Wavelet evaluation of the Stokes and Vening Meinesz integrals

The wavelet transform is a powerful tool in evaluating some singular geodetic integrals. Due to its localization properties in both the time (space) and frequency (scale) domains, and because the kernels of some geodetic integrals have singular points and decay smoothly and quickly away from the singularities, many wavelet transform coefficients of the kernels become zeros or negligible, and only a small number of wavelet transform coefficients are significant. It is thus possible to significantly compress the kernels of these integrals on a wavelet basis by neglecting the zero coefficients and the small coefficients below a certain threshold. Therefore, wavelets provide a convenient way of efficiently evaluating these integrals in terms of fast computation and savings of computer memory. A modified algorithm for the wavelet evaluation of Stokes' integral is presented. The same modified algorithm is applied to the evaluation of the Vening Meinesz integral, whose kernel has a stronger singularity than does Stokes' kernel. Numerical examples illustrate the efficiency and accuracy of the wavelet methods.Acknowledgments.The author express their sincere thanks to Dr. Salamonwicz for providing his PhD thesis. E-mail correspondence between the authors and Dr. Barthelmes and Dr. Benciolini contributed to the work. R. Benciolini and the other two anonymous reviewers are thanked for their constructive comments. Support for this research was provided by research grants to Dr. Sideris from the Natural Sciences and Engineering Reserch Council of Canada (NSERC) and the Geomatics for Informed Decisions (GEOIDE) Network of Centres of Excellence. The MATLAB Wavelet Toolbox package was used as the platform to develop the software in this project.     

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.     

Groebner-basis solution of the three-dimensional resection problem (P4P)

The three-dimensional (3-D) resection problem is usually solved by first obtaining the distances connecting the unknown point P{X,Y,Z} to the known points P_i{X_i,Y_i,Z_i}i=1,2,3 through the solution of the three nonlinear Grunert equations and then using the obtained distances to determine the position {X,Y,Z} and the 3-D orientation parameters {,, }. Starting from the work of the German J. A. Grunert (1841), the Grunert equations have been solved in several substitutional steps and the desire as evidenced by several publications has been to reduce these number of steps. Similarly, the 3-D ranging step for position determination which follows the distance determination step involves the solution of three nonlinear ranging (`Bogenschnitt') equations solved in several substitution steps. It is illustrated how the algebraic technique of Groebner basis solves explicitly the nonlinear Grunert distance equations and the nonlinear 3-D ranging (`Bogenschnitt') equations in a single step once the equations have been converted into algebraic (polynomial) form. In particular, the algebraic tool of the Groebner basis provides symbolic solutions to the problem of 3-D resection. The various forward and backward substitution steps inherent in the classical closed-form solutions of the problem are avoided. Similar to the Gauss elimination technique in linear systems of equations, the Groebner basis eliminates several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of a univariate polynomial whose roots can be determined by existing programs e.g. by using the roots command in Matlab.Acknowledgments.The first author wishes to acknowledge the support of JSPS (Japan Society of Promotion of Science) for the financial support that enabled the completion of the write-up of the paper at Kyoto University, Japan. The author is further grateful for the warm welcome and the good working atmosphere provided by his hosts Professors S. Takemoto and Y. Fukuda of the Department of Geophysics, Graduate School of Science, Kyoto University, Japan.     

Nonlinear filtering of continuous systems: foundational problems and new results

The previous work of Xu on discrete nonlinear filtering is extended to continuous systems. The new results are summarized as follows: (1) a second-order unbiased prediction of the true state governed by a vector stochastic differential equation is worked out; (2) a set of coupled differential equations for a new truncated second-order nonlinear filter and its variance–covariance matrix are derived from the frequentist point of view. The new filter is proved to be unbiased to the second-order approximation; and, most importantly, (3) comparison of the new filtering and accuracy results with the literature on nonlinear filtering has indicated that more than 40 years of nonlinear filtering of continuous systems may have foundational problems.Acknowledgments.This work is supported by a Grant-in-Aid for Scientific Research (C13640422). The author thanks Prof J.A.R. Blais, Prof A. Dermanis and Prof B. Schaffrin for their constructive comments.     

GPS vector configuration design for monitoring deformation networks

 The performance of geodetic monitoring networks is heavily influenced by the configuration of the measured GPS vectors. As an effective design of the GPS measurements will decrease GPS campaign costs and increase the accuracy and reliability of the entire network, the identification of the preferred GPS vectors for measurement has been highlighted as a core problem in the process of deformation monitoring. An algorithm based on a sensitivity analysis of the network, as dependent upon a postulated velocity field, is suggested for the selection of the optimal GPS vectors. Relevant mathematical and statistical concepts are presented as the basis for an improved method of vector configuration design. A sensitivity analysis of the geodetic geodynamic network in the north of Israel is presented, where the method is examined against two deformation models, the Simple Transform Fault and the Locked Fault. The proposed method is suggested as a means for the improvement of the design of monitoring networks, a common practice worldwide. Received: 30 July 2001 / Accepted: 3 June 2002 Acknowledgments. It is my pleasant duty to thank the Survey of Israel and Dr. E. Ostrovsky for providing the variance–covariance matrix of the G1 network in northern Israel. I would like to thank the reviewers of this paper for their constructive and helpful remarks.     

