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________________________________ Notes on microwave radiation from snow samples and emission of layered snowpacks Christian Mätzler

_______________________________

Research Report No. 96-09 1996, updated Feb. 2004 “Notes 1 to 14”

Institute of Applied Physics

Dept. of Microwave Physics

__________________________________________________________ Sidlerstr. 5, 3012 Bern Switzerland

Tel. : +41 31 631 89 11 Fax. : +41 31 631 37 65 [email protected]

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Notes on microwave radiation from snow samples and emission of layered snowpacks (Notes on the development of a microwave emission model of layered snowpacks, 1996-1998)

Contents A sandwich model for radiometry of snow slabs.............................................................................................. 4 A six-flux model for internal reflectivity and transmissivity of snow slabs ................................................... 7 Two-Flux Results from Weise's Data .............................................................................................................. 11 Six-Flux Results from Weise's Data ................................................................................................................ 13 Extract from MICROWAVE PROPERTIES OF ICE AND SNOW ............................................................ 25 A microwave emission model for a layered snowpack .................................................................................. 28 A polarimetric microwave emission model for a layered snowpack............................................................. 32 A Four-Flux Model for the internal reflectivity and transmissivity of a snow slab, including polarization mixing................................................................................................................................................................ 37 Improved Born Approximation for scattering of radiation in a granular medium ....................................... 40 Coherent scattering effects in a snowpack ...................................................................................................... 51 Effective propagation angle and polarization mixing in a snowpack............................................................ 56 Extension of the Microwave Emission Model of Layered Snowpacks to Coarse- Grained Snow ............. 59

See also the following publications related to or derived from these notes: C. Mätzler, "Applications of the Interaction of Microwaves with the Natural Snow Cover", Remote Sensing Reviews, Vol. 2, pp. 259-392 (1987). C. Mätzler, "Autocorrelation functions of granular media with free arrangement of spheres, spherical shells or ellipsoids", J. Applied Physics, Vol. 81 (3), pp.1509-1517 (1997). C. Mätzler, "Microwave properties of ice and snow", in B. Schmitt et al. (eds.) ”Solar System Ices”, Astrophys. and Space Sci. Library, Vol. 227, Kluwer Academic Publishers, Dordrecht, pp. 241-257 (1998). J. Pulliainen, K. Tigerstedt, W. Huining, M. Hallikainen, C. Mätzler, A. Wiesmann, and C. Wegmüller, "Retrieval of geophysical parameters with integrated modeling of land surfaces and atmosphere (models/inversion algorithms)", Final Report, ESA/ESTEC, Noordwijk, Netherlands, Contract No. 11706/95/NL/NB(SC), Oct. (1998). A. Wiesmann, C. Mätzler and T. Weise, "Radiometric and structural measurements of snow samples", Radio Science, Vol. 33, pp. 273-289 (1998). C. Mätzler, "Improved Born Approximation for scattering in a granular medium", J. Appl. Phys., Vol. 83, No. 11, pp. 6111-6117 (1998). A. Wiesmann, C. Mätzler, "Microwave emission model of layered snowpacks", Remote Sensing of Environment, Vol. 70, No. 3, pp. 307-316 (1999). A. Wiesmann and C. Mätzler, ”Technical Documentation and Program Listings for for MEMLS 99.1, Microwave Emission Model of Layered Snowpacks”, Research Report No. 99-4, Microwave Dept., Institute of Applied Physics, University of Bern, Sept. (1999). C. Mätzler and A. Wiesmann, "Extension of the Microwave Emission Model of Layered Snowpacks to CoarseGrained Snow", Remote Sensing of Environment, Vol. 70, No. 3, pp. 317-325 (1999). A. Wiesmann, C. Fierz and C. Mätzler, "Simulation of microwave emission from physically modeled snowpacks", Annals of Glaciology, Vol. 31, pp. 397-405 (2000). C. Mätzler, A. Wiesmann, J. Pulliainen and M. Hallikainen, "Development of microwave emission models of snowpacks", IEEE Geoscience and Remote Sensing Society Newsletter, pp. 18-25, June (2000). C. Mätzler, “Relation between grain size and correlation length of snow”, J. Glaciology, 48(152), 461-466 (2002)

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A sandwich model for radiometry of snow slabs Note 1, April 11, 1996 (with corrections, Jan. 1997) C. Mätzler, NSIDC-CIRES, Campus Box 449, University of Colorado, Boulder CO, 80309

Introduction This note is related to the Thesis of Thomas Weise "Radiometric and structural measurements of snow", Faculty of Natural Philosophy, University of Bern, Switzerland, Feb. 1996. Weise made radiometric measurements with natural snow slabs of about 10 cm thickness. The brightness temperatures of the snow slabs were measured in two different situations, (a) slab on a microwave absorber, and (b) slab on a flat aluminum plate. In case (a) the snow temperature and the absorber temperature were kept the same. In both cases the slab was mechanically supported by a styrofoam plate of 3 cm thickness; its transmissivity was perfect under all conditions. However, due to its low relative permittivity of about 1.03 (which is close enough equal to the air permittivity) an additional reflection occurs at the interface, snow -styrofoam. In order to account for this reflection, and in order to distinguish between the internal scattering and the reflections at the interfaces, a simple sandwich model, based on radiative transfer, is proposed and developed here. Coherent effects, arising from interference between the reflections at different interfaces, cannot be treated by this model. However, it was noted by Weise that these effects were almost completely eliminated by the averaging taking place over the angular range of the antenna beam and over the frequency range of the radiometer channels.

The model Consider a plane-parallel slab of snow with internal emissivity, e, transmissivity, t, and reflectivity, r, at any given frequency, incidence angle and polarization (Here we assume that there is no coupling between radiation at different polarization; this effect will later be considered in Notes 7 and 8.). Due to the principle of detailed balance and Kirchhoff's law we have e+r+t=1

(1)

At the upper slab boundary we assume a surface reflectivity ra, and at the bottom of the slab a surface (or interface) reflectivity rb. Now, since the Rayleigh-Jeans Approximation of Planck's radiation is linear, we can define the slab temperature to be zero by subtracting the absolute snow temperature from all absolute temperatures to follow. Then we have T=0, i.e. the emission terms in Equations (2) to (7) disappear. The remaining terms are due to reflection and transmission. The brightness temperatures, T1 to T8, are as defined in Figure 1.

z=d

T2 T1 esa, rsa, tsa ______________________________________________ ra T4 T3 T, e , r , t

z=0

T6 T5 ______________________________________________ rb T8 T7 esb, rsb, tsb

Figure 1: Sandwich model of a plane parallel snow slab with thickness d, illuminated by brightness temperatures T1 from above and by T8 from below, emitting T2 upwards and T7 downwards.

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A set of linear equations relates the four internal (T3 to T6) and the four external (T1, T2, T7, T8) brightness temperatures: T2 = ra T1 + (1-ra) T4 T3 = (1-ra) T1 + ra T4 T4 = r T3 + t T6 T5 = t T3 + r T6 T6 = rb T5 + (1-rb)T8 T7 = (1-rb)T5 + rb T8

(2) (3) (4) (5) (6) (7)

The total reflectivity rsa and the total transmissivity tsa of the slab when illuminated from above (T8=0, T1>0) are given by the ratios rsa = T2 / T1 ; tsa = T7 / T1

(8)

Equivalently, the total reflectivity rsb and the total transmissivity tsb of the slab when illuminated from below (T1=0, T8>0) are given by the ratios rsb = T7 / T8 ; tsb = T2 / T8

(9)

Due to reciprocity we should find that tsb=tsa; this is, indeed, true, and we call this transmissivity ts. Solving the system of linear equations (2) to (7) for rsa, rsb and ts we find: rsa = ra + (1-ra)2 roa /(1- ra roa) rsb = rb + (1- rb)2 rob/(1- rb rob) ts = t (1- ra)(1- rb) / ((1- r ra)(1- r rb) - ra rb t2 )

(10) (11) (12)

roa = r + rb t2 /(1 - r rb) ;

(13)

where rob = r + ra t2 /(1 - r ra)

Note that Indices a and b are interchanged in the two Equations (13). Also note that (12) is symmetrical with respect to a and b. The sandwich emissivities are now given by (1): esa = 1 - ts - rsa esb = 1 - ts - rsb

(14) (15)

The assumption T=0 is no longer needed. We can write the observable brightness temperatures as T2 = rsa T1 + esa T + ts T8 ; T7 = rsb T8 + esb T + ts T1

(16)

where T is the physical snow temperature.

Applying the model A) Slab on an absorber Here we have T8 = T, since the absorber is a black body at the physical snow temperature. The snow slab was selected as a homogeneous slab, therefore the reflectivities ra and rb are equal, ri = ra = rb, where ri stands for interface reflectivity. From (14) and (16) we get, by calling T2=TA: TA = (1 - r_abs) T + r_abs T1

(17)

where r_abs = rsa for the case ra=rb=ri. This quantity is just 1-e_abs of Weise (1996).

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B) Slab on a metal plate The reflectivity of the metal plate is with good approximation equal to 1. Therefore we can write T7=T8 which also means that T5=T6, so that we can simply assume that rsb=1. For rsa we have still the same value ri as above. From (10), (15) and (16) we find, by calling T2=TM: TM = (1 - r_met) T + r_met T1

(18)

where r_met = rsa for the case ra=ri and rb=1. This quantity is just 1-e_met of Weise (1996). Equations (17) and (18) can be used to determine r_abs and r_met from the measured brightness temperatures and from T. T1 is the sky brightness temperature. On the other hand, the reflectivities, r_abs and r_met, are related to the internal values, r and t, by (10) to (14): r_abs = ri + (1-ri)2 Rabs

(19)

Rabs = (r + ri t2/(1- r ri)) / (1 - r ri - (ri t)2/(1- r ri) )

(20)

where

and equivalently r_met = ri + (1- ri)2 Rmet

(21)

Rmet = (r + t2/(1- r)) / (1 - r ri - ri t2/(1- r) )

(22)

where

How to solve for r and t ? The values of Rabs and of Rmet can be computed from the measured r_abs and r_met using Equations (20) and (22) if ri is known. For ri we can assume the Fresnel reflectivity at the given incidence angle, polarization and dielectric slab permittivity . Therefore the problem is to find r and t with the two equations. Here we propose an iterative procedure. Rewriting (20) and (22) in the form r = Rabs (1 - r ri - (ri t)2/(1-r ri)) - ri t2/(1-r ri) t2 = Rmet ((1-r ri)(1-r) - ri t2) - r(1-r)

(23) (24)

we find an iterative solution for small values of ri, starting with ri=0: then r = Rabs ;

t2 = (Rmet - Rabs)(1- Rabs)

(25)

Using these values on the right-hand side of (23) and (24) leads to new values of r and t2. The new values can then be inserted on the right-hand side of the two equations to yield better and better values of r and t2, until we are satisfied. Using 7 iterations I got errors of r and t2 smaller than 0.0005 for all cases found in Weise (1996).

