Simplified Noise Modeling of GPS Measurements for a Fast and Reliable Cycle Ambiguity Resolution
* Byungwoon Park 1, Changdon Kee 2
1Mechanical and Aerospace Engineering, Seoul Nat’l Univ.(E-mail: [email protected] ) 2 Mechanical and Aerospace Engineering, Seoul Nat’l Univ.(E-mail: [email protected] )
Abstract
The relationship between the observable noise model and the satellite elevation angle can be modeled quite well by an exponential function.[Jin, 1996] Noise size and dependence on the elevation angle are, however, different for each observation and receiver type. Therefore, the coefficient determination of this model is an issue, and various methods including PR-CP, single difference, and time difference have been suggested. The limitations of them are difficulty to model the carrier phase noise and to eliminate bias.
To overcome these disadvantages for using Jin’s model, we suggest zero baseline double difference (DD) and noise sorting algorithm. Data DD technique in zero baseline is useful to eliminate all the troublesome GPS biases, and the remaining error is the sum of GPS measurement noises from two satellites. These DD residuals for hours should be sorted by the combination of satellite elevation angles, and then variance value of the residual for each combination can be estimated. Using these values, we construct an over-determined linear equation whose solution is a set of noise variance for each satellite elevation angle.
With 24hr Trimble 4000ssi data, we easily worked out the coefficients of the noise model not only for pseudorange but also for carrier phase. We estimated the standard deviation of the measurement DD using our model, and plotted 1 and 3 sigma lines for every epoch to verify the representation of the residual error. 63.3% of pseudorange residual and 65.9% of phase error did not exceed the 1 sigma lines. Additionally, 99.2% and 99.5% of them lied within 3sigma line. These figures prove that the Gaussian property of measurement noise, and that the suggested model by our algorithm corresponds to the observable noise information.
Keywords: GNSS, Noise Model, Double Difference, Elevation angle, Ambiguity resolution
1. Introduction
GNSS receiver is a kind of measuring device, so its observable has always certain level of noise. To describe position’s quality and monitor the data’s integrity, stochastic model for the measurement noise is essential. Recently there has been interest in using two types of data, Carrier-to-Noise Ratio (CNR) and satellite elevation as quality indicators for GPS observations. The standard deviation of tracking loop noise estimated by the former indicator is occasional different from that of measurement noise. On the other hand, elevation angle and statistical property in measurement noise are very familiar to user and close to real world, so researchers in many scientific and commercial GPS software packages are interested in it.
These techniques have their own merits and demerits in the reality and accessibility. It is necessary to compare their characteristics and performance, and if possible to propose a new algorithm or improved one. The noise statistics well-modeled by the suggested method is essential for the description of the resulting position’s quality and for monitoring the incoming data’s integrity. The covariance matrix calculated by the stochastic noise model is useful to Kalman filter and the weight matrix in the least square method.
2. Existing Algorithms
All GPS observables produced in a receiver will contain noise.
The noise is caused by some factors external to the receiver, such as those included by the satellite and introduced during the trans- atmospheric signal propagation, and other factors internal to the receiver / antenna system itself, commonly called as a receiver
noise.
There have been several techniques to model the noise, and many standard formats recommend two kinds of parameters, Signal-to-Noise Ratio (SNR) and elevation angle. According to MOPS (Minimum Operational Performance Standards) for GPS/WAAS Airborne Equipment, accuracy of the receiver given by the current signal-to-noise ratio is used for the purpose of defining WAAS protection level. Another standard format, the ICAO (International Civil Aviation Organization) SARPs (Standard and Recommended Practices) recommends that the RMS of the total aircraft receiver contribution to the error for GNSS shall be less than a0 a1 e 0
θ
−θ
+ × (θ is an elevation angle, anda0,a1, and θ0 are defined in SARPs). Generally, Carrier-to-Noise Ratio (CNR) has been selected for the SNR estimation. To find the variation of noise with satellite elevation, several methods have been proposed. It is important to understand these methods better in order that an improved algorithm superior to the existing ones may be suggested.
