GENERATION OF DIGITAL TERRAIN MODELS FROM MULTIPLE SAR AIRBORNE SURVEYS OVER FORESTED AREAS Stefano Tebaldini, Guido Gatti , Mauro Mariotti d’Alessandro and Fabio Rocca. Dipartimento di Elettronica e Informazione, Politecnico di Milano, via Ponzio 34/5, 20133 Milan, Italy, Email:
[email protected]
ABSTRACT This paper considers the retrieval of ground elevation in presence of volume scattering, basing on multibaseline airborne SAR data. Once the interferometric phases related to ground-only contributions have been estimated through Algebraic Synthesis technique, the problem is similar to the one of generating a Digital Elevation Model (DEM) basing on multiple repeat-pass spaceborne SAR surveys. Nonetheless, airborne SAR systems present a significant difference with respect to spaceborne SAR system: since the Radar sensor is much closer to the target, the incidence angle sensitivity undergoes a dramatic increase with respect to the spaceborne case, resulting in a high spatial variability of the normal baselines. In order to properly cope with this issue, a real case has been analyzed taking explicitly into account the spatial variation of the normal baselines. The test site selected is the Krycklan catchement, in northern Sweden, chosen for the ESA BIOSAR 2008 campaign. 1.
INTRODUCTION
The aim of this work is the estimation of the ground elevation in presence of volume scattering, exploiting multi-baseline airborne SAR data. Two problems have to be considered. The first one is to obtain the interferometric phases relative to ground-only contributions. Such a problem has been considered in [1], where it has been shown that, if multi-polarimetric acquisitions are available, the ground phases can be provided basing on the concepts of Algebraic Synthesis [2] and Phase Linking [3]. The Sum of Kronecker Products (SKP) Decomposition, on which is based the Algebraic Synthesis, is a technique recently introduced in literature for the analysis of forested areas, whose aim is to separate ground and volume contributions within the SAR signal. The Phase Linking Algorithm, instead, has been introduced as a tool for the optimal estimation of the interferometric phases in presence of distributed targets. The second problem to deal with, which constitutes the main focus of this paper, is to split the estimated ground phases into two terms, one associated with ground elevation and the other with propagation disturbances, mainly deriving from uncompensated platform motion. Cast in
_____________________________________________________ Proc. ‘Fringe 2009 Workshop’, Frascati, Italy, 30 November – 4 December 2009 (ESA SP-677, March 2010)
this form, the problem is no different from the one of generating a Digital Elevation Model (DEM) basing on multiple repeat-pass spaceborne SAR surveys, where the interferogram phases are affected by slow-varying random oscillations due to the propagation through the atmosphere. As such, this problem can be dealt with by processing the estimated ground phases with techniques from Permanent Scatterers Interferometry (PSI) [3], [4], resulting in an accurate topography reconstruction. Nevertheless, airborne SAR systems introduce a significant difference with respect to spaceborne SAR system: since the Radar sensor is much closer to the target, the incidence angle sensitivity undergoes a dramatic increase with respect to the spaceborne case, resulting in a high spatial variability of the normal baselines. This phenomenon results in the phase difference between two nearby points that becomes dependent on the absolute ground elevation. This dependence, which can usually be neglected in the spaceborne case, prevents the application of standard satellite techniques. In order to properly cope with this issue, a theoretical analysis has been carried out taking into account the spatial variation of the normal baselines. Experimental results will be shown basing on a data-set of multipolarimetric and multi-baseline SAR images, at both P and L band, acquired by DLR’s E-SAR. This paper is organized as follows. Section 2 presents the selected test site, the available data-sets and the data processing needed to obtain ground interferometric phases. In Section 3 the Phase Locking problem related to the airborne case is introduced and a solution is proposed in the form of a system of linear equations. Section 4 contains an algebraic analysis of the Phase Locking solution, researching the existence and the properties of its intrinsic nullspace. Section 5 shows the results obtained on the data-sets under analysis. Finally, conclusions are drawn in Section 6. 2.
