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

J. LapazaranA. Martín-EspañolJ. OteroF. Navarro

International Symposium on Radioglaciology9-13 September 2013, Lawrence, Kansas, USA

On the errors involved in the estimate of glacier ice volume from ice thickness

data

Photo: J. Lapazaran

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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.

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Step1: Thickness error in georadar dataData error ԐHdata can be split in 2 independent errors

being GPR or other georadar type

Error in thickness measurement, ԐHGPR

Error in thickness positioning, ԐHGPS

being positioned by GPS or other positioning system

2 2i i iHdata HGPR HGPS

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Step1: Thickness error in georadar dataԐHGPR : 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 errorsError in RWV, Ԑc Error in TWTT, ԐƬ

/ 2H c 22 /ad c

2 2 2 212i i iHGPR c c

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Step1: Thickness error in georadar dataԐHGPR : Error in thickness measurementԐc : 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/µs171.4 m/µs/2H c

2 2 20.022i iHGPRc

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Step1: Thickness error in georadar dataԐHGPR : Error in thickness measurementԐƬ: 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 → 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).

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Step1: Thickness error in georadar dataԐHGPS : 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:

• Odometer7

2

1

11i i

n

HGPS k ik

d dn

( )

𝑑𝑖 → 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.

D

2 2H k k Hinterp k HGPRx x x ( ) ( ) ( )

Data errorstransmitted to grid points (xk)

Thickness in DEM grid points (xk)

Interpolation errors in grid points (xk)

Step2: Thickness error in DEMErrors in DEM construction

ԐHdata i

can be considered

independent 8

Transmission toDEM grid points.

I N T E R P O L A T I O N

Georadar thicknessdata (xi)

k Hinterpx( ) k HGPRx( )

Step2: Thickness error in DEMԐ(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:

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1

n

k i ii

H x H x

( ) ( )

1

n

k HGPR i i HGPRi

x x

( ) ( )

points with georadar measurement

grid points

Step2: Thickness error in DEMԐ(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". 10

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Step2: Thickness error in DEMԐ(xk)Hinterp : Thickness interpolation errorDistance-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.

R1R2

R3R4

-300

-200

-100

0

100

200

0 200 400 600 800 1000 1200 1400

Discrepancy (m) vs Distance (m)

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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.

y = 3E-08x3 - 7E-05x2 + 0,0027x - 5,4091R² = 0,9998

-70-60-50-40-30-20-10

00 500 1000

DBF

y = -6E-09x3 - 1E-05x2 + 0,0586x + 17,584R² = 0,9998

010203040506070

0 500 1000

DEF

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Distance (m)

Bia

s (m

)Fr

eque

ncy

- 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.


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 ԐV can be split in 2 independent errors

Error in volume due to error in thickness, ԐVH

Error in volume due to boundary error, ԐVB

2 2V VH VB

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1

N

k kk

V A H

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Step3: Error in volumeԐVH : 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

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Step3: Error in volumeԐVH : 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.

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NR: Number of independent values =Number of points separated the independence distance (Range)

2Area

RangeRN

Step3: Error in volumeԐVH : Error in volume due to error in thicknessWe consider the glacier to have an independency degree derived from the number of range-size subsets.

2 2

1i

N

VH c HiR

N AN

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Step3: Error in volumeԐVB : Error in volume due to boundary error

HA12 = 24 m !!

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• Glacier covered by moraines.• Rocks covered by snow. fA (%)

glacier

bedrock20


glacier

bedrock

debris

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glacier

bedrockHGPR

debris

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glacier

bedrockHGPR

debris

error

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H m

de

vol er r or L e H = / 2m

H m = V / A

H m

L

earea error, f A

L e = f Avol er r or f V = / 2

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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:

2 2

12bc

A c

N

Vpix bc A

A

H

NEGLIGIBLECan 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).

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Uncertainty inDEM boundary

Error involume

Step3

NR

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Summary

DEM construction: Transmission to grid

Thickness data

Interpolation

• Thickness DEM (grid)• Data error Interpolation error = = (grid)Thickness error in DEM

Step2

HGPR

Hdata

HGPS

c RWV ( )

TWTT ( )

Step1

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Results

Weren. 1 Weren. 2

Werenskioldbreen

G la c ie r Vol. in te rp . E rro r Error as in d e p en d . Kriging Varian c e Real Vo l. (km 3 ) (km 3 ) (km 3 ) (km 3 ) (km 3 )

Weren. 1 2.80 0.10 0.001 0.21 2.88

Weren. 2 2.76 0.13 0.001 0.27 2.88

Aldg. 1 0.46 0.02 0.0001 0.06 0.45

Aldg. 2 0.44 0.02 0.0001 0.05 0.45

Aldg. 3 0.43 0.03 0.0001 0.08 0.45

Paiel. 1 1.24 0.09 0.0001 0.12 1.28

Paiel. 2 1.30 0.04 0.001 0.13 1.27

Hans. 1 0.991 0.05 0.001 0.10 1.04

Hans. 2 0.993 0.05 0.001 0.16 1.04

Hans. 3 1.02 0.05 0.0001 0.08 1.04

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Results


data

Thank you !

29Photo: J. Lapazaran

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for your attention...Photo: J. Lapazaran

Thank you !


data

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Javier [email protected]

s

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

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