2.1 Carrier-to-Noise Ratio (CNR)
Receiver noise, which mainly consists of thermal noise, increases as temperature rises due to the excitation of electrons in the device, and so the receiver with a bigger bandwidth suffers from larger noise. Generally, a higher CNR will produce lower noise observables in the presence of white noise, because it would means higher gain and a lower noise figure. [Large 2001]
CNR parameter and receiver bandwidth are, therefore, representatives of GPS measurement quality, and it has been widely used to construct the stochastic model for high-accuracy application. The observation precision in terms of standard
deviation in delay-lock loop (DLL) and phase-lock loop (PLL), which provides the pseudorange and carrier phase measurement, are expressed in equation (1) and (2) [Parkinson, 1996].
2 / 0 L
c c
B
σ = c n λ ………..(1)
2 / 0 2
P P
B c n σ λ
= π ….………(2)
where and σc and σP the standard deviation of pseudorange and carrier phase, BLand BPare the code and carrier tracking loop bandwidth, c n/ 0 the carrier-to-noise density ratio(=
/ 0
10 10 C N
for C N expressed in dB-Hz), / 0 λcand λ are the PRN code and carrier wavelength.
These equations express tracking error due to thermal noise very well, so they are good indicators of the receiver noise. The statistics by the equation (1) and (2), however, are generally different from those in measurement domain, since they do not consider the factors external to the receiver. All the receivers were treated essentially as black boxes which have their own data filters, so the measurement noise property after filtering is far different from that before filtering. Additionally,c n/ 0 does not have consistent characteristics for the same receiver types, and occasional sudden drop in c n/ 0 may misrepresent of the measurement noise statistics [Satirapod, 2000]. Moreover, most commercial receiver would not provide c n/ 0 value, so the GPS solution software developers are not able to apply this equation to their tools.
2.2 PR-CP Method and Elevation-dependent Model
The satellite elevation is a potent influence on the measurement precision for most receivers. Measurement noise increases at a lower elevation angle, and the relationship can be modeled quite well by an exponential function [Jin, 1996]described as equation (3).
2
( )
0 1
( )
El
El x x e x
σ = + ⋅ − …………(3)
, where σ is the noise error standard deviation, x0, x1, and x2 are coefficients dependent on the receiver brand and the observation type, and El is the satellite elevation angle in degrees.
Researchers in many scientific and commercial GPS software packages have been interested in this function of elevation.
The elevation angle, the input of this function, is generally given by a receiver, and the output is the statistical property in a measurement domain. Noise size and dependence on the elevation angle are, however, different for each observation and receiver type. Therefore, the coefficient determination of the equation (3) is an issue, and various methods including the difference between pseudorange and carrier phase (PR-CP), single difference, and time difference have been suggested.
Observable noise on L1 pseudorange can be estimated by equations (4) and (5), assuming that noise in L1 and L2 carrier phase,φ2 and φ1, is negligible compared with pseudorange noise.[Kee, 1996] Using
2 1
2
( L 1.65)
γ =⎛⎜L ⎞⎟ ≈
⎝ ⎠ for ionospheric
delay estimation, the divergence term in equation (4) is canceled, and then only pseudorange noise ερremains by the ambiguity integer property. According to Kee’s paper, carrier phase noise was regarded as 1/1000 of pseudorange noise.
2 1
1 1
( ) 2
ρ φ φ1
ε ρ φ γ
= − − −
− …………(4)
ρ mean(ρ)
ερ =ε − ε …………(5)
Following this algorithm, noise statistics are free from time correlation, and the relationship between the deviation and elevation angle can be easily analyzed. But it is a big defect that this PR-CP method can not be applied to the single frequency receiver because of the unfeasibility to estimate the ionospheric divergence term in equation (4). Moreover, the noise of carrier phase is so small that it is impossible to pick out the carrier noise model from the bundle of pseudorange noise.
