We consider the Global Navigation Satellite System Reflectometry (GNSS-R) for land applications. A distinct feature of land is that the topography has multiple elevations. The rms of elevations is in meters causing random phases between different elevations, which affect the coherent wave that has definite phase and the Fresnel zone effects as shown previously by a Kirchhoff numerical simulator (KA simulator). In this paper, we develop a physical patch model that is computationally efficient. The entire area within the footprint is divided intopatches. Each patch is small enough to satisfy the plane wave incidence and is large enough to ignore mutual wave interactions between patches. The bistatic scattering cross section of each patch for the coherent and incoherent field is computed. The bistatic cross section of plane wave incidence is obtained from lookup tables (LUTs) of the numerical 3D solution of Maxwell equations (NMM3D). The SWC represents the summation of weighted coherent fields over patches. The SWICI represents the summation of weighted incoherent intensities over patches. The formula of the received power is the sum of powers from the SWC and SWICI (the SWC/SWICI formula). The weighting factor of each patch is based on thegeometry, spherical waves, and the considerations of field amplitudes and phase variations. We also present an alternative formula, the ``correlation'' formula, using the summation of power from each physicalarea and correlations of SWCs from areas. The SWC/SWICI formula and the ``correlation'' formula are shown analytically to be the same. Results are compared with the KA simulator and two common models (the coherent model and the incoherent model). Results of the patch modelare consistent with the KA simulator. For the simulation cases, the results fall between the coherent model and the incoherent model. The patch model is much more computationally efficient than the KA simulator and the results are more accurate. In examples of this paper, the patch model results are independent of patch size as long as the patch size smaller than 50 m and much larger than the wavelength of GNSS-R frequency.
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