Volume 1, Issue 1 -

Foreword

A Numerical Study of the Analytical Downward Continuation Error in Geoid Computation by EGM08 Today the geoid can be conveniently determined by a set of high-degree spherical harmonics, such as EGM08 with a resolution of about 5'. However, such a series will be biased when applied to the continental geoid inside the topographic masses. This error we call the analytical downward continuation (DWC) error, which is closely related with the so-called topographic potential bias. However, while the former error is the result of both analytical continuation of the potential inside the topographic masses and truncation of a series, the latter is only the effect of analytical continuation. This study compares the two errors for EGM08, complete to degree 2160. The result shows that the topographic bias ranges from 0 at sea level to 5.15 m in the Himalayas region, while the DWC error ranges from -0.08 m in the Pacific to 5.30 m in the Himalayas. The zero-degree effects of the two are the same (5.3 cm), while the rms of the first degree errors are both 0.3 cm. For higher degrees the power of the topographic bias is slightly larger than that for the DWC error, and the corresponding global rms values reaches 25.6 and 25.3 cm, respectively, at n max =2160. The largest difference (20.5 cm) was found in the Himalayas. In most cases the DWC error agrees fairly well with the topographic bias, but there is a significant difference in high mountains. The global rms difference of the two errors clearly indicates that the two series diverge, a problem most likely related with the DWC error.

Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral Data Spherical Harmonic Transforms (SHTs) which are non-commutative Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known global strategies for discrete SHTs of band-limited spherical functions are Chebychev quadratures and least squares for equiangular grids. With proper numerical preconditioning, independent of latitude, reliable analysis and synthesis results for degrees and orders over 3800 in double precision arithmetic have been achieved and explicitly demonstrated using white noise simulations. The SHT synthesis and analysis can easily be modified for the ordinary Fourier transform of the data matrix and the mathematical situation is illustrated in a new functional diagram. Numerical analysis has shown very little differences in the numerical conditioning and computational efforts required when working with the two-dimensional (2D) Fourier transform of the data matrix. This can be interpreted as the spectral form of the discrete SHT which can be useful in multiresolution and other applications. Numerical results corresponding to the latest Earth Geopotential Model EGM 2008 of maximum degree and order 2190 are included with some discussion of the implications when working with such spectral sequences of fast decreasing magnitude.

Planar, spherical and ellipsoidal approximations of Poisson's integral in near zone Planar, spherical, and ellipsoidal approximations of Poisson's integral for downward continuation (DWC) of gravity anomalies are discussed in this study. The planar approximation of Poisson integral is assessed versus the spherical and ellipsoidal approximations by examining the outcomes of DWC and finally the geoidal heights. We present the analytical solution of Poisson's kernel in the point-mean discretization model that speed up computation time 500 times faster than spherical Poisson kernel while preserving a good numerical accuracy. The new formulas are very simple and stable even for regions with very low height. It is shown that the maximum differences between spherical and planar DWC as well as planar and ellipsoidal DWC are about 6 mm and 18 mm respectively in the geoidal heights for a rough mountainous area such as Iran.

The Integration of TLS and Continuous GPS to Study Landslide Deformation: A Case Study in Puerto Rico Terrestrial Laser Scanning (TLS) and Global Positioning System (GPS) technologies provide comprehensive information on ground surface deformation in both spatial and temporal domains. These two data sets are critical inputs for geometric and kinematic modeling of landslides. This paper demonstrates an integrated approach in the application of TLS and continuous GPS (CGPS) data sets to the study of an active landslide on a steep mountain slope in the El Yunque National Forest in Puerto Rico. Major displacements of this landslide in 2004 and 2005 caused the closing of one of three remaining access roads to the national forest. A retaining wall was constructed in 2009 to restrain the landslide and allow the road reopen. However, renewed displacements of the landslide in the first half of 2010 resulted in deformation and the eventual rupture of the retaining wall. Continuous GPS monitoring and two TLS campaigns were performed on the lower portion of the landslide over a three-month period from May to August 2010. The TLS data sets identified the limits and total volume of themoving mass, while the GPS data quantified the magnitude and direction of the displacements. A continuous heavy rainfall in late July 2010 triggered a rapid 2-3 meter displacement of the landslide that finally ruptured the retaining wall. The displacement time series of the rapid displacement is modeled using a fling-step pulse from which precise velocity and acceleration time series of the displacement are derived. The data acquired in this study have demonstrated the effectiveness and power of the integrating TLS and continuous GPS techniques for landslide studies.

Least Squares Spectral Analysis for Detection of Systematic Behaviour of Digital Level Compensator Levelling is the most precise technique for height difference measurements in geomatics engineering. Various systematic errors affect precise levelling observations and reduce the precision of the observed height differences. This study investigates digital levels residual compensator error and observational method for its elimination. For this purpose the levelling data, which was collected with Zeiss DiNi 12 digital levels, was analysed. There are different statistical and spectral methods that can reveal the presence of systematic errors in levelling results. In this study, the Least Squares Spectral Analysis (LSSA) method is used. The analysis confirmed that using alternating pointing method ( B FFB, F BBF) instead of usual observation routine ( B FFB) will eliminate the Zeiss DiNi 12 digital levels residual compensator error from section height differences and discrepancies. In this way, it does not matter using different instruments in the forward and backward section runs and the discrepancies can be used to investigate other systematic errors.

