A Comprehensive Comparison of Far-Field and Near-Field Imaging Radiometry in Synthetic Aperture Interferometry

Abstract

Synthetic aperture interferometry (SAI) is a signal processing technique that mixes the signals collected by pairs of elementary antennas to obtain high-resolution images with the aid of a computer. This note aims at studying the effects of the distance between the synthetic aperture interferometer and an observed scene with respect to the size of the antenna array onto the imaging capabilities of the instrument. Far-field conditions and near-field ones are compared from an algebraic perspective with the aid of simulations conducted at microwave frequencies with the Microwave Imaging Radiometer by Aperture Synthesis (MIRAS) onboard the Soil Moisture and Ocean Salinity (SMOS) mission. Although in both cases the signals kept by pairs of elementary antennas are cross-correlated to obtain complex visibilities, there are several differences that deserve attention at the modeling level, as well as at the imaging one. These particularities are clearly identified, and they are all taken into account in this study: near-field imaging is investigated with spherical waves, without neglecting any terms, whereas far-field imaging approximation is considered with plane waves according to the Van–Citter Zernike theorem. From an algebraic point of view, although the corresponding modeling matrices are both rank-deficient, we explain why the singular value distributions of these matrices are different. It is also shown how the angular synthesized point-spread function of the antenna array, whose shape varies with the distance to the instrument, can be helpful for estimating the boundary between the Fresnel region and the Fraunhofer one. Finally, whatever the region concerned by the aperture synthesis operation, it is shown that the imaging capabilities and the performances in the near-field and far-field regions are almost the same, provided the appropriate modeling matrix is taken into account.

1. Introduction

Regardless of the immediate environment of a receiving or radiating aerial, the space surrounding an antenna is usually subdivided into two regions: the near-field and the far-field regions [1], also known as the Fresnel and the Fraunhofer regions, respectively. While there is no clear boundary between these regions, a common and acceptable criterion is that $2D^2/\lambda$ represents a safe far-zone distance, where D is the maximum dimension of an antenna aperture and $\lambda$ is the operating wavelength [2]. At such a Fraunhofer distance, the maximum path-length difference between the contribution from the edge of the aperture and that from the center corresponds to the fraction $\lambda /16$ or to a phase $\pi /8$ radians, as very well illustrated by Figure 1.13 in [3]. However, as suggested by Equation (9) in [4], this criterion might not be sufficiently strict for aperture synthesis imaging applications. This is what is observed with the study reported in this note, which aims at comparing the well-known far-field conditions to the near-field ones in detail and to estimate if this criterion, which has been historically established for elementary antennas outside any imaging context, has to be revisited, or not, for antenna arrays when they are used for synthesis imaging with interferometric data.

Such a comparison between near-field and far-field imaging has already been addressed in synthetic aperture imaging radiometry. However, among the differences that can be found between these two situations, one or the other, or several particularities, are very often neglected. This is not the case in this study, where near-field imaging is considered with spherical waves and without neglecting any term in the modeling equation, whatever the price to pay for taking them all into account in the numerical processing conducted with the aid of a computer. On the other hand, far-field imaging approximation is considered with plane waves according to the Van Citter-Zernike theorem [5,6]. The question of the boundary between these two regions arises in synthetic aperture interferometry when complex visibilities of a scene observed at a given distance from the antenna array are inverted in order to retrieve the brightness temperature distribution of the synthesized field of view. Indeed, as soon as these complex visibilities have been obtained in the near-field region, they have to be inverted with the appropriate resolving matrix, namely the inverse of the modeling matrix of the instrument in near-field imaging conditions, not that in the far-field approximation; otherwise, errors might be expected in the retrieved map.

Regarding synthesis imaging operations with an array of elementary antennas, it is often suggested in the literature to convert near-field visibilities to far-field ones and then to invert the latter with a standard far-field resolving matrix, thus making the calculation and the inversion of the near-field imaging matrix unnecessary. However, the conditions of validity and reliability of this kind of pre-processing are not always mentioned. Although it obviously has many advantages in terms of computational effort, it is well suited only for unresolved point sources, but it is definitively not for extended ones. As argued in this note, these approaches are therefore useless in many cases. This provides clear support for the approach adopted in this study, which makes no approximation in the modeling and no compromise in the processing as far as near-field conditions are concerned. It aims at inverting near-field complex visibilities with the appropriate resolving matrix obtained from the near-field modeling matrix without neglecting any term in the corresponding equation and whatever the computational effort to calculate it and to invert it.

This note is devoted to the comparison between near-field and far-field imaging radiometry in synthetic aperture interferometry from a modeling point of view, from an algebraic point of view, and from an imaging point of view with the aid of numerical simulations. Section 2 gives an overview of the main differences between far-field and near-field synthesis imaging with an antenna array, as illustrated by a pair of elementary antennas with geometrical considerations. The modeling of a synthetic aperture interferometric radiometer is discussed in Section 3, where the integral relation between the brightness temperature distribution T of an observed scene and the complex visibility V for a pair of elementary antennas ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$ is established for near-field imaging conditions when considering spherical waves as well as for far-field ones within the frame of the approximation with plane waves, as it is assumed in the Van Citter–Zernike theorem. At this level of the modeling equations, three differences between these two regions are clearly identified, and their potential impacts on the synthesis imaging are reported. The regularized reconstruction procedure that will be implemented for data processing is summarized in Section 4. To support the comparison, numerical simulations conducted at microwave frequencies with the Microwave Imaging Radiometer by Aperture Synthesis (MIRAS) onboard the Soil Moisture and Ocean Salinity (SMOS) mission are presented in Section 5. (In order to establish some consistency between this note and a previous one addressing a similar comparison under the same algebraic angle but between two observational paradigms in the far-field region only [7], the same array of elementary antennas have been used, although it is well known that MIRAS does not operate in the near-field region.) Although the two imaging conditions use the same interferometric paradigm and share the same goal, there are several differences that deserve attention while performing the comparison. From an algebraic point of view, this is the case of the corresponding modeling matrices, which are both rank-deficient with the same effective rank but with different distributions of their singular values. The impacts of the three differences identified in Section 3 are illustrated and quantified. Complex visibilities of point sources are simulated and inverted, both in the near-field region and in the far-field one. It is shown that the angular resolution [8] does not depend on the region concerned by synthesis imaging. On the contrary, as soon as near-field visibilities are inverted with the far-field resolving matrix, it turns out that the shape of the angular synthesized point-spread function varies with the distance between the point-source and the phase center of the antenna array. A similar analysis is conducted with extended sources to quantify the so-called reconstruction floor error [9]. Finally, the radiometric sensitivity [10] is addressed, and simulations do not reveal any significant difference in the propagation of random Gaussian noise from the complex visibilities to the retrieved brightness temperature maps. For the sake of transparency and pedagogy, this note is completed with an annex, where the complex cross-correlation between two quasi-monochromatic electromagnetic fields is established for spherical waves. Although this study does not claim to define a boundary between the near-field region and the far-field one for synthetic aperture imaging radiometers, it is shown that for the antenna array of SMOS, the common criterion $2D^2/\lambda$ is under-estimating the extent of the near-field, or conversely over-estimating that of the far-field. Indeed, the safe far zone would probably be around $10D^2/\lambda$ for this antenna array when it is used for synthesis imaging with interferometric data.

