Fast space-domain evaluation of geodetic surface integrals

Geodetic surface integrals play an important role in the numerical solution of geodetic boundary-value problems. In many cases they can be evaluated using fast methods in the frequency domain (FFT). However, this is not possible in general, because the domain of integration may be non-trivial (as is the surface of the Earth), the kernel function may not be of convolution type, or the data distribution may be heterogeneous. Therefore, fast evaluation strategies are also required in the space domain. They are more difficult to design because only one property is left where a more or less fast evaluation strategy can be built upon: the potential type of the kernel function. Consequently, the idea is not to replace well-established frequency domain techniques, but to supplement them. Our approach to this problem goes in two directions: (1) we use advanced cubature methods where the integration nodes automatically densify in the vicinity of the evaluation points; (2) we use powerful computer hardware, namely MIMD computers with distributed memory. This enables us to evaluate geodetic surface integrals of any practical complexity in reasonable time and accuracy. This is shown in a numerical example. Received: 7 May 1996 / Accepted:17 March 1997     

ON THE EVALUATION OF DEFLECTIONS OF THE VERTICAL USING FFT TECHNIQUE

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Fourier-series representation and projection of spherical harmonic functions

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10^?15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.     

FFT-Evaluation and applications of gravity-field convolution integrals with mean and point data

Due to the fact that the spectrum of a convolution is the product of the spectra of the two convolved functions, the convolution integrals of physical geodesy can be evaluated very efficiently by the use of the fast Fourier transform (FFT) provided that gravity and/or terrain data are available on a regular grid. All Fourier transform-based methods usually treat the gridded data as point values despite the fact that these discrete values may have been obtained by averaging and they represent mean values over the whole area of a grid element. In the frequency domain, this fact can be taken into account very easily, because the spectra of mean and point data are related via a two-dimensional (2D) sinc function. The paper shows explicitly this relationship using the convolution integrals for the evaluation of geoid undulations, deflections of the vertical, and gravity and gradiometry terrain effects. Numerical tests are presented, indicating that the differences in the two approaches may become significant when highly accurate results are wanted. The application of the2D sinc function in the evaluation, update, and inversion of other convolution integrals is briefly discussed as well.     

动态空间正图像透视投影正反解

卫星图像都是在动态情形下获取的。瞬间曝光获取的图像投影性质符合透视投影。本文针对卫星动态获取的正图像,建立其平面透视投影,利用矢量解法研究其正反解变换和星下点坐标计算方法,最后给出了算例。     

Local relationships between the disturbing density, the disturbing potential and the disturbing gravity of the Earth's gravity field

A function having some properties of a wavelet and being harmonic around a given point in R ³ is defined, and three models showing the local relationships between the disturbing density, the disturbing potential and the disturbing gravity are established by using the function as the kernel function of the integrals in the models. The local relationship has two meanings. One is that we can evaluate with a high accuracy the integrals in the models by using mainly high-accuracy and high-resolution data in a local area. The other is that we can obtain a stable solution with high resolution when we invert the integrals in the models because of the rapid decrease of the kernel function of the integrals. As a result, with these models we evaluate one quantity with high resolution, in a band limited by the maximum degree of a set of geopotential coefficients or by the resolution (spacing) of the local data, from another quantity (or quantities) in a local area, and the resulting solution is stable. Received: 6 April 1998 / Accepted: 16 June 1999     

On the evaluation of deflections of the vertical using FFT technique

There exist three types of convolution formulae for the efficient evaluation of gravity field convolution integrals, i.e., the planar 2D convolution, the spherical 2D convolution and the spherical 1D convolution. The largest drawback of both the planar and the spherical 2D FFT methods is that, due to the approximations in the kernel function, only inexact results can be achieved. Apparently, the reason is the meridian convergence at higher latitudes. As the meridians converge, the ??,?λ blocks do not form a rectangular grid, as is assumed in 2D FFT methods. It should be pointed out that the meridian convergence not only leads to an approximation error in the kernel function, but also causes an approximation error during the implementation of 2D FFT in computer. In order to meet the increasing need for precise determination of the vertica deflections, this paper derives a more precise planar 2D FFT formula for the computation of the vertical deflections. After having made a detailed comparison between the planar and the spherical 2D FFT formulae, we find out the main source of errors causing the loss in accuracy by applying the conventional spherical 2D FFT method. And then, a modified spherical 2D FFT formula for the computation of the vertical deflections is developed in this paper. A series of numerical tests have been carried out to illustrate the improvement made upon the old spherical 2D FFT. The second part of this paper is to discuss the influences of the spherical harmonic reference field, the limited capsize, and the singular integral on the computation of the vertical deflections. The results of the vertical deflections over China by applying the spherical 1D FFT formula with different integration radii have been compared to the astro-observed vertical deflections in the South China Sea to obtain a set of optimum deflection computation parameters.     

