**1. Introduction**

Interferometric radiometers are passive imaging instruments whose operation is based on the Van Cittert–Zernike theorem. Their main advantage with respect to other kinds of radiometers is that they do not need moving elements to produce images, as these are entirely formed through data processing of the raw measurements. This technique was firstly proposed for earth observation in Le Vine et al. [1] and Ruf et al. [2]. One of the most representative examples is the Microwave Imaging Radiometer with Aperture Synthesis (MIRAS) [3,4] embarked on board the SMOS (Soil Moisture and Ocean Salinity) satellite [5], launched by the European Space Agency in 2009 and still providing useful geophysical data to the scientific community.

As originally derived for optical signals, the Van Cittert–Zernike theorem states that the mutual intensity of a radiation is the two-dimensional Fourier Transform of the intensity distribution across the source. In microwave radiometry terms, the visibility function is the two-dimensional Fourier transform of the brightness temperature image. An inverse Fourier transform should then allow recovering of the brightness temperature from the calibrated visibility measurements. Nevertheless, in wide field of view instruments, those imaging an extended source covering most of the space in front of the antenna, there are non-negligible effects such as antenna patterns differences, obliquity factor, decorrelation, crosstalk and others that alter substantially this basic relation [6,7].

The visibility function is measured by cross-correlating all pairs of analytic signals collected by individual antennas. Assuming these ones evenly distributed on a fixed structure (as in the MIRAS case, shown in Figure 1) the visibility function becomes sampled at discrete points (*u*, *v*) on a space-limited regular grid [8]. Since the visibility equation is ultimately a Fourier transform, the recovered brightness temperature is affected by aliasing in case the antenna separation fails to meet the Nyquist rate, which is usually the case. In addition, the fact that the visibility is limited in space (due to the instrument's finite size) is equivalent to having a spatial filter that sets the spatial resolution of the recovered map and produces ripples in sharp transitions (Gibbs effect). Moreover, the combined effects of antenna pattern differences and spatial decorrelation make the visibility equation depart from a simple Fourier transform, inducing errors even in the alias-free field of view [9].

**Figure 1.** Microwave Imaging Radiometer with Aperture Synthesis (MIRAS) instrument layout and coordinate definition. Courtesy of AIRBUS Defence and Space [formerly EADS CASA Espacio].

The objective of this paper is to present an alternative image reconstruction algorithm for 2D interferometric radiometers. The algorithm is tested on a set of real measurements acquired by the MIRAS sensor, from which good results consistent with those obtained through the SMOS nominal processing chain are obtained. The paper is organized as follows: The main steps of the algorithm as well as MIRAS characteristics relevant to its application are described in Section 2; the results of the image reconstruction relying on the proposed algorithm are presented in Section 3; the discussion about these latter and those used by the SMOS nominal processing is given in Section 4; and finally the main conclusions are summarized in Section 5.

#### **2. Methods**

The main steps of the proposed image reconstruction algorithm are here presented by referring to the case of a single polarization measurement, and later extended to the case of a full polarimetric one. The algorithm is applied to the specific layout of the MIRAS radiometer (Figure 1). After the introduction of the visibility Equation (Section 2.1) and its conversion into a linear system of equations (Section 2.2), the algorithm develops through regularization of the matrix associated to the linear system of equations (Section 2.3), its inversion (Sections 2.4 and 2.5) and image reconstruction (Section 2.6). Lastly, the case of including the apodization into the processing chain is evaluated in Section 2.7 and the case of full polarimetric measurements is assessed in Section 2.8. Application to the MIRAS radiometer is illustrated throughout all the sections when needed.

#### *2.1. Visibility Equation*

The visibility equation to be used in aperture synthesis radiometry, derived in Corbella et al. [6], is a modified version of the Van Cittert–Zernike theorem to include the effect of the coupling between receivers and to fulfill the principle of energy conservation. In the single polarization case, after canceling the contribution of the receivers' physical temperature (i.e., approach 2 of Corbella et al. [8]) it is given by:

$$V(u,v) = \iint\limits\_{\xi^2 + \eta^2 < 1} \mathcal{T}'(\xi,\eta)e^{-j2\pi(u\xi + v\eta)}d\xi d\eta,\tag{1}$$

where *T* (*ξ*, *η*) is the so-called "modified Brightness Temperature", expressed as:

$$T'(\xi,\eta) = T(\xi,\eta)\frac{F\_k(\xi,\eta)F\_j^\*(\xi,\eta)}{\sqrt{1-\xi^2-\eta^2}\sqrt{\Omega\_k\Omega\_j}}\overline{\mathbf{F}}\_{kj}\left(-\frac{u\_k^z+v\eta}{f\_0}\right),\tag{2}$$

