F-24 INFORME INDIVIDUAL DE PARENTESCOS Y OTROS VINCULOS

There is a wide variety of imaging algorithms that have been used to reconstruct reflectivity images of a scene by using SAR techniques. A 2-D reflectivity image can be

formed by synthesizing a 1-D aperture with a wide-band radar. Accordingly, a 3-D reflectivity image is formed by synthesizing a 2-D aperture.

An imaging algorithm with an arbitrary spatial and frequency diversity can be formulated as in (2.11). The radar instrumentation provides frequency diversity by varying the irradiating frequency and spatial diversity by varying the relative orientation between object and radar. If g(x,y) is the image to be determined from the complex object response measured as a function of frequency and a spatial or angular coordinate θ, the image can be formed by the expression [25]:

( , ) ( , ) exp( 4 / ).

f

g x y G f j rf c

θ θ π

=∑ ∑ ⋅ (2.11)

The exponential function in (2.11) is unique to each x, y, where the dependence on θ,

x and y is implicit in the range r. In addition, the exponential function represents the set of

basis functions which are the complex conjugate of the response to a unit-strength scatterer (i.e. the impulse response) located at a particular (x,y). When the object consists of a single scatterer located at (x,y), the measured data and the basis function for that coordinate have a conjugate phase relation, and therefore sum to a maximum in (2.11). All other basis functions produce a lesser value, ideally zero.

Computational considerations aside, the imaging algorithm for any form of frequency and spatial diversity is simple: the image is formed by the integrated product of the measured data multiplied by the conjugate phase history postulated for a point located at each pixel in the image space. This process can accommodate all forms of frequency and spatial diversities, including the various types of synthetic apertures and near-field geometries. The quality of the image is thus determined by the accuracy to which the diversities are known.

Practical aspects of SAR imaging involve the application of efficient algorithms to numerically evaluate (2.11). Nearly every imaging scheme in current use formulates the basis functions in terms of two-dimensional Fourier transforms that can be implemented using the FFT algorithm and thus be computationally efficient.

As regards the working principles of the existing radar imaging algorithms for satellite and air-borne sensors, a general classification is the following [23]: Polar Format Algorithm, Range Migration Algorithm and Chirp Scaling Algorithm. In all three types of SAR imaging algorithms, when the radar is within the far-field zone of the target, the illuminating wavefront can be considered to be planar and, hence, the processing reduces to an interpolation plus an inverse Fourier transform. Nevertheless, a common requirement for these algorithms is that of producing imagery with the higher possible resolution, particularly in the cross-range direction. In practice, a criterion to assess the optimality of a SAR system is to compare the achieved cross-range resolution in the imagery with the physical dimension of the radar’s antenna, where (2.7) dictates the best achievable resolution of a linear SAR in stripmap mode.

Unfortunately the optimal resolution in cross-range cannot be typically achieved with GB-SAR systems, and in particular in the scenarios monitored during the two field campaigns carried out as part of this PhD thesis’ work. In these cases the radar system has an aperture length of a few metres and is illuminating a scene spanning a few square kilometres located within the far-field of the radar aperture. This is quite a different

scenario than for satellite- and air-borne SAR, which usually have aperture lengths of several kilometres.

In the following sections three focusing algorithms suitable for the image formation in the typical GB-SAR scenarios are described.

2.3.1. Time-domain back-propagation

The first algorithm presented is the time-domain back-propagation algorithm (TDBA), which consists of a direct implementation of (2.11) in the time domain without any kind of optimization in terms of the basis functions before mentioned.

Let us assume a scene consisting of a single point scatterer located at a point p, with cylindrical coordinates (ρ,θ) with respect to the radar position. Let us assume also that a linear radar array acquires the backscatter signal G(f,xa) as a function of two parameters:

the frequency of the signal, f, and the position of the antenna set in the linear array, xa.

Finally, let us assume that the backscatter data is sampled uniformly both in the frequency and the space domain. A measurement with this radar array will give as a result the following two-dimensional matrix of complex values:

,

( _m, _{a n}) ( _c / 2 , _a / 2 _a ),

G f x =G f = f −BW + Δ ⋅f m x = −L + Δ ⋅x n (2.12)

where m = 0, 1, ..., M−1; n = 0, 1, ..., N−1; fc is the radar central frequency; BW is the

frequency bandwidth in the measurement; Δf is the frequency step; M is the number of frequencies measured; Δxa is the spacing of the physical radar array elements or the

movement step used in the linear scan and N is the number of array elements or points measured within the linear scan.

