1

Microwave Emission Model of Layered Snowpacks

Andreas Wiesmann and Christian Mätzler

Institute of Applied Physics - University of Bern

Sidlerstr. 5, 3012 Bern, Switzerland

Phone ++ 41 31 631 45 90 - Fax ++ 41 31 631 37 65

wiesmann@iap.unibe.ch, matzler@iap.unibe.ch

Paper 1, submitted to Remote Sensing of Environment, December, 1998

Revised, March 1999

Abstract

A thermal Microwave Emission Model of Layered Snowpacks (MEMLS) was developed for

the frequency range, 5 to 100 GHz. It is based on radiative transfer, using six-flux theory to

describe multiple volume scattering and absorption, including radiation trapping due to total

reflection and a combination of coherent and incoherent superpositions of reflections between

layer interfaces. The scattering coefficient is determined empirically from measured snow

samples, whereas the absorption coefficient, the effective permittivity, refraction and reflection

at layer interfaces are based on physical models and on measured ice dielectric properties. The

number of layers is only limited by computer time and memory. A limitation of the empirical

fits and thus of MEMLS is in the range of observed frequencies and correlation lengths (a

measure of grain size). First model validation for dry winter snow was successful. An extension

to larger grains is given in Paper 2 (Mätzler and Wiesmann, 1999). The objective of the present

paper is to describe and illustrate the model and to pave the way for further improvements.

MEMLS has been coded in MATLAB. It forms part of a combined land-surface-atmosphere

microwave emission model for radiometry from satellites (Pulliainen et al., 1998).

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1. Introduction

The observation of passive microwave signatures of different types of snowpacks [e.g. Mätzler

1994] and the need to simulate their microwave emission for future applications was the moti-

vation for the development of a Microwave Emission Model of Layered Snowpacks

(MEMLS). Since the penetration depth of microwaves and the scattering coefficient depend on

frequency and snow properties, a reasonable model should account for realistic snow-profiles of

the sensitive parameters, such as correlation length, density and temperature, and in case of wet

snow also the liquid-water content. A successful model might then be used for the development

of remote sensing tools for observing processes on and within the snowpack. An important

internal process, being responsible for the release of large avalanches, is temperature-gradient

metamorphism, often taking place near the bottom of the snowpack. This process is based on

large water-vapor gradients; it leads to the formation of cohesionless depth hoar (Armstrong,

1977, 1985). Microwave emission of depth-hoar layers are very special due to the enhanced

volume-scattering by the large crystals (Weise, 1996). On the other hand, snowpack

stratification has important implications for physical processes, such as avalanche formation by

weak layers (Föhn et al. 1998; Fierz, 1998), reduced vertical diffusion of water vapor and heat

transfer, and percolation of liquid water in case of wet snow (Arons and Colbeck, 1995, and

references therein). Snow layers also mark meteorological events in the history of snowpacks,

and thus they contain information about past weather conditions. Specific microwave signatures

related to stratification are expressed, for instance, by polarization features of microwave

emission, and sometimes by special spectral properties. Since the earliest radiometer observa-

tions, interest has been paid to layering of polar ice sheets (Gurvich et al. 1973; Zhang et al.

1989; Rott et al. 1993; Surdyk and Fily, 1993; Steffen et al. 1999), and layer effects were also

found in snow-covered sea ice (e.g. Mätzler et al. 1984). Microwave emission of layered media,

such as firn, was computed in the past by several authors (e.g. Gurvich et al. 1973; Djermakoye

and Kong 1979; West et al. 1996). Their models are applicable at frequencies below about 20

GHz where volume scattering by the granular snow structure is not dominant.

When both volume scattering by snow grains and by the stratification are relevant, the models

become more complex. An example is the strong fluctuation theory (Stogryn 1986). In the

implementation by Surdyk (1992), this model was tested by the experimental snow data of

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Weise [1996] to be used also in the present work. The model validation clearly failed, probably

because multiple scattering is ignored. Indeed, West et al. (1993) showed on the basis of the

dense-medium radiative transfer theory applied to backscattering from snow that multiple

scattering by snow grains is important.

