Least-squares Adjustment Method Research Articles

Integration of multi-photos and laser scanner data to form a complete 3d model

Nowadays, laser scanners are widely used to get three-dimensional modeling, but sometimes some obstacles on the site or choosing the wrong scan width and the shadow areas due to different planes lead to the appearance of some gaps in the objective to be scanned. Mobile phone images (Conventional terrestrial photogrammetry technique) can be used to compensate for these missing data after transforming the image coordinates to ground coordinates. So, the main objective is to develop a technique to transform image coordinates to ground ones through the Direct Linear Transformation model (DLT) depending on a computer algorithm using the least-squares adjustment method. Four field experiments were made on AL-ASHRAFI QAITBAY . The obtained results showed that the putative technique can be applied to form a full 3D model with precision (11–22 mm) and it is compatible with the precision of a laser scanner.

Estimating Arctic Sea Ice Thickness with CryoSat-2 Altimetry Data Using the Least Squares Adjustment Method.

Satellite altimeters can be used to derive long-term and large-scale sea ice thickness changes. Sea ice thickness retrieval is based on measurements of freeboard, and the conversion of freeboard to thickness requires knowledge of the snow depth and snow, sea ice, and sea water densities. However, these parameters are difficult to be observed concurrently with altimeter measurements. The uncertainties in these parameters inevitably cause uncertainties in sea ice thickness estimations. This paper introduces a new method based on least squares adjustment (LSA) to estimate Arctic sea ice thickness with CryoSat-2 measurements. A model between the sea ice freeboard and thickness is established within a 5 km × 5 km grid, and the model coefficients and sea ice thickness are calculated using the LSA method. Based on the newly developed method, we are able to derive estimates of the Arctic sea ice thickness for 2010 through 2019 using CryoSat-2 altimetry data. Spatial and temporal variations of the Arctic sea ice thickness are analyzed, and comparisons between sea ice thickness estimates using the LSA method and three CryoSat-2 sea ice thickness products (Alfred Wegener Institute (AWI), Centre for Polar Observation and Modelling (CPOM), and NASA Goddard Space Flight Centre (GSFC)) are performed for the 2018–2019 Arctic sea ice growth season. The overall differences of sea ice thickness estimated in this study between AWI, CPOM, and GSFC are 0.025 ± 0.640 m, 0.143 ± 0.640 m, and −0.274 ± 0.628 m, respectively. Large differences between the LSA and three products tend to appear in areas covered with thin ice due to the limited accuracy of CryoSat-2 over thin ice. Spatiotemporally coincident Operation IceBridge (OIB) thickness values are also used for validation. Good agreement with a difference of 0.065 ± 0.187 m is found between our estimates and the OIB results.

Calculating the Vertical Displacement using the Method of Least Squares Adjustment and 3D Objects Fitting

In this research, the Iraqi flagpole in Baghdad University, which is the longest flagpole in Baghdad, with a height of 75m, was monitored. According to the importance of this structure, the calculation of the displacement (vertical deviation) in the structure monitored using the Total Station device, where several observation were taken at different times for a period of two years the monitoring started from November 2016 until May 2017, at a rate four observations for one year. And the observation processed using the least square method, and the fitting of circles, then the data was processed and the deviation was calculated using Matlab program with calculating the values of corrections, where the mathematical laws have been programmed in a format that suits the program, observation have been entered and correction were made on them, calculating corrected values and the amount of error between the observed and calculated values. The deviation was between (0.720 to 0.759)m during the observation period. The Auto CAD program and the 3D MAX program used to product the two-dimensional (2D) and (3D) Models of the structure.

