# Image Simulator for One Dimensional Synthetic Aperture Microwave Radiometer

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2 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August where v T (θ) is the voltage seen at the antenna terminals, θ is the direction of incident radiation and ω is the observation frequency. <, > is the expectation operator. Noting that v T (θ)and v T (θ ) are independent since they are generated by independent thermal sources of zero mean, the expectation collapse to a single integral. Moreover, T(θ) is proportional to (v T (θ, t)v T (θ, t d sin θ /c)) and separate equation () into its real and imaginary components gives (where d is the baseline distance) V I (d) = V Q (d) = T(θ)cos(2πfd sinθ c )dθ T(θ)sin(2πfd sinθ c )dθ V I, V Q are respectively the real and imaginary components of complex visibility functions. If it is thinned arrays of antenna contained many pairs of interferometer viz. d = n u(n=,, 2... N, N is corresponding to maximal antenna separation), equation (2) becomes the Fourier harmonics of the FOVs. But this analysis assumes identical antenna patterns and identical gain and phase parameters through each correlator channel. Actually, the instrument doesn t meet such assumptions. Differences in correlator transfer functions cause two correlators measuring the same visibility function to have different outputs. In the whole, considering all the effects of the instrument, substitute a Dirac delta function for T b (θ) to evaluate for all values of θ [3]. The result is the impulse response of the system for a given baseline n and angle θ. g(n, θ ) = c ij δ(θ θ )p i (n, θ)p j(n, θ)exp( 2π nd sin θ)dθ (3) λ where p i (θ) and p j (θ) are respectively antenna patterns of antenna i and j. c ij = c ij e jϕij is complex gain of correlator between channel i and channel j. Phase errors between channels is critical in correlation radiometer and it can be divided in two kinds. One is phase errors between channels, the other is phase errors on the quadrature in a particular channel.to discriminately treated the effects of antenna arrays and receiver to the visibility functions, the equation (2) becomes, (2) V I (n) = V Q (n) = g a (n, θ)g I r(n, θ)t(θ)dθ g a (n, θ)g Q r (n, θ)t(θ)dθ (4) where g a (n, θ) represents antenna pattern, g r (n, θ) represents complex gain between channels, superscript I, Q represent odd and even lines G matrix. Measuring g(n, θ) in discrete angle θ, then V (n) = G a G r T (5) where V is visibility function samples, T is m-d column vector, G r = (g r, g r2,..., g rm ), G a = diag(g a, g a2,...g am ). Variable m is determined by special sample and generally let m>2n+, we take m>3n. The effect of antenna arrays is invariable with time and only depend on the structure of the antenna arrays and can be measured. So next we only consider the effect of the receiver. Rewriting equation (6) with lumping in-phase and quadrature-phase together yields V = GT (6) where V is a vector with p elements, p=2n+ is results of all correlators, T is vector with m elements, G is (p*m) matrix. Brightness temperature image reconstructed is as follows T = G V (7) where G = G t [ GG t ] is pseudo-inverse of G and T is the estimated brightness image.

4 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August d= d=deltau d=2deltau d=3deltau Figure 2: basis functions for complex correlators Figure 3: spatial impulse response of the system Figure 4: response of a point source T δ = G G (8) where the measured visibilities act as both the G-matrix and the visibilities that are to be inverted. The measured result closely resembles the identity matrix shown as Figure 3. The nadir spatial impulse response is the center row or column of the impulse response matrix. Figure 4 shows the nadir spatial impulse response of the synthetic aperture radiometer. 3. Two Dimensional G-matrix For two dimensional synthetic aperture microwave radiometer, the visibility function and the brightness temperature are related as [], [2]: V (u, v) = K T(ξ, η)e j2π(uξ+vη) dξdη (9) ξ 2 +η 2 T B (ξ, η) T(ξ, η) = ξ2 η F n(ξ, η) 2 () 2 where (u, v) is the spacing between the two antennas in wavelengths; T B (ξ, η) is the brightness temperature; T(ξ, η) will be called the modified brightness temperature; / ξ 2 η 2 is the obliquity factor for a thermal source lying on a plane; F n (ξ, η) is the normalized antenna voltage pattern; ξ = sin θ cosφ, η = sin θ sinφ, are the directing cosines. In the same way, image simulator can also simulate a point source in ξ η plane. Assume there are pixels in the ξ η plane, G-matrix is G(u n, v n, ξ m, η m ) where n is baseline number corresponding to spatial frequency sample; m = = 496 is pixel number in the FOV, i.e. it is equivalent

5 528 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August to 64 one dimensional G-matrixes, and there are 64 samples in each one dimensional direction. Hence, it is obvious that the computation is huge. Conclusion Calibration of the synthetic aperture radiometer is the most important issue after the instrument is developed. It reflects the true relation between the output (visibility) and the brightness temperature at the input. It also effects the image reconstruction algorithm. In this paper, a measurement set up called image simulator is present. This subsystem can expediently simulate one point noise source in arbitrary direction in the FOV and measure the G-matrix to describe the instrument exclude antenna arrays and using for image reconstruction. The simulator was tested with X band 8 channel synthetic aperture microwave radiometer, the experiment testify the theory in section two. REFERENCES. Ruf, C. S, C. T. Swift, A. B. Tanner and D. M. Le Vine., Interferometric Synthetic Aperture Microwave Radiometry for the Remote Sensing of the Earth, IEEE Trans. GRS, Vol. 26, 597-6, Le Vine, D. M, Andrew, J. Griffis, Calvin T. Swift, Thomas J. Jackson., ESTAR: A Synthetic Aperture Microwave Radiometer for Remote Sensing Applications, Proceedings of the IEEE., Vol. 82, No. 2, 787-8, Tanner, Alan B., Calvin T. Swift., Calibration of a Synthetic Aperture Radiometer, IEEE. Transactions on Geoscience and Remote Sensing, Vol. 3, No., , Le Vine, D. M., et al., Passive Microwave Remote Sensing with the Synthetic ApertureRadiometer ESTAR during the Southern Great Plane Experiments, Proceedings of IGRASS 98, Liu, Hao, Wu Ji, Wu Qiong, Some Further Consideration for The Image Retrieving of Synthetic Aperture Radiometer, Proceedings of 24 th Asian Conference on Remote Sensing (ACRS), Wu, Qiong, Wu Ji, Liu Hao, Development of Image Simulator for Synthetic Aperture Microwave Radiometer, Journal of Electronics and Information Technology, Vol. 26, Suppl (Chinese), 24.

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