in which *T*(*ξ*, *η*) is the scene brightness temperature, *Fk*,*<sup>j</sup>* are the complex field antenna patterns for the two elements *k* and *j*, Ω*k*,*<sup>j</sup>* is their corresponding antenna solid angles and r˜*kj*( ) is the normalized fringe washing function [6], which depends on the receivers' frequency responses. Only in the case of having identical antennas and neglecting the fringe washing function, the modified brightness temperature (Equation (2)) becomes independent of the specific antenna pair and Equation (1) reduces to a two-dimensional Fourier transform *V*(*u*, *v*) = F [*T* (*ξ*, *η*)].

The domain variables for visibility (*u*, *v*) and brightness temperature (*ξ*, *η*) are defined as

$$\begin{array}{ll} \mu = (\mathbf{x}\_{j} - \mathbf{x}\_{k}) / \lambda\_{0} & v = (y\_{j} - y\_{k}) / \lambda\_{0} \\ \xi = \mathbf{x} / r & \eta = y / r, \end{array} \tag{3}$$

where (*x*, *y*) are the Cartesian coordinates of the observation point located at a distance *r* from the instrument. This latter is assumed to be centered on the origin of coordinates and aligned with the *z* = 0 plane, with the antennas at coordinates (*xk*,*j*, *yk*,*j*) (see discussion below). Finally, *λ*<sup>0</sup> is the wavelength at the center frequency *f*0, and *ξ* and *η* are the director cosines of the observation point with respect to axes *x* and *y* respectively, often expressed as a function of the elevation and azimuth angles (*θ*, *φ*) as *ξ* = sin *θ* cos *φ* and *η* = sin *θ* sin *φ*.

The antenna pattern of a given element *Fk*(*ξ*, *η*) characterizes the electromagnetic field radiated by the whole structure when this particular element is active and no signal is applied to the rest. The antenna position (*xk*, *yk*) in Equation (3) is the point at which its phase pattern is referenced to. To measure the embedded antenna pattern, the whole structure must rotate around a mechanical center of coordinates, so the radiated field becomes proportional to *Fk*0(*ξ*, *η*)*e*−*jkr*/*r* where *k* = 2*π*/*λ* is the wave number and *Fk*0(*ξ*, *η*) the pattern referenced to the coordinate center. The antenna phase pattern can then be referenced to any arbitrary position (*xk*, *yk*, *zk*) by expressing *r* as *r* = *rk* + (*r* − *rk*), with *rk* the distance from this position to the observation point. For large distances, the differential

length can be approximated by *r* − *rk* ≈ *ξxk* + *ηyk* + *γzk* where *γ* = *z*/*r* = cos *θ* is the third director cosine. The antenna pattern with phase referenced to coordinates (*xk*, *yk*, *zk*) is then

$$F\_k(\emptyset, \eta) = F\_{k0}(\emptyset, \eta)e^{-jk(\mathbf{x}\_k \xi + y\_k \eta + z\_k \gamma)}.\tag{4}$$

Using this equation, the "antenna position" (*xk*, *yk*, *zk*) can be chosen arbitrarily as long as the pattern phase is referenced to it. The position of the center of a sphere on which the phase variation of *Fk*(*ξ*, *η*) is minimum is the antenna phase center, but this is not necessarily the best choice. In what follows, without loss of generality, the antenna positions are assumed to be equal to the nominal values and patterns are referenced to them. If antennas are properly designed, these positions should not be far away from their respective phase centers. And this is the case for SMOS.

In consequence the visibility (Equation (1)) is sampled at the the (*u*, *v*) coordinates corresponding to the nominal antenna positions (Equation (3)) using the regular distribution of antennas within the instrument in the *z* = 0 plane, as shown in Figure 1 for the MIRAS case.

### *2.2. Discretization and G-Matrix*

To solve Equation (1) the director cosine domain (*ξ*, *η*) must also be discretized. The visibility equation becomes then a linear system of equations *V* = *GT*, where *G* is a complex matrix whose elements are function of the individual antenna patterns [8], *V* is the vector of visibilities in the (*u*, *v*) space and *T* the vector of brightness temperatures in the director cosine space (*ξ*, *η*). The G-matrix, defined as linear operator relating visibility to brightness temperature, was originally proposed in Tanner et al. [10]. In principle, recovering the brightness temperature requires only inverting the system of equations: *T* = *G*−1*V*. However, the G-matrix just defined happens to be ill-conditioned [11], so the solution is not straightforward.