GPS network monitors the Western Alps' deformation over a five-year period: 1993–1998

  The Western Alps are among the best studied collisional belts with both detailed structural mapping and also crustal geophysical investigations such as the ECORS and EGT seismic profile. By contrast, the present-day kinematics of the belt is still largely unknown due to small relative motions and the insufficient accuracy of the triangulation data. As a consequence, several tectonic problems still remain to be solved, such as the amount of N–S convergence in the Occidental Alps, the repartition of the deformation between the Alpine tectonic units, and the relation between deformation and rotation across the Alpine arc. In order to address these problems, the GPS ALPES group, made up of French, Swiss and Italian research organizations, has achieved the first large-scale GPS surveys of the Western Alps. More than 60 sites were surveyed in 1993 and 1998 with a minimum observation of 3 days at each site. GPS data processing has been done by three independent teams using different software. The different solutions have horizontal repeatabilities (N–E) of 4–7 mm in 1993 and 2–3 mm in 1998 and compare at the 3–5-mm level in position and 2-mm/yr level in velocity. A comparison of 1993 and 1998 coordinates shows that residual velocities of the GPS marks are generally smaller than 2 mm/yr, precluding a detailed tectonic interpretation of the differential motions. However, these data seem to suggest that the N–S compression of the Western Alps is quite mild (less than 2 mm/yr) compared to the global convergence between the African and Eurasian plate (6 mm/yr). This implies that the shortening must be accomodated elsewhere by the deformation of the Maghrebids and/or by rotations of Mediterranean microplates. Also, E–W velocity components analysis supports the idea that E–W extension exists, as already suggested by recent structural and seismotectonic data interpretation. Received: 27 November 2000 / Accepted: 17 September 2001     

Present-day crustal deformation in China relative to ITRF97 kinematic plate model

 A global plate motion model is established based on the ITRF97 velocity fields and geological model NUVEL1. Sub-plate models are estimated by using the velocity fields derived from 45 global positioning system (GPS) sites under the ITRF97 reference frame in China. Comparisons between space geodesy and geological models are given. It is found that the Euler vector of the AFRC–EURA pair has an obvious discrepancy between space geodetic and geological models. The motion patterns of tectonic blocks in China predicted by GPS are consistent with those of geological data on the whole. Received: 9 November 2000 / Accepted: 17 September 2001     

Surface deformation analysis of dense GPS networks based on intrinsic geometry: deterministic and stochastic aspects

In this contribution, the regularized Earth’s surface is considered as a graded 2D surface, namely a curved surface, embedded in a Euclidean space . Thus, the deformation of the surface could be completely specified by the change of the metric and curvature tensors, namely strain tensor and tensor of change of curvature (TCC). The curvature tensor, however, is responsible for the detection of vertical displacements on the surface. Dealing with eigenspace components, e.g., principal components and principal directions of 2D symmetric random tensors of second order is of central importance in this study. Namely, we introduce an eigenspace analysis or a principal component analysis of strain tensor and TCC. However, due to the intricate relations between elements of tensors on one side and eigenspace components on other side, we will convert these relations to simple equations, by simultaneous diagonalization. This will provide simple synthesis equations of eigenspace components (e.g., applicable in stochastic aspects). The last part of this research is devoted to stochastic aspects of deformation analysis. In the presence of errors in measuring a random displacement field (under the normal distribution assumption of displacement field), the stochastic behaviors of eigenspace components of strain tensor and TCC are discussed. It is applied by a numerical example with the crustal deformation field, through the Pacific Northwest Geodetic Array permanent solutions in period January 1999 to January 2004, in Cascadia Subduction Zone. Due to the earthquake which occurred on 28 February 2001 in Puget Sound (M _w > 6.8), we performed computations in two steps: the coseismic effect and the postseismic effect of this event. A comparison of patterns of eigenspace components of deformation tensors (corresponding the seismic events) reflects that: among the estimated eigenspace components, near the earthquake region, the eigenvalues have significant variations, but eigendirections have insignificant variations.     

Monitoring the Deformation of Cableways

One type of aerial cableway consists of a continuous cable,which works at the same time as carrying cable and drawing cable.This kind of cableway is supported by many towers,which divide the total length in linear sections.even small deviations from the planed layout can increase the strain of the cable and the danger of derailments.The deviations of the position of the towers from the planed layout are due to local sliding or sinking of the foundation or other mechanical movements.Up to now the axis of the cableways were periodically measured by traditional methods. The goal of this project is to demonstrate the possibility of measuring the axis of a cableway with GPS techniques,while the cable is moving.The results show that the proposed measuring device provides precise results in a simple and reliable way.     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

A spectral optimally interpolated mean sea surface from satellite radar altimetry