TEST SITE AND DATA PROCESSING
The test site analyzed in this work is the forested area within the Krycklan River catchment, Northern Sweden, selected for the ESA BIOSAR 2008 campaign. The data available is represented by 4 sets of Single Look Complex (SLC), fully polarimetric, SAR images in Radar Geometry acquired by DLR’s E-SAR
Table 1. Parameters of the acquisition systems in P-Band and L-Band. Parameter Carrier Frequency
Symbol f0
P-Band 350 [Mhz]
L-Band 1300 [Mhz]
Carrier Wavelength Sampling Frequency
λ fs
0.86 [m] 100 [MHz]
0.23 [m] 100 [MHz]
Slant Range Resolution
dr
2.12 [m]
2.12 [m]
Azimuth Resolution
da
1.6 [m]
1.2 [m]
Horizontal Baseline Spacing
db
8 [m]
6 [m]
Total Horizontal Baseline Aperture Look Angle Flight Height
A ϑ H
40 [m] 25° (Near Range) – 55° (Far Range) 3900 [m]
30 [m] 25° (Near Range) – 55° (Far Range) 3900 [m]
over a 2.3 × 9.5 km 2 area. Each data-set is constituted by 6 interferometric tracks (5 baselines) for each polarimetric channel. The test site has been imaged from two opposite points of view at both P-Band and LBand, as the SAR sensor has been flown with look direction South-West (SW) or North-East (NE). The parameters of the acquisition system are briefly summarized in Tab. 1. The acquisition campaign has been carried along the same day for a single band, so we can neglect targets motion in the acquisition period. The multi-polarimetric information has been fully exploited to separate ground and volume contributions within the SAR signal through the use of the SKP Decomposition. Afterwards, ground interferometric phases has been estimated by the Phase Linking Algorithm. Successively, through a two-dimensional Phase Unwrapping step, unwrapped ground phases ϕip has been obtained for each interferogram i ( i = 1… 5 ), formed with respect to a common master, and each point p in the slant range-azimuth plane (for both PBand and L-Band). These phases (flattened), together with coefficients for height-to-phase conversion Kzip (estimated by DLR with a motion compensation processing), form the starting point of the main aim of this work. A detailed description about the mentioned polarimetric and tomographic techniques can be found in [5] and in [6], presented at this conference.
3.
THE PHASE LOCKING PROBLEM
It is well known that the unwrapped interferometric phase ϕ of a target p is ideally connected to its height h through the following linear equation:
ϕip = Kzip h p in which Kz can be expressed as [7]:
(1)
Kz =
4π 1 B⊥ λR sin ϑ
(2)
where R is the sensor-target distance, B⊥ is the interferometric normal baseline and ϑ is the looking angle. Eq. 1 is not verified, apart from target motion and noise, in case of propagation disturbances, mainly caused by atmosphere in the spaceborne case or by an uncompensated platform motion for an airborne system. In this case, Eq. 1 have to be changed in:
ϕip = Kzip h p + α ip
(3)
in which α expresses the phase term due to the change of the acquisition conditions. Without neither ground reference points nor a model that accounts for the physics of propagation disturbances, we can just outline the problem compensating a constant offset for each interferogram, assuming α ip = α i . Usually, with spaceborne systems, the solution is easily retrieved subtracting from each interferogram the phase of a reference point P0 . In this way, the constant term is automatically removed and the terrain heights can be estimated with respect to P0 . This kind of solution results from the fact that, for the spaceborne case, Kz can be assumed, for limited areas, constants along a single interferogram and, consequently, a phase difference is directly related to a proportional difference of altitude:
∆ϕi = Kzi ∆h
(4)
Nevertheless, this solution is impracticable for the airborne case. Here, because of the nearness of the sensor with the ground and the high variability of the looking angle, the height-to-phase conversion factors are highly variable too. This means that the same value
of interferometric phase implies a different height for a different location of the target. So, the phase locking effect will change according to the chosen reference point. In this case, Eq. 4 becomes:
coefficients vector Kz i relative to all the N P points of the i-th interferogram:
(5) ∆ϕi = Kzi ∆h + ∆Kzi h in which appears the dependence whit the absolute elevation of the target and, consequently, phase differences are no longer directly proportional to heights differences. However, since in our work we are using unwrapped phases, the terrain topography can be retrieved directly by solving the linear system composed by the following equations:
Kzi2
ϕip = Kzip h p + αi
(6)
with i = 1… N I and p = 1… N P . The solution of this problem, if it is well posed, allows to estimate the absolute heights with respect to the constant phase offset that models at the best the change of the acquisition conditions in each interferogram. 4.