2. Far-Field vs. Near-Field: A Geometrical Comparison

Since the very first successful experiment of interferometry in radio astronomy conducted after World War II by Ryle and Vonberg [11] with a two-element radio interferometer for studying the Sun at 175 MHz, synthetic aperture interferometry (SAI) techniques have evolved well, but the observing principle has not changed [12,13].

Shown in Figure 1 is the geometry of a one-dimensional SAI radiometer with two elementary antennas ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$ observing a source under far-field conditions, as well as under near-field ones. The two antennas point in the same (Nadir) direction and are separated by the baseline vector ${\mathbf{b}}_{pq}$. In the far-field region, owing to the distance from the source, the incident wavefronts are assimilable to plane waves coming from the same direction $\theta$ for both elementary antennas. As a consequence, the differential path $r_q-r_p$ is equal to $b_{pq}\sin \theta$, where $b_{pq}$ is the magnitude of the baseline vector ${\mathbf{b}}_{pq}$. On the contrary, in the near-field region, incident wavefronts are spherical waves coming from different directions, ${\theta }_p$ for ${\mathcal{A}}_p$ and ${\theta }_q$ for ${\mathcal{A}}_q$, which are no longer equal to $\theta$. The differential path $r_q-r_p$ is here equal to $b_{pq}\sin \mathrm{½}({\theta }_p+{\theta }_q)/\cos \mathrm{½}({\theta }_p-{\theta }_q)$ which tends to $b_{pq}\sin \theta$ as soon as ${\theta }_p$ and ${\theta }_q$ both tend towards $\theta$ whenever far-field conditions are approached, as expected.

As a consequence of this very short preliminary comparison, SAI imaging radiometry might be more complicated under near-field conditions than under far-field ones because, for every baseline ${\mathbf{b}}_{pq}$, it requires a precise calculation of the vectors ${\mathbf{r}}_p(\theta ,\varphi )$ and ${\mathbf{r}}_q(\theta ,\varphi )$ between the elementary antennas and any radiation coming from a direction $(\theta ,\varphi )$. It is also necessary to have an accurate estimation of the local incoming directions $({\theta }_p,{\varphi }_p)$ and $({\theta }_q,{\varphi }_q)$ to evaluate the appropriate attenuation by the elementary voltage patterns of ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$.

3. Instrument Modeling

According to Equation (20) page 573 in [12] and Equation (15.6) page 770 in [13], when neglecting decorrelation effects that can be reduced with spectral sub-banding [14] if narrow-band conditions are not satisfied [15], the complex visibility V measured with a pair of elementary antennas ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$ separated by the baseline vector ${\mathbf{b}}_{pq}$, like those in Figure 1, is given by the integral relation established in the Appendix A at the end of this note in spherical coordinates $\theta$ and $\varphi$:

$${\displaystyle V\left({\mathbf{b}}_{pq}\right)=\underset{source}{\int \int }\frac{{\mathcal{F}}_p^{\phantom{*}}({\theta }_p,{\varphi }_p)}{\sqrt{{\mathsf{\Omega }}_p}}\frac{{\mathcal{F}}_q^{*}({\theta }_q,{\varphi }_q)}{\sqrt{{\mathsf{\Omega }}_q}}T(\theta ,\varphi )\frac{{\mathrm{e}}^{{\displaystyle {\displaystyle \mathrm{j}{k}_o\left(r_p(\theta ,\varphi )-r_q(\theta ,\varphi )\right)}\phantom{\frac{a}{\lambda }}}}}{r_p(\theta ,\varphi )r_q(\theta ,\varphi )}\mathrm{d}S,}$$

(1)

where ${\mathcal{F}}_p$ and ${\mathcal{F}}_q$ are the voltage patterns of the two elementary antennas with equivalent solid angles ${\mathsf{\Omega }}_p$ and ${\mathsf{\Omega }}_q$, T is the brightness temperature distribution of an observed scene, $k_o\equiv 2\pi /{\lambda }_o$ is the central angular wavenumber, and ${\lambda }_o$ is the central wavelength of observation. The angular coordinates $({\theta }_p,{\varphi }_p)$ and $({\theta }_q,{\varphi }_q)$ are local spherical angles (the colatitude and the azimuth, respectively) from which the elementary brightness temperature contribution $T(\theta ,\varphi )\mathrm{d}S$ is seen by the antennas ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$, as illustrated in Figure 1. They are completed by the two distances $r_p$ and $r_q$ between the two antennas and the source element $\mathrm{d}S$ responsible for that elementary contribution to the complex visibility. On introducing the solid angle $\mathrm{d}\mathsf{\Omega }\equiv \mathrm{d}S/r^2$, we therefore have the general expression for V:

$${\displaystyle V\left({\mathbf{b}}_{pq}\right)=\underset{source}{\int \int }\frac{{\mathcal{F}}_p^{\phantom{*}}({\theta }_p,{\varphi }_p)}{\sqrt{{\mathsf{\Omega }}_p}}\frac{{\mathcal{F}}_q^{*}({\theta }_q,{\varphi }_q)}{\sqrt{{\mathsf{\Omega }}_q}}T(\theta ,\varphi )\frac{r^2(\theta ,\varphi )}{r_p(\theta ,\varphi )r_q(\theta ,\varphi )}{\mathrm{e}}^{{\displaystyle {\displaystyle \mathrm{j}{k}_o\left(r_p(\theta ,\varphi )-r_q(\theta ,\varphi )\right)}\phantom{\frac{a}{\lambda }}}}\mathrm{d}\mathsf{\Omega },}$$

(2)

where $r(\theta ,\varphi )$ is the distance from the origin of the reference frame of the antenna array (generally taken at the phase center of the array) to the source element $\mathrm{d}S\equiv r^2\mathrm{d}\mathsf{\Omega }$ responsible for the elementary contribution $T(\theta ,\varphi )\mathrm{d}S$, as illustrated by Figure 1 in [4], Figure 2 in [16], Figure 3 in [17] and Equation (2) in [16] and Equation (2) in [18].

