Wireless Sensor Network, 2011, 3, 334-340
doi:10.4236/wsn.2011.310037 Published Online October 2011 (http://www.SciRP.org/journal/wsn)
3D DOPs for Positioning Applications Using Range
Measurements
Binghao Li1, Andrew G. Dempster1, Jian Wang2
1School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, Australia
2School of Environment and Spatial Informatics, China University of Mining & Technology, Xuzhou, China
Received August 22, 2011; revised September 20, 2011; accepted September 30, 2011
Abstract
For terrestrial positioning, some applications require three dimensional coordinates. The Dilution of preci-
sions (DOPs) for position systems using range measurement is reviewed and the average values of DOPs for
different deployments of base station geometries are examined. It is shown that to obtain the lowest DOPs,
the base stations for different types of positioning systems need to be deployed differently. Changing the
N-sided regular polygon to an (N – 1)-sided polygon with one base station in the centre of the polygon can
decrease the value of DOP in general for a pseudorange time of arrival (TOA) system but not for an absolute
range TOA system. The height of the base station in the centre can also change the DOP significantly. The
finding can be used to optimize the deployment of the base stations for range measurement positioning sys-
tems.
Keywords: Positioning, 3D DOP, TOA, TDOA
1. Introduction
Using range measurements for positioning is the method
most popular in satellite navigation systems. The typical
example is the Global Positioning System (GPS) which
uses the Time of Arrival (TOA) to estimate the receiver’s
position [1]. Similar examples can be found in terrestrial
positioning systems [2-5]. Since, in these systems, the
clock error in the receiver is unknown, pseudorange
rather than the absolute range is measured. If the trans-
mitters and receiver are well synchronized or a round trip
time (RTT) can be obtained, the absolute range meas-
urement can be used for positioning [6]. To distinguish
these two technologies, the former is called pseudorange
processing and the latter is named absolute range proc-
essing in this paper. Time Difference of Arrival (TDOA)
is also applied in many positioning systems such as the
widely used Loran-C and Omega [7], primarily used be-
fore the satellite positioning era, and the global cellular
network [8]. A TDOA system is also known as a hyper-
bolic multilateration system as it relies on the fact that all
the points where the difference in the TOA radio signals
from different stations is constant, its “line of position”,
forms a hyperbola.
No matter what range measurement is used, to obtain
an accurate user position requires a good geometrical
distribution of the base stations or beacons. Dilution of
precision (DOP) is used to describe the effect of geome-
try on the relationship between measurement error and
position determination error. DOP has been well investi-
gated for Global Navigation Satellite Systems (GNSS) [9]
[10-12]. However, originally it was discussed in hyper-
bolic multilateration systems [13,14]. In terrestrial posi-
tioning systems, the 2D coordinate can be of more of
interest [15,16]. But in some applications such as ma-
chine guidance, warehouse management and emergency
services, 3D coordinates are often required. As an exam-
ple, fire fighters enter a building that they are not famil-
iar with. In an environment full of smoke, with poor
visibility and unknown dangers, 3D position is needed to
navigate to the correct floor. Base stations can be in-
stalled on the fire engines and deployed quickly around
the building in fire to facilitate this requirement [3].
Several potential positioning systems for the emer-
gency applications have been reported, such as a mobile
wireless localisation network known as WASP (Wireless
Ad-hoc Self Positioning) which can achieve better than
half-metre accuracy in non-line-of-sight (NLOS) envi-
ronments. The Worcester Polytechnic Institute (WPI)
Precision Personnel Locator (PPL) project demonstrated