2.3 Single Difference Method
The basic GPS measurements consist of biased and noisy estimates of ranges to satellites. The principal source of bias is the unknown receiver clock offset, and the remaining errors are induced by the satellite clocks and ephemeris modeling, atmosphere, multipath and receiver noise.
i i i i i i i
u du Bu b Iu Tu R ρu
ρ = + − + + +δ +ε
………..(5)
i i i i i i i
r dr Br b Ir Tr R ρr
ρ = + − + + +δ +ε
………..(6)
i i i i i i i i
u du Bu b Iu Tu R Nu φu
φ = + − − + +δ + λ ε+
………..(7)
i i i i i i i i
r dr Br b Ir Tr R Nr φr
φ
= + − − + +δ
+λ ε
+………..(8) : code measurement from the i-th satellite to user site
: carrier phase measurement from the j-th satellite to reference site : wavelength of the carrier phase
: integer ambiguity : distance
i u j r
i r
N d ρ ϕ λ
from the reference station to the i-th satellite : satellite clock bias of satellite
: clock bias of receiver : ionospheric delay
: troposheric delay : orbit error
: user pseudorange noise of
i u
b B I T R
ρ
δ
ε i-th satellite
: reference carrier phase noise of j-th satellite
j φr
ε
A method commonly employed to estimate observation noise is the zero baseline test, where two receivers are connected to the same antenna via a splitter. A number of external error sources of the two devices become almost perfectly correlated, so there remains no significant residual effect except for the clock offset between the two receivers after single differencing (SD).
i i
rΔuρ = Δr uB+ Δr uερ
………..(9)
i i i
rΔuφ = Δr uB+ Δr uNλ+ Δr uεφ
………..(10)
rΔ : single difference between reference and user measurements u
The adjustment with these equations implies averaging the m observations to determine the only one unknown-namely, the
differential receiver clock error, rΔuB . In order to computerΔuB , the single-differenced ambiguities should be resolved to their integer values and consequently kept fixed in this adjustment [Tiberius, 2000].
1
ˆ 1 m i
r u r u
i
Bρ m ρ
=
Δ =
∑
Δ……….(11)
1
ˆ 1 ( )
m
i i
r u r u r u
i
B N
φ m φ λ
=
Δ =
∑
Δ − Δ……….(12) , where
Bˆρ and
Bˆφ
are the receiver clock error values estimated by code and carrier phase measurements
1
ˆ 1 m
i i j
r u r u r u r u
j
Bρ ρ m ρ
ρ ε ε
=
Δ − Δ = Δ −
∑
Δ……….(13)
1
ˆ 1 m
i i j
r u r u r u r u
j
Bφ φ m φ
φ ε ε
=
Δ − Δ = Δ −
∑
Δ……….(14) From equations (13) and (14), variance (V) components can be estimated as (15) and (16) if we assume that statistical properties of all the current measurement noises are same to estimate the single channel variance. [Tiberius 2003]
2 [ ˆ ]
1
i i
r u r u
m V B
ρ m ρ
σ = Δρ − Δ
− ……….(15)
2 [ ˆ ]
1
i i
r u r u
m V B
φ m φ
σ = Δφ − Δ
− ……….(16)
The model estimated by this SD algorithm is very close to the real. However, it would not reflect the sensitive variation with the elevation angle, because the noise statistics of the residuals, which were assumed same in the averaging process, are different from each other in a strict sense.
3. Suggested Noise Modeling Algorithms 3.1 Double Difference in Zero Baseline
The remaining receiver clock difference between reference and user in equation (9) and (10) can be excluded by the double- difference (DD) technique, which subtracts again the single- difference measurement of i-th satellite from that of j-th. The zero baseline residual processed by DD can be expressed by equation (17) and (18).
i j i j i i j j
rΔ ∇u ρ= Δ ∇r u ερ =ερu+ερr−ερu−ερr
……..(17)
i j i j i j
r u r u r u
i j i i j j
r u u r u r
N N
φ
φ φ φ φ
φ λ ε
λ ε ε ε ε
Δ ∇ = Δ ∇ + Δ ∇
= Δ ∇ + + − −
………..(18)
i j
rΔ ∇u
: double difference between the two satellites i and j, and receivers r and u
The GPS pseudorange double difference observations are contaminated by several kinds of errors, namely, residual orbital errors, residual ionospheric and tropospheric delays, and system noise. There’s no problem to consider the total sum of the residual errors as a receiver noise. The carrier phase observation, however, is a little different from the code measurement, because of the integer ambiguity. After the DD process, the integer ambiguity (rΔ ∇ui jNλ ), which is biased error, still remains in equation (19). Using the integer property,
we can separate the carrier noise from the DD measurement, and then statistical analysis in equation (20) can be done in the same method as the code’s.