On introducing the angular position variable $\mathit{\xi }$ whose components ${\xi }_1=\sin \theta \cos \varphi$ and ${\xi }_2=\sin \theta \sin \varphi$ are direction cosines in the reference frame of the antenna array, the solid angle $\mathrm{d}\mathsf{\Omega }\equiv \mathrm{d}S/r^2\equiv \sin \theta \mathrm{d}\theta \mathrm{d}\varphi$ turns out to be equal to $\mathrm{d}\mathit{\xi }/\sqrt{1-{\parallel \mathit{\xi }\parallel }^2}$ when accounting for the Jacobian of the transformation from spherical coordinates $(\theta ,\varphi )$ to direction cosines $({\xi }_1,{\xi }_2)$, so Equation (2) now reads

$${\displaystyle V\left({\mathbf{b}}_{pq}\right)=\underset{\parallel \mathit{\xi }\parallel \le 1}{\int \int }\frac{{\mathcal{F}}_p^{\phantom{*}}\left({\mathit{\xi }}_p\right)}{\sqrt{{\mathsf{\Omega }}_p}}\frac{{\mathcal{F}}_q^{*}\left({\mathit{\xi }}_q\right)}{\sqrt{{\mathsf{\Omega }}_q}}T\left(\mathit{\xi }\right)\frac{r^2\left(\mathit{\xi }\right)}{r_p\left(\mathit{\xi }\right)r_q\left(\mathit{\xi }\right)}{\mathrm{e}}^{{\displaystyle {\displaystyle -2\mathrm{j}\pi \frac{r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)}{{\lambda }_o}}\phantom{\frac{a}{\lambda }}}}\frac{\mathrm{d}\mathit{\xi }}{\sqrt{1-{\parallel \mathit{\xi }\parallel }^2}}.}$$

(3)

It is often read in the literature that expression (3) is valid only in near-field imaging conditions, but it is exactly the opposite: this expression is the most general one. Indeed, so far, no assumption has been made on the region of the electromagnetic field to be concerned so that it is valid in the near-field region as well as in the far-field one. However, from a numerical point of view, it is admitted that Expression (3) is not the easiest one to implement. This is why it is often used in the literature with approximations or assumptions to make it simpler.

Referring back to Figure 1, let us now assume that the source under observation is far enough from the antenna array so that $r_p\equiv r_q\equiv r$, ${\theta }_p\equiv {\theta }_q\equiv \theta$ and ${\varphi }_p\equiv {\varphi }_q\equiv \varphi$ (or equivalently ${\mathit{\xi }}_p\equiv {\mathit{\xi }}_q\equiv \mathit{\xi }$). The differential path $r_q-r_p$ is therefore nothing but the scalar product ${\mathbf{b}}_{pq}·\mathit{\xi }$ and as a consequence of these assumptions, expression (3) reduces to

$${\displaystyle \tilde{V}\left({\mathbf{b}}_{pq}\right)=\underset{\parallel \mathit{\xi }\parallel \le 1}{\int \int }\frac{{\mathcal{F}}_p^{\phantom{*}}\left(\mathit{\xi }\right)}{\sqrt{{\mathsf{\Omega }}_p}}\frac{{\mathcal{F}}_q^{*}\left(\mathit{\xi }\right)}{\sqrt{{\mathsf{\Omega }}_q}}T\left(\mathit{\xi }\right){\mathrm{e}}^{{\displaystyle {\displaystyle -2\mathrm{j}\pi \frac{{\mathbf{b}}_{pq}·\mathit{\xi }}{{\lambda }_o}}\phantom{\frac{a}{\lambda }}}}\frac{\mathrm{d}\mathit{\xi }}{\sqrt{1-{\parallel \mathit{\xi }\parallel }^2}}.}$$

(4)

Expression (4) is the far-field expression of the complex visibility, also known in the literature as the Van Cittert–Zernike theorem [5,6]. The reasons for deriving such an approximation of the general expression (3) under some assumptions listed in page 774 of [13] are due to the observing conditions experienced in radio astronomy, which might be different from those encountered in Earth observation. This is particularly true with ground or airborne instruments that may sometimes need to be used in the Fresnel area, whereas spaceborne ones operate naturally in the Fraunhofer region. Moreover, contrary to (3), Expression (4) is very simple to implement as it is easier to know the baseline ${\mathbf{b}}_{pq}$ with some accuracy than the distances $r_p$ and $r_q$ for every viewing direction $\mathit{\xi }$.

When closely inspecting Expressions (3) and (4), it turns out that near-field conditions have three impacts on the modeling equation with respect to far-field approximation:

• The very first dissemblance between (3) and (4) is found in the phase term of the exponential where the difference $r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)$ is approximated by the scalar product ${\mathbf{b}}_{pq}·\mathit{\xi }$ in the far-field approximation;
• The second difference is the significant presence of the fraction $r^2\left(\mathit{\xi }\right)/r_p\left(\mathit{\xi }\right)r_q\left(\mathit{\xi }\right)$ in (3), whereas it is absent in (4) since it tends toward 1 in the far-field approximation;
• The third and last dissemblance is found in the antenna voltage patterns ${\mathcal{F}}_p$ and ${\mathcal{F}}_q$ where local directions ${\mathit{\xi }}_p$ and ${\mathit{\xi }}_q$ of the local spherical angles $({\theta }_p,{\varphi }_p)$ and $({\theta }_q,{\varphi }_q)$ are assumed to be equal to $\mathit{\xi }$ in the far-field approximation.

As a consequence of the last item of this list, it is important to note that with an array of identical elementary antennas with the same voltage pattern $\mathcal{F}$, if synthesis imaging under far-field conditions can take into account this property to simplify the imaging process, it is no longer possible under near-field conditions. Indeed, for every viewing direction $(\theta ,\varphi )$, the attenuation of the brightness temperature $T(\theta ,\varphi )$ by the voltage pattern $\mathcal{F}$ varies from one elementary antenna to another because the local incoming directions $({\theta }_p,{\varphi }_p)$ and $({\theta }_q,{\varphi }_q)$ are no longer equal. In the words of Zhang et al. [19] “that means for different baselines the source distribution is different, thus the Fourier transform cannot be used for the inversion.” From a modeling point of view, it is as if each antenna had a different diagram from the other one, according to the identities ${\mathcal{F}}_p(\theta ,\varphi )\equiv \mathcal{F}({\theta }_p,{\varphi }_p)$ and ${\mathcal{F}}_q(\theta ,\varphi )\equiv \mathcal{F}({\theta }_q,{\varphi }_q)$. It would therefore not be surprising to see the effect of this disparity of the elementary voltage patterns [20] on the properties of the modeling matrix of the instrument, as well as on the qualities of the inversions.

The previous effect cannot be seen in [21] nor in [22] as a “simplify visibility function ignoring the antenna voltage pattern” is used. Taking into account the antenna voltage pattern at the modeling level is not a guarantee of being able to capture this effect. This is the case in the recent studies reported in [23,24], where Equation (4) in the former and Equation (1) in the latter are both suffering from the same assumption that is not justified nor documented by the authors: the local incoming directions ${\mathit{\xi }}_p$ and ${\mathit{\xi }}_q$ corresponding to the local spherical angles $({\theta }_p,{\varphi }_p)$ and $({\theta }_q,{\varphi }_q)$ are assumed to be identical (and equal to $\mathit{\xi }$). This is in contradiction with expression (2), which is often referred to as the near-field complex visibility in the literature, like in Equation (1) of [16], which is very frequently cited in this imaging domain. Moreover, it is clear from Figure 1 that it cannot be the case when one claims to address near-field imaging. What is true with local spherical angles must remain true with direction cosines, as is the case in the present study with expressions (2) and (3). Finally, unlike the assumption made in [17,23] where the product of the distances between the two antennas and the source element $\mathrm{d}S$ is approximated by $r^2$ and therefore the corresponding fraction is neglected in the expression of the complex visibilities, the present note aims at using the complete expression (3) without neglecting any of the three terms listed previously and without introducing any reductive hypothesis, as is often the case when expending the difference $r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)$ in Taylor series [25] like in [19,24,26].