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A six-flux model for internal reflectivity and transmissivity of snow slabs Note 2: April 12, 1996 C. Mätzler, NSIDC-CIRES, Campus Box 449, University of Colorado, Boulder CO, 80309 Version C: Feb. 4, 1997 (cos included in Equations (3-4), (10-11), (14-15), and (20), and a correction in (9)); IAP Univ. Bern

Introduction In Note 1 (C. Mätzler, A sandwich model for radiometry of snow slabs, April 11, 1996) it was shown how the internal reflectivity r and transmissivity t are related to the measured brightness temperatures, assuming additional reflectivities at the upper and lower interface of the slab. The present note is dedicated to the question of modeling a natural snowpack with the data found from Weise's Thesis, e.g. how we can determine the e, r, and t for a variable snowpack thickness, and also how we can treat the variable temperature inside the pack. If we can answer these questions we may be able to relate the snow-structure information of Weise's samples to predict the microwave emission of any given snowcover. For this purpose we have to develop the links between the r and t of Weise's samples on the one hand, and the internal scattering and absorption coefficients on the other. In addition the approach has to be concerned with the question of how to deal with the total reflection occurring for radiation approaching the snow boundary from inside the pack at certain angles. This trapped radiation can have an influence on the observable brightness temperature. The model is a simplified radiative transfer model, reducing the radiation directions to 6 fluxes streaming along and against the 3 principal axes of the slab. It will be shown that in case of a plane-parallel slab the model reduces to the well-known Two-Flux Model, however, the absorption and scattering coefficients, being functions of the 6-flux parameters. The horizontal fluxes represent the trapped radiation, and the vertical fluxes represent the radiation, which interacts with the space above the snowpack. Considering the Two-Flux Model alone would not allow including the trapping effect. Scattering by angles near 90 degrees causes a coupling between the vertical and the horizontal fluxes. For the 2-Flux Model there exist analytical solutions, and since the 6-Flux Parameters can be related to the internal scattering and absorption coefficient, it is possible to give analytical expressions for the brightness temperature for simulated snowpacks of any depth and composition using scattering and absorption coefficients determined from Weise's data.

The model Consider a plane-parallel snow medium with internal emissivity, e, transmissivity, t, and reflectivity, r, at any given frequency, incidence angle 0 and polarization. It is recalled that e+r+t=1

(1)

How the effects of reflection at the interfaces have to be eliminated is the topic of Note 1. Here we consider the radiation propagating in a given direction to belong to one of 6 fluxes. Since the fluxes are proportional to the brightness temperatures Ti of the corresponding radiation, we describe the fluxes by the six brightness temperatures T1 to T6 (s. Figure 1). z=d

z=0

--------------------------------------------------------------------------T1 z T4 T3 T6 T5 T2 y----------------------------------------------------------------------- x

Figure 1: The 6 fluxes of radiation inside the snow slab.

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Let us denote the refractive index of the slab by n. Radiation with an internal incidence angle larger than c = arcsin(1/n) (2) will suffer total reflection when reaching the slab boundary. This radiation is trapped; the probability that it will be absorbed is large. The radiation belongs to one of the fluxes expressed by T3 to T6, depending on its horizontal direction. On the contrary, radiation at an incidence angle smaller than c can be transmitted through the slab. This radiation belongs to T1 for downwelling radiation or to T2 for upwelling radiation. The transfer equations for the 6 fluxes can be written as

dT1 cos = -a(T1 - T) - b(T1 -T2) - c(4T1 - T3 - T4 - T5 - T6) dz dT + 2 cos = -a(T2 - T) - b(T2 - T1) - c(4T2 - T3 - T4 - T5 - T6) dz

+

dT3 dx dT4 dx dT5

+

dy

dT6 dy

(3) (4)

= -a(T3 - T) - b(T3 - T4) - c(4T3 - T1 - T2 - T5 - T6)

(5)

= -a(T4 - T) - b(T4 - T3) - c(4T4 - T1 - T2 - T5 - T6)

(6)

-a(T5 - T) - b(T5 - T6) - c(4T5 - T3 - T4 - T1 - T2)

(7)

= -a(T6 - T) - b(T6 - T5) - c(4T6 - T3 - T4 - T1 - T2)

(8)

=

where a is the absorption coefficient, b the backward-scattering coefficient, and c is the scattering coefficient for around the "corner" describing scattering near 90°. Now, for a plane-parallel medium there is no variation in the x- and y- direction, which means that Equations (5) to (8) are zero. In addition if the medium is isotropic around the z axis, all horizontal fluxes are the same: then it follows from (5) to (8) that T3 = T4 = T5 = T6 = (a T + c(T1 + T2)) / (a + 2c) (9) Inserting (9) in (3) and (4) gives

dT1 cos = -a' (T1 - T) - b' (T1 -T2) dz dT + 2 cos = -a' (T2 - T) - b' (T2 - T1) dz

(10) (11)

which are the Two-Flux Equations with modified coefficients a' = a ( 1 + 4c /(a +2c)) b' = b + 4c2 /(a +2c)

(12) (13)

For constant T and constant coefficients T1 and T2 can be written as (Mätzler, 1987, Section 3.6) T1 = T + A exp(z') + B exp(-z') T2 = T + r0 A exp(z') + B exp(-z')/ r0

(14) (15)

where z'=z/ cos . The coefficients A and B are determined by boundary conditions at z=0 and z=d; and r0 is the reflectivity at infinite thickness, given by r0 = b' / (a' + b' + )

(16)

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and where the damping (attenuation) coefficient is given by = a ' ( a '+2 b ' )

(17)

For a snow layer of thickness d the values for the internal reflectivity and transmissivity are then found to be (18) r = r0 (1 - t02)/(1 - r02t02) (19) t = t0 (1 - r02)/(1 - r02t02) where t0 = exp(-d/ cos )

(20)

With Equations (12), (13) and (16) to (20) we can express r and t by 4 parameters: the 6-flux parameters, a, b, c and by the layer thickness d. Or else, we can express r and t by 3 parameters: the 2-flux parameters, a', b' and by the layer thickness d. Together with the results of Note 1 we can simulate the brightness temperature of a snow slab of any thickness by these parameters and together with the snow temperature T and the interface reflectivities. The effect of the total reflection is just to modify the coefficients as expressed by (12) and (13). It is seen that the effective absorption coefficient is enhanced by the interaction with the trapped radiation. The same is true for the back-scattering coefficient b'.

How to derive a' and b' from r and t? The basic equations relating a' and b' , and r and t are Equations (16) to (20). An iterative procedure to find a' and b' from r and t is indicated. First we have to find r0 and t0 from r and t. Writing according to (18) and (19)

r0 =

1 + 1 + 4 G 2 ; 2G

G

r0 r t0 = 2 t 1 t02 1 r0

(21)

and t0 = t ( 1 + r02 (1 - t02) / (1 - r02))

(22)

we get a first estimate of t0 by assuming t0 = t. Inserting this value in (21) gives a first estimate of r0. With this value of r0 we can obtain an improved value of t0 by insertion in (22). The procedure leads to a rapid convergence for the values of Weise's data. Next we get from inverting (20), and the other parameters a' and b' from manipulating (16) and (17): a' =

1 r0

(23)

1 + r0

b' = ( + a ')

r0

(24)

1 r0

How to derive a , b and c from a' and b'? Only if a special relationship is known between a and b can we derive all three parameters of the 6-Flux Model. The two coefficients b and c are part of the total scattering coefficient s. How they are related is determined by the internal scattering phase function and by the critical angle c given by (2). This critical angle can be used to determine the solid angles for scattering in one or in the other beam, respectively. For upwelling radiation the solid angle of radiation of the beam for T2 is easily seen to be 9

1 = 2 (1 -

1

)

(25)

and the solid angle 2 of trapped radiation is the complement 2 - 1 , i.e. 2 = 2

1

(26)

where the refractive index has been replaced by the square root of the relative permittivity. The larger 1 the larger will be b , and the larger 2 the larger will be c . Assuming symmetry in the solid angles for up- and downwelling radiation, 1 is also the solid angle of the flux T2. For isotropic scattering we have the total scattering coefficient s given by the sum s = 2b + 4c

(27)

where the first term stands for the scattering in the backward beam b plus the same value for scattering in the forward beam, and the factor 4 results from the fourfold scattering into the 4 beams at right angles. For isotropic scattering the proportionality between b and 1 and of the respective proportionality between 2c and 2 leads to the ratio 2c/b F = 2c/b = x/(1-x) ;

x=

1

(28)

With this additional equation we can determine the 6-Flux Coefficients and s from the snow permittivity and from the Two-Flux coefficients a' and b', using (12) and (13). The inversion of these two equations, however, is not straightforward. After some algebraic manipulations we find a quadratic equation for b of the form a b2 + b b + c = 0

(29)

with a = 2 + F - F2, b = - (a' + 4b' + Fb'),

c = b' (a' + 2b ')

(30)

This equation can be solved for b, and a then follows from a = F b ( -1 + F b/(b'- b))

(31)

Using these equations we get the 6-Flux Coefficients from the Two-Flux Coefficients for the case of isotropic scattering inside the snowpack.

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Two-Flux Results from Weise's Data Note 3, April 16, 1996 C. Mätzler, NSIDC-CIRES, Campus Box 449, University of Colorado, Boulder CO, 80309 In the following some of Weise's data are analyzed with the Two-Flux Model. f=35 GHz, vertical polarization: Table 1: Density in g/cm3, slab thickness d in cm, snow temperature T in K, slope-correlation length in mm, Gamma = , Ga = a' and Gb = b' all in neper/m of the snow data from Weise (1996) at f = 35 GHz, vertical polarization.