GEOMETRIC ASPECTS EVALUATION OF GNSS CONTROL NETWORK FOR DEFORMATION MONITORING IN THE JATIGEDE DAM REGION

Many factors led to dam construction failure so that deformation monitoring activities is needed in the area of the dam. Deformation monitoring is performed in order to detect a displacement at the control points of the dam. Jatigede Dam deformation monitoring system has been installed and started to operate, but there has been no evaluation of the geometry quality of control networks treated with IGS points for GNSS networks processing. Therefore, this study aims to evaluate the geometric quality of GNSS control networks on deformation monitoring of Jatigede Dam area. This research data includes the GNSS measurements of five CORS Jatigede Dam stations (R01, GG01, GCP04, GCP06, and GCP08) at doy 233 with network configuration scenarios of 12 IGS points on two quadrants (jat1), three quadrants (jat2), and four quadrants (jat3 and jat4). GNSS networks processing was done by GAMIT to obtain baseline vectors, followed by network processing usingparameter method of least squares adjustment. Networks processing with least squares adjustment aims to determine the most optimal by precision and reliability criterion. Results of this study indicate that network configuration with 12 IGS stations in the two quadrants provides the most accurate coordinates of CORS dam stations. Standard deviations value of CORS station given by jat1 configuration are in the range of 2.7 up to 4.1 cm in X-Z components, whereas standard deviations in the Y component are in the range 5.8 up to 6.9 cm. An optimization assessment based on network strength, precision, and reliability factors shows optimum configuration by jat1.

SHAVEL: A program for the spherical harmonic analysis of a horizontal vector field sampled in an equiangular grid on a sphere

A method for performing a spherical harmonic analysis, using observed horizontal components of a tangent vector on a sphere, is presented. The vector data samples are assumed to be provided in an equiangular grid, which essentially simplifies the least-squares analysis by making use of (1) the block diagonal structure of the normal equations of least squares, (2) the even–odd symmetry of the associated Legendre functions, and (3) the fast Fourier transform of mix-radix. The correct function of the program and its numerical precision is verified by applying it to a data set, derived by evaluating a given set of vector spherical harmonic coefficients. That the program works correctly is demonstrated by the excellent agreement between the input and output spherical harmonic coefficients. Program summaryProgram Title: SHAVELProgram Files doi:http://dx.doi.org/10.17632/nppz4y7wg7.1Licensing provisions: GPLv3Programming language: Fortran 2003, LinuxExternal routines: FFTPACK5.1, CPC program library SPHANClassification: 4.9, 4.10, 4.11Nature of problem: The least-squares analysis of horizontal vector field sampled in an equiangular grid on a sphere in terms of horizontal vector spherical harmonics.Solution method: The vector spherical harmonic coefficients of a horizontal vector field are estimated by the method of least-squares adjustment of data samples distributed in an equiangular grid on a sphere. For such a regular grid the normal matrix is sparse and allows the system of the normal equations to be decomposed into a series of subsystems according to azimuthal order m. The solution of each subsystem is sought by the Gauss elimination. The fast Fourier transform of mix-radix is implemented in (i) setting up the right-hand sides of the normal equations, and (ii) performing the spherical harmonic synthesis where the series of spherical harmonics are summed.

INVESTIGATION OF HORIZONTAL AND VERTICAL CONTROLS ON KNUST CAMPUS

Controls are coordinated horizontal or vertical positional data for land and geographic information systems forming a framework to which surveys are started, referenced and adjusted. Controls on KNUST campus were established over three decades ago with new ones being added as and when t h e y a r e required. Conventional survey methods like trilateration, triangulation and traversing were used and are still being used for densification of controls. These approaches involve the use of low precision instruments such as analogue/optical theodolites and leveling equipment. Since the establishment of these controls, t h e y have not been investigated but t h e y are being used for p r o j e c t development. A modern method that uses satellite positioning techniques such as GPS is currently in operation and has numerous advantages in the establishment of control networks. GPS control surveys were carried-out on seven controls and ellipsoidal coordinates were obtained in the World Geodetic System 84 reference frame. The Cartesian coordinates were projected onto the Universal Transverse Mercator frame. A two dimensional conformal transformation was done using existing KNUST boundary coordinates to Ghana National coordinates based on the War Office ellipsoid. Precise Level routines were carried out on the seven controls for the computations of orthometric heights. The method of least squares adjustment, root mean square errors (RMSE), standard errors (SE) and residuals derived were used to analyse the differences in horizontal positions and heights of the controls. The linear displacement between the computed and the existing coordinates were within the range of 0.015m to 0.014m. The RMSE were 0.048 and 0.106, whiles the SE were also 0.057 and 0.125 for the northings and eastings respectively. The computed and existing heights differed between -0.075m and -0.004m with a mean downward movement of 0.011m. Keywords: Surveying; coordinates; control points; GPS; precise level

Sinusoidal Wave Estimation Using Photogrammetry and Short Video Sequences.