The usefulness of approximations or reductive assumptions in the near-field modeling is questionable, especially since there is no reason not to consider all the terms found in the general expression (3). This is also true for antenna arrays with an aperture so wide [27] that some baselines are in the near-field region while the others are in the far-field one, according to the criteria provided by the Fraunhofer distance [2]. As a consequence, this study focuses only on near-field and far-field synthesis imaging as illustrated by the modeling Equations (3) and (4) and by the corresponding linear operators $\mathbf{G}$ and $\tilde{\mathbf{G}}$ introduced in the next section.

4. Regularized Inversion

According to a recent revisit of (4), accounting for the mutual coupling between elementary antennas, the brightness temperature distribution $T\left(\mathit{\xi }\right)$ has to be substituted with the difference $T\left(\mathit{\xi }\right)-T_r$, where $T_r$ is the physical temperature of the receivers [28]. In any case, this constant term does not affect the zero-spacing baselines, and it can be removed by measuring a flat target [29]. As a consequence, without any loss of generality, after the discretization of the integral found in (4) over an appropriate sampling grid in the direction of the cosine domain [30], the relationship between the complex visibilities and the brightness temperature distribution of the scene under observation can be written in algebraic form:

$$\tilde{V}=\tilde{\mathbf{G}}T$$

(5)

where $\tilde{\mathbf{G}}$ is the linear modeling matrix of the imaging radiometer using SAI in the far-field approximation. Likewise, the general relationship (3) can be written in a very similar algebraic form:

$$V=\mathbf{G}T$$

(6)

where $\mathbf{G}$ is the general linear modeling matrix of the imaging radiometer using SAI, contrary to what is implied in the title of [31] as the difference $r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)$ is there expended in Taylor series, thus resulting in an approximated linear modeling matrix, not a general one.

It is now well established that these modeling matrices are rank-deficient [32] as a consequence of a number of unknowns (the values of the brightness temperature distribution over the sampling grid) being larger than the number of data (the complex visibilities). In both paradigms, the effective rank of the modeling matrix is equal to the number of Fourier frequencies in the experimental frequency coverage $\mathcal{H}$ of the antenna array. Indeed, such imaging radiometers are band-limiting devices as the extent of the baselines ${\mathbf{b}}_{pq}$ is limited to $\mathcal{H}$. As a consequence, the corresponding inverse problems have to be regularized in order to provide unique solutions to the linear systems (5) and (6). For obvious reasons of comparison, in this study, the same regularization is implemented in either case. Otherwise, the differences that might be observed could be attributed to the regularization itself and not to the modeling matrix (and, finally, not to the concerned electromagnetic region). Among the many regularization methods that can be found in the literature, the minimum-norm one is widely used in SAI [33]:

$${\tilde{T}}_r=\underset{T}{min}{\parallel T\parallel }^2\mathrm{s}.\mathrm{t}.\tilde{\mathbf{G}}T=\tilde{V}.$$

(7)

The extension to the general case does not raise any difficulty:

$$T_r=\underset{T}{min}{\parallel T\parallel }^2\mathrm{s}.\mathrm{t}.\mathbf{G}T=V.$$

(8)

Numerical implementations are available in many algorithmic libraries and with many programming languages. In both cases, the reconstructed map is obtained through the computation of the pseudo-inverse [34] of the corresponding modeling matrix:

$${\tilde{T}}_r={\tilde{\mathbf{G}}}^{+}\tilde{V},$$

(9)

$$T_r={\mathbf{G}}^{+}V.$$

(10)

In either case, the pseudo-inverse is computed with the aid of a truncated singular value decomposition [35], where only the largest significant singular values are kept during the inversion.

Finally, in order to filter out the Gibbs effects [36] caused by the sharp cut-off in the Fourier domain due to the limited extent of the frequency coverage $\mathcal{H}$, these maps have to be damped by an appropriate apodization (or windowing) function W [37]:

$${\tilde{\mathcal{T}}}_r=W★{\tilde{T}}_r,$$

(11)

$${\mathcal{T}}_r=W★T_r,$$

(12)

where ★ is the convolution operator. These maps have to be compared to

$$\mathcal{T}=W★T,$$

(13)

which is the “ideal” temperature map to be reconstructed and apodized with the same window W. Indeed, they cannot be compared to T because they are not at the same angular resolution.

5. Numerical Simulations and Comparison

The numerical simulations discussed in this section have been performed with the antenna array of the imaging radiometer MIRAS [38] onboard the SMOS [39] mission. The reader will surely notice that this work and the conclusions drawn from the results presented in this section have evidently no impact on the operational processing of SMOS data as MIRAS is always operating in the far-field region and never in the near-field one. This antenna array has been chosen for feeding these numerical simulations just because it is a concrete and tangible one with available voltage patterns measured for every elementary antenna [40], as well as for establishing some consistency with a previous note [7], as already mentioned in the Section 1. It should also be noted that the study reported in this note is missing real interferometric data from both electromagnetic regions with the same instrument to make the comparison complete. Such a possibility is currently not available, but it will be in the near future, as the French space agency (CNES) and the CESBIO are together developing an L-band antenna array equipped with 32 elementary antennas and with a longest baseline D of about 1 m intended to be used in a wide range of distances h (typically from 5 m to 40 m) suspended from a crane over an experienced and controlled field, as was performed few years ago with MIRAS breadboard [41]. Although the main goal of this forthcoming antenna array will not be to compare far-field and near-field imaging conditions but rather to confront synthetic aperture interferometry and digital beam forming [7], the flexibility of the instrument will allow such a comparison.

