This study deals with the calculation of the distribution of radiometric errors (“noise maps”) in the images formed by a two-dimensional aperture synthesis radiometer, specifically the MIRAS and the HUT-2D instruments. MIRAS is a spaceborne Y-shaped interferometric radiometer that will be boarded in the European Space Agency's SMOS Earth Explorer Mission. The HUT-2D is a U-shaped airborne interferometric radiometer under development by the Helsinki University of Technology. The study focuses on the known fact that the cross-correlation of the visibility noise does depend on the shape of the sensor. We investigate this dependence for the Y and the U configurations, taking into account also the results of previous studies on redundancy and tapering.
 Total power radiometers estimate the antenna temperature by integrating the received signal during a certain time τ. Due to the noisy nature of the source and the receiver's own noise, as well as the finite integration time, the uncertainty in the estimate is given by [Tiuri, 1964]:
where B is the pre-detection noise bandwidth, TA is the antenna temperature and TR is the receiver noise temperature.
 Correlation radiometers consist of two receiver channels whose outputs are multiplied and averaged. If the receivers are connected to the same antenna and the same polarization, they perform a total power measurement at that polarization. If they are connected to the same antenna, but in two orthogonal polarizations, they perform a measurement of the third and fourth Stokes parameters. However, if they are connected to antennas separated a certain distance, they perform interferometric measurements. If the receivers perform in-phase i and quadrature q demodulation, there are two cross-correlation results available: 〈i1i2〉 and 〈q1i2〉, that can be interpreted respectively as the real and imaginary parts (Vr, Vi) of the “visibility” sample [Camps, 1996]. This measurement represents a sample of the Fourier transform of the target's radiation intensity distribution at a spatial frequency (u12, v12) = (x2 − x1, y2 − y1)/λ.
 The spatial frequency of the Fourier sample is given by the distance between the antennas relative to the wavelength. As a matter of fact, by placing two antennas at appropriate distances in a plane, a set of visibilities can be gathered and the source's radiation distribution can be recovered by a two-dimensional inverse Fourier transform. The uncertainty on the correlator's output in each separate visibility measurement is [Ruf et al., 1988; Camps, 1996]
where Δf = f0 − fLO, f0 being the center frequency, fLO the local oscillator frequency and Λ() is the triangular function: Λ(x) = 1 − ∣x∣ for ∣x∣ < 1 and 0 elsewhere. Note that σr,i2 is scene-dependent through Vr,i. However, in the case Δf ≥ B/2, the variance σ2 = σr2 + σi2 will recover (1) and will be equal on each visibility sample.
 The amount of noise in the brightness temperature image estimate is the same as the total noise on the visibility samples since the DFT preserves the second norm. If the noise is uncorrelated in each visibility sample, the rms noise in the image will be uniformly distributed over the image pixels.
 Aperture synthesis radiometers consist of several correlation receivers at various antenna distances providing a complete set of visibilities. In practice this is implemented by connecting a set of receivers to a switch which connects them to a correlator bank so that each receiver's output is cross-correlated with that of every other receiver. The inverse discrete Fourier transform (DFT) of these measured visibilities is the modified brightness temperature which is defined as the brightness temperature multiplied by the antenna pattern and divided by cos(θ), the so called obliquity factor. The brightness temperature distribution is given in directing cosine coordinates (ξ, η) = (sinθ cosϕ, sinθ sinϕ) where ϕ is the azimuth and θ is the angle measured from normal with respect to the plane of the instrument. Assuming that the errors in the visibility samples (2) are uncorrelated, the standard deviation in the modified brightness temperature map is uniformly distributed and is given by
where A is the area of one pixel in the image recovered by the two-dimensional inverse DFT of the measured visibilities ( for hexagonal sampling and A = d2 for rectangular sampling, with d being the distance between antennas). The other parameters are as follows: NV is the number of measured visibility samples, and αW is a coefficient that takes into account the noise reduction due to window Wmn = W (umn, vmn) used to weight the visibility samples [Camps, 1996]
where the window coefficients are such that max Wmn = W(0, 0) = 1, and 0 ≤ Wmn ≤ 1. However, since all the visibilities are estimated from the same wavefront, their noise is cross-correlated. Taking any group of four receivers k, l, m and n, the level of cross-correlation between the p-th visibility measured by first two V(k,l) receivers and the q-th visibility V(m,n) measured by the other two receivers is given by [Bará et al., 2000]