Driven by the need for a realistic snowpack emission model, we started with the development

of MEMLS, including multiple scattering both by stratification and by the granular snow

structure, and a tuned combination of coherent and incoherent superpositions of different scat-

tering contributions. First, the detailed behavior of single layers was investigated experimentally

by Weise [1996] and later by Wiesmann et al. [1998]. The work lead to a physical model of the

emissivity of homogeneous snow slabs in the frequency range from 10 to 100 GHz and for

correlation lengths from 0.05 to 0.3 mm. With the exception of the empirical determination of

the scattering coefficient, the parameters are physically based. In MEMLS this information is

applied to describe the behavior of individual layers. Here, the snowcover is considered as a

stack of n horizontal, planar layers, characterized by thickness, correlation length, density,

liquid-water content and temperature. The sandwich model of Wiesmann et al. [1998] is

extended to couple internal scattering and reflections at the interfaces of all layers. Internal

volume scattering is accounted for by a six-flux model (streams in all space directions). Homo-

geneity in the horizontal directions reduces to the well-known two-stream model (up and down

welling streams) where the two-stream absorption and scattering coefficients are functions of

the six-flux parameters. The absorption coefficient depends on density, frequency and tem-

perature, and the scattering coefficient depends on the correlation length, density and frequency

[Wiesmann et al. 1998]. The purpose of this paper is to describe and illustrate the model and to

pave the way for further improvements. Details on the realization are given by Wiesmann and

Mätzler [1998].

2. Microwave Emission Model of Layered Snowpacks

The snowcover is assumed to be a stack of n horizontal snow layers (j = 1,...n) with planar

boundaries between air-snow and snow-snow (Figure 1), characterized at frequency f, polari-

zation p, and incidence angle θ = θn by

− the snowpack brightness temperature Tb,

− the ground-snow interface reflectivity s0 and the ground temperature T0,

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− the interface reflectivity sj on top of each layer j,

− the thickness dj, temperature Tj, internal reflectivity rj, emissivity ej and transmissivity tj of

each layer,

− and the downwelling (sky) radiation, expressed by the brightness temperature Tsky.

The snow-ground interface is allowed to be rough; we only specify its reflectivity s0, based on

observation or theory.

2.1 Radiation in multi-layered media

Considering layer j in Figure 2, we can derive the following equations relating the up- and

downwelling brightness temperatures at the layer boundaries:

A r B t C e Tj j j j j j j= + + (1)

( )B s A s Dj j j j j= + −− − −1 1 11 (2)

( )C s A s Dj j j j j= − ++1 1 (3)

D t B r C e Tj j j j j j j= + + (4)

Note that for j = 1 on the left-hand side of (2) we have a term with D0 on the right-hand side

which is identified as the ground temperature

D T0 0= (5)

and s0 is the ground-snow reflectivity. Similarly for j = n we have in (3) a term with An+1,

namely the sky brightness

A Tn sky+ =1 (6)

Solving the system of equations (1) to (4) for j = 1 to n, and using (5) and (6) leads to Dn and

thus to the brightness temperature Bn+1 = Tb of the snowpack according to

( )T s D s Tb n n n sky= − +1 (7)

Sometimes the emissivity esp or the transmissivity tsp of the whole snowpack is of more interest

than the brightness temperature Tb. The computation of these parameters is not obtained by

simply dividing Tb by a physical temperature because the snowpack is not regarded as isother-

mal. Nevertheless by computing Tb under various illumination conditions the required quantities

can be found. As an example we can get the total emissivity (snowpack and background) from

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one minus the reflected fraction of the downwelling sky radiation by choosing two choices of

Tsky (0K and 100K),

( ) ( )e

T T T Tsp

b sky b sky= −

= − =1

100 0

100

K K

K(8)

Likewise tsp can be obtained from

( ) ( )( )t

T T T T

ssp

b b== − =0 0273 173

100

K K

K 1- 0(9)

If only one layer is present, Tb follows from (7) where Dn=D1 is given by

( ) ( ) ( ) ( )

01

2110

11

111100101

111100101

1

11

111

11

sr

tsssr

TeTsrTstsr

TeTstTsrst

Dsky

sky

−−−

+−+−+−

+−+−

= (10)

2.2 Radiative transfer in a refractive medium - the six flux model

In order to compute realistic values of rj, ej, and tj, volume scattering and absorption within each

layer j are to be described by a radiative transfer model. An important distinction between

radiative transfer in the atmosphere and in snow is that refraction plays a role in snowpacks.

Even more important is the fact that total internal reflection occurs for certain rays. Our model

takes these processes into account. We use a simplified radiative transfer model, reducing the

radiation at a given polarization to six fluxes streaming along and opposed to the three principal

axes of the slab (Figure 3). The horizontal fluxes represent trapped radiation, i.e. radiation

whose internal incidence angle θ is larger than the critical angle θc for total reflection:

( )'/1arcsin nc =θ>θ (11)

where n' is the real part of the refractive index of the slab. Only the vertical fluxes (θ < θc) can

escape from the snowpack. Scattering by angles near 90° causes a coupling between the vertical

and the horizontal fluxes. The trapped horizontal fluxes are represented by T03 to T06. Radiation

at θ < θc is represented by T01 for downwelling radiation and by T02 for upwelling radiation. We

assume that the radiation of the vertical fluxes can be represented by an effective propagation

direction represented by the angle θ with respect to the z axis. Then the transfer equations for

the six fluxes, expressed by brightness temperatures, are of the type

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( ) ( )− = − − − −dT

dzT T T Ta b

0101 01 02cosθ γ γ ( )− − − − −γc T T T T T4 01 03 04 05 06 (12)

