J. Lapazaran A. Martín-Español J. Otero F. Navarro International Symposium on Radioglaciology 9-13 September 2013, Lawrence, Kansas, USA On the errors involved in the estimate of glacier ice volume from ice thickness data Photo: J. Lapazaran 1

Objectives Analyze the error sources & transmit them to the volume estimate. Which are the sources? Evaluate each error value. Combining errors. DATA: georadar ice thickness DEM of glacier ice thickness Glacier ice volume estimate Involved processes Steps on error estimation Step1 Thickness error in georadar data. Step2 Thickness error in DEM. Step3 Error in volume. 2

Step1: Thickness error in georadar data Data error Hdata can be split in 2 independent errors being GPR or other georadar type Error in thickness measurement, HGPR Error in thickness positioning, HGPS being positioned by GPS or other positioning system 3

Step1: Thickness error in georadar data HGPR : Error in thickness measurement Hypothesis: Zero offset profiling: (Dyn. corr. ). Migrated radargram. Only picking where bed is clearly identified. HGPR can be split in 2 independent errors Error in RWV, c Error in TWTT, Ƭ 4

Step1: Thickness error in georadar data HGPR : Error in thickness measurement c : Error in RWV RWV is measured (CMP) or estimated by experience. We look for the mean RWV of the profile. Bias: Error in the mean value of RWV chosen for the profile. Rnd. error ( c ): Variability around the mean RWV along the profile. Bias: Unknown sign. 2% in CMP (Barret et al, 2007) 2% of 168 = 3.36 m/µs, so ±2% of c means ± 2% of H. It must be considered separately. Rnd. Error ( c ): About another 2% m/µs m/µs 5

Step1: Thickness error in georadar data HGPR : Error in thickness measurement Ƭ : Error in TWTT Frequency of the radar Threshold for vertical resolution Widess (1973) ʎ / 8 → 1 / 4f in TWTT (in absence of noise). Yilmaz (2001) ʎ / 4 → 1 / 2f in TWTT. Reynolds (1997) ʎ / 4 (theoretical) not realistic in real media. Barret et al (2007) an error of ʎ is not impossible → 2 / f in TWTT. We conservatively take ʎ / 2 → Ƭ = 1 / f in TWTT. Resolution of the recording Sampling resolution. Much smaller than 1 / f. NEGLIGIBLE Migration Profile must be migrated. CAUTION with profiles close to lateral walls (or 3D migration). Moran et al (2000) found 15% of error in a small sample of 100x340 m. Picking error DO NOT PICK if not sure where the bed is (scattering, clutter). 6

Step1: Thickness error in georadar data HGPS : Error in thickness due to bad positioning Grows with the steepness of the thickness field. Negligible in DGPS. GPS in autonomous → XY = 5 m. We build the thickness DEM and evaluate its steepness in n directions around each measuring point: Odometer 7 → mean value of the n differences of thickness between the n surrounding points ( k ) and the evaluated point ( i ) Using the same method but we must estimate D. Is there any GPS track? Who have done the profile? 5-20% of the length, at the centre of the profile.

Data errors transmitted to grid points ( x k ) Thickness in DEM grid points ( x k ) Interpolation errors in grid points ( x k ) Step2: Thickness error in DEM Errors in DEM construction Hdata i can be considered independent 8 Transmission to DEM grid points. I N T E R P O L A T I O N Georadar thickness data ( x i )

Step2: Thickness error in DEM ( x k ) HGPR : Data errors transmitted to grid points We have interpolated the measured data H ( x i ) in the grid points x k : Now, data error are propagated into the grid using the same interpolation weighting: 9 points with georadar measurement grid points

Step2: Thickness error in DEM ( x k ) Hinterp : Thickness interpolation error Georadar data: High concentration of data in several lines. Huge spaces without data. Evaluation of the interpolation error: Cross-validation evaluates the error in data-concentrated zones but not in data-free zones. Useless for georadar data interpolating. Kriging variance (if interp. with kriging) has been criticized (Rotschky et al, 2007; Journel, 1986; Chainey and Stuart, 1998) as "been ineffective and poor substitute for a true error", "the kriging variance, depending only on the geometrical arrangement of the sample data points, simply states that accuracy decreases with growing distance from input data". 10

11 Step2: Thickness error in DEM ( x k ) Hinterp : Thickness interpolation error Distance-Error & Distance-Bias Functions (DEF & DBF) Take (e.g.) 10 values of distance, between 0 and the maximum distance between grid point and measured point. For each distance value, center a blanking circumference of this radio on each data point and interpolate with remaining data -one at a time-. Mean discrepancies (biases) and their standard deviations (errors) are calculated for each distance.

12 Step2: Thickness error in DEM ( x k ) Hinterp : Thickness interpolation error Distance-Error & Distance-Bias Functions (DEF & DBF) DBF & DEF are the mean squared adjusted curves. DBF shows how the bias has negative values that grows with increasing the distance to the nearest measurement. DEF shows how the error grows with increasing the distance to the nearest measurement.

13 Distance (m) Bias (m) Frequency -A bias value is applied to every cell in the grid, modifying the kriging prediction. -Every cell in the grid receives an error value from the DEF. Step2: Thickness error in DEM ( x k ) Hinterp : Thickness interpolation error Distance-Error & Distance-Bias Functions (DEF & DBF) A bias value and an error value are extracted from DBF and DEF and assigned to each node in the DEM grid, depending on its distance to the nearest measurement.

Step3: Error in volume Volume error V can be split in 2 independent errors Error in volume due to error in thickness, VH Error in volume due to boundary error, VB 14

15 Step3: Error in volume VH : Error in volume due to error in thickness Can thickness errors be considered independent? Are they linearly dependent? There is a spatial dependency among ice thickness measurements due to the surface continuity and thus their errors are correlated too

16 Step3: Error in volume VH : Error in volume due to error in thickness Error correlation The Range is the greatest distance to consider correlation. Semivariogram relates the spatial correlation between pairs of points and the distance separating them.

17 N R : Number of independent values = Number of points separated the independence distance (Range) Step3: Error in volume VH : Error in volume due to error in thickness We consider the glacier to have an independency degree derived from the number of range-size subsets.

18 Step3: Error in volume VB : Error in volume due to boundary error

H A12 = 24 m !! 19 Step3: Error in volume VB : Error in volume due to boundary error Glacier covered by moraines. Rocks covered by snow. fA (%)

20 Step3: Error in volume VB : Error in volume due to boundary error

21 Step3: Error in volume VB : Error in volume due to boundary error

22 Step3: Error in volume VB : Error in volume due to boundary error

23 Step3: Error in volume VB : Error in volume due to boundary error

24 Step3: Error in volume VB : Error in volume due to boundary error

What about the pixelation errors? Related to the software used to mask the ice thickness map. ArcGis 9.3: - Inner cells are error free. - Frontier cells: NEGLIGIBLE Can be considered included in the boundary uncertainty error. At each boundary cell, it can be approximated by the standard deviation of an uniform random variable between plus and minus half the cell area times the mean boundary-cell thickness (being zero the boundary thickness). 25 Step3: Error in volume VB : Error in volume due to boundary error

26 Summary

27 Results Weren. 1 Weren. 2 Werenskioldbreen

28 Results

On the errors involved in the estimate of glacier ice volume from ice thickness data Thank you ! 29 Photo: J. Lapazaran

30 for your attention... Photo: J. Lapazaran Thank you ! On the errors involved in the estimate of glacier ice volume from ice thickness data

31 Javier Lapazaran