## 1. Introduction

[2] 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, *T*_{A} is the antenna temperature and *T*_{R} is the receiver noise temperature.

[3] 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: 〈*i*_{1}*i*_{2}〉 and 〈*q*_{1}*i*_{2}〉, that can be interpreted respectively as the real and imaginary parts (*V*_{r}, *V*_{i}) 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 (*u*_{12}, *v*_{12}) = (*x*_{2} − *x*_{1}, *y*_{2} − *y*_{1})/λ.

[4] 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 Δ*f* = *f*_{0} − *f*_{LO}, *f*_{0} being the center frequency, *f*_{LO} the local oscillator frequency and Λ() is the triangular function: Λ(*x*) = 1 − ∣*x*∣ for ∣*x*∣ < 1 and 0 elsewhere. Note that σ_{r,i}^{2} is scene-dependent through *V*_{r,i}. However, in the case Δ*f* ≥ *B*/2, the variance σ^{2} = σ_{r}^{2} + σ_{i}^{2} will recover (1) and will be equal on each visibility sample.

[5] 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.

[6] 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) = (*sin*θ *cos*ϕ, *sin*θ *sin*ϕ) 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 = *d*^{2} for rectangular sampling, with *d* being the distance between antennas). The other parameters are as follows: *N*_{V} is the number of measured visibility samples, and α_{W} is a coefficient that takes into account the noise reduction due to window *W*_{mn} = *W* (*u*_{mn}, *v*_{mn}) used to weight the visibility samples [*Camps*, 1996]

where the window coefficients are such that *max W*_{mn} = *W*(0, 0) = 1, and 0 ≤ *W*_{mn} ≤ 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)} + *T*_{Rk} and similarly when *l* ≡ *n*: *V*_{(l,l)} = *V*_{(0,0)} + *T*_{Rl}. In the case that receiver *k* ≡ *m* and also *l* ≡ *n*, since *V*_{(0,0)} = *T*_{A}(5) reduces to (1) and it is consistent with σ^{2} = σ_{r}^{2} + σ_{i}^{2} 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:

[7] 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).

[8] 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).