1. Estimation of ice basal reflectivity
12/01/2016
EECS - 823

2. Images: www.skepticalscience.com/graphics.php

3. Reasons ?
Natural:
Earth’s orbital change – ice age cycles ? | sun – solar activity ?| Volcanic activity – Co2 emissions ?
Man-made:
Deforestation – more sunlight reflection-cooling effect? | Ozone (Natural & pollution made)- ? | Aerosol – cooling ? |Greenhouse gases ?
Images:

4. Sea level rise Why this parameter ? Major sources:
93.4% of global warming goes into Oceans (section IPCC 4th Assessment report)
density of sea water depends on the temperature
It represents many aspects of hydrological cycles
used as a key observational constraint on climate models
Major sources:
Added water from melting land ice
Expansion of sea water as it warms
Estimated increase by 2100: cm (for RCP8.5 –with medium confidence- IPCC 5th Assessment report) could effect more than 100 million people[1].

5. Glaciers Mass loss ∝ sea level rise Antarctica [2][3]
extends almost 14 million sq. km
could potentially contribute 57 m
↓ 118 (+ 79) Gt/yr
Greenland [2][3]
Extends almost 1.7 million sq. km
Could potentially contribute 7.3 m
↓ 281 (+ 29) Gt/yr
Sub glacial Lakes
Effects ice sheets dynamics
Modeling
Predict future rise
Basal conditions are much need for accurate predictions
Habitat loss (image:shscreative.com/students/ebernal/dsg/images/whatisit3.jpg )

6. Methods Ice cores drilling Seismic sounding Radiometry
Direct evidence
Contamination of Ecosystems
Cant drill complete glaciers
Seismic sounding
More penetration depth
More time to setup
Radiometry
Upwelling radiation from ice sheet is ~ 2.1K (assumptions: Operating frequency < 700 MHz, water as the basal condition at 283k, average englacial attenuation is assumed to 2.99 Np/km and no scattering, lossless atmosphere- above ice sheet)
Galactic noise and man made noise dominates
Radar (airborne)
Much feasible
Have to deal with englacial attenuation and scattering losses.

7. Radar equation (for specular surfaces with small scale roughness)[12]
𝑃 𝑟 = 𝑃 𝑡 𝜆 4𝜋 𝐺 𝑡 𝐺 𝑟 2 ℎ+ 𝑧 𝑛 𝑖 𝐿 𝑖 2 𝜌 𝑅 𝑏 2
𝑃 𝑟 - Received power; 𝑃 𝑡 - Transmitted Power;
ℎ - height of aircraft above ice; 𝑧 - Ice thickness;
𝐺 𝑡 - Receiver Gain; 𝐺 𝑡 - Transmitter Gain;
𝜆 – wavelength of radar signal (in air) ; 𝑛 𝑖 - index of refraction for ice
𝐿 𝑖 -Englacial attenuation
𝜌 accounts for small roughness scattering effects
R b is the averaged basal reflection coefficient over the imaged resolution cell, given by (nadir looking)
𝑅= 𝜀 1 − 𝜀 𝜀 𝜀 Where 𝜀 1 and 𝜀 2 are complex dielectric permittivity of two media

8. 𝜌 – power reduction due to roughness
The vertical root mean square (RMS) displacement of surface from its mean plane 𝜎 produces RMS variation in TWTT, 𝜎 𝑡 ,given by 𝜎 𝑡 = 2𝜎 𝑐 Corresponding RMS phase variation,∅,is given by ∅= 4𝜋𝜎 𝜆 𝑠 𝜆 𝑠 is the wavelength at the reflecting surface. power reduction due to roughness is given by [10] 𝜌= 𝑒 − ∅ 2 𝐼 𝜊 2 𝜙 2 2 𝐼 0 is the zeroth-order modified Bessel function of the first kind

