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Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
A Tropospheric Delay Modelfor the user of the
Wide Area Augmentation System
J. Paul Collins and Richard B. Langley
1st October 1996
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
+0641&7%6+10
• OBJECTIVES➡ Develop and test a tropospheric propagation delay model for the
WAAS user.
➡ Determine the bounds of the tropospheric delay contribution to thepseudorange error budget.
• LIMITATION➡ Lack of “real-time” meteorology to quantify the state of the
atmosphere.
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick. Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
• Refractive index in neutral atmosphere greater than unity.
• Decrease in velocity increases propagation time andequivalent path length (the “delay”).
• Refraction bends raypath - significant at low elevationangles.
• Hydrostatic zenith delay accurately determined withatmospheric pressure measurement.
• Ignoring horizontal gradients and assuming azimuthalsymmetry allows use of mapping functions (ratios of zenithdelay to angle of elevation delay).
6412152*'4+%�&'.#;
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
&'.#;�(14/7.#6+10
N kP
Tk
e
Tk
e
Td= + +1 2 3 2
d d m d mtrop hydz
hyd wetz
wet= ⋅ + ⋅
d Ndrtropz
r
r
= − ∫10 6
0
1
Refractivity,
k1, k2, k3 - refractivity constantsPd, T, e - atmospheric parameters
Zenith Delay,
N - refractivityr0, r1 - radial distance from centre of earth
Total Delay,
d d hydrostatic and wet zenith delaym m hydrostatic and wet mapping function
hydz
wetz
hyd wet
, ,,
−−
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
0#8+)#6+10�6;2'�0#8��/1&'.5
ElevationAngle
Latitude Height Met.Parameters
Altshuler ✓ ✓ ✓ NWAAS ✓ ✓ ✓ NNATO†
✓ ✗ ✓ NUNB1‡
✓ ✓ ✓ P, T, e, β, λ
† Central Radio Propagation Laboratory Reference Atmosphere 1958+ Chao ‘dry’ Mapping Function
‡ Saastamoinen Zenith Delay models + Niell Mapping Functions +
f ( T0 = 288.15K, P
0 = 1013.25mbar, e
0 = 11.7mbar ) ≈ 324.8 N units
T T H P PT
Te e
T
T
g
R
g
R= − ⋅ =
=
0 0
00
0
4
β , , ,β β
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
#�&'5%4+26+10�#0&�241%'55+0)
• Frizzle ‘95 Experiment, Newfoundland, Canada, March1995.
• 12, 3-5 hour flights recording dual frequency GPS andmeteorological data at aircraft and ground station.
• P(Y) code on L1 processed for optimum positions.
• IGS precise orbits used.
• Ionosphere largely removed with single differences.
• Benchmark solutions computed using UNB1 and thesimultaneously recorded meteorological data.
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
(4+<<.'�a���(.+)*6�2#6*5
300˚
300˚
310˚
310˚
45˚ 45˚
50˚ 50˚
55˚ 55˚
-55-54
-53-52
48
49
50
51
0
1000
2000
3000
4000
5000
6000
7000
LongitudeLatitude
Hei
ght
(m)
March 10th Flight Path
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
ALTSHULER
Latitude Difference (m)
Hei
ght
(m)
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
WAAS
Latitude Difference (m)
Latitude Position Differences for March 10
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
NATO
Latitude Difference (m)-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
UNB1
Latitude Difference (m)
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
ALTSHULER
Longitude Difference (m)
Hei
ght
(m)
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
WAAS
Longitude Difference (m)
Longitude Position Differences for March 10
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
NATO
Longitude Difference (m)-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
Longitude Difference (m)
UNB1
-1 0 10
1000
2000
3000
4000
5000
6000
ALTSHULER
Height Difference (m)
Hei
ght
(m)
-1 0 10
1000
2000
3000
4000
5000
6000
WAAS
Height Difference (m)
Height Position Differences for March 10
-1 0 10
1000
2000
3000
4000
5000
6000
NATO
Height Difference (m)-1 0 1
0
1000
2000
3000
4000
5000
6000
UNB1
Height Difference (m)
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
51.76+10�&+(('4'0%'5
Time (minutes)0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
Hei
ght
Diff
ere
nce
(m)
-1.0
0.0
1.0
Height (m
)
0
1500
3000
4500
6000
Lon
gitu
deD
iffer
en
ce(m
)
-0.3
0.0
0.3 PD
OP
2
4
6
La
titud
eD
iffer
en
ce(m
)
-0.3
0.0
0.3
Ba
seline
(km)
100200300400500
Baseline Length
PDOP
Aircraft Height
Altshuler WAAS NATO UNB1
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
0#8�)25�4'57.65
Position Difference Ranges
Altshuler WAAS NATO UNB1
Po
sitio
n D
iffer
ence
(m
)0.0
0.5
1.0
1.5
2.0
Latitude Longitude Height
Position Difference Meanand Standard Deviation
Altshuler WAAS NATO UNB1
Po
sitio
n D
iffer
ence
(m
)
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
Latitude Longitude Height
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
0#8�)25�57//#4;
• Worst case height differences:➡ Altshuler - 1.38m.
