Error Modeling

1. Error Modeling Thomas Herring Room 54-611; 617-253-5941 tah@mit.edu http://geoweb.mit.edu

2. Overview • Notations • Error modeling of GPS phase measurements • Error modeling in time series analyses Unavco GG Lec 03

3. 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. Unavco GG Lec 03

4. 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). Unavco GG Lec 03

5. 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) Unavco GG Lec 03

6. 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 <1 mm horizontal and <3 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) Unavco GG Lec 03

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9. Spectrum for Non-stable GPS site California frame:Non-Stable site Unavco GG Lec 03

10. PSD:North America Frame Unavco GG Lec 03

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15. Example Power Spectral Density Southern California reference frame Very stable sites Unavco GG Lec 03

16. PSD of same site in North America frame Unavco GG Lec 03

17. “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. Unavco GG Lec 03

18. 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) Unavco GG Lec 03

19. Unavco GG Lec 03 Raw times series (error bars not shown)

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21. Unavco GG Lec 03 Red Lines are realistic sigma of rate estimates

22. Realistic Sigma chi-squares for East component Unavco GG Lec 03

23. Unavco GG Lec 03 Site with post-2005 data. Notice realistic error estimate

24. Chi-square with averaging for pre-2005 and post-2005 data Unavco GG Lec 03

25. How well to estimate agree? • Rate comparisons: Comp <2005 mm/yr >2005 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. Unavco GG Lec 03

26. 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 Unavco GG Lec 03