4. Baseline Error Estimation 43
4.3. Mathematical Model
4.2.7. Robustness
A principal requirement for robust estimates are redundant observations. Whereas redundancy itself already involves some degree of resistance with respect to outliers, the resistance can be further enhanced by dedicated robust estimation techniques. These generally confine the influence of individual outliers on the final estimates. For the application of robust techniques, two levels can be distinguished:
1. Pixel level. Shirzaei and Walter (2011) use a dedicated reweighting scheme that iteratively downweights contributions of pixels with outlying phase values. B¨ahr and Hanssen(2012) propose to apply classical data snooping (Baarda, 1968), rejecting one outlying observation per iteration completely (see section 4.5.2).
2. Interferogram level. Robustness of the estimates can also be enhanced by the inclusion of redundant interferograms in a network of linearly dependent interferometric combinations (see chapter 5;
Kohlhase et al., 2003;Biggs et al., 2007;Pepe et al., 2011). A systematic rejection scheme of interferograms with outlying baseline estimates has been proposed byB¨ahr and Hanssen(2012) and will be revisited in detail in section 5.3.
4.3. Mathematical Model
= 4π λ
nh#»rM,ref,#»ehiδxh,M+h#»rM,ref,#»eaiδxa,M+h#»rM,ref,#»eviδxv,M
− h#»rS,ref,#»ehiδxh,S− h#»rS,ref,#»eaiδxa,S− h#»rS,ref,#»eviδxv,S
o +ϕ0
=: −ah,Mδxh,M−aa,Mδxa,M−av,Mδxv,M
+ah,Sδxh,S+aa,Sδxa,S+av,Sδxv,S+ϕ0,
(4.5)
where #»rM,ref := #»rM(t, RM,ref) and #»rS,ref := #»rS(t, RS,ref) are unit vectors describing the assumed line of sight (see figure 3.1b). #»eh(t), #»ea(t) and #»ev(t) form a Frenet frame as defined in eq. (3.2). From the virtual acquisition geometry for zero-Doppler focussed data (see figure 2.1b) followsaa,M=h#»rM,ref,#»eai= 0, and also aa,S = h#»rS,ref,#»eai ≈ 0 holds due to the high degree of orbit collinearity in spaceborne SAR. Hence, the interferometric phase is not sensitive to orbit errors in along-track direction, and their contributions are neglected in the following. Instead, variations of orbit errors in time are allowed for by introducing polynomials of degree d:
E{ϕ}= −
d
X
i=0
ah,Mti
·δxh,M−
d
X
i=0
av,Mti
·δxv,M
+
d
X
i=0
ah,Sti
·δxh,S+
d
X
i=0
av,Sti
·δxv,S+ϕ0.
(4.6)
This most general observation equation has 4(d+ 1) + 1 parameters. But as orbit trajectories are very smooth curves, errors in their determination can adequately be described with a polynomial of low degree.
In the following,d= 1 will be assumed, since a linearly varying baseline error is considered an appropriate approximation for most applications (see also sections 4.4.2 and 4.4.4).
Moreover, the coefficients in eq. (4.6) are almost identical for master and slave due to the very small divergence between #»rM and#»rS. This makes the joint estimation of individual orbit errors for both master and slave an ill-posed problem that can only be solved in a network of interferograms (see chapter 5).
Considering one interferogram on its own, only components of a relative errorδB#»=δ#»xS−δ#»xM can be robustly estimated. In this case, it must be decided if the estimated error is heuristically attributed to inaccuracies in the master orbit, the slave orbit or to errors in both of them. In the following relation, the error is attributed in equal proportions to master and slave in order to avoid an arbitrary discrimination of one of the two acquisitions:
E{ϕ}= ah,M+ah,S
2
δBh+t δB˙h
+av,M+av,S
2
δBv+t δB˙v
+ϕ0
=:ah
δBh+t δB˙h +av
δBv+t δB˙v +ϕ0.
(4.7)
Considering the residual interferometric phasesϕT = (··· ϕi···) ofnϕ pixels that are regularly arranged on a grid spanning the whole interferogram, baseline error parameters bT = (δBhδB˙hδBvδB˙v) can be estimated in a functional model of the following kind:
E{ϕ}=
... ... ... ... ah,i ah,iti av,i av,iti
... ... ... ...
δBh
δB˙h
δBv
δB˙v
+
ϕ0
... ϕ0
=:Abb+1ϕ0, (4.8)
where 1T = (1 1··· 1). The stochastic model is generically defined by some covariance matrix:
D{ϕ}=σ02Qϕ, (4.9)
(a)Constant componentδB.#» (b)Linear componentδB.#»˙
Figure 4.3.: Anisotropic estimation quality of the baseline error, visualised by error ellipses. The grey area represents the sensor’s field of view. The orientation angles of the ellipses with respect to the nadir (¯θ0 and ¯θ1, respectively) can be computed from the eigenspaces of the covariance matrix D{b}. It follows that the estimability ofˆ δB⊥(¯θ0) andδB˙k(¯θ1) is best, whereasδBk(¯θ0) andδB˙⊥(¯θ1) are most weakly determined.
where D{·} denotes the dispersion. The choice of D{ϕ} is discussed in more detail in section 4.6. As ϕ0is of no further interest, this parameter can be eliminated from eq. (4.8) yielding (Teunissen, 2000, p. 91 et seqq.;Niemeier, 2008, p. 307 et seqq.;J¨ager et al., 2006, p. 37 et seqq.):
E{ϕ}= ¯Abb (4.10)
with:
A¯b= I−1(1TQ−1ϕ 1)−11TQ−1ϕ
Ab, (4.11)
where I is the identity matrix.