The SMOS mission of the European Space Agency (ESA) is devoted to Soil Moisture and Ocean Salinity remote sensing from space [39]. It was launched in November 2009, and since then, it has been operational and has provided accurate L-band brightness temperature maps [42] with a spatial resolution between $\sim 25$ km and $\sim 60$ km, depending on the position in the field of view. These maps are used for retrieving surface soil moisture (SM) [43], as well as sea surface salinity (SSS) [44]. The very principle of the imaging radiometer MIRAS is that of aperture synthesis, a technique initially developed for radio astronomy where the signals collected by each pair of interferometric arrays of elementary antennas are cross-correlated [13]. It is therefore not surprising that the antenna array of MIRAS was inspired by the Very Large Array (VLA) [45] for the geometry of the array and by the Cosmic Background Imager (CBI) [46] for the regular spacing between the antennas. As shown in Figure 2, it is a Y-shaped array equipped with 69 equi-spaced elementary antennas. As a consequence of the longest baseline $D=6.75$ m and of the central wavelength of the instrument $\lambda =21.21$ cm, the Fraunhofer distance $2D^2/\lambda$ is here about 430 m [2]. Unless it is explicitly precised, in the remainder of this note, this antenna array will be placed in the near-field region at a distance $h=100$ m from a source. However, unlike SMOS, which is pointing at the Earth with a constant tilt angle of $32.5^{\circ }$ in the orbital plane [38,39], here, this offset is reduced to zero so that the number of pixels of the sampling grid in the synthesized field domain belonging to the Earth surface is constant over a wide range of values for h. This is essential for conducting studies where this distance is a key parameter; otherwise, the differences that might be observed could be attributed to this varying number of pixels and not to the modeling equations themselves. Referring back to the geometry of Figure 1 and to illustrate it concretely, shown in Figure 3 are the distances $r_p\left(\mathit{\xi }\right)$ from every antenna ${\mathcal{A}}_p$ of this array to that source in the Nadir direction $\mathit{\xi }=(0,0)$, as well as the corresponding local angles ${\theta }_p\left(\mathit{\xi }\right)$. For this particular direction pointed at by every antenna, the distance to the source can be increased here by 8 cm, that is to say, larger than $\lambda /3$, which is not negligible. Likewise, whereas in far-field conditions, the colatitude angle corresponding to that direction is by definition equal to $0^{\circ }$ for every antenna, at a distance $h=100$ m, it varies from one antenna to the other and can be as large as $2.3^{\circ }$. As a consequence, these conditions are clearly not those of far-field imaging.

5.1. Singular Values

Referring back to (3) and (4), in both cases, the number of complex visibilities is equal to the number of baselines ${\mathbf{b}}_{pq}$. For SMOS, it is equal to $69\times (69-1)=4692$. In addition to these interferometric measurements, 3 out of the 69 receivers are also operating as reference radiometers to provide precise measurements of the average brightness temperature of the observed scene [47]. The total number of measurements is therefore equal to $4692+3=4695$. However, when accounting for the redundant baselines along the three arms of the Y-shaped array, the number of Fourier frequencies in the corresponding star-shaped frequency coverage $\mathcal{H}$ is only equal to $2790+1=2791$, including the zero frequency.

The distributions of the 4695 singular values of the modeling matrices $\mathbf{G}$ and $\tilde{\mathbf{G}}$ are shown in Figure 4. According to the floating point relative accuracy of Matlab ($\epsilon \approx 2·{10}^{-16}$), the 1904 smallest singular values of $\tilde{\mathbf{G}}$ could be considered equal to zero [48]. The number of remaining strictly positive singular values is therefore equal to $4695-1904=2791$, which is also the number of Fourier frequencies in the frequency coverage $\mathcal{H}$. This property remains true in the distribution of the singular values of the modeling matrix $\mathbf{G}$; however, the gap between the two groups of singular values is much smaller: here less than one order of magnitude instead of 13 in the far-field conditions.

As a consequence, when computing the pseudo-inverse with the aid of the truncated singular value decomposition, the impact of the singular values discarded prior to inversion might not be the samem and a larger floor error might be observed in the near-field reconstructions compared to that in the far-field ones. Finally, the propagation of random noise is governed by the conditioning of the modeling matrix, which is the ratio between the largest and the smallest singular values kept in the inversion. It turns out that here, this ratio is about $11.2$ for $\tilde{\mathbf{G}}$ and $12.6$ for $\mathbf{G}$. Although this ratio is an upper bound for the actual amplification factor of random noise, one can expect very similar behaviors of the two matrices.

5.2. Far-Field vs. Near-Field

Referring back to Section 3, near-field conditions have three impacts on the modeling equation when comparing expressions (3) and (4). Shown in Figure 5 and Figure 6 are examples of the spatial distribution of the error made when the difference $r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)$ is approximated by the dot product ${\mathbf{b}}_{pq}·\mathit{\xi }$. If the shape of this error is quite simple for pairs of elementary antennas located on the same arm, this is no longer the case when the two antennas are located on two different arms. With regards to the magnitude of this error, it is a not negligible fraction of the wavelength as high as $7.5$ cm compared to the 3 mm accuracy required for the baselines coordinates at the central wavelength of SMOS $\lambda =21.21$ cm. As a consequence, such an approximation should have on strong impact on the brightness temperature retrievals.

Likewise, shown in Figure 7 are examples of the spatial distribution of the multiplicative real-valued fraction $r^2\left(\mathit{\xi }\right)/r_p\left(\mathit{\xi }\right)r_q\left(\mathit{\xi }\right)$ of (3). In the far-field approximation (4), this fraction is assumed to be equal to 1. In the near-field case shown here, the deviation from 1 could be as high as $4\%$ for the shortest baselines at the extremities of the arms, but it decreases as soon as the antennas are closer to the phase center of the array. However, this is a not negligible fraction as the brightness temperature seen by the antenna array for these baselines from the corresponding directions is directly affected by this deviation. Although this term does not affect the phase of the complex visibilities, the impact on the magnitude is strong enough to affirm that it should not be neglected in near-field synthesis imaging.

Finally, shown in Figure 8 are examples of the spatial distribution of the magnitude and the phase of the difference ${\mathcal{F}}_p\left(\mathit{\xi }\right){\mathcal{F}}_q^{*}\left(\mathit{\xi }\right)-{\mathcal{F}}_p\left({\mathit{\xi }}_p\right){\mathcal{F}}_q^{*}\left({\mathit{\xi }}_q\right)$. The error introduced when ignoring local viewing directions ${\mathit{\xi }}_p$ and ${\mathit{\xi }}_q$, assuming $\mathit{\xi }={\mathit{\xi }}_p={\mathit{\xi }}_q$ in that part of the modeling equation involving the voltage pattern of the elementary antennascould be as high as $0.4$ dB and $4^{\circ }$. Here again, this is large enough to affirm that local viewing directions at the level of the antenna’s voltage pattern should be taken into account in near-field synthesis imaging.

Keeping in mind that the objective of this study is to determine when the far-field approximation can be used and when near-field synthesis imaging has to be performed, shown in Figure 9, Figure 10 and Figure 11 are the variations of the largest value of the previous spatial distributions with the observing distance h from the phase center of the antenna array. With regards to the difference ${\mathbf{b}}_{pq}·\mathit{\xi }-(r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right))$, it turns out that for the Franhaufer distance $2D^2/\lambda \simeq 430$ m, the largest value is here about $\lambda /16$, as expected, according to the common criterion for separating the Fresnel region from the Fraunhofer one. With respect to the fraction $r^2\left(\mathit{\xi }\right)/r_p\left(\mathit{\xi }\right)r_q\left(\mathit{\xi }\right)$, the largest deviation from the value 1 for the same Franhaufer distance is slightly less than $1\%$. Finally, regarding the impact of the local viewing directions ${\mathit{\xi }}_p$ and ${\mathit{\xi }}_q$ onto the antenna voltage patterns ${\mathcal{F}}_p$ and ${\mathcal{F}}_q$, the largest error introduced at the amplitude level is less than $0.1$ dB for the same Franhaufer distance.