The objective of the work is to model the shape of the sinusoidal shape of regular water waves generated in a laboratory flume. The waves are traveling in time and render a smooth surface, with no white caps or foam. Two methods are proposed, treating the water as a diffuse and specular surface, respectively. In either case, the water is presumed to take the shape of a traveling sine wave, reducing the task of the 3D reconstruction to resolve the wave parameters. The first conceived method performs the modeling part purely in 3D space. Having triangulated the points in a separate phase via bundle adjustment, a sine wave is fitted into the data in a least squares manner. The second method presents a more complete approach for the entire calculation workflow beginning in the image space. The water is perceived as a specular surface, and the traveling specularities are the only observations visible to the cameras, observations that are notably single image. The depth ambiguity is removed given additional constraints encoded within the law of reflection and the modeled parametric surface. The observation and constraint equations compose a single system of equations that is solved with the method of least squares adjustment. The devised approaches are validated against the data coming from a capacitive level sensor and on physical targets floating on the surface. The outcomes agree to a high degree.

Non-isostatic effects on crustal thickness: A study using CRUST2.0 in Fennoscandia

The crustal thickness can be determined based on gravimetric–isostatic and seismic models. Modelling crustal thickness by a gravimetric–isostatic model suffers from some problems. The isostatic assumption for compensating the topographic potential is incomplete, as there are other effects which should be considered. Using the isostatic hypothesis for determining the depth of crust causes some disturbing signals, non-isostatic effects, which influence the crustal thickness determination. Isostatic and non-isostatic compensations are the main issues in this paper. We present three methods to overcome the problem due to the disturbing signals, namely the approach by truncating the spherical harmonic approach, determination of non-isostatic correction using a seismic crustal thickness model (e.g., CRUST2.0) and combination of isostatic and seismic models by applying a least-squares adjustment method. The estimated results of the non-isostatic effects varies between −65.2 and 391.8mGal in Fennoscandia. The root mean squares difference of the crustal thickness obtained from the gravimetric–isostatic model and CRUST2.0 is improved up to six times (from 6.15 to 0.97km) when the non-isostatic effects are considered.

Measuring positional error of circular curve features in Geographic information systems (GIS)

This paper addresses the task of modeling positional uncertainty of circular features. The specific interest provided by this study is the consideration of the actual curve function in the measurement of circular curve error. This approach differs from the existing methods based on line segment approximation. A least-squares adjustment method, composed of a series of discrete points and the covariance of these parameters, is adopted to estimate the parameters of a circular curve. Two geometric measures are developed, based on the error ellipses of the points on the circular curve: the standard deviation-based measure and the maximum distance-based measure. A comparison of the two measures for circular curves was conducted by means of a simulation experiment. The implementation of the two geometric measures is further illustrated by two case studies; one describes the positional error of digital shorelines and the other digitizes circular road curves.

Parameter Estimation from Uncertain Data in Geometric Algebra

We show how standard parameter estimation methods can be applied to Geometric Algebra in order to fit geometric entities and operators to uncertain data. These methods are applied to three selected problems. One of which is the perspective pose estimation problem. We show experiments with synthetic data and compare the results of our algorithm with standard approaches. In general, our aim is to find multivectors that satisfy a particular constraint, which depends on a set of uncertain measurements. The specific problem and the type of multivector, representing a geometric entity or a geometric operator, determine the constraint. We consider the case of point measurements in Euclidian 3D-space, where the respective uncertainties are given by covariance matrices. We want to find a best fitting circle or line together with their uncertainty. This problem can be expressed in a linear manner, when it is embedded in the corresponding conformal space. In this space, it is also possible to evaluate screw motions and their uncertainty, in very much the same way. The parameter estimation method we use is a least-squares adjustment method, which is based on the so-called Gauss-Helmert model, also known as mixed model with constraints. For this linear model, we benefit from the implicit linearization when expressing our constraints in conformal space. The multivector representation of the entities we are interested in also allows their uncertainty to be expressed by covariance matrices. As a by-product, this method provides such covariance matrices.