35V dens

D

Tsnow

corl_m Gamma

Ga

-------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.417 0.212 i4 93.0 10.2 266.7 0.062 0.339 0.170 i8 159.0 11.1 267.1 0.067 0.290 0.212 i10 162.0 13.0 268.2 0.070 0.329 0.205 i10 162.0 9.5 268.2 0.070 0.385 0.248 i7 231.0 5.6 261.6 0.071 0.382 0.364 i9 367.0 8.2 269.3 0.092 0.539 0.508 i1 260.0 8.1 266.7 0.123 0.426 0.356 i5 381.0 7.0 266.2 0.127 0.664 0.473 i5n 384.0 10.4 267.7 0.127 0.603 0.507 i2 332.0 12.6 264.2 0.203 1.351 0.773 i12 335.0 9.5 273.0 0.223 1.292 0.833 i13 345.0 8.6 271.6 0.233 3.137 1.624 i6 279.0 9.7 270.8 0.325 5.348 2.298

Gb 0.304 0.255 0.092 0.161 0.174 0.018 0.032 0.077 0.230 0.106 0.793 0.585 2.217 5.074

Page &P

10.00 35 GHz, V-pol. 1.00 2-Flux coeff. (1/m)

Absorption Scattering 0.10

0.01 0.05

0.10

0.2

0.3 0.4 0.5

correlation length (mm) Fig. 2: Coefficients a' (squares) and b' (diamonds) and fitted functions of the Two-Flux model in (1/m) at 35 GHz, vertical polarization, versus slope-correlation length of the snow-slab data of Weise (1996). Note that logarithmic scales are used on both axes. Shown are all data.

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10.00 35 GHz v-pol.

1.00

Ga Gb 15*X**1.68 340*X**3.73

1/m 0.10

0.01 0.05 0.06 0.080.10

0.2

0.3

0.4 0.5

correlation length (mm) Fig. 3: Coefficients a' (squares) and b' (diamonds) and fitted functions of the Two-Flux model in (1/m) at 35 GHz, vertical polarization, versus slope-correlation length of the snow-slab data of Weise (1996). Note that logarithmic scales are used on both axes. Shown are the data of snow samples with density > 200 kg/m3.

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Six-Flux Results from Weise's Data Note 4, April 16-17, 1996 C. Mätzler, NSIDC-CIRES, Campus Box 449, University of Colorado, Boulder CO, 80309 In this Note, Weise's data are analyzed with the Six-Flux Model assuming isotropic scattering in the snow slab. First the absortion and scattering coefficients are shown versus correlation length as determined by the slope at zero displacement (see Weise 1996). A plot is shown for each frequency and polarization. In addition, for each Figure, a Table is added to show the numerical data including additional information. Then, in order to make a first test of the 6-Flux Model, the absorption coefficients are compared with predictions from mixing theory (Figures 11 and 12).

10.000 11 GHz, V-pol.

1.000

Absorption Scattering

1/m 0.100

0.010

0.001 0.05 0.06 0.080.10

0.2

0.3

0.4 0.5

correlation length (mm) Fig. 1: Coefficients a (squares) and s (diamonds) of the Six-Flux Model in (1/m) at 11 GHz, vertical polarization, versus slope-correlation length of the snow-slab data of Weise (1996). Note that logarithmic scales are used on both axes. Shown are all data. Table 1: Density in g/cm3, slab thickness d in cm, snow temperature T in K, slope-correlation length in mm, Ga6 = a in 1/m, Gb6 = b in 1/m, Gc6 = c in 1/m and Gs6 = s in 1/m of the snow data from Weise (1996); same frequency and polarization as above.

11V dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

---------------------------------------------------------------i4 93.0 10.2 266.7 0.062 0.033 0.20 0.06 i1 260.0 8.1 266.7 0.123 0.120 0.00 0.00 i5 381.0 7.0 266.2 0.127 0.033 0.04 0.04 i5n 384.0 10.4 267.7 0.127 0.060 0.01 0.01 i2 332.0 12.6 264.2 0.203 0.042 0.08 0.06 i6 279.0 9.7 270.8 0.325 0.064 0.29 0.19

13

Gs6 0.64 0.01 0.23 0.05 0.40 1.35

10.00 11 GHz, H-pol. 1.00 Absorption Scattering

1/m 0.10

0.01 0.05 0.06 0.080.10

0.2

0.3

0.4 0.5

correlation length (mm) Fig. 2: Coefficients a (squares) and s (diamonds) of the Six-Flux Model in (1/m) at 11 GHz, horizontal polarization, versus slope-correlation length of the snow-slab data of Weise (1996). Note that logarithmic scales are used on both axes. Shown are all data. Table 2: Density in g/cm3, slab thickness d in cm, snow temperature T in K, slope-correlation length in mm, Ga6 = a in 1/m, Gb6 = b in 1/m, Gc6 = c in 1/m and Gs6 = s in 1/m of the snow data from Weise (1996); same frequency and polarization as above.

11H dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

---------------------------------------------------------------i4 93.0 10.2 266.7 0.062 0.030 0.49 0.14 i1 260.0 8.1 266.7 0.123 0.028 0.36 0.23 i5 381.0 7.0 266.2 0.127 0.048 0.15 0.14 i5n 384.0 10.4 267.7 0.127 0.036 0.12 0.11 i2 332.0 12.6 264.2 0.203 0.042 0.28 0.22 i6 279.0 9.7 270.8 0.325 0.063 0.66 0.44

14

Gs6 1.54 1.63 0.85 0.65 1.44 3.08

10.000 21 GHz, V-pol. 1.000 Absorption Scattering

1/m 0.100

0.010

0.001 0.05 0.06 0.080.10

0.2

0.3

0.4 0.5


21V dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.044 0.19 0.06 i4 93.0 10.2 266.7 0.062 0.039 0.25 0.07 i8 159.0 11.1 267.1 0.067 0.041 0.08 0.03 i10 162.0 13.0 268.2 0.070 0.039 0.14 0.06 i10 162.0 9.5 268.2 0.070 0.058 0.15 0.06 i7 231.0 5.6 261.6 0.071 0.211 0.00 0.00 i9 367.0 8.2 269.3 0.092 0.243 0.00 0.00 i1 260.0 8.1 266.7 0.123 0.125 0.03 0.02 i5 381.0 7.0 266.2 0.127 0.111 0.06 0.05 i5n 384.0 10.4 267.7 0.127 0.141 0.03 0.03 i2 332.0 12.6 264.2 0.203 0.105 0.18 0.14 i12 335.0 9.5 273.0 0.223 0.146 0.13 0.10 i13 345.0 8.6 271.6 0.233 0.177 0.26 0.21 i6 279.0 9.7 270.8 0.325 0.291 0.80 0.53

15

Gs6 0.62 0.80 0.29 0.52 0.56 0.01 0.01 0.13 0.32 0.19 0.94 0.65 1.38 3.73


1/m 0.100

0.010

0.001 0.05

0.10

0.2

0.3

0.4 0.5


21H dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

Gs6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.032 0.09 0.03 i4 93.0 10.2 266.7 0.062 0.034 0.27 0.08 i8 159.0 11.1 267.1 0.067 0.027 0.19 0.08 i10 162.0 13.0 268.2 0.070 0.040 0.16 0.07 i10 162.0 9.5 268.2 0.070 0.048 0.21 0.09 i7 231.0 5.6 261.6 0.071 0.068 0.19 0.11 i9 367.0 8.2 269.3 0.092 0.193 0.00 0.00 i1 260.0 8.1 266.7 0.123 0.052 0.42 0.26 i5 381.0 7.0 266.2 0.127 0.211 0.00 0.00 i5n 384.0 10.4 267.7 0.127 0.073 0.17 0.16 i2 332.0 12.6 264.2 0.203 0.087 0.40 0.31 i12 335.0 9.5 273.0 0.223 0.117 0.16 0.12 i6 279.0 9.7 270.8 0.325 0.233 0.96 0.64 Page &P

16

0.30 0.84 0.68 0.60 0.78 0.81 0.01 1.87 0.01 0.99 2.05 0.81 4.47

100.00

10.00

1/m

35 GHz, V-pol. Absorption Scattering

1.00

0.10

0.01 0.05

0.10

0.2

0.3

0.4 0.5


35V dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.097 0.22 0.07 i4 93.0 10.2 266.7 0.062 0.079 0.19 0.05 i8 159.0 11.1 267.1 0.067 0.129 0.07 0.03 i10 162.0 13.0 268.2 0.070 0.105 0.11 0.05 i10 162.0 9.5 268.2 0.070 0.130 0.13 0.05 i7 231.0 5.6 261.6 0.071 0.328 0.02 0.01 i9 367.0 8.2 269.3 0.092 0.423 0.03 0.02 i1 260.0 8.1 266.7 0.123 0.242 0.06 0.04 i5 381.0 7.0 266.2 0.127 0.241 0.12 0.11 i5n 384.0 10.4 267.7 0.127 0.324 0.07 0.06 i2 332.0 12.6 264.2 0.203 0.337 0.39 0.31 i12 335.0 9.5 273.0 0.223 0.394 0.31 0.25 i13 345.0 8.6 271.6 0.233 0.668 1.02 0.84 i6 279.0 9.7 270.8 0.325 0.893 2.47 1.65 Page &P

17

Gs6 0.72 0.60 0.27 0.42 0.47 0.07 0.15 0.27 0.69 0.39 2.02 1.61 5.39 11.55

100.000 10.000

35 GHz, H-pol.

1.000


1/m 0.100 0.010 0.001 0.05

0.10

0.2

0.3 0.4 0.5


35H dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

Gs6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.101 0.20 0.06 i4 93.0 10.2 266.7 0.062 0.102 0.39 0.11 i8 159.0 11.1 267.1 0.067 0.092 0.31 0.13 i10 162.0 13.0 268.2 0.070 0.116 0.22 0.09 i10 162.0 9.5 268.2 0.070 0.151 0.23 0.10 i7 231.0 5.6 261.6 0.071 0.238 0.08 0.04 i9 367.0 8.2 269.3 0.092 0.688 0.03 0.02 i1 260.0 8.1 266.7 0.123 0.140 0.66 0.41 i5 381.0 7.0 266.2 0.127 0.514 0.00 0.00 i5n 384.0 10.4 267.7 0.127 0.283 0.12 0.11 i2 332.0 12.6 264.2 0.203 0.253 0.73 0.58 i12 335.0 9.5 273.0 0.223 0.344 0.42 0.33 i6 279.0 9.7 270.8 0.325 0.606 2.75 1.84 Page &P

18

0.65 1.23 1.13 0.81 0.85 0.33 0.14 2.95 0.01 0.66 3.77 2.16 12.86

100.0

48 GHz, V-pol.

10.0


1/m 1.0

0.1 0.05 0.06 0.080.10

0.2

0.3

0.4 0.5


48V dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

Gs6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.196 0.23 0.07 0.7 i4 93.0 10.2 266.7 0.062 0.136 0.28 0.08 0.8 i7 231.0 5.6 261.6 0.071 0.447 0.06 0.04 0.2 i1 260.0 8.1 266.7 0.123 0.356 0.20 0.13 0.9 i2 332.0 12.6 264.2 0.203 0.709 0.94 0.74 4.8 i12 335.0 9.5 273.0 0.223 1.104 0.99 0.79 5.1 i6 279.0 9.7 270.8 0.325 1.113 4.90 3.27 22.8

19

100.0

48 GHz, H-pol. 10.0 Absorption Scattering

1/m 1.0

0.1 0.05

0.10

0.2

0.3

0.4 0.5


48H dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

Gs6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.162 0.33 0.10 1.06 i4 93.0 10.2 266.7 0.062 0.137 0.58 0.16 1.82 i7 231.0 5.6 261.6 0.071 0.341 0.21 0.12 0.89 i1 260.0 8.1 266.7 0.123 0.261 0.76 0.48 3.42 i2 332.0 12.6 264.2 0.203 0.564 1.21 0.95 6.23 i12 335.0 9.5 273.0 0.223 0.893 0.96 0.77 5.00 i6 279.0 9.7 270.8 0.325 1.007 4.86 3.24 22.68

20

100.0 94 GHz, V-pol. 10.0 Absorption Scattering

1/m 1.0

0.1 0.05

0.10

0.2

0.3

0.4 0.5


94V dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.910 1.02 0.32 i8 159.0 11.1 267.1 0.067 0.988 0.96 0.40 i10 162.0 13.0 268.2 0.070 0.809 0.72 0.30 i10 162.0 9.5 268.2 0.070 1.050 0.72 0.31 i7 231.0 5.6 261.6 0.071 1.494 0.60 0.34 i9 367.0 8.2 269.3 0.092 2.937 0.39 0.34 i12 335.0 9.5 273.0 0.223 3.973 5.50 4.39 i13 345.0 8.6 271.6 0.233 3.775 9.13 7.51

21

Gs6 3.3 3.5 2. 2.6 2.5 2.1 28. 48.


