A Tropospheric Delay Model for the user of the Wide Area Augmentation System

A Tropospheric Delay Model for the user of the Wide Area Augmentation System J. Paul Collins and Richard B. Langley 1st October 1996

+641&7%6+1 OBJECTIVES Develop and test a tropospheric propagation delay model for the WAAS user. Determine the bounds of the tropospheric delay contribution to the pseudorange error budget. LIMITATION Lack of real-time meteorology to quantify the state of the atmosphere.

6412152*'4+%&'.#; Refractive index in neutral atmosphere greater than unity. Decrease in velocity increases propagation time and equivalent path length (the delay ). Refraction bends raypath - significant at low elevation angles. Hydrostatic zenith delay accurately determined with atmospheric pressure measurement. Ignoring horizontal gradients and assuming azimuthal symmetry allows use of mapping functions (ratios of zenith delay to angle of elevation delay).

Zenith Delay, &'.#;(14/7.#6+1 d z trop = 1 6 N - refractivity r, r 1 - radial distance from centre of earth r r 1 Ndr Refractivity, N k P T k e T k e d = + + T 1 2 3 2 k 1, k 2, k 3 - refractivity constants P d, T, e - atmospheric parameters Total Delay, d = d m + d m trop z hyd hyd z wet wet z z d hyd, dwet hydrostatic and wet zenith delay, m, m hydrostatic and wet mapping function hyd wet

#8+)#6+16;2'#8/1&'.5 Elevation Angle Latitude Height Met. Parameters Altshuler N WAAS N NATO N UNB1 P, T, e, β, λ Central Radio Propagation Laboratory Reference Atmosphere 1958 + Chao dry Mapping Function Saastamoinen Zenith Delay models + Niell Mapping Functions + T T H P P T Rβ T Rβ = β, = e = e T, T, f ( T = 288.15K, P = 113.25mbar, e = 11.7mbar ) 324.8 N units g 4g

&#6#&'5%4+26+1#&241%'55+) Frizzle 95 Experiment, Newfoundland, Canada, March 1995. 12, 3-5 hour flights recording dual frequency GPS and meteorological 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 the simultaneously recorded meteorological data.

(4+<<.'a(.+)*62#6*5 55 3 31 55 5 5 45 45 3 31

March 1th Flight Path Height (m) 7 6 5 4 3 2 1 51 5 Latitude 49 48-55 -54-53 Longitude -52

ALTSHULER Latitude Position Differences for March 1 WAAS NATO UNB1 6 6 6 6 5 5 5 5 4 4 4 4 Height (m) 3 3 3 3 2 2 2 2 1 1 1 1 -.4 -.2.2.4 Latitude Difference (m) -.4 -.2.2.4 Latitude Difference (m) -.4 -.2.2.4 Latitude Difference (m) -.4 -.2.2.4 Latitude Difference (m)

ALTSHULER Longitude Position Differences for March 1 WAAS NATO UNB1 6 6 6 6 5 5 5 5 4 4 4 4 Height (m) 3 3 3 3 2 2 2 2 1 1 1 1 -.4 -.2.2.4 Longitude Difference (m) -.4 -.2.2.4 Longitude Difference (m) -.4 -.2.2.4 Longitude Difference (m) -.4 -.2.2.4 Longitude Difference (m)

ALTSHULER Height Position Differences for March 1 WAAS NATO UNB1 6 6 6 6 5 5 5 5 4 4 4 4 Height (m) 3 3 3 3 2 2 2 2 1 1 1 1-1 1 Height Difference (m) -1 1 Height Difference (m) -1 1 Height Difference (m) -1 1 Height Difference (m)

51.76+1&+(('4'%'5 Latitude Difference (m) Longitude Difference (m) Height Difference (m).3. -.3.3. -.3 1.. -1. Baseline Length Altshuler WAAS NATO UNB1 PDOP Aircraft Height 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 Time (minutes) 5 4 3 2 1 6 4 2 6 45 3 15 Baseline (km) PDOP Height (m)

#8)254'57.65 Position Difference Mean and Standard Deviation Position Difference Ranges Position Difference (m).5.4.3.2.1. -.1 -.2 -.3 -.4 -.5 Latitude Longitude Height Altshuler WAAS NATO UNB1 Position Difference (m) 2. 1.5 1..5. Latitude Longitude Height Altshuler WAAS NATO UNB1

#8)2557//#4; Worst case height differences: Altshuler - 1.38m. WAAS - 1.7m. NATO - 1.9m. UNB1 -.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'&'%; Hyd. Zenith Delay d P = P T T P κ d z hyd z hyd = f( P) z = τ P hyd g Rβ hyd z hyd T = T β H = τ κ P d hyd At user s altitude (H) Wet Zenith Delay d e T z wet = z wet ( e T βλ) = f,,, z e = τwet T e T e T T T κ λ g 1 Rβ wet z e = τwet κwet T

7$/1&'.57//#4; Require values for: Surface pressure (P ), Surface temperature (T ), Surface water vapour pressure (e ), 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.

