5. Snow height retrieval
5.2. Single layer, dry snow model
Remote sensing has been used successfully for snow monitoring. Passive microwave sensors have a long history in snow monitoring. Several investigations have demonstrated the capability of multi-frequency microwave radiometer systems to locally map snow extent, snow depth, and snow water equivalent [2][3]. Experiments with satellite-borne synthetic aperture radars have demonstrated the capability of SAR to distinguish wet snow from bare ground and estimate snow liquid water content [7].
For water resources planning and management the snow water equivalent (SWE) is the most important variable. It is defined as the amount of water contained within the snowpack and can be thought of as the depth of water that would theoretically result after melting the entire snowpack. Snow water equivalent can be presented in units of [kg/m2] or metres of depth of liquid water that would result from melting the snow. SWE is the product of depth and density, so both values are directly related as follows:
- SWE (units [kg/m2]) = depth [m] × snow density [kg/m3]
- SWE (units [m]) = depth [m] × snow density [kg/m3] / density of water [kg/m3] The problem of retrieving the snow cover depth, thus, is directly related to the SWE mapping. Because of the high spatial variability of SWE, in-situ point measurements are not a suitable basis for SWE mapping. Presently no remote sensing method exists which enables reliable observation and mapping of SWE in complex terrain: retrieving SWE from SAR backscattering data, for example, suffers from the insensitivity of long wavelengths (L- and C-band) to dry winter snow, whereas at short wavelength the snow metamorphic state becomes important. On the other hand, the interferometric phase shift in snow due to differences in the propagation constant relative to the atmosphere offers a direct method for retrieving SWE and consequently for snow height retrieval, as first proposed in [42]. A more recent study [43] has demonstrated the feasibility of combining backscattering and InSAR correlation observations for retrieving snow accumulation parameters by means of a model that considers both volume scatter and surface scatter from hoar layers at depth. Nevertheless, the before mentioned model works under very restrictive conditions and a- priori knowledge of some ground-truth parameters, so the use of the interferometric phase remains the more promising technique to infer snow cover depths with SAR imagery.
Concerning the modelling of the snow cover, two main trends have been explored. A multi-layer model, including volume scatter and the incoherent addition of echoes from hoar layer interfaces in which the scattering from a single layer is found by small- perturbation methods was used in [43]. This approach entails precise knowledge of the different snow pack stratums and is particularly suited to estimating the backscattering properties of a given snow pack. On the other hand, the reduction of the snow cover into a single layer has proved to be effective both under dry and wet snow conditions for the retrieval of snow covered areas [44] and the creation of SWE maps [42]. A single layer
model is again used in [45] to infer the SWE, snow depth and particle size with a complex scattering model that takes into account the scattering of the dry snow layer at particle level and its interactions with the air and ground boundaries.
Backscattering measurements have been the only input to most of the before mentioned studies mainly because of lack of sufficient and adequate images to produce interferograms. GB-SAR overcomes this limitation because of its high image acquisition rate, where repeat-pass interferometry can be applied several tens of times a day to accurately generate snow cover height evolution plots.
5.2.2. Phase-to-height conversion
The propagation of radar waves in snow is governed by the complex permittivity of snow which is strongly dependent on its liquid water content. The penetration depth dp of
microwaves at the wavelength in free space λ0 can be estimated from the real ε' and
imaginary ε'' parts of the complex permittivity of snow according to [46]:
0 '. 2 '' p d λ ε πε = (5.1)
The imaginary part of the permittivity, ε'', of dry snow at C- and L-band is of the order of 0.001 to 0.0001, whereas the real part, ε', depends only on the snow density ρs
[46], where ρs is specified in g/cm3:
3
' 1 1.60 s 1.86 .s
ε = + ρ + ρ (5.2)
The result is a typical penetration depth of dry winter snow of dp ≈ 20 m at C-band
[47]. The dielectric losses of wet snow, on the other hand, are large and the typical penetration depth of wet snow with liquid water of several per cent by volume is of the order of few centimetres only.
The DInSAR snow depth retrieval algorithm presented in this chapter exploits the large penetration depth in dry snow, taking into account that the main contribution of backscattering from ground covered by dry winter snow stems from the ground surface. It is assumed that the dielectric properties are constant all over the snow cover (single layer model). In the case in which this is not true, an equivalent single dielectric constant should be computed considering the geometry and properties of the different layers. This is outside the scope of this work since the calibration procedure for the interferometric phase- to-snow depth conversion presented below directly assumes a single snow layer (see Figure 5.1) with a single dielectric constant for all the period in which the calibration parameters are valid (typically one day).
