Trilateration Network Research Articles

On the exact and efficient solution of the Huber function for measurement applications

The findings of this paper are summarized as follows: (1) We point out that the recently published conclusion that Iteratively Reweighted Least-Squares (IRLS) cannot guarantee the global solution of the well-known Huber function is incorrect. We present that the Huber function has only one minimum and IRLS will obtain the exact minimum unless it is incorrectly implemented. (2) We revisit the variants of the IRLS algorithm and compare them with the exact IRLS algorithm in the geodetic free trilateration networks adjustment. Notably, the pointwise differences between the variants of IRLS and the exact IRLS are even beyond the differences between the Least-Squares solution and the exact IRLS for over 75% points. (3) For rigorously and efficiently solving the Huber M−estimation problem, we propose the Quasi-Newton approach and its limited memory version. The time expense of our proposed algorithm is quite low compared with the time expense of the existing exact IRLS in the cases of a large number of unknown parameters and observations. Our results show that the computation efficiency is improved by one or two orders of magnitude.

Sensitivity and antenna pattern for an interplanetary laser trilateration network

An array of laser transponders, moving along orbits centered on Venus, Earth, and Mars, could provide significant new information about the distribution of mass in the solar system and beyond. In this report we begin an analysis of the sensitivity and effective antenna pattern for such a device. While the main source of changes in the lengths of the three legs of the Venus-Earth-Mars triangle is motion along the nearly circular heliocentric orbits of those bodies, measurable perturbations can be caused by other masses, either inside or outside the triangle. Our main focus here is on sources which are close to the ecliptic plane, and farther from the Sun than Jupiter. We present results on simulations of the ability of the transponder array to measure accelerations, and use that to describe the ability of the array to characterize sources of perturbing mass. Measurements of time-of-flight for short laser pulses between pairs of detectors directly constrains distance between them. Fitting a quadratic model to a daily collection of range measurements yields an estimate of the average daily difference in accelerations at those points, projected onto the baseline between them. Range measurements along a single baseline, with 1 ​cm accuracy, and a 1 ​Hz sampling cadence, will yield acceleration sensitivity of 10−15 ​m/s2, for each daily acceleration normal point. We consider two different types of sources: one is an isolated point mass, and the other is a sinusoidal variation in mass density along a circular arc. In both cases, we are able to characterize the performance of the transponder array. Simultaneous projections of acceleration differences onto the 3 baselines, at a single epoch, provide unique values for the mass and position of a single point source. Characterization of distributed sources requires a range of baseline orientations.

Control of Laser Scanner Trilateration Networks for Accurate Georeferencing of Caves: Application to El Castillo Cave (Spain)

There is a growing demand for measurements of natural and built elements, which require quantifiable accuracy and reliability, within various fields of application. Measurements from 3D Terrestrial Laser Scanner come in a point cloud, and different types of surfaces such as spheres or planes can be modelled. Due to the occlusions and/or limited field of view, it is seldom possible to survey a complete feature from one location, and information has to be acquired from multiple points of view and later co-registered and geo-referenced to obtain a consistent coordinate system. The aim of this paper is not to match point clouds, but to show a methodology to adjust, following the traditional topo-geodetic methods, 3DTLS data by modelling references such as calibrated spheres and checker-boards to generate a 3D trilateration network from them to derive accuracy and reliability measurements and post-adjustment statistical analysis. The method tries to find the function that best fits the measured data, taking into account not only that the measurements made in the field are not perfect, but that each one of them has a different deviation depending on the adjustment of each reference, so they have to be weighted accordingly.

Adjustment of Planned Surveying and Geodetic Networks Using Second-Order Nonlinear Programming Methods

Due to the huge amount of redundant data, the problem arises of finding a single integral solution that will satisfy numerous possible accuracy options. Mathematical processing of such measurements by traditional geodetic methods can take significant time and at the same time does not provide the required accuracy. This article discusses the application of nonlinear programming methods in the computational process for geodetic data. Thanks to the development of computer technology, a modern surveyor can solve new emerging production problems using nonlinear programming methods—preliminary computational experiments that allow evaluating the effectiveness of a particular method for solving a specific problem. The efficiency and performance comparison of various nonlinear programming methods in the course of trilateration network equalization on a plane is shown. An algorithm of the modified second-order Newton’s method is proposed, based on the use of the matrix of second partial derivatives and the Powell and the Davis–Sven–Kempy (DSK) method in the computational process. The new method makes it possible to simplify the computational process, allows the user not to calculate the preliminary values of the determined parameters with high accuracy, since the use of this method makes it possible to expand the region of convergence of the problem solution.