i j i j i i j j
rΔ ∇ = Δ ∇u φ r u Nλ ε+ φu+εφr−εφu−εφr
………..(19)
( )
i j
i j r u i i j j
r u φ roundoff φ λ εφu εφr εφu εφr
λ
Δ ∇ − Δ ∇ = + − −
(20) Generally, the receiver noise is considered as a white noise, so the standard deviation is the representative of their statistical characteristics. If we use the two receivers of the same model, it is proper to assume that the noises of them have the same statistical property. It means that the standard deviations of the noises in user and reference receiver are equal as expressed in equation (21) and (22).
i i i
u r
ρ ρ ρ
σ =σ =σ
………..(21)
i i i
u r
φ φ φ
σ =σ =σ
………..(22)
Therefore, the variance of the pseudorange code and carrier phase residuals in equation (17) and (20) can be summarized as (23) and (24) if the measurement precision of each satellite does not vary. In these equations, we assume that there is no correlation between the measurement residuals of different satellite,
2 2 2 2 2 2
[r ui j ] iu ir ju jr 2 i 2 j V Δ ∇ ρ =σρ +σρ +σρ +σρ = σρ + σρ
(23)
2 2 2 2 2 2
[ ( ) ]
2 2
i j
i j r u
r u
i i j j i j
u r u r
V roundoff
φ φ φ φ φ φ
φ φ λ
λ
σ σ σ σ σ σ
Δ ∇ − Δ ∇
= + + + = +
………..(24)
3.2 Sorting Data by the Elevation
We need to sample enough data for the statistical analysis of the receiver noise. In contrast with the assumption for the equations (23) and (24), the precision varies continuously, because the elevation does not stay constant as the satellite moves. Therefore, we need to divide all the elevation angles into several sections whose length is ElΔ and determine the representative value of each group. Then the noise properties of all the data in each elevation group are treated as the same. In other words, all the elevations (El ) of the i-th satellite which i belongs to the inequality equation (25) are considered as El , ir and El as j El . rj
2 2
r r
i i i
El El
El −Δ <El<El +Δ ………..(25)
Extracting the data set whose elevation angle combination is a pair of El and ir El from the whole measurement, we can get rj the reliable variance values for the particular elevation combination. We assume the exponential functions in (3) for code and phase, σρ(El)andσφ(El), can model the receiver noise precision quite well, then the variances in (23) and (24) are stated as (26) and (27).
2 2
[r ui j ] 2 ( ir) 2 ( rj) V Δ ∇ ρ = σρ El + σρ El
………..(26)
2 2
[ ( ) ]
2 ( ) 2 ( )
i j i j
El El
El El r u
r u
r r
i j
V roundoff
El El
φ φ
φ φ λ
λ
σ σ
Δ ∇ − Δ ∇
= +
………..(27)
3.3 Obtaining Set of Standard Deviations by Least- square Method
Finally, we can construct linear equations in (26) and (27) when the variance values of the DD observation residual for all the elevation combinations are calculated and arranged.
1
1 2
1
2
2
2
2
{ }
2 2 0 0
( )
{ }
2 0 2 0
( )
0 2 0 2 { }
0 0 2 2
( )
{ }
r
r r
r i
r r i j
r j
r r
n n
r n El
El El
r u El
El El
r u El
El El
r u El
V
z H x
V
V
ρ
ρ
ρ ρ ρ
ρ
ρ
σ
ρ σ
ρ σ
ρ σ
−
⎡ ⎤
⎡ ⎤ ⎢ ⎥
⎢ Δ ∇ ⎥ ⎡ ⎤⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
= = =
⎢ Δ ∇ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢⎣ ⎥⎦⎢ ⎥
⎢ ⎥ ⎢ ⎥
Δ ∇
⎢ ⎥
⎣ ⎦ ⎢ ⎥
⎣ ⎦
" #
"
# # # # # # #
"
# " #
...(28)
1 2
1 2
1
1
( ( ) )
( ( ) )
( ( ) )
{
2 2 0 0
2 0 2 0
0 2 0 2
0 0 2 2
r r
r r
r r i j r r
i j
n n
i j
r
El El
El El r u
r u
El El
El El r u
r u
El El
El El r u
r u
El
V round
z
V round
V round
φ
φ
φ φ λ
λ
φ φ λ
λ
φ φ λ
λ σ
−
⎡ ⎤
⎢ Δ ∇ ⎥
⎢ Δ ∇ − ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
=⎢ Δ ∇ − Δ ∇ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
Δ ∇
⎢ Δ ∇ − ⎥
⎢ ⎥
⎣ ⎦
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
=⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
#
#
"
"
# # # # #
"
"
2
2
2
2
}
{ }
{ }
{ }
r i
r j
r n El
El
El
H x
φ
φ φ φ
φ
σ
σ
σ
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥ =
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
#
#
#
………(29) , where n is the number of representative elevation angle.