B. H. LI ET AL.335
better than 1m accuracy in high multipath environments
[3]; and a Ultra Wide Band (UWB) system was reported
to have achieved 15cm location accuracy in an open en-
vironment [17].
How to deploy the base stations to achieve a better
DOP for the area of interest must be investigated as the
DOP value has a great impact on the positioning accu-
racy. In the following sections, the theoretical issues of
DOP are reviewed first in Section 2, and then the de-
ployment of the base stations is discussed in Section 3.
Finally, the conclusion is given.
2. Dilution of Precession
In 1975 Lee published the first paper [14] to discuss
Geometric DOP (GDOP) of hyperbolic multilateration
systems. GDOP is defined as the ratio of the root mean
square (rms) position error to the rms ranging error. Al-
though only TDOA was discussed, the fundamental work
can be easily applied to absolute range or pseudorange
systems. Similarly, Position DOP (PDOP), Horizontal
DOP, Vertical DOP (VDOP) can be defined.
A range measurement can be expressed as:

,,
L
fxyz
where L is a measured value, and x, y and z are unknown
parameters (the coordinate of the user). To linearize the
equation, Taylor’s theorem is normally applied




00
000
00
00
dd
(, ,)1! 1!
d
d
1! !
dd
!!
nnn
nnnn nn
n
L
xx Lyy
Lfxyz
Lx x
Lz z
n
Ly yLxz z
R
nn
 




 



(1)
Hence



000
00
,,,,
dd
fxyz fxyz
0
d
L
xx Lyy Lz

  z
r
(2)
Assume there are n observations, matrix notation can
be used:
xH (3)
where Δx is the vector of offset of the true position of the
user from the linearization point, Δr is the vector offset
of the true range to the range values corresponding to the
linearization point, and H can be present as



11
00
22
00
00
nn
Lx Ly
Lx Ly
Lx Ly
  

 



 






H
2.1. Absolute Range Processing
If absolute range measurements are available (the user is
synchronized with the base stations or the round trip of
the signal is available), the range from the user to the ith
base station is:
 
22
ii i
2
i
R
xx yy zz (5)
where (xi, yi, zi) is the coordinate of ith base station.
There are n measurements

1
T
n

R
rr (6)
The vector of unknowns is

T
xyzX

(7)
Assuming the errors in the measurements are random,
independent, have zero mean and have an identical rms
σr, the error covariance matrix is
2
r
I
 Q (8)
The H matrix in Equation (4) can be expressed as
11
111
nn
nnn
xx yy zz
RRR
xx yy zz
RRR

1
n







 









H (9)
When least squares is used (n ≥ 3)

1
TT
xH Hr


H

d
(10)
It can also be expressed as



1
dd
TT
TL
xx xHHr

 
TL
Hrr (11)
where xT is the error-free position, xL is the position de-
fined as the linearization point and dx is the position er-
ror. rT represents the vector of true range values, rL is the
vector of range values computed at the linearization
point and dr is the range measurement error. Hence

1
d
TT
xH Hr

H (12)
DOP is defined as
DOP
x
r


 (13)
since




 




11
11
12
covdd d
dd
covd
T
TTTT
TT T
T
r
xExx
EHHrr H
HH rH
H 













HHH
HH
H
H
(14)

(4)

B. H. LI ET AL. 336
where

2
covdr rI






22
11
1,1 2,2
HDOP xy TT
r
HH




  HH
(15)


21
3,3
VDOP zT
r
H



 H (16)






222
11
1,12,23,3
PDOP
xyz
r
TT T
HH H





HH H
1
(17)



1
GDOP T trace H H (18)
In this case, PDOP is the same as GDOP. Obviously,
when range measurements are used, DOP is unitless,
whereas in angle of arrival, there are units of meters [18].
2.2. Pseudo r ange Pr oc e ss ing
When pseudorange measurements are used, there is one
more unknown—the receiver clock error. Hence the
range from the user to the ith base station is
 
222
ii ii
R
xx yy zzt (19)
Similarly to the case for absolute range, there are n
measurements but the vector of unknowns is slightly
different

T
xyz tX

 (20)
Then the H matrix can be expressed as
111
111
1
1
nnn
nnn
xxyy zz
RRR
xx yy zz
RRR




 




H








 (21)
All the DOPs are similar except the GDOP since there
is one more diagonal element in (HTH)−1 which is TDOP