March 15th Flight Path Height (m) 7 6 5 4 3 2 1 52 Latitude 5 48-58 -56 Longitude -54-52

UNB1 Latitude Position Differences for March 15 UNB2 UNB3 UNB4 6 6 6 6 5 5 5 5 4 4 4 4 Height (m) 3 3 3 3 2 2 2 2 1 1 1 1 -.1.1 Latitude Difference (m) -.1.1 Latitude Difference (m) -.1.1 Latitude Difference (m) -.1.1 Latitude Difference (m)

UNB1 Longitude Position Differences for March 15 UNB2 UNB3 UNB4 6 6 6 6 5 5 5 5 4 4 4 4 Height (m) 3 3 3 3 2 2 2 2 1 1 1 1 -.1.1 Longitude Difference (m) -.1.1 Longitude Difference (m) -.1.1 Longitude Difference (m) -.1.1 Longitude Difference (m)

UNB1 Height Position Differences for March 15 UNB2 UNB3 UNB4 6 6 6 6 5 5 5 5 4 4 4 4 Height (m) 3 3 3 3 2 2 2 2 1 1 1 1 -.4 -.2.2.4 Height Difference (m) -.4 -.2.2.4 Height Difference (m) -.4 -.2.2.4 Height Difference (m) -.4 -.2.2.4 Height Difference (m)

/'6'141.1)+%#.241(+.'5 March 15 6 UNB1 UNB2 UNB3 5 UNB4 UNB1 UNB2 UNB3 UNB4 6 5 Height (m) 4 3 4 3 Height (m) 2 2 1 1-4 -3-2 -1 1 2 Temperature ( C) 2 4 6 8 1 12 Water Vapour (mbar)

7$)254'57.65 All data Position Difference Mean and Standard Deviation Position Difference Ranges Position Difference (m).2.1. -.1 -.2 Latitude Longitude Height UNB1 UNB2 UNB3 UNB4 Position Difference (m) 1.5 1..5. Latitude Longitude Height UNB1 UNB2 UNB3 UNB4

7$)254'57.65 Data upto 1 km in altitude (above msl) Position Difference Mean and Standard Deviation Position Difference Ranges Position Difference (m).3.2.1. -.1 -.2 -.3 Latitude Longitude Height UNB1 UNB2 UNB3 UNB4 Position Difference (m) 1..5. Latitude Longitude Height UNB1 UNB2 UNB3 UNB4

'4414/1&'..+) 1 1 9 9 8 8 7 7 Height (km) 6 5 4 6 5 4 Height (km) 3 3 2 2 1 1 -.12 -.8 -.4..4 -.12 -.8 -.4..4 Hydrostatic Zenith Delay Error (m) Wet Zenith Delay Error (m) (P ) = +.25 mbar (T ) = -38.9 K (β) = -2.6 K/km Total (β) = -2.6 K/km (T ) = -38.9 K (λ) = -1.75 (e ) = -1.98 mbar Total

7$)2557//#4; Error model shows potential impact of: incorrect water vapour pressure at surface, incorrect scaling of total surface pressure. Improvement in position solutions follows improved representation 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 by UNB4.

Height Position Differences for March 1 WAAS UNB4 6 6 5 5 Height (m) 4 3 Height (m) 4 3 2 2 1 1-1 1 Height Difference (m) -1 1 Height Difference (m)

Height Position Differences for March 15 WAAS UNB4 6 6 5 5 Height (m) 4 3 Height (m) 4 3 2 2 1 1-1 1 Height Difference (m) -1 1 Height Difference (m)

24+%+2.'51(4#;64#%+) Application of Snell s Law to a spherically stratified medium. nr+ ( R+ h hi i) sin( φ R+ ) hi = n ( R+ h ) sin( φ ) R+ hi+ 1 i+ 1 R+ hi+ 1

4#&+151&'.#7%*5+6'5 Alert Kotzebue Iqaluit Whitehorse The Pas Landvetter St. John s Denver Grand Jnc. Oakland Nashville Guam San Juan

4#;64#%'4'57.65 (Ray-trace minus model prediction) Residual Distribution Mean and Standard Deviation Residual Distribution Maxima and Minima 1.5 3 1. 2.5 1 Error (m). -.5-1. -1.5 UNB1 UNB2 UNB3 UNB4 WAAS 5 1 15 2 25 3 9 Error (m) -1-2 -3 UNB1 UNB2 UNB3 UNB4 WAAS 5 1 15 2 25 3 9 Elevation Angle (deg) Elevation Angle (deg)

14/#.&+564+$76+16'56 4 Gaussian Plot of Zenith Delay Residuals 3 Normalised Residual Distribution 2 1-1 -2-3 UNB1 UNB2 UNB3 UNB4 WAAS -4-4 -3-2 -1 1 2 3 4 Theoretical Normal N(,1) Distribution

6412152*'4+%&'.#;'4414.+/+6 x + 2σ Range Error (from ray-trace data) 1σ Range Error for AFCRL model 2.5 2.5 Delay Error (m) 2. 1.5 1. UNB1 UNB2 UNB3 UNB4 WAAS ALTS NATO Delay Error (m) 2. 1.5 1. Curve A Curve B Curve C Curve D.5.5. 1 2 3 4 5 6 7 8 9. 1 2 3 4 5 6 7 8 9 Elevation Angle (deg) Elevation Angle (deg)

':64'/'%1&+6+15 Hydrostatic Delay: Nominal: 2.3m @ zenith 23.32m @ 5 deg. Wet Delay (UNB3, Tropics): Nominal:.28m @ zenith 3.m @ 5 deg. SLP = 9 mbar, delay = 2.6m range error = -.26m @ zenith -2.65m @ 5 deg. Height Bias = -2.4m max PW = 8mm, delay =.51m range error = +.23m @ zenith +2.46m @ 5 deg. Height Bias = +2.24m SLP = 184 mbar, delay = 2.47m range error = +.17m @ zenith +1.73m @ 5 deg. Height Bias = +1.57m min PW = mm, delay = m range error = -.28m @ zenith -3.m @ 5 deg. Height Bias = -2.73m

%1%.75+15 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 delay error 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 wet delay

#%-19.'&)'/'65 Transport Canada Aviation (TCA). Federal Aviation Administration (FAA). National Research Council (NRC). Atmospheric Environment Service (AES). Virgilio Mendes.