On the aliasing problem in the spherical harmonic analysis

To apply the least squares method for the interpolation of harmonic functions is a common practice in Geodesy. Since the method of least squares can be applied only to overdetermined problem, the interpolation problem which is always under-determined, is often reduced to an overdetermined form by truncating a series of spherical harmonics. When the data points are the knots of a regular grid it is easy to see that the estimated harmonic coefficients converge to the correct theoretical values, but when the observation density is not constant a significant bias is introduced. The result is obtained by assuming that the number of observations tends to infinity with points sampled from a given distribution. Under the same conditions it is shown that quadrature and “collocation-like” formulas displays a statistically consistent behaviour.     

Spherical harmonic analysis and synthesis for global multiresolution applications

 On the Earth and in its neighborhood, spherical harmonic analysis and synthesis are standard mathematical procedures for scalar, vector and tensor fields. However, with the advent of multiresolution applications, additional considerations about convolution filtering with decimation and dilation are required. As global applications often imply discrete observations on regular grids, computational challenges arise and conflicting claims about spherical harmonic transforms have recently appeared in the literature. Following an overview of general multiresolution analysis and synthesis, spherical harmonic transforms are discussed for discrete global computations. For the necessary multi-rate filtering operations, spherical convolutions along with decimations and dilations are discussed, with practical examples of applications. Concluding remarks are then included for general applications, with some discussion of the computational complexity involved and the ongoing investigations in research centers. Received: 13 November 2000 / Accepted: 12 June 2001     

Spherical harmonic analysis of the CRUST 2.0 global crustal model

The spherical harmonic analysis of the 2×2 density and stratification information contained in the global crustal model CRUST 2.0 is presented from the viewpoint of gravity field recovery and interpretation. Using a standard Airy/Heiskanen (A/H) isostatic hypothesis and a radially distributed compensation mechanism, two models of topographic/isostatic (t/i) potential harmonic coefficients are obtained up to degree and order 90. The CRUST-derived coefficients are compared with the spectrum of uncompensated topography, with the EGM96 observed gravity field, and with the t/i spectrum based on an A/H hypothesis with a constant compensation depth of 30 km. The signal degree variances of both CRUST models decrease quite smoothly towards degree 90, while the seven-layer model approaches the EGM96 spectrum for degrees 80–90. The significant deviation of the CRUST spectra from the A/H combined spectrum may prove of relevance to local and regional applications investigating the validity of current isostatic hypotheses.Acknowledgments. Sincere thanks go to Nikolaos Pavlis and three unknown referees for their thoughtful comments. Figure 1 has been produced using the mapping package m_map by R. Pawlowicz, which is a MATLAB toolbox that can be freely downloaded from http://www2.ocgy.ubc.ca/~rich/map.html     

Spherical cap harmonic model for mapping and predicting regional TEC

An approach to modeling the regional ionospheric total electron content (TEC) based on spherical cap harmonic analysis is presented. This approach not only provides a better regional TEC mapping accuracy, but also the capability for ionospheric model prediction based on spectrum analysis and least squares collocation. Unlike conventional approaches, which predict the immediate TEC with models using current observations, the spherical cap harmonic approach utilizes models using past observations to predict a model which will provide future TEC values. A significant advantage in comparison with conventional approaches is that the spherical cap harmonic approach can be used to predict the long-term TEC with reasonable accuracy. This study processes a set of GPS data with an observation time span of 1 year from two GPS networks in China. The TEC mapping accuracy of the spherical cap harmonic model is compared with the polynomial model and the global ionosphere model from IGS. The results show that the spherical cap harmonic model has a better TEC mapping accuracy with smoother residual distributions in both temporal and spatial domains. The TEC prediction with the spherical cap harmonic model has been investigated for both short- and long-term intervals. For the short-term interval, the prediction accuracies for the latencies of 1-day, 2-days, and 3-days are 2.5 TECU, 3.5 TECU, and 4.5 TECU, respectively. For the long-term interval, the prediction accuracy is 4.5 TECU for a 2-month latency.     

Spherical harmonic synthesis and least squares computations in satellite gravity gradiometry

The computational requirements in the simulations of geopotential estimation from satellite gravity gradiometry are discussed. Fast algorithms for spherical harmonic synthesis and least squares accumulation on a vectorizing supercomputers are presented. Using these methods, in a test case estimation of 2595 coefficients of a degree and order 50 gravity field, sustained program execution speeds of 275 Mflops (87 % peak machine speed) on a single processor of a CRAY Y-MP were achieved, with spherical harmonics computation accounting for less than 1 % of total cost. From the results, it appears that brute-force estimation of a degree and order 180 field would require 537 Million Words of memory and 85 hours of CPU time, assuming mission duration of 1 month, and execution speed of 1 Gflops. Both memory size and execution speed requirements are within the capabilities of modern multi-processor supercomputers.     

Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies

The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30?km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).     

Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation

Spherical cap harmonic analysis is the appropriate analytical technique for modelling Laplacian potential and the corresponding field components over a spherical cap. This paper describes the use of this method by means of a least-squares approach for local gravity field representation. Formulations for the geoid undulation and the components ξ, η of the deflection of the vertical are derived, together with some warnings in the application of the technique. Although most of the formulations have been given by another paper, these were confusing or even incorrect, mainly because of an improper application of the spherical cap harmonic analysis. Received: 16 January 1996 / Accepted: 17 March 1997     

Polar motion modeling,analysis, and prediction with time dependent harmonic coefficients

A time dependent amplitude model was proposed for the analysis and prediction of polar motion time series. The formulation was implemented to analyze part of the new combined solution, EOP (IERS) C 04, daily polar motion time series of 14 years length using a statistical model with first order autoregressive disturbances. A new solution approach, where the serial correlations of the disturbances are eliminated by sequentially differencing the measurements, was used to estimate the model parameters using weighted least squares. The new model parsimoniously represents the 14-year time series with 0.5 mas rms fit, close to the reported 0.1 mas observed pole position precisions for the x and y components. The model can also predict 6 months into the future with less than 4 mas rms prediction error for both polar motion components, and down to sub mas for one-step ahead prediction as validated using a set of daily time series data that are not used in the estimation. This study is dedicated to the memory of Prof. Urho Uotila (1923–2006) whose teaching of “Adjustment Computations” over the years influenced so much, so many of us who had the privilege of being his students.     

球谐分析及合成并行计算

本文介绍了球谐合成和分析问题的数值计算原理,研究分析了其内在的并行性,提出了区域划分和聚合通信的并行计算策略。基于傅立叶变换技术,应用MPI(消息传递接口)并行编程模式,实现了球谐分析及合成计算的并行化。实验结果表明,较之串行实现,并行实现大大缩短了计算时间,获得了近似线形加速比,取得很高的并行效率。     

Object-based image analysis through nonlinear scale-space filtering

In this research, an object-oriented image classification framework was developed which incorporates nonlinear scale-space filtering into the multi-scale segmentation and classification procedures. Morphological levelings, which possess a number of desired spatial and spectral properties, were associated with anisotropically diffused markers towards the construction of nonlinear scale spaces. Image objects were computed at various scales and were connected to a kernel-based learning machine for the classification of various earth-observation data from both active and passive remote sensing sensors. Unlike previous object-based image analysis approaches, the scale hierarchy is implicitly derived from scale-space representation properties. The developed approach does not require the tuning of any parameter—of those which control the multi-scale segmentation and object extraction procedure, like shape, color, texture, etc. The developed object-oriented image classification framework was applied on a number of remote sensing data from different airborne and spaceborne sensors including SAR images, high and very high resolution panchromatic and multispectral aerial and satellite datasets. The very promising experimental results along with the performed qualitative and quantitative evaluation demonstrate the potential of the proposed approach.     

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially arbitrary degrees and all co-latitudes in the interval (0°, 180°). The routines use extended-range floating-point arithmetic, in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time; for degree 3,000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart. The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [-89°57.5′, 89°57.5′] and maximum degree 1,800. Higher degrees significantly limit the range of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2,700 for almost the whole latitude range. The errors increase towards the North pole and exceed 1 mm at latitude 82° for degree 2,700. For a maximum degree 3,000, HARMONIC SYNTH produces errors exceeding 1 mm at latitudes of about 60°, whereas GEOGFG18 is limited to latitudes below 45°. Further extending the latitudinal band towards the poles may produce errors of several metres for both programs. A SHA of a uniform random signal on the sphere shows significant errors beyond degree 1,700 for the SHA program SHA by Heck and Seitz.     

