M. Niang 1, JP. Dedieu 2, Y. Durand 3, L. Mérindol 3, M. Bernier 1, M. Dumont 3

1 INRS-ETE, 490 de la Couronne, Québec city, G1K 9A9, Canada 2CNRS / Edytem, Université de Savoie. Le Bourget-du-Lac, 73376, France

3CEN / Météo-France, Domaine Universitaire, 38406, St. Martin d’Hères, France

ABSTRACT

This study presents a new method to analyse some parameters of snow in alpine region: density, wetness. Here are developed the methodology and first results obtained from a set of ASAR-ENVISAT images registered during 2004 spring period over an experimental river basin located in the French Alps (N 45° 05’/ E 6° 10’). For ENVISAT/ASAR data, a topographic processing is done using a Digital Elevation Model (DEM) in order to correct the strong slope effects and determine the local incidence angles. During each ENVISAT data acquisition, intensive field measurements of snow pack properties (snow lines, snow pits) and weather conditions were gathered on 7 representative test sites. We propose a new inversion method for snow wetness and snow density retrievals by assimilating times series of ENVISAT/ASAR Alternating Polarization data. We use a forecast snow parameters and the covariance of the forecast error in order to produce inferred parameters compatible with the radar satellite measurements. An iterative method is set to find the maximum probably solution to a non-linear retrieval problem. This performs an inversion of the backscattering signal in VV and VH polarizations to retrieve simultaneously the snow liquid water content and snow density. This method provides a powerful tool since it consists in: 1) a fully physical method, involving calculations for each snow profile of corresponding backscattering signal by the ‘forward model’ in VV and VH polarization (and of derivatives of backscattering with respect to profile parameters); and 2) a statistical method, since it uses the covariance of forecast error as a constraint. The theory of this new approach is presented and the application to ENVISAT/ASAR data is discussed. Error characteristics of this method are investigated. The results of using as ‘the first guess’ as the snow characteristics profile given by the distributed SAFRAN/CROCUS snow model (Météo-France) are presented. The results are very satisfactory. Furthermore, for the wet snow condition in alpine environment, an appropriate methodology to have

accurate radar measurements and simulated backscattering is proposed.

1. INTRODUCTION

For many applications in snow hydrology, knowledge of the state and development of snowpack wetness and density are essential. Most of the time, liquid-water content and density are measured by manual snow sampling and/or are estimated with snow models at the basin scale. The manual method is rather costly, particularly when it comes to large and remote areas and is not adequate to achieve representative values for natural snow cover with large spatial variability.

The well-known snow model ‘CROCUS’ [1] and [2] is used for operational avalanche hazard forecasting as well as for research and studies. This is proved to be an efficient way to simulate in real time the snow cover evolution as function of elevation, slope and aspect (snow temperature, liquid-water content, grain size and shape, and density profiles) . This model needs hourly meteorological information on the snow-atmosphere interface at every computation point, which is given by SAFRAN model [3].

However, the ‘CROCUS’ model works at massif scale (500 to 1000 km²) and during the winter, between two dates, the forecaster never takes into account the snow ground-based observation to update the evolution of the snow cover, because resolution of manual observations is weak and temporally discontinuous. Furthermore, the spatial variability of snow properties in high relief-regions is large, particularly for snow depth, and it is difficult to obtain a representative value or a perfect statistical indicator. Therefore, the model makes a succession of errors which accumulates over the entire winter cycle but without systematic bias. Active microwave remote sensing has long promised the advantage of sensitivity to many snow parameters, ability to provide day or night imaging in all weather condition, a spatial resolution comparable with the topographic variation in alpine regions [4]. Many studies can be found for the detection of the wet snow

NEW INVERSION METHOD FOR SNOW DENSITY AND SNOW LIQUID WATERCONTENT RETRIEVAL USING C-BAND DATA FROM ENVISAT/ASAR

ALTERNATING POLARIZATION IN ALPINE ENVIRONMENT

_____________________________________________________

Proc. ‘Envisat Symposium 2007’, Montreux, Switzerland 23–27 April 2007 (ESA SP-636, July 2007)

condition [5]; [6]and [7]. Conversely, only few studies focus on the inversion model from active microwave data. The relationship between the backscattering coefficients and snow properties depends on a large number of parameters (snow, soil, and configuration of sensor) makes difficult to develop a statistical inversion model. In C-band, we can apply the algorithm of Shi and Dozier [4] for only snow wetness retrieval. They also developed an algorithm for snow density and grain size retrievals using a combination of L and X band [8].

Our inversion model performs a simultaneous retrieval of the snow wetness and snow density using ENVISAT SAR alternating polarization. It is based on the Newtonian iteration approach to the nonlinear inversion problem described by [9]. It is fully physical method involving calculations for each snow parameter of corresponding backscattering signal in VV and VH polarization and of derivatives of backscattering signal with respect to snow parameters. It is fully statistical method, since it uses the covariance of the guess-field error. The guess-field is computed from ‘Crocus’ model and it will be updated as function to the date of satellite overpass. The promising approach for the extraction of snow density and snow liquid water content started with the theoretical investigation of the inversion and the forward models. The method for the processing of The ENVISAT ASAR data and the snow field measurements coinciding with the satellite overpasses are presented.

