Combination of regional and global geoid models at continental scale: application to Iranian geoid

High precision geoid determination is a challenging task at the national scale. Many efforts have been conducted to determine precise geoid, locally or globally. Geoid models have different precision depending on the type of information and the strategy employed when calculating the models. This contribution addresses the challenging problem of combining different regional and global geoid models, possibly combined with the geometric geoid derived from GNSS/leveling observations. The ultimate goal of this combination is to improve the precision of the combined model. We employ fitting an appropriate geometric surface to the geoid heights and estimating its (co)variance components. The proposed functional model uses the least squares 2D bi-cubic spline approximation (LS-BICSA) theory, which approximates the geoid model using a 2D spline surface fitted to an arbitrary set of data points in the region. The spline surface consists of third- order polynomial pieces that are smoothly connected together, imposing some continuity conditions at their boundaries. In addition, the least-squares variance component estimation (LS- VCE) is used to estimate precise weights and correlation among different models. We apply this strategy to the combined adjustment of the high-degree global gravitational model EIGEN-6C4, the regional geoid model IRG2016, and the Iranian geometric geoid derived from GNSS/leveling data. The accuracy of the constructed surface is investigated with five randomly selected subsamples of check points. The optimal combination of the two geoid models along with the GNSS/leveling data shows a reduction of 21 mm (~20%) in the RMSE values of discrepancies at the check points.

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Total Least Squares Spline Approximation

Mathematics ◽

10.3390/math7050462 ◽

2019 ◽

Vol 7 (5) ◽

pp. 462 ◽

Cited By ~ 3

Author(s):

Frank Neitzel ◽

Nikolaj Ezhov ◽

Svetozar Petrovic

Keyword(s):

Least Squares ◽

Spline Approximation ◽

Functional Model ◽

Total Least Squares ◽

Deformation Monitoring ◽