Boris Yurchak1) and Waleed Abdalati2) 1)University of Maryland, Baltimore County, Goddard earth sciences & Technology Center, MD 20771, [email protected] 2)Earth Science and Observation Center/CIRES, University of Colorado, Boulder, CO 80309, [email protected] An assessment of the snow accumulation rate over the remote regions of Greenland and Antarctic by satellite radars is an issue of a significant importance in the assessment of snow balance. Based on the experimental studies, it is commonly accepted that the radar backscatter at cm wavelengths is inversely proportional to the snow accumulation rate. Known backscatter models interpret this fact as being primarily due to a decrease in snow grain size with accumulation rate, which is a result of the snow densification processes driven by weather conditions. Our analysis of the radar backscatter, from the layered snow pack under the "slice" approach, shows that the backscatter from dry and thick snow/firn slab is governed primarily by the magnitude of variation in the variable jumps of the firn density at the internal layers' interfaces (which are both physical interfaces and dielectric interfaces), and by the fine-scale fluctuations in the concentration and sizes of the snow grains within the layers. The inverse trend of the return signal strength, with respect to the accumulation rate, can be caused by the relatively low values of the snow density variations within the scattering volume that are inherent, in accordance with the field data, in areas with high accumulation rates. It is well known that in low-accumulation regions the density variation in the subsurface firn increases, due to the length of the time a deposited firn layer remains near the surface and is more strongly exposed (in comparison with the high-accumulation regions) to the variable weather conditions. It is this phenomenon that is largely responsible for the corresponding increase in the radar backscatter in low accumulation areas.

I. SUMMARY OF APPROACH A. initial conditions The scheme of radar sounding of a thick snow pack is depicted in Fig.1.1. It is considered that the continuous wave (CW) radar is located stationary at the height R0 and illuminates a snow pack in the close to nadir direction with the spherical wave. The adjoining layers inside the pack are assumed to have a random thickness and a planar interface with a finite radial spatial scale. Fig.1.1. Simplified scheme of radar sounding of a layered thick snow pack We also assume that the r.m.s. of the height deviation from a mean interface profile is much less that the wavelength (smooth surface), and the snow medium has uniform properties in the horizontal direction within the scattering volume. For this case only an area (effective area) with the size equals to approximately of 3.5 radii of the first Fresnel zone contributes into backscatter. We consider that the interfaces between layers have transitions zones of a finite thickness.

B. Principal (slice) approach We apply the so-called slice approach (Yurchak, 2009). It means that the volume backscattering occurs from a set of slices which are elementary adjoining volumes coincided with the wavefront configuration and the radial size much less than the wavelength. Here we extend this approach for a planar surface, as well, by using the effective area as a slice's boundary surface. In accordance with the slice approach the total mean radar cross section of the scattering volume for point scatterers is

= Var(b) M eff

(1)

where b is the slice radar equivalent length (SREL) which is equal to the sum of all backscatters from individual particles within a slice, Meff is the effective number of slices within the scattering volume. Basing on the previous assumptions, the SREL for the layered medium can be determined by the radar cross section L of a layer's interface:

b= L

(2)

If an interface surface is partially reflected with the given Fresnel coefficient (0) and perfectly flat, its radar cross section for the specular reflection is equal to (e.g., Atlas, 1960)

L = R022 (0) (3) with the corresponding scattering coefficient:

0 L

=2

2 (0)

(4)

where is the proportional

coefficient. Correspondingly,

the total backscattering

coefficient is:

0

= 2Var ()M eff

(5)

where Meff can be assumed here as a number of layers

within the effective scattering

volume.

D. Density Deviation Factor

Using (6) and (7), the reflection coefficient variance can be expressed as :

Var ( )

=

g

1 16

2

(8)

where

2

=

2

Var

( 2

)

(9)

C. Reflection coefficient for a layer with transition zone of a finite thickness

is the Permittivity Deviation Factor (DF). Assuming:

=1+q

(10)

For discontinuous ("sharp") change of the snow permittivity , the square of the reflection coefficient is (e.g., Paren, 1981) :

02

=

1 4

2

(6)

where q=1.9-2.2 is a fitted coefficient, the DF (9) can be written in the form:

where

2

=

q

2

2

(11)

2

=

V ar( ) (1+ q )2

(12)

where <> is the mean snow permittivity of adjacent layers.

