A Two-Dimensional Spectrum for General Bistatic SAR Processing

This letter derives a 2-D point target spectrum for general bistatic synthetic aperture radar (SAR). For the bistatic configuration, the contributions of the transmitter and the receiver to the overall instantaneous Doppler are unequal due to the different slant range histories. In this letter, an instantaneous Doppler contribution ratio is proposed to represent the difference between the instantaneous Doppler contributions of the transmitter and the receiver, which varies with instantaneous Doppler and range frequency. Then, the 2-D spectrum is obtained by using the stationary phase principle and Taylor series expansion for general bistatic SAR. The accuracy of the spectrum is verified with a point target simulation of different general bistatic configurations.      

Investigating the propagation mechanism of unmodelled systematic errors on coordinate time series estimated using least squares

The propagation of unmodelled systematic errors into coordinate time series computed using least squares is investigated, to improve the understanding of unexplained signals and apparent noise in geodetic (especially GPS) coordinate time series. Such coordinate time series are invariably based on a functional model linearised using only zero and first-order terms of a (Taylor) series expansion about the approximate coordinates of the unknown point. The effect of such truncation errors is investigated through the derivation of a generalised systematic error model for the simple case of range observations from a single known reference point to a point which is assumed to be at rest by the least squares model but is in fact in motion. The systematic error function for a one pseudo-satellite two-dimensional case, designed to be as simple but as analogous to GPS positioning as possible, is quantified. It is shown that the combination of a moving reference point and unmodelled periodic displacement at the unknown point of interest, due to ocean tide loading, for example, results in an output coordinate time series containing many periodic terms when only zero and first-order expansion terms are used in the linearisation of the functional model. The amplitude, phase and period of these terms is dependent on the input amplitude, the locations of the unknown point and reference point, and the period of the reference point's motion. The dominant output signals that arise due to truncation errors match those found in coordinate time series obtained from both simulated data and real three-dimensional GPS data.     

InSAR基线对干涉图的影响分析

基于InSAR技术的基本原理,在h=0处进行了泰勒级数展开,推导了干涉相位组成。根据干涉相位组成论证了基线对影像相干性、高度灵敏度、平地效应及干涉SAR系统距离分辨率等的影响,利用数据进行了仿真实验,指出了基线长度对干涉图的影响程度,对合理选择干涉像对具有指导意义。     

一种基于泰勒级数展开的电离层延迟改正模型

针对电离层延迟误差对定位精度的影响,本文将泰勒级数展开应用于电离层延迟的求解过程中,提出了一种基于泰勒级数展开的电离层改正模型（TSE模型）。将该模型与传统的Klobuchar模型和NeQuick模型在VTEC的求解精度和定位精度上进行了对比分析,用IGS网站提供的测量数据,通过仿真试验验证,得出了TSE模型在提高定位精度上具有较好的可行性与时间适应性的结论。     

机载双基地SAR实测数据成像

建立双基地SAR的单基地等效模型,分析了系统时间同步误差的机理;提出了双基地SAR回波中的直达波数据进行时间同步误差校正的算法;在双基地SAR单站等效模型的基础上,利用时变阶梯变换算法进行成像处理。经过理论分析,实测数据处理验证,这一算法是有效的,能够校正双基地SAR时间同步误差,较好地进行实测数据的成像处理。     

机载合成孔径雷达图像几何纠正方法研究

机载合成孔径雷达图像几何处理是机载合成孔径雷达图像后续处理的基础,同时也是机载合成孔径雷达大规模测绘应用的基础。本文根据机载合成孔径雷达图像的成像机理,基于距离多普勒模型,考虑到合成孔径雷达成像时斜视角与多普勒频率的关系,在影像纠正时,将斜视角作为一个变量处理来减小雷达成像时零多普勒误差的影响。试验证明,此种方法比将斜视角作为零处理时的纠正精度有明显提高。详细地分析了利用此方法进行几何纠正时控制点的个数和分布对纠正精度的影响。     