N-dimensional B-spline surface estimated by lofting for locally improving IRI N-dimensional surfaces are defined by the tensor product of B-spline basis functions. To estimate the unknown control points of these B-spline surfaces, the lofting method also called skinning method by cross-sectional curve fits is applied. It is shown by an analytical proof and numerically confirmed by the example of a four-dimensional surface that the results of the lofting method agree with the ones of the simultaneous estimation of the unknown control points. The numerical complexity for estimating v n control points by the lofting method is O(v n+1 ) while it results in O(v 3n ) for the simultaneous estimation. It is also shown that a B-spline surface estimated by a simultaneous estimation can be extended to higher dimensions by the lofting method, thus saving computer time. An application of this method is the local improvement of the International Reference Ionosphere (IRI), e.g. by the slant total electron content (STEC) obtained by dual-frequency observations of the Global Navigation Satellite System (GNSS). Three-dimensional B-spline surfaces at different time epochs have to be determined by the simultaneous estimation of the control points for this improvement. A four-dimensional representation in space and time of the electron density of the ionosphere is desirable. It can be obtained by the lofting method. This takes less computer time than determining the four-dimensional surface solely by a simultaneous estimation.

Orientation of the Geometrically Best fitting Triaxial Lunar Ellipsoid with Respect to the Mean Earth/Polar Axis Reference Frame This study provides new estimates for the orientation of a geometrically best fitting lunar triaxial ellipsoid with respect to the mean Earth/polar axis reference frame calculated from the footprint positions of the Chang'E-1 (CE-1), SELenological and ENgineering Explorer (SELENE) laser altimetry measurements and Unified Lunar Control Networks 2005, (ULCN 2005) station coordinates. The semi-principal axes of the triaxial ellipsoid and the coordinates of its geometric center are also calculated simultaneously. All the estimated parameters from all three data sets are found to be consistent. In particular, the RMS differences of the semi-principal axes of the triaxial ellipsoids and the locations of their geometric centers from solutions with and without modeling Euler angles (orientation of the triaxial ellipsoid) using uniformly distributed laser altimetry (LAL) footprints are 29 and 31 m respectively. The misclosures of all the solutions indicate a better fit for the triaxial ellipsoid to the footprint and station coordinates if the Euler angles are included in the models.

Comprhensive Approach to the Analysis of the 3D Kinematics Deformation with appliction to the Kenai Peninsula The problem of analyzing surface deformation of the Earth's crust in three-dimensions is discussed. The isoparametric and Lagrangian formulations of deformation are extended from 2D to 3D. Analytical and numerical investigation of problem conditioning proves that analyzing the 3D kinematics of deformation can be an ill-posed problem. The required mathematical elements for solving this problem, including sensitivity analysis of the deformation tensor and regularization, are proposed. Regularized deformation tensors were computed using the method of truncated singular value decomposition (TSVD). The optimal regularization parameter was attained by minimizing regularization errors. Regularization errors were assessed using the corresponding 2D results of deformation analysis. The proposed methods were applied to the GPS network in the Kenai Peninsula, south-central Alaska, in order to compute the 3D pattern of postseismic crustal deformation in this area. Computed deformation in the vertical direction is compared to the existing pattern of vertical deformation obtained from the combination of precise leveling, gravity and GPS measurements from other studies on this area.

Poisson Downward Continuation Solution by the Jacobi Method Downward continuation is a continuing problem in geodesy and geophysics. Inversion of the discrete form of the Poisson integration process provides a numerical solution to the problem, but because the B matrix that defines the discrete Poisson integration is not always well conditioned the solution may be noisy in situations where the discretization step is small and in areas containing large heights. We provide two remedies, both in the context of the Jacobi iterative solution to the Poisson downward continuation problem. First, we suggest testing according to the upward continued result from each solution, rather then testing between successive solutions on the geoid, so that choice of a tolerance for the convergence of the iterative method is more meaningful and intuitive. Second, we show how a tolerance that reflects the conditioning of the B matrix can regularize the solution, and suggest an approximate way of choosing such a tolerance. Using these methods, we are able to calculate a solution that appears regular in an area of Papua New Guinea having heights over 3200 m, over a grid with 1 arc-minute spacing, based on a very poorly conditioned B matrix.

A contemporary perspective of geoid structure The present paper reviews the contemporary state of definition and theory of the geoid. Key features are:quasigeoid, external gravitational field from satellites and its analytical downward continuation to the Earth's interior, data combination by least-squares collocation, and a new view of gravity reduction. This is done under the modern systematic perspective provided by the possibility of a purely geometric satellite determination of the Earth' surface by GPS combined with satellite altimetry.