Tomás Soler

A new matrix development of the potential and attraction at exterior points as a function of the inertia tensors

In this work a novel tensor-matrix notation is first introduced and later applied to develop a new general expression to compute the potential of a body at exterior points as a function of the full tensors of inertia. As a corollary the general matrix development of the gravitational attraction in function of the inertia tensors is also established. For clarity the first terms in both expansions are given explicitly in a simplified matrix form. Some classical particular cases still used in geophysical and geodetic literature are pointed out and discussed.

Using OPUS for Measuring Vertical Displacements in Houston, Texas

AbstractThe Houston area has been suffering from subsidence for several decades. Therefore, continuously operating reference stations (CORS) in this area may have experienced considerable vertical displacements. The Online Positioning User Service (OPUS), provided by the National Geodetic Survey (NGS), uses CORS as references in its data processing. This study investigated what effects, if any, the subsidence experienced by these CORS around Houston contributes to the accuracy of OPUS vertical results. Our OPUS results were determined from three long-term (over 10-year) blocks of continuous data using Global Positioning System (GPS) stations located in different parts of the Houston area. The OPUS results were compared with the vertical measurements from the precise point positioning with single receiver phase ambiguity (PPP-SRPA) solution implicit in the GIPSY/OASIS 6.1.2 software. This particular study indicates that OPUS achieves 1.0-cm vertical accuracy for daily sessions (24 h) in the region around H...

Measuring Land Subsidence Using GPS: Ellipsoid Height versus Orthometric Height

AbstractGlobal positioning system (GPS) technology has been frequently used to monitor geological hazards associated with ground deformations, such as long-term landslides and subsidence. When GPS data are processed, they yield ellipsoid heights, which are the distances above a smooth ellipsoid surface. However, orthometric heights are often used in practical surveying and engineering applications. Orthometric height is a physical quantity that refers to the surface of the geoid. In this study, a more practical alternative definition of orthometric height is used. This approximation is the one commonly implemented in practical surveying and engineering applications to compute relative orthometric height values. This well-known procedure computes orthometric heights by combining GPS-measured ellipsoid height and a geoid model. Any type of orthometric height is a physically based quantity. GPS alone, which is a geometric technique, cannot directly measure orthometric heights. This study investigates the ver...

Densifying 3D GPS Networks by Accurate Transformation of Vector Components

A primary output of Global Positoning System (GPS) post-processed data is a set of non-trivial (independent) vector components and their full covariance information referred to a specific local Cartesian terrestrial frame (e. g., ITRF, WGS84) and epoch. It is important to recognize that when GPS-determined vector components are simultaneously combined into 3D geodetic network adjustments, they should always refer to a common coordinate frame and epoch. This paper uses geometric concepts to formulate rigorous matrix transformations to correct vector components for changes in coordinate systems, secular displacements due to plate rotations, and antenna centering and/or height measuring errors. Finally, the associated variance-covariance matrix of the transformed vector components is derived.

Accurate determination of Cartesian coordinates at geodetic stations using the global positioning system

Comparison of Cartesian coordinates determined at collocated sites using two independent space techniques, very long baseline interferometry (VLBI) and Global Positioning System (GPS), shows remarkable agreement even when the points in question span transcontinental distances. The results corroborate the capabilities of commercial dual-frequency GPS receivers to perform geodetic work at the highest available accuracy. Adjusted geocentric coordinates of a configuration of GPS stations well distributed along the eastern half of the United States were accurately determined (better than 10−8) in the rigorously defined International Earth Rotation Service (IERS) terrestrial reference frame ITRF 89.

Alternative transformation from Cartesian to geodetic coordinates by least squares for GPS georeferencing applications

The inverse transformation of coordinates, from Cartesian to curvilinear geodetic, or symbolically (x,y,z)->(@l,@f,h) has been extensively researched in the geodetic literature. However, published formulations require that the application must be deterministically implemented point-by-point individually. Recently, and thanks to GPS technology, scientists have made available thousands of determinations of the coordinates (x,y,z) at a single point perhaps characterized by different observational circumstances such as date, length of occupation time, distance and geometric distribution of reference stations, etc. In this paper a least squares (LS) solution is introduced to determine a unique set of geodetic coordinates, with accompanying accuracy predictions all based on the given sets of individual (x,y,z) GPS-obtained values and their variance-covariance matrices. The (x,y,z) coordinates are used as pseudo-observations with their attached stochastic information in the LS process to simultaneously compute a unique set of (@l,@f,h) curvilinear geodetic coordinates from different observing scenarios.

Comparisons of OPUS and PPP Solutions for Subsidence Monitoring in the Greater Houston Area

AbstractGlobal positioning system (GPS) techniques have been applied to study land subsidence in the greater Houston area for over two decades (1993–2016). The free Online Positioning User Service ...

On rotation of frames and physical vectors: an exercise based on plate tectonics theory

The mathematical interaction between the simultaneous rotation of both a coordinate frame and a set of physical vectors in that frame is covered and theoretically and empirically explained. A practical example related to the secular motion of the pole determined using recent GPS results is addressed. A least-squares adjustment is introduced to determine a possible displacement of the geodetic north pole of the frame caused by plausible changes in the coordinates of the observing stations defining the frame due to the rotation of the plates on which these stations are located. Two examples of GPS networks are investigated both referred to the latest definition of the IGS08 geodetic frame. The positioning and velocities of the points were exclusively obtained using GPS data as published by the International GNSS Service (IGS). The first case comprises the complete GPS/IGS network of global stations; the second one assumes the closest GPS/IGS stations to the now discontinued International Latitude Service network. The results of this exercise hints at the possibility that the secular global rotation of the frame caused by plate rotations should be accounted for in order to rigorously determine the true absolute velocities referred to the IGS frame before the actual velocities of the rotation of the plates using GPS observations are published.

On Deflection of the Vertical Components and Their Transformations

AbstractThe deflection of the vertical is an important parameter that combines both physical (astronomic) and geometric (geodetic) determined quantities. This paper introduces an alternative didactic approach based on rotations of three-dimensional right-handed local frames that simplifies such concepts as the derivation of Laplace’s equation and facilitates understanding of vertical deflection components and their transformations under different assumptions. Variations in deflections of the vertical as a consequence of geometric changes introduced by the future replacement of the North American Datum of 1983 (NAD 83) are also investigated.

Variance–Covariance Matrix of Transformed GPS Positions: Case Study for the NAD 83 Geodetic Datum

AbstractThis investigation introduces a rigorous mathematical transformation of variance–covariance matrices between a global geocentric frame and a plate-fixed geodetic frame. A practical example between the geocentric frame of International GNSS Service 2008 (IGS08) epoch 2005.00 and the geodetic frame North American Datum of 1983 (NAD 83) (2011) epoch 2010.00 was implemented. Although the theory is general, the transformation used here is controlled by the assumptions implicit in the definition of NAD 83. However, the same approach could be extended to future definitions of fixed-plate datums used by geodetic organizations for charting and mapping applications. Consequently, because the transformation between these two specific frames is assumed by definition to be a one-to-one errorless transformation, the uncertainties for the 14 Helmert transformation parameters between the two frames are assumed to be zero. Nevertheless, the formulation is complete and applicable to other specific datum-transformat...