Spaceborne Bistatic SAR Processing Using the EETF4 Algorithm

In this letter, the fourth-order extended exact transfer function (EETF4) is adapted for spaceborne bistatic synthetic aperture radar (SAR) processing. The problems with high squint and large bistatic Doppler centroid variations are analyzed, and it is shown that if both transmitter and receiver are highly squinted in the same direction, then it may be demanding to achieve perfect quality of the point targets. It is shown that if both transmitter and receiver have high squint in opposite directions with small Doppler centroid variations, then the SAR image can be processed very precisely.     

POS与DEM辅助机载SAR多普勒参数估计

由于机载SAR多普勒参数受载机不稳定飞行和复杂地形影响明显,目前仍缺少有效的矢量估计法。本文对POS和DEM数据及其误差对机载多普勒参数的影响进行了分析,以距离-共面方程构建的SAR几何关系为基础,在POS和DEM数据支持下,建立了一种多普勒参数矢量估计法,并通过仿真对提出的方法进行了验证和分析。理论和仿真表明,本文方法在高山地形、不稳飞行及大斜视角等复杂条件下,亦能够为机载SAR瞬时多普勒中心提供高精度估计。     

A method for computing the coefficients in the product-sum formula of associated Legendre functions

The product of two associated Legendre functions can be represented by a finite series in associated Legendre functions with unique coefficients. In this study a method is proposed to compute the coefficients in this product-sum formula. The method is of recursive nature and is based on the straightforward polynomial form of the associated Legendre function's factor. The method is verified through the computation of integrals of products of two associated Legendre functions over a given interval and the computation of integrals of products of two Legendre polynomials over [0,1]. These coefficients are basically constant and can be used in any future related applications. A table containing the coefficients up to degree 5 is given for ready reference.     

Motion Analysis in SAR Images of Unfocused Objects Using Time–Frequency Methods

Synthetic aperture radar (SAR) image formation processing assumes that the scene is stationary, and to focus an object, one coherently sums a large number of independent returns. Any target motion introduces phases that distort and/or translate the target's image. Target motion produces a smear primarily in the azimuth direction of the SAR image. Time-frequency (TF) modeling is used to analyze and correct the residual phase distortions. An interactive focusing algorithm based on TF modeling demonstrates how to correct the phase and to rapidly focus the mover. This is demonstrated on two watercraft observed in a SAR image. Then, two time-frequency representations (TFRs) are applied to estimate the motion parameters of the movers or refocus them or both. The first is the short-time Fourier transform, from which a velocity profile is constructed based on the length of the smear. The second TFR is the time-frequency distribution series, which is a robust derivative of the Wigner-Ville distribution that works well in this SAR environment. The smear is a modulated chirp, from which a velocity profile is plotted and the phase corrections are integrated to focus the movers. The relationship between these two methods is discussed. Both methods show good agreement on the example.     

A comparison of the tesseroid,prism and point-mass approaches for mass reductions in gravity field modelling

The calculation of topographic (and iso- static) reductions is one of the most time-consuming operations in gravity field modelling. For this calculation, the topographic surface of the Earth is often divided with respect to geographical or map-grid lines, and the topographic heights are averaged over the respective grid elements. The bodies bounded by surfaces of constant (ellipsoidal) heights and geographical grid lines are denoted as tesseroids. Usually these ellipsoidal (or spherical) tesseroids are replaced by “equivalent” vertical rectangular prisms of the same mass. This approximation is motivated by the fact that the volume integrals for the calculation of the potential and its derivatives can be exactly solved for rectangular prisms, but not for the tesseroids. In this paper, an approximate solution of the spherical tesseroid integrals is provided based on series expansions including third-order terms. By choosing the geometrical centre of the tesseroid as the Taylor expansion point, the number of non-vanishing series terms can be greatly reduced. The zero-order term is equivalent to the point-mass formula. Test computations show the high numerical efficiency of the tesseroid method versus the prism approach, both regarding computation time and accuracy. Since the approximation errors due to the truncation of the Taylor series decrease very quickly with increasing distance of the tesseroid from the computation point, only the elements in the direct vicinity of the computation point have to be separately evaluated, e.g. by the prism formulas. The results are also compared with the point-mass formula. Further potential refinements of the tesseroid approach, such as considering ellipsoidal tesseroids, are indicated.     

