Interferometric coherence is an important indicator of reliability 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 classi cation indicator for various coverage types. Coherence magnitude can be calculated as an absolute value of the correlation coe cient 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 coherence estimation algorithm is proposed that is based on the 2-FFT peak height assessment without calculation of the correlation coe cient. It is shown that such estimate has signi cantly less dependence on the topographic phase slope and provides satisfactory results in InSAR data quality assessment.
Interferometric data processing (InSAR) for extraction of information about the Earth terrain and its changes becomes one of the general guidelines in development of contemporary space-based radar systems together with the implementation 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 (noncoherent summation); phase noise supression; phase unwrapping (the elimination 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 solving the nal problem, i.e. the DEM generation or terrain changes mapping. The coherence factor is calculated as an absolute value 0 of the correlation coe cient between samples of two complex SAR images (single-look data complex, SLC) taken in the local windows (1)
j P z_1(m; n) z2(m; n)j pP jz_1(m; n)j2
P jz2(m; n)j2
;
where z1(2)(m; n) are the SLC samples (z1(2)(m; n) are the complex-conjugate
samples) [1{4, 6], ^0 takes values in interval [
So, the coherence value may be used as an interferometric phase quality indicator, 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 simpli es this processing step because some unwrapping 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 lters. A commonly used Goldstein-Baran
adaptive frequency lter adapts a frequency response in dependence on local
coherence [
F (k; l) = jS(k; l)j1 ^ S(k; l); (2) where S(k; l) and F (k; l) are the spectra of raw and ltered interferograms relatively, taken in a local window; ^ is the local coherence estimate. So, the lower coherence leads to extension of lter'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 con guration) lead to the degradation
of estimate (1). Its value depends on the slope and tends towards the value
0 0:2:::0:3, i.e. a bias of the estimate for independent Gaussian values of the
correlation coe cient [
Thus, coherence loses its properties as measure of the interferogram quality because its value becomes independent on the relation between topographic and uctuating components of the phase.
The problem of coherence estimate degradation was described by [
To eliminate the e ect 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
[!x; !y] = argmaxfF2[z_1 z2]g;
(3)
where !x and !y 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
performed according to the rule:
that eliminates in uence of the phase slope on the estimate value. After
substitution of (4) into (1), instead of z2 the correlation coe cient would give a correct
estimate for coherence magnitude [
Here, the main problem is that such way has a low computational e ciency:
the advantages of the FFT application are negated by the need of a correlation
coe cient 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
[
jz1j = 1; jz2j = 1
M = maxfF2[z1 z2]g:
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 uctuating near 1.
Dependence of the spectrum peak value P and scene decorrelation may be retrieved by simulation of Gaussian homogenous scenes with di erent correlations (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:
Estimate ^2 is less dependent of the phase slope value than a standard one ^0, which a ects only peak location within a spectrum. Figure 3 demonstrates the estimate behaviour by the simulation of homogenous scenes with di erent simulated correlation coe cients.
As can be seen, that for high correlations, the estimate tends to fall down near the slopes of radians to the level about 0.71 of ^2, which is related to sampling e ects. 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 uctuations.
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 [
As it is seen from Fig. 4, ^2 estimate has more wide band for the standard deviation than ^0 and it has a linear decreasing section for 0.2-0.45 coherence values, so it better re ects the quality of the InSAR data; ^0 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
A coherence estimation algorithm is proposed based on 2-FFT normalized peak height assessment. It is shown that such estimate has a signi cantly less dependence 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 (^) dependency that correctly characterizes 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.