( ) ( )01020202 cos TTTT

dz

dTba −γ−−γ−=θ+ ( )− − − − −γc T T T T T4 02 03 04 05 06 (13)

and similar equations for T03 to T06, where γa is the absorption coefficient, γb the scattering

coefficient in the backward direction and γc the coefficient for coupling between the vertical and

horizontal fluxes. Since the medium is plane parallel and isotropic in the (x, y) plane, the fluxes

T03 to T06 are the same with vanishing derivatives in x and y directions, leading to

T T T T03 04 05 06 = = = ( )( )( )= γ γ γ γa c a cT T T+ + + −01 02

12 (14)

Inserting (14) in (12) and (13) leads to two-flux equations with modified coefficients:

( ) ( )− = − ′ − − ′ −dT

dzT T T Ta b

0101 01 02cosθ γ γ (15)

+ ( ) ( )dT

dzT T T Ta b

0202 02 01cosθ γ γ= − ′ − − ′ − (16)

The two-flux absorption coefficient aγ′ and scattering coefficient sγ′ can be written in terms of

the six-flux parameters:

( )( )′ = + + −γ γ γ γ γa a c a c1 4 21

(17)

( )′ = + + −γ γ γ γ γb b c a c4 22 1(18)

The typical difference between γ'a and γa is about a factor 2, and γa agrees well with the theo-retical absorption coefficient (Wiesmann et al. 1998). For constant T and constant coefficientswithin the given layer, T01 and T02 can be written as

( ) ( )T T A d B d01 = + ′ + − ′exp expγ γ (19)

( ) ( )dBrdArTT ′γ−+′γ+= − expexp 10002 (20)

where

d' = d/|cos θ|. (21)

The reflectivity r0 at infinite layer thickness is given by

( )r b a b01= ′ ′ + ′ + −γ γ γ γ (22)

and the damping coefficient γ (eigenvalue) is)2( baa γ′+γ′γ′=γ (23)

For a snow layer of thickness d the internal r and t, considering all multiple reflections, are

( )( )r = r - t - r t0 1 102

02

02 1−

(24)

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( )( )t t r r t= − −−

0 02

02

02 1

1 1 (25)

where

( )t d0 = − ′exp γ (26)

is the one-way transmissivity through the slab. The layer emissivity is obtained from e=1-r-t.

Note that e, r, t are to be indexed by the layer number j.

The way γb and γc are related is determined by the internal scattering phase function and by the

critical angle θc given by (11). The solid angle of the beams for T01 and T02 is

√√

↵

−−π=Ω'

1'12

2

1n

n(27)

and the solid angle Ω2 of trapped radiation is the complement 2π - Ω1; that is,

'

1'2

2

2n

n −π=Ω (28)

The scattering coefficient γs is the sum over all fluxes

cbs γ+γ=γ 42 (29)

and using (27) and (28) we get for isotropy of scattering

1

22

ΩΩ=

γγ

b

c(30)

With Equations (27) - (30) we can determine the six-flux coefficients γa, γb, γc and γs from n' and

the two-flux coefficients aγ′ and bγ′ , using (17) and (18).

2.3 Reflectivity at layer interfaces

The interface reflectivity sj between two regular layers is given by sj = Fj 2 where Fj is the

Fresnel reflection coefficient of the interface between layers j and j+1. By regular we mean that

a simple transition occurs in the permittivities εj to εj+1, and that coherent superpositions due to

reflections at other interfaces can be ignored because if the layers are deep enough, the natural

variability in thickness, local incidence angle and also the different frequencies allowed within

the radiometer bandwidth tend to cancel the interference effects. This is the usual situation.

However, for layers thinner than about half a wavelength the variability is not large enough to

cancel the coherent effects, and thus they cannot be ignored; in other words, the quarter-wave-

length resonance must be taken into account. The one-way phase through layer j is given by

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λθπ

= −1cos'2 jjjj

ndP . (31)

which is a function of the thickness dj, of layer j, the real part jn' of the refractive index, the

propagation angle θj-1 and vacuum wavelength λ. Defining

jiPj eE 2… , (32)

the reflection coefficient R2 of a coherent double layer (Figure 4) can be written as

RF F E

F F E22 1 2

2 1 21=

++ (33)

and the reflectivity s is given by

s RF F E

F F E= =

++2

2 2 1 2

1 2 2

2

1 . (34)

For a loss-less medium

( )( )s

F F F F P

F F F F P=

+ ++ +2

21

21 2 2

12

22

1 2 2

2 2

1 2 2

cos

cos(35)

The coherent effect is represented by the terms containing the phase P2. If they were neglected,

the equation would correspond to the incoherent treatment of the reflections. A continuous

transition from the coherent to the incoherent behavior is possible at every zero of cos(2P2).