Vector zρ and zφ contain observations, double differences of measurements, and x xρ, φ are sets of code and phase variance for each representative elevation angle. The over-determined systems of linear equation (28) and (29) have approximate solution in the least square sense.
( T )1 T
x= H H − H z ………(30)
3.4 Fitting the Solution to Model
Similar to the PR-CP technique, this algorithm is also based on the assumption that the satellite elevation angle information is a good indicator of the precision of each satellite. The last work to finish modeling measurement noise is fitting the result of (30) to the equation (3). We should optimize only three parameters in equation (3), x x0, 1, and x2,and they minimize the cost function J, the sum of error.
2
1
( ( ))
n
r
i i
i
J x σ El
=
=
∑
−………(31)
3.5 Advantage over Other Noise Estimation Methods
There are a few advantages of our suggested noise estimation algorithm over the existing ones. First of all, it is a very simple algorithm, so users need not to have many considerations or any technique. As described before, DD technique in zero-baseline eliminates all the external error sources of the two receivers, so there remains no significant residual effect except for the measurement noise. Even if there is a multi-path for a certain satellite, it is common to the measurements of both receivers.The ambiguity of carrier phase is easily resolved every epoch because of integer characteristic, and the receiver noise of DD process can be picked out in spite of occasional cycle slips. In other words, it does not need any cycle slip detector nor memory for the ambiguity of carrier phase. Therefore it is such a simple algorithm as expressed in [Figure 1] that even a non-expert can apply it to every GPS receiver including a single-frequency device.
[Figure 1] Block Diagram of the Noise Model Algorithm Using DD in Zero Baseline
Secondly, it is a mathematically clear algorithm, which means that little assumptions are included in it. Existing algorithms have their own limitations, and so they add some hypotheses or another estimator. Their effects are not seriously big, but derivation of the theory is not perfect. Occasionally, the standard deviation is estimated wrong or bigger than the true one.
The residual error remained after DD in our suggested algorithm, however, is caused by nothing other than receiver noise, so there is no jump of logic in our development of theory.
Besides them, it does not mix code and phase or L1, L2, and L5 for the estimation, so it can make accurate noise models not only of pseudorange but also of carrier phase for each measurement type, L1, L2 or L5. Moreover, it deals with observables in a distance domain, not with parameters in a process module such as DLL or PLL. Therefore, it is very practical and familiar to GPS users and application developers.
4. Field Test
4.1 Field Test Constructions
We organized an experiment test set, which would prove our simulation results. Two Trimble 4000ssi receivers, reference and user station, are connected to the same computer, and the data collected in zero baseline during static tests for 24hrs is used in quantifying the observation noise characteristics. The mask angle was 10 degree, and DD residual errors from 10 degree to 83 degree were obtained by the data logged every second in both
receiver.
[Table 1] Summary of the Field Test
Location SNU,Seoul(37°26’58.903”N, 126°57’09.887”E, 281.925m)
Date Feb. 28, 2006 20:00~Mar. 1, 2006 20:00 (24hr, Static Test)
GPS Receiver Trimble 4000ssi
Sampling Time 1 sec Elevation Mask 10°
4.2 Field Test Results
[Figure 2] shows the noise models for each observable and measurement type of Trimble 4000ssi receiver, which is estimated by the suggested algorithm. Its tendency to be dependent on the elevation angle is not so clear as that of the previous simulation results, but it is obvious that the estimated standard deviation of each observation is in inverse proportion to satellite elevation. L2 observables are noisier than L1 measurements. The result of carrier phase, in spite of occasional cycle-slips, follows the exponential model better than that of pseudorange, because this method is free from the cycle-slip, and because unexpected residual error still remains in pseudorange.