9. 𝜎 RMS height Method 1: C. S. Neal [4]
using statistics of power variance 𝜎 𝑝 and fading length 𝑥 𝑓
𝜎 𝑝 = 〈P(x)P(x)〉 P x 2 −1
X represents the long track position
Auto correlation function, 𝜌 𝑝 𝜇 , is given as
𝜌 𝑝 𝜇 = P x P x+μ − P x 𝑃(𝑥)𝑃(𝑥) − P x 2
Fading length is given as distance over which the correlation function drop to value of 1/e
𝑥 𝑓 = 𝜇| 𝜌 𝑝(𝜇) =1/𝑒

10. 𝜎 RMS height (Method 1 .…)
Estimate phase shift ∅ and correlation length 𝐿 using a theoretical model given by the equations
𝜎 𝑝 =2 𝜙 2 ( 1 1+2Γ − 1 1+ 𝑅 2 )
𝜌 𝑝 𝜇 = 𝑒 − 𝜇 𝐿 − Γ − 1+2Γ 1+ 𝑅 2 𝑒 − 𝜇 𝐿 𝑅 cos 𝜇 2 𝑅 𝐿 𝑅 𝑅 sin 𝜇 2 𝑅 𝐿 𝑅 (1− 1+2Γ 1+ 𝑅 2 )
𝑅 is the Fraunhofer region of the sounding geometry given by 𝑅= 𝜆(ℎ+ 𝑧 𝑛 )/(𝜋 𝐿 2 )
Γ= 𝜆 2 ℎ+ 𝑧 𝑛 /(4 𝜋 2 𝐿 2 𝜏), where 𝜆 is the wavelength, 𝑧 is the depth, ℎ is the terrain clearance, 𝑛 is the refractive index of ice and 𝜏 is the pulse length in terms of distance in air.
Valid only for 𝜙 2 <0.3 – not always the case for surface roughness

11. 𝜎 RMS height Method 2: convolution model for waveform analysis [5]
Received echo signal is modeled as multiple convolution of
radar point target 𝑃 𝑃𝑇𝑅 (𝑡)
flat surface impulse response 𝑃 𝐹𝑆𝐼𝑅 𝑡 and surface height density function 𝑃 𝑝𝑑𝑓 (𝑡)
𝑃 (𝑡) = 𝑃 𝑃𝑇𝑅 (𝑡) ∗ 𝑃 𝐹𝑆𝐼𝑅 (𝑡) ∗ 𝑃 𝑝𝑑𝑓
Same echo shape can be obtained for numerous bed and ice conditions [6]
Method 3: using statistics
Received signal power at antenna- summation of backscattered radiation – coherent (or deterministic) component and diffused (or random) component
Smooth surface – specular reflection – coherent component with negligible incoherent component

12. 𝜎 RMS height (Method 3…) [11]
Approximating the distribution of power values with 2- dimensional RICE distribution, given by
𝑃 𝑅𝑖 𝐴 𝑠,𝜎 = 𝐴 𝜎 2 𝐼 0 𝑠 𝜎 2 𝐴 exp − 𝑠 2 + 𝐴 𝜎 2
𝑠 - NonCentrality parameter (Coherent signal component of Rice distribution )
𝑠 2 - coherent signal power; 𝐴 2 - signal Intensity
𝜎 – scale parameter and 𝐴 - signal Amplitude;
2𝜎 2 - Diffuse signal power of Rice distribution
( 𝑠 2 + 2𝜎 2 ) is the total signal power of Rice distribution
𝐼 0 is zero-order modified Bessel function of the first kind

13. 𝜎 RMS height (Method 3…) [11]
Coherent power component is given by
𝑃 𝑐 = 𝑟 2 exp − 2𝑘 𝜎 ℎ 2
𝑟 - surface Fresnel coefficient (𝑟=( 1− 𝜀 1+ 𝜀 ) ); 𝜀 - dielectric constant of ice/firn 𝑘 - wavenumber(𝑘= 2𝜋 𝜆 ) ; 𝜎 ℎ - RMS vertical height
Incoherent component is given by
𝑃 𝑛 = 4 𝑘 2 𝑟 2 𝜎 ℎ 2 erf 𝜋 𝑙 𝑥 2𝐿 erf⁡(𝑘 𝑙 𝑦 𝑐 ℎΔ𝑓 )
𝑙 𝑥 and 𝑙 𝑦 are correlation lengths in 𝑥 and y directions, 𝐿 is the synthetic aperture length, ℎ is range to surface, Δ𝑓 is the signal bandwidth
For a given footprint dimension, a threshold exist beyond which correlation length can be neglected.

