Towards repeat-track measurements of elevation change using ICESat/GLAS

B. E. Smith

Funded by the ICESat science teamWith 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

3E3F

Elevation rate estimates: repeat track analysisTechniques

• 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

Czz … …

Czx Cxx …

Czy Cxy Cyy

Covariance between my 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

Example

Elevation rate estimates: repeat track analysisTechniques

• 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

Czz … …

Czx Cxx …

Czy Cxy Cyy

Covariance between my and dz/dt

Variance in dz/dt

Interpretation of the model covariance matrixTechniques

• Interpretation of the model covariance matrix• Expected errors in model

parameters given by

• dz/dt= Czz1/2

• x-slope= Cxx1/2

• y-slope= Cyy1/2

• Expected correlations between errors determined by off-diagonal components• E[y / dz/dt] given by Czy/Cyy

• Correlation between cross-track slope and dz/dt given by Czy/(Czz Cyy)1/2

dz/dt

y

dz/dt

dz/dt

tan()=C zy/C yy

correlation=C zy/(C zz

C yy)1/2

Czz … …

Czx Cxx …

Czy Cxy Cyy

Covariance between my and dz/dt

Variance in dz/dt

Covariance between mx and dz/dt

Interpretation of the model covariance matrixTechniques

• 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?

Czz … …

Czx Cxx …

Czy Cxy Cyy

Covariance between my and dz/dt

Variance in dz/dt

Covariance between mx 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 termsTechniques

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 termsTechniques

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 estimatesTechniques

x

y

Evaluating slope estimatesTechniques

• 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 magnitudesTechniques

• 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 – R2 between dz/dt and slope > 0.5

AND

– ddz/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 ratesResults

• 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

Cyz is small compared to Czz and Cyy