A Comparison of A Second-order Snow Model With
Field Observations
Jinyang Du
1
, Jiancheng Shi
1,2
, Shengli Wu
1
1.The Institute of Remote Sensing Applications, Chinese Academy of Sciences
P.O.Box9718,Beijing 100101,China
sunnydjy@hotmail.com
2. Institute for Computational Earth System Science, University of California, Santa Barbara, U.S.A
Abstract—A microwave scattering model based on second-order
solution of radiative transfer equation has been developed for dry
snow. Advanced integral equation model (AIEM) and a semi-
empirical model were included in the model to account for
ground contribution. Also, ellipsoid grain shape was adopted to
describe ice particle. This model was compared with the ground-
based scatterometer data (frequencies are 1.25GHz and 15.5 GHz)
from NASA Cold-land Processes Field Experiment (CLPX).
Inputs to the model were from Local-Scale Observation Site
(LSOS) snow pit measurements, except that particle size and
shape were computed as free parameters. The comparison shows
that the model agrees well with the field data. Also from the
comparison, it could be seen that particle shape had a significant
effect on the cross-polarization signals.
Keywords- snow; second-order model; CLPX
I. INTRODUCTION
Active microwave remote sensing can provide useful
information on snow parameters, such as snow cover extent
and snow water equivalent (SWE) or the product of snow
density and depth, for hydrological, climatological, and
meteorological applications. In order to understand dry snow
scattering behavior, theoretical models, many of which are
based on radiative transfer theory, have been developed. For
an optically thin layer of snow, single scattering model, which
is based on first-order solution of radiative transfer equation,
is applicable. However, as the optical depth of snow layer
increases, multiple scattering models should be considered.
Another factor affecting theoretical simulation results is the
description of particle shape. Ice particle is generally modeled
as spherical particles. However, the actual shape of ice particle
is generally non-spherical and non-spherical particles can lead
to strong depolarization return. In this paper, a second-order
model, with ellipsoid scatterer considered, is described and
compared with measured data.
II. M
ODEL DESCRIPTION
The proposed scattering model for dry snow was based on
second-order solution of radiative transfer equation. And
ellipsoid scatterer, which has the same volume as the sphere
defined by input radius, was assumed when calculating phase
matrix of snow. The advanced Integral Equation Model
(AIEM) [1] is used in the calculation of the backscattering of
subsurface and air-snow interface as well as the interactions
between ground and snow. It is noted that the surface
backscattering signals for cross-polarization were estimated by
an empirical model [2]. The received signal calculated by the
model contains the following scattering mechanisms: (a) direct
backscattering from air-snow interface, (b) direct
backscattering from ground, (c) second-order volume
scattering, (d) downward scattering by the particles followed
by the coherent scattering by the ground, (e) coherent
scattering by the ground followed by the scattering of the
particles, (f) coherent scattering by the ground, followed by
the scattering of the particles and another coherent scattering
of the ground, (g)scattering from the particles followed by
non-coherent scattering by the ground, (h) non-coherent
scattering by the ground followed by scattering from the
particles. The last five mechanisms could be regarded as
interaction term T
gv
and the first three mechanisms belong to
two surface terms T
a
, T
g
and one volume term T
v
respectively.
The mathematically expressions of T
a
, T
g
and T
gv
could be
expressed as the following equations.
),;,(
000001 iiiiR
a
ST
φµφπµ
−+=
(1)
[]
rei
irirRr
g
dT
STT
µκµ
φµφπµµ
/2exp)(
),;,()(
001
1210
−⋅⋅
−+⋅=
(2)
[
+−⋅
−
⋅⋅=
),;,(
2
exp
)()(
1
0110
irir
r
e
ir
i
gv
d
T
φµφπµ
µ
κ
µµ
µ
P
TT
(3)
),;,()[(
)(
2
)/2exp(1
12
2
12
irirr
r
e
re
d
φµφπµµ
µ
κ
µκ
−+−
+
−−
⋅
PR
R
]
)],;,()1)(exp(
22 irir
d
φµφπµκκ
+⋅−+ P
∫∫
′
′
−
+
π
φµ
µ
κ
2
0
1
0
1
'
)exp(
dd
d
2
2
)exp(1
κ
κ
d−−
),;','()',';,(
12 irRii
φµφµφµφπµ
SP ⋅+⋅
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}
),;','(
)',';,(
1)exp(
12
2
2
ir
irR
d
φµφµ
φµφπµ
κ
κ
−−⋅
+
−
+
P
S
where S is the phase matrix of surface calculated by AIEM, T
is the transmitivity matrix, R is the reflectivity matrix, P is the
phase matrix of snow, the subscripts 0,1,2 in S ,T and R denote
air, snow and ground medium respectively, d is the depth of
snow layer and )'/1/1(
1
µµκκ
+=
ie
, )'/1/1(
2
µµκκ
−=
ie
.
Volume term T
v
could be derived from two iterative processes
when solving the radiative transfer equation [3], [4]. It is noted
that Rayleigh Approximation was applied to ellipsoid particles
when calculating the phase matrix of snow [4].
To calculate scattering effects from surface, surface
scattering model AIEM is applied. AIEM is the a recent
development of Integral Equation Method (IEM) [5], which
was verified by laboratory measurements of bistatic scattering
from surfaces with small, intermediate and large scale
roughness. AIEM kept the absolute phase term in Greens
function, meaning better accuracy than the old IEM. According
to AIEM, the bistatic scattering coefficient can be expressed as
the sum of Kirchhoff term, complementary term and cross term,
as (4) shows.
!