Using piecewise linear interpolations in spherical harmonic analysis

In this paper an alternative approach to spherical harmonic analysis is investigated which uses piecewise linear interpolations (PLIs) to replace conventional block mean values (BMVs). Detailed formulations for using PLIs in spherical harmonic analysis are systematically developed. Research shows that BMVs give rise to larger errors than PLIs. Error estimations for the two approaches are given, convincingly showing that PLIs yield much better results than BMVs. Many numerical examples are illustrated and numerically show that PLIs are more reliable and applicable than BMVs. Received: 8 October 1996 / Accepted: 6 March 1998     

顾及降水因子的卡尔曼滤波在滑坡变形数据处理中的应用

滑坡体在受到内部及外部相关因素作用后逐渐失稳,促使滑坡体发生变形.本文利用卡尔曼滤波对滑坡体上的监测点进行变形分析,估计出监测点的状态参数以全面地反映滑坡体的运动状态.普通卡尔曼滤波没有考虑其他控制因子的影响,如降水量、河流的水位等,但这些因素对某些滑坡的变形有时起了决定性作用.本文探讨了将降水因子作为卡尔曼滤波的控制输入及控制输入系数确定的方法,利用该方法可以初步判断出滑坡体对降水是否敏感.     

综合滤波InSAR干涉图去噪实验分析

本文在分析均值滤波、圆周均值滤波、中值滤波、自适应中值滤波、二维自适应滤波和不规则窗口自适应中值滤波方法对模拟数据干涉图纹图进行滤波去噪处理的效果的基础上,引入综合滤波去噪方法,使用该滤波去除方法对真实数据InSAR干涉图进行滤波去噪处理,并对结果从定性和定量两个方面进行分析讨论。实验结果分析表明,InSAR干涉图综合滤波去噪方法是有效和可行的。     

Flood detection from multi-temporal SAR data using harmonic analysis and change detection

Flood mapping from Synthetic Aperture Radar (SAR) data has attracted considerable attention in recent years. Most available algorithms typically focus on single-image techniques which do not take into account the backscatter signature of a land surface under non-flooded conditions. In this study, harmonic analysis of a multi-temporal time series of >500 ENVISAT Advanced SAR (ASAR) scenes with a spatial resolution of 150 m was used to characterise the seasonality in backscatter under non-flooded conditions. Pixels which were inundated during a large-scale flood event during the summer 2007 floods of the River Severn (United Kingdom) showed strong deviations from normal seasonal behaviour as inferred from the harmonic model. The residuals were classified by means of an automatic threshold optimisation algorithm after masking out areas which are unlikely to be flooded using a topography-derived index. The results were validated against a reference dataset derived from high-resolution airborne imagery. For the water class, accuracies > 80% were found for non-urban land uses. A slight underestimation of the reference flood extent can be seen, mostly due to the lower spatial resolution of the ASAR imagery. Finally, an outlook for the proposed algorithm is given in the light of the Sentinel-1 mission.     

椭球谐和球谐系数之间一个简单的转换关系

本文推导的椭球谐系数和球谐系数相互之间转换关系的核心思想是在ε~2量级下利用Legendre函数的正交性,从球谐系数求解的积分表示出发,将积分中的椭球坐标变量与球坐标变量相互转换,从而得出椭球谐系数与球谐系数之间的转换关系。本文导出的转换关系有以下优点:①对于第二类Legendre函数的计算采用Laurent级数表示,使计算第二类Legendre函数更为简单;②保留了ε~2量级下,导出的转换关系相比文献[2]的形式更简单,满足物理大地测量边值问题线性化的要求;③顾及了余纬和归化余纬的区别。     

状态约束卡尔曼滤波的导航定位精度分析

针对满足一些状态约束的线性系统,通常的卡尔曼滤波未能有效利用状态约束信息,从而限制了导航定位解算性能的问题,该文将线性系统的状态约束条件融入卡尔曼滤波中,充分利用状态约束信息,对比分析了准确测量法、估计投影法、系统投影法和滑动时域估计法4种状态约束下的卡尔曼滤波方法,以提高卡尔曼滤波的导航状态估计精度。以陆地车辆导航定位的状态估计为对象,对比分析了非约束卡尔曼滤波和带有状态约束的卡尔曼滤波的状态估计精度,结果表明,状态约束卡尔曼滤波可以明显提高导航定位精度。     

基于杨赤中滤波推估法的地形剖面分析的设计与实现

应用杨赤中滤波推估法,将地形碎部点高程观测值经插值转换为格网DEM数据,并利用VB和MapInfo来实现其立体等高线的绘制功能。在此基础上,阐述其地形剖面分析功能的设计与实现。最后通过实验比较,验证了基于杨赤中滤波推估法的地形剖面分析能取得较好的效果。     

海洋局域地磁场球冠谐分析建模方法研究

本文研究了将球冠谐分析用于海洋局域地磁场建模的理论和方法,并对建模数据的质量控制、数据的代表性、球冠极点和半角的选择、模型截至阶数的确定、球冠谐分析建模中边界效应的影响和削弱等建模中关键问题进行了深入研究,最后提出了分区建摸的思想,并构建了高精度的地磁场模型。