➡ WAAS - 1.07m.
➡ NATO - 1.09m.
➡ UNB1 - 0.71m.
• Step in WAAS solutions.
• NATO and UNB1 perform similarly.
• Altshuler and WAAS:➡ Poor modelling of height change of troposphere delay.
➡ Poor elevation angle modelling.
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
6412152*'4+%�*'+)*6�&'2'0&'0%;
( )d e Te
T
wetz
wetz
= ′
= ⋅
f , , ,β λ
τ
P PTT
gR
P hyd
=
≡ ⋅
00
0
β
κ
e
T
e
T
T
T
gR
e
T wet
=
′ −
≡ ⋅
0
0 0
0
0
1λ
β
κ
T T H= − ⋅0 β
d Phydz
hydz
hyd= ⋅ ⋅τ κ 0 deTwet
zwetz
wet= ⋅ ⋅τ κ 0
0
At user’saltitude (H)
Wet Zenith DelayHyd. Zenith Delay
d P
Phydz
hydz
== ⋅
f ( )
τ
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
70$�/1&'.�57//#4;
• Require values for:➡ Surface pressure (P0),
➡ Surface temperature (T0),
➡ Surface water vapour pressure (e0),
➡ Temperature lapse rate (β), and
➡ Water vapour “lapse rate” (λ).
• UNB1 - Global constant values (STP).
• UNB2 - Latitudinal averages (various sources).
• UNB3 - U.S. Standard Atmosphere Supplements, 1966.➡ (temporal variation based on Niell mapping function concepts).
• UNB4 - Modification to UNB3 temperature profile.
-58-56
-54-52
48
50
52
0
1000
2000
3000
4000
5000
6000
7000
LongitudeLatitude
Hei
ght
(m)
March 15th Flight Path
-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB1
Latitude Difference (m)
Hei
ght
(m)
-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB2
Latitude Difference (m)
Latitude Position Differences for March 15
-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB3
Latitude Difference (m)-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB4
Latitude Difference (m)
-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB1
Longitude Difference (m)
Hei
ght
(m)
-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB2
Longitude Difference (m)
Longitude Position Differences for March 15
-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB3
Longitude Difference (m)-0.1 0 0.10
1000
2000
3000
4000
5000
6000
UNB4
Longitude Difference (m)
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
UNB1
Height Difference (m)
Hei
ght
(m)
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
UNB2
Height Difference (m)
Height Position Differences for March 15
-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
UNB3
Height Difference (m)-0.4 -0.2 0 0.2 0.40
1000
2000
3000
4000
5000
6000
UNB4
Height Difference (m)
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
/'6'141.1)+%#.�241(+.'5
Temperature (°C)
-40 -30 -20 -10 0 10 20
Hei
ght
(m)
0
1000
2000
3000
4000
5000
6000 UNB1 UNB2 UNB3 UNB4
Water Vapour (mbar)
0 2 4 6 8 10 12
Hei
ght
(m)
0
1000
2000
3000
4000
5000
6000UNB1 UNB2 UNB3 UNB4
March 15
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
70$�)25�4'57.65���
All data
Position Difference Ranges
UNB1 UNB2 UNB3 UNB4
Po
sitio
n D
iffer
ence
(m
)0.0
0.5
1.0
1.5Latitude Longitude Height
Position Difference Meanand Standard Deviation
UNB1 UNB2 UNB3 UNB4
Po
sitio
n D
iffer
ence
(m
)
-0.2
-0.1
0.0
0.1
0.2
Latitude Longitude Height
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
70$�)25�4'57.65���
Data upto 1 km in altitude (above msl)
Position Difference Ranges
UNB1 UNB2 UNB3 UNB4
Po
sitio
n D
iffer
ence
(m
)0.0
0.5
1.0Latitude Longitude Height
Position Difference Meanand Standard Deviation
UNB1 UNB2 UNB3 UNB4
Po
sitio
n D
iffer
ence
(m
)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
Latitude Longitude Height
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
'4414�/1&'..+0)
Hydrostatic Zenith Delay Error (m)
-0.12 -0.08 -0.04 0.00 0.04
Hei
ght
(km
)
0
1
2
3
4
5
6
7
8
9
10
∆(P0) = +0.25 mbar
∆(T0) = -38.9 K
∆(β) = -2.6 K/kmTotal
Wet Zenith Delay Error (m)
-0.12 -0.08 -0.04 0.00 0.04
Hei
ght
(km
)
0
1
2
3
4
5
6
7
8
9
10
∆(β) = -2.6 K/km∆(T0) = -38.9 K
∆(λ) = -1.75∆(e0) = -10.98 mbar
Total
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
70$�)25�57//#4;
• Error model shows potential impact of:➡ incorrect water vapour pressure at surface,
➡ incorrect scaling of total surface pressure.
