=Paper= {{Paper |id=Vol-2534/14_short_paper |storemode=property |title=Estimation of Parameters of a Snow Cover on a Ground Surface without a Relief and with Relief by the Radar Interferometry Method |pdfUrl=https://ceur-ws.org/Vol-2534/14_short_paper.pdf |volume=Vol-2534 |authors=Pavel N. Dagurov,Aleksey V. Dmitriev,Sergey I. Dobrynin,Tumen N. Chimitdorzhiev }} ==Estimation of Parameters of a Snow Cover on a Ground Surface without a Relief and with Relief by the Radar Interferometry Method== https://ceur-ws.org/Vol-2534/14_short_paper.pdf

Estimation of Parameters of a Snow Cover on a Ground Surface
without a Relief and with Relief by the Radar Interferometry
Method

Pavel .N.Dagurov1,2, Aleksey V. Dmitriev1, Sergey I. Dobrynin2, Tumen N. Chimitdorzhiev1
1 Institute of Physical Materials Science of SB RAS, Ulan-Ude, Russia, pdagurov@gmail.com
2
Buryat State University, Ulan-Ude, Russia, pdagurov@gmail.com
3 Buryat Institute of Infocommunications of SibSUTIS, Ulan-Ude, Russia, wmdumb@gmail.com

Abstract. The remote sensing of snow cover is studied using radar interferometry. An
approximate model of interferometric sounding based on the small perturbation method is
proposed. The contribution of the scattering from the snow surface to the amplitude and
interferometric phase was estimated. The analysis of the effect of the relief on the
assessment of snow cover parameters was performed. The results of numerical estimates
are given.

Keywords: snow water equivalent; radar interferometry; small perturbation method;
relief.

1 Introduction
Seasonal snow cover in the regions of temperate and northern latitudes is an important natural factor. Snow has a
great impact on climate, hydrological and soil processes, plant and animal life, and human life.
The main characteristics of the snow cover that determine its impact on the environment are its thickness and the
snow water equivalent (SWE) [1]. The snow water equivalent determines the water content in the snow cover. In
particular, in the case of homogeneous snow with a constant height, the snow water equivalent is defined as the
product of the depth of snow cover d and its density ρs, referred to the density of water ρw, and is expressed in units of
length.
Using the method of radar interferometry has shown that it is an effective tool for the diagnosis and monitoring of
various changes in the earth's cover [3 - 6]. Interferometric methods were also used to analyze the snow cover and
estimate the snow water equivalent [7 - 11]. The possibility of direct measurements of SWE using differential
interferometry was first considered in [7]. In [8], the results of experiments in the C-band using the remote sensing
satellites ERS-1 and ERS-2 are presented, which show agreement with the calculated dependences. The theoretical
dependence for the interferometric phase on the wind farm was also used in [8] for comparison with experimental
data obtained from span SAR flights in the L-band. A comparison of the calculation formula for determining SWE of
dry snow and experimental results was carried out in [9]. In [10], Sentinel-1 radar data were used to estimate the wind
farm. In [11], similar estimates were performed by analyzing the ALOS PALSAR 2 data obtained at a test site on the
shores of Lake Baikal.
Earlier in [12], a model of backscattering from snow cover on a flat average Earth's surface was proposed.
Backscattering occurs due to small-scale roughness. In this paper, we present some numerical results for this situation
and consider the more general case of scattering from snow cover on a surface with a relief..

2 Backscattering from snowpack without relief
When constructing a model for the backscattering of microwaves from a dry snowpack on the soil, we assume that
snow is a continuous homogeneous medium and that there is no volumetric scattering. This is true in the C- and
especially in the L-bands when the size of the snow particles is much smaller than a wavelength. Figure 1 shows the
geometry of the problem and the paths of microwaves along which they propagate in the absence and presence of a
snowpack. In the absence of snow, the wave designated as 1, falling from the air onto the soil at an angle of θi, is
scattered back by the roughness of the soil in the form of the wag wave (red dashed line). Note that Figure 1shows
trajectories of the incident waves by solid lines and those of the scattered waves using dashed ones. In the presence of
a snowpack, when wave 2 falls onto the snowpack from the air at an angle of θi, it refracts and propagates in the snow

Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
layer. Afterward, wave 2 falls onto the “snow–ground” interface at an angle of θt, and the same roughness of the soil
scatters the wave back as the wsg wave (green dashed line). The single-wave model of Guneriussen et. al. [30] takes
into account only this scattered wave. Let us consider a more general model that takes into account the backscattering
wave from the “air–snow” interface. Assume that radar wave 2 falls from the air at the angle θi onto the snow layer
covering the ground (Figure 1). The radar is in the far zone and the incident wave can be considered as a plane wave
E E e
i
p 0p (k is a wavenumber in the air). Here and below, the p index describes the polarization of the radiation:
jk ( x sin i  z cos  i )

p = h when the polarization of the radiation is horizontal and p = v for vertical polarization.
_______________________________________________
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Figure 1. Geometry of backscattering from soil in the absence and presence of a snowpack.