Here, we choose the second zero at

2P2 = 3π/2 ≅ 4.712. (36)

In MEMLS the layer is considered to be coherent for phases below the limit (36), i.e. for P <

2.36 and incoherent if P is larger. If it is coherent, the effect a layer j is taken into account by

using the coherent reflectivity s of this thin layer as the interface reflectivity sj between the two

adjacent (incoherent) 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 s. This is a valid approximation because volume scattering

and absorption are negligible in layers thinner than about half a wavelength. This procedure has

to be done at each frequency and incidence angle. Care must be taken when two ore more

adjacent layers are thin.

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2.4 Effective propagation angle and polarization mixing

So far it has been assumed that at any position within the snowpack the effective propagation

angle θj (in layer j+1, see Figure 1) with respect to the surface normal is given by Snell's law for

both vertically and horizontally polarized (up- and downwelling) radiation:

1'

sinsin

+

θ=θj

njs

n(37)

Here the index s in θjs refers to Snell. In addition to refraction, deviations in propagation

direction occur due to volume scattering. The proposed change leads to a first-order correction

in effective propagation path and polarization mixing for radiation originating at the radiometer

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).

Effective propagation angle

For a radiometer observing at the incidence angle θn the effective propagation angle is well

approximated by (37) as long as volume scattering is not too important. This is true just below

the smooth snow surface. 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 may differ from (37). Assuming that, in the limit of strong scattering, the forward

(downward) and backward (upward) fluxes are diffuse then the effective propagation angle θjd

can be represented by

coscos

θθ

jdjc=

+1

2 (38)

where θjc is the critical angle for total reflection, i.e. the maximum propagation angle of radia-

tion belonging to the vertical fluxes; it is given by

21 )'(1cos −

+−=θ jjc n (39)

For a dilute medium ( 1' +jn = 1) we get cosθjd = 0.5; this is the usual value of the two-stream

model Ishimaru [1978]. In our case of six streams or "fluxes", the streams have more narrow

beams, therefore we have to use the modified form given by Equation (38).

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The actual value of the effective propagation angle θj in layer j+1 is somewhere between θjs and

θjd. 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 cos ( ) cosθ θ θj z js z jdt t= + −1 (40)

and tz is taken to be

tz = θγ

− nz

z js

s dzz

)'cos2

)'(exp( (41)

where γs(z') is the scattering coefficient at height z'. For negligible scattering tz ⇒ 1, leading to θj

= θjs, whereas for strong scattering we get tz = 0 and thus θj = θjd.

Polarization mixing

After diffuse scattering the photons loose their memory about where they came from. As a con-

sequence 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 θjs and

θjd. This mixing can be modeled by modifying the polarization-dependent parts of the

snowpack model, i.e. the reflectivities sjv and sjh at the interface between Layer j and j+1. The

computation of these reflectivities without polarization mixing is described in Section 2.3. Here

we describe the modification due to the mixing effect.

Let us denote the difference ∆sj by

∆sj = sjh - sjv (42)

This difference is reduced due to scattering, leading to effective interface reflectivities sjveff and

sjheff between Layer j and j+1 at position z = zj:

sjheff = [ ]0 5. s s t sjh jv z j+ + ∆ (43)

sjveff = [ ]0 5. s s t sjh jv z j+ − ∆ (44)

In (43)and (44) the values of sjv and sjh are the ones computed for the angles θjs. In a further

improvement, these coefficients could be averaged over the range of incidence angles. However,

except near the Brewster angle, the effects are small because they tend to cancel.

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2.5 Primary model parameters (input data)

At a given frequency f, polarization p and incidence angle θ the following snow physical

parameters are needed for each layer j (j = 1,… n):

− density ρj

− temperature Tj

− liquid water content Wj

− correlation length pcj

− vertical extent zj

Furthermore the ground temperature T0 and the snow-ground reflectivity s0 are needed. These

primary parameters define the derived secondary model parameters to be described below.

2.6 Secondary model parameters

Dielectric properties of dry snow

For dry snow the real part of the microwave permittivity can be well expressed according to

Mätzler [1996] in terms of the density ρ in g/cm3:

′ = + +ε ρ ρd 1 15995 1861 3. . ; 0 ≤ ρ ≤ 0.4 g/cm3 (45)

( )( )′ = − +ε ν ε νεd h s13

; ρ > 0.4 g/cm3 (46)

where εh = 1.0, εs = 3.215 and ν = ρ/0.917 g/cm3.

