## 4 Observation Simulations of GMR, GeoSTAR, and GIMS
### Simulation of the Observed Brightness Temperature TA of GMR, GeoSTAR, and GIMS
#### 4.2.2 Geostationary Synthetic Thinned Aperture Radiometer (GeoSTAR)  
GeoSTAR is the first SAIR concept for GEO-MW atmospheric sounding. As illustrated in Figure 6a, GeoSTAR consists of a Y-shape array of 312 50-60 GHz antennas and receivers, where three linear arrays are offset 120\({}^{\circ}\) from each other. All of the antennas are pointed toward Earth. Unlike RA, SAIR measures the upwelling brightness temperature TB in the spatial frequency domain. This is achieved by the complex cross-correlations of the signals received by receiver pairs (so-called baseline pair) in a sparsely filled 2-D array, which is called the visibility function. Each baseline pair forms an interferometer, which  
Figure 5: Simulated 2-D antenna pattern of RA and synthetic AF of SAIR at 50.3 GHz. **(a)** GMR, **(b)** GeoSTAR with a rectangular window, **(c)** GeoSTAR with a [PERSON] window, **(d)** GIMS with a rectangular window, and **(e)** GIMS with a Blackman window. The simulated 1-D antenna pattern and synthetic AF of SAIR. **(f)** GMR, **(g)** GeoSTAR with a rectangular window, **(h)** GeoSTAR with a [PERSON] window, **(i)** GIMS with a rectangular window, and **(j)** GIMS with a [PERSON] window. measures a particular spatial harmonic of the brightness temperature distribution across the field of view. The spatial harmonic depends on the spacing between the antennas and the wavelength of the measured radiation. The visibility function is related to the brightness temperature of the observed scene as follows [44]:  
\[V(u,v)=\iint\limits_{\xi^{2}+\eta^{2}\leq 1}T^{\prime}(\xi,\eta)r\left(-\frac{u \xi+v\eta}{f_{0}}\right)e^{-j2\pi(u_{\xi}^{2}+v\eta)}d\xi d\eta \tag{4}\]  
where \(V\) is visibility function, \((\xi,\eta)=(\sin\theta\cos\varphi,\sin\theta\sin\varphi)\) are the direction cosine coordinates, \((u,v)=\left(D_{x}/\lambda_{0},D_{y}/\lambda_{0}\right)\) are the baseline components normalized to the wavelength \(\lambda_{0}=c/f_{0}\), with \(f_{0}\) being the center frequency of the instrument, and \(r(\tau)\) is the fringe washing function accounting for spatial decorrelation effects. \(T^{\prime}(\xi,\eta)\) is the modified brightness temperature, which is defined as follows when assuming identical antenna patterns:  
\[T^{\prime}(\xi,\eta)=\frac{TB(\xi,\eta)}{\sqrt{1-\xi^{2}-\eta^{2}}}\frac{ \left|\text{F}_{n}(\xi,\eta)\right|^{2}}{\Omega} \tag{5}\]  
where \(TB\) is the scene's actual upwelling brightness temperature distribution, \(\text{F}_{n}\) is the modified antenna electric field pattern, and \(\Omega\) is the antenna equivalent solid angle. The modified brightness temperature \(T^{\prime}(\xi,\eta)\) and the visibility function \(V(u,v)\) are related by the Fourier transform in the ideal case, as indicated by Equation (4). Each baseline pair can sample two visibility points in the spatial frequency domain (so-called UV plane) by virtue of the conjugate symmetry property. Figure 6a shows the distribution of elements in GeoSTAR's aperture plane, and Figure 6c shows the resulting samples in the UV plane. Figure 6b,d is the corresponding partial zoomed-in views. The smallest element spacing is 3.5 wavelengths (2.1 cm) for the 50-60 GHz band. The visibility function is essentially the 2-D Fourier transform of the 2-D brightness temperature field. The measurements made at these UV plane samples determine the coefficients of a 2-D Fourier series expansion of the brightness temperature field, and the field can then be reconstructed by an inverse Fourier transform as follows:  
\[T^{\prime}(\xi,\eta)=\iint W(u,v)V(u,v)e^{j2\pi(u_{\xi}^{2}+v\eta)}dudv \tag{6}\]  
where \(W(u,v)\) is a window function used for weighting the visibility samples prior to inverse Fourier transformation for sidelobe suppression. In the actual case, the visibility functions are sampled at discrete points, and the scene brightness temperature distribution can be reconstructed by general inverse discrete Fourier transform (DFT) or inverse fast Fourier transform (FFT) routines.