The relative estimation quality of the least squares estimates of ˆbis given by their covariance matrix:
D{b}ˆ =σ02( ¯ATbQ−1ϕ A¯b)−1=
σB2h σB
hB˙h σBhBv σB
hB˙v
σB˙hBh σ2˙
Bh σB˙hBv σB˙hB˙v
σBvBh σB
vB˙h σB2
v σB
vB˙v
σB˙vBh σB˙vB˙h σB˙vBv σ2˙
Bv
(4.12)
and can be visualised qualitatively by the error ellipses in figure 4.3. Their orientation can be ob-tained from the eigenvalues of D{b}. The constant component has its largest variance at an orient-ˆ ation β = ¯θ0±90◦ with:
tan ¯θ0= σBˆhBˆv
σ2ˆ
Bh−λ0
. (4.13)
λ0 is the major eigenvalue of D
(BˆhBˆv)T . The linear component has maximum variance forβ = ¯θ1
and β = ¯θ1 + 180◦, respectively, where
tan ¯θ1+ 90◦
= σˆ˙
BhBˆ˙v
σ2ˆ
B˙h−λ1 . (4.14)
λ1 is the major eigenvalue of D
(Bˆ˙hBˆ˙v)T , and ¯θ1 ≈ θ¯0 usually holds. The strong elongation of the error ellipses shows that the baseline is determined best perpendicular to the line of sight, whereas its rate of change has maximum precision in look direction of the sensor. This basically confirms the previously drawn conclusions identifying δB˙k and δB⊥ as the components with the most significant impact on the interferometric phase.
4.3. Mathematical Model
Even though the complementary components δBk andδB˙⊥ are theoretically estimable, their estimates would be too weakly determined to be considered reliable. This can be seen from figure 3.5a, where a relatively huge error inBk induces only a very faint error signal in the phase. Conversely, a faint atmo-spheric signal that matches by chance this phase pattern, would result in unrealistically large estimates ofδBk in the order of metres (see figure 4.9a). Analogous considerations apply toδB˙⊥ (see figure 3.5d).
Therefore, it is preferable to constrain these two components to zero. This can be achieved by narrowing the parameter space from four parametersbto two parametersbTθ = (δB˙kδB⊥), yielding:
E{ϕ}= ¯AbTTbθ (4.15)
with:
T= 0 sin(θ0) 0 −cos(θ0) cos(θ0) 0 sin(θ0) 0
!
. (4.16)
The mean look angle θ0, which is required for the decomposition into parallel and perpendicular com-ponent here, is heuristically defined by:
θ0:=
θ¯0+ ¯θ1
2 . (4.17)
The deviation between ¯θ0 and ¯θ1 depends on the spatial distribution of phase observations and is usu-ally small, i. e., on the 0.1◦ level.
Least squares adjustment yields:
bˆθ= TA¯TbQ−1ϕ A¯bTT−1
TA¯TbQ−1ϕ ϕ (4.18)
D{bˆθ}=σ02Qθ=σ02 TA¯TbQ−1ϕ A¯bTT−1
(4.19) with an estimable variance factor:
ˆ
σ02=vTϕQ−1ϕ vϕ
nϕ−u , (4.20)
where u= 3 is the number of unknowns (δB˙k,δB⊥ andϕ0). vϕare the predicted corrections:
vϕ= ¯AbTTbˆθ−ϕ. (4.21)
Note that the here addressed corrections denominate updates to the observations and are not identical with residuals. The notion of residuals rather refers to the remainder of the observations after subtraction of their predictions and thus implies an opposite sign.
4.3.2. Gridsearch Estimator
A major shortcoming of the least squares estimator outlined in the previous subsection is that it re-quires unwrapping. However, there are many applications in which unwrapping is cumbersome or even infeasible. In these cases, an alternative gridsearch approach can be pursued. It consists in minimising an objective function of the wrapped phase by incrementally searching the parameter space spanned by δB˙k and δB⊥.
From eqs. (4.15) and (4.21) follows E{ϕ −A¯bTTbθ} = 0. By analogy to the ensemble coher-ence from (Ferretti et al., 2001), a dedicated coherence measure can be defined as a function of bTθ = (δB˙kδB⊥):
γ(bθ) = 1 nϕ
nϕ
X
j=1
ei(ϕj−¯ab,jTTbθ), (4.22)
where i is the imaginary unit and ¯ab,j is the jth row of ¯Ab. Considering 0≤ |γ| ≤ 1 and E{|γ|}= 1, bˆθ is defined as the set of parameters that maximises |γ|. Note thatγ(bθ) can also be interpreted as a discrete integral transform of the two-dimensional signaleiϕ to the (δB˙k, δB⊥)-domain.
Asγis insensitive to arbitrary cycle jumps of individual phasesϕj, application of the gridsearch estimator does not require explicit phase unwrapping. The computational load is higher than for the least squares method but still negligible compared to other InSAR processing steps. Whereas thegridsearch approach does not provide any intrinsic quality measures for the estimates, heuristic, peak-to-noise ratio-like in-dicators can be defined. A noteworthy drawback is that the estimates turn out to be unreliable in some cases, in particular when|γ|(bθ) has more than one distinct local maximum (see figure 6.5b).