The imaginary part of the dielectric constant of dry snow is given by:

′′ = ′ ′′ε ε νεd d iK2(47)

where K2 is the squared field factor (Figure 1 in Paper 2); for dry snow K2≅ 0.44 for ν≤0.33,

then increasing to K2=0.54 at ν=0.5. A similar expression, optionally used by MEMLS, is the

one proposed by Tiuri et al. [1984].

( )′′ = ′′ ? + ?ε ε ρ ρd i 0 52 0 62 2. . (48)

For the imaginary part εi" of the dielectric permittivity of ice, the formula of Mätzler (1998b)

(see also Equations (61) to (64) of Wiesmann et al. 1998) is used:

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′′ = + ?εα

βi ff (49)

θ = −300

1K

T(50)

( ) ( )α θ θ= + ? ? − ?0 00504 0 0062 221. . exp . (51)

( )( )( )

β = ?−

+B

T

b T

b TB f1

2 22

1

exp

exp( )( )+ − + ? −exp . .10 02 0 0364 273T K (52)

where f is the frequency in GHz, B1=0.0207 K⋅GHz-1, b=335 K and B2=1.16⋅10-11 GHz-3.

Dielectric properties of wet snow

Wet snow exhibits a distinct Debye relaxation spectrum in the microwave range. According to

physical mixing models (e.g. Mätzler [1987]), for small values of the volumetric liquid water

content W, the effective permittivity ε = ε' + iε" is linear in W and given by 4 terms

ε = εa + εb + εc + εd (53)

where εd is the permittivity of dry snow, and εa, εb, εc are Debye terms expressed as

ε εε ε

k ksk k

kif f= +

−−×

×1 0/ ; k = a,b,c (54)

The Debye parameters εsk, ε∞k, f0k are given by the following three relations

√√↵

−

εε?+

ε−ε=ε11

3

d

swk

dswsk

A

W ; k = a,b,c (55)

√√↵

−

εε?+

ε−ε=ε×

××

113

d

wk

dwk

A

W ; k = a,b,c (56)

( )

ε−ε?+ε

ε−ε+=×

×

dwkd

wswkwk

A

Aff

)(100 ; k = a,b,c (57)

where the static εsw and infinite permittivity ε∞w and the relaxation frequency f0w of liquid water

at T = 0°C are: εsw = 88, ε∞w = 4.9, f0w = 9 GHz (Ulaby et al. 1986). The depolarization factors

are taken from experimental data: Aa = 0.005, Ab = Ac = 0.4975 (Mätzler,1987).

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Absorption coefficient

The absorption coefficient γa follows from the imaginary part n" of the refractive index, and

thus from the complex permittivity, ε = ε'+iε" = (n'+in")2, according to

λπ=γ "4 n

a (58)

where λ is the vacuum wavelength.

Scattering coefficient

According to Wiesmann et. al. [1998], the six-flux scattering coefficient can be obtained from

snow physical properties. Several possible fits were found among which the following two

were of equal quality with regard to the available snow-sample data:

5.25.2

3 GHz5054.0

cm/g123.1

mm12.9 √√

↵

√√↵

+ρ−=γ fpec

s (59)

5.25.2

GHz50mm1295

mm116.3 √√

↵

?√

√

↵

+=γ fpp ecec

s (60)

where pec is the exponential correlation length, i.e. a length obtained by fitting the measured

correlation function to exp(-x/pec). The range of validity includes snow densities from 0.1 to 0.4

g/cm3, pec from 0.05 to 0.3 mm and frequencies from 5 to 100 GHz. An extension to larger

correlation lengths is given in Paper 2 (Mätzler and Wiesmann, 1998). For pec<0.05mm Equa-

tion (60) should be used. In the following examples we will use γs according to (59).

3. Examples

3.1 Emissivity of a snowpack

MEMLS was written in Matlab 5 [MathWorks, 1996]. All subroutines are commented, and a

help text is available with help subroutine. Many simulations have been made so far, and the

data seem in general to be realistic as long as the limited parameter ranges are respected. For

illustration purposes we present an example of an emissivity simulation. Table 1 shows the

input data, i.e. the snow profile of Dec. 21, 1995 found at Weissfluhjoch-Davos, Switzerland. It

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is a winter snow situation with a snow height of 60.3 cm. The top layer (j=4) was 20cm of new

snow, below was a thin crust, and the bottom layer consisted of coarse-grained snow.

The measured and simulated emissivities as functions of frequency and incidence angle are

shown in Figure 5. The lines indicate computed data while the symbols indicate measurements.

