=Paper=
{{Paper
|id=Vol-1814/paper-12
|storemode=property
|title=InSAR Data Coherence Estimation Using 2D Fast Fourier Transform
|pdfUrl=https://ceur-ws.org/Vol-1814/paper-12.pdf
|volume=Vol-1814
|authors=Andrey V. Sosnovsky,Viktor G. Kobernichenko,Nina S. Vinogradova,Odhuu Tsogtbaatar
}}
==InSAR Data Coherence Estimation Using 2D Fast Fourier Transform==
https://ceur-ws.org/Vol-1814/paper-12.pdf
InSAR Data Coherence Estimation Using 2D
Fast Fourier Transform
Andrey V. Sosnovsky1 , Viktor G. Kobernichenko1 , Nina S. Vinogradova1 ,
Odhuu Tsogtbaatar1,2
1
Ural Federal University, Yekaterinburg, Russia
and 2 Ulan-Baator Technical Unisersity, Ulan-Baator, Mongolia
sav83@e1.ru
Abstract. Interferometric coherence is an important indicator of relia-
bility for interferograms obtained by interferometric synthetic aperture
radar (Interferometric SAR, InSAR). Areas with low coherence values are
unsuitable for interferometric data processing. Also, the coherence may
be used as a classification indicator for various coverage types. Coher-
ence magnitude can be calculated as an absolute value of the correlation
coefficient between two complex SAR images with averaging in a local
window. The problem of coherence estimation is in its dependence on
the phase slope caused by relief topography (topographic phase). A co-
herence estimation algorithm is proposed that is based on the 2-FFT
peak height assessment without calculation of the correlation coefficient.
It is shown that such estimate has significantly less dependence on the
topographic phase slope and provides satisfactory results in InSAR data
quality assessment.
Keywords: Synthetic aperture radar images, InSAR systems, coherence
estimation
1 Introduction
Interferometric data processing (InSAR) for extraction of information about the
Earth terrain and its changes becomes one of the general guidelines in develop-
ment of contemporary space-based radar systems together with the implemen-
tation modes of ultra-high spatial resolution (1–3 meters) and full-polarimetric
processing [1–3]. The InSAR processing for building the digital elevation models
(DEM) includes the following steps: synthesis of the pair of complex synthetic
aperture radar (SAR) images and their coregistration; forming the interferogram
by the element-wise complex multiplication of these SAR images; compensation
of the phase slope caused by side-looking imaging geometry; multilooking (non-
coherent summation); phase noise supression; phase unwrapping (the elimina-
tion of phase ambiguities); and conversion of the absolute phase interferogram
in elevation grid data and its projection.
Interferometric coherence is an important indicator of suitability of the data
scene obtained by a radar remote sensing system for the further processing and
99
solving the final problem, i.e. the DEM generation or terrain changes mapping.
The coherence factor is calculated as an absolute value γ0 of the correlation co-
efficient between samples of two complex SAR images (single-look data complex,
SLC) taken in the local windows
P
| ż1 (m, n) · z̄2 (m, n)|
γˆ0 = pP P , (1)
|ż1 (m, n)|2 · |z̄2 (m, n)|2
where z1(2) (m, n) are the SLC samples (z̄1(2) (m, n) are the complex-conjugate
samples) [1–4, 6], γˆ0 takes values in interval [0,1], and near-zero values corre-
spond to areas of high or full decorrelation, which are not suitable for interfero-
metric data processing. An intrinsic radar signal decorrelation is caused by the
radar looking angle difference (geometric decorrelation) and by the Earth sur-
face variability (temporal decorrelation). Strong decorrelation occurs due to loss
of echo-signal (typical for water surfaces), volume scattering (forest vegetation),
signal layover, and shadowing, etc. Commonly, the areas with coherence lower
than 0.2–0.3 are unsuitable for conversion into DEM.
So, the coherence value may be used as an interferometric phase quality indi-
cator, which gives an opportunity to reject the areas, where the phase is unstable
and is not related to the Earth topography. Also, rejection of the decorrelated
areas before phase unwrapping simplifies this processing step because some un-
wrapping algorithms become slower and unstable while processing such areas.
