Following these publised projects in the jounal IEEE-TGRS [65], I count on applying this algorithm to the fragmentation of radar images acquired on snowpack. Starting from this fragmentation in homogeneus zones, it will be interesting to calculate the mean backscattering integrated from each zone, and to modify differently the parameters of the physical model of evolution of the snow Surfex/Crocus. This procedure is described in chapter 5 of this manuscript. Another aspect to examine is that of phase information. It has been recently shown, through in-situ multi-temporal measures, that it is possible to obtain the equivalent of dry snow through water (SWE, Snow Water Equivalent), from the evolution of the phase in bande X.
Chapitre 4
Estimators and statistics of RSO images for the measurement of deplacements
The emergence of spatial high resolution in imagery RSO leads us to attempt to model radar backscattering, according to different statistic laws of the gaussian laws. This concept is explored in the section4.1. The idea developed in the section 4.2 proceeds to further exploit the texture of several RSO images from the same zone, to attempt to follow the deplacement of geophysical objects. Finally, we apply this method regarding the glaciers in the section 4.3.
4.1 SIRV Model.
A jumble model PoISAR based on the use of Spherical Invariant Random Vectors (SIRV) -studied [66, 67]. The vector of complex polarmetric backscatteringk can be defined as the product of the root of a random positive scalar variableτ with an independent gaussian vector z :
k =√
τ z, (4.1.1)
where τ represents the texture - that is to say, a process potentially non-homogenous. We attempt to separate the texture from the strength. The vector z that represents the speckle is complex, gaussian, of zero-mean and of a matrix of covariance.M = E{zzH} (H is the operator of united complex transposition).
Estimated texture. τ
For a matrix of given covariance. [M], estimated ˆτi in the sense of maximum probability, of parameter of texture τ for the pixel i is given by :
ˆ
τi=kiH[M]−1ki
p , (4.1.2)
where p is the dimension of the backscattering vector k (p = 3 in the monostatic case, because the emission in horizontal polarization H and the reception in vertical polarization V is equivalent to VH (HV = VH)).
With the case of a determining texture, the estimator in the sense of maximum probability of the matrix of normalized covariance is the solution of the following recursive equation [68][69] :
[M]ˆ F P = f ( ˆ[M]F P) = p N
XN i=1
kikHi kHi [M]ˆ −1F Pki
= p N
XN i=1
zizHi zHi [M]ˆ −1F Pzi
. (4.1.3)
Concerning the case of a non-determining textur, the estimator (4.1.3) is only an approached version of the estimator of the matrix of covariance, in the sense of the maximum of probability. The estimator in the sense of maximum of probability of the matrix of normalized covariance depends, thus, on the distribution of τ . Its expression is :
[Mˆ M L] = 1 N
XN i=1
hp+1
kHi [Mˆ M L]−1ki
hp
kHi [Mˆ M L]−1ki
kikHi (4.1.4)
where the expression of the generating function of density hp(x) est donn´ee par [70] [71] :
hp(x) =
+∞Z
0
1 τpexp
−x τ
fτ(τ ) dτ (4.1.5)
Pascal et al. established existence and uniqueness in a factor close to the matrix of normalized covariance of the Fixed Point [ ˆM ]F P as well as the convergence of the recursive algorithm which allows to estimate ˆτ and [ ˆM ]F P regardless of the initialization of [ ˆM ]F P [72, 73].
The recursive algorithm of estimation consists of calculating, in a first setting, an estimate of the matrix of covariance ˆM with (4.1.3) or (4.1.4) then, estimating the texture ˆτ with (4.1.2).
In our case, the trace of the matrix of covariance is normalized by p. It is of importance that we insist upon the fact that absolutely no hypothesis had been made concerning the probability density of the texture in th definition of SIRV’s. The SIRV’s describe a whole class of stochastic processes.
This class includes the conventional models of clutter, like the gaussian laws, K, G0 et KummerU respectively corresponding to textures distributed according to the laws Dirac, Gamma, Inverse Gamma and Fisher [74, 75, 62].
In practice, the estimator (4.1.4) is only used in cases in which the distribution of τ is known, during which the analytical expression of hpcan be calculated. In other cases, this is the approached estimator (4.1.3) which is used.
Simulation of the texture.
The laws of Fisher are commonly used to simulate the data RSO in an urban context [56]. In our case, for natural ice and glacier fields, the law of Fisher more effectively simulates data than other laws, thus we have implemented it. The distribution of Fisher is a law of three parameters, defined by :
fτ(τ ) =F [m, L, M] = Γ(L + M) Γ(L)Γ(M)
L Mm
Lτ Mm
L−1
1 + Lτ Mm
L+M (4.1.6)
SIRV Model.
47
o`u m > 0 is a scale parameter,L > 0 and M > 0 are two parameters of form
The particularity of this family of laws is its hybrid behavior, being ”head-heavy” and
”tail-heavy” at the same time. In this sense, it generalizes the laws Gamma and Gamma Inverse. The first three log-cumulatives of the law of Fisher are written :
˜
κx(1)= log(m) + Ψ(L) − log(L) − (Ψ(M) − log(M)).
˜
κx(2)= Ψ(1,L) + Ψ(1, M).
˜
κx(3)= Ψ(2,L) − Ψ(2, M).
(4.1.7)
The plan formed by the log-cumulatives of order 2 and 3, ˜κ2− ˜κ3, allow to represent the defined laws on R+. The plan ˜κ2− ˜κ3 is represented in the figure 4.1.1 with the repartition of different families of laws, with which the Gamma and Gamma Inverse laws cut the plan in three. Here, we limit ourselves to the representation of the laws of monovaried random variable, however this type of representation could be expanded to a multivaried case [76].
../figures/chap4/Classes_k2_k3.png
Figure 4.1.1 – Representation of the laws derived from the Pearson system.
The illustration 4.1.2 shows an extract of image TerraSAR-X (28/01/2009) from the Argentiere Glacier. This illustrates the changes of distribution between intensity (here, from canal HH) and the texture τ extracted from a clear cut database TSX dula-pol, by the processes SIRV (previously presented). Although very similar, these two images cannot be modelled by the same fashion, as the associated histograms suggest.
If we use the plan ˜κ2− ˜κ3 to represent these distributions, we notice that the cloud of points (κ3, κ2) of the image of intensity traced in the illustration 4.1.3 corresponds more to the Gamma laws, whereas the cloud of the image of texture is more distributed in the region of the laws of Fisher (Figure 4.1.3).
0 0.1 0.2 0.3 0.4 0.5 0.6 0
1000 2000 3000 4000 5000 6000
0 0.1 0.2 0.3 0.4 0.5 0.6
0 1000 2000 3000 4000 5000 6000 7000