21
4,4
TDOP tT
r
H



 H
2.3. TDOA
The TDOA measurement can be presented as


222
111
222
2,3,
i
iii
Rxxyyzz
xxyyzz i
 

(22)
The measurements are

2
T
n

R
rr (23)
The vector of unknowns is the same as Equation (7),
but the H matrix is different
12 1 21
12 1212
111
11 1
nn
nn
xx xxyyyyzz zz
RRRRRR
xxyy zz xx yyzz
RRR RRR
  
2
n
n








 


  







 H
(24)
And the error covariance matrix is no longer an iden-
tity matrix
2
21 1
12 1
11 1 2
r










Q

(25)
The DOPs are expressed in the same way as equations
(15) to (18). GDOP and PDOP are identical.
3. 3D DOPS Associated with Different
Deployments of Base Stations
In real applications, there are many restrictions on the
deployment of the base stations. To discuss the 3D DOPs,
the scenarios are simplified as follows:
The projection of the base stations on the x-y plane
forms an N-sided regular polygon where the vertices are
the locations of the projection of the base stations. The
radius of the circumscribed circle is a (see Figure 1).
The heights of the base stations can be configured to set
up different scenarios. To make a fair comparison of the
scenarios with different numbers of base stations, the
area of interest is that inside the circumscribed circle on
the x-y plane (the shadowed area in Figure 1). The aver-
age DOP in this area is calculated. The square area out-
side the circumscribed circle is used to show the mesh
plot of GDOP results. It is reported in [19] that at the
centre of a regular polygon the lowest “GDOP”, more
precisely PDOP (note the GDOP in pseudorange proc-
essing includes TDOP element but otherwise is the same
as PDOP), can be achieve in 2D scenarios (2N)
when absolute range and pseudorange are used for posi-
tioning. In [13], it is shown that for TDOA systems the
minimum GDOP is 2N and 3N (N is the num