As a consequence of this preliminary analysis, none of the three terms identified in Section 3 can be seen as a clear indicator of the boundary between the near-field region and the far-field one. Although the approximation of $r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)$ by ${\mathbf{b}}_{pq}·\mathit{\xi }$ leads to an error very close to that associated with the Fraunhofer distance, according to Equation (9) in [4], the threshold of $\lambda /16$ is questionable for imaging purposes with an antenna array: so why would a phase error of $\pi /8$ at complex visibilities level be acceptable? Referring back to the principle of synthesis imaging, the appropriate threshold has to be found at the level of the imaging process in the comparison made between the reconstructed maps ${\tilde{\mathcal{T}}}_r$ and ${\mathcal{T}}_r$ with the ideal one $\mathcal{T}$. This is precisely what is carried out in the next subsections, first with a point source, then with an extended one.

5.3. Point Source

Complex visibilities V and $\tilde{V}$ have been simulated according to (3) and (4) for the same brightness temperature distribution $T\left(\mathit{\xi }\right)$ of a point-source, which is equal to 0 everywhere except in the boresight direction of the antenna array. They are shown with polar plots on Figure 12. Whatever the pair of elementary antennas ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$, in such a special case of brightness temperature distribution, the far-field visibilities $\tilde{V}\left({\mathbf{b}}_{pq}\right)$ are real-valued, as expected according to (4) as soon as the only contribution to the integral is for $\mathit{\xi }=(0,0)$. This is no longer the case with the near-field visibilities $V\left({\mathbf{b}}_{pq}\right)$ simulated when the antenna array is placed at a distance $h=100$ m from the point source. Although the instrument is still the same, the phase of these complex visibilities varies in the range $\pm {130}^{\circ }$, as a consequence of the effects of the geometry illustrated in Figure 1. Furthermore, simulations conducted with (3) for different values of h have shown that this range of phases is larger as the distance between the antenna array and the point-source gets closer. On the contrary, it tends toward zero as this distance increases and reaches the far-field region, as expected and as shown in Figure 13. For the Fraunhofer distance of $2D^2/\lambda =430$ m, it turns out that this range of phases is about $\pm {30}^{\circ }$. Referring back to the far-field approximation, such a phase translates into a baseline error as large as $\lambda /12\simeq 18$ mm. This is 6 times larger than the 3 mm baseline uncertainty of SMOS, which corresponds to a phase error of the order of $5^{\circ }$ that is reached here only from a distance h greater than 2 Km, that is to say, about $10D^2/\lambda$. As a consequence, there is no doubt that inverting the near-field complex visibilities V with the pseudo inverse ${\tilde{\mathbf{G}}}^{+}$ of the far-field modeling operator will lead to large errors and artifacts not only for $h=100$ m, which is evidently in the near-field region, but also for any value of $h\lesssim 10D^2/\lambda$, including of course for the Fraunhofer distance $2D^2/\lambda$.

The previous complex visibilities $\tilde{V}$ and V have been inverted according to (9) and (10). Shown in Figure 14 is the reconstructed point-spread function of SMOS when the antenna array is placed at a distance $h=100$ m from the source in the near-field region, and the visibilities V shown in red in Figure 12 are inverted with the pseudo-inverse of the near-field modeling operator $\mathbf{G}$ or with that of the far-field one $\tilde{\mathbf{G}}$. Both have to be compared to the point-spread function obtained when the array is in the far-field region and the corresponding visibilities $\tilde{V}$ shown in green in Figure 12 are inverted with ${\tilde{\mathbf{G}}}^{+}$. In every case, a rectangular windowing function W has been used so that the reconstructed point-spread function is not apodized and the angular resolution, as estimated by the full-width at half-maximum of the main lobe, is the finest one that could be obtained from this antenna array.

As soon as the complex visibilities are inverted with the appropriate operator, the one corresponding to the concerned electromagnetic region, the retrieved point-spread functions exhibit the expected imaging qualities. More precisely, ${\mathbf{G}}^{+}V$ in the near-field case and ${\tilde{\mathbf{G}}}^{+}\tilde{V}$ in the far-field one are perfectly identical, and they evidence the predicted values for many figures of merit (full-width at half-maximum of the main lobe, level of the side lobes, etc.). On the contrary, when inverting near-field visibilities with a far-field operator, it turns out that ${\tilde{\mathbf{G}}}^{+}V$ is completely different from the expected point-spread function. Clearly, the far-field imaging operator is responsible for large aberrations in the near-field point-spread function. This is exactly what happens when far-field imaging conditions are assumed to be met when they are not. As the energy transported by the electromagnetic field is the same because it does not depend on the distance but only on the brightness temperature of an observed scene, the norm of the retrieved maps is also the same. However, it is clear that the spatial distribution of this energy is distorted so that the shape of the point-spread function has significantly changed. It is no longer that of a main lobe surrounded by six-fold patterns with an alternating succession of positive and negative side lobes, but it is now similar to the shape of the antenna array itself. Finally, as illustrated by Figure 15, this shape also depends on the location of the point source in the field of view synthesized by the instrument.

Shown in Figure 16 are reconstructed point-spread functions of SMOS when the antenna array is placed at a distance $h=2D^2/\lambda$, $3D^2/\lambda$, $4D^2/\lambda$, $5D^2/\lambda$, $10D^2/\lambda$ and $20D^2/\lambda$ from a source and when the complex visibilities V are inverted with the pseudo-inverse of the far-field modeling operator $\tilde{\mathbf{G}}$. As predicted earlier in this subsection when looking at the range of the phase of the near-field visibilities V reported in Figure 13, large errors and artifacts are observed here at the level of the reconstructed point-spread functions for any value of $h\lesssim 10D^2/\lambda$. Although the Mean Squared Error (MSE) has many attractive features, it is not well suited for quantifying the sameness, or on the contrary the difference, between images [49]. The Structural SIMilarity (SSIM) index [50] is a well-known method for measuring the similarity, or the dissimilarity, between two images. This index can be viewed as a quality measure of one of the images being compared, provided the other image is considered of perfect quality. Here, the reference image is the reconstructed point-spread function shown in Figure 14 when the antenna array is in the far-field region and the visibilities $\tilde{V}$ are inverted with the far-field modeling operator ${\tilde{\mathbf{G}}}^{+}$. It is compared to every reconstructed point-spread function, like those shown in Figure 16, obtained when the array is placed at a distance of h from the source and when the corresponding visibilities V are nevertheless inverted with the far-field modeling operator ${\tilde{\mathbf{G}}}^{+}$. Shown in Figure 17 are the variations in the SSIM index obtained for this comparison. For the Fraunhofer distance $2D^2/\lambda =430$ m, this index is only about $0.82$, but it is larger than $0.99$ for any value of $h\gtrsim 10D^2/\lambda$.