Geodetic Network Geometry versus Reliability Measures in Examples of the Trilateration Network

AbstractThe article describes selected case studies of the trilateration network design. The analyzed case studies may be of interest to the geodetic community but may also be potentially interesti...

Modeling ground deformation associated with the destructive earthquakes occurring on Mt. Etna's southeastern flank in 1984

Abstract. The Timpe Fault System is the source of very shallow but destructive earthquakes that affect several towns and villages on the eastern flank of Mt. Etna (Italy). In 1984, several seismic events, and specifically on 19 and 25 October, caused one fatality, 12 injuries and produced serious damage in the Zafferana and Acireale territories. This seismicity was mainly related to the activity of the Fiandaca Fault, one of the structures belonging to the Timpe Fault System. We inverted ground deformation data collected by a geodimeter trilateration network set up in 1977 at a low altitude along the eastern side of the volcano in order to define the Timpe Fault System faulting mechanisms linked to the seismicity in 1984. We have found that in the period May 1980–October 1984, the Fiandaca Fault was affected by a strike-slip and normal dip-slip of about 20.4 and 12.7 cm respectively. This result is kinematically consistent with field observations of the coseismic ground ruptures along the fault but it is notably large compared to displacements estimated by seismicity, then suggesting that most of the slip over the fault plane was aseismic. The results once again confirm how seismicity and its relation with ground ruptures and creep displacement represent a very high hazard to the several towns and villages situated along the Timpe Fault System.

19 years of tilt data on Mt. Vesuvius: state of the art and future perspectives

<p>Mt. Vesuvius, located along the SW border of the Campania Plane graben, is one of the most studied volcanoes worldwide, from both the volcanological and the geophysical, geochemical and geodetic point of view. In order to better understand its dynamics, the deformation of the volcano has been already studied since the early ’70s by setting up levelling lines and, since a few years later, through trilateration networks, whereas ground tilt monitoring started in 1993. Tilt variations were recorded by an automatic surface station set up at the Osservatorio Vesuviano (O.V.) bunker (OVO) and data recorded were transmitted to the O.V. Surveillance Centre in Naples. Afterwards, in 1996 two more identical stations were set up close to Torre del Greco (CMD), and close to Trecase (TRC). In 2002 the data acquisition system was replaced, while at the end of 2011 a Lily borehole sensor was set up at 26 m depth, replacing the old TRC tilt station. The paper describes in details the tilt network of Mt. Vesuvius, its development over time and the data processing procedure; moreover, the ground deformation pattern is discussed, as inferred from the study of 19 years of data and its change during the seismic crises of 1995-1996 and 1999-2000. From the information obtained from the tiltmetric monitoring, a complex deformation pattern can be deduced, strongly dependent on the position of the sites in which the sensors were set up with respect to the morphology of the volcanic edifice and its structural outlines. If we consider the signals as they were recorded, although previously corrected for the influences of the thermo-elastic strain on the sensors, the tilting occurs mainly in the SW direction with rates of about 11 µradians/year on both the western and eastern flanks and of about 13 µradians/year on the southern one. Because tilt vectors point in the long term outward from the summit and towards the subsiding area, this supports the hypothesis of a southern areas subsidence, according with a spreading effect of Vesuvius, taking into account geological, structural, geophysical and geodetical (optical levelling, InSAR) data. The SW tilting occurs therefore irregularly and shows some seasonalities, consistent with the solar thermal radiation whose removal by statistical procedure outlines a different but equally interesting deformation field as it shows interruptions with changes in both trend and amplitude during two periods of strong seismic activity that affected Mt. Vesuvius in the periods 1995-1996 and late 1999-2000, marked by an average rate of energy release of at least one order of magnitude greater than the previous and following periods. Another change in intensity and direction of the deformation detected by tiltmeters since 2000, connected with the variations of the phase shift between the tilt components and the temperature recorded, compared to previous years, occurs during a strong decrease of the energy released by Vesuvius earthquakes.</p>