2. INVERSION MODEL

The inversion algorithm proposed is based on the adjoin method. Having the SAR backscattering measurements, the method consist to predict the snow density and the snow liquid water content representing (X ) the analysis vector only by knowing their guess-field Xg. Let Y be a vector of the observed SAR backscattering measurements in VV and VH polarizations. Let J a cost-function defined as follows:

)))(())(()()()( 11 XHYOXHYXXAXgXXJ obst

obsgt −−+−−= −−

(1) where A is the guess-field errors covariance matrix; O is the observation error covariance matrix; H is the forward backscattering coefficient model for wet snow; Yobs is the observed backscattering coefficients; and superscripts t and -1 denote matrix transpose and inverse respectively. We may minimize (Eq.2) and assuming there are no multiple minima and find where the gradient of J(X) with respect to X is zero

0))((2)(2 1*1 =−−−=Δ −− XHYOhXXAJ obsg

(2) where h* is linear adjunct of H.

h* contains the partial derivatives of H with respect to the elements of X. The problem for the retrieval of snow liquid water content and snow density from ENVISAT data is assumed in this paper to be nonlinear case. Therefore an iterative approach is used to solve (eq.2). Newtonian iterative yields the following solution: if we guess a vector Xn, the solution is found through the iteration

)(121 nnn XJJXX ∇∇−= −+ α (3)

Where by differentiation of hOhAJ 1*12 22 −− +=Δ (4)

The coefficient α is the retrieval increment and it is used to yield a fine iterative descent. Experiments have shown that J in Eq.1 varies differently according to the elements of X. α is a coefficient. It will be used in order to adjust the slope of J for each element of X. In this paper, it is assumed to be constant for ω and ρ.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

5.0002.0

α (5)

The iteration of Eq.3 proceed until convergence i.e. until the increment ( nn XX −+1 ) is acceptably small. At this point we can estimates the covariance analysis by [9] which is the inverse matrix of the Hessian.

3. FORWARD MODEL

The forward model, H, computes backscattering coefficients in VV and VH polarization. Due to the low penetration depth of the backscattering signal in the wet snow at C-band, the main components of H are snow volume-backscattering and the surface backscattering [4].

),,,,(),,,,(),,,,,( rsLsrLs rpq

VpqS

pqT θρωσθρωσθρωσ +=

(6)

The subscripts T, S and V indicate the total, the surface scattering at the air-snow interface and the volume scattering coefficients respectively. ω is the snow liquid water, ρ is the snow density, θ is the incidence angle, s is the standard deviation of the random surface height and L its correlation length, r is the grain radius. For VV polarization ),,,,( LsVV

S θρωσ are computed from the integral equation method (IEM) model [10] and the Dense Medium Radiative Theroy [11] is used for ),,,,( rsr

VVV θρωσ . For cross-polarization VH,

we used a simplified method proposed by (Shi et al., 2003) is used:

{ }vvsurfrsvvhh

volrvhisvv

ishhvht sfsTT σθεσθαθε

θεσ ⋅+⋅⋅= ),,(),(),(),(

(7) Tvv and Thh are the power transmission coefficients at the air-snow interface in VV and HH polarization

respectively. The parameter vhα is the degree of correlation between co-polarized channels VV and HH given by Oh’s formulae [12]. vhα is a constant depending only on the refractive angle ( rθ ) into the snowpack obtained by the Snell’s law, and the standard deviation of the snow surface height (s). [ ]*Re hh

volvvvol

vvhhvol σσσ ⋅= is the real part of the cross

product of VV and HH*. The function ),,( rs sf θε is also given by Oh et al. (1992). To improve the simulation results, we introduced the relationship between snow wetness and snow density [13]

ωωρ

ρ−−

=0.1

wsds (8)

where ρds and ρws are the densities of dry and wet snow respectively in g.cm-3 and ω the snow liquid water content.

4. MATERIAL AND METHOD

The study area is the Romanche river basin (45°.05’ N, 6°.10’ E) located in the northern French Alps. This high relief-mountain is about 120 km2, 50% of which surfaces are situated over 2000 m elevation. During the spring and the summer 2004, snow field measurements were conducted by Météo-France in April 08, and in May 13 coinciding with ENVISAT overpass. Four test sites were selected and a snow-pit was made for each test site. Snow properties measurements in the vertical profile included snow density, snow liquid water content, snow depth, grain size and, temperature. Due to the variations of local meteorological conditions and of the solar radiations caused by the topography, the snowpack is an inhomogeneous medium, characterized by highly variable scattering elements mainly: snow

depth, grain size and water inclusions. Then, we construct an equivalent snowpack corresponding to the multilayer snowpack by a weight function f expressed as function to the extinction coefficient eK for snow liquid water and snow density:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−= ∫ ))(cos(

