Evaluation of the GPS Positioning Error
due to the Inhomogeneous Distribution ofAtmospheric Delayby a Numerical Weather Model Data
Hiromu Seko

（Meteorological Research Institute/JMA）

Hajime Nakamura（Numerical Prediction Department / JMA ）Sei-ichi Shimada（National Research Institute for Earth Science and Disaster Prevention）-Contents-

1. Motivation
2. Methodology
3. Positioning error caused by Mountain lee wave
Comparison with observed results (Shimada, 2002)

4. Summary11. Motivation
・Large positioning errors were caused by atmosphere.
・Atmospheric model with gradient does not always improve
positioning error.
→ Small scale variation existed?
・To discuss only influence of atmosphere, ‘clean’ delay wascalculated by the numerical weather model.
・‘Clean’ delay was used as input data ofanalysis program andpositioning error was estimated.
・Small scale variation(mountain lee wave case)causes the positioning error?

40km
Cloud images observed bygeostationary meteorological satellite32. Method
2.a Flowchart of estimation of positioning error
Numerical
weather model

Initial & boundary conditions
Upper sounding data

Ray tracing program
Simulatedslant total delay(O)(only atmospheric effect)

GPS analysis
Least-squares fitting
GPS atmospheric model
ZTD and gradient

Residual(O-C)
Beutler et al 1988

Positioning error

Estimatedslant delay (C)Analysis process4

2.b Atmospheric model(a) Constant modelτest = τzenm(θ)(b) Linear gradient model (MacMilan, 1995)
τest = τzenm(θ)+ m(θ)/tan(θ)(GNcosφ+GEsinφ).
(c) Second order function fitting modelτest = τzenm(θ)+ m(θ)(GNy+GEx+GN2y2+GNExy+GE2x2),
where x=1.0/tan(θ)*sinφ, y=1.0/tan(θ)*cosφ
2.c Estimation of positioning error
When estimated slant delay (τest ) is larger (smaller) thansimulated delay (τ), the receiver position was shifted toopposite (same) side of satellite.
Positioning error (Beutler, 1988):
δx=－Σ{(τ－τest)cosθsinφ}/N、
δy=－Σ{(τ－τest)cosθcosφ}/N、
δz=－Σ{(τ－τest)sinθ}/N

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Analysis process

Numerical weather model

Zenith delay

Ray tracingv

・Non-hydrostatic model ofMeteorological ResearchInstitute・Variables: U V ρ P θzqv qc qr qs qh qi etc.
・Horizontal grid : 250m
x/ｙ・Initial data: upper sounding

1. Actual positions of GPS receivers
and satellites were used. (cutoffangle 15°)
2. The path was found by using theray tracing method.
3. Slant delays were obtained byintegrating the delay along the path.

skymap6

Topography relative to GPS receiver
Cartesian grid
relative to a receiver・Curvature of the earth wastaken into account.
・NHM model
Latitude,Longitude,Height→ Cartesian orthogonal
coordinate relative toa receiver opposition∆x, ∆y=250m, ∆z=100m
・Refractivity(N) on theCartesian grid wascalculated fromsimulated data(Thayer,1974).
・Delay was calculated fromrefractivity (N).

Outside of domain
Horizontal distribution of N at z=3km7

Outline of ray tracing method
Derivation of equation（１）・ d/ds(n dX/ds) = ∇n ...(1)where、X：position of tracer, n：refractivity,s：increment of tracing distance（100m）
・ When Y = n dX/ds is introduced,
a equation (1) becomedX/ds = Y/n ...(3)dY/ds = ∇n ...(2)・ Furthermore, dτ = ds/n is introduced.
Equation (2) and (3) became
dY/dτ = n∇n ...(2)’ dX/dτ = Y...(3)’
Ray tracing technique（２）・At the starting point of tracing,

Reference：
１Principles of Optics,ⅠBorn, Max /Wolf, Emil

２Direct assimilation of GPS/MET refraction

angle measurements, X. Zou et al.

Tracers are traced in the three
dimensions space. Traces that
arrived at the upper boundaryof the domain are used.
GPS sat.
upper boundary

X0=0, Y0 = n0 dX/ds
・X is calculated as follows;
Xm+1 = Xmn∇n+ dτ * Ym+0.5

Ym+1.5 = Ym+0.5 + dτ * (n∇n) at Xm+1・Y0.5 is estimated, implicitly.
X1 = X0 + dτ * Y0.5

xm

xm+2yxm+1 m+1.5

ym+0.5

GPS receiver

Y0.5 = Y0 + 1/2 * dτ * (n∇n) at X0.25 = Y0 + 1/2 * dτ * (n∇n) at (X0 + dτ*Y0.5/4)

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3. Mountain lee wave simulated bynumerical model (7 March 1997) ・Line-shaped cloud bands
1036JST

GMS-Orientation：
north-south direction-Location：
East of Izu peninsula-Wave length：～15km
・Numerical model
simulated well mountainlee wave.