1/m 1.0

0.1 0.05

0.10

0.2

0.3

0.4 0.5


94H dens

D

Tsnow

corl_m Ga6

Gb6

Gc6

Gs6

---------------------------------------------------------------i11 107.0 8.9 267.0 0.062 0.834 1.12 0.35 3.64 i8 159.0 11.1 267.1 0.067 0.946 1.39 0.58 5.12 i10 162.0 13.0 268.2 0.070 0.763 0.90 0.38 3.32 i10 162.0 9.5 268.2 0.070 1.112 0.73 0.31 2.70 i7 231.0 5.6 261.6 0.071 1.366 0.76 0.43 3.25 i9 367.0 8.2 269.3 0.092 3.901 0.05 0.04 0.28 i12 335.0 9.5 273.0 0.223 3.558 5.00 4.00 26.00 & As an assessment of the 6-Flux Model we compare the absorption coefficient with the computed absorption based on mixing theory and from the dielectric data of pure ice. The following figures show the average values of measured 6-flux absorption coefficients using all snow samples with density between 0.159 and 0.162 (Fig. 11) and for densities between 0.33 and 0.36 g/cm3 (Fig. 12). For the computation we assumed densities of 0.160 and 0.350 g/cm3, respectively, both at T=268 K. The computed ratio between the absorption coefficient of snow to the absorption coefficient of ice is 0.157 for the lower density and 0.359 at the higher density. These values follow from Tiuri et al. (1984), for the selected density (see also Eqs. (2.2) and (2.20) of Mätzler, 1987). The absorption coefficient of ice was determined with the new formula of Mätzler developed for pure water ice (Procs. Solar Systems Ices, to be published 1996), a slight improvement of the formula of Hufford (1991). In Note 5, the new formulas are given.

22

10.00

2.0 1.5

1.00

r a 1.0 t i o

Absorption (1/m) 0.10 T = 268 K den.=0.16 g/cc 0.01 10

20

50

0.5

absnow computed absnow6 measured Computed/ measured

0.0 100

Frequency (GHz) Figure 11: Absorption coefficients (neper/m) of dry snow at low density: Comparison of theoretical values with measured values based on the 6-Flux Model. Also shown is the ratio between the computed and measured absorption coefficient. Logarithmic scales are used.

10.00

1.2 1.0

1.00

0.8

r a 0.6 t i o 0.4

Absorption (1/m) 0.10 T = 268 K den.=0.35 g/cc 0.01 10

20

50

absnow computed absnow6 measured Computed/ measured

0.2 0.0 100

Frequency (GHz) Figure 12: Absorption coefficients (neper/m) of dry snow at high density versus frequency: Comparison of theoretical values with measured values based on the 6-Flux Model. Also shown is the ratio between the computed and measured absorption coefficient. Logarithmic scales are used.

The results of Figures 11 and 12 are surprisingly good, especially at the higher density. They support the effects described by the 6-Flux Model. It should be noted that the 2-flux absorption coefficients are clearly larger than the 6-flux coefficients due to the effects of the trapped radiation as described in Note 2. Table 11: Absorption coefficients (neper/m) of dry snow at low density: Comparison of theoretical values with measured values based on the 6-Flux Model. Numerical data of Figure 11.

23

0 0.16 g/cc 1 f (GHz) 2 absnow 3 absnow6 4 Com 268 K computed measured measured ----------------------------------------------------------1 data 11 0.0187 2 from 21 0.0663 0.0422 1.571090 3 Weise 35 0.1830 0.1210 1.512397 4 (1996) 48 0.3430 5 94 1.3150 0.9450 1.391534 Table 12: Absorption coefficients (neper/m) of dry snow at high density: Comparison of theoretical values with measured values based on the 6-Flux Model. Numerical data of Figure 12.

0 0.35 g/cc 1 f (GHz) 2 absnow 3 absnow6 4 Computed/ 268 K computed measured measured ----------------------------------------------------------1 data 11 0.042721 0.042 1.017167 2 from 21 0.151498 0.153 0.990183 3 Weise 35 0.417517 0.440 0.948902 4 (1996) 48 0.783697 0.820 0.955728 5 94 3.003753 3.630 0.827480

24

Extract from MICROWAVE PROPERTIES OF ICE AND SNOW Note 5, April 17, 1996 C. Mätzler, NSIDC-CIRES, Campus Box 449, University of Colorado, Boulder CO, 80309 preprint, Jan. 1996, published in B. Schmitt et al. (eds.) ”Solar System Ices”, Astrophys. and Space Sci. Library, Vol. 227, Kluwer Academic Publishers, Dordrecht, C. Mätzler, "Microwave properties of ice and snow", pp. 241-257 (1998).

3. Dielectric properties of water ice: imaginary part 3.1 PURE WATER ICE The imaginary part ", also called loss factor, of the relative permittivity is responsible for wave absorption, and for emission of thermal radiation. For ice let " be denoted by i". At frequencies f above 10 KHz and up to 1 GHz i" is governed by the high-frequency tail of the orientational Debye relaxation (Auty and Cole 1952; see also the excellent review of Petrenko 1993) with its relaxation frequency in the Hz to KHz range. Above 100 KHz the relaxation tail can be represented by a 1/f term, i.e. like a conductive component. This term is quite sensitive to ionic impurities, even at the level of a few ppm (see e.g. Moore et al. 1993). The low-frequency tail of the vibrational bands peaking in the infrared (Mishima et al. 1983) is a term proportional to f. The sum of these two terms leads to a function with a deep minimum in the microwave range. At T=250K the minimum of i"10-4 is found near 1 GHz. With decreasing temperature the minimum and its frequency decrease, because the change of i" with changing temperature is mostly due to the relaxation term. Additional losses due to unexplained relaxation effects in the MHz range were proposed in the literature. For a long time they were assumed to exist; however, they could not be quantified, and thus were not confirmed. Therefore Hufford (1991) tried to explain i"(f,T) from 100 KHz to 1 THz by the two terms only. The formula was derived from a critical analysis of published data. The mentioned data of Auty and Cole (1952) and of Mishima et al. (1983) formed the basis. Unfortunately, these two datasets correspond to non-overlapping temperature ranges. Therefore, additional information was sought to extend Mishima's data at T>200K. Hufford found suitable measurements of Lamb (1946) at 10 GHz, and of Wegmüller (1986) at frequencies just above the transition region (2-10 GHz, see Figure 4). Hufford concluded from these data that i" can be written as

i " =

+ f f

(2)

where the first term is the unaltered relaxation term of Auty and Cole, and the second term is based on Mishima, corrected by the additional loss observed at temperatures above 200K. It is to be noted that Equation (2) was already proposed by Walford (1968); only the parameters were inaccurate at that time. Using the modified inverse temperature , defined by =T0/T-1, where T0=300K, Hufford (1991) obtained the following expressions for the parameters and :

= (0.00504 + 0.0062 ) exp(221 . )

(GHz)

(3)

2

1+ 0.502 0.131

(GHz-1) (4) = 10 4 + 0.542 10 6 + 0.0073 1+ where the first term in (4) is an approximation to the expression of Mishima et al. (1983), and the second term is the mentioned correction. At very low temperatures (below 100K) and at very high frequencies (above 500 GHz) the more accurate formula of Mishima should be used for the first term, including a third term increasing with f 3 (see Appendix, Equation (A1)). To test the fit of Equations (2) to (4) the computed absorption coefficient a

a =

25

2" f c '

(5)

(where c is the vacuum speed of light) is compared with the independent data measured at T=-5C, and 15C (f=21, 35 and 94 GHz) by Mätzler and Wegmüller (1987), and at T=-24C (f=30 and 40 GHz) by Fujita et al. (1995) (see also Surdyk and Fujita, 1995).

Appendix: An improved formula for Mishima et al. (1983) derived and tested a theoretical expression for the absorption coefficient a of cold water ice (T200K we propose to use a new correction for T>200K, as derived by the author from Wegmüller's (1986) data:

The total value of is the sum

= exp(10.02 + 0.0364 (T 273K ))

(A2)

= M +

(A3)

This applies to a larger range than (4). Differences are small except for temperatures below 80K where the temperature dependence of Hufford's approximation starts to deviate from (A1), and for wavelengths 0, we have: I < Q. This means that the polarization difference is more strongly damped in the snowpack than the intensity. This result may be of interest if we want to retrieve geophysical information from a scattering medium. The penetration depth, i.e. the inverse of the damping coefficient, is smaller for the polarization difference than for the intensity. For a snow layer of thickness d the values for the internal reflectivity and transmissivity are rI = r0I(1 - t0I2)/(1 - r0I2t0I2) rQ = r0Q(1 - t0Q2)/(1 - r0Q2t0Q2) tI = t0I(1 - r0I2)/(1 - r0I2t0I2) tQ = t0Q(1 - r0Q2)/(1 - r0Q2t0Q2)

(23) (24) (25) (26)

t0I = exp(-Id) t0Q = exp(-Qd)

(27) (28)

where

38

With Equations (7) to (10), and (19) to (28) we can express the internal reflectivities and transmissivities at like- (rp=rii, tp=tii) and at cross- (rx,tx) polarization for the linearly polarized fluxes T1 to T4. The result is: rp = rhh = rvv = 0.5(rI + rQ) rx = 0.5(rI - rQ) tp = thh = tvv = 0.5(tI + tQ) tx = 0.5(tI - tQ)

(29) (30) (31) (32)

Then the total internal reflectivity rh at h polarization is, according to (2) rh = rhh + rx = rI

(33)

The same result appears at v polarization (rv = rh). And for the total transmissivity th = tv, we get th = tv = thh + tx = tI

(34)

Thus we can drop the polarization index from the internal reflectivity r and transmissivity t in (1), and thus also from the emissivity e. The fact that these parameters are the same at v and h polarization is a result of the special symmetry assumptions made here. These assumptions should hold for the Rayleigh phase function. The polarization-dependence of the reflectivity is determined by the boundary conditions at the upper and lower layer interface where Fresnel reflectivities are assumed to hold (see Note 7). The internal scattering reduces the polarization difference due to the coupling (rx and tx). In a more general case we could allow different values for rvv and rhh. In that case, however, the 4 equations for the Stokes Parameters would not decouple, leading to a more complex situation. The final parameters of each layer j can be applied to the layered snow-pack model of Note 7 in conjunction with Equations (1) and (2).