B. H. LI ET AL.337
Figure 1. Six-sided regular pol ygon and the area of interest
to calculate the DOPs.
ber of transmitters) in 2D and 3D scenarios respectively.
Shin and Sung [20] prove that for TDOA and TOA
processing
TDOA TOA
PDOP PDOP
and the equality also holds for HDOP and VDOP. This
conclusion is consistent with the results in [13,14,19]. It
has also been confirmed by calculating the average
DOPs for different scenarios. Hence further comparisons
are only for absolute range processing and pseudorange
processing.
Obviously, if the base stations are all in the same plane
as that of the user, e.g. the x-y plane, DOPs are infinite.
The height of the base stations should be different from
the user. If the maximum height of the base stations is
restricted by the application to 0.1 times of a (for exam-
ple, the base stations on a fire engine cannot be lifted
more than 5 m, say, but the distance between two fire
engines can be easily more than 50 m). Table 1 shows
the average DOPs with different combinations of the
base station height when N is 3 and 4 using absolute
range. The digits separated by semicolons in the pair of
square brackets are the height (z value) of the base sta-
tions. To make the calculation meaningful, when the
value of GDOP at a specific location is too large or the
DOP could not be calculated, a value of 1000 is assigned.
Similarly, the results of four-sided and five-sided regular
polygons for TOA pseudorange processing are presented
in Table 2. For TOA pseudorange processing, at least
four measurements are required for 3D positioning, so no
result for a three-sided polygon is reported. Figures 2
and 3 give four examples of the GDOPs with pseudo-
range processing and absolute range processing. One can
see several interesting things from these results. Firstly
as expected, the more transmitters, the better (smaller)
Table 1. The average DOPs for three-sided and four-sided
regular polygons with different combinations of base station
heights using absolute range measurements (five-sided regu-
lar polygon with heights = [5;5;5;5;5] is also listed).
Heights GDOP PDOP HDOP VDOP
[0;5;0] 44.3 44.3 9.8 44.2
[0;5;5] 14.5 14.5 2.0 14.4
[5;5;5] 6.1 6.1 1.4 6.0
[0; 5; 0; 5] 7.7 7.7 1.1 7.6
[0; 0; 5; 5] 28.8 28.8 8.6 28.7
[0;0;0;5] 16.5 16.5 1.3 16.5
[0;5;5;5] 7.0 7.0 1.2 6.8
[0;1.7;3.3;5] 11.1 11.1 1.3 11.0
[5;5;5;5] 4.5 4.5 1.1 4.3
[5;5;5;5;5] 3.7 3.7 0.9 3.6
Table 2. The average DOPs for four-sided and five-sided
polygon with different combinations of base station heights
using pseudorange measurement.
Heights GDOPPDOP HDOP VDOPTDOP
[0; 5; 0; 5] 11.8 11.8 1.6 11.6 1.2
[0; 0; 5; 5] 401.1400.6 217.5 399.6211.9
[0;0;0;5] 23.9 23.9 2.1 23.8 1.3
[0;5;5;5] 34.0 33.7 7.8 33.2 7.7
[0;1.7;3.3;5] 47.5 47.2 8.6 46.9 8.3
[5;5;5;5] 210.0208.8 92.1 208.3 99.6
[0; 5; 0; 5; 5] 10.4 10.3 1.5 10.2 1.2
[0; 5; 0; 5; 0] 10.8 10.8 1.4 10.7 0.9
[0;0;5;5;0] 26.5 26.4 2.7 26.2 1.8
[0;0;5;5;5] 31.4 31.2 4.0 30.9 3.3
[0;0;0;0;5] 17.8 17.8 1.5 17.7 0.8
[5;5;5;5;0] 16.9 16.7 2.3 16.5 2.3
[0;1.25;2.5;3.75;5]25.8 25.7 2.4 25.5 2.3
[5;5;5;5;5] 113.9113.2 31.4 112.8 36.9
average DOPs can be achieved. Secondly, with similar
configurations (N − 1 base stations with absolute range
processing and N base stations with pseudorange proc-
essing are regarded a fair comparison pair), absolute
range processing tends to provide better DOPs. Thirdly,
in absolute range processing, the lowest GDOP and
VDOP can be achieved (HDOP is slightly worse than the
lowest value) when all the base stations have the same

B. H. LI ET AL. 338
Figure 2. GDOP of Pseudor ange proces sing when N = 4, height = [0;5;0;5] (top) and [5;5;5;5] (bottom).
Figure 3 . GDOP of absolute range processing w hen N = 3, heights = [0;5;0] (top) and [5;5;5] (bottom).
maximum height (5 m), while in pseudorange processing
the base stations should be deployed with minimum and
maximum height alternately (e.g. [0;5;0;5] when N is 4).
If absolute range processing is possible, clearly it is
much better to be utilized. Unfortunately, neither syn-
chronization nor measuring the RTT is a simple task.
Finally, VDOP is significantly worse than HDOP. For
instance, the best VDOP in the case of four base stations
with pseudorange processing, VDOP is 11.6 - 7 times
worse than HDOP (1.6). This is expected because of the
reduced vertical diversity in base station position. It
suggests that for a range measurement error is only 25
cm (and, considering the poorly behaved indoor envi-
ronment, 25 cm error is very small) the minimum 2D
error is as small as 40 cm which is accurate enough for
most applications, but the error in height is 2.9 m, which
may locate a fire fighter on the wrong floor of the build-
ing.
As the base stations are all deployed on the border of
the area of interest, it means the centre of the polygon
has a bad DOP. Intuitively, moving one of the base sta-
tions to inside the polygon may change the situation. In
fact, for pseudorange processing, a three-sided regular
polygon with one base station located at 5 m (0.1 times
of a) above the centre improves the DOPs (compared
with the four-sided regular polygon). The new GDOP,
PDOP, HDOP, VDOP and TDOP are 9.3, 9.3, 1.7, 9.1
and 0.9 respectively (refer to Tables 2 and 3). All the
DOPs except HDOP are improved (although not signifi-
cantly). When N = 4 (plus one base station in the centre),
a similar result can be found. However, in the case of
absolute range processing, the DOPs are not improved
(see Tables 1 and 3). As pseudorange processing is
widely used, the new deployment is of interest. The extra
base station can be deployed anywhere inside the poly-
gon. Figure 4 shows clearly that the location above the
centre of the polygon generates the best average DOPs.
This figure was generated by setting a four-sided regular
polygon with a base station which is projected inside the
polygon. The height of the transmitter was set to 2a.
When this transmitter located differently, different DOPs
can be obtained. The further the transmitter moves from
the centre, the worse the average GDOP can be calcu-
ated. l