Especially, it is hard to model L2-L1 code measurement, because Trimble 4000ssi is a codeless receiver. If P code is available to the receiver, our algorithm may also estimate noise model of L2 code as well as others.
10 20 30 40 50 60 70 80 90
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Elevation Angle(deg)
Standard Deviation of Pseudorange Noise(m)
Estimated Noise Statistics Estimated Noise Model
10 20 30 40 50 60 70 80 90
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28
Elevation Angle(deg)
Standard Deviation of Carrier Phase Noise(mm)
Estimated Noise Statistics Estimated Noise Model
10 20 30 40 50 60 70 80 90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Elevation Angle(deg)
Standard Deviation of Pseudorange Noise(m)
Estimated Noise Statistics Estimated Noise Model
10 20 30 40 50 60 70 80 90
0 0.5 1 1.5 2 2.5 3
Elevation Angle(deg)
Standard Deviation of Carrier Phase Noise(mm)
Estimated Noise Statistics Estimated Noise Model
[Figure 2] Estimation of Standard Deviation Model for Trimble 4000ssi Measurement (Left Above: L1 code, Right Above: L1 phase, Left Below: L2-L1 code, Right Below: L2 phase) [Table 2] Estimated Model Parameters
x0 x1 x2
ρ1
σ 0.05571 0.33690 12.09805
φ1
σ 0.09728 1.56384 11.61057
dρ
σ 1.3358e – 4 2.3843e - 4 20.16905 φ2
σ 3.0217e – 4 6.5182e - 4 9.84810 The data of PRN 26 and 29 for 19,596 epochs were taken for the validation of the estimated model. The DD residual consists of the measurement noise error of each station from these two satellites, and their stochastic characteristics is close to white noise whose standard deviation depends on the elevation angle.
The standard deviation of DD is calculated by the
equation 2 (σ Elsv# 26)2+2 (σ Elsv# 29)2 , and it represents the statistical characteristics of DD. Recall that for a Gaussian distribution the ± 3σ includes 99% of all randomly distributed values, while 68% of the values lie within ± 1σ. Using the previous model we estimated the standard deviation of pseudorange and carrier phase DD, and plotted 3σ lines to prove the representation of the residual error in [Figure 3]. 99.2% of pseudorange DD residual errors, and 99.5% of phase errors did not exceed the 3σ line. Additionally, the percentage of DD error’s coming into 1σ area was 63.3% for pseudorange and 65.9% for carrier phase. These figures prove that the Gaussian property of measurement noise, and that the suggested model by our algorithm corresponds to the observable noise information.
2.86 2.88 2.9 2.92 2.94 2.96 2.98 3 3.02 3.04
x 105 0
20 40 60 80
elevation angle(deg) SV#26
SV#29
2.86 2.88 2.9 2.92 2.94 2.96 2.98 3 3.02 3.04
x 105 -1
-0.5 0 0.5 1
DD pseudorange residual(m)
residual error 3 σ of DD
2.86 2.88 2.9 2.92 2.94 2.96 2.98 3 3.02 3.04
x 105 -2
-1 0 1 2
DD carrier phase residual(mm)
gps time(s)
residual error 3 σ of DD
[Figure 3] DD Residual and 3-sigma Error Estimation (Up:
Elevation Angle Variation, Middle: DD Pseudorange Residual, Bottom: DD Carrier Phase Residual)
[Table 3] Suggested Model’s Representation of the Gaussian Property of Measurement Noise
Pseudorange Code
Carrier Phase Gaussian Property
1σ 63.3% 65.9% 68%
3σ 99.2% 99.5% 99%
4.2 Performance Comparison with Existing Algorithms
To compare the performance of the new algorithm with those of existing algorithms, we applied PR-CP and SD techniques to estimation of the parameters in exponential noise model, and those estimated by DD algorithm are smallest of all in [Figure 4].In other words, our algorithms can estimate the residual error more compactly than the existing algorithms.
The compact estimation of DD residual error is useful to the ambiguity resolution for the precise positioning. Real valued ambiguities are mapped onto integers, and an optimal one should be selected from a search space. Length of each axis is number of candidate for each ambiguity, so the tightly estimated noise property makes the search space small. The smaller the search
space is, the lower the computational burden is.
