longitude measurements are normally
(Gaussian) distributed and unbiased (with mean being the true value). Although
measurements close in time are correlated, as equal weighting would be applied to these in
what follows, the results from assuming the measurements are independent generally can
be applied.
It is easily derived that the “maximum likelihood estimator” for measurements of the mean
from unbiased Gaussian distributions having the same mean but possible different standard
deviations σi is given by:
n
xˆ = ∑ λi xi
i =1
where:
1
λi =
σ i2
n
1
∑σ
j =1
2
j
(This is different from the simple average in which all the λi are 1/n.) The above formulas
would both be applied to separately calculate weighted averages for the latitude and
longitude.
If we assume that σi is proportional to HDOPi (that is, σi = HDOP⋅σ1), a little simple
algebra will give:
1
HDOPi 2
λi = n
1
∑
2
j =1 HDOPj
The predicted ratio of the RMS of horizontal error from this weighted average to the RMS
error of horizontal error from simple averaging is then:
1
n
RMS ( xˆ )
RMS ( x )
=
1
∑ HDOP
i =1
2
i
n
∑ HDOP
2
j
j =1
n
When the HDOP distribution obtained from 6 days using a Garmin 12XL were inserted
into the above formula, the result obtained was 0.89. In other words, under the
assumptions, one should obtain an 11% reduction in RMS error of latitude and longitude
from averaging if one uses the above HDOP weighted average rather than the simple
average. However, in looking at actual data, the author has failed to see evidence of this
predicted small improvement by using the weighting described above.
As the relationship between HDOP and error is only approximate, inaccuracies due to that
modeling might be the cause of the failure to observe the expected improvement in
accuracy by weighting using HDOP in the above effort. One is then tempted to try using
the apparently better model, described elsewhere in this work, relating HDOP and error
based on measured Garmin 12XL data:
In this case, the desired weighting of coordinates would be given by:
and the predicted ratio of RMS of horizontal error from this new weighted average to the
RMS error of horizontal error from simple averaging is given by:
Using the 6 days of Garmin12XL data again, one obtains a result of 0.96. In other words,
under the assumptions, one should obtain a 4% reduction in RMS error of latitude and
longitude from averaging if one uses the new HDOP weighted average rather than the
simple average. Although this is smaller than the earlier 11%, it is believed to more
accurately reflect the possible improvement in horizontal accuracy by weighting the
average. It is a very small improvement and probably too small to detect.
In conclusion, evidence suggests that in general, weighting horizontal position
measurements using HDOP in averaging to improve accuracy is of minimal value. The
exception is probably in the case where a few values are being averaged with very different
HDOP, rather that the distribution of HDOP usually seen in continuously recording GPS
data. Finally, some people eliminate fixes with large HDOP when simple averaging. A
better approach in the same vein for large samples would be to eliminate fixes with any
coordinate being an outlier (say perhaps more than 3 standard deviations from the sample
mean).
A COMPARISON
OF DIFFERENTIAL AND NON-DIFFERENTIAL GPS
HORIZONTAL ACCURACY
The plot below shows several DPGS and non-DGPS configurations; only horizontal errors
are considered here. There were no thunderstorms (which can make receiving the beacon
difficult) in the area during the tests. A few non-differentially corrected fixes were
discarded. The Garmin 12XL curves are "jagged" due to the 0.001-minute latitude and
longitude resolution; all other receivers in the plot had 0.0001-minute latitude and longitude
resolution. For both the Oncore and Garmin 12XL, two days with DGPS corrections were
interleaved with two days without DGPS corrections to attempt to partially remove any
time-varying other errors. The same external GPS antenna (Micropulse) was used for the
Oncore and Garmin 12XL tests. The USCG differential corrections were provided by a
Lowrance receiver using an 8-foot whip monitoring the USCG beacon station at Driver,
Virginia that is 105 US miles (169 km) away. The Garmin 12XL is a consumer-grade12channel handheld receiver. The Motorola Oncore VP is a discontinued OEM 8-channel
board capable of providing pseudo-range and carrier-phase information for post-processing
giving survey-grade accuracy. Four days of Starlink Invicta data was collected in late 1999
using the Driver beacon for differential corrections. The Starlink Invicta is an integrated
DGPS system with an H-field beacon receiver and GPS antenna in a surprisingly small unit
feeding a control/interface box that provides RS-232 output. In regard to the Invicta,
Starlink states its "DGPS receivers are designed for professional users where performance
is more important than cost. Our engineers have worked very hard to make the receiver
provide the best possible performance." The Omnistar was an Omnistar 7000 with five
days of data collected in 1997; newer Omnistar receivers/data may perform better.
Omnistar provides subscription C-band differential corrections using a combined antenna
(for both GPS and their C-band service) and receiver. Omnistar has the advantage of
providing differentially corrected fixes at locations not serviced by USCG (or other agency)
low-frequency beacon differential corrections.
0.70
0.60
0.50
0.40
95%
Legend
50%
DIFFERENTIAL
1.00
0.90
0.80
0.30
Starlink
Invicta
cmlinv
cmlom
Omnistar
Oncore
cmlvpd VP
cmlg12d 12XL
Garmin
cmlvp VP
Oncore
0.20
Garmin
cmlg12 12XL
0.10
0.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
NONDIFF.
Probability(Error < Distance)
DISTRIBUTION OF DGPS AND NON-DGPS
HORIZONTAL ERRORS
Distance [meters]
The table below shows some error statistics for the data used in the above plot. Distances
are in meters. Mean error is the arithmetic mean or average error.
Receiver/mode
Garmin 12XL/non-diff.