Multilateration with the wide-angle laser ranging system: ranging performance and first ground-based validation experiment

A wide-angle laser ranging system has been developed for high-precision multilateration. A laser transmits short pulses in a widely diverging beam toward a network of cube-corner retroreflectors (CCRs). Returned echoes are detected, amplified, digitized, and timed. Impact of instrumental error sources, atmospheric refraction, and scintillation are discussed. The precision of the correlation method for the estimation of times of arrival and a least-squares adjustment method for the estimation of relative CCR coordinates is studied. Data from several ground-based experiments are analyzed. The positioning precision achieved is of 1-2.3 mm in relative radial coordinates and 1.8-3.8 cm in relative transverse coordinates. Results are consistent with a covariance analysis, demonstrating that, as follows: 1) systematic instrumental errors are well corrected; 2) outliers arising from coincident echoes are properly detected; 3) models for forward and inverse problems and estimation techniques are correct. A numerical simulation program based on these models and techniques can be used for optimizing instrumental aspects, providing extensive simulations in various conditions and processing data from future airborne experiments to which the system is devoted.

Marine gravity and geoid determination by optimal combination of satellite altimetry and shipborne gravimetry data

. Satellite altimetry derived geoid heights and marine gravity anomalies can be combined to determine a detailed gravity field over the oceans using the least-squares collocation method and spectral combination techniques. Least-squares collocation, least-squares adjustment in the frequency domain and input-output system theory are employed to determine the gravity field (both geoid and anomalies) and its errors. This paper intercompares these three techniques using simulated data. Simulation studies show that best results are obtained by the input-output system theory among the three prediction methods. The least-squares collocation method gives results which are very close to but a little bit worse than those obtained using input-output system theory. This slightly poorer performance of the least-squares collocation method can be explained by the fact that it uses isotropic structured covariance (thus approximate signal PSD information) while the system theory method uses detailed signal PSD information. The method of least-squares adjustment in the frequency domain gives the poorest results among these three methods because it uses less information than the other two methods (it ignores the signal PSDs). The computations also show that the least-squares collocation and input-output system theory methods are not as sensitive to noise levels as the least-squares adjustment in the frequency domain method is.

Performance of Wielandt-Streckeisen STS-1 seismometers in the tidal domain—preliminary results

Abstract The aim of this work is to show that STS-1 seismometers are able to provide a signal with a very long period vertical component which can be used for Earth tide studies provided a number of recording conditions are improved. We use data from the GEOSCOPE network for BNG (Bangui, Central African Republic), MBO (M'bour, Senegal), CRZF (Crozet, TAAF (Terres Australes et Antarctiques FranDc¸aises), Indian Ocean), ECH (Echery, France) and PAF (Port-aux-FranDc¸ais, Kerguelen, TAAF, Indian Ocean) stations operated by the Ecole et Observatoire de Physique du Globe de Strasbourg (EOPGS) and Institut FranDc¸ais de Recherche Scientifique pour le De´veloppement en Coope´ration (ORSTOM). These data are analysed with a least-squares adjustment method to fit the Earth tidal waves, and the results for these stations are compared. Special attention is paid to the comparison of results from ECH with those from the superconducting gravimeter of Strasbourg (J-9). For several stations used in this study (CRZF, MBO and ECH) the results are promising, but they are disappointing for others (PAF and BNG). We did not find any clear reason for such a difference in quality. We propose some improvements that will enable researchers to achieve a better recording of this long-period signal and obtain higher-quality data allowing gravimetric studies from global seismological networks.

A STRATEGY FOR DETECTION OF OUTLYING OBSERVATIONS IN MEASUREMENTS OF HIGH PRECISION

Undetected outliers may significantly influence the results and interpretation of high precision engineering surveys. The least-squares adjustment method smooths out the effect of outliers and make...