Modified ambiguity function approach for GPS carrier phase positioning

This paper presents a new strategy for GPS carrier phase data processing. The classic approach generally consists of three steps: a float solution, a search for integer ambiguities, and a fixed solution. The new approach is based on certain properties of ambiguity function method and ensures the condition of integer ambiguities without the necessity of the additional step of the integer search. The ambiguities are not computed explicitly, although the condition of “integerness” of the ambiguities is ensured in the results through the least squares adjustment with condition equations in the functional model. An appropriate function for the condition equations is proposed and presented. The presented methodology, modified ambiguity function approach, currently uses a cascade adjustment with successive linear combinations of L1 and L2 carrier phase observations to ensure a correct solution. This paper presents the new methodology and compares it to the three-stage classic approach which includes ambiguity search. A numerical example is provided for 25 km baseline surveyed with dual-frequency receivers. All tests were performed using an in-house developed GINPOS software and it has been shown that the positioning results from both approaches are equivalent. It has also been proved that the new approach is robust to adverse effects of cycle slips. In our opinion, the proposed approach may be successfully used for carrier phase GPS data processing in geodetic applications.     

On the accurate numerical evaluation of geodetic convolution integrals

In the numerical evaluation of geodetic convolution integrals, whether by quadrature or discrete/fast Fourier transform (D/FFT) techniques, the integration kernel is sometimes computed at the centre of the discretised grid cells. For singular kernels—a common case in physical geodesy—this approximation produces significant errors near the computation point, where the kernel changes rapidly across the cell. Rigorously, mean kernels across each whole cell are required. We present one numerical and one analytical method capable of providing estimates of mean kernels for convolution integrals. The numerical method is based on Gauss-Legendre quadrature (GLQ) as efficient integration technique. The analytical approach is based on kernel weighting factors, computed in planar approximation close to the computation point, and used to convert non-planar kernels from point to mean representation. A numerical study exemplifies the benefits of using mean kernels in Stokes’s integral. The method is validated using closed-loop tests based on the EGM2008 global gravity model, revealing that using mean kernels instead of point kernels reduces numerical integration errors by a factor of ~5 (at a grid-resolution of 10 arc min). Analytical mean kernel solutions are then derived for 14 other commonly used geodetic convolution integrals: Hotine, Eötvös, Green-Molodensky, tidal displacement, ocean tide loading, deflection-geoid, Vening-Meinesz, inverse Vening-Meinesz, inverse Stokes, inverse Hotine, terrain correction, primary indirect effect, Molodensky’s G1 term and the Poisson integral. We recommend that mean kernels be used to accurately evaluate geodetic convolution integrals, and the two methods presented here are effective and easy to implement.     

A new approach for airborne vector gravimetry using GPS/INS

A new method for airborne vector gravimetry using GPS/INS has been developed and the results are presented. The new algorithm uses kinematic accelerations as updates instead of positions or velocities, and all calculations are performed in the inertial frame. Therefore, it is conceptually simpler, easier, more straightforward and computationally less expensive compared to the traditional approach in which the complex navigation equations should be integrated. Moreover, it is a unified approach for determining all three vector components, and no stochastic gravity modeling is required. This approach is based on analyzing the residuals from the Kalman filter of sensor errors, and further processing with wavenumber coefficient filterings is applied in case closely parallel tracks of data are available. An application to actual test-flight data is performed to test the validity of the new algorithm. The results yield an accuracy in the down component of about 3–4 mGal. Also, comparable results are obtained for the horizontal components with accuracies of about 6 mGal. The gravity modeling issue is discussed and alternative methods are presented, none of which improves on the original approach. Received: 18 April 2000 / Accepted: 14 August 2000     

GIS中面向对象时空数据模型

由于当前的地理信息系统软件难以处理时态现象，时态数据模型已忧为ＧＩＳ领域的一个研究热点。许多学者提出了多种时态数据模型。本文作者在提出了矢量栅格一体化的面向对象数据模型之后，再一次对时态问题进行了分析研究，净面向对象的数据模型扩充到时间维。有三种方法表达空间对象的历史变化。第一种是将版本信息记录在关系表上；第二种是将版本信息标记在记录上；第三种是将版本信息标记在属性上。本文采用面向对象的方法，将版     

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.     

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.     