The synthesis of a radar image can be achieved by coherently summing the signal contributions relative to different radar positions and different continuous wave frequencies as in (2.11). This technique is known as wavefront back-propagation. Thus, with the assumed imaging geometry the radar reflectivity at the point p(ρ,θ) can be calculated as follows: 1 1 , 0 0 ( , ) M N ( _m, _{a n}) exp( 4 _{n m}/ ), m n g ρ θ − −G f x j πρ f c = = = ∑ ∑ ⋅ (2.13)

where c is the propagation velocity and

2 2

,

( sin ) ( cos ) .

n xa n

ρ = ρ θ− + ρ θ (2.14)

The synthesis of an entire reflectivity image using (2.13) has associated a high computational cost, which is O(MNM’N’), where M’ and N’ denote the number of pixels in the y and x directions respectively. The algorithm described in (2.13) is a frequency- domain back-propagation algorithm. The formulation of the same algorithm in the time domain, TDBA, is usually preferred as it is computationally more efficient than that in the frequency domain. The algorithm in the time domain estimates the radar reflectivity at

1 , 0 ( , ) N _t( 2 _n/ , _{a n}), n g ρ θ − G t ρ c x = = ∑ = (2.15)

where Gt(t,xa,n) denotes the time domain backscatter data. The computational cost of this

algorithm is approximately O(NM’N’). Furthermore, when the reflectivity is calculated at a limited number of points (and not the whole image area with M’N’ points), the computational cost is significantly reduced and becomes O(NJ), where J is the number of points where the reflectivity is calculated.

2.3.2. Polar format focusing

The polar format algorithm, also known as Range-Doppler algorithm, originates from optical signal processing. It is based on the polar nature of the frequency domain backscatter data, works with motion compensation to a point and as such needs to be used under the far field condition. It formulates the basis functions in terms of two-dimensional Fourier transforms that can be implemented using the FFT algorithm, but requires interpolation prior to the Fourier transform and only partially compensates the range curvature [23, pp.81-111].

Although this algorithm was initially developed to be used with satellite- and air- borne SAR, it can be also be used with GB-SAR scenes under certain constraints. Since the motion compensation is applied to a single point in the image, usually the central one, the image extents need to be smaller than the range to the centre of the scene in order to minimize the geometrical distortions originated in the rest of points in the focused image. Anyhow, a certain amount of geometrical distortion will be present in the focused image and the larger are the image extents, the larger also will be the distortion.

The basic polar formation itself is simply a modification in the data storage format. In effect, the polar formatting operation means recording successive pulses not side-by- side, but in an annular geometry as depicted in Figure 2.2.

The procedure for the image formation starts with a motion compensation block that completely removes the range curvature for the scene central point, while leaving residual range curvatures for points at other distances. Then, range and azimuth interpolation (polar formatting) is applied to reduce those residual range curvatures. A weighting function is afterwards applied to the data in the range dimension and the FFT into that dimension computed. At this point, an autofocus algorithm can be applied in order to reduce the phase errors present after conventional motion compensation and data formatting procedures. Next, a weighting function is applied to the data in the azimuth dimension and the FFT into that dimension computed. The result is a complex focused image in rectangular coordinates.

2.3.3. Pseudo-polar format focusing

Besides the constraint of the image extent with respect to the centre’s scene range, the weakest point of the polar format algorithm is the interpolation prior to the Fourier transform, which is particularly time-consuming.

This interpolation can be avoided by defining a pseudo-polar coordinate system ρ',θ' such that [26]: ' 2 / , ' 2sin / ,_c c ρ ρ θ θ λ = = (2.16)

where λc is the central wavelength transmitted by the radar system. The radar reflectivity

according to the far-field pseudo-polar format algorithm (FPFA) can be then calculated as a series expansion of 2-D Fourier transforms. Under the condition of having a range resolution comparable to the aperture length (L/ΔR ≈ 1, as is the case of the two field campaigns described in Chapter 4), the series expansion can be approximated by the zeroth order term, since higher orders present reflectivity values typically 30 to 40 dB below the zeroth-order term: 0 0 ( ', ') _p( ', ') ( ', '). p g ρ θ ∞ g ρ θ g ρ θ = = ∑ ≈ (2.17)

Considering the two-dimensional data matrix defined in (2.12), the reflectivity image obtained with the FPFA at the point p(ρ',θ') can be expressed as [26]:

0( ', ') FFT2[ ( m, a n, )],

g ρ θ = G f x (2.18)

where FFT2[.] is the two-dimensional Fast Fourier Transform operator and the transformation from the pseudo-polar to the polar grid or to the rectangular grid can be easily done by inverting (2.16). The computational cost of this algorithm is O(NM log2M).

The pseudo-polar format algorithm requires the image scene to be located in the far- field or the radar’s aperture. However, provided this condition is satisfied, the extent of the

scene is only limited by the field of view of a single antenna array element (which implies a highly sub-optimal cross-range resolution, typical of GB-SAR scenes). This condition overcomes the limitation of the polar format algorithm of considering the image extents smaller than the range to the centre of the scene. In addition, the FPFA does not introduce any geometrical distortion at any point of the image, which is only true in the central point of the scene in the case of the polar format algorithm.