Homogeneous ground is characterized by a complex dielectric constant  g   g  j g ; dry snow is assumed to be a non-
absorbing medium with a dielectric constant εs, and the dielectric constant of air is equal to1. The boundary surfaces
of “air–snow” and “snow–soil” are statistically rough ones with random irregularities. Their heights are uncorrelated
with each other and described by stationary random functions zs(x, y) and zg(x, y), with mean values equal = d
and = 0 (where d is the average depth of the snowpack), standard deviations equal to bs and bg, and correlation
radii of ls and lg, respectively. If we let irregularities be small compared with the wavelength, then their slopes are
insignificant and the small perturbation method (SPM) applicability conditions, such as kbs, kbg < 0,3 and kls, klg < 3
[13] , are satisfied. It is assumed that irregularities do not influence the coherent field (Born approximation). The
incident wave passes through the snow layer at the angle θt, determined by the Snell–Descartes law. When leaving the
snow, the scattered wave also refracts according tothe Snell–Descartes law. Fresnel formulas for a flat interface are
used to calculate the reflection and transmission coefficients of coherent waves. The backscattered field is the
coherent sum of the waves scattered by the irregularities of boundaries. The first wave is the was wave scattered by
the “air–snow” interface (blue dashed line in Figure 1), and second one is the wsg wave backscattered by the “snow–
ground” interface that occurs after the wave passes through the layer and emerges from it (main scattered wave). In
addition to these waves, weaker backscattering waves appear because of reflections from the layer interfaces and
scattering by irregularities. However, we can neglect the influence of these waves because the values of the reflection
coefficient from the “dry snow–air” interface is insignificant (see the next subsection).We do not take into
consideration phase fluctuations of waves caused by small irregularities at media interfaces. Let us suppose that phase
values are due to the paths of the waves and their interaction with averaged flat interfaces in the framework of the
small perturbation method. Then, a random electromagnetic field with an amplitude Ep and phase Φ backscattered by
rough boundaries can be expressed by the following sum:
j
Ep e j  Esp e js  TspTsp Egp e g , (1)

where the summands on the right-hand side of the formula describe the was and wsg wave fields, respectively. In (1),
Tsp and T'sp are Fresnel transmission coefficients of the wave that propagates through the averaged flat “air–snow”
interface in the forward and backward directions, respectively; Esp is the amplitude of the field scattered by snow
irregularities in Figure 1; Egp is the amplitude of the field scattered by the uneven ground. Φs is the interferometric
phase of the wave backscattered by snow. Φg is the interferometric phase of the wave backscattered by the ground
after passing through the snowpack in the forward and backward directions

Фs =  0 , Фg =   0 , (2)

where  =2k  2 d / coss ,   2kd (tgi  tgt )sin i , 0 =2kd / cosi
After averaging relation (1), we obtain
j
Ep e j  Esp e js  TspTsp Egp e g (3)

where the expressions in angled brackets represent the average amplitudes of the scattered fields. We assume that the
squares of these average amplitudes are proportional to the mean squares of the amplitudes. This is true, for example,
for random values distributed according to the Rayleigh law. Since the mean square of the backscattering field
amplitude is proportional to the backscattering coefficient, it follows from expression (3) that

 0p e j   sp0 e j  TspTsp  gp0 e j
s g
(4)

where  0p is the resulting backscattering coefficient,  sp0 is the backscattering coefficient from the “air–snow”
interface, and  gp0 is the backscattering coefficient from the “snow–ground” interface.
The value  0p e j can be called a complex amplitude-backscattering coefficient. Taking into consideration the
relations between transmission coefficients and a reflection coefficient Rsp from the“air–snow” interface Tsp = 1 + Rsp,
Tsp' = 1 − Rsp, we can rewrite equation (4) in the following form:

 op e j   sp0 e j  (1 Rsp2 )  gp0 e j .
s g
(5)

Further, for calculations, we will use the known expressions of the backscattering coefficients in the approximation of
the small perturbation method [12]. The dielectric constant of dry snow is determined by the expression [14]

 s  1 1.6 1.86 3 (6)

where ρ is the numerical value of the snow density and is expressed in g/cm3. This ratio is valid for frequencies in the
range from 100 MHz to 10 GHz and for a snow density of less than 0.5 g/cm3.
Figure 3 shows the relative interferometric phase δΦ = ∆Φ/Φg variations when the snow depth changes. The
presented dependencies show that the variations can exceed 10% in the situations under consideration. Notice that as
the depth of snow increases, variations in the relative phase decrease. This result is due to the fact that when the depth
of the snow increases, the value of Φg grows faster than ∆Φ. Ifthe snow depth is greater than 40 cm, then δΦ < 4%.
This means that in the considered hypothetical case, in which the root-mean-square values and correlation radius of
irregularities on the surfaces of the soil and the snow are equal, the impact of backscattering from the “air–snow”
interface on the interferometry phase is rather weak.

