ON THE PERFORMANCE OF FOREST VERTICAL STRUCTURE ESTIMATION VIA
POLARIZATION COHERENCE TOMOGRAPHY
Anna Fontana1,2, Konstantinos P. Papathanassiou1, Antonio Iodice2, Seung-Kuk Lee1
1
German Aerospace Center (DLR), Microwaves and Radar Institute (DLR-HR), Wessling, Germany.
2
Università di Napoli “Federico II”, DIBET, Via Claudio 21, 80125 Napoli, Italy.
ABSTRACT
In this paper, we assess potentials and limitations of
vertical structure estimation via polarization coherence
tomography (PCT) for the processing of multibaseline SAR
data, by using simulated data. We discuss the quality of the
profile retrieval for natural environment in presence of
several distortions as system errors and inaccurate a priori
information, by means of simulations.
1. INTRODUCTION
Vertical profiles (representing the variation of
backscatter as a function of height) of forest density are
potentially robust indicators of forest biomass, fire
susceptibility and ecosystem function. Synthetic aperture
radar imagery (SAR) is inherently a 2-D imaging process.
Since the radar cross-section observed for a given pixel is
the sum or integral of contributions from all scatterers at the
same range, encompassing all heights, the information about
vertical structure is lost. To recover such information we
need to image in the third dimension. In the case of remote
sensing of vegetated land surface many method for the
processing of multibaseline data to reconstruct vertical
profiles have been proposed. These methods often provide
good results but require large number of operational
baselines limited for any future potential spaceborne
applications.
Recently a new approach called polarization coherence
tomography (PCT) [1] has emerged in which a small
number of baselines or interferograms is used.
However, the price to pay for this simplification is that a
priori information, namely vegetation top height and true
surface topographic phase, is required. These two can be
obtained by a variety of methods, such as field
measurements, other sensors (such as TLiDAR [2]) or can
themselves be estimated from the radar data in advance of
application of PCT by using Pol-InSAR [3] algorithms (as
supposed in this work).
We mainly propose a performance analysis of
multibaseline SAR interferometry for the retrieval of
vertical profile via PCT, by using simulated data. For
obtaining simulated values of volume coherence and
information for the analysis, we use, as reference function,
terrestrial light detection and ranging (TLiDAR) data.
Then we show the trade-off between the stability of the
reconstruction related to the condition number and the
resolution of the vertical profile, and the quality of the
reconstruction in presence of system errors (decorrelation,
random fluctuations of phase and amplitude coherence) and
inaccurate a priori information (inaccurate results from PolInSAR inversion).
2. COHERENCE TOMOGRAPHY
The key radar observable in PCT for a penetrable volume
scattering is the complex interferometric coherence . This
coherence can be formulated for a random distribution of
scatterers, due for example to the presence of vegetation
cover, as shown in (1)
(1)
where
is the ground phase,
is the position of the
bottom of the scattering layer and
is the vertical
structure function and kz is the vertical wavenumber, related
to the baseline of the considered interferometric pair. The
reconstruction of the function
from
at each point
in the image is then termed coherence tomography. We first
note that the function
is bounded (by the underlying
surface and top of the vegetation layer for example) and so
can be expanded efficiently in terms of the FourierLegendre series. If we approximate
with a finite
number n of series terms, then the unknowns of our
problem (1) are the n Legendre coefficients, and
tomography reduces to estimation of this set of coefficients
from data. Of course, to retrieve the n coefficients we need n
image pairs (i.e., n baselines), so that a (linear) system of n
equations of the kind of (1) in n unknowns can be obtained
and solved. Hence the problem is the solution of a set of
(complex) linear equations which can be reformulated in a
compact form as shown in (2):
(2)
The evaluation of these coefficients is related to a matrix
inversion problem. The complexity of this operation can be
overcome using the singular value decomposition (SVD)
that allows a simple measurement of the condition number.
This can be calculated independently of the actual profile
and it depends only on the vertical wavenumber
.
High condition number of the F matrix acts to amplify
any noise on the measurement vector b and this leads to a
wrong coefficients estimation and then an inaccurate
retrieval profile. Increasing the number of baselines or
interferograms the condition number increases.
In fig.1 we show this effect on the stability of the
solution and then of the reconstruction in presence of small
phase noise, selecting several values of vertical wavenumber
at increasing conditioning.
To guarantee accuracy in the profile retrieval we analyze the
reconstruction only in the case of dual-baseline which is
more robust to noise.
Fig.2 Diagram of the proposed performance analysis
Fig.1. Dual (left) and Triple (right) Baseline reconstructed profile for
different values of the vertical wavenumber.
3. PERFORMANCE ANALYSIS
In fig.2 we propose the conceptual scheme to analyze the
PCT performance.
