A Tropospheric Delay Model for the user of the Wide Area

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


+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+%�&'.#;


&'.#;�(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

, ,,

−−


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


&#6#�&'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.


(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 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


-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 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


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 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



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


0#8�)25�4'57.65

Position Difference Ranges


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


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



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.


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


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


Hei

ght

(m)

-0.1 0 0.10

1000

2000

3000

4000

5000

6000

UNB2


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


-0.1 0 0.10

1000

2000

3000

4000

5000

6000

UNB1


Hei

ght

(m)

-0.1 0 0.10

1000

2000

3000

4000

5000

6000

UNB2


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


-0.4 -0.2 0 0.2 0.40

1000

2000

3000

4000

5000

6000

UNB1


Hei

ght

(m)

-0.4 -0.2 0 0.2 0.40

1000

2000

3000

4000

5000

6000

UNB2



-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



/'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


70$�)25�4'57.65��

All data


UNB1 UNB2 UNB3 UNB4

Po

sitio

n D

iffer

ence

(m

)0.0

0.5

1.0

1.5Latitude Longitude Height


UNB1 UNB2 UNB3 UNB4

Po

sitio

n D

iffer

ence

(m

)

-0.2

-0.1

0.0

0.1

0.2



70$�)25�4'57.65��

Data upto 1 km in altitude (above msl)


UNB1 UNB2 UNB3 UNB4

Po

sitio

n D

iffer

ence

(m

)0.0

0.5

1.0Latitude Longitude Height


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



'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


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


Hei

ght

(m)

WAAS


-1 0 10

1000

2000

3000

4000

5000

6000

Hei

ght

(m)


UNB4


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


4#&+1510&'�.#70%*�5+6'5

Alert Kotzebue Iqaluit

Whitehorse Landvetter The Pas St. John’sDenver Grand Jnc.

Oakland Nashville

San Juan

Guam


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


5 10 15 20 25 30 90

Err

or (

m)

-3

-2

-1

0

1

2

3

UNB1UNB2UNB3UNB4WAAS

(Ray-trace minus model prediction)


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


6412152*'4+%�&'.#;�'4414�.+/+6

1σ Range Errorfor AFCRL model


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)


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


':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


%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.


(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


#%-019.'&)'/'065

• Transport Canada Aviation (TCA).

• Federal Aviation Administration (FAA).

• National Research Council (NRC).

• Atmospheric Environment Service (AES).

• Virgilio Mendes.

A Tropospheric Delay Model for the user of the Wide Area

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