Repeat-pass differential interferometry is based on the phase comparison of a pair of complex coherent radar images of the same scene taken in different instants of time. Let ds
be the depth of the whole snow cover at some instant as shown in Figure 5.1-a. After some minutes, during which the snow depth may have changed, the radar sends another pulse. Let the new snow cover depth be ds’ as shown in Figure 5.1-b. The two-way propagation
path difference, 2Δr = 2Δrs−2(Δrs’+Δra), in the form of a phase shift, Δφs, is actually the
a) b)
Fig. 5.1. Propagation path of microwaves with a snow layer of depth a) ds; and b) ds’.
The propagation path difference Δr can be expressed from the geometry in Figure 5.1 as a function of the change in the snow cover depth Δds = ds’−ds and the incidence and
refraction angles θi and θs:
cos( ) 1 ( ’ ) . cos i s s s a s s r r r r d θ θ θ − − Δ = Δ − Δ + Δ = Δ (5.3)
Taking into consideration the different propagation constants (λs =λ0 ε') and the relationship between θi and θs according to the Snell’s law, the interferometric phase shift
in dry snow related to Δds can be written as [42]:
(
2)
0 0 cos( ) ' 4 4 cos ' (sin ) . cos i s s s s i i s d θ θ ε d π π ϕ θ ε θ λ θ λ − − Δ = − Δ = − Δ − − (5.4)Thus, the phase shift Δφs and the differential snow depth Δds can be related by a
very simple linear relationship depending only on three variables: λ0, θi and ε'. When the
area under study is small enough, the incidence angle and the snow permittivity can be considered constant, allowing the reduction of the above-mentioned three variables to a single constant α [m/rad]:
0 2 1 . 4 cos ' (sin ) i i λ α π θ ε θ ⎛ ⎞ ⎜ ⎟ = − ⎜ − − ⎟ ⎝ ⎠ (5.5)
The differential snow depth, thus, can be simply expressed as: .
s s
d α ϕ
Δ = ⋅ Δ (5.6)
Because the phase shift values come from differential measurements, an offset constant β [m] needs to be added to the previous equation in order to calibrate the results to
.
s s
d = ⋅ Δ +α ϕ β (5.7)
The constant and offset parameters α and β, also named calibration parameters, are
directly retrieved from a single ground-truth control point and are treated in more detail in Section 5.3.6.
Provided that continuous image acquisitions are available for a given test site, a single offset constant β would be required to calibrate the absolute snow height, ds, during
the first interferometric pair formation. For the rest of interferograms, the differential snow depth Δds as expressed in (5.6) would be simply added to the preceding values of snow
depth. The parameter α, instead, depends of the central radar wavelength and the incidence
angle, that can be assumed constant over the field campaign. But it also depends of the snow permittivity, which may change over time depending on the snow density as shown in (5.2). A ground-truth control point is, hence, necessary every time the snow density varies to accurately obtain the parameter α and consequently the snow cover depth.
5.2.3. Permittivity and density retrieval
Another interesting application is that of the permittivity and snow density retrieval from the snow depth measurements, always based on the single layer, dry snow model presented in the preceding sections.
As stated in (5.7), the absolute snow depth retrieval can be done through the interferometric phase and a single ground-truth calibration point. This calibration technique, and in particular the parameter α as defined in (5.5), comprises all the geometric algebra involved in the scene analyzed and depicted in Figures 5.1 and 5.2.
If the parameter α, instead, is retrieved by fitting a time-series of interferograms as
described in (5.7) to the snow depth evolution available from the data logger of an automatic meteorological station, the snow permittivity and density can be retrieved through (5.5) and (5.2) respectively.
Fig. 5.2. Geometry of the acquisition scenario for the permittivity and density retrieval.
From the field campaign carried out in the Sion Valley in Switzerland, the differential height between the LISA instrument and the monitored test point, Δh, the slant-
θn, as described in Figure 5.2 are know. From these parameters, the incidence angle θi can
be directly expressed as:
/ 2 , i h n θ θ= +π −θ (5.8)
(
)
asin / . h h r θ = Δ (5.9)The snow permittivity is finally obtained by inverting (5.5):
2 2 0 ' (sin ) cos . 4 i i λ ε θ θ πα ⎛ ⎞ = +⎜ + ⎟ ⎝ ⎠ (5.10)
The snow density inversion ρs, instead, requires solving the third order equation
(5.2). However, the typical dry snow densities (~0.1 g/cm3, [52]) allow (5.2) to be approximated by the linear term only.