39

Improved Born Approximation for scattering of radiation in a granular medium Note9r.doc

Version 4, Jan. 1998

Christian Mätzler, Institute of Applied Physics, University of Bern Sidlerstr. 5, CH-3012 Bern, Switzerland, Fax: +41-31-631-3765, email: [email protected] Submitted to J. Appl. Physics, Paper Ref. JR-1044, revised Jan. 1998 Publ. in J. Appl. Phys., Vol. 83, No. 11, pp. 6111-6117 (1998). Abstract An improved Born Approximation for volume scattering and absorption of electromagnetic radiation by a non-magnetic, granular medium of small grains with large dielectric contrast is formulated. In this lowfrequency approximation it is assumed that the permittivity of the medium is given by the EffectiveMedium Theory of Polder and van Santen which also provides an expression for the internal field. The model is used to derive formulae for the scattering and absorption coefficients of freely arranged spheres and spherical shells. In addition measured dielectric properties are used to derive relevant model data of dry snow. In contrast to the conventional Born Approximation, the results reproduce Rayleigh scattering for a single particle, and for a dense medium, the scattering coefficient shows a non-linear behavior with increasing density, comparable to the Dense-Medium Radiative Transfer Theory. Characteristic differences of the spectra of the scattering coefficients result for the different particle types.

Introduction Measured signatures of microwave emission and backscattering of snowpacks motivated the advanced theoretical modeling of volume scattering in low-loss media. For a dense medium, such as snow, the quantitative treatment has been a difficult task. The conventional Born Approximation is inaccurate because of the relatively large difference between the dielectric permittivities of air and ice. Also the consideration of snow as an agglomeration of discrete ellipsoidal or spherical scatterers cannot be handled as in the case when the medium is dilute. Models to cope with these situations have been found, in principle, by the Strong-Fluctuation Theory (SFT) (Tsang and Kong, 1981, Stogryn, 1986) and by the Dense-Medium Radiative Transfer (DMRT) Theory (Tsang and Kong, 1980; Tsang 1987), respectively. For a textbook reference to these models, see Tsang et al. (1985). Their application to real situations has been hampered by the complexity of the models, and by more inherent difficulties encountered when a natural medium is being modeled. As an example it has been assumed that the medium can be described by a given autocorrelation function, usually of exponential shape. Recent results indicate that the actual shape depends on snow type (Weise, 1996). In a simpler approach, using a quasi-static model for the internal field, Mätzler (1987) formulated an improved Born Approximation. It is a simplified SFT where the mean field is assumed to be given by a quasi-static dielectric mixing theory. Due to an incorrect treatment of the variance describing the internal field variation, Mätzler (1987) was unable to recover Rayleigh scattering. In the present paper, the model is reformulated to correct for this error. The reformulation is also indicated because Sihvola and Kong (1988) clarified the connection between the different physical mixing formulae. The advantage of the model is its simplicity and its applicability to natural media consisting of complex particles. Furthermore, the model is able to describe scattering from a dense agglomeration of individual scatterers, such as spheres, i.e. the situation of the DMRT Theory, if the autocorrelation function of the medium is known. For a medium with free arrangement of the particles, it was shown, that this function is identical to the autocorrelation function of its particles (Mätzler, 1997). With free arrangement it is meant that the probability to find a particle anywhere outside of a test particle is equal to the volume fraction of particles. The autocorrelation function of the particle can be determined simply from the ratio, A(x)=Vc/V, where Vc is the common volume of two identical particles whose centers are separated by the displacement vector x, and V is the volume of the particle.

40

The improved Born Approximation will be applied to the volume element considered in the derivation of the radiative transfer equation in a refractive medium (Bekefi, 1966). In this way, scattering and absorption coefficients and phase functions are obtained. Integral representation of the scattered field Following Ishimaru (1978), an incident plane wave of unit-amplitude electric field êi with harmonic time variation exp(-it) propagates in direction î in a non-magnetic granular medium with relative permittivity eff. Here, eff is considered to be real. The wave is scattered in a limited volume V (Figure 1) of heterogeneous permittivity (r') where r' is the position vector. We will assume that the medium in V consists of grains of permittivity 2 embedded in a host medium of permittivity 1. The scattered field at the far-field distance r from the scattering center in direction of unit vector ô is given by a spherical wave (the time factor being omitted in the following) (1) Es(r) = f(ô,î) exp(ikeffr)/r where r = rô, keff =k( eff)1/2 is the wave number of the medium with effective permittivity eff, k is the vacuum wave number, and f(ô,î) is the scattering amplitude, determined by f(ô,î) = -

2 keff

4

ô

{

ôF

}

(2)

and F is given by the volume integral F=

E(r')[ (r')/ eff-1] exp(-ikeffr'ô) dV'

(3)

V

where E(r') is the electric field inside the scatterer at position r'.

ô

Es Ei î

V Figure 1: Geometry for scattering in a heterogeneous medium from incident direction î to ô. Scattering and absorption cross sections and coefficients The bistatic scattering cross section bi(ô,î) - being related to the differential scattering cross section d = bi/(4) - is determined from f(ô,î) by 4 2 keff $ F 2 sin 2 (ô,î) = 4 f (o$ , i ) = 4 bi

(4)

where is the angle between ô and F. The bistatic scattering coefficient is bi(ô,î)=bi(ô,î)/V. Furthermore, the backscattering cross section b is given by b = bi(-î,î), and the scattering cross section s is obtained by averaging (4) over all directions ô. The scattering coefficient s to be used in the radiative transfer equation is related to s by s = s/V, and the scattering phase function follows from bi(ô,î). In the backscatter direction we can define the volume-backscatter coefficient b = bi(-î,î)/V. Similarly, the absorption coefficient a is obtained from a=a/V, where the absorption cross section a is given by (Ishimaru, 1978) 2

a = k "(r' ) E(r' ) dV'

(5)

V

and "(r') is the imaginary part of the relative permittivity at r'.

41

The internal field In the conventional Born Approximation, also called Rayleigh-Gans Approximation (e.g. Jones 1961; Ishimaru 1978; for application to snow, see Vallese and Kong, 1981), E(r') is replaced by the incident field, Ei = êiexp(ikeffr'î). This approximation is useful for very weak fluctuations of the permittivity, such as in atmospheric turbulence. In case of strong fluctuations we need a better approximation. The simplest solution is to assume proportionality between E(r') and Ei. Expressing the proportionality in tensor form leads to (6) E(r') = Kêi exp(ikeffr'î) where the tensor K relates the internal field to the incident field outside the particle. An analytic expression for K can be found for small ellipsoidal scatterers; their internal field is homogeneous. If ellipsoids are oriented with their main axes parallel to the selected Cartesian coordinates, K is diagonal

K11 K= 0 0

0 K 22 0

0 0 K33

(7)

with Kjj =

a ; a + ( 2 1 )A j

j=1, 2, 3

(8)

where Aj is the depolarization factor of axis j. Here, the so-called apparent permittivity a, has been introduced (Sihvola and Kong, 1988). It is the permittivity of the ambient background which appears locally to a given particle of permittivity 2. For a single ellipsoid embedded in a large volume, a is given by the host permittivity 1. With increasing volume fraction of ellipsoids distributed in V, different dielectric mixing theories give more and more different values of a, ranging from 1 to eff (Sihvola and Kong, 1988). The depolarization factors can be computed from elliptic integrals (Stratton, 1941); they depend only on the axial ratios of the particle and obey (9) Aj0 and A1 + A2 + A3 = 1. An improved Born Approximation For computational reasons, let us introduce at vector function y(r') consisting of a spatial mean value y0 and a fluctuating part yf with zero mean y(r') = y0 + yf(r') = K e$ i (r' ) (10) where (r') is the relative susceptibility with respect to eff: (11) (r') = 0 + f(r') = [(r')/eff-1] and where 0=m/eff-1 is the spatial mean susceptibility and f is its fluctuating part. Here m is the spatial mean permittivity which is obtained from the mean value of (r'). In general, mixing theory dictates that eff is not larger than m; therefore 0 is a small non-negative number. The role of m is just mathematical; the quantity will disappear later, see Equation (20). Inserting (10) and (11) in (6) and (3), we get the following improved form of the Born Approximation: F( k) = F0(k)+ Ff( k) (12) with F0(k) and Ff( k) given by F0(k)= y0 exp( ix k)dV (13)

V Ff(k) = y f ( x ) exp( ix k)dV V

(14)

where x = r' and k = (ô-î)keff. Note that for V, the integral in Equation (13) converges to a delta function. Therefore, F0(k) describes a coherent wave scattered in the forward direction, and (14) describes volume scattering due to the heterogeneity of the permittivity. This is what we are looking for. We can find the volume-scattered field by inserting Ff(k) instead of F( k) in Equations (2) and (4). The improvement of (14) with respect to the conventional Born Approximation is a better description of the amplitude of the scattered wave. The improvement is due to K. This quantity may be estimated from either (a) the orientation and shape of the particles, using the quasi-static approximation, Equations (7) to (9), or (b) the Strong-Fluctuation Theory, or else (c) empirically from measurements of the effective