Kz1i 0 Ai = 0
0
0
0 0 ⋱ KziN P
(10)
φi represents the column vector of the ground phases for the interferogram i; α is a column vector containing the N I unknown offsets; 0 ( 1 ) denotes a column vector of N P zeros (ones). Our aim is to determine if the problem of Eq. 7 is well posed, that is to study the existence, the dimension and the shape of its nullspace (kernel). According to the rank-nullity theorem, the dimension of the nullspace of G can be obtained as follows:
dim[Null(G )] = ( N P + N I ) − Rank(G )
(11)
ALGEBRAIC APPROACH
It is useful to analyze the problem expressed by Eq. 6 under its algebraic point of view. We can define the direct problem as: d = Gm
(7)
where d is the data vector ( (N P ⋅ N I ) elements) constituted by the interferometric ground phases; m denotes the model vector ( (N P + N I ) elements) formed by the heights of the targets and by one phase offset for each interferogram; G is the forward operator ( (N P ⋅ N I )×(N P + N I ) ) describing the explicit relationship between data and model parameters. It is easy to rewrite the components of Eq. 7 in the following extended notation exploiting block matrices:
A1 A 2 G= ⋮ A N I
φ1 φ d= 2 ⋮ φ NI
0 0 1 0 ⋱ 0 0 1
To solve Eq. 11 we need to calculate the rank of G . With a first analysis, from Eq. 8 can be derived that the operator has at least N P + N I − 1 linearly independent rows1. This means that the forward operator admits at most an unitary nullspace. To verify the existence of the nullspace we need to determine the absence of a further linear independent row. Taking advantage of the iterative Gaussian-elimination algorithm, it is possible, starting from Eq. 8, to recast the system of Eq. 7 as follows:
dɶ = Gm with: A1 0 G= 0
1 0
h1 ⋮ ; m = hN P α
(8)
where (9)
Ai is a matrix containing, on its principal diagonal, the
(12)
0 0 1
(13)
Kz i denotes an element-by-element Kz j
vector
1 Kz 2 − Kz1
0 1
⋱ 0
⋱ −
Kz N P Kz N P −1
division and 0 an all-zeros matrix of the same dimension of Ai . 1
obtained from the N P rows belonging to a block matrix A i and from one row relative to every one of the remaining N I − 1 blocks.
The linear system of Eq.12 has the same model m of ) = Rank(G ) . Moreover, with the Eq. 7 and Rank(G homogeneous system the data vector dɶ corresponds to a null vector as well as d . After Eq. 13 it can be easily seen the condition under which it doesn’t exist another linear independent row. Consequently, we can state: Lemma 1: The operator G allows an unitary nullspace
if and only if:
Kzip Kz jp
=
Kziq Kz qj
∀(i, j ),( p, q ) : (i ≠ j ) ∧ ( p ≠ q) .
Lemma 1 explains that an unitary nullspace exists if the Kz coefficients are characterized on all the interferograms by the same spatial variation along the slant range, azimuth coordinates, up to a scale factor. From Eq. 12, assuming dɶ equal to a null vector, it is also possible to obtain the shape of the nullspace, corresponding to the eigenvector relative to the zero eigenvalue: 1 − Kz1 i 1 − 2 Kzi m0 = ⋮ 1 − NP Kzi α 0
Kz1p p Kzi p Kz2 p ; α 0 = Kzi ⋮ p Kz N I p Kzi
(14)
1 Kz1p 1 Kz p m0 = ⋮ ; α 0 = 2 ⋮ 1 Kz p α NI 0
(15)
This means that from the linear system of Eq. 7 the terrain topography can be retrieved with respect to a constant offset. This is the typical case of spaceborne acquisitions, in which height-to-phase conversion factors undergo no (relevant) spatial variation over a limited area and the unknown constant is fixed by the phase locking to a reference point. 4.2.
Kz distributed with the same spatial variation
If the phase-to-height coefficients present the same spatial variation upon all the interferograms (up to a constant), according to Lemma 1 a nullspace still exists. This instance can represents the case of airborne acquisitions not affected by differential platform motion (or of spaceborne acquisitions over extended areas with ideal orbits). Fig. 1(a) shows a profile of the Kz coefficients along the slant range direction calculated through Eq. 2 using the parameters concerning to the PBand of Tab. 1. The solution can be retrieved with respect to the relative nullspace heights shown in Fig. 1(b).
which, for the targets heights, depends directly by the inverse of the height-to-phase coefficients. It is clear that the effect of the nullspace is known up to a constant c: m = G + d + c ⋅ m0
(15)
Since the well posing of the problem is dependent just on the Kz, it make sense to analyze three typical cases of their spatial distribution. 4.1.
Kz with constant distribution
Upon the condition that Kz coefficients are characterized by no spatial variations, the same phase value is converted in the same height for each target. Since Lemma 1 is satisfied, the solution of the problem is affected by a nullspace derivable from Eq. 13. Assuming, for instance, Kzip = −1 ∀ p , the nullspace will be:
Figure 1. a) Simulated profile of Kz coefficients along the slant range direction for P-Band. b) Corresponding heights of the nullspace. 4.3.
