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Stilla, Uwe

In: Stilla U, Rottensteiner F, Paparoditis N (Eds) CMRT09. IAPRS, Vol. XXXVIII, Part 3A/V4 — Paris, France, 3-4 September, 2009
identify automatically, while layover areas still carry useful
information, even if this is embedded in clutter (see the example
of a repeat-pass TerraSAR-X interferogram in Figure 1).
Moreover, the incorporation of additional knowledge about
buildings may help to separate useful signal from clutter,
especially as buildings have regular shapes and, often, digital
maps indicating the building footprints are available.
Multi-baseline and multi-aspect approaches show great
potential to reconstruct buildings with high accuracy and level
of detail. However, the time needed to acquire the necessary
images is usually too long for rapid mapping, especially in the
context of providing crisis information. Consequently, the
analysis in Section 3 concentrates on accuracy aspects of single
pass interferometry. The building heights are expected to be
computed from the interferometric signal of layover regions.
The inherent contribution of clutter in these areas is
accommodated by some loss of interferometric coherence,
which is also taken into account for the final height accuracy.
Figure 1 : Interferometric fringes in layover area of tall buildings
computed from dual-pass TerraSAR-X interferogram (courtesy
M.Eineder, DLR).
In this section we revise the mathematical theory for relating
specific sensor and scene parameters with the desired height
accuracy for the case of space-borne SAR. A detailed derivation
of the formulae can be found in (Bamler & Schattler, 1993;
Bamler & Haiti, 1998; Gumming & Wong, 2005). Figure 2
(left) depicts the typical geometric configuration of across-track
interferometry. The phase values of the two acquisitions can be
derived from the well-known two-way range equation
0i=—— 2/? + 0 scatli (1)
where (f)^ and are the SAR phases at a certain pixel, A is the
wavelength, R is the range between one antenna and the point
on ground in viewing direction 0, and AR is the range
difference induced by the baseline vector B and its component
perpendicular to the viewing direction B ± , respectively. Under
the assumption that the unknown phase contributions caused by
random scattering (/> scatt x and t scatt, 1 ft scat t,2
one can express the interferometric phase (J) for a certain point
0 = fi-0 2 = — AR (4)
In order to convert the this phase into height values Z , it is
useful to first formulate the functional relationship between
AR and the direction perpendicular to R on ground, £ (see
Figure 2 (right)):
RR s A
Ç = --^-AR = —-$- — (/>
B ± An
Solving for (/) and projecting into the vertical direction Z yields
AnB L z
A R sin 0
Equation (6) is the basis to calculate the so-called phase-to-
height sensitivity:
d(f) _ An B ±
dz A R sin#
Figure 2: Geometric layout of across-track interferometry (left) and
definition of local co-ordinate system on ground, C, , (right).
Figure 3 illustrates the influence of varying incidence angle and
baseline length on the phase-to-height-sensitivity. As can be
seen, the interferometric measurement gets more and more
sensitive the longer the baseline and the smaller (steeper) the
incidence angle is.
Figure 3: Influence of incidence angle and baseline on phase-to-height-