Finally, it is frequently read in the literature (Equation (5) in [16], Equation (9) in [19] or Equation (2) in [51], for example) that near-field complex visibilities V can be corrected from their distortion to obtain far-field visibilities $\tilde{V}$ by simply subtracting the near-field phase and adding the far-field one for every baseline ${\mathbf{b}}_{pq}$ according to

$$V\left({\mathbf{b}}_{pq}\right){\mathrm{e}}^{{\displaystyle {\displaystyle +2\mathrm{j}\pi \frac{r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)}{{\lambda }_o}}\phantom{\frac{a}{\lambda }}}}{\mathrm{e}}^{{\displaystyle {\displaystyle -2\mathrm{j}\pi \frac{{\mathbf{b}}_{pq}·\mathit{\xi }}{{\lambda }_o}}\phantom{\frac{a}{\lambda }}}}\simeq \tilde{V}\left({\mathbf{b}}_{pq}\right).$$

(14)

However, referring back to Equations (3) and (4), only a point source can be concerned by such a correction because these modeling equations are integral relations so that the complex visibility for any antenna pair ${\mathcal{A}}_p$ and ${\mathcal{A}}_q$ does not depend on the variable $\mathit{\xi }$. It should be read between the lines in [16,19] that $\mathit{\xi }$ is here nothing but the viewing direction from a single point source; more precisely, the brightness temperature distribution $T\left(\mathit{\xi }\right)$ in (3) and (4) is assumed here to be null everywhere except for that particular value of $\mathit{\xi }$. To be complete with this correction, it is also assumed that the second and the third impact out of the three listed in Section 3 are neglected. This is exactly what should be understood in the words of [16] “in the case that only path delay phase errors are taken into account…if we take into account a single punctual source.” Such an approach is therefore useless since observed scenes are not always reduced to a single point source. Moreover, applying this phase correction to an extended source or to a collection of point sources is simply not possible, unless the same phase correction is applied for every viewing direction in the field of view, which would assume that it does not change from one direction to another. This is exactly what is suggested in [16], where “near-field visibilities from an extended source can be partially corrected just by focussing the near-field visibilities to the pixel at boresight.” This is not always true as illustrated here by Figure 5, where, for three pairs of elementary antennas, the difference between ${\mathbf{b}}_{pq}·\mathit{\xi }$ and $r_q\left(\mathit{\xi }\right)-r_p\left(\mathit{\xi }\right)$ can be as large as $\pm \lambda /3$ between the boresight direction and other viewing directions in the field of view. Thus, for the corresponding baselines, if the phase correction (14) is correctly introduced at boresight, an error as large as $\pm 2\pi /3$ is not corrected in the complex visibility for the contributions from other directions. The same remark holds for Figure 6, where, for three other pairs of elementary antennas, this phase error is as large as $\pm 2\pi /9$, which is still not negligible. As a consequence, owing to the role played by the phase of the complex visibilities in imaging synthesis operations, large errors should be expected in the regions, or for the point sources, far from the boresight direction. This is exactly what is shown in Figure 18, where the brightness temperature distribution of three point sources located at $\mathit{\xi }=(0,0)$ and at $\mathit{\xi }=(\pm 0.5,0)$ is observed when the antenna array is in the near-field region at a distance $h=100$ m from the first source at boresight (the two other sources are therefore at a distance $h/cos{30}^{\circ }\simeq 115$ m from the phase center). The near field visibilities V are corrected according to (14) with $\mathit{\xi }=(0,0)$, and the corresponding “almost” far-field visibilities $\tilde{V}$ thus obtained are inverted with the far-field modeling operator ${\tilde{\mathbf{G}}}^{+}$. As expected, the reconstructed point-spread function at boresight is identical to the expected one already shown in Figure 14. On the contrary, as soon as the point source moves away from the very center of the field of view, the reconstructed point-spread function is still suffering from strong aberrations, although the situation is obviously slightly better than that shown in Figure 15 without the phase correction (14). Accounting for the previous considerations and referring back to Section 4, it is recommended to invert near-field complex visibilities V with the appropriate imaging operator ${\mathbf{G}}^{+}$ rather than trying to convert them in far-field ones $\tilde{V}$ to be inverted with ${\tilde{\mathbf{G}}}^{+}$, contrary to what is suggested in [51] where it is claimed that “after near-field correction, far-field image retrieval tools can be used without any additional modification.”

5.4. Extended Source

Near-field imaging is no exception to field aliasing; as a consequence, unlike what is often found in the literature, it has to be studied and compared to what is observed and very well known when far-field imaging conditions are met. Here again, complex visibilities V and $\tilde{V}$ have been simulated according to (3) and (4) for the same brightness temperature distribution $T\left(\mathit{\xi }\right)$ of an extended source with a constant temperature. Both have been inverted with the appropriate operator: ${\mathbf{G}}^{+}$ for V according to (10) and ${\tilde{\mathbf{G}}}^{+}$ for $\tilde{V}$ according to (9).

Two cases have been considered, depending on the extent of the source. The first one corresponds to a situation that has been widely simulated with a circular [16] or a square [24] flat target that never extends beyond the field of view synthesized by the antenna array. The brightness temperature distribution $T\left(\mathit{\xi }\right)$ of an extended source set to 100 K in the field of view synthesized by the antenna array of Figure 2 and to 0 K outside this hexagon is shown in Figure 19, together with the corresponding complex visibilities V and $\tilde{V}$.

Shown in Figure 20 are the brightness temperature maps ${\tilde{\mathcal{T}}}_r$ and ${\mathcal{T}}_r$ retrieved after inverting the complex visibilities $\tilde{V}$ and V of Figure 19 according to (9) and (10). In every case, a Blackman windowing function W has been used to damp the reconstructed maps ${\tilde{T}}_r$ and $T_r$. As the brightness temperature T of the observed scene is here set to 0 K outside the field of view synthesized by the antenna array, field aliasing is not an issue because it is not expected. Indeed, there is no reason to observe it in such conditions where the brightness temperature from the six neighbors to the synthesized field of view does not contribute to the complex visibilities. As a consequence, in the absence of any modeling error or noise, there is no significant difference observed between $\mathcal{T}=W★T$ and the two retrieved maps ${\tilde{\mathcal{T}}}_r$ and ${\mathcal{T}}_r$, as expected. In both cases, the floor error as illustrated here by the Root Mean Squared Error (RMSE) [52] in the whole synthesized field of view is much less than $0.01$ K, and it does not vary with the distance h between the antenna array and the source in the boresight direction.

The second case considered here corresponds to a situation that has not been widely simulated nor reported so far: this is the case where the brightness temperature distribution of the observed scene extends beyond the field of view synthesized by the antenna array. In such a situation, field aliasing has to be observed, depending on the geometry of the antenna array and on the spacing between the elementary antennas. The brightness temperature distribution $T\left(\mathit{\xi }\right)$ of an extended source set to 100 K in the whole space in front of the antenna array of Figure 2 is shown in Figure 21, together with the corresponding complex visibilities V and $\tilde{V}$.