Garmin 12XL/diff.
Motorola Oncore VP/non-diff.
Motorola Oncore VP/diff.
Omnistar/diff.
Starlink Invicta/diff.
RMS
error
5.49
4.48
4.87
5.01
3.56
1.51
Mean
error
4.45
3.44
4.11
3.40
2.51
1.05
CEP (50%)
error
3.96
2.60
3.57
2.40
1.91
0.81
95%
error
9.62
8.65
9.02
9.30
5.91
2.62
The analysis presented here is not meant to be definitive; however, some tentative
conclusions can be derived. At the approximately 100 mile distance from the USCG
differential corrections beacon site, no significant benefit by using differential corrections is
noted in the 95% error distance level (the distance which will includes 95% of the error)
with either a Garmin 12XL or a Motorola Oncore. (For smaller baselines, that is, shorter
distance between the GPS and the differential correction site, the differential corrections
certainly should have a greater impact in reducing the error.) At the 50% error distance
level (CEP), when using the Garmin 12XL or Motorola Oncore VP, some benefit with
differential is noted; however, the amount of benefit is probably of little practical value.
For all practical purposes, the performance of the Garmin 12XL and Motorola Oncore VP
were essentially the same in these tests. The Omnistar unit gave about half the CEP error
of the non-differentially corrected Garmin 12XL and Motorola Oncore VP and also gave a
better 95% error distance (about 6 meters versus about 9 meters). Finally, the Starlink
Invicta gave the best performance-even though it used the same distant differential
correction site as the Garmin 12XL and Motorola Oncore VP differential tests. It is clear
that there are different levels of accuracy obtainable by using differential beacons. As
higher-end receivers that perform better depend on proprietary methods, it is difficult to
further analyze the reasons for the differences in performance.
In conclusion, DGPS should better improve accuracy when using consumer-grade
GPS receivers on shorter baselines than the long one tested here. With higher-end
survey grade GPS/DGPS receivers, DGPS gives very good accuracy even on long
baselines. Note that caution should be used in comparing the above numbers to
manufacturers' specifications. In some circles, manufacturers do not use the
mathematical definition of RMS error but instead use the error probability (63%)
that would correspond to RMS error if the distribution were exactly Rayleigh. In
those circles that number has become a reference value for comparison that has some
benefits over the mathematically defined RMS error since the error distribution may
not be exactly Rayleigh. Do not forget that the error depends on the latitude, due to
its influence on HDOP. The test position is near latitude 38 degrees and the
horizontal error is believed to perhaps be at maximum at something a little over 40
degrees latitude. Finally, the accuracy of the source of the DGPS corrections will
affect the accuracy of those using it. All these factors can cause differences in
measured error statistics.
Addendum
Differential and non-differential data was later collected using an Eagle Explorer. The plot
and table below summarize the measured errors.
Probability (Error < Distance)
EAGLE EXPLORER
DIFFERENTIAL AND NON-DIFFERENTIAL
1.00
0.90
0.80
Differential
Non-differential
0.70
0.60
0.50
0.40
Error
Non-diff. Diff.
RMS
4.78
4.49
Mean
4.04
3.68
CEP (50%)
3.49
3.19
95%
8.94
8.27
Units: meters
0.30
0.20
0.10
0.00
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
Distance [meters]
As these measurements were at a later time, extreme caution should be applied in
comparing these Eagle Explorer errors to the Garmin 12XL errors; however, they are of the
same magnitudes. Again, on the long baseline, there was little difference between
differential and non-differential.
COMPARISON OF HORIZONTAL ERRORS
WITH AND WITHOUT SELECTIVE AVAILABILITY
When SA was turned off in May 2000, GPS measurement of position significantly
improved. SA, or Selective Availability, was the intentional degradation of the SPS, or
Standard Positioning Service, navigation data in order to deprive a military adversary of
real-time more precise positioning.
The plot below compares a day with and without SA using the same GPS receiver.
Actually such a plot as the above is misleading as one cannot see the point plotted on top of
each other. The plot and embedded table below show that the difference is much greater
than the above plot might make one think.
HORIZONTAL ERROR DISTRIBUTION
WITH & WITHOUT SA
Typical day - Garmin 12XL
Probability (Error < Distance)
1.00
0.90
SA "Off"
0.80
0.70
0.60
SA "On"
0.50
0.40
0.30
0.20
0.10
0.00
Errors [m] for these two days:
SA "On"
SA "off"
RMS
29.4
5.5
Mean
25.1
4.5
CEP (50%) 22.2
4.0
95%
54.9
10.1
0
10
20
24 hours w/wo SA
Measurement every 2 sec.
30
40
50
60
Distance [m]
ASSORTED MATHEMATICS
The following are an assortment of equations and derivation for the more mathematically
advanced and interested reader:
Mean or expectation:
RMS (Root Mean Squared):
70
80
Standard deviation:
Variance:
Covariance:
Correlation:
Bivariate normal distribution density function:
Density function for bivariate normal distribution with zero correlation and equal variances:
Probability that the error is less than D using the above (the result is Weibull with shape
parameter β=2 or Rayleigh distribution):
Inverse formula:
RMS Error (RMS_R):
Median (50% error or CEP):
95% Error:
Mean (average) Error:
Mean absolute error using normal distribution (altitude):
Sketch of weighting by HDOP analysis:
∂ ln L n xi − µ
=∑
2
∂µ
i =1 σ i
Setting:
∂ ln L
=0
∂µ
and solving for µ gives the following weighted estimator for µ:
n
xˆ =
xi
∑σ
i =1
n
1
∑σ
j =1
2
i
2
j