The other way of the interferometric coherence utilization implies its usage as
a parameter in adaptive phase noise filters. A commonly used Goldstein-Baran
adaptive frequency filter adapts a frequency response in dependence on local
coherence [5]
F (k, l) = |S(k, l)|1−γ̂ · S(k, l), (2)
where S(k, l) and F (k, l) are the spectra of raw and filtered interferograms rel-
atively, taken in a local window; γ̂ is the local coherence estimate. So, the lower
coherence leads to extension of filter’s bandwidth and vice versa. For this reason,
the coherence map is usually calculated before the phase noise supression stage
in the InSAR processing chain.
However, this approach entails some problems because, in fact, a random
variable is estimated here, but not a random process. So, any phase slopes caused
by both natural topography variability and by point-of-view geometry (remote
sensing radar systems have a side-looking configuration) lead to the degradation
of estimate (1). Its value depends on the slope and tends towards the value
γ0 ≈ 0.2...0.3, i.e. a bias of the estimate for independent Gaussian values of the
correlation coefficient [3], which in practice takes the value about 0.1–0.3 (Fig
1).
Thus, coherence loses its properties as measure of the interferogram quality
because its value becomes independent on the relation between topographic and
fluctuating components of the phase.
The problem of coherence estimate degradation was described by [3, 4, 7], and
some approaches for its correction were proposed. The basic approach involves a
100
Fig. 1. Dependencies of γ̂0
coherence estimate on the phase slope angle for different simulated correlation
coefficients (a) and local window (sample) sizes (b).
reference digital elevation map (with lower spatial sampling frequency, as a rule)
for phase compensation prior to the coherence estimation. But such reference
DEM with a sufficient sampling factor is often unavailable for a specific terri-
tory (especially for the Northern ones). Another aprroach implies elimination
of the phase components from estimate [3] andcalculation of an estimate based
on images magnitude. But as it is shown [8], such estimates have a large bias
and they can not be used for detection of low-coherent areas. So, a reasonable
approach may lie in an adaptive phase slope estimation and its compensation, as
it offered in [3, 9], but some additional measures should be considered to improve
the efficiency of such approach.
2 A 2-FFT interferometric coherence estimate
To eliminate the effect of coherence estimates degradation, we construct the
estimate in a way, which makes it immunable to phase slope value. Since a
constant phase slope can be considered as the interferogram spatial frequency
modulation, it is reasonable to use a 2D fast Fourier transform (2-FFT) to
determination of modulation frequencies, which can be found as
[ωx , ωy ] = argmax{F2 [ż1 · z̄2 ]}, (3)
where ωx and ωy are the spatial frequencies in both spatial dimensions, which
are estimated in an interferogram sample of M × N size. A common idea for
2-FFT coherence estimation is that further spatial frequency demodulation is
101
performed according to the rule:
ωx ωy
z2c = z2 · exp[−j( + )], (4)
M N
that eliminates influence of the phase slope on the estimate value. After substi-
tution of (4) into (1), instead of z2 the correlation coefficient would give a correct
estimate for coherence magnitude [3, 9].
Here, the main problem is that such way has a low computational efficiency:
the advantages of the FFT application are negated by the need of a correlation
coefficient computation, and, so, a computational time for this estimate exceeds
the same one for the usual estimate. From the other side, the images’ magnitude
information gives minor contribution to the estimate value, as it is shown in
[8]. Moreover, radar brilliant points significantly degrade the estimate in their
neibourhood. So, it is reasonable to normalize images magnitudes taking
|z1 | = 1, |z2 | = 1 (5)
and calculate a peak value of the discrete spectrum for a normalized interfero-
gram (in the local window) as follows
M = max{F2 [z1 · z2 ]}. (6)
In this case, taking into account the Parseval’s identity, for a scene sample
M × N with a constant phase slope, one can get that the following:
1) for fully coherent scenes a 2-FFT spectrum has a single peak P of an
M × N height in a position matching to a spatial frequency value;
2) fully incoherent scenes may be considered as a discrete white noise with
the spectrum fluctuating near 1.
Dependence of the spectrum peak value P and scene decorrelation γ may be
retrieved by simulation of Gaussian homogenous scenes with different correla-
tions (Fig. 2).