The G-matrix has as many rows as visibility samples, including those corresponding to zero spacing (single antenna). For an instrument having N antennas there are *N*(*N* − 1)/2 complex rows (MIRAS, with *N* = 69, has 2346) and as many real rows as number of antennas used to measure the antenna temperature (visibility at zero spacing). The current nominal SMOS processing uses only one, but there is a backup mode that uses all or a selected set of antennas for the visibility at the origin [12]. The number of columns of the G-matrix is the total number of grid points (*ξ*, *η*) that fall inside the unit circle defined as *ξ*<sup>2</sup> + *η*<sup>2</sup> < 1.

Using Equations (1) and (2), the elements of the G-matrix are written as

$$\mathbf{G}\_{lm} = \Delta\_{\mathbf{s}} \Delta \eta \frac{F\_{\mathbf{k}}(\mathbf{\tilde{s}}\_{\nu} \eta) F\_{\mathbf{j}}^{\*}(\mathbf{\tilde{s}}\_{\nu} \eta)}{\sqrt{1 - \mathbf{\tilde{s}}^{2} - \eta^{2}} \sqrt{\Omega\_{\mathbf{k}} \Omega\_{\mathbf{j}}}} \widetilde{\mathbf{r}}\_{k\bar{j}} \Big( -\frac{u \mathbf{\tilde{s}} + v \eta}{\gamma\_{0}} \Big) e^{-j2\pi(u \mathbf{\tilde{s}} + v \eta)} \tag{5}$$

where the values of (*u*, *v*) and (*ξ*, *η*) are those of their respective grids and Δ*ξ*Δ*η* is the elementary area (see Section 2.4).

#### *2.3. Hermiticity and Redundant Baselines*

Given the hermiticity property of the visibility *V*(−*u*, −*v*) = *V*∗(*u*, *v*), for each complex row of the G-matrix an additional one can be added by changing the signs of *u* and *v*, provided the corresponding row of the visibility vector is conjugated. This operation has to be performed before dealing with the redundant baselines.

Redundant baselines are those having identical (*u*, *v*) values. Even though they correspond to the same visibility sample, they provide slightly different measurements with respect to each other because of the diverse antenna patterns of the involved elements. Considering two redundant baselines, the corresponding two distinct rows of the discretized visibility equation are:

$$V\_l(\boldsymbol{u}, \boldsymbol{v}) = \sum\_m \mathcal{G}\_{lm} T\_m \quad ; \quad V\_n(\boldsymbol{u}, \boldsymbol{v}) = \sum\_m \mathcal{G}\_{nm} T\_m \tag{6}$$

where the subscripts *l* and *n* refer to two (*k*, *j*) pairs of redundant baselines and the subscript *m* ranges all (*ξ*, *η*) pairs in the unit circle. Note that (*u*, *v*) is the same in both by definition of redundant baselines. These two equations can be averaged to form a third one relating to the visibility of the same (*u*, *v*) point to the scene brightness temperature

$$
\bar{V}\_l(\mu, \upsilon) = \sum\_m \bar{G}\_{lm} T\_{lm} \tag{7}
$$

where *V*¯ *<sup>l</sup>*(*u*, *v*)=(*Vl*(*u*, *v*) + *Vn*(*u*, *v*))/2 and *G*¯ *lm* = (*Glm* + *Gnm*)/2. This last equation can be used for inversion without any loss of information. As a matter of fact, different complete sets of visibility samples *V*(*u*, *v*) are obtained by randomly choosing unique sets of non-redundant baselines. For each one, the corresponding visibility function becomes related to the same brightness temperature image, so the image reconstruction algorithm for each of them should yield the same result in the absence of noise and errors. So only one set of non-redundant visibilities is enough to fully recover the brightness temperature. Averaging all measurements of the same (*u*, *v*) is not needed in the ideal case but has the effect of thermal noise reduction in practice.

The averaging operation must also be performed for the zero spacing visibility, which has a redundancy order equal to the number of antennas used to measure the antenna temperature.

After hermiticity extension and averaging of redundant visibilities, the number of rows of the G-matrix becomes equal to the total unique points in the (*u*, *v*) domain. In MIRAS it is equal to 2791, of which one is real and the rest are complex. These two operations notably improve the G-matrix condition number, acting as a regularization method to make image reconstruction feasible. This method was used in both references [8] and [13], although these references do not mention it explicitly.