• Improvement in position solutions follows improvedrepresentation of water vapour profile.
• Most near-surface biases reduced by UNB3 and UNB4.
• Altitude bias introduced by UNB3.
• Most consistent results throughout flight paths provided byUNB4.
-1 0 10
1000
2000
3000
4000
5000
6000
Height Difference (m)
Hei
ght
(m)
WAAS
Height Position Differences for March 10
-1 0 10
1000
2000
3000
4000
5000
6000
Hei
ght
(m)
Height Difference (m)
UNB4
-1 0 10
1000
2000
3000
4000
5000
6000
Height Difference (m)
Hei
ght
(m)
WAAS
Height Position Differences for March 15
-1 0 10
1000
2000
3000
4000
5000
6000
Hei
ght
(m)
Height Difference (m)
UNB4
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
24+0%+2.'5�1(�4#;�64#%+0)
Application of Snell’s Law toa spherically stratified medium.
n R hn R h
R hi i R hi
R hi i R hi
+ +
+ + + + +
⋅ + ⋅ ′= ⋅ + ⋅ ′
( ) sin( )( ) sin( )
φφ
1 1 1
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
4#&+1510&'�.#70%*�5+6'5
Alert Kotzebue Iqaluit
Whitehorse Landvetter The Pas St. John’sDenver Grand Jnc.
Oakland Nashville
San Juan
Guam
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
4#;�64#%'�4'57.65
Residual DistributionMean and Standard Deviation
Elevation Angle (deg)
5 10 15 20 25 30 90
Err
or (
m)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
UNB1 UNB2 UNB3 UNB4 WAAS
Residual DistributionMaxima and Minima
Elevation Angle (deg)
5 10 15 20 25 30 90
Err
or (
m)
-3
-2
-1
0
1
2
3
UNB1UNB2UNB3UNB4WAAS
(Ray-trace minus model prediction)
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
014/#.�&+564+$76+10�6'56
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4Gaussian Plot of Zenith Delay Residuals
Theoretical Normal N(0,1) Distribution
Nor
mal
ised
Res
idua
l Dis
trib
utio
n
UNB1
UNB2
UNB3
UNB4
WAAS
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
6412152*'4+%�&'.#;�'4414�.+/+6
1σ Range Errorfor AFCRL model
Elevation Angle (deg)
0 10 20 30 40 50 60 70 80 90
De
lay
Err
or (
m)
0.0
0.5
1.0
1.5
2.0
2.5
Curve A Curve B Curve C Curve D
x + 2σ Range Error(from ray-trace data)
Elevation Angle (deg)
0 10 20 30 40 50 60 70 80 90
De
lay
Err
or (
m)
0.0
0.5
1.0
1.5
2.0
2.5
UNB1UNB2UNB3UNB4WAASALTSNATO
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
':64'/'�%10&+6+105
• Hydrostatic Delay:➡ Nominal: 2.30m @ zenith
23.32m @ 5 deg.
• SLP = 900 mbar, delay = 2.06m➡ range error = -0.26m @ zenith
-2.65m @ 5 deg.
➡ Height Bias = -2.40m
• SLP = 1084 mbar, delay = 2.47m➡ range error = +0.17m @ zenith
+1.73m @ 5 deg.
➡ Height Bias = +1.57m
• Wet Delay (UNB3, Tropics):➡ Nominal: 0.28m @ zenith
3.00m @ 5 deg.
• max PW = 80mm, delay = 0.51m➡ range error = +0.23m @ zenith
+2.46m @ 5 deg.
➡ Height Bias = +2.24m
• min PW = 0mm, delay = 0m➡ range error = -0.28m @ zenith
-3.00m @ 5 deg.
➡ Height Bias = -2.73m
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
%10%.75+105
• Navigation-type models unsuitable for aircraft navigation.➡ Poor height dependency.
➡ Poor mapping functions.
• Models based on physical principles are superior.
• Provision of “correct” meteorological values important.
• All models susceptible to extreme conditions.➡ Only real-time measurements will overcome these problems.
➡ Even so, wet delay error can still be large.
• Under “normal” conditions UNB3 and UNB4 limit delayerror to ~1.5m at 5 degrees elevation angle.
• UNB3 model provides best results near surface.
• UNB4 consistent at all altitudes represented by GPS data.
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
(7674'�914-
• Investigation of environmental extremes➡ frequency
➡ duration
➡ magnitude
• Assessment of methods to cope➡ ground monitoring of atmospheric conditions
➡ provision of “real-time” atmospheric parameters• barometric pressure — accurate hydrostatic delay
• temperature — computation of saturation profile - upper limit of wetdelay
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
#%-019.'&)'/'065
• Transport Canada Aviation (TCA).
• Federal Aviation Administration (FAA).
• National Research Council (NRC).
• Atmospheric Environment Service (AES).
• Virgilio Mendes.