Remainder Reliability and Robust Estimation for Surveying Networks when Two Gross Errors Included in the Observations

Robust estimation methods can efficiently eliminate or weaken the effects of gross errors on parameter estimation when such errors exist in observations. Remainder reliability can essentially show whether gross errors may be eliminated or weakened by robust estimation. Taking 8 leveling networks and 4 trilateration networks with different remainder reliability adjustments as examples, simulation experiments are used to compare 4 frequently used robust estimation methods. The results indicate that when observations simultaneously contain two gross errors and if the remainder reliability RR20.45, all the four robust estimation methods may effectively eliminate or weaken these effects.

Extended S-transformation as a tool for deformation analysis

In geodetic monitoring networks, the deformation parameters can be estimated only if the datum of the network has not been changed between measurement epochs. The measurements of a monitoring network must be adjusted so that the datum remains undisturbed. In deformation analysis, similarity transformation is used intensively in order to bring all monitoring campaigns to the same common datum. However, there are geodetic measurements which contain datum definition components. When part of the datum definition contained in the measurements is not constant in time, the estimated deformation parameters could be erroneous. An appropriate approach should be taken to prevent the inclusion of these components in the adjustment of a 4D network. If not, the result is an inevitable mixture between the deformation parameters and the datum components of the measurements. The extended S-transformation allows us to strip the datum content from a set of coordinates that contain the complete contribution of the geodetic measurements to a datumless set of coordinates. Based on the datumless sets of coordinates, the variations in the network geometry can be modelled by means of a physical model without the influence of a datum definition inherent in geodetic measurements. Following a short introduction of the concept of extended free net adjustment constraints, the development of an extended S-transformation is presented. Two examples of a trilateration network and a GPS network are presented for implementing the method.

Basic Concepts of Optimization and Design of Geodetic Networks

This contribution reviews a few basic concepts of optimization and design of a geodetic network. Proper assessment and analysis of networks is an important task in many geodetic-surveying projects. Appropriate quality-control measures should be defined, and an optimal design should be sought. The quality of a geodetic network is characterized by precision, reliability, and cost. The aim is to present a few case studies that have been designed to meet optimal precision and reliability criteria. Though the case studies may be of interest to the geodetic community in their own right, the aim is to gain insight into the general optimization problem of a geodetic network. This is also potentially of interest for educational purposes. The case studies include a zeroth-order design to improve the precision of the network points in a traverse network and a first-order design to meet the high reliability and maximum precision criteria in a geodetic network. It is shown that not only the configuration of the network but also the type of the observations used can affect the design criteria. For example, the case studies presented show that the optimal shape of the trilateration network (intersection with distances) can result in a weak network in the sense of reliability and precision if the observations are replaced by angles rather than distances (triangulation network). In close relation to the optimization problem of a geodetic network, the global positioning system satellite configuration is also optimized for a particular case that provides the minimum value of the geometric dilution of precision.

INDEPENDENT CONTROL OF GPS NETWORKS ABOVE LONG TUNNELS BY ASTRONOMICALLY DETERMINED AZIMUTHS OR BEARING ANGLES

At the Faculty of Geodesy, University of Zagreb, the automatic astronomical method of determining the azimuth by observing stars and the Sun has been developed within the frame of the scientific work. The classical astronomical method of azimuth determination is rather uneconomical because of extensive field measurements and long-lasting processing. The automation of measuring procedures and processing has enabled the application of the astronomical method in everyday surveying practice, and also in determining the tunnel cut direction, in the independent control of microtrilateration and GPS networks above tunnels. The importance of such an independent control of geodetic networks above tunnels lies in the fact: <list list-type="order"> <list-item>That we control the relative positional relationships of the points at the entrance, and the exit of a tunnel, respectively,</list-item> <list-item>That we control whether the exact value the deflection of the vertical have been taken into account. Namely, the deflection of the vertical at the points for traverse orientation used for entering the tunnel can be remarkable which can largely influence the traversing orientation, especially if the orientation sighting lines are steep.</list-item> </list>Applying the astronomical determination of azimuth as an independent control of trilateration or GPS networks over long tunnels, the surveyors can expect with more confidence that the tunnel be cut without any problems.