In the natural snow medium

this change occurs within the

finite transition zone (e.g.,

Atlas, 1964). The space

finiteness of the permittivity

variation

causes

the

decreasing of the reflection

coefficient. This reduction can

be accounted by the correction

coefficient g that depends on

the law of the permittivity

change within the transition

zone. Thus

2 = g 02 (7) For simplicity, the coefficient g is assumed to be roughly the same for all layers.

is the Density Deviation Factor (DDF). Combining (11), (8) and (5) we have the following expression for the volume layered component of the backscatter coefficient:

0

=

1 2

q 2

2

g

M 2

2

eff

Therefore the backscatter coefficient of the layered snow medium depends on the number of layers through parameter Meff and on the density fluctuation statistics of these layers through the DDF.

References for part I Atlas,D., 1960. "Possible key to the dilemma of meteorological "angel" echoes," Journal of Meteorology, vol. 17, No. 2, pp. 95-103. Atlas, D., Advances in radar meteorology. Advances in Geophysics, v.10. Academic press, New York, pp. 317-478. Paren, J.G. 1981. Reflection coefficient at a dielectric interface. Journal of Glaciology, vol. 27, No. 95, pp. 203-204. Yurchak, B.S., 2009. Radar volume backscatter from spatially extended geophysical targets in a "slice" approach. IEEE Trans. Geosci. Remote Sens., vol. 47, issue 11, pp.3690-3696.

II. DENSITY DEVIATION FACTOR VS MEAN DENSITY (FIELD DATA)

Density deviation factor, dB

Fig.2.1. Statistical relationship between

-25

-30

y = -0.0722x - 7.4155

All

density deviation factor (DDF) and the mean

-35

R2 = 0.7301

Rtch density calculated after data of several H

-40

We authors. (Legend: Rtch Rotschky et al.,

Wi

-45

Gow 2006; H- Holmlund et al., 2000; We - West et

-50

Sch Linear (All)

al.,

1996;

Wi

Winebrenner

et

al.,

2001;

-55

300

400

500

600

mean density, kg/m^-3

Gow Gow, 1968; Schl Schlosser and Oerter, 2002)

III. DENSITY IRREGULARITIES AND RADAR BACKSCATTER

IV. DENSITY DEVIATION FACTOR AND BACKSCATTER VS ACCUMULATION RATE (FIELD DATA)

Density deviation factor, dB

-30

Fig.4.1. Density deviation

-32

y = -0.0323x - 32.218

factor (DDF) versus

-34

R2 = 0.8707

accumulation

rate.

-36

Calculation is based on

-38

the 2-4-cm-scale variation

-40

of firn density data (by

-42

-44

Winebrenner et al., 2001)

50

100

150

200

250

300 for Greenland and

Accumulation rate, kg m^-2 a^-1

Antarctic.

(EXAMPLES OF FIELD DATA)

Density gradient (abs value), arbitrary units

0.2

6

0.18

5.5

0.16

5

0.14

4.5

0.12

4

0.1

3.5

0.08

3

0.06

2.5

0.04

2

0.02

1.5

0

1

131 121 112 102 91 81 72 57 46 36 27

Depth from the bottom, cm

density_grad

backscatter

Amplitude, dB Amplitude, dB

6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0

Density Gradient-Amplitude scatterplot

y = 15.908x + 2.7419 R2 = 0.538

0.02 0.04 0.06 0.08

0.1

0.12 0.14 0.16 0.18

0.2

Modul of density gradient, arbitrary units

Fig. 3.1. Snow density gradient and the corresponding backscatter. Processing of profile data after Ellerbruch and Boyne (1980). FM-CW radar in the Frequency range 8-12 GHz. Space resolution is ~3 cm. correlation coefficient between the backscatter amplitude and the snow density is 0.09; the same for the density gradient is about 0.73. Fig. 3.2. Snow density profile and the radar backscatter for the dry-snow zone of Greenland. Taken from Hawley et al., (2006). Data from the ESA Airborne SAR / Interferometric radar altimeter System (ASIRAS). Ku-band (13.5 GHz), bandwidth is 1 GHz , range resolution is ~8.8 cm (in air). The red diagonal line display a travel time computed for snow using the measured densities. There is an obvious correspondence between the density gradient and the backscatter. For example: the lower backscatter amplitude at level -6 m corresponds with higher density fluctuation that has the lower gradient in comparison with the same at level -8 m (peak-like fluctuation). The data demonstrate the backscatter's dependence on the density irregularities that can be parameterized by the DDF.