Note the significant polarization difference at frequencies below 35 GHz where Layer 3 acts as

a coherent reflector in agreement with the measurements. The increase of the measured emis-

sivity at small incidence angles is due to the shadowing of the snowcover by the radiometers.

Whereas the emissivity at vertical polarization is slightly underestimated, the agreement is

excellent at horizontal polarization. The broad minimum in the h-polarized spectrum around 25

GHz is the result of the 3 mm crust of Layer 3 acting there as a quarter-wave resonance. The

transition from coherent to incoherent interaction takes place at 35 GHz where a kink is recog-

nized. Table 2 shows the intermediate model data of the simulation at 25 GHz. The indices h

and v stand for horizontal and vertical polarization, respectively. Layer 3 disappeared here

because it is considered a coherent layer. Its effect is expressed by the large value of sh for j=2.

3.2 Simulation of spatial snowpack emissivity variations

Sturm and Holmgren [1994] describe a vertical cross section through a snowpack covering a

distance of several meters at Imnavit Creek, Alaska, for November 18, 1989. Five layers were

observed; they consisted of depth hoar and wind crusts in alternation. The situation was

characterized by type, thickness of each layer and soil temperature. The corresponding layer

temperatures were interpolated from the soil and air temperatures. The correlation lengths and

densities were specified using typical values from our snow sample catalog (Wiesmann, 1998).

The spatial variation of the emissivity at 37 GHz for an incidence angle of 50° and the snow

height are shown in Figure 6. The expected anticorrelation between these two parameters is

apparent. Due to the strong scattering in the depth hoar layers, the emissivity is below 0.7. At a

displacement of 2m, the depth hoar is very shallow, leading to an increase of the emissivity up

to 0.9 and thus to a pronounced spatial variation in phase with the microtopography. The mean

emissivity agrees quite well with SSM/I data (assuming Tsky=30K), as shown in the figure.

4. Discussion and outlook to further development

The purpose of the present paper was mainly to describe MEMLS and all its elements. First

applications were used to show that reasonable results can be obtained. In order to further test

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the model and in order to assess its parameter range and errors, MEMLS should be applied to

many kinds of snowpacks and firn with well-known physical and structural properties. Since

the number of parameters can be large, it is difficult at present to give a perspective on the fea-

sibility to retrieve special snow parameters from microwave measurements. However, this

potential can now be studied by simulations. First sensitivity studies of MEMLS were done by

Pulliainen et al. (1998). Some possible steps for future considerations are proposed in the fol-

lowing:

• Weighting functions, defined by wij = ∂ep/∂xij, where xij is the ith physical parameter of layer

j, can be computed in order to assess the sensitivity of the model to any parameter.

• The assumption of an isotropic and exponential correlation function is not valid in general,

but it is a reasonable simplification. Furthermore it leads to a representative definition of the

structure parameter, i.e. the correlation length pec, which is obtained from fitting the

observed correlation function to exp(-x/pec) as described by Weise (1996). The index e in pec

is meant to indicate this exponential fit. Wiesmann et al. (1998) used the slope of the cor-

relation function at zero displacement x=0 to define a slightly different correlation length pc.

This lead to noisier and slightly different values, especially in case of fine-grained snow.

• By including the occasional snow-structure anisotropy (e.g. observed in new snow depos-

ited under calm wind condition, surface hoar, depth hoar) the model can be improved,

especially its polarization behavior at frequencies above 30 GHz.

• Occasionally, we observed clear deviations of the spatial correlation function from simple

exponentials. Improvements can be expected by choosing more suitable functions. Using the

improved Born Approximation and measured correlation functions, it can be tested how

these functions are related to snow types and how they influence the microwave properties.

• Further improvements of the polarization behavior of the model is possible by using a fully

polarimetric radiative transfer code. However, the expected gain from this more elaborate

approach is limited, because volume scattering in snow tends to be insensitive to polariza-

tion (Wiesmann et al. 1998). The most obvious polarization effects arise from reflections at

layer interfaces which are treated by the present model.

16

• A limitation of MEMLS at very high frequencies results from the assumption of smooth

interfaces between layers and of a smooth snow surface. Improvements can be expected by

including roughness effects.

• A limitation of the present MEMLS version is given by the limited range of correlation

lengths. The range covers the snow structure observed at Weissfluhjoch during two winter

seasons. Larger grains are observed for old wet snow; when it refreezes the scattering leads

to extreme microwave signatures which cannot be accounted for by the present model.

Paper 2 will be dedicated to this situation.

The complexity of MEMLS can be large. In fact, arguments were raised for preferring single-

layer snow emission models as they may be more practical and thus more feasible. This state-

ment can also be tested by simulations using MEMLS for different values of n where the single-

layer situation (n =1) is met as well. Nevertheless it should be pointed out that layering is

inherent to most snowpacks as described in the introduction and found in the examples.