From TLiDAR result we extract the information about
the height (the "reference function"). Looking on the leftside of the scheme, the reference function is used as an input
in the Legendre decomposition, so that we obtain the
reference values of the Legendre coefficients,
. On the
other side of the scheme, the reference function is used to
simulate volumetric coherence data (via eq.(1));
disturbances due to different error sources are also
simulated. These simulated noisy data are then processed
via PCT to obtain the retrieved coefficients vector
. To
3.1 Effect of decorrelation phase noise
In general, for SAR systems where the size of the
resolution cell is larger than the radar wavelength, many
scatterers contribute to the response of one resolution cell,
and it is not possible determine the response of individual
scatterers within a resolution cell [4]. The received signal
will be related to a sum of many scatterers. Since the
complex coherence is related to two observations, recorded
from the master and the slave antennas, it has to be
considered as a random variable. We treat the coherence
phase as a Gaussian random variable in which the standard
deviation depends on the amplitude coherence and
multilook level. In fig. 3a we note how the fractional error
increases at low values of second baseline corresponding to
high condition number.
To quantify the upper bound of the fractional error in
which it is possible to reconstruct the profile we show in fig.
3b three different profiles for three different variations at
low second baseline (high condition number). The
reconstructed profile degenerates with 40% fractional error
achieved at high phase variation.
measure the amount of error made, we finally calculate the
fractional error defined as:
.
(3)
Fig.3 a) Fractional error against the second baseline for different
phase variation. b) Profiles reconstructed at low second baseline.
Fig. 5 Fractional error against height under and over estimation
.
Fig.4 Fractional error against the condition number for different
multilook levels.
Phase noise, and hence the fractional error, can be reduced
by using multilook. In fig.4 we shown how the sensitivity to
the error increases at low second baseline (high condition
number) and at low numbers of looks.
3.2. Effect of height and ground phase errors
We suppose to use, as a priori knowledge, the height and
the ground phase from the Pol-InSAR results. For this
reason we account the possible inaccurate estimation of the
height and/or the ground phase from the Pol-InSAR
inversion due to several decorrelation sources as the
temporal decorrelation for non-simultaneous acquisitions.
In fig.5 we show the fractional error in function of the
height error percent. We note how the overestimation leads
to acceptable values of fractional error (in particular if the
reference profile is an high tree).
Conversely, the effect of DEM error in which
(for the TLiDAR profile case
=0 is
supposed) is more critical. The inaccurate estimation of the
ground phase leads to larger variations of the coefficients
than the height error.
We assume an upper bound for the variance of DEM
error for accurate retrieval around 0.5m. This result is due to
a single look. Hence it can be improved increasing the
number of looks (for example with N = 9 the upperbound
for accurate retrieval is assumed around 5m).
3.3. Temporal decorrelation effects
The coefficients evaluation does not account for other
decorrelation source except of volume decorrelation. The
coherence level affected by temporal decorrelation is
interpreted, in the Pol-InSAR inversion, as volume
coherence caused by a higher top layer and it leads to an
overestimation of the forest height.
The amplitude coherence variation due to notsimultaneous acquisitions leads to high fractional error that
increases for low value of second baseline (high condition
number) and quite high decorrelation factor. Hence, we
conclude that the temporal decorrelation is the most critical
aspect of the analysis.
Fig.6 Reconstructed profile for several decorrelation factors
To keep the fractional error below 40% a temporal
correlation coefficient as high as 0.99 is needed for low
condition number, as we show in fig.6.
4. CONCLUSIONS
The main aim of this work has been to assess potentials
and limitations of PCT vertical structure estimation,,by
using simulated data. A possible upper bound of coefficients
error in order to have reasonable profile retrieval is given. It
turns out that:
1) The fractional error on the estimated coefficients due
to the stochastic nature of the volume coherence can reach
high values. Effects of amplitude and phase variation can be
compensated by low condition numbers and/or multilook.
An upper bound around 50% of fractional error for accurate
retrieval of vertical profile has been fixed.
2) Height errors are less critical. Height overestimation
leads low fractional error in particular for high trees.
3) DEM errors are by far more critical. DEM variance
can be compensated by increasing multilook level.
4) Temporal decorrelation provides high fractional error.
The solution here may be to accept a higher conditioning
number with the benefit of having larger baselines or the use
of single-pass mode.
5. REFERENCES
[1] S. R. Cloude, “Polarization coherence tomography,” Radio
Science, Vol.41, RS4017, 2006.
[2] J. F. Coté, J. L. Widlowski, R. A. Fournier, and M. M.
Verstaete, “The structural and radiative consistency of threedimensional tree reconstructions from terrestrial lidar,” Remote
Sensing of Environment, 113, 1067-1081, Jan. 2009.
[3] K. P.Papathanassiou and S. R. Cloude, “Polarimetric SAR
Interferometry”, IEEE Trans. Geosci.Remote Sensing ,36,, Sept.
1998.
[4] R. F. Hanssen, “Radar Interferometry Data Interpretation and
Error Analysis”, Klewer Academic Publishers, 2001.