Electron Paramagnetic Resonance Studies of X-Irradiated KH2AsO4, KD2AsO4, RbH2AsO4, RbD2AsO4, CsH2AsO4, NH4H2AsO4, and ND4D2AsO4 (Ferroelectrics and Antiferroelectrics)

Electron paramagnetic resonance (EPR) studies of defect centers produced by x or γ irradiation of the hydrogen-bonded ferroelectrics KH2AsO4, KD2AsO4, RbH2AsO4, RbD2AsO4, and CsH2AsO4 and of the antiferroelectrics NH4H2AsO4 and ND4D2AsO4, are presented. The spectra, observed over 300–4.2°K, are characterized by a very large hyperfine interaction of the unpaired electron with the 75As (I=32) nucleus, and exhibit orthorhombic symmetry in general. The analysis thus required an exact diagonalization of the spin Hamiltonian. Such a procedure, coupled with the use of the method of least-squares adjustment, has resulted in accurate values for the hyperfine parameters describing the observed spectra, both for single crystal and powder samples. A comparison of the hyperfine parameters with those reported previously for the AsO44− in KH2AsO4 has enabled us to identify lines from this center in all the above crystals. The parameters are found to be sensitive to change of the cation and also to a variation of the sample temperature. This indicates that EPR can be used as a microscopic probe for studying hydrogen bonding and the proton-lattice motion in these compounds. An observed hydrogen-deuterium isotope effect on the hyperfine parameters has also been discussed. Absence of proton superhyperfine for AsO44− in RbH2AsO4, and CsH2AsO4 has been discussed in the light of preliminary ENDOR experiments on RbH2AsO4. Accurate values have also been obtained for the spin Hamiltonian parameters of several other centers which show fairly similar spectra, for both the high and low temperature phases. Here also the 75As hyperfine parameters show isotope effects, both for the change of the cation and for the substitution of D in place of H. All these centers are assigned to a family of AsO32− centers. A previously reported proton superhyperfine structure, which cast strong doubt on the identification of these centers, was not observed. This lends further support to our assignment. Possible mechanisms for the formation of these centers are discussed, but no definite answer could be obtained through these studies alone.

LEAST SQUARES ADJUSTMENT OF E.D.M. TRAVERSES

Abstract Now that it is becoming commonplace to measure the lengths of traverse legs by means of Electronic Distance Measurement equipment, several surveyors have asked the author how best the method of least squares adjustment of a traverse, suggested in a previous article, [2], can be modified so that weights assigned to distance measurements are sensitive to the zero error of the E.D.M. instrument used. This article suggests a means whereby this may be done, and illustrates the method by a numerical example. Eleven commercial traverses are analysed, in which the E.D.M. instrument used was a Wild DI.10 Distomat, and the variations between the Least Squares solution and the Bowditch method are shown.

LEAST SQUARES ADJUSTMENT OF NETWORKS

AbstractThe method of least squares adjustment described in this paper was developed for large scale networks of height and gravity data. This data is obtained by observers with instruments travelling from point to point taking readings which reduce to the differences in height or gravity between adjacent points. Thus the paths of the observers define a network in which the observation points are nodes and the paths between them represent the measured differences.

On the question of dispersion in the first preliminary seismic waves1

AbstractSummaryBy use of the Byerly-Jeffreys travel-time curve for P, and Geiger's method of least-squares adjustment, the epicenter of the Alaskan earthquake of October 24, 1927, was placed at5 7 ° 26 ' ± 5 0 ' N . 13 7 ° 03 ' ± 1 9 ' W .and the time of occurrence was placed at 15h 59m 55s ± 2s, G.M.C.T.A second solution was obtained using Mohorovičić's multiple travel-time curves for P. The co-ordinates of the epicenter were the same as those given above, but the time of occurrence was found to be 16h00m, G.M.C.T.It has been held by some seismologists that anomalous dispersion can be observed in the first preliminary waves; i.e., that shorter periods travel faster than long ones. Investigations of periods were made with a view to testing this hypothesis, with the following results:The general conclusion is that observation of periods gives no evidence for dispersion in waves of longitudinal type.It is shown that, if dispersion did exist, the travel time of the beginning would be a continuous function of epicentral distance, and, therefore, Mohorovičić's curves are not evidence for dispersion. The observations of the epicentral distances at which P2, P1, and Pn are most frequently recorded first are contrary to dispersion.In the Alaskan earthquake the distribution of first motion (condensation or rarefaction) is very complicated. Dispersion offers no explanation for this fact, and it is believed that complex movements at the source are responsible for the observed distribution.