COSMIC

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New Developments in Ground Based Meteorology
John BraunUCARCOSMIC Program
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Outline

• Review of Concepts in Previous Lectures
• GPS Observation Equation
• Estimation of Parameters
• Parameterization of Troposphere delay in GPS
  Software
• Atmospheric Gradients
• Parameterization Description of Gradients
• Slant Water Vapor (SW)
• Combination of parameter estimate and residuals
• Obtaining the most accurate SW
• Information contained within SW
• Tomography
• Design of the problem
• Linear representation of SW
• Addition of Constraints to Assist Solution
• Improvement of Vertical Resolution

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GPS Station and Observation Description

• Ground based GPS station measures carrier phase
  (and pseudorange) to all visible satellites.
  Typically this is between 6-12 satellites
• Carrier phase observations are a function of
  range to satellite, clock errors, initial
  ambiguities, ionosphere, neutral atmosphere, and
  noise.
• GPS analysis software uses the observations to
  estimate parameters that are a function of the
  observation equation.

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GPS Carrier Phase Observation Equations
Eliminate Ionosphere (Iik(t))
Eliminate Satellite Clock (cdk(t))
Eliminate Receiver Clock ((cdk(t)) and (cd(t))
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Review of GPS Analysis Software
Linearize the observation equation with Taylor
expansion
Create (O-C) values about apriori
Dimension of A is number of observations x number
of parameters
LSQ estimate
Update a-priori state with estimate
Residuals represent (O-C) of updated state
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Simplified Example of ZWD Estimate
swd4239
swd3286
swd2462
swd11210
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Solve For ZWD Estimate
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(No Transcript)
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Computation of Residuals
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Zenith Delay Estimation in Software
or
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Removal of Dry Delay with Surface Pressure
Surface pressure measurements remove almost all
hydrostatic delay. The remaining error is less
than 1 mm in PW/SW.
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Variability of Water Vapor as Observed by GPS
Homogeneous zenith model accurately removes the
large scale variation of water vapor. The
horizontal variability represents 0-20 of the
zenith value.
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Isolate Neutral Atmosphere From Observation
Equation
Parameterization of station-satellite delay is
not possible without additional constraints. The
degree of freedom would be too large (i.e. there
would be more parameters than linearly
independent observations).
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Atmospheric Gradients
Davis, J. L., G. Elgered, et al. (1993).
"Ground-based measurement of gradients in the
"wet" radio refractivity of air." Radio Science
28 1003-1018. Bar-Sever, Y. E., P. M. Kroger,
et al. (1998). "Estimating horizontal gradients
of tropospheric path delay with a single GPS
receiver." Journal of Geophysical Research 103
5109-5035. Aonashi, K., T. Iwabuchi, et al.
(2004). "Statistical study on precipitable water
content variations observed with ground-based
microwave radiometers." Journal of the
Meteorological Society of Japan 82(1B) 269-275.
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Atmospheric Gradients (Tilting Mapping)
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Slant Water Vapor

• Residuals are the observed minus computed (O-C)
  difference of the observations and estimated
  state.
• They represent the error in the model
  parameterization and observation noise.
• The largest error in the modeled parameters is
  the atmospheric delay.
• Combining the residuals with the ZWD estimate
  results in the SW value.

or
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Precise Point Positioning and SW

• Precise Point Positioning (PPP) provides single
  station single satellite residuals. GOOD
• These residuals contain satellite and receiver
  clock errors, ambiguities, and range errors. This
  makes PPP less sensitive to slant delays. BAD

Can not be solved in PPP. Need to compute double
differences of ambiguities
Can not remove exactly. Have to estimate or use
IGS product (0.1 ns 30 mm delay)
Can not remove exactly. Have to estimate (0.1 ns
30 mm delay)
Observation error - multipath, phase center
variations, measurement noise
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Noise in DD and PPP Residuals
RMS noise of PPP residuals is twice as large as
unwrapped DD residuals.
Braun, J. J., C. Rocken, et al. (2001).
"Validation of single slant water vapor
measurements with GPS." Radio Science 36 459-472.
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Slant Water Vapor (Double Difference Level)
A SW value derived from a combination of DD
residuals and ZWD from two stations can be
computed, but the combination of stations and
satellites makes their interpretation more
difficult.
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Double Difference Observable

• A double difference observable is very precise.
  All satellite clocks and receiver errors are
  completely removed from observation equation.
• Why not relate the double difference residual to
  the original one way carrier phase measurement?
• For M satellites and N stations, there are only
  (M-1)(N-1) linearly independent combinations.
• This makes exact mapping of double difference
  residuals to one way measurements impossible.

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Unwrapping DD Residuals

• For large numbers of observations, the residuals
  from the estimate are gaussian and unbiased (zero
  mean).
• Assuming that the estimation process resulted in
  precise estimates of all parameters, the
  residuals should represent the error in the the
  modeling of the observations and the observation
  noise.
• The largest component of the residuals is due to
  the mis-modeling of atmospheric delay.

