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.
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 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 Eip E0pejk(xsinizcos i) (k is a wavenumber in the air). Here and below, the p index describes the polarization of the radiation: p = h when the polarization of the radiation is horizontal and p = v for vertical polarization.
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Homogeneous ground is characterized by a complex dielectric constant g g j g ; dry snow is assumed to be a nonabsorbing 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 <zs> = d and <zg>= 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:
Ep e j Esp e js TspTsp Egp e jg ,
(
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 (
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.
After averaging relation (
Ep e j Esp e js TspTsp Egp e jg
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 (
p o e j s0p e js (1 Rs2p ) g0p e jg .
p
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
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 (
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.
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 sin0jcos0k
(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 nˆ is determined from equation
cosil kˆi nˆ
tan sini cosi .
1 tan2 tan2
It follows from (
gl 2kd cos cos2 (0 ) 1.6 0.86 3 cos(0 ) 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 gl 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
(
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
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).