WHAT IS THE RELIABILITY OF CONVENTIONAL OUTLIER DETECTION AND ROBUST ESTIMATION IN TRILATERATION NETWORKS?

There are two fundamental approaches to determine outliers; these are, the Conventional Outlier Detection Test Procedures (e.g., data-snooping of Baarda, τ-test of Pope etc.) and Robust Estimation Methods.As is known, the Least Squares Estimation (LSE) method is very sensitive to outliers and it spreads the corrupt influence of the outliers upon the good observations. Therefore the Baarda and Pope test methods derived from the LSE are thought of as unsuccessful methods on outlier detection. It is asserted that in case of more than one outlier, these Conventional Outlier Detection Test Procedures become inefficient. In this situation, Robust Estimation Methods are proposedfor application. In this study these possibilities are investigated in trilateration networks structured artificially.

WHAT IS THE RELIABILITY OF CONVENTIONAL OUTLIER DETECTION AND ROBUST ESTIMATION IN TRILATERATION NETWORKS?

There are two fundamental approaches to determine outliers; these are, the Conventional Outlier Detection Test Procedures (e.g., data-snooping of Baarda, τ-test of Pope etc.) and Robust Estimation Methods.As is known, the Least Squares Estimation (LSE) method is very sensitive to outliers and it spreads the corrupt influence of the outliers upon the good observations. Therefore the Baarda and Pope test methods derived from the LSE are thought of as unsuccessful methods on outlier detection. It is asserted that in case of more than one outlier, these Conventional Outlier Detection Test Procedures become inefficient. In this situation, Robust Estimation Methods are proposedfor application. In this study these possibilities are investigated in trilateration networks structured artificially.

Salton Trough Regional Deformation Estimated from Combined Trilateration and Survey-Mode GPS Data

The Salton Trough in southeastern California, United States, has one of the highest seismicity and deformation rates in southern California, including 20 earthquakes M 6 or larger since 1892. From 1972 through 1987, the U.S. Geological Survey (USGS) measured a 41-station trilateration network in this region. We re-measured 37 of the USGS baselines using survey-mode Global Positioning System methods from 1995 through 1999. We estimate the Salton Trough deformation field over a nearly 30-year period through combined analysis of baseline length time series from these two datasets. Our primary result is that strain accumulation has been steady over our observation span, at a resolution of about 0.05 μstrain/yr at 95% confidence, with no evidence for significant long-term strain transients despite the occurrence of seven large regional earthquakes during our observation period. Similar to earlier studies, we find that the regional strain field is consistent with 0.5 ± 0.03 μstrain/yr total engineering shear strain along an axis oriented 311.6° ± 23° east of north, approximately parallel to the strike of the major regional faults, the San Andreas and San Jacinto (all uncertainties in the text and tables are standard deviations unless otherwise noted). We also find that (1) the shear strain rate near the San Jacinto fault is at least as high as it is near the San Andreas fault, (2) the areal dilatation near the southeastern Salton Sea is significant, and (3) one station near the southeastern Salton Sea moved anomalously during the period 1987.95-1995.11. Online material: Tables of stations used in this study and of estimated strain fields. Manuscript received 8 January 2003.

L 1 -norm pre-analysis measures for geodetic networks

Several pre-analysis measures which help to expose the behavior of L1 -norm minimization solutions are described. The pre-analysis measures are primarily based on familiar elements of the linear programming solution to L1-norm minimization, such as slack variables and the reduced-cost vector. By examining certain elements of the linear programming solution in a probabilistic light, it is possible to derive the cumulative distribution function (CDF) associated with univariate L1-norm residuals. Unlike traditional least squares (LS) residual CDFs, it is found that L1-norm residual CDFs fail to follow the normal distribution in general, and instead are characterized by both discrete and continuous (i.e. piecewise) segments. It is also found that an L1 equivalent to LS redundancy numbers exists and that these L1 equivalents are a byproduct of the univariate L1 univariate residual CDF. Probing deeper into the linear programming solution, it is found that certain combinations of observations which are capable of tolerating large-magnitude gross errors can be predicted by comprehensively tabulating the signs of slack variables associated with the L1 residuals. The developed techniques are illustrated on a two-dimensional trilateration network.