ρ = 0.2 ρ = 0.4

Figure 2. The relative interferometric phase variations in the radar signal versus the snow depth.

The phase difference of the main backscattering wave that has passed the snow cover and the backscattering wave
in the absence of snow cover is determined according to the relation [7].

1  2kd (  s  sin 2 i  cosi ). (7)

Estimates show that the interferometric phase of the main scattered wave is linearly related to the
snow density with good accuracy and, accordingly, it is directly proportional to the WES (SWE).
This important result for practice was obtained for the first time in [7]. The relationships between
the interferometric phase and the density of snow and wind farm (SWE) are obtained in the form
1  1.5kd  cosi SWE  1 cosi (1.5k ) (8)

An estimate of the relative error in determining the interferometric phase due to the influence of a
wave scattered by the air – snow boundary and the nonlinearity of formula (7) shows that the
resulting error of formula (8) does not exceed 8% for incidence angles of 20 ° –45 ° and snow
density (0.2 - 0.3) g / cm3.

3 Assessment of the effect of the relief on the interferometric phase
The above backscattering model from snow cover considers the case of on average flat and horizontal surfaces of
ground and snow. However, the real earth's surface has relief. In order to estimate the influence of topography on the
interferometric phase let us assume that the known digital elevation model (DEM) describes the relief. We use the
tangent plane and geometric optics approximations when considering the interaction of waves with the surface of
snow and soil (see Fig. 13). These planes are the result of roughness averaging of the earth and snow surfaces.

Figure 3. The geometry of the wave incidence on the tangent plane, with local slopes.

Electromagnetic wave falls on the snow layer, bounded by the planes "air - snow" and "snow - soil" at a local
incidence angle θil and scattered back after refraction in the snow at an angle θtl and the passage of the snowpack. The
incidence angle (angle of view) is equal to θi with respect to the vertical axis z. The wave vector ki is in the yz plane
and the unit wave vector has the form
kˆi  sin0 j  cos0k (j, k – unit vectors of the y and z axes).
The "air – snow” and “snow - soil" planes are parallel to each other and the distance between them vertically (the
depth of the snow cover) is equal to d. The local inclination of these planes is determined from the DEM. The angle α
is measured with respect to the radar coordinate "range" (y-axis) and angle β with respect to the coordinate “azimuth”
(x-axis). Then the local incidence angle θil at the point with the unit normal n̂ is determined from equation

tan  sin i  cosi
cosil  kˆi  nˆ  . (9)
1  tan 2   tan 2 

It follows from (9) that relief inclination in “range” direction most influence on local incidence angle. Let us estimate
influence of relief in assumption that β = 0. The local interferometric phase of wave, reflected by soil can be
expressed as follows
gl  2kd cos  cos ( ) 1.6  0.86  cos( )
2
0
3
0 (10)

Expression (23) valid both for positive values of α (front slope) and negative α values (back slope). Let us estimate
the relative changes in the interferometric phase due to the influence of relief with help of the formulas (18) and (23)

gl  g
gl  . (11)
g

Figure 4 shows dependence of  from the angle α at different values of θi and . These plots prove that the influence
of terrain slopes on the relative phase can be quite significant. Relative phase changes reach 40% for steep slopes with
α values of about 45°. However, with gentle slopes, these changes are small. The slope of the relief did not exceed
1.5° changes the interferometric phase do not exceed 2% at these values, which confirms the possibility of using the
model of the averagely horizontal surface of the earth in this case.

a) b)

Figure. 4. Dependences of the interferometric phase relative changes from the angle of inclination of the terrain along
the "range" radar coordinate: a) at different θ0,  = 0.3 g/cm3; b) at different , θ = 30°.

3 Conclusion
An approximate model is constructed for determining the interferometric phase, which is the phase difference of
radar signals in the absence of snow and after snowfall. The model is based on the small perturbation method. An
analysis is made of the effect of the backscattering wave on the roughness of the snow cover on the phase of the radar
signal. The model is generalized to the general case of backscattering from snow cover on the earth’s surface with a
relief, and the influence of the slope angles of the relief on the interferometric phase is estimated.

Acknowledgements. This work was supported by Russian Foundation for Basic Research (Grant No. 18-05-01051
A).

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