Recurrence relations for integrals of Associated Legendre functions

Recurrence relations for the evaluation of the integrals of associated Legendre functions over an arbitrary interval within (0°, 90°) have been derived which yield sufficiently accurate results throughout the entire range of their possible applications. These recurrence relations have been used to compute integrals up to degree 100 and similar computations can be carried out without any difficulty up to a degree as high as the memory in a computer permits. The computed values have been tested with independent check formulae, also derived in this work; the corresponding relative errors never exceed 10⁻²³ in magnitude. Contribution from the Earth Physics Branch No. 719     

On the use of heterogeneous noisy data in spectral gravity field modeling methods

Spectral methods have been a standard tool in physical geodesy applications over the past decade. Typically, they have been used for the efficient evaluation of convolution integrals, utilizing homogeneous, noise-free gridded data. This paper answers the following three questions:

1. Can data errors be propagated into the results?
2. Can heterogeneous data be used?
3. Is error propagation possible with heterogeneous data?

The answer to the above questions is yes and is illustrated for the case of two input data sets and one output. Firstly, a solution is obtained in the frequency domain using the theory of a two-input, single-output system. The assumption here is that both the input signals and their errors are stochastic variables with known PSDs. The solution depends on the ratios of the error PSD and the signal PSD, i.e., the noise-to-signal ratios of the two inputs. It is shown that, when the two inputs are partially correlated, this solution is equivalent to stepwise collocation. Secondly, a solution is derived in the frequency domain by a least-squares adjustment of the spectra of the input data. The assumption is that only the input errors are stochastic variables with known power spectral density functions (PSDs). It is shown that the solution depends on the ratio of the noise PSDs. In both cases, there exists the non-trivial problem of estimating the input noise PSDs, given that we only have available the error variances of the data. An effective but non-rigorous way of overcoming this problem in practice is to approximate the noise PSDs by simple stationary models.     

Calculation of strongly singular and hypersingular surface integrals

Efficient numerical computation of integrals defined on closed surfaces in ℝ³ with non-integrable point singularities that arise in physical geodesy is discussed. The method is based on the use of polar coordinates and the definition of integrals with non-integrable point singularities as Hadamard finite part integrals. First the behavior of singular integrals under smooth parameter transformations is studied, and then it is shown how they can be reduced to absolutely integrable functions over domains in ℝ². The correction terms that usually arise if the substitution rule is formally applied, in contrast to absolutely integrable functions, are calculated. It is shown how to compute the regularized integrals efficiently, and, numerical efforts for various orders of singularity are compared. Finally, efficient numerical integration methods are discussed for integrals of functions that are defined as singular integrals, a task that typically arises in Galerkin boundary element methods. Received: 15 April 1997 / Accepted: 7 May 1998     

基于 Lidar 数据和倾斜摄影的城市三维模型构建

针对传统建模方式建模效率低、模型纹理不够丰富真实等缺点,阐述一种以机载Lidar技术来获取地面点高精度的三维坐标、用倾斜摄影测量为三维建模提供纹理的一种新兴的三维建模方式。实验表明,该建模方式不仅工作效率高、真实反应城市空间布局,而且定位精度高、具备可量测分析功能等,同时提供DOM,DEM,点云数据,倾斜影像等数据。     

稳健加权总体最小二乘方法

加权总体最小二乘没有考虑观测数据中可能存在的粗差,本文基于IGG权函数,采用选权迭代法求解加权总体最小二乘。结合模拟数据和真实数据,系统地比较了加权总体最小二乘方法、基于Huber权函数的稳健加权总体最小二乘方法和基于IGG权函数的稳健加权总体最小二乘方法的系数估计和误差估计,通过对比分析表明,两种稳健加权总体最小二乘方法的参数估计结果比加权总体最小二乘方法更加可靠,且以基于IGG权函数的稳健加权总体最小二乘方法为最优。     

Colour Hue and Texture Evaluation for 3D Symbolization of Indoor Environments Using RGB-D Data

In this work an evaluation procedure of the use of colour hue and texture for 3D symbolization of indoor environments using RGB-D data is presented. The main characteristics of the proposed method involve RGB-D cameras to collect colour and depth data and the evaluation of the visual variables colour hue and texture combined with visual aspects such as lighting and camera configuration for the symbolization of 3D indoor environments. High quality coloured point clouds of indoor environments are obtained with the Kinect sensor and then registered and processed in order to create 3D models. These sets of point clouds can be detected, recognized and classified under one of the three following main categories: ceiling, wall or floor. After classification is done, the resulting point clouds are properly treated as 3D triangular meshes in the MeshLab software. These meshes are then filtered to form complete surfaces which are then refined before being considered for visual variable application. We evaluated the proposed method on several indoor environments to show that it efficiently combines the colour and depth data obtained from RGB-D cameras with different parameters defined for the visual variables colour hue and texture for 3D symbolization of indoor environments.