where the receiver noise must be added whenever k ≡ m: V(k,k) = V(0,0) + TRk and similarly when l ≡ n: V(l,l) = V(0,0) + TRl. In the case that receiver k ≡ m and also l ≡ n, since V(0,0) = TA(5) reduces to (1) and it is consistent with σ2 = σr2 + σi2 of (2). This nonzero cross-correlation will cause the Fourier transformed noise in the image to be unequally distributed. This study deals with the analysis of noise maps in two different sensor configurations Y- and the U-shaped, which provide different (u, v) spatial frequency samples and different level of redundancy:
 1. MIRAS (Microwave Imaging Radiometer by Aperture Synthesis) is an L-band instrument, that will be the single payload of ESA'S SMOS Earth Explorer Opportunity Mission. It consists of three arms arranged in Y shape, each arm accommodating 21 receivers plus 3 spare ones (Figure 1a). Visibility samples are arranged in a hexagonal grid. Since the antenna spacing is 0.875 λ, larger then the Nyquist criterion for hexagonal sampling there is some level of aliasing and six aliases partially overlap in the brightness temperature image [Camps et al., 1997]. Receivers positioned along the same arm and spaced the same distance measure the same visibility sample. In principle, these redundant measurements can be averaged to reduce noise (Figure 1b).
 2. HUT-2D is an L-band air-borne instrument under development at the Laboratory of Space Technology of the Helsinki University of Technology, Finland. The sensor consists of 36 receiving elements arranged in a U-shape. Each of the segments accommodates 12 receiving elements (Figure 1c). Visibility samples are arranged in a rectangular grid. Since the antenna spacing is 0.7λ, larger than the Nyquist criterion for rectangular sampling (0.5λ), there is also some level of aliasing and four aliases partially overlap in the brightness temperature image. The samples produced by the receiver pairs lying along the same arm produce redundant measurements. In the case of a U-shaped array also the measurements by the receiver pairs located at the two parallel arms are redundant (Figure 1d).
2. Noise Cross-Correlation Matrix
 The noise cross-correlation in the visibility samples (5) can be arranged in the cross-correlation matrix CV = [σpσqρpq] = [〈ΔVkl ΔVmn*〉] of size NR2 × NR2 where NR is number of receivers. To evaluate this matrix, knowledge of the configuration (Y or U) of the particular sensor is necessary: the cross-correlation between the visibilities measured by the kl and the mn receiver pairs can be found if the visibilities of receiver pairs km and ln are known.
 Based on the previous result, the variance in the estimate of the brightness temperature can be calculated. The measured visibilities are related to the brightness temperature by the two-dimensional discrete Fourier transform where F is the Fourier transform operator. In actual calculations, the G-matrix [Ruf et al., 1988] can be used in place of F and similarly to the procedures used in spectra estimation, the results are improved by tapering the visibilities prior to the transformation. For slowly varying targets like soil moisture or ocean salinity, tapering with Blackman function is likely to bring best results. In Bará et al.  some other tapers were also studied.
 The ordering of the visibilities in the C matrix is dictated by the array configuration, whereas the ordering in the F matrix is dictated by the samples in the spatial frequency plane. Nevertheless, the columns of F can be interchanged so that they conform to those of C. Let these permutations be represented by a T. Because both configurations (Y and U) are redundant the T matrix is not square and the redundant measured baselines must be averaged: the columns of the matrix corresponding to redundant baseline are multiplied by 1/r where r is the level of redundancy (Figures 1b and 1d). Otherwise each row of T contains ones. These values will be substituted by the appropriate tapering coefficients and the matrix will be marked as W.
 If a visibility noise vector is a realization of independent unit-variance complex circular Gaussian noise, it can be transformed into cross-correlated noise vector by the Cholesky factorization of the cross-correlation matrix RHR = C so that
 Now, the corresponding realization of noise on image pixels is
The visibility samples are Hermitian, so half of the visibility samples are not measured, but substituted by their conjugate values. The cross-correlations in the image can finally be calculated as
because (identity matrix), as it is independent, circular and with unit variance. The noise map of the image pixels can be found by placing the the diagonal values of CT on the (ξ, η) grid.