Schematic illustration ofmountain lee wave andwater vapor distribution(Shimada et al. 2001)
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Positions estimated with ’Constant model ’
Zenith wet delay

7th 9JST

6th 9JST8th 9JST

Movement of position with model of τest = τzenm(θ).
Positioning error （observation）

observation Westward /Deviation of PWV and・ Numerical
positioning error（simulation）
Westward/
Eastward

Eastward

Eastward/
Westward(7th)-(6th)Eastward/
Westward(8th)-(7th)(7th)-(6th)(m)(8th)-(7th)model simulated
well the featureof positioning
error.
・Positioning
error wascaused bymountain leewave.
10Gradient estimated with 'Linear gradient model'

Linear gradient model: τest=τzenm(θ)+ m(θ)/tan(θ)(GNcosφ+GEsinφ).
Observed gradient at el.=10°Simulated gradient at el.=10°

5105
5105

2108

2108

・Simulated gradients pointed
to large PWV region fromsmall PWV region.
・Simulated directions ofgradient are consistent withobserved ones, except KWN
and 3042.

10cm(m)

Large

Small

Small

Large

Total delay converted in the zenith direction (m)11Improvement of positioning error by using the gradientpositioning error

Delays in the zenith direction (m)Constant model

Large

Small

Small

Large

51052108(m)(m)Residual (m) (=τ - (τzenm(θ)+ m(θ)/tan(θ)(GNcosφ+GEsinφ)))Linear gradient model

8mm51052108(m)・Positioning error was reduced by using 'Linear gradient model'.
・Large positioning error remained at 2108 where delay did not varylinearly. ←Large residual in red circle causes the large error.
12Number of wave in Skymap is a half at most→Atmospheric Model is extended to'Second-ordermodel'
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τest = τzenm(θ)+ m(θ)(GNy+GEx+GN2y +GNExy+GE2x ),where x=1.0/tan(θ)*sinφ, y=1.0/tan(θ)*cosφ
Residual (m) (=τ - (τ m(θ)+ m(θ)/tan(θ)(G cosφ+G sinφ)))positioning error
8mm
Linear gradient modelzenNE・ Large positioning
error remained
where delay did notvary uniformly(red circle).

5105
2108(m)Residual (m)(=τ - τzenm(θ)+m(θ)(GNy+GEx+GN2y2+GNExy+GE2x2))4mm・ Second-order model

Second-order model
5105

reduces positioning
error.

2108(m)・Residual in a red
circle was decreased.
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Number of wave in Skymap is a half at most→Atmospheric Model is extended 2to Second-ordermodel2τest = τzenm(θ)+ m(θ)(GNy+GEx+GN2y +GNExy+GE2x ),where x=1.0/tan(θ)*sinφ, y=1.0/tan(θ)*cosφ
Positioning error

Positioning error in east-west direction (m)Linear gradient modelLinear gradient model

6mm

5105
2108
(m)

Second-order model

Second-order model

2mm

5105

・ E-W componentof residual (O-C )is expressed as-(τ-τest)cosθsinφ.
・ Second order model
reduces
E-W component ofresidual (O-C)
・ Large residual ina red circle wasdecreased.

2108(m)14

Improvement of vertical positioning error at 2107
Zenith total delayδz :－Σ{(τ－τest)sinθ}/Nwith linear gradient model
+1cm

0.8
0.6
0.4
0.2
0-0.2-0.4-0.6

2106
12107
2Constantmodel
without
gradient(cm)+0.4cmδz= 0.187cm

δz=0.679cm

δz= 0.687cm

2108
33042
4

Linear gradientwith gradient(cm)model

δz :－Σ{(τ－τest)sinθ}/Nwith second-order model

5105
54111
6

Second-orderwithquadrant (cm)model

・When the delays isestimated withsecond ordermodel, vertical
positioning errors
were reduced.
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Vertical component of residualImprovement of vertical positioning error－(τ－τest)sinθ

•'Constant model', 'Linear gradient model':Large residual O-C existed at the large elevation angles.
・Vertical component : Large residual was multiplied by sinθ,⇒ Large vertical positioning error remained.
•'Second-order function fitting model' :Residual at the large elevation angle is small⇒Vertical positioning error was reduced

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5. Summary
・The positioning error was evaluated with the delays simulatedby the numerical weather model.
Simulated refractivity distributions are useful for evaluation of
positioning error.
・Small scale variation of delay-Positioning error is greatly reduced by using gradient model.
-Second order curve fitting model improves the positioningerror further.
-Second order term is essential for the improvements of vertical
positioning error.
Acknowledgement
This study is the results of the GPS meteorology / Japan. Authors thankProf. Heki and Dr. Iwabuchi for advising estimation method ofthe positioning error.

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