三维坐标转换严密方法的研究及VB程序实现

传统的基于泰勒级数展开的线性模型转换方法仅适用于小角度的空间直角坐标转换,角度较大时产生误差较大,需要进行复杂的三角函数和迭代计算,增加了数据处理的难度。而基于Rodrigo矩阵的方法不需要复杂的三角函数计算和迭代计算,且计算速度快,易于程序实现。实验表明,采用Rodrigo矩阵是进行三维坐标转换的一种比较严密简便的方法。     

On the computation and approximation of ultra-high-degree spherical harmonic series

Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.     

On the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses

This paper is devoted to the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses in the most rigorous way. Such an expansion can be used to compute gravimetric topographic effects for geodetic and geophysical applications. It can also be used to augment a global gravity model to a much higher resolution of the gravitational potential of the topography. A formulation for a spherical harmonic expansion is developed without the spherical approximation. Then, formulas for the spheroidal harmonic expansion are derived. For the latter, Legendre’s functions of the first and second kinds with imaginary variable are expanded in Laurent series. They are then scaled into two real power series of the second eccentricity of the reference ellipsoid. Using these series, formulas for computing the spheroidal harmonic coefficients are reduced to surface harmonic analysis. Two numerical examples are presented. The first is a spherical harmonic expansion to degree and order 2700 by taking advantage of existing software. It demonstrates that rigorous spherical harmonic expansion is possible, but the computed potential on the geoid shows noticeable error pattern at Polar Regions due to the downward continuation from the bounding sphere to the geoid. The second numerical example is the spheroidal expansion to degree and order 180 for the exterior space. The power series of the second eccentricity of the reference ellipsoid is truncated at the eighth order leading to omission errors of 25 nm (RMS) for land areas, with extreme values around 0.5 mm to geoid height. The results show that the ellipsoidal correction is 1.65 m (RMS) over land areas, with maximum value of 13.19 m in the Andes. It shows also that the correction resembles the topography closely, implying that the ellipsoidal correction is rich in all frequencies of the gravity field and not only long wavelength as it is commonly assumed.     

Total zero Doppler Steering-a new method for minimizing the Doppler centroid

This letter presents a new method, called total Zero Doppler steering, to perform yaw and pitch steering for spaceborne synthetic aperture radar (SAR) systems. The new method reduces the Doppler centroid to theoretically 0 Hz, independent of the range position of interest. Residual errors are only due to pointing inaccuracy or due to approximations in the implementation of the total zero Doppler steering law. This letter compares the new method with currently applied methods. The attitude angles and the residual Doppler centroid frequencies are calculated and depicted exemplarily for the parameters of TerraSAR-X, for which the new method will be implemented and used. The new method provides a number of advantages. The low residual Doppler centroid and the reduced variation of the Doppler centroid over range allow a more accurate Doppler centroid estimation. Due to the low residual Doppler centroid, the synthetic aperture radar (SAR) processing can be alleviated, since the range cell migration is reduced and the Doppler frequencies are low. This facilitates the use of very efficient processing algorithms, which are based on approximations whose quality is better for low Doppler frequencies. The new method will furthermore optimize the overlap of the azimuth spectra of SAR image pairs for cross-track interferometry. Low Doppler centroids will also reduce the impact of coregistration errors on the interferometric phase. Furthermore, scalloping corrections in the ScanSAR processing are alleviated due to the low variation of the Doppler centroid over range.     