B. H. LI ET AL. 339
Table 3. The average DOPs when a transmitter is locate d in the centre of the polyg on.
Heights GDOP PDOP HDOP VDOP TDOP
[0;0;0] + 5 9.3 9.3 1.7 9.1 0.9
Pseudo range
[0;0;0;0] + 5 8.6 8.6 1.3 8.4 0.7
[5;5;5] + 5 5.4 5.4 1.3 5.2
Absolute range
[5;5;5;5] + 5 3.9 3.9 1.0 3.8
Figure 4. The average GDOP of different locations of the
base s tation inside a four-sided poly gon.
If the height of the one in the centre changed, the
DOPs may also change. Figu re 5 shows when the height
of the base station in the centre increases, the HDOP and
GDOP decrease dramatically at the beginning (height <
a), then become flat quickly when the height reaches a.
VDOP and TDOP are always the same (presented as a
flat line). Obviously, to achieve a good VDOP, the
height should have a similar magnitude to the range be-
tween the base stations on the x-y plane (e.g. a). This
requirement sometimes is hard to meet. Using the fire
fighter’s example again, it is impossible to place a base
station above field of fire, especially about 50 m high
from the ground. Hence the height estimation of the user
is always significant worse than the 2D coordinates. If a
more accurate height is needed, alternative methods such
as a barometer [21] must be considered. Where alterna-
tives are not possible, the deployment of the base stations
is restricted by the height of the transmitters above the
ground—that is the transmitters on the ground (or close
to the ground) cannot be significantly separated.
4. Concluding Remarks
DOP is an important factor for positining when range or
angle measurement is used. This paper investigates the
DOPs in 3D positioning applications using range meas-
012345678910
0
1
2
3
4
5
6
7
8
9
10
Height of the transmiter in the centre
Aver age DOP
GDOP
HDOP
VDOP
TD OP
Figure 5. The average DOPs of the three-sided regular
polygon w ith one base stati on in the centre (the height unit
is a).
urements. The performance of two types of measure-
ments—absolute range and pseudorange has been dis-
cussed.
It has been found that to achieve the best (lowest) av-
erage GDOP, the deployment of the base stations for
absolute range and pseudorange processing are different.
In the case of the N points located at the vertices of a
N-sided regular polygon, the transmitters should be de-
ployed with minimum and maximum height alternatively
for pseudorange processing and with the same maximum
height (the maximum height can be up to a, if it is larger
than a, a should be chosen) for absolute range processing.
Changing the setup from N-sided regular polygon to (N −
1)-sided polygon plus one station located in the centre
with maximum height decreases the DOPs for pseudo-
range processing.
General speaking, using absolute range processing re-
quires fewer transmitters and can guarantee lower DOPs.
However, to make absolute measurements is not a simple
task. Pseudorange processing is still most widely used in
GNSS. For pseudorange processing, putting a base sta-
tion above the area of interest can lead to better DOPs.
But for some applications, it is impossible to deploy
transmitters in that way, especially because the magni-
tude of the height is significant. Alternative methods must

B. H. LI ET AL. 340
be chosen to avoid the much lower accuracy in height.
When TDOA is used, the conclusions drawn from TOA
pseudorange are also true.
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