Error Modeling

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Error Modeling

• Thomas Herring
• Room 54-611 617-253-5941
• tah_at_mit.edu
• http//geoweb.mit.edu

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Overview

• Notations
• Error modeling of GPS phase measurements
• Error modeling in time series analyses

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Notations

• Terms that are used in gamit/globk
• Weighted-root-mean-square (WRMS) scatter square
  root of the mean of residuals squared weighted by
  the estimated variance of the residuals.
• Normalized-root-mean-square (NRMS) scatter
  square root of mean of sum of residuals divided
  by their standard deviations squared. Square-root
  of chi-squared per degree of freedom. Should be
  near unity.

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GPS phase noise modeling

• A common assumption in GPS analyses is to assume
  that all phase data has the same random noise
  level. Examination of phase residuals
  (phs_res_root option in autcln.cmd) shows this is
  not the case.
• Default gamit processing uses a site dependent
  elevation angle dependent model
• The model is computed in autcln and passed to
  solve through the n-file (noise file).
• The noise levels are increased so that gamit NRMS
  of phase residuals is 0.2-0.3. Short-term
  scatter of position estimates suggest sigma 1-2
  times too small. (Positions estimates do not
  depend on sampling rate to about 4 minutes).

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Globk re-weighting

• There are methods in globk to change the standard
  deviations of the position (and other parameter)
  estimates.
• Complete solutions
• In the gdl files, variance rescaling factor and
  diagonal rescaling factors can be added.
• First factor scales the whole covariance matrix.
  Useful when
• Using SINEX files from different programs
• Accounting for different sampling rates
• Second factor is not normally needed and is used
  to solve numerical instability problems. Scales
  diagonal of covariance matrix.
• Needed in some SINEX files
• Large globk combinations (negative chi-square
  increments) Large combinations are best done
  with pre-combinations in to weekly or monthly
  solutions
• Individual sites with sig_neu command. Wild
  cards allowded in site names (both beginning and
  end)

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Time series noise

• The most complicated aspect of GPS timeseries
  analysis Basic problem Errors in GPS position
  estimates are both spatially and temporally
  correlated
• Spatial correlations
• Handled in glorg by using a local definition of
  reference frame. Stable sites in the area are
  used in the determining translation and rotation
  of local frame. Generally, the smaller the area
  the smaller the RMS scatter of position
  estimates. (Median WRMS scatter for sites in
  Southern California lt1 mm horizontal and lt3 mm
  vertical Same sites for North America frame 1
  mm and 3 mm.
• Temporal correlations
• More difficult because stochastic process of
  noise(s) not known and for velocity sigma
  estimates lowest (least well determined part of
  spectrum needed)

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Spectrum for Non-stable GPS site California
frameNon-Stable site
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PSDNorth America Frame
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Example Power Spectral Density
Southern California reference frame Very stable
sites
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PSD of same site in North America frame
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Realistic Sigma Algorithm

• Motivation was to develop an algorithm that could
  quickly estimate standard deviations of
  parameters estimated from time series that could
  have gaps and variable quality data
• Concept Common practice is to scale standard
  deviations of parameter estimates by the scatter
  of the post-fit residuals Valid if residuals are
  white noise. Scatter of residuals is
  characterized by chi-squared per degree of
  freedom should near unity.
• If data are averaged over some interval (e.g., 1
  week or month), then fit of parameters (such as
  rate) rarely affected but the chi-squared of
  averaged residuals will increase if residuals are
  not white noise Hence scale by new chi-squared
  But if longer averages are used , chi-squared
  keeps increasing so scale standard deviations by
  chi-squared obtained with infinite time
  averaging. Later step requires as method for
  extrapolating chi-squared estimates from short
  averaging times to long ones. We use a
  first-order Gauss Markov model to this.

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Algorithm

• Time series parameters are estimated using
  standard weighted least squares
• Postfit residuals from this fit are averaged from
  successive longer intervals and chi-squared
  computed for the averaged residuals.
• A first-order Gauss Markov process model
  (characterized by a variance and correlation
  time) is fit to the estimates of chi-squared.
• The chi-squared value for infinite averaging time
  predicted from this model is used to scale the
  white-noise sigma estimates from the original
  fit.
• Example CVHS (part of the Baldwin Park event)

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Raw times series (error bars not shown)
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Red Lines are realistic sigma of rate estimates
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Realistic Sigma chi-squares for East component
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Site with post-2005 data. Notice realistic error
estimate
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Chi-square with averaging for pre-2005 and
post-2005 data
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How well to estimate agree?

• Rate comparisons
• Comp lt2005 mm/yr gt2005 mm/yr ??mm/yr
• North 23.81 0.09 23.51 0.18 0.30
• East -26.08 0.13 -25.04 0.69 -1.04
• Height -4.52 0.13 -3.92 0.40 -0.60
• Adding the additional data after 2005, make rate
  sigma increase factor 5. In this case, algorithm
  seems to do a reasonable job of computing sigma.

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Summary

• All algorithms for computing estimates of
  standard deviations have various problems
  Fundamentally, rate standard deviations are
  dependent on low frequency part of noise spectrum
  which is poorly determined.
• Assumptions of stationarity are often not valid
  (example shown)
• Realistic sigma algorithm implemented in tsview
  and enfit/ensum sh_gen_stats generates mar_neu
  commands for globk based on the noise estimates