Three-dimensional modeling of the behavior of the oblique convergent boundary of southeast Taiwan: friction and strain partitioning

The Longitudinal Valley Fault (LVF) of eastern Taiwan is the plate boundary between the Philippine Sea plate and Eurasian plate. Analyses of triangulation networks showed that two distinct deformation zones coexist: thrusting prevails between the Pinanshan Conglomerate massif and the Central Range to the west, while strike-slip dominates between the Pinanshan massif and the Coastal Range to the east. Crustal deformation and displacement partitioning along this oblique convergence boundary is investigated by using a 3D distinct-element model. To constrain our models, we take into account the present-day deformation of southern LVF based on geodetic data. Particular attention is paid to the role of the major faults that play an important role in the distribution of the regional strain patterns.The overall deformation is examined by separating the transverse and lateral components of the regional deformation. Our models predict that both the dip-slip and strike-slip increase as the friction coefficient decreases in the eastern fault between Coastal Range and Pinanshan Conglomerate massif. For the western fault between Central Range and Pinanshan Conglomerate massif, the dip-slip increases when the friction coefficient decreases. Conversely the strike-slip decreases as the friction coefficient decreases. The model that provides the best fit in the eastern fault has an effective coefficient in the range of 0.1–0.4. The friction coefficient of 0.27 predicts a strike-slip slip of 20.5mm/yr, similar to the rate of 22mm/yr towards N353°E calculated from geodetic data. Along the western fault, the models that provide the best fit have a friction coefficient of 0.8–1.0. These models predict the average fault-normal shortening of about 15.7mm/yr. The displacement mentioned above is a 30% larger than that of geodetic measurements. Internal deformation by numerous minor faults inside the Pinanshan Conglomerate massif accounts for the discrepancy between the model-predicted and the observed displacement rate. Our models predict an average displacement rate of about 33mm/yr towards N318°E for reasonable values of the coefficient of friction. These results are consistent with the displacement rate of 28±8mm/yr towards N329±10°E between the Coastal Range and the Central Range based on the trilateration networks around the southern Longitudinal Valley. The friction parameters do affect the orientation and magnitude of slip along faults by a series of parametric studies. We conclude that the strain partitioning of crustal deformation of southern LVF strongly depends on the orientation of oblique convergence, the geometry of faults, and their friction coefficients. Taking into account both the simplified assumptions of the 3D numerical simulation and the uncertainties of geodetic measurements, our estimates of friction parameters along major discontinuities are only valid to first order.

Ground deformation at Merapi Volcano, Java, Indonesia: distance changes, June 1988–October 1995

Edifice deformations are reported here for the period 1988–1995 at Merapi volcano, one of the most active and dangerous volcanoes in Indonesia. The study period includes a major resumption in lava effusion in January 1992 and a major dome collapse in November 1994. The data comprise electronic distance measurements (EDM) on a summit trilateration network, slope distance changes measured to the upper flanks, and other data collected from 1988 to 1995. A major consequence of this study is the documentation of a significant 4-year period of deformation precursory to the 1992 eruption. Cross-crater strain rates accelerated from less than 3×10 −6/day between 1988 and 1990 to more than 11×10 −6/day just prior to the January 1992 activity, representing a general, asymmetric extension of the summit during high-level conduit pressurization. After the vent opened and effusion of lava resumed, strain occurred at a much-reduced rate of less than 2×10 −6/day. EDM measurements between lower flank benchmarks and the upper edifice indicate displacements as great as 1 m per year over the four years before the 1992 eruption. The Gendol breach, a pronounced depression formed by the juxtaposition of old lava coulées on the southeast flank, functioned as a major displacement discontinuity. Since 1993, movements have generally not exceeded the 95% confidence limits of the summit network. Exceptions to this include 12 cm outward movement for the northwest crater rim in 1992–1993, probably from loading by newly erupted dome lava, and movements as much as 7 cm on the south flank between November 1994 and September 1995. No short-term precursors were noted before the November 1994 lava dome collapse, but long-term adjustments of crater geometry accompanied lava dome growth in 1994. Short-term 2-cm deflation of the edifice occurred following the November 1994 dome collapse.