42

permittivity of the scattering volume, by applying a physical mixing theory. In (14) it is also assumed that the incident wave propagates through the inhomogeneous volume with the same effective wave number as outside V. By this assumption, we allow a smooth transition of the incident and scattered waves into and out of the scattering volume without refraction. Therefore we can allow any shape for the scattering volume. The actual value of the propagation constant is computed in the quasi-static limit from dielectric mixing theory (Sihvola and Kong, 1988, 1989) or from the Dyson Equation for the mean field in the Strong-Fluctuation Theory (Tsang and Kong, 1981; Jin, 1989). Since f(r') is zero outside V, Equation (14) remains unchanged if the integration is extended to infinity. Then (14) is the 3-dimensional Fourier Transform of yf(r'). One way to compute Ff is to use Equation (14). However this requires knowledge of the actual arrangement, size and shape of all scatterers. Another way is to use statistical information of the medium, as expressed by the autocorrelation function. Such a formulation is more practical for natural media. Application of the Autocorrelation Theorem (e.g. Bracewell, 1965) to (14) leads to the following expression for the spectral power density 2

F f ( k) = V2 A( x ) exp( ix k)dV

(15)

where A(x) is the autocorrelation function normalized to A(0)=1, and the variance 2 is given by 2 =

2 2 1 1 y f (r' ) dV' = K e$ i f (r' ) dV' V V

(16)

A(x) is computed from

A( x ) =

1 y f (r' ) y f (r' x ) dV' V 2

(17)

For isotropic functions, A(x)=A(x), where x=x, the integration in (15) over directions yields

1 (18) A( x )x sin(x )dx 0 sin( / 2) , and is the scattering angle (see Figure 1), with 0. Due to

2

F f ( k) = 4V2 where = k = 2 keff

decorrelation, A(x) converges to zero for large x. The main decrease takes place in the interval from 0 to pc, the correlation length, which is an effective size of the particles. Equation (18) gives the dependence of scattering on dielectric and structural properties of the medium as expressed by the autocorrelation function and by the variance. Application to a granular medium Let us consider a granular medium, consisting of particles with the same shape, and allowing different sizes. The particles have permittivity 2, and they are embedded in a host medium with permittivity 1.

r' in host 1 ; 2 ; r' in particle

(r') =

(19)

For ellipsoids and other particles with a similar behavior, it is assumed that K is the Tensor (7) with elements given by (8). Since K e$ i

2

only depends on direction, it can be separated from the volume

integral (16), leading to 2

2

< K e i > (r' ) m v(1 v)( 2 1 )2 2 $ < K e > = = dV ' i eff 2 V eff V 2

(20)

where v is the volume fraction of ellipsoids, and the brackets < > mean averaging over orientation of the ellipsoids due to isotropy. From (7) and (8) we get the mean-squared field ratio K = < K e$ i 2

2

2 a 1 3 1 3 >= K jj = 3 j =1 3 j =1 a + ( 2 1 )A j

43

2

(21)

For eff the Effective Medium formula of Polder and van Santen (1946) has been shown to be a good approximation, especially for snow (Mätzler, 1996). The mixing formula can be written in a generalized form (Sihvola and Kong, 1988): 3

a j = 1 a + A j ( 2 1 ) eff = 1 + 3 Aj 3 v ( 2 1 ) j = 1 a + A j ( 2 1 ) v ( 2 1 )

(22)

In case of Polder and van Santen a and eff are related by (23) a=eff(1-Aj)+1Aj It is to be noted that a is different for each Index (j=1,2,3). So far dielectric losses have been ignored, i.e. all permittivities have been assumed to be real. Dielectric losses can be introduced by a small imaginary part 2" of the permittivity of particles to get the absorption coefficient of the low-loss medium from Equation (5): a = vk2"K2 (24) For a given autocorrelation function and given values of v, 1, 2, 2 " and Aj, the Equations (4), (18) and (20) to (24) are complete to compute the bistatic scattering cross section, the absorption and scattering coefficient. In (4), F is to be replaced by Ff. By eliminating the variance, the formulae can be rearranged to yield bi(ô,î)=

4 keff

4V

Ff

2

sin 2 = v(1 v)( 2 1 )2 K 2 I k 4 sin 2

(25)

where K2 is the mean-squared field ratio (21), and I=

1 A( x )x sin(x )dx 0

(26)

Small spherical scatterers Let us apply the formulae to a medium consisting of freely arranged spheres of volume V0, radius R, diameter D. The depolarization factors are Aj=1/3, the effective permittivity of Polder and van Santen is eff =

2 1 2 + 3v( 2 1 ) +

(21 2 + 3v( 2 1 ))2 + 81 2

(27)

4

and the mean-squared field ratio becomes 2

K =

2 eff + 1

2

(28)

2 eff + 2

The normalized autocorrelation function of freely arranged spheres is (Mätzler, 1997)

3x x3 + ; 0xD 1 A(x) = 2 D 2 D3 0 ; else

(29)

The correlation length, pc=2D/3, is the inverse coefficient of the linear term of A(x). Approximating the sine function in (26) for small arguments by sin(x) x[1-(x)2/6] leads to

R3 I [1-(R)2/5] 3

(30)

Neglecting the term (R)2/5 and inserting I in (25), we get 2

( 2 1 )(2 eff + 1 ) R3k 4 (ô,î) = v(1 v ) sin 2 2 eff + 2 3 bi

The scattering coefficient s is obtained by averaging (31) over directions:

44

(31)

( 2 1 )(2 eff + 1 ) 3 pc 3 k 4 s v(1 v) 2 eff + 2 32

2

(32)

Here the correlation length is used as effective size parameter. If absorption losses are limited to 2, the absorption coefficient is given by a = vk 2 "

2 eff + 1

2

(33)

2 eff + 2

For a single sphere, where v=V0/V1) to the surrounding medium is taken into account (see e.g. Bekefi, 1966, and Section 3.8 of Mätzler, 1987). The dependencies of b and s on k and pc, respectively, are clearly different for small spheres and for thin shells. In snow both types of scattering may occur. Crusts of refrozen snow consist of rounded and sintered grains. On the other hand, fresh snow consists of plate-like particles, and depth hoar consists of cup-like or hexagonal grains which may be approximated by hollow spheres. The quantitative test of the present model with natural snow-sample data (Wiesmann et al. 1998) will be the topic of a forthcoming paper. Acknowledgments The author likes to thank Jaroslav Ricka, Ari Sihvola and Dale Winebrenner for valuable comments on the subject of this paper. The work was supported by an ESA-ESTEC study, Contract No. 11706/95/NL(PB) and by the Swiss contribution to COST Action 712. References 1. 2. 3. 4.

Bekefi, G. "Radiation Processes in Plasmas", John Wiley, New York, 377pp. (1966). Bracewell, R. "The Fourier Transform and its Applications", McGraw-Hill, New York, 381pp. (1965). Chandrasekhar, S. "Radiative Transfer", Dover Publications, New York, 393pp. (1960). Hufford G. "A model for the complex permittivity of ice at frequencies below 1 THz", Int. J. Infrared and Millimeter Waves, Vol. 12, pp. 677-682 (1991). 5. Ishimaru, A. "Wave Propagation and Scattering in Random Media", Vols. 1 and 2, Academic Press, Orlando, 572pp. (1978). 6. Jin, Y.Q., "The radiative transfer equation for strongly-fluctuating, continuous random media", J. Quant. Spectrosc. Radiat. Transfer, Vol. 42, pp.529-537 (1989). 7. Jones, D.S. "Theory of Electromagnetism", Pergamon Press, New York, 807pp. (1964). 8. Mätzler C. "Applications of the Interaction of Microwaves with the Natural Snow Cover", Remote Sensing Reviews, Vol. 2, pp. 259-392 (1987). 9. Mätzler, C. "Microwave permittivity of dry snow", IEEE Trans. Geosci. Remote Sensing, Vol. 34, No. 2, pp. 573-581 (1996). 10. Mätzler, C. "Autocorrelation functions of granular media with free arrangement of spheres, spherical shells or ellipsoids", J. Applied Physics, Vol. 81 (3), pp.1509-1517 (1997). 11. Polder, D. and J.H. van Santen, "The effective permeability of mixtures of solids", Physica, Vol. 12, (5), pp. 257271 (1946). 12. Pulliainen, J., K. Tigerstedt, W. Huining, M. Hallikainen, C. Mätzler, A. Wiesmann, U. Wegmüller and J. Noll, "Retrieval of geophysical parameters with integrated modelling of land surfaces and atmosphere", (Models/Inversion Algorithms) Draft Final Report, ESTEC Contract No. 11706/95/NL/NB(SC), Dec. (1997). 13. Shih, K.-H. Ding, J.A. Kong and Y.E. Yang, Modeling mm wave backscatter of time-varying snowcover, Prog??ress in Electromagnetics Research, PIER 16, pp. 305-330, (1997). 14. Sihvola A. and J.A. Kong, "Effective permittivity of dielectric mixtures", IEEE Trans. Geosc. Rem. Sens. Vol. 26, pp. 420-429 (1988); Errata, Vol. 27, pp. 101-102 (1989).

49

15. Stogryn, A., "A Study of the Microwave Brightness Temperature of Snow from the Point of Strong Fluctuation Theory", IEEE-Trans. GE-24, 2, pp. 220-231, (1986). 16. Stratton, J.A. "Electromagnetic Theory", McGraw Hill, New York, 615pp. (1941). 17. Tsang, L. "Passive remote sensing of dense nontenuous media", J. Electromagnetic Waves and Applications, Vol. 1, pp. 159-173 (1987). 18. Tsang, L. and J.A. Kong, "Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formulism", J. Appl. Phys. Vol. 51, pp.3465-3485 (1980). 19. Tsang, L. and J.A. Kong, "Scattering of electromagnetic waves from random media with strong permittivity fluctuations", Radio Science, Vol. 16, pp. 303-320 (1981). 20. Tsang, L., J.A. Kong, and R.T. Shin, "Theory of Microwave Remote Sensing", Wiley-Interscience, New York, 613pp. (1985). 21. Vallese, F., and J.A. Kong, "Correlation function studies for snow and ice", J. Appl. Phys. 52, 4921-4925 (1981). 22. Weise, T. "Radiometric and structural measurements of snow", Doctoral Thesis, Faculty of Natural Philosophy, University of Bern, Switzerland, Feb. (1996). 23. Wen, B., L.Tsang, D.P. Winebrenner and A.K. Ishimaru, "Dense medium radiative transfer theory: Comparison with experiment and application to microwave remote sensing of snow", IEEE Trans. Geosci. and Remote Sensing, Vol. 28, pp. 46-58 (1990). 24. West, R., L. Tsang and D.P. Winebrenner, " Dense Medium Radiative Transfer theory for two scattering layers with a Rayleigh Distribution of particle sizes", IEEE, Trans. Geoscience and Remote Sensing, Vol. 31, pp. 426437 (1993). 25. Wiesmann, A., C. Mätzler and T. Weise, "Radiometric and structural measurements of snow samples", Radio Science, in press (1998).