2.86 2.88 2.9 2.92 2.94 2.96 2.98 3 3.02 3.04 x 105 -3
-2 -1 0 1 2 3
gpstime(s)
L1 pseudorange DD residual(m)
residual by DD 3σ line modeled by PR-CP 3σ line modeled by SD 3σ line modeled by DD
2.86 2.88 2.9 2.92 2.94 2.962.98 3 3.02 3.04 x 105 -3
-2 -1 0 1 2 3
gpstime(s)
L2-L1 pseudorange DD residual(m)
residual by DD 3σ line modeled by SD 3σ line modeled l by DD
2.86 2.88 2.9 2.92 2.94 2.96 2.98 3 3.02 3.04 x 105 -3
-2 -1 0 1 2 3
gpstime(s)
L1 carrier phase DD residual(mm)
residual by DD 3σ line modeled by CNR 3σ line modeled by SD 3σ line of SD and correlation 3σ line modeled by DD
2.86 2.88 2.9 2.92 2.94 2.962.98 3 3.02 3.04 x 105 -6
-4 -2 0 2 4 6
gpstime(s)
L2 carrier phase DD residual(mm)
residual by DD 3σ line modeled by CNR 3σ line modeled by SD 3σ line of SD and correlation 3σ line modeled by DD
[Figure 4] DD Residual Estimation (Left Above: L1 code, Right Above: L2-L1 code, Left Below: L1 carrier, Right Below: L2 carrier)
Using the logged data we examined how much the computing load reduced. Three among seven observable satellites were at high elevation angle at GPSTime 294,000 second, and the number of candidate for L2 measurement is reduced much more than that of L1. If we set 5 sigma(99.999%) for the ambiguity search, the number of candidates for each L2 measurement is reduced by 2 to 6, so the size of the search space becomes 34.5%
of that made by the SD algorithm.
The compact noise model of DD method for carrier phase is useful to make a tight cut-off criterion, so it easily gets rid of the fault candidate by means of the residual test. Therefore, it can increase the speed of searching true ambiguity and enhance the reliability and efficiency of ambiguity resolution in carrier phase positioning. Moreover, the accurate estimate of carrier phase noise will enhance the utility of the frequency combination, because some of them suffer from enlarged measurement noise after combining.
5. Conclusions
Zero-baseline double difference (DD) excludes all the significant error, so there remains each receiver's residual error from two observable satellites. Sorting them by a pair of elevation angles, we can obtain an over-determined linear equation. In the sense of least square, we can get the noise standard deviation for each elevation angle. Fitting them to the exponential noise model, we can finally estimate the stochastic noise model of each measurement.
This algorithm has several advantages over the existing modeling method. First of all, it is so simple because it does not need cycle slip detector or memory for phase ambiguity.
The development of this algorithm does not need any assumption, so it is mathematically clear. Besides them it is very practical and familiar to GPS users and application developers.
We applied this algorithm to real data of the 24hr Trimble 4000ssi receiver, and we easily calculated the coefficients of the noise model not only for pseudorange but also for carrier phase.
We found its advantage of performance as well as simplicity.
The residual error of each measurement lied in the anticipated range according to the Gaussian rule; therefore, the nose standard deviation estimated by our method corresponds to the observable noise information and represents the Gaussian property
Compared with the performance of the existing algorithms,
our method models the DD residual error of each observable, and it is especially powerful in the modeling of L2-related measurement. It can estimate the standard deviation of the L2 noise from high elevation more tightly than SD method by 35%.
This algorithm is useful to estimate a very practical and exact stochastic noise model. It would enhance the reliability and efficiency of ambiguity resolution in carrier phase positioning without consideration of the cross-correlation between satellites because of the reduced search space and tight criteria. Besides ambiguity resolution issue, it is expected that this modeling algorithm is essential to various applications including monitoring data integrity, high-precision static/kinematic positioning techniques, and the estimation of the position quality.
Acknowledgement
The study was supported in part by in part by the Brain Korea 21 (BK-21) Program for Mechanical and Aerospace Engineering Research, the Institute of Advanced Machinery and Design, Institute of Advnaced Aerospace Technology at Seoul National University. We would like to thank for their support.
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