Modified Newton identities

The sums of powers with identical exponents of natural, real, or complex numbers, considered as roots of algebraic equation, are expressed directly through the products of coefficients of that equation, starting from the well-known Newton identities. The final Eq. (6) includes the same power of sum of all numbers ± a sum over all partitions of the exponent. Each term of the last sum is the equation coefficients product with the net power keeping the “dimensionality” of the exponent and having a numerical factor, equal to a proper polynomial coefficient, built of exponents of equation coefficients entering the product. The revers Eq. (43) for equation coefficients is also a sum over all partitions of the same exponent with known numerical coefficients. The entering products are built of “commutators-anticommutators of power of sum and sum of powers” (C-A) of the initial sum addends. The numerous identities Eq. (44) for a C-A with an exponent, exceeding the number of C-A sum terms by 2, and similar C-A-s with lower exponents are established.

Feasibility of Replacing the Range Doppler Equation of Spaceborne Synthetic Aperture Radar Considering Atmospheric Propagation Delay with a Rational Polynomial Coefficient Model

Usually, the rational polynomial coefficient (RPC) model of spaceborne synthetic aperture radar (SAR) is fitted by the original range Doppler (RD) model. However, the radar signal is affected by two-way atmospheric delay, which causes measurement error in the slant range term of the RD model. In this paper, two atmospheric delay correction methods are proposed for use in terrain-independent RPC fitting: single-scene SAR imaging with a unique atmospheric delay correction parameter (plan 1) and single-scene SAR imaging with spatially varying atmospheric delay correction parameters (plan 2). The feasibility of the two methods was verified by conducting fitting experiments and geometric positioning accuracy verification of the RPC model. The experiments for the GF-3 satellite were performed by using global meteorological data, a global digital elevation model, and ground control data from several regions in China. The experimental results show that it is feasible to use plan 1 or plan 2 to correct the atmospheric delay error, no matter whether in plain, mountainous, or plateau areas. Moreover, the geometric positioning accuracy of the RPC model after correcting the atmospheric delay was improved to better than 3 m. This is of great significance for the efficient and high-precision geometric processing of spaceborne SAR images.