Alber, C., R. H. Ware, et al. (2000). "Obtaining
single path phase delays from GPS double
differences." Geophysical Research Letters 27
2661-2664.
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DD Residuals to SD Residuals
Constraint is applied to each baseline (station
pair) Error in assumption is evenly distributed
to all single difference residuals for this
station pair.
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DD Residuals to SD Residuals
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Unwrap SD Residuals into ZD Residuals
Constraint is applied to each satellite. Error is
distributed to all stations that observe the
satellite. Can be minimized by increasing number
of stations in network.
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Errors in Coordinates
Vertical Coordinate Errors
These predominately affect the ZWD estimate and
therefore would have a systematic error in all
retrieved SWD
Horizontal Coordinate Errors
Tightly constraining the coordinates of a station
to incorrect horizontal coordinates will make the
resulting residuals in error. The SW will then
have an error that is a function of the satellite
elevation and azimuth.
Satellite Position Errors
The use of IGS rapid or final satellite orbits,
and double differencing will make satellite orbit
errors negligible for most continental size
networks and smaller.
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Station Position Errors
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Error Sources in Umodeled Residuals
Antenna PCV
Multipath
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Stacked Multipath Maps
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Atmospheric Delays During Squall Line
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SW Fluctuations at Dry-Line
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GPS and Pointing WVR
WVR
GPS
Braun, J. J., C. Rocken, et al. (2003).
"Comparisons of line-of-sight water vapor
observations using the global positioning system
and a pointing microwave radiometer." Journal of
Atmospheric and Oceanic Technology 20 606-612.
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Slant Water Provide Information on Relative
Variability of Water Vapor
GPS Sik (red line) and WVR Sik (blue line).
Elevation angle plotted as reference.
PW (circles) and SW (lines) scaled to zenith.
Elevation angle plotted as black line.
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SW Examples
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SW within Convective System
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Correlation of Slant Inhomogeneities
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Profile of Water Vapor Density
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Length of Path Through Boundary Layer
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Applications of SW

• PW accurately measures the large scale (gt60 km)
  water vapor variations.
• SW enhances the PW measurement by including the
  water vapor variations on small scales (lt60 km)
• SW can be directly used to study the variability
  of water vapor within the boundary layer.
• Should improve numerical forecasting. Results so
  far are promising, but inconclusive.
• SW collected from a network of GPS stations can
  be combined to estimate 3D distribution of water
  vapor.

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Examples of Tomography
Seismic Tomography Relates the propagation speed
of seismic signals through the Earths crust to
determine its density. The Earths crust can be
assumed to be invariant in time. This allows for
the combination of data collected over many
years.
http//www.geof.ruu.nl/bijwaard/abstracts/vakidio
ot/vak_uk.html
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Tomography Example - Medical Imaging
Magnetic Resonance Imaging Relates the
integrated attenuation of an RF signal that has
passed through a body to the composition of the
tissue. In this example both the source and the
receiving instrument can be positioned at nearly
any geometry. Combining these optimal observation
geometries and numerous precise measurements
allows for detailed reconstruction of the body.
http//www.gemedicalsystemseurope.com/euen/rad/mri
/products/signa/excite-neuro.html
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Tomography Observation Equation
In the tomography problem, each SW value is
approximated as a riemann sum. The atmosphere is
divided into voxels with constant density within
each voxel.
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Bending of Signal at Low Elevation Angles
The rays are assumed to follow the straight line
path to the satellite. For elevation angles above
5 degrees, this is essentially true
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Tomography Observation Equation
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Organization of Linear System of Equations
In the tomography problem, each SW value is
approximated as a riemann sum. The atmosphere is
partitioned into voxels (three dimensional
pixels) with constant density within each voxel.
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Density of Independent Observations
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Horizontal Constraints
Horizontal constraints provide linear
independence of observation equations through
smoothing.
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Addition of Vertical Profile
If the density profile is known at a location, it
can be included into the tomography solution as a
constraint or additional observation. Profile
information from an external source will improve
the tomography solution.
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Solution of Tomography With Constraints
The solution of the constrained tomography system
can be computed with regular LSQ (or Kalman
filter) estimation parameters.
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Simulated Tomography Solution
Simulated
Tomography
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June 12 Storm
Radar
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June 12 Surface Met and SW
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June 12 Surface Met and Tomography
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June 12 Surface Met and Tomography
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Summary

• Review of Concepts in Previous Lectures
• SW is an extension of standard analysis software
• Atmospheric Gradients
• Gradients provide an improved atmospheric model
  but still rather crude.
• Slant Water Vapor (SW)
• Combination of parameter estimate and residuals
• Double difference analysis provides most precise
  SW.
• Tomography
• Allows for 3D retrieval of atmosphere water vapor
• Horizontal constraints allow for less dense GPS
  geometry
• Weakest geometry is in vertical, using additional
  profile information improves geometry