J. Lapazaran A. Martín-Español J. Otero F. Navarro

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

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

Step1: Thickness error in georadar data • Data error ԐHdata can be split in 2 independent errors Error in thickness positioning, ԐHGPS Error in thickness measurement, ԐHGPR being positioned by GPS or other positioning system being GPR or other georadar type

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

Step1: Thickness error in georadar data • ԐHGPR: Error in thickness measurement • Ԑc: 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 (Ԑc): 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 (Ԑc ): • About another 2% 164.6 m/µs 171.4 m/µs

Step1: Thickness error in georadar data • ԐHGPR: Error in thickness measurement • ԐƬ: 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 ʎis not impossible→ 2 / f in TWTT. • We conservatively take ʎ / 2 → ԐƬ = 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).

Step1: Thickness error in georadar data • ԐHGPS: Error in thickness due to bad positioning • Grows with the steepness of the thickness field. • Negligible in DGPS. • GPS in autonomous → ԐXY= 5 m. • We build the thickness DEM and evaluate its steepness in n directions around each measuring point: • Odometer • → 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.

Step2: Thickness error in DEM • Errors in DEM construction Georadar thickness data (xi) ԐHdatai Transmission to DEM grid points. I N T E R P O L A T I O N Thickness in DEM grid points (xk) Interpolation errors in grid points (xk) Data errors transmitted to grid points (xk) can be considered independent

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

Step2: Thickness error in DEM • Ԑ(xk)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".

Step2: Thickness error in DEM • Ԑ(xk)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.

Step2: Thickness error in DEM • Ԑ(xk)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.

Step2: Thickness error in DEM • Ԑ(xk)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. - 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. Frequency Bias(m) Distance (m)

Step3: Error in volume • Volume error ԐV can be split in 2 independent errors Error in volume due to boundary error, ԐVB Error in volume due to error in thickness, ԐVH

Step3: Error in volume • ԐVH: 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

Step3: Error in volume • ԐVH: 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.

Step3: Error in volume • ԐVH: 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. NR: Number of independent values = Number of points separated the independence distance (Range)

Step3: Error in volume • ԐVB: Error in volume due to boundary error

Step3: Error in volume • ԐVB: Error in volume due to boundary error HA12 = 24 m !! fA (%) • Glacier covered by moraines. • Rocks covered by snow.

Step3: Error in volume • ԐVB: Error in volume due to boundary error

Step3: Error in volume • ԐVB: 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: 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). NEGLIGIBLE Can be considered included in the boundary uncertainty error.

Summary

Results Werenskioldbreen Weren. 1 Weren. 2

Results

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

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

Javier Lapazaran javier.lapazaran@upm.es

J. Lapazaran A. Martín-Español J. Otero F. Navarro