 The cross-correlation of two visibilities (5) can be decomposed as
so the C matrix can be divided into four parts
where the CVV depends only on the target, the CRR only on the receiver noise (and is diagonal) and only the CVR + CRV depends on both and is affected by the configuration of the sensor (U- or Y-shaped).
3. Point Source Case
 A point source located at the instrument boresight would ideally produce equal visibilities. Then, by (8): the CVV component, after Fourier transformation, would cause in the reconstructed image high variance at the position of the point source and the CRR a constant variance across the image due to the uncorrelated receiver noise. The mixed components cause an increase of variance along the symmetry axes of the sensor (every 60° for the Y-shaped sensor and each 90° for the U-shaped sensor Figure 2). Both images on Figure 2 were calculated with the Blackman window [Bará et al., 1998].
4. Extended Sources
Figure 3 shows the aliased version of a uniform source and the variance when imaged by a Y- and a U-shaped sensor. The visibilities were tapered with a two-dimensional Blackman function of circular symmetry for Y and by the cross-product of two one-dimensional Blackman windows in the case of the U-shaped configuration. The result is that on top of a strong constant term the variance basically follows the intensity distribution of the source. This confirms the result obtained for the one-dimensional case which was presented by Bará et al. . The discrepancy between the profile of the variance and that of the source is caused by the sensor's configuration expressed by terms CVT and CTV of the cross-correlation matrix.
 An example of more realistic radiation intensity profile is given in Figure 4, showing a region around the Baltic sea, as seen from 790 km height by a Y-shaped interferometric radiometer tilted 32°. Therefore there are significant aliases only along the three upper edges of the hexagon. The ∼3 K aliases of the cold sky along the lower hexagon edges are not noticeable in the figures. The changes in noise distribution apparently follow the colder sea regions and the hotter continental areas.
 From (8) one can see the behavior when the ratio TB/TR in the sensors' channels decreases as the image is getting overshadowed by the receivers' noise. When the TB/TR ratio is low, the receiver noise dominates, and the image variance tends to be evenly distributed over the pixels. On the other extreme, when TR → 0, it is the source's noisy character which dominates and the variance in the image follows the distribution of the source's radiation intensity (Figure 5).
 The redundancy existing in the Y- or U-configurations offers a way to reduce the noise on some samples by averaging. However, the improvement depends on the correlation level between the averaged measurements. In Section 1 an approximation for the expected improvement was presented in form of the αW coefficient (equation (4)). Under the assumption of uncorrelated visibility noise, if the m − n baseline, tapered by Wmn coefficient is rmn-fold redundant, the noise improvement is
Tables 1 and 2 show the approximated values and the exact values based on the target with constant radiation intensity distribution. The best predicted improvements are ∼22% for MIRAS with Blackman taper and ∼13% for the HUT-2D. The rather modest improvements and the excellent agreement between the exact values and the estimated values under the assumption of uncorrelated noise are due to the relatively low redundancy as indicated in Table 3.
 The effect of cross-correlation of the visibilities in the two-dimensional aperture synthesis radiometers is difficult to follow analytically due to the fact that it is determined by the sensor configuration. In this work numerical calculations have been performed to evaluate the effect in two cases: the Y-shaped MIRAS sensor and the U-shaped HUT-2D sensor.
 The combination of real (noisy) sensor and extended target yields in the image: (1) a large uniform noise distribution with standard deviation σΔT due to the uncorrelated noise content, plus (2) a nonconstant distribution of noise with standard deviation σΔT (ξ, η) due to the correlated noise content that follows closely, but not exactly, the radiation intensity distribution of the source. Some deviations between the two distributions are due to mixing of the noise originating from the target and that of the instrument (CVR, CRV terms in equation (8)).
 The approximated formulas for improvement by averaging redundant baselines are in excellent agreement with the values computed taking into account noise cross-correlation. The improvement is rather modest, about ∼22% for MIRAS and ∼13% for HUT-2D, which opens a possibility for a trade-off in design of the cross-correlator board by leaving the redundant baselines unmeasured. High accuracy image reconstruction algorithms (from the visibilities to brightness temperature maps) will need to take into account the cross-correlation of the visibilities in their noise models.
 This work has been partially supported by grant of the Spanish government CICYT TIC99-1050-C03-01.