As can be seen from Fig. 2, the dependence between a normalized spectrum
peak value P/(M × N ) and a simulated correlation is slightly biased. So, a
possible way for coherence estimation is to recalculate a peak value (or peak to
mean value ratio) into a coherence magnitude in the following way:
P
γ̂2 = . (7)
M ×N
Estimate γ̂2 is less dependent of the phase slope value than a standard one
γ̂0 , which affects only peak location within a spectrum. Figure 3 demonstrates
the estimate behaviour by the simulation of homogenous scenes with different
simulated correlation coefficients.
As can be seen, that for high correlations, the estimate tends to fall down
near the slopes of π radians to the level about 0.71 of γ̂2 , which is related to
sampling effects.
102
Fig. 2. Depedencies of a normalized spectrum peak value of the 2-FFT spectrum on
the simulated correlation coefficients of different local window (sample) sizes.
3 Experimental results
An experimental assesment of the proposed method of interferometric coherence
estimation was carried out within application of a reference DEM. Since the
coherence value indicates, first of all, the fluctuations level of the absolute phase
around the topographic phase, and the latter is closely related with the reference
DEM heights [2], it is obviously to use a statistics of phase deviations to assess
the quality of the coherence estimates. Such assesment was performed in the
following way:
1) the reference DEM was reprojected into a flight vehicle coordinate system
(range-azimuth), and then the heights were recalculated to reference absolute
phase values using an InSAR height ambiguity factor; the reminder shift was
fixed through the cross-correlation between the reprojected DEM and the scene
absolute phase obtained after phase unwrapping stage;
2) coherence estimates γ̂0 and γ̂2 were calculated with the window size 19×19;
3) coherence values interval [0,1] were splitted into subintervals with a fixed
step 0.05;
4) for each subinterval i the standard deviation of the absolute phases and
reference absolute phases σψi were counted in corresponding image points.
The obtained dependence of σψ on γ̂ acts as a quality indicator for the inter-
ferometric coherence estimate. The better estimates should give a less-fluctuating
103
Fig. 3. Dependencies of γ̂2 coherence estimate on the phase slope angle for different
simulated correlation coefficients.
decreasing dependence at least for low- and medium-valued coherences (except
extreme-low values) with a possible wider range for both coherence and standard
deviation values. This means that a better estimate is more sensitive to the level
of phase noise fluctuations.
The reference DEM covered a territory 8 × 5 km, which contained average
hills and river valleys and had the vertical height accuracy about 1.7 m, which is,
at least, triple times better than the expected ALOS PALSAR DEM accuracy
(about 7–8 m [10]. An interferogram has a size of 2000 × 750 elements. The
obtained dependecies σpsi (γ̂) are shown in Fig. 4. The range of values for γ̂0
was about 0.25–0.75, and 0.2–0.6 for γ̂2 . Extremely high values (γ̂0 > 0.7 and
γ̂2 > 0.55) were excluded because they were, as a rule, corresponded to terrain
elements that are not joined with the Earth relief (buildings, facilities, roads,
engineer communications, etc).
As it is seen from Fig. 4, γ̂2 estimate has more wide band for the standard
deviation than γ̂0 and it has a linear decreasing section for 0.2-0.45 coherence
values, so it better reflects the quality of the InSAR data; γ̂0 has an abnormal
behaviour, the standard deviation increases within the coherence value. Both
estimates have an abnormal behaviour for extremely low and extremely high
coherences.
4 Conclusions
A coherence estimation algorithm is proposed based on 2-FFT normalized peak
height assessment. It is shown that such estimate has a significantly less depen-
104
Fig. 4. Dependency of the absolute phase standard deviation on the coherence esti-
mates.
dence on the topographic phase slope. A method of coherence estimates quality
assesment is proposed, based on calculation of InSAR absolute phase deviation
from a reference DEM. It is also shown, that proposed coherence estimate has a
quazi-linear decreasing section on the σψ (γ̂) dependency that correctly charac-
terizes it as a quality measure for InSAR data.
Acknowledgments. The work was supported by Act 211 Government of the
Russian Federation, contract 02.A03.21.0006.
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