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Abstract-A semi-empirical model of the ensemble-averaged differential Mueller matrix for microwave backscattering from bare soil surfaces is presented. Based on existing scattering models and data sets measured by polarimetric scatterometers and the JPL AirSAR, the parameters of the co-polarized phase-difference probability density function, namely the degree of correlation and the co-polarized phase-difference , in addition to the backscattering coefficients 0 , 0 and 0 , are modeled empirically in terms of the volumetric soil moisture content and the surface roughness parameters and , where = 2 , is the rms height and is the correlation length. Consequently, the ensemble-averaged differential Mueller matrix (or the differential Stokes scattering operator) is specified completely by 0 , 0 , 0 , , and Index Terms-Bare soil surfaces, differential Mueller matrix, microwave backscattering, semi-empirical model.

The recent interest in radar polarimetry has led to the development of several calibration techniques to retrieve the Mueller matrix of a distributed target from the multipolarization backscatter measurements recorded by a radar system. Because a distributed target is regarded as a statistically uniform random medium, the measurements usually are conducted for a large number of independent samples (usually spatially independent locations), from which the appropriate statistics characterizing the elements of the Mueller matrix can be derived. Existing calibration methods rely on two major assumptions. The first is that the illuminated area of the distributed target is regarded as a single equivalent point target located along the antenna's boresight direction, and that the statistics of the scattering from all of the measured equivalent point targets (representing the spatially independent samples observed by the radar) are indeed the same as the actual scattering statistics of the distributed target. The second assumption pertains to the process by which the actual measurements made by the radar for a given illuminated area are transformed into the scattering matrix of that area. The process involves measuring the radar response of a point calibration target of known scattering matrix, located along the boresight direction of the antenna, and then modifying the measured response by a constant, known as the illumination integral, when observing the distributed target. The illumination integral accounts for only magnitude variations of the illuminating fields. Thus, possible phase variations or antenna crosstalk variations (between orthogonal polarization channels) across the beam are totally ignored, which may compromise the calibration accuracy. To rectify this deficiency of existing calibration techniques, a new technique is proposed with which the radar polarization distortion matrix is characterized completely by measuring the polarimetric response of a sphere over the entire main lobe of the antenna, rather than along only the boresight direction. Additionally, the concept of a "differential Mueller matrix" is introduced, and by defining and using a correlation-calibration matrix derived from the measured radar distortion matrices, the differential Mueller matrix is accurately calibrated. Comparison of data based on the previous and the new techniques shows significant improvement in the measurement accuracy of the copolarized and cross-polarized phase difference statistics.

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REFERENCES 632Vorperian, V. (1990) Simplified analysis of PWM converters using the model of the PWM switch: Parts I and 11. A unified analysis of PWM converters in discontinuous modes. IEEE Transaction onPower Electronics, 6, 3 (July 1991). A general unified approach to modelling switching dc-to-dc converters in discontinuous conduction mode. IEEE Power Electronics Specialists Conference Record, 1977, pp. 36-57; IEEE Publication 77 CH 12-13-8AES. Current injected equivalent circuit approach to modeling of switching dc-dc converters in discontinuous inductor conduction mode. One cycle control of switching converters. IEEE Paver Electronics Specialists Conference Record, 1990, pp. 88%8% IEEE Publication 91CH3008-0. Modelling, analysis, and design of switching converters. Ph.D. dissertation, California Institute of Technology, Pasadena, Nov. 1976.A general unified approach to modelling switching converter power stages.On the use of averaging for the analysis of power electronic systems.Polarimetric radar backscattering measurements of a variety of powerline cables are presented. The objective of the first part of the investigation was to study the effect of braiding of the cables on the backscattering at skew incidence. The measurements were performed for four different actual size powerline samples at C-, X-, and Ka-band over a wide range of incidence angles.The data were collected over a 500 MHz bandwidth at C-and Xband with a 1.25 MHz increment and a 1 GHz bandwidth at Ka-band with a 2 5 MHz increment. Also the effect of nonuniform illumination and measurement in the near field of the cables were studied. Experimental data shows a significant radar backscatter for W-polarization (U"") at angles away from normal incidence. This backscatter is proportional to the number and diameter of the strands on the surface of the cables. There is also noticeable backscatter for the HH and MI components of the scattering matrices. Their magnitudes, relative to that of the W component, are proportional to the pitch angle of the helix.Since detection of these cables is an important safety issue for low-flying airplanes a detection algorithm using polarimetric synthetic aperture radar (SAR) images was developed using the knowledge gleaned from the measurements.The detection algorithm was tested on a simulated image and worked well, detecting a power line whose backscatter power was 6 dB below the average background power.

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