),(
)](exp[
2
)(
2
1
2
222
2
n
kkkkW
Is
kks
k
ysyxsx
n
n
pq
n
n
szz
c
qp
kc
qp
k
qp
s
qp
−−
+−=++=
∑
∞
=
σσσσ
(4)
where
k
qp
σ
denotes Kirchhoff term,
c
qp
σ
for complementary
tem,
kc
qp
σ
for cross term, pq denotes polarization state, s is the
standard deviation of the surface height, k is wave number,
φθ
cossinkk
x
= ,
φθ
sinsinkk
y
= ,
sssx
kk
φθ
cossin= ,
sssy
kk
φθ
sinsin= ,
θ
coskk
z
= ,
ssz
kk
θ
cos= ,
ϕ
θ
,
are for
incident direction
，
ss
φθ
, are for the scattered direction,
n
W
is the Fourier transform of the nth power of a known surface
correlation function. And detailed expression of
n
pq
I could be
referred to [1]. Bistatic scattering coefficient
0
σ
calculated
by AIEM could be linked with surface phase matrix as (5)
shows.
Ω=
∫
dII
i
s
ss
ss
s
),(]
cos
);,(
[
4
1
),(
0
φθ
θ
φφθθσ
π
φθ
(5)
where
I
s
and I
i
are scattering and incident intensity respectively.
III.
FIELD MEASUREMENTS
In order to verify the proposed model, data from NASA
Cold-land Processes Field Experiment (CLPX) were used for
comparison. The data were measured by University of
Michigan truck-mounted L band (1.25 GHz) and Ku band
(15.5 GHz) scatterometers during the third Intensive
Observation Period (IOP3, dry snow), February 19 -25, 2003
at the Local Scale Observation Site (LSOS) test site. The test
site consisted of a small (0.8-ha) clearing surrounded by trees
and located within the CLPX Fraser Intensive Study Area
(ISA), near the Fraser Experimental Forest Headquarters
Facility, Colorado. The incidence angles are 20, 35 and 50
degrees. Snow input parameters were from snow pits
measurements, which were taken at the same time as the
ground-based scatterometer measurements, at LSOS except
that particle size and shape were computed as free parameters.
Actually, measurements on particle size along the snow-pack
vertical profile were available, but the proposed one-layer
model needs an effective radius, which should be the radius
having particle size distribution and near-field effects
considered. Similarly, the shape used in simulation could be
regarded as an effective shape. The contribution from the air-
snow surface is assumed to be negligible since the
discontinuity between air and dry snow is normally small.
Also, since there is a lack of ground information and snow
contribution is quite small at L-Band in this case, comparisons
between simulated data with variant ground situations and
measured data were made in order to estimate ground
parameters. The ground and snow inputs could thus be
concluded as the following:
• Ground rms-height 0.5 cm
• Ground surface correlation length 10cm
• Soil moisture 25%
• Snow depth 0.99 meter
• Snow volume fraction 22.4%.
IV.
SIMULATION RESULTS AND DISCUSSION
As stated above, comparison with the measurement at L-
band is used to find proper ground parameters. Fig. 1 shows the
result of comparison between the model and the measurement
on the morning of Feb 23, 2003 at L-Band. Grain radius is
0.46mm and a prolate ellipsoid with 0.7 as the ratio of short
axis to long axis is considered. Fig. 2 shows the comparison
result at Ku band. Particle size and shape were best-fit
parameters in this case. A 0.42 mm radius and 0.1 short axis to
long axis ratio were selected in the simulation.
Figure 1. Comparison between proposed model and measured data at L-
Band (scatter points are for measured data)
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Figure 2. Comparison between proposed model and measured data at Ku-
Band (scatter points are for measured data. short axis to long axis ratio is 0.1)
Figure 3. Comparison between proposed model and measured data at Ku-
Band (scatter points are for measured data. short axis to long axis ratio is 0.7)
Figure 4. Backscattering coefficients trend as particle shape changes (x-axis
is the short axis to long axis ratio of ellipsoid)
It can be seen from Fig.1 that simulation agrees well with
the measured data at L-band with the selected ground
parameters. The selected ground parameters were then used in
Ku-band simulations. Fig.2 shows that by selecting proper
particle size and shape, which were the best-fit parameters to
minimize the Mean Square Error (MSE) between simulated
and measured backscattering coefficients, the simulation
agrees quite well with measured data in terms of magnitudes
for co-polarizations and cross-polarizations as well as the
difference between VV and HH polarizations.
In order to illustrate the effects of particle shape on snow
backscattering, simulation was made by considering a slightly
larger but more sphere-like snow particle as Fig. 3 shows.
Fig.3 only differs from Fig. 2 in that a 0.7 short axis to long
axis ratio and 0.46 mm radius were adopted. By comparing the
two figures, it can be seen that particle shape have a
significant effect on snow backscattering, especially for cross-
polarization. The needle-like particles can produce much more
cross-polarization return than sphere-like particles and this is
more evident as Fig. 4 shows, which shows how
backscattering coefficients change with ellipsoid shape.
V.
CONCLUSION
A second-order model for snow was described and
comparisons were made between the model and measured data
from CLPX. Model inputs were represented by the average
values of measured snow parameters, with the exception of the
particle radius and shape, which were regarded as effective
values to the model. The model shows good agreement with
the measured data. Also from the comparison, it can be seen
that particle shape has a significant effect on snow
backscattering signals, especially for the cross-polarization.
R
EFERENCES
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41, no. 3, pp. 90-101, 2003.
[2] Y. Oh, K. Sarabandi, and F.T. Ulaby, “An empirical model
and an inversion technique for radar scattering from bare soil
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[3] J. Shi and J. Dozier, “Estimation of Snow Water Equivalence
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