S AND
oral SAR
he Gauss-
ties of the
systematic
p. and the
depending
ources on
sed on er-
(1)
estimated
; are avail-
Observa-
| therefore
vice versa.
servations
ed covari-
of a over-
is derived
(2)
yields the
racies ex-
Lr (3)
(4)
aining ap-
known to-
estimated.
ace move-
stermined.
ent of sur-
ita source.
. Unfortu-
can not be
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
arbitrarily chosen. At is above all limited by temporal decorre-
lation. Especially in snow covered polar regions changing wind
conditions, temperature variations, and precipitation result in a
strong decrease of correlation with time. To warrant interfer-
ograms with sufficient quality, only interferograms originating
from the ERS tandem mission are considered, which comprise
a temporal baseline of only a single day. The ambiguous in-
terferometric phase values are unwrapped based on a minimum
spanning tree approach before implementing them into the ad-
justment. In addition a reference phase screen is subtracted from
the interferograms in beforehand using ERS D-PAF precision or-
bit information.
23 Functional model
As described above, the functional model comprises the deter-
ministic relations between observations an unknowns. For solv-
ing the proposed problem, three different sub-models are nec-
essary. The formulation of the sub-models and their particular
characteristics are derived in the following.
2.3.1 Interferometric model Although the phase ¢ of an in-
terferogram acquired over glaciated terrain is influenced by many
parameters, ¢ is dominated -by influences from surface topogra-
phy h, coherent senor motion v in line-of-sight of the sensor, the
difference of the slant-atmospheric delay Asd between the two
acquisitions, and the penetration depth d of the RADAR signal
into the glacier surface. The unwrapped interferometric phase at
position (7, 7) of an interferogram can be written as
Gi ruin ae. bei h3 ^ 4 As e)
À r*^J sin(0*3)
9 - Ve q3 . pi
rJ A tan(0*)
The notation used in the equation is in accordance with (Hanssen,
2001). The four different parts of Equation (5) show the mathe-
matical description of the above mentioned influences onto the
interferometric phase. The geometric reference phase is already
corrected in this representation. According to Equation (5) each
interferometric phase observation induces 4 unknown parameters
(h, v, Asd, d). Thus, the inversion of the model is a highly un-
derdetermined problem. A solution can be found if i) additional
observations are incorporated on a pixel by pixel basis, or if ii)
prior information is integrated into the equation system. The sec-
ond strategy might be employed if one or more parameters of the
equation system are known (e.g. external DEM's, or knowledge
about surface deformation). Such information is mostly not avail-
able in the arctic environment. Thus, a solution has to be found
by a combination of a series of interferograms in consideration of
additional assumptions about the time evolution of some param-
eters.
23.0 Temporal model To guarantee a successful separation
of the phase components in Equation (5) functional relations de-
scribing the connection between unknowns in different data sets
have to be established. Such models are only found for determin-
istic processes, i.e. signals that do not arise from a stochastic pro-
cess. In principle, this holds only for the evolution of topography
and surface displacement. As topography changes are usually
slow, and because of the limited sensitivity of the interferometric
phase with respect to topography variations, a time independent
description of surface topograpy ^ has been chosen. Introducing
this model reduces the amount of topography-related unknowns
from N - i. j unknowns to 7 - j unknowns.
1005
As described in (Fatland and Lingle, 1998) and (Frolich and Doake,
1998) the assumption of constant glacier flow is doubtful espe-
cially if ERS tandem interferograms are used. For modeling a
time-dependent flow behavior v(t) a mathematical model is em-
ployed. We refrain from using physical flow models, because of
their high complexity, significant non-linearity, and limited qual-
ity. As least-squares adjustments are better suited for solving lin-
ear problems, linear models for describing the glacier flow are
favored. Considering the usually uneven distribution of the data
sets over time a piecewise Lagrange polynom is selected. The
maximum polynomial order à is equal to à — N — u, — 1,
where u, is the number of parameters not related to surface mo-
tion. The term —1 warrants a redundant equation system. Thus,
the surface motion v(t) is modeled by
N—ùu
vit) = y ar’ (6)
g=1
2.3.3 Spatial model The unknown parameters are not solved
in each pixel but rather in the nodes of a regular spatial grid.
This step is allowed if the sampling rate of the digital data sets
is higher than necessary for the representation of their informa-
tion content. The restriction of calculating the desired parameters
only in a coarser grid entails several advantages. On one hand,
it reduces processing time, on the other hand, it increases redun-
dancy and, by this, the ability of the adjustment to detect gross
errors in the observations. The mesh size has to be chosen prop-
erly to avoid undersampling. Bilinear planes have been selected
for approximating the spatial correlation of topography and mo-
tion. The functional relation between an observed phase value in
an arbitrary position 9^7 and an unknown value in a node of the
corresponding bilinear raster Q^ is given by
oH = a + (i oe o dr 4- (ott = o^ dec 4-
(gan ce Qe = peut + d^! )drdc (7)
where dr = $! — $^ and de 5 $? — ot.
Although using the proposed models allows to reduce the num-
ber of unknowns, the equation system is still underdetermined.
This is due to the un-modeled atmospheric artifacts and the un-
known penetration depth. In (Hanssen, 2001) a stochastic model
for approximating the influence of the atmosphere on SAR inter-
ferograms is proposed, which is based on the spatial correlation
of the atmospheric signal. As the atmosphere in the arctic area
can not hold much water vapor and is usually characterized by
a stable stratification, atmospheric effects are neglected in this
study. The penetration depth of C-band SAR signals into firn and
ice was studied in detail in (Hoen, 2001). Maximal penetration
depth into dry snow is shown to be up to 30 m. In this paper
penetration depth is considered constant in time. Influences by
constant penetration depth is considered as part of the topogra-
phy component.
2.4 Stochastic model
Weighting of observations is done by considering the coherency
of the observed phase values. The probability density function
(PDF) of the interferometric phase for each resolution cell is cal-
culated from the coherency using the theory described in (Bamler
and Hartl, 1998) and (Lee et al., 1994). The standard deviations
of the observed phase values are derived from the PDF function
by
o
ere f (à — óo) PDF(o)dt (8)
—¢