Shown in Figure 22 are the brightness temperature maps ${\tilde{\mathcal{T}}}_r$ and ${\mathcal{T}}_r$ retrieved after inverting the complex visibilities $\tilde{V}$ and V of Figure 21 according to (9) and (10). Here again, in every case, a Blackman windowing function W has been used to damp the reconstructed maps ${\tilde{T}}_r$ and $T_r$. As the brightness temperature T of the observed scene is not set to 0 K outside the field of view synthesized by the antenna array, field aliasing is expected from the six neighbors to that hexagon. This is exactly what can be observed in Figure 22, where only the central part of the hexagonal field of view is free from any aliasing artifact. However, even in the absence of any modeling error or noise, small differences can be observed between $\mathcal{T}=W★T$ and the two retrieved maps ${\tilde{\mathcal{T}}}_r$ and ${\mathcal{T}}_r$. More precisely, here, the floor error in that alias-free region of the synthesized field of view is about $0.14$ K for the far-field case, whereas it is of the order of $0.53$ K for the near-field one. In real situations, this floor error can be efficiently reduced down to a much lower level with the introduction of an artificial scene [53], as close as possible to the observed one, whose visibilities are removed prior to inversion and whose brightness temperature distribution is added back after inversion. However, referring back to Figure 4, such a difference at this raw level of reconstruction was already expected when comparing the singular values of $\mathbf{G}$ and $\tilde{\mathbf{G}}$ and especially those discarded prior to the pseudo-inversion, which are quite different. Finally, shown in Figure 23 are the variations of this floor error with the observing distance h from the phase center of the antenna array. Although it decreases very rapidly to reach a plateau with a value very similar to that obtained in far-field conditions, it is still very difficult to find a value that makes a clear separation between near-field and far-field regions. With this figure of merit, one can again at most consider a transition zone from the Fresnel region to the Fraunhofer one between $2D^2/\lambda$ and $10D^2/\lambda$.

5.5. Radiometric Sensitivity

The real and the imaginary parts of the complex visibilities V and $\tilde{V}$ have been blurred by an additive Gaussian noise $\Delta V$ and $\Delta \tilde{V}$ with standard deviation ${\sigma }_{\Delta V\phantom{\tilde{V}}}={\sigma }_{\Delta \tilde{V}}=0.1$ K. This is a typical value for SMOS, which should translate in the brightness temperature domain with a radiometric sensitivity on the order of $2.5$ K [10]. This is what is observed in Figure 24 where the radiometric sensitivity in the boresight direction $\mathit{\xi }=(0,0)$ is equal to $2.55$ K in far-field imaging conditions and to $2.57$ K in near-field ones. Moreover, the radiometric sensitivity map does not depend on the distance between the antenna array and the scene observed by the instrument. Referring back to Figure 4, this is an expected behavior when comparing the singular values of $\mathbf{G}$ and $\tilde{\mathbf{G}}$, and especially the role played by those kept in the pseudo-inversion, which are almost identical.

6. Conclusions

A comprehensive comparative study between near-field and far-field imaging radiometry in synthetic aperture interferometry has been conducted. At the modeling level, the integral relation between the brightness temperature distribution of an observed scene and the corresponding complex visibilities has been established for near-field imaging conditions when considering spherical waves as well as for far-field ones with plane waves. Three differences between the two regions have been clearly identified and their potential impacts on the synthesis imaging have been reported. The first dissemblance is at the level of the difference between the distances from any pixel in the field of view to any pair of elementary antennas that is estimated in the far-field approximation by the scalar product between the corresponding baseline and the viewing direction. The second difference is the significant presence of a multiplicative fraction that tends towards 1 as soon as the far-field conditions are met. The third dissemblance is at the level of the antennas’ voltage pattern where local viewing directions vary from one antenna to another in near-field imaging, whereas they are assumed to be equal in the far-field approximation. All these differences have been taken into account throughout this study, and their impact has been quantified. From an algebraic point of view, the properties of the corresponding linear operators have been compared with an emphasis on their singular values. As expected, the modeling matrices are both rank-deficient with the same effective rank. However, they exhibit significantly different distributions of their singular values: those kept in the inversion are very similar, whereas those discarded prior to the inversion are not. As a consequence, the propagation of random noise does not vary from far-field to near-field conditions, as it is governed by the former singular values. This is not the case for the reconstruction floor error, which depends on the latter ones. At the imaging synthesis level, numerical simulations have been conducted for the antenna array of SMOS in order to compare near-field and far-field imaging performances when observing a point source or when looking at an extended source. In the first situation, no significant difference has been observed provided the visibilities are inverted with the appropriate resolving matrix. Nevertheless, when inverting near-field visibilities with the far-field operator, strong errors have been observed in the reconstructed point-spread functions, as expected. In the second case, when the extended source is limited to the field of view synthesized by the antenna array, no difference has been found between near-field and far-field imaging: the reconstruction floor error is close to zero in both imaging conditions. On the contrary, when the brightness temperature distribution of an observed scene extends beyond the synthesized field of view, field aliasing is observed and the reconstruction floor error is at a higher level for near-field imaging than it is for the far-field one. Finally, with regard to the propagation of random noise, no difference has been found between near-field and far-field inversions, as expected.

Although it is admitted that there is no clear boundary between the Fresnel and the Fraunhofer regions, the well-known distance $2D^2/\lambda$ is often used as a criterion for separating the near-field region from the far-field one. This criterion has been historically established for elementary antennas with diameter D and operating at a wavelength $\lambda$ outside any imaging context. It was not therefore unreasonable to take advantage of this comparative study to check how this distance is consistent with a collection of elementary antennas working all together for synthesis imaging from interferometric data and when D is now the longest baseline of the antenna array. This is why parametric studies with the distance between the phase center of the antenna array and an observed source in the boresight direction have been conducted with the idea of confronting different figures of merit to this historical criterion. In any case, no sharp separation has been observed between the near-field and the far-field regions: one can at most consider a transition zone between the Fresnel and the Fraunhofer regions. However, as already suggested in the literature, it turns out that $2D^2/\lambda$ is not sufficiently strict for aperture synthesis imaging applications since it is always found to be among the lowest values of this transition interval between the two regions. Referring back one last time to Figure 13, Figure 17 and Figure 23, a criterion around $10D^2/\lambda$ might be more appropriate to preserve the imaging performances. However, it would be very presumptuous to claim to change the definition of this criterion solely on the basis of a comparative study conducted with a single antenna array. The lesson learned from this parametric study should be to always invert the complex visibilities of a synthetic aperture imaging radiometer with the most appropriate resolving matrix and to consider the far-field approximation only after quantifying its impact on the imaging performances such as the angular resolution, the floor error and the radiometric sensitivity.