Kz distributed with different spatial variations
The problem of Eq. 7 is well posed just under the hypothesis of which the height-to-phase coefficients present a different spatial variation between two
interferograms. In this case no nullspace exists and, ideally, absolute terrain elevation can be estimated. This is the case under analysis, which is a set of real airborne acquisitions. The different spatial variation of the coefficients is due to differential platform motion in different tracks and the consequent corrections. Nonetheless, the retrieval of the solution has to cope with the problem conditioning. 5.
NULLSPACE COMPENSATION AND EXPERIMENTAL RESULTS
In the case under analysis the problem is well posed. Even if the nullspace is not algebraically defined, however, the last eigenvalue decrease strongly. As a consequence, the problem is ill-conditioned , hindering the retrieval of absolute terrain elevation. Fig.2 shows the least squares solution of the problem with respect to the LIDAR DEM for P-Band and South-West look direction data-set. On this topographic profile is possible to notice a slope along the slant range coordinate not detectable in the reference DEM and consequently due to the ill-conditioning. In order to avoid this problem, terrain topography has been retrieved by allowing a 1-dimensional nullspace, ensuring the robustness of the solution. According to Fig. 2, the effect of the imposed nullspace, however, can not be described as a simple high offset, but rather
To fix this artefact the nullspace component is reintroduced in the solution in order to give the best match with the LIDAR DEM. This kind of approach is the best way to remove the trend turned out from the truncated solution and we name it as Algebraic Altitude Estimation (ALGAE). Fig. 3 shows the allowed nullspace for P-Band SW data-set. It’s evident how the use of this component, conveniently weighted, is more accurate than other de-trend methods, since it takes account of the proper ambiguity of the problem. It can be easily understood from Fig. 4, where it is displayed the same nullspace up to its rectilinear component. Final results are shown, for each band, in Fig. 5 and Fig.6. All panels report the estimation error with respect to the LIDAR DEM. In all cases errors are visible, whose dynamic may be assessed in about ± 15 m at PBand and ± 30 m at L-Band. Such errors are characterized by a large spatial decorrelation length and are correlated with the flight direction. These features clearly indicate that terrain elevation errors are the result of uncompensated phase terms, due to the residual platform motion.
Look Direction
Figure 3. Allowed nullspace component for P-Band SW data-set. 6. Figure 2. Terrain elevation error [m] with respect to the LIDAR DEM for P-Band SW data-set. Least squares solution. as a slowly deformation along the slant range direction, which represents the intrinsic problem ambiguity.
CONCLUSION
In this work has been shown that the problem of the terrain topography retrieval is well posed only upon the condition that each phase to height conversion factor is characterized by a different spatial variation along the slant range, azimuth, baseline coordinates. If the height to phase conversion coefficients undergo no (relevant) spatial variation (as it may be the case of spaceborne
acquisitions relative to limited areas), the nullspace is simply given by a constant elevation offset and terrain topography is retrieved by phase locking to a reference point. The problem is well posed in case of airborne acquisitions, due to differential platform motion in different tracks. As a matter of fact, however, it is illconditioned. For this reason terrain topography has been retrieved by allowing a 1-dimensional nullspace in order to ensure the robustness of the solution. Since the effect of the nullspace is a slowly varying deformation, it has been reintroduced in the solution in way of achieving the best match with the reference DEM. We have named this method as Algebraic Altitude Estimation (ALGAE). Certainly, better results can be obtained by considering ground reference points or modeling the propagations disturbances. Otherwise, this is the best solution that can be retrieved in absence of any kind of a-priori information. Future work will be centred on the development of the same approach at the differential case, estimating the terrain topography through phase differences. This issue is characterized by a further ill-conditioned linear system. Nevertheless, it will let us to exploit directly wrapped phases, avoiding potential errors in the phase unwrapping step.
Figure 4. Allowed nullspace non-rectilinear component for P-Band SW data-set.
Figure 5. Terrain elevation error [m] with respect to the LIDAR DEM. ALGAE solution. Left panel: P-Band SW dataset. Right panel: P-Band NE data-set.
Figure 6. Terrain elevation error [m] with respect to the LIDAR DEM. ALGAE solution. Left panel: L-Band SW dataset. Right panel: L-Bamd NE data-set.
7.
ACKNOWLEDGEMENTS
The authors wish to thank Dr. Malcolm Davidson (ESA), along with Dr. Irena Hajnsek (DLR), Dr. Kostas Papathanassiou (DLR), Dr. Thuy Le Toan (CESBIO), Prof. Lars Ulander (FOI), and the whole BioSAR 2008 team. 8.
REFERENCES
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