14. 𝜎 RMS height (Method 3…) [11]
RMS value can be obtained by ratio fitting a value for
𝑃 𝑐 𝑃 𝑛 = exp − 2𝑘 𝜎 ℎ [4 𝑘 2 𝑟 2 𝜎 ℎ 2 erf 𝜋 𝑙 𝑥 2𝐿 erf⁡(𝑘 𝑙 𝑦 𝑐 ℎΔ𝑓 ) ] −1
for correlation length greater than threshold, erf 𝑥 ≈1

15. Radar Equation (dB) [𝑃] 𝑑𝐵 = [𝑆] 𝑑𝐵 − [𝐺] 𝑑𝐵 + [𝑅] 𝑑𝐵 − 𝐿 𝑑𝐵
[𝑃] 𝑑𝐵 = [𝑆] 𝑑𝐵 − [𝐺] 𝑑𝐵 + [𝑅] 𝑑𝐵 − 𝐿 𝑑𝐵
P – Received power; (after correcting for roughness)
S – System parameters
G – Geometric spreading loss; ( [𝐺] 𝑑𝐵 = 2[2 ℎ+ 𝑧 𝑛 𝑖 ] 𝑑𝐵 )
R – Ice Bed Reflectivity
L – Englacial Attenuation given by
𝐿 𝑑𝐵 =2 𝑁 𝑎 z
𝑧 is Ice depth and 𝑁 𝑎 is Englacial attenuation rate
Power after geometric loss correction, [ 𝑃 𝑐 ] 𝑑𝐵 is
[𝑃] 𝑑𝐵 + [𝐺] 𝑑𝐵 = [ 𝑃 𝑐 ] 𝑑𝐵 = [𝑆] 𝑑𝐵 + [𝑅] 𝑑𝐵 − 𝐿 𝑑𝐵

16. Methods to estimate Englacial Attenuation 𝐿
Ice attenuation depends on [7]
dielectric permittivity
Imaginary part depends on
frequency and
Conductivity
density
impurity concentrations (increases with volcanic eruptions)
Temperature (Increases with depth of ice sheet)
Method 1:
laboratory grown pure ice and measuring conductance and dielectric permittivity for different impurity concentrations, pressures and temperature
Method 2:
to use Ice core data to calculate ice attenuation rate
due to differences in physical properties of ice, can’t be applied for complete glacier

17. Methods to estimate Englacial Attenuation 𝐿 (Method 3)
Relative geometrically corrected bed-echo power
[ 𝑃 𝑟 𝑐 ] 𝑑𝐵 = [ 𝑃 𝑐 − 𝑃 𝑐 ] 𝑑𝐵 = [𝑅− 𝑅 ] 𝑑𝐵 − 𝐿− 𝐿 𝑑𝐵
(Assuming [𝑆] 𝑑𝐵 are constants along a flight line) and
[ 𝑅 𝑟 ] 𝑑𝐵 = [𝑅− 𝑅 ] 𝑑𝐵 = [ 𝑃 𝑟 𝑐 ] 𝑑𝐵 + 𝐿− 𝐿 𝑑𝐵
𝐿− 𝐿 𝑑𝐵 = 2 𝑁 𝑎 z− 𝑧
z is the average ice depth
𝑃 𝑐 - Average geometrically corrected bed echo power
𝑅 - Average bed reflectivity
𝐿 - Average Englacial attenuation loss
𝑅 𝑟 - Variations in the material & geometric properties of the bed