 A technique is presented for the development of a high-precision and high-resolution mean sea surface model utilising radar altimetric sea surface heights extracted from the geodetic phase of the European Space Agency (ESA) ERS-1 mission. The methodology uses a cubic-spline fit of dual ERS-1 and TOPEX crossovers for the minimisation of radial orbit error. Fourier domain processing techniques are used for spectral optimal interpolation of the mean sea surface in order to reduce residual errors within the initial model. The EGM96 gravity field and sea surface topography models are used as reference fields as part of the determination of spectral components required for the optimal interpolation algorithm. A comparison between the final model and 10 cycles of TOPEX sea surface heights shows differences of between 12.3 and 13.8 cm root mean square (RMS). An un-optimally interpolated surface comparison with TOPEX data gave differences of between 15.7 and 16.2 cm RMS. The methodology results in an approximately 10-cm improvement in accuracy. Further improvement will be attained with the inclusion of stacked altimetry from both current and future missions. Received: 22 December 1999 / Accepted: 6 November 2000     

特殊情况下深基坑围护测斜及变形浅析

结合工程实例具体阐述当测斜管不能正常使用情况下运用全站仪代替测斜仪进行深基坑围护结构的倾斜测量,同时运用岩土工程相关知识对所得监测数据表现的变形及突变原因作了深入分析,获得了预期的安全监测成果,确保了深基坑工程的顺利实施。为今后类似情况的处理提供借鉴。     

电子水平仪在桥梁变形监测中的应用

提出电子水平仪与计算机之间的通信连接应用于桥梁变形监测的方法。通过对简支梁受力变形理论的分析与总结,提出电子水平仪与计算机串口连接,尝试对桥梁进行实时连续的变形监测,编写一套VB平台上的实时监测软件,并以简支梁为例做了桥梁变形监测的模拟试验。     

Combining the geodetic models of vertical crustal deformation

Different kinds of analytical models of vertical curstal deformation have both advantages and disadvantages, and are appropriate to different deformation patterns. It is difficult to identify which deformation model is most suitable for a particular deformation area. In order to obtain a more precise and reliable analytical result, the combined model based on forecast theory has been approached. As a result, the fitting ability of single models is significantly improved. The combined model still possesses the same important features as the single models, which enable transferal of the derived crustal deformation information, within or even outside the time span of the data coverage, from geodetic sites to any user-specified locations where geodetic data may not exist. Examples are presented of both numerical simulation and a real-life situation, and the results are analysed in detail to demonstrate the efficiency of the proposed methodology. Received: 26 January 1998 / Accepted 14 August 1998     

Detection of impervious surface change with multitemporal Landsat images in an urban–rural frontier

Mapping and monitoring impervious surface dynamic change in a complex urban-rural frontier with medium or coarse spatial resolution images is a challenge due to the mixed pixel problem and the spectral confusion between impervious surfaces and other non-vegetation land covers. This research selected Lucas do Rio Verde County in Mato Grosso State, Brazil as a case study to improve impervious surface estimation performance by the integrated use of Landsat and QuickBird images and to monitor impervious surface change by analyzing the normalized multitemporal Landsat-derived fractional impervious surfaces. This research demonstrates the importance of two-step calibrations. The first step is to calibrate the Landsat-derived fraction impervious surface values through the established regression model based on the QuickBird-derived impervious surface image in 2008. The second step is to conduct the normalization between the calibrated 2008 impervious surface image with other dates of impervious surface images. This research indicates that the per-pixel based method overestimates the impervious surface area in the urban-rural frontier by 50%-60%. In order to accurately estimate impervious surface area, it is necessary to map the fractional impervious surface image and further calibrate the estimates with high spatial resolution images. Also normalization of the multitemporal fractional impervious surface images is needed to reduce the impacts from different environmental conditions, in order to effectively detect the impervious surface dynamic change in a complex urban-rural frontier. The procedure developed in this paper for mapping and monitoring impervious surface area is especially valuable in urban-rural frontiers where multitemporal Landsat images are difficult to be used for accurately extracting impervious surface features based on traditional per-pixel based classification methods as they cannot effectively handle the mixed pixel problem.     

利用SBAS技术监测洛杉矶地区地表形变

以InSAR技术为基础发展起来的小基线集法(SBAS)有效克服时空去相干和大气效应的影响,可长时间、高精度监测地表形变。文中以美国南加州洛杉矶地区为研究区域,利用34景ENVISAT雷达影像开展地面沉降监测,并通过时间序列分析的方法获取2003年9月到2009年8月时间段内高精度地表形变结果。研究发现多处由于地下水过度开采引起的地表形变明显的区域,该成果与同时段内数据资料比较,吻合较好。因此,该技术可广泛应用于城市地表形变监测,为相关部门决策提供科学可靠的数据参考。     

变形分析与预报方法综述

简要介绍国内外现有的变形分析与变形预报的理论和方法,并对诸方法进行比较和评价。     

区域性变形数据序列变异特性分析

对单点变异函数模型及变异函数的套合理论进行介绍,以某矿沉降变形数据为例,建立区域内单个变形监测点的变异函数,借助套合理论建立区域性通用变异函数。实例分析表明该方法可用于区域变形的时变特征分析。为变形预报提供了重要理论依据。