In accordance with Li and Zwally (2004, pp. 312-313), "...the major cause of the large variability in density in the low-temperature, lowaccumulation region is the time of exposure to temperature gradients. This period depends on the accumulation rate. At low-accumulation sites, each deposited firn layer remains near the surface and exposed to the extreme temperature gradient for a longer period...". This is a favorable condition for creation the fine scale layers during the accumulation period due to crust and hoar formation (Alley, 1998; Goodwin, 1988).

Backscatter coefficient, (dB)

Fig.4.2. Backscatter coefficient versus

-5

-7

y = -0.0662x - 5.6991 R2 = 0.4494

accumulation rate after Zahnen et al.,

-9

(2002) data. Dronning Maud Land

-11

(Antarctica). The result demonstrate

-13

satisfactory positive correlation (R0.5)

-15

40 50 60 70 80 90 of the backscatter coefficient with the

Accumulation rate, mm w.e. (kg/m^2)

Standard deviation of dielectric constant

and low negative correlation (R=-0.1...-0.28) with the mean dielectric

constant. It is exactly in accordance with behavior of the DF, (9). The

same trend has been reported by Forster et al., (1999), Drinkwater et al,

(2001), and Rotschky et al, (2006) but this feature was interpreted

basing on the snow grain size variation.

V. CONCLUDING REMARKS 1. The backscatter from a layered snow pack can be expressed by the slice approach using the snow density inhomogeneity term Density Deviation Factor (DDF). 2. The observed inverse dependence of radar backscatter on snow accumulation rate is due to the same (inverse) dependence of the DDF on the accumulation rate.

References for parts II-IV

Drinkwater, M.R., et al. 2001. Greenland snow accumulation estimates from satellite radar scatterometer data. J. Geoph. Res., v. 106, No. D24, pp. 33,935-33,950. Ellerbruch, D.A., and Boyne, H.S. 1980. Snow stratigraphy and water equivalence measured with an active microwave system. Journal of Glaciology, vol. 26, No. 94, pp. 225-233. Forster, R.R., et al. 1999. Relationships between radar backscatter and accumulation rates in the Greenland ice sheet. Int. J. remote sensing, vol. 20, No. 15 & 16, pp. 3131-3147. Gow, A.J., 1968. Deep core swtudies of the accumulation and densification of snow at Byrd Station and Little America V., Antarctica. CRREL Research Report 197, 45 p. Li, J., and J. Zwally. 2004. Modeling the density variation in the shallow firn layer. Annals of Glaciology, vol. 38, pp. 309-313. Hawley, R.L., et al. 2006. ASIRAS airborne radar resolves internal annual layers in the dry-snow zone of Greenland. Geoph. Res. Lett., v. 33, L04502, doi:10.1029/2005GL025147. Holmlund, P., et al. 2000. Spatial gradients in snow layering and 10 m temperatures at two EPICA-Dronning Maud Land (Antarctica) pre-site-survey drill sites. Annals of Glaciology, vol. 30, pp. 13-19. Rotschky, G., et al. 2006. Retrieving snowpack properties and accumulation estimates from a combination of SAR and scatterometer measurements. IEEE Trans. Geosci. Remote Sens., vol. 44, issue 4, pp.943-956. West, R.D., et al. 1996. Microwave emission from density-stratified Antarctic firn at 6 cm wavelength. Journal of Glaciology, vol. 42, No. 140, pp. 63-76. Winebrenner, D.P., Arthern, R.J., Shuman, C.A. 2001. Mapping Greenland accumulation rates using observations of thermal emission at 4.5-cm wavelength. J. Geoph. Res., vol. 106, No. D24, 2001, pp. 33,919-33,934. Zahnen, N., et al. 2002. Correlation between Antarctic dry snow properties and backscattering characteristics in Radarsat SAR imagery. Proceedings of EARSeL, Bern, March 11-13, 2002, pp. 140-148.

Acknowledgements This work was supported by NASA's Cryospheric Sciences Program and NASA Goddard Space Flight Center.

B Yurchak, W Abdalati

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