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Towards repeat-track measurements of elevation change using ICESat/GLAS B. E. Smith

Towards repeat-track measurements of elevation change using ICESat/GLAS B. E. Smith. Funded by the ICESat science team With thanks to Charlie Bentley, Charlie Raymond, Ian Joughin, and Howard Conway. Repeat track elevation change. Outline. Three questions Data Estimating elevation changes

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Towards repeat-track measurements of elevation change using ICESat/GLAS B. E. Smith

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  1. Towards repeat-track measurements of elevation change using ICESat/GLAS B. E. Smith Funded by the ICESat science team With thanks to Charlie Bentley, Charlie Raymond, Ian Joughin, and Howard Conway

  2. Repeat track elevation change

  3. Outline • Three questions • Data • Estimating elevation changes • Evaluating elevation change estimates • Three answers

  4. Three questions • I have a study site on an ICESat track. Can I find the elevation rate? • Will cross-track slopes confuse the elevation rate estimates? • When should I believe that an along-track elevation rate is accurate?

  5. Elevations from GLA12 L2A, L2B and L3A, and L3D campaigns have the best pointing models. Data filtering for apparent reflectivity, pulse shape rejects 20% of all data Errors estimated from same-period crossovers over flat terrain and apparent surface roughness Data 3E 3F 3D 2A 2B 3A 3C 3B

  6. Techniques y x Variance in dz/dt Czz … … Czx Cxx … Czy Cxy Cyy Covariance between my and dz/dt Elevation rate estimates: repeat track analysis • Applied to 1 km segments of track • Assume elevations follow: • as a matrix equation: • Generalized inverse: • Elevation rate and slope estimates: • Model covariance matrix:

  7. Pick a point on a ground track Fit nearby ICESat data with a plane Estimate dz/dt from residuals to the plane Example

  8. Example

  9. Techniques y x Variance in dz/dt Czz … … Czx Cxx … Czy Cxy Cyy Covariance between my and dz/dt Elevation rate estimates: repeat track analysis • Applied to 1 km segments of track • Assume elevations follow: • as a matrix equation: • Generalized inverse: • Elevation rate and slope estimates: • Model covariance matrix:

  10. Techniques Variance in dz/dt Covariance between mx and dz/dt Covariance between my and dz/dt ey tan(q)=Czy/Cyy sdz/dt edz/dt Czz … … Czx Cxx … Czy Cxy Cyy sdz/dt correlation=Czy/(Czz Cyy)1/2 Interpretation of the model covariance matrix • Interpretation of the model covariance matrix • Expected errors in model parameters given by • sdz/dt= Czz1/2 • sx-slope= Cxx1/2 • sy-slope= Cyy1/2 • Expected correlations between errors determined by off-diagonal components • E[ey / edz/dt] given by Czy/Cyy • Correlation between cross-track slope and dz/dt given by Czy/(Czz Cyy)1/2

  11. Techniques Variance in dz/dt Covariance between mx and dz/dt Covariance between my and dz/dt edz/dt dedz/dt/d(eslope) =Czy/Cyy sdz/dt eslope Czz … … Czx Cxx … Czy Cxy Cyy sslope correlation=Czy/(Czz Cyy)1/2 Interpretation of the model covariance matrix • Estimates of dz/dt may be problematic if • sdz/dt is large • -Or- • there is a large slope error • -And- • There is a strong correlation between slope and dz/dt • -And- • errors dz/dt depend strongly on errors in the slope • How often is this a problem?

  12. Techniques edz/dt dedz/dt/d(eslope) =Czy/Cyy sdz/dt eslope sslope correlation=Czy/(Czz Cyy)1/2 Magnitude of covariance terms

  13. Techniques Magnitude of covariance terms • For most segments, there is a moderate dependence of dz/dt errors on slope errors • BUT • The correlation between the two is usually weak edz/dt d edz/dt/d eslope=Czy/Cyy sdz/dt eslope sslope correlation=Czy/(Czz Cyy)1/2

  14. Techniques y x Evaluating slope estimates • DEMs provide slope estimates on a 10+ km scale • Local slopes from dynamically supported topography are much larger. • Can derive slope estimates from crossing tracks

  15. Techniques • Red: slope estimates from ascending tracks • Blue: slope estimates from descending tracks • Purple: slope estimates from both ascending and descending tracks Evaluating slope estimates • Derive slopes at cross-over points • Compare ascending slope, descending slope, and cross- over slope

  16. Techniques m/km Slope error magnitudes • Derive slopes at cross-over points • Compare ascending slope, descending slope, and cross- over slope • Assume that the cross-over slope is correct.

  17. Interpret only the best models: Segments ignored if sdz/dt > 0.2 ma-1 RMS residual > 0.4 m OR R2 between dz/dt and slope > 0.5 AND dedz/d/d(slope) > 0.05 (m/a)/(m/km) Of the remaining points, 68% have sdz/dt < 0.06 ma-1 Cross-over analysis on dz/dt estimates is consistent with formal errors 68% of differences < 0.09 ma-1 Implies sdz/dt≈ 0.07 ma-1 Along-track elevation rates Errors

  18. Results • Lines: along track dz/dt estimates • Points: dz/dt estimates at crossings Elevation rates • Ross embayment ice streams

  19. Three questions • I have a study site on an ICESat track. Can I find the elevation rate? Usually • Will cross-track slopes confuse the elevation rate estimates? Usually not • How can I know that an along-track elevation rate is accurate? When sdz/dt is small and Cyz is small compared to Czz and Cyy

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