The Effect of Turbulence on GPS Signals: Theory and Measurements

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The Effect of Turbulence on GPS SignalsTheory
and Measurements

• Larry Cornman and Rod Frehlich
• National Center
• for
• Atmospheric Research
• USA

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Measurements of Turbulence from GPS Signals

• Is there signal in the signal?
• or
• What is one persons noise is anothers signal

GPS/MET occultation through turbulence?
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Theoretical Background for GPS Occultations
Through Turbulence

• As a microwave signal passes through a field with
  fluctuating index of refraction, the signal
  amplitude and phase will fluctuate.
• The statistics of the fluctuations can be
  calculated for certain conditions.

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• Occultation Geometry

R is the distance from the GPS to the LEO, and z1
is the distance from the GPS to the turbulence.
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Assumptions

• Narrow angular scattering by turbulence.
• Parabolic wave equation is valid.
• Background gradient in refractive index has
  linear dependence over scattering region (tilt
  only no focusing).
• Locally homogeneous turbulence.

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Assumptions, cont.

• The transmitter and receiver velocities are
  constant over the sampling time.
• Ionospheric effects are negligible
• Effects of attenuation can be removed by
  detrending.

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Phase Screen Approximation

• Turbulence occupies small region compared to path
  length
• GPS signal experiences only phase fluctuations
  after passing through turbulence
• Intensity fluctuations grow with distance from
  the turbulence from interference of scattered
  waves

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GPS Measurements

• Electric field at the receiver
• A(x,R) - amplitude
• S(x,R) - phase

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Accuracy of the Mean Estimates

• Variance of mean amplitude over measurement
  length L
• Leff - effective correlation length
• Defined by correlation of A
• is the number of independent samples

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Effective Correlation Length

• For homogeneous turbulence
• ltAgt - ensemble average of A
• s,r - distance along receiver track

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Scintillation Statistics

• Normalized amplitude fluctuations (modulation
  index)
• normalized amplitude spectrum

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Weak Scattering Theory

• Normalized amplitude fluctuations are small mlt0.5
• Phase screen statistics are isotropic
• Amplitude and phase spectrum have simple form

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• Occultation Geometry

R is the distance from the GPS to the LEO, and z1
is the distance from the GPS to the turbulence.
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• Theoretical amplitude frequency spectrum in the
  coordinate system of the GPS
• Where, the effective velocity is

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• Spectrum of phase fluctuations under the phase
  screen approximation
• is the local spectrum of the
  
• refractive index
  fluctuations.
• Using a von Karman spectrum gives

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Theoretical phase frequency spectrum in the
coordinate system of the GPS
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• Therefore, the proportionality between the
  measured amplitude spectrum and the turbulence
  spectrum given all the inherent assumptions
  is
• Where is the integral from above.
• The phase spectrum has a similar structure.
• The turbulence intensity information is contained
  in the term , which unfortunately is a
  function of the unknown size of the turbulent
  region.

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EXAMPLES OF GPS TURBULENCE

• GPS/MET data from upper troposphere, lower
  stratosphere
• Negligible humidity effects.
• Altitude of commercial transport aircraft in
  cruise.
• Amplitude time series (SNR) was detrended
  (normalized by best-fit polynomial).
• Phase time series was detrended (subtracted best
  fit polynomial).
• Amplitude and phase spectrum compared to theory.

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• Occultation 546 Raw Amplitude and Altitude (via
  phase)

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• Occultation 546 Raw and detrended amplitude

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• Occultation 546 Raw and detrended phase

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• Occultation 546 Amplitude and Phase Spectra,
• Measured and Model

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Occultation 586 Raw Amplitude and Altitude (via
phase)
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Occultation 586 Amplitude and Phase Spectra,
Measured and Model
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Occultation 641 Raw Amplitude and Altitude (via
phase)
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Occultation 641 Amplitude and Phase Spectra,
Measured and Model
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Application to Aircraft Platforms

• Unfortunately, the number of GPS-LEO occultations
  is limited and hence not very useful in an
  operational sense.
• What about using GPS receivers on commercial
  aircraft?

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Turbulence Scales of Motion
Turbulent eddies
Large eddies
Small eddies
10s km
cm
Aircraft responds to scales from few m few km
energy cascade
Energy
turbulence
no turbulence
Mesoscalemodelresolution
NWPresolution
1 eddy size
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Theoretical Amplitude and Phase Spectra for an
Aircraft 50 km from Turbulence
Noise Floor From GPS-Met
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Aircraft Flight Track with GPS Occultations
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Aircraft Flight Track with GPS Occultations
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Elevation angle from Aircraft to Intersection
Point

Angle, In Deg.
-
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3 Hours of MDCRS Data (Met reports from
commercial aircraft)
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Summary

• Occultation data from GPS-MET is consistent with
  theoretical turbulence and scattering models
  especially amplitude data.
• Aircraft platform seems to be feasible more
  work is required to fully understand the issues.