5. References

Armstrong R.L., “Continuous monitoring of metamorphic changes of the internal snow struc-

ture as a tool in avalanche studies”, J. Glaciol., Vol. 19, pp. 325-334 (1977).

Armstrong R.L., “Metamorphism in subfreezing, seasonal snow cover: the role of thermal and

vapor pressure conditions”, PhD Thesis, University of Colorado, Fac. Graduate School,

Dept. Geography (1985).

Arons E.M and S.C. Colbeck, “Geometry of heat and mass transfer in dry snow: a review of

theory and experiment”, Reviews of Geophys. Vol. 33, pp. 463-493 (1995).

Djermakoye B., and J.A. Kong, “Radiative-transfer theory for the remote sensing of layered

random media”, J. Appl. Phys. Vol. 50, pp. 6600-6604 (1979).

Fierz C. “Field observations and modelling of weak-layer evolution”, Annals Glaciol. 26, pp. 7-

13 (1998).

Föhn P.M., C. Camponovo and G. Krüsi, “Mechanical and structural properties of weak snow

layers measured in situ”, Annals Glaciol. Vol. 26, pp. 1-6 (1998).

17

Gurvich A.S., V.I. Kalinin, and D.T. Matveyev, “Influence of the internal structure of glaciers

and their thermal radio emission”, ATM. Oceanic Phys. Vol. 12, pp. 713-717 (1973).

Ishimaru A., "Wave propagation and scattering in random media", Vols. 1 and 2, Academic

Press, Orlando (1978).

Looyenga H., "Dielectric constant of heterogeneous mixtures", Physica, Vol. 21, pp. 401 - 406

(1965).

MathWorks The, Inc., "Matlab, the language of technical computing", 24 Prime Park Way,

Natick, MA 01760-1500 (1996).

Mätzler C., R.O. Ramseier, and E. Svendsen, "Polarization effects in sea-ice signatures", IEEE

J. Oceanic Engin., Vol. OE-9, pp. 333-338 (1984).

Mätzler C., "Applications of the Interaction of Microwaves with the Seasonal Snow Cover",

Remote Sensing Reviews, Vol. 2, pp. 259 - 387 (1987).

Mätzler, C. "Passive microwave signatures of landscapes in winter", Meteorology and Atmos-

pheric Physics, Vol. 54, pp. 241-260 (1994).

Mätzler C.: "Microwave permittivity of dry snow", IEEE Transactions Geosci. and Rem.

Sens., Vol. 34, No. 2, pp. 573 - 581 (1996).

Mätzler C., "Improved Born Approximation for scattering of radiation in a granular media", J.

Appl. Phys., Vol. 83, pp. 6111-6117 (1998a).

Mätzler C., "Microwave properties of ice and snow", in B. Schmitt et al. (eds.), Solar System

Ices, pp. 241 - 257, Kluwer Academic Publ. Dordrecht (1998b).

Mätzler C. and A. Wiesmann, "Extension of the Microwave Emission Model of Layered

Snowpacks to Coarse-Grained Snow", to be published in Remote Sensing of Environment,

Paper (1999).

Mishima O., D. Klug, and E. Whalley, "The far-infrared spectrum of ice in the range 8-25cm-1:

Sound waves and difference bands, with application to Saturn's rings", J. Chem. Phys., 78,

6399 - 6404 (1983).

18

Polder D., and J.H. van Santen, "The effective permeability of mixtures of solids", Physica, Vol.

12, pp. 257 - 271 (1946).

Pulliainen J., 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).

Reber B., C. Mätzler, E. Schanda, "Microwave Signatures of Snow Crusts: Modelling and

Measurements", Internat. J. of Remote Sensing, Vol. 8, pp. 1649 - 1665 (1987).

Rott H., K. Sturm and H. Miller, “Active and passive microwave signatures of Antarctic firn by

means of field measurements and satellite data”, Annals Glaciol. Vol. 17, pp. 337-343

(1993).

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region”, J. Glaciol. (in press 1999).

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of strong fluctuation theory”, IEEE Trans. Geosci. Remote Sens., GE-24, pp. 220-231

(1986).

Sturm M., and J. Holmgren, "Effects of microtopography on texture, temperature and heat flow

in Arctic and sub-Arctic snow", Annals of Glaciol., Vol. 19, pp. 63 - 68, 1994.

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calotte polaire Antarctique”, Thèse de doctorat de l’institut national polytechnique de

Grenoble (1992).

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Antarctic ice sheet with ground traverse data”, Annals Glaciol., Vol. 17, pp. 161-166 (1993).