18. Methods to estimate Englacial Attenuation 𝐿 (Method 3) [8]
From RES data, considering a linear relationship between depth and attenuation rate (neglecting the contributions due to internal layers)
i.e... Assuming 𝑁 𝑎 as constant
𝐿 𝑑𝐵 =2 𝑁 𝑎 z
Then [ 𝑅 𝑟 ] 𝑑𝐵 can be calculated from
[ 𝑅 𝑟 ] 𝑑𝐵 = [𝑅− 𝑅 ] 𝑑𝐵 = [ 𝑃 𝑟 𝑐 ] 𝑑𝐵 + 2 𝑁 𝑎 z− 𝑧
can’t be applied over the large areas due to variations in ice conditions and temperature profile

19. Methods to estimate Englacial Attenuation 𝐿 (Method 4) [9]
To account for spatially variable englacial attenuation rate – modeling the englacial attenuation rate as
𝑁 𝑎 = 𝑁 𝑎 + 𝜕 𝑁 𝑎 𝜕𝑥 (𝑥− 𝑥 )
𝑁 𝑎 is the average attenuation rate
𝑥 is the along track position
𝑥 is the mean position
𝜕 𝑁 𝑎 𝜕𝑥 is the long track derivative of attenuation rate

20. Surface Roughness estimation for byrd glacier
Using Rice Distribution for power values
Limiting to 5% of wavelength

21. Reflectivity Map - Using averaging – about every 200m (color axis scaled) after roughness correction with (using method 3) and using modeled attenuation rate (method 4)

22. References
[1] Douglas, Bruce C., and W. Richard Peltier, “The puzzle of global sea-level rise,” Physics Today, pp , March 2002.
[2] L. Allison, R. Alley, H. Fricker, R. Thomas, and R. Warner, “Ice sheet mass balance and sea level,” Antarctic Sci., vol. 21, no. 5, pp. 413– 426, Oct
[3] Lythe, M.B., D.G. Vaughan, and BEDMAP Consortium BEDMAP: A new ice thickness and subglacial topographic model of Antarctica. Journal of Geophysical Research 106(B6):
[4] C. S. Neal, Radio echo determination of basal roughness characteristics on the Ross Ice Shelf, Ann. Glaciol., 3, 1982, pp. 216– 221
[5] G.S. Brown, The average impulse response of a rough surface and its applications, IEEE Trans. Antennas Propagat. 25, 1977, pp. 67–74
[6] Paden John, “SYNTHETIC APERTURE RADAR FOR IMAGING THE BASAL CONDITIONS OF THE POLAR ICE SHEETS”, Ph. D. Thesis, University of Kansas, 2006
[7] Kanagaratnam, Pannirselvam, “Airborne radar for high-resolution mapping of internal layers in glacial ice to estimate accumulation rate,” Ph. D. Thesis, University of Kansas, 2002.
[8] Jacobel, R. W., B. C. Welch, D. Oserhouse, R. Pettersson, and J. A. Mac-Gregor, 2009, Spatial variations of radar-derived basal conditions on Kamb Ice Stream, West Antarctica: Annals of Glaciology, 50, 10–16, doi: /
[9] Dustin M. Schroeder, Cyril Grima, and Donald D. Blankenship (2016). ”Evidence for variable grounding-zone and shear-margin basal conditions across Thwaites Glacier, West Antarctica.” GEOPHYSICS, 81(1), WA35-WA43. doi: /geo
[10] Peters, M. E., D. D. Blankenship, and D. L. Moore, 2005, Analysis techniques for coherent airborne radar sounding: Application to West Antarctic ice streams: Journal of Geophysical Research, 110, B06303, doi: /2004JB
[11] Cyril Grima, , Dustin M. Schroeder, Donald D. Blankenship, 2014, Duncan A. Young, Planetary landing-zone reconnaissance using ice- penetrating radar data: Concept validation in Antarctica, Planetary and Space Science, 103:191–204,
[12] Sasha Peter Carter, Evolving Subglacial Water Systems in East Antarctica from Airborne Radar Sounding, Ph. D. Thesis, University of Texas at Austin, 2008