Helmert扰动位及其积分核函数的椭球实用公式

借助以地心参考椭球面为边界面的第二大地边值问题的理论,基于Helmert空间的Neumann边值条件,给定Helmert扰动位的椭球解表达式,并详细推导第二类勒让德函数及其导数的递推关系、Helmert扰动位函数的椭球积分解以及类椭球Hotine积分核函数的实用计算公式,便于后续椭球域第二大地边值问题的实际研究。     

Fourier transform summation of Legendre series and D-functions

The relation between D- and d-functions, spherical harmonic functions and Legendre functions is reviewed. Dmatrices and irreducible representations of the rotation group O(3) and SU(2) group are briefly reviewed. Two new recursive methods for calculations of D-matrices are presented. Legendre functions are evaluated as part of this scheme. Vector spherical harmonics in the form af generalized spherical harmonics are also included as well as derivatives of the spherical harmonics. The special dmatrices evaluated for argument equal to/2 offer a simple method of calculating the Fourier coefficients of Legendre functions, derivatives of Legendre functions and vector spherical harmonics. Summation of a Legendre series or a full synthesis on the unit sphere of a field can then be performed by transforming the spherical harmonic coefficients to Fourier coefficients and making the summation by an inverse FFT (Fast Fourier Transform). The procedure is general and can also be applied to evaluate derivatives of a field and components of vector and tensor fields.     

Harmonic analysis of the Earth's gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP

An algorithm for the determination of the spherical harmonic coefficients of the terrestrial gravitational field representation from the analysis of a kinematic orbit solution of a low earth orbiting GPS-tracked satellite is presented and examined. A gain in accuracy is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high precision, in the range of a few centimeters. In particular, advantage is taken of Newton's Law of Motion, which balances the acceleration vector with respect to an inertial frame of reference (IRF) and the gradient of the gravitational potential. By means of triple differences, and in particular higher-order differences (seven-point scheme, nine-point scheme), based upon Newton's interpolation formula, the local acceleration vector is estimated from relative GPS position time series. The gradient of the gravitational potential is conventionally given in a body-fixed frame of reference (BRF) where it is nearly time independent or stationary. Accordingly, the gradient of the gravitational potential has to be transformed from spherical BRF to Cartesian IRF. Such a transformation is possible by differentiating the gravitational potential, given as a spherical harmonics series expansion, with respect to Cartesian coordinates by means of the chain rule, and expressing zero- and first-order Ferrer's associated Legendre functions in terms of Cartesian coordinates. Subsequently, the BRF Cartesian coordinates are transformed into IRF Cartesian coordinates by means of the polar motion matrix, the precession–nutation matrices and the Greenwich sidereal time angle (GAST). In such a way a spherical harmonic representation of the terrestrial gravitational field intensity with respect to an IRF is achieved. Numerical tests of a resulting Gauss–Markov model document not only the quality and the high resolution of such a space gravity spectroscopy, but also the problems resulting from noise amplification in the acceleration determination process.     

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the \(4 \pi \) fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as \(2^{30}\,{\approx }\,10^9\). The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.     

辅助纬度与大地纬度间的无穷展开

利用无穷级数理论和拉格朗日反演定理,详细推导了大地测量和制图学中常用的辅助纬度与大地纬度间的无穷展开,主要表现为参考椭球第一偏心率的幂级数形式。通过建立一系列严格的系数递推公式,得到了等量纬度反解展开式和等角纬度反解展开式;同时,推导了古德曼函数的泰勒展开式,进而得到了等角纬度正解展开式;利用级数除法公式,得到了等距离纬度正解展开式系数的行列式表示。通过比较本文方法与计算机代数系统Mathematica直接推导求得的辅助纬度正反解展开式e^0～e⁴⁰阶系数和相应的程序用时,表明本文算法是正确的、快速的。以CGCS2000参考椭球为例,对辅助纬度正反解进行了算例分析,也进一步验证了本文公式的正确性。