)(2exp)(0

zdzzKzf

r

z

e θ

(9) where θr is refraction angle in the snow, it is related to the incidence angle of the sensor by the Snell’s law. It is used in order to take account the wave attenuation when it propagates through the snowpack. 0z is the descending vertical axis. For the grain radii, we use the formalism of dendricity and sphericity [2] in which for a ‘dentritic’ snow if d is the dentricity and s the sphericity, the equivalent grain radii is :

⎟⎠⎞

⎜⎝⎛ −+⎟⎠⎞

⎜⎝⎛ ++−= )

991(4

993

991

99ssddreq

(10) For ‘not dentritic’ snow if p is the sphericity and t the grain size, the equivalent grain radius is :

⎟⎠⎞

⎜⎝⎛ −+=

991)2/,4max(

99ptptreq (11)

The ENVISAT-ASAR raw SLC images were first converted into a multi-looked precision image (PRI) format. A Digital Elevation Model (DEM) from IGN-France with a 50 m vertical resolution was used for local incidence angle computing. We calculated the calibrated backscattering coefficients by applying the local incidence angle values. The general method used in this study is summarized by the diagram set in Fig.1.

Figure 1: General method used for the snow water liquid content and snow depth retrievals

4. SIMULATION RESULTS We first examine the theoretical behaviour of the backscattering coefficient of snowpack as function of wetness and incidence angle for three values of the

snow surface roughness Lsm ⋅= 2 (very rough: m =

0.028, rough: m = 0.075 and smooth: m = 0.131 wet snow surface). In dry snow condition (volumetric liquid water content ≤ 2%), Fig. 2a shows that the snow surface roughness has little effect on the total backscattering coefficient. Therefore, more wet is the snowpack, more significant is the difference of backscattering coefficient for the three values of m. When the snow liquid water content is set at 4%, figure 2b shows the increasing difference of backscattering coefficient of the three values of the slope surface m as function of the incidence angle. That denotes the magnitude of the local incidence angle variations and the change in snow liquid water content on the wet snow backscattering signal with regard to the surface roughness condition. When the snow liquid water increases, the surface scattering becomes the major scattering phenomenon [14]; [4] contributing on the backscattering coefficient and then differences between the surface roughness properties can be observed. Fig. 2c shows that the proposed cross-polarization model agrees well with theoretical behavior of VH in comparison with VV.

Figure 2: Simulated backscattering coefficient in VV polarization for three values of surface slope; a) as

function of the snow liquid water content. Snow density = 450 kg.m-3; b) as function of the incidence angle. Snow liquid water content at 4% m. c) Comparison

between VV and VH.

4. INVERSION RESULTS

The two important points to emphasize in this section is that the guess-field is computed from ‘Crocus’ model and that the snow roughness parameters (s: standard deviation of the random surface height and L its correlation length) are obtained from a best fit procedure between the wet snow signal simulated from the forward models and the ENVISAT ASAR data measured in VV and VH polarization. With guess-field, we notice variations of the cost-function around the measured values of snow liquid water (Fig. 3a) and of snow density (Fig.3b).

Figure 3: The cost-function J; a) as function of the snow liquid water content. Snow density is fixed at 400 kg.m-3; b) J as function of the snow density. The snow

liquid water is fixed at 4%.

a)

b)

c)

a)

b)

Since no secondary minima are observed, this is promising for the gradient descent method used for the snow liquid water content and snow density retrievals.

Figure4: Comparison between guess-field, computed and measured values for a) snow density and b) snow

liquid water content

Table 1: the absolute error εω and ερ between inferred and measured snow density ρ and inferred

and measured ω .

The results show that the absolute errors between the inferred and measured snow liquid water content are inferior to 10-3 % with a maximum for the snow density at 36 kg/m3 (tab. 1). The values of the guess-field are distant of the regression axis.

5. CONCLUSION

The retrieved values are well in agreement with measured snow liquid water content and snow density. For the wet snow condition, the numerical simulations demonstrate that the surface scattering mechanism controls the relationship between snow wetness (and snow density) and the total backscattering coefficient. The surface roughness and the incidence angle also play an important role. Then, a best fit between SAR measurements and forward model is required in order to derive the roughness when they are not available. The equivalent snowpack model is an interesting new approach to build a representative mono-layer snowpack from model run. It reduces also the complexity of scattering mechanisms. Therefore, in order to optimize the iteration for gradient descent, the parameter α has been adjusted for certain values of the snow water content for a fast convergence of the two parameters. The physical and statistical inversion method can be used to update the CROCUS model. It would be a useful tool for the mapping of other snow parameters (i.e. grain size) when multi-polarization SAR data such as RADARSAT will become available.

ACKNOWLEDGMENTS

This study was funded by the « Programme National de Télédétection Spatiale »(PNTS), France, 2003-2005 (#56) and supported by ESA ENVISAT-CAT 1 Program (project (#2528). We are grateful to Météo-France/CEN and EDF for fieldwork snow measurements.

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