Spatial - temporal modeling

Spatial - temporal modeling

Crustal deformation of the Marlborough Fault Zone in the South Island of New Zealand: Geodetic constraints over the interval 1982–1994

Crustal deformation across the Marlborough fault zone in the South Island of New Zealand has been investigated by resurveying with the Global Positioning System (GPS) a triangulation and trilateration network across part of the zone. The principal strain rates, which mostly have errors between 5% and 10%, vary systematically across the region. The principal axis of horizontal contraction gradually swings from SE‐NW southeast of the Hope fault to almost east‐west in the west. The component of velocity parallel to the strike of the faults can account for 95% of the relative motion between the Australian and Pacific plates. This component exhibits little variation west of the Alpine (Wairau) fault, and shows an almost linear variation from the Wairau fault to the east coast. The horizontal strain rate northwest of the Wairau fault is a small east‐west uniaxial contraction (∼0.1 ppm yr−1). If it is assumed that the crustal blocks bounded by the major faults are primarily driven by basal shear tractions, with stresses on the faults being relatively insignificant, then the slip rates expected on the faults may be derived from the observed velocity variations across the fault zone. These predicted slip rates are in good agreement with geologically observed slip rates. The larger slip rates on the Hope fault are related to the observation that the strain field extends southeast of the fault over a distance large compared with the spacing between the faults. The agreement between predicted and geological slip rates lends support to the idea that the observed pattern of deformation reflects that in the lower lithosphere.

Plate‐boundary strain partitioning along the sinistral collision suture of the Philippine and Eurasian plates: Analysis of geodetic data and geological observation in southeastern Taiwan

Crustal deformation and strain partitioning of oblique convergence between the Philippine Sea plate and the Eurasian plate in the southern Longitudinal Valley of eastern Taiwan were characterized, based on geodetic analysis of trilateration network and geological field investigation. The Longitudinal Valley fault, one of the most active faults on Taiwan, branches into two individual faults in the southern Longitudinal Valley. These two active faults bound the Plio‐Pleistocene Pinanshan Conglomerate massif between the Coastal Range (the Luzon island arc belonging to the Philippine Sea plate) and the Central Range (the metamorphic basement of the Eurasian plate). A geodetic trilateration network near the southern end of the valley shows a stable rate of the annual length changes during 1983–1990. The strain tensors for polygonal regions (including triangular regions) of the Taitung trilateration network reveal that there are two distinct zones of deformation: a zone of shortening (thrusting) between the Pinanshan massif and the Central Range on the west and a strike‐slip movement between the Pinanshan massif and the Coastal Range on the east. The analysis of a discontinuity model consisting of three rigid blocks separated by two discontinuities has been carried out. The results show that the deformation in this region can be characterized by two major faults. A reverse fault is located between the Plio‐Pleistocene Pinanshan massif and the metamorphic basement of the Central Range, with a shortening rate of about 12 mm/yr in the direction N280°E. A strike‐slip fault is located principally along the river between the Pinanshan massif and island arc system of the Coastal Range with a purely strike‐slip component of about 22 mm/yr in the direction N353°E. The analysis of the geodetic data further suggests that substantial deformation (probably strike slip in type) occurs within the Pinanshan massif. Geological evidence of deformation in the Plio‐Pliestocene Pinanshan Conglomerate includes regional folding, a conjugate set of strike‐slip fractures at the outcrop scale, and morphological lineaments related to fracturing, all indicating that the Pinanshan massif is being deformed within a transpressive stress regime. Regional kinematic data indicate that a significant portion of the 82 mm/yr of motion between the Eurasian plate and the Philippine Sea plate is absorbed in the southern Longitudinal Valley by the decoupling of two distinct major faults. The geometry of the oblique convergence and the rheology of the rock units (the well‐consolidated Plio‐Pleistocene conglomerate and the sheared mélange formation) play the two important roles in the partitioning of crust deformation.