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 Evaluating elevation change estimates Three answers

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?

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 2A 2B 3A 3B 3C 3D 3E 3F

Elevation rate estimates: repeat track analysis Techniques Applied to 1 km segments of track Assume elevations follow: –as a matrix equation: Generalized inverse: Elevation rate and slope estimates: –Model covariance matrix: x y C zz … … C zx C xx … C zy C xy C yy Covariance between m y and dz/dt Variance in dz/dt

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

Elevation rate estimates: repeat track analysis Techniques Applied to 1 km segments of track Assume elevations follow: –as a matrix equation: Generalized inverse: Elevation rate and slope estimates: –Model covariance matrix: x y C zz … … C zx C xx … C zy C xy C yy Covariance between m y and dz/dt Variance in dz/dt

Interpretation of the model covariance matrix Techniques Interpretation of the model covariance matrix Expected errors in model parameters given by  dz/dt = C zz 1/2  x-slope = C xx 1/2  y-slope = C yy 1/2 Expected correlations between errors determined by off-diagonal components E[  y /  dz/dt ] given by C zy /C yy Correlation between cross-track slope and dz/dt given by C zy /(C zz C yy ) 1/2  dz/dt yy  dz/dt tan(  )=C zy /C yy correlation=C zy /(C zz C yy ) 1/2 C zz … … C zx C xx … C zy C xy C yy Covariance between m y and dz/dt Variance in dz/dt Covariance between m x and dz/dt

Interpretation of the model covariance matrix Techniques Estimates of dz/dt may be problematic if  dz/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? C zz … … C zx C xx … C zy C xy C yy Covariance between m y and dz/dt Variance in dz/dt Covariance between m x and dz/dt  slope  dz/dt  dz/dt  slope d  dz/dt /d(  slope ) =Czy/Cyy correlation=C zy /(C zz C yy ) 1/2

Magnitude of covariance terms Techniques  slope  dz/dt  dz/dt  slope d  dz/dt /d(  slope ) =Czy/Cyy correlation=C zy /(C zz C yy ) 1/2

Magnitude of covariance terms Techniques  slope  dz/dt  dz/dt  slope d  dz/dt /d  slope =C zy /C yy correlation=C zy /(C zz C yy ) 1/2 For most segments, there is a moderate dependence of dz/dt errors on slope errors BUT The correlation between the two is usually weak

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 Evaluating slope estimates Techniques x y

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

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

Interpret only the best models: Segments ignored if –  dz/dt > 0.2 ma -1 – RMS residual > 0.4 m OR – R 2 between dz/dt and slope > 0.5 AND –d  dz/d /d(slope) > 0.05 (m/a)/(m/km) Of the remaining points, 68% have  dz/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  dz/dt ≈ 0.07 ma -1 Along-track elevation rates Errors

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

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  dz/dt is small and C yz is small compared to C zz and C yy