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snow at microwave frequencies", IEEE J. Oceanic Eng., OE-9, pp. 377 - 382 (1984).

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2025 (1986).

19

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samples", Radio Sci. Vol. 33, No. 2 , pp. 273-289 (1998).

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of Layered Snowpacks", Research Report No. 98-2, Microwave Dept., Institute of Applied

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20

Tables

Table 1: Snow profile of data from Dec. 21, 1995 where Tsnow is the snow temperature, W

the liquid water content, ρ the snow density, d the layer thickness, and pec the expo-

nential correlation length. The layer number increases with height above ground (from

Wiesmann et al. 1996).

Date/layer No.

Tsnow

[K]W

[%]ρ

[kg/m3]d

[cm]pec

[mm]21.12.95

1 273.0 0.00 259.0 25.0 0.16162 272.0 0.00 177.0 15.0 0.07203 271.8 0.00 400.0 0.3 0.06004 271.4 0.00 70.0 20.0 0.0550

Table 2: Intermediate model data of the simulation at 25 GHz and 50° incidence angle.The upwelling brightness temperatures in each layer are given by Dh and Dv.

j θ[°]

d'[cm]

sh sv r t Dh

[K]Dv

[K]0 39.5 0.096

50.0535

1 42.3 32.4 0.0024

0.0001

0.0696

0.8358

252.9 255.9

2 46.6 20.3 0.0972

0.0041

0.0110

0.9582

251.8 254.8

4 50.0 29.1 0.0036

0.0000

0.0180

0.9615

235.9 251.7

Air 235.4 251.7

21

Figure CaptionsFigure 1: A multi-layer system with a wave incident from above at an angleθn.

Figure 2: Vertical fluxes at the boundaries of layer j.

Figure 3: The six fluxes within a selected layer j.

Figure 4: Thin layer j between two thicker, scattering layers j-1 and j+1, here for j=2.

Figure 5: Measured (circles, triangles) and simulated (curves) data of the snowpack of

Dec. 21, 1995. Left: Emissivity versus frequency at θn = 50°. Right: Emissivity versus

incidence angle at f = 21 GHz.

Figure 6: Emissivity (upper lines) and snowheight (lower line) versus distance. Observa-

tion frequency is 37 GHz and incidence angle θn=50°. Input data are taken from Sturm

and Holmgren [1994]. The horizontal dashed line indicates the emissivity of the nearest

SSM/I pixel at 37 GHz ( v-pol.) assuming Tsky=30K.

22

Figures

s0

s1

s2

s3

s4

sn-1

sn

e1 r1

θn

θn-1

T0

t1 T d 1 1

e2 r2 t2 T d 2 2

e3 r3 t3 T d 3 3

e4 r4 t4 T d 4 4

en-1 rn-1 tn-1 T d n-1 n-1

en rn tn T d n n

b T sky T

θ0

θ1

θ2

Figure 1: A multi-layer system with a wave incident from above at an angleθn.

sj

sj-1

e , r , t , T

A

C

A

C

j+1

j

j

j-1

B

D

B

Dj-1

j

j

j+1

z

zj-1

zj

j j j j

Figure 2: Vertical fluxes at the boundaries of a selected layer j.

23

Figure 3: The six fluxes within a selected layer j.

n2

n3

n1

F1

F2

θ0

θ1

θ2

d2j

j-1

j+1

Figure 4: Thin layer j between two thicker, scattering layers j-1 and j+1, here for j=2.

10 100F [GH ]

0.5

0.6

0.7

0.8

0.9

1.0

emis

sivit

y e

v - pol

h - pol

v - pol

h - pol

10 20 30 40 50 60 70 8incidence angle [degrees]

0.5

0.6

0.7

0.8

0.9

1.0

emis

sivi

t y e

v - pol

h - pol

v - pol

h - pol

24

Figure 5: Measured (circles, triangles) and simulated (curves) data of the snowpack ofDec. 21, 1995. Left: Emissivity versus frequency at θn= 50°. Right: Emissivity versusincidence angle at f=21 GHz.

0.00 1.00 2.00 3.00 4.00 5.00distance [m]

0.40

0.50

0.60

0.70

0.80

0.90

1.00

emis

sivi

t y

0 .00

0.20

0.40

0.60

0.80

sno

whei

ght [cm

]

h-pol

v-pol

snow height

Figure 6: Emissivity (upper lines) and snowheight (lower line) versus distance. Observa-tion frequency is 37 GHz and incidence angle θn=50°. Input data are taken from Sturmand Holmgren [1994]. The horizontal dashed line indicates the emissivity of the nearestSSM/I pixel at 37 GHz ( v-pol.) assuming Tsky=30K.