50

Coherent scattering effects in a snowpack Note11.doc, August 97 Andreas Wiesmann and Christian Mätzler, Institute of Applied Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland, Fax: +4131 631 37 65, E-Mail: [email protected]

Abstract In Note 6 [Mätzler, 1996] a model for microwave emission of a layered snowpack was presented. In this model the layers were considered as sufficiently thick, so that coherent effects can be neglected. Sometimes thin layers (ice lenses, surface hoar, onset of snowfall, melt or refreezing condition) occur within or at the surface of the snowpack. These layers can show broadband coherent effects. In this note we present an adaptation of the previous model to account for coherent effects in layers thinner than about half a wavelength.

Introduction Different meteorological events can lead to thin layers in a snow cover. Wind densifies the snow surface so that wind crusts are formed, melt conditions lead to wet layers and they, when refrozen, transform into ice lenses or crusts. These layers can have a large spatial extent and strongly affect the microwave behavior of the snow. For layers thinner than /2 the coherent reflection dominates the microwave behavior. Strong coherent effects are expected for layers of the size /4 (see Figure 3). Due to the spatial variability in the thickness of layers, the coherent effects are smeared out for thicker layers by averaging over lateral distances, and also by averaging over the beamwidth and bandwidth of the radiometers; then the coherent effects disappear, and thus they can be ignored. These layers are handled as explained in Note 6 (Mätzler 1996). The thinner layers are dealt with in an improved way as explained in this note. The thin layers are identified from the input table describing the snow profile; i.e. the layers, of the snowpack to be modeled. The effective reflection of these layers is described here. This addition can be applied to the layered snowpack emission model presented in Note 6. The following chapters explain the theoretical approach, the implementation and an example of the modifications needed before the model of Note 6 can be applied.

Reflectivity on a double layer According to the invariant embedding method by Adams and Denman [Adams and Denman, 1966] the coherent reflection coefficient Rn of a system containing n coherent layers (Figure 1) can be written as:

Rn =

Fn + Rn 1e 2ipn 1 + Fn Rn 1e 2ipn

; n1

(1)

where Fn is the Fresnel reflection coefficient for the interface between layer n and n+1, Rn-1 is the reflection coefficient for the system containing n-1 layers (R0 = F0) and pj is the one-way phase through layer j (here for j=n), given by

pj =

2d j n j cos j

.

(2)

which is a function of the thickness dj, of layer j, the refractive index nj, the propagation angle j and the vacuum wavelength .

51

Figure 1: A multi layer system with a Figure 2: Thin layer j of coherent reflecwave incident from above at an anglen+1. tion between two scattering layers j-1 and j+1, here for j=2. To simplify (1) we define

Ej e

2 ip j

.

(3)

Using (1) and (3) the reflection coefficient of a double layer (Figure 2) R2 (with R1 = F1) can be written as

R2 =

F2 + F1 E 2 1 + F2 F1 E 2

(4)

The power reflectivity r of a double layer is given by

r = R2

2

F + F1 E 2 = 2 1 + F1 F2 E 2

2

.

(5)

If the interface reflection coefficients are real and p2 = p2’ + ip2’’ then

r=

F2 2 + F12 e 4 p2 + 2 F1 F2 e 2 p2 cos(2 p2 )

1 + F12 F2 2 e 4 p2 + 2 F1 F2 e 2 p2 cos(2 p2 )

(6)

and

2 p2 =

4 d 2 n2 cos 2

(7)

2 p2 =

4 d 2 n2 . cos 2

(8)

If p2” « 1, then (6) simplifies to

r=

F2 2 + F12 + 2 F1 F2 cos(2 p2 )

(9)

1 + F12 F2 2 + 2 F1 F2 cos(2 p2 ) 52

and

2 p2 =

4 d 2 n2 cos 2 .

(10)

The coherent power reflectivity of layer j, r depends on dj by a cosine function cos(2pj):

1.0

cos(2pj)

0.5

0.0 /2

3/2

2pj

-0.5

-1.0 Figure 3: Dependence of r with d: cos(2pj) as a function of 2pj. Figure 3 presents cos(2pj) as a function of 2pj. To take the first two extrema into account and to get a smooth transition from coherent to incoherent reflection, it is necessary to handle a layer as coherent as long as 2pj < 3/2. I.e. a layer is defined as thin and handled as a coherent layer if

2pj < 3/2 4.7.

(11)

Otherwise the layer is considered as incoherent. Then the snowpack model of Mätzler [Mätzler, 1996] applies; especially the interface reflectivity is given by sj = Fj2. Condition (11), and using (10) with j=2 ensures that the effective path dj cosj in the coherently reflecting slab is less than about half a wavelength. The effect of the coherent layer j is taken into account by using the coherent reflectivity r of the thin layer as the interface reflectivity sj between the two scattering layers, j+1 and j-1, separated by the thin layer. Afterwards the coherent layer is removed from the input table of the layered snowpack emission model, i.e. the coherent layer is represented exclusively by r. Some restrictions apply. The snowpack emission model works only if there is at least one scattering layer. Because coherent interaction with a rough ground surface cannot be described within our model, it is also required that the lowest snow layer is incoherent. However, a coherent surface layer is allowed by the model. In this case, the top layer in Figure 2 is the air above the snow surface.

Model implementation and programming Given a special snow profile, then at each frequency f, the following tasks have to be performed: 1. Check if there is a volume scattering layer at all (2pj > 4.7). If there is no such layer, the snowpack emission model cannot be applied to this situation. 2. Remove thin layers from the bottom until a scattering layer is found. The thickness of the scattering layer increases by the extension of the removed layers.

53

3. 4. 5. 6.

Locate thin layers elsewhere from the table of the snow profile. Locate the thickest layer if more than one subsequent layer is found to be thin and ignore the others. Repeat until all thin layers have been identified. Compute the coherent reflectivities (at h and v polarization) of the thin layers and replace the interface reflectivities of the adjacent layers by the coherent reflectivities. 7. Ignore volume scattering within a thin layer, e.g. remove it from the input table of scattering layers. 8. The new input table is now ready to be used by the layered snowpack emission model.

Example In Table 1 a possible input snow profile «test01» is shown. Starting from the ground, there is a thin bottom layer followed by two layers of densified snow. Above, we have three thin crusts followed by a layer of moderately densified snow. On top two crusts can be observed. Table 2 shows the profile after step 4 using a frequency f of 2 GHz and a 50° incidence angle n . Table 3 shows the adapted data after step 7 (also for f = 2 GHz, = 50°) ready to be used as input for the snowpack emission model. In Figure 4 the calculated brightness temperature versus the frequency from 2 to 100 GHz is shown.

Table 1: Properties of the input snow profile. #: layer number starting at the bottom; d: layer thickness;: density of the layer; Tsnow,: snow temperature; pc: correlation length; : local incidence angle; 2p: given by (10). #

1 2 3 4 5 6 7 8

d [m] 0.030 0.170 0.085 0.035 0.010 0.020 0.100 0.005

[kg/m3] 0.300 0.380 0.280 0.380 0.200 0.200 0.200 0.200

Tsnow [K] 265.5 263.5 256.2 253.1 253.0 253.0 256.0 256.0

[deg] 40.4 38.2 40.9 38.2 43.2 43.2 43.2 43.2

pc [mm] 0.200 0.100 0.300 0.100 0.100 0.050 0 0

2p 2.36 14.63 6.56 3.01 0.70 1.41 7.05 0.35

Table 2: Properties of the input snow profile. #: layer number starting at the bottom; d: layer thickness;: density of the layer; Tsnow,: snow temperature; pc: correlation length; Fh, Fv: Fresnel reflection coefficient at h and v polarization, respectively; : local incidence angle. #

2 3 4 7 8

d [m] 0.200 0.085 0.035 0.100 0.005

[kg/m3] 0.380 0.280 0.380 0.200 0.200

Tsnow [K] 263.5 256.2 253.1 256.0 256.0

pc [mm] 0.100 0.300 0.100 0 0

Fh

[deg] 38.2 40.9 38.2 43.2 43.2

Fv

0.0579 -0.0588 0.1060 0 0.1473

-0.0113 0.0104 -0.0175 0 0.0040

Table 3: Properties of the input snow profile. #: layer number starting at the bottom; d: layer thickness;: density of the layer; Tsnow,: snow temperature; pc: correlation length; sh, sv: interface reflectivity on top of the layer at h and v polarization, respectively;: local incidence angle. #

2 3 7

d [m] 0.200 0.085 0.100

[kg/m3] 0.380 0.280 0.200

Tsnow [K] 263.5 256.2 256.0

pc [mm] 0.100 0.300 0

54

sh

sv

0.0033 0.0267 0.0217

0.0001 0.0008 0

[deg] 38.2 40.9 43.2

Figure 4: Calculated Brightness Temperature from snow profile «test01» versus frequency with vertical pol (upper) and horizontal pol. (lower curve), both at = 50°.

References 1. Adams R.N. and E.D. Denman: “Wave Propagation in Turbulent Media”, New York, Elsevier, 1966. 2. Mätzler C.: “A microwave emission model for a layered snowpack”, Note 6, Institute of Applied Physics, University of Bern, 1996.

55

Effective propagation angle and polarization mixing in a snowpack Note12, Nov. 1997 Christian Mätzler and Andreas Wiesmann, Institute of Applied Physics, University of Bern,

1) Introduction In Note 6 [Mätzler, 1996] a model for microwave emission of a n-layer snowpack was presented. The model is based on a simplified radiative transfer, described by six "fluxes" streaming along and against the principal axes. This 6-Flux Model and the determination of experimentally fitted model parameters were described by Wiesmann et al. (1997). In Note 6 it was assumed that the effective propagation angle i with respect to the surface normal is given by Snell's law of refraction at any position of the snowpack for both vertically and horizontally polarized radiation. Horizontal snow layers with smooth interfaces are assumed throughout the model. Deviations in propagation direction occur due to diffuse volume scattering. This does not necessarily mean that the effective propagation angle of the fluxes changes. Especially for an incidence angle near 50 degrees the effective propagation angle is similar to the one computed from Snell's law. However at near vertical incidence scattering leads to an increase of the effective propagation angle making it necessary to correct this angle. The model proposed is a first- order correction in effective propagation path and polarization mixing, considering single-scattering of radiation originating at the radiometer antenna and propagating into the snowpack (i.e. we describe the reciprocal and equivalent path of photons as if they were emitted by the radiometer antenna). The consequences of a modified propagation angle are • a different length of the propagation path within each snow layer, leading to a modified transmissivity, t0i = exp(-d/ cos i ) where is the damping coefficient and d the thickness of Layer i (Wiesmann et al. 1997), • change of the effective reflectivities at layer interfaces at v and h polarization.

Figure 5: n- layer snowpack with a wave incident from above at an anglen+1.

56

2) Effective propagation angle In Note 6 it was assumed that the effective propagation angle i with respect to the surface normal is given by Snell's law of refraction, using the refractive index ni of Layer i and the incidence angle n+1 on the snowpack for a refractive index in air of nn+1=1 (see Figure 1), i.e.

sin is =

sin n +1 ni

(1)

Here we use the index s in is to indicate that we mean the angle computed according to Snellius. For a radiometer observing at the incidence angle n+1 the effective propagation angle is well approximated by (1) for smooth and horizontal surfaces and layer interfaces as long as volume scattering is not too important. This is true just below the smooth snow surface. Equation (1) was applied to the 10 cm thick snow samples used by Wiesmann et al. (1997). However with increasing snow depth more and more of the forward streaming flux consists of forward scattered radiation leading to an effective propagation angle which differs from is. Assuming that, in the limit of strong scattering, the forward (downward) and backward (upward) fluxes are isotropic (or diffuse) then the effective propagation angle id can be represented by

cos id =

1 + cos ic 2

(2)

where ic is the critical angle for total reflection, i.e. the maximum propagation angle of radiation belonging to the vertical fluxes; it is given by

cos ic = 1 (ni ) 2

(3)

For a diluted medium where ni=1 we get cosid = 0.5; this is the usual value of the 2-Stream Model (Ishimaru, 1978). In our case of 6 streams or "fluxes", the streams have more narrow beams, therefore we have to use the modified form given by Equation (2). The actual value of the effective propagation angle i in layer i is somewhere between is and id. We assume that the two cosines are weighted according to the transmissivity of tz for non-scattered radiation from the snow surface (at height zn) to height z< zn, i.e.

cos i = t z cos is + (1 t z )cos id

(4)

and tz is given by z

tz = exp( z n

s ( z' ) dz' ) cos is

(5)

where s(z') is the scattering coefficient at height z'. For negligible scattering tz 1, leading to i = is, whereas for strong scattering we get t z = 0 and thus i = id. In the snowpack model the effective incidence angle is changed only at the layer interface, assuming constant values within each layer. Thus the value of z in Equation (5) is limited to zi; i = 1, 2, ..., n.

3) Polarization mixing After diffuse scattering the photons loose their memory about where they came from. As a consequence they cannot remember their plane of incidence which is needed to define the state of h or v polarization. Therefore the v- and h-polarized fluxes are mixed in a similar way as is and id. This mixing can be modeled by modifying the polarization-dependent parts of the snowpack model, i.e. the reflectivities siv and sih at the interface between Layer i and i+1. The computation of these reflectivities without polarization mixing is described in Note 11 (Wiesmann and Mätzler, 1997). Here we describe the modification due to the mixing effect. Let us denote the difference si by (6) si = sih - siv This difference is reduced due to scattering, leading to effective interface reflectivities siveff and siheff between Layer i and i+1 at position z = zi:

57

[ ] siveff = 0.5[sih + siv t z si ]

siheff = 0.5 sih + siv + t z si

(7) (8)

In (7) and (8) the values of siv and sih are the ones computed for the angles is. In a further improvement, these coefficients could be taken at the effective incidence angle i. However the effects of such a further correction are small because they tend to cancel.

4) Model implementation The modifications described here are implemented in the layered snowpack model in form of a (optional) subroutine. For a given frequency, incidence angle and for a given snowpack, the basic model parameters are first computed for each layer. Then the number of layers are reduced to incoherently scattering layers as described in Note 11. Then we apply the modifications described here, followed by the computation of the matrices and vectors and by the matrix calculus to obtain the emitted brightness temperature at h and v polarization, respectively.

References 1. Ishimaru, A. "Wave propagation and scattering in random media", Vol. 1 and 2, Academic Press, Orlando (1978). 2. Mätzler C. “A microwave emission model for a layered snowpack”, Note 6, in C. Mätzler, "Six notes on microwave radiation from snow samples and from a layered snowpack", Research Report No. 96-9, Microwave Dept., Institute of Applied Physics, University of Bern (1996). 3. Wiesmann, A., C. Mätzler and T. Weise, "Radiometric and structural measurements of snow samples", Radio Science, in press (1997). 4. Wiesmann, A. and C. Mätzler, "Coherent scattering effects in a snowpack", Note 11, Research Report No. 97-7, Microwave Dept., Institute of Applied Physics, University of Bern (1997).

58

Extension of the Microwave Emission Model of Layered Snowpacks to Coarse- Grained Snow Note 14, August 1998 , publ. in Remote Sensing of Environment, Vol. 70, No. 3, pp. 317-325 (1999) Christian Mätzler and Andreas Wiesmann, Institute of Applied Physics, University of Bern

Abstract The Microwave Emission Model of Layered Snowpacks (MEMLS) is a multi-layer and multiple-scattering model for a refractive medium, originally developed for dry winter snow (Pulliainen et al. 1998; Wiesmann and Mätzler, 1998). The model is based on radiative transfer, using a 6-Flux Theory to describe multiple volume scattering and absorption, including radiation trapping due to total internal reflection. The scattering coefficient was determined empirically from measured snow samples (Wiesmann et al. 1998a), whereas the absorption coefficient, the effective permittivity, refraction and reflection at layer interfaces were based on physical models. A limitation of the empirical fits is in the applicable range of the observed frequencies and correlation lengths. In order to extend the model, a physical determination of the scattering coefficients, describing the coupling between the six fluxes, is developed here, based on the improved Born Approximation for freely arranged snow grains. An exponential spatial autocorrelation function was selected. With this addition, the microwave emission model of a layered snowpack obtains a complete physical basis. The extended model is void of any free parameter. Model validation presented here was done with two types of experiments made at the alpine test site, Weissfluhjoch: (1) radiometry at 11, 21, 35, 48 and 94 GHz of winter snow-samples on black-body and on metal plate, respectively (Wiesmann et al. 1998a), and (2) radiometric monitoring at 4.9, 10.4, 21, 35 and 94 GHz of coarse-grained crusts growing and decaying during melt-and-refreeze cycles (Reber et al. 1987). Digitized snow sections were used to measure snow structure in both experiments. The coarsest grains were found in the refrozen crusts with a correlation length up to 0.71 mm; the winter snow samples had smaller values, from 0.035 mm for new snow to about 0.33 mm for depth hoar. Satisfactory results were obtained in all situations, indicating the importance of the multi-layer and multiple-scattering approach.

1. Introduction Microwave emission spectra of snowpacks show characteristic signatures depending on snow type and snow properties (Mätzler, 1994). In order to better understand these signatures, and in order to quantitatively use them in microwave remote sensing, a tool is needed to compute the microwave brightness temperature for any given snowpack under any sky illumination. Therefore a Microwave Emission Model of Layered Snowpacks (MEMLS) was developed by Wiesmann and Mätzler (1998) within an ESA study (Pulliainen et al. 1998), see also Wiesmann et al. 1998b). It is a multi-layer and multiple-scattering radiative transfer model applicable to the refractive snow medium, using a 6-Flux Theory to describe volume scattering and absorption, including radiation trapping due to total internal reflection. The scattering coefficient was determined empirically from measured snow samples (Wiesmann et al. 1998a), and the absorption coefficient, the effective permittivity, refraction and reflection at the plane layer interfaces were physically described. The empirical model for the scattering coefficient (Wiesmann et al. 1998a) has a similarity to Rayleigh scattering, however, with a weaker frequency dependence. The scattering increases by a power law of the microwave frequency times the correlation length with a power of approximately 2.5. Above a certain frequency or above a certain correlation length, the increase saturates in a similar way as Mie scattering does for spheres. The snow-sample experiments of Wiesmann et al. (1998a) were made with winter snow without very large grains (correlation length 0.035 to 0.33 mm). Thus the saturation effect was not observed. If the model is applied to refrozen crusts of coarse-grained firn, the simulated brightness temperatures become too low at mm wavelengths. As an example, at 35 GHz the brightness temperature is underestimated by more than 30K for a refrozen coarse-grained crust observed on June 14, 1984 with a correlation length of 0.71mm. The error is even larger at 94 GHz. Therefore a model is needed to better describe the scattering behavior beyond the power-law regime, taking into account the reduced scattering in the backward direction. A suitable model to be applied is one 59

which describes the snow by statistical properties without the need for defining a grain size. The improved Born Approximation (Mätzler, 1998) was used; there the snow structure is described by the two-point correlation. Although characteristic spatial autocorrelation functions are obtained for different grain shapes (Mätzler, 1997), the correlation length characterizes the smallest extent of the particles, not the typical grain size. For natural snow an exponential function is assumed, as this is often a good approximation.

2. Microwave scattering according to the improved Born Approximation In the improved Born approximation the bistatic volume-scattering coefficient (bistatic scattering cross section per unit volume, also the phase function if divided by the extinction coefficient, ext) for scattering from direction î to ô of isotropically oriented, freely arranged ice particles in air is given by (Mätzler, 1998): bi(ô,î) = v (1 v )( i 1) 2 K 2 I k 4 sin 2 (1) where v is the volume fraction of grains, i is the ice permittivity, k is the vacuum wave number, K2 is the ratio of the mean-squared electric fields inside and outside of the particles. This ratio depends on the shape of the ice particles. An analytic expression for K2 was found for the case of ellipsoids: 2

a 1 3 K = 3 j =1 a + ( i 1)A j 2

(2)

Here Aj is the depolarization factor of principal axis j of the ellipsoidal grains, and a is the apparent permittivity (Sihvola and Kong, 1988), given by a=eff(1-Aj)+Aj (3) For the effective permittivity eff the Effective Medium formula of Polder and van Santen (1946) can be used, Equation (22) in Mätzler (1998). Measurements of the effective permittivity of dry snow made with high accuracy near 1 GHz showed that the effective Medium Theory, using Equations (2) and (3), provides a good quantitative description of natural snow (Mätzler, 1996). The factor I in (1) is a Fourier integral of the autocorrelation function. This function was determined experimentally for all snow experiments to be presented here. An exponential function, exp(-x/pc), gave a good fit in most cases. The correlation length pc is the only parameter. For this function, the factor I is the following integral: I=

2 pc3 1 x / pc = sin( ) e x x dx 0 (1 + 2 pc2 ) 2

(4)

where is the scattering parameter, defined by 2 2 2 2 = 4 eff k sin ( / 2) = 2 eff k (1 cos ) (5) and is the scattering angle, and is the polarization angle (between the incident electric field and the direction ô of the scattered radiation). The correlation length pc was determined from the e-folding distance of the exponential function fitted to the measured correlation function. The analysis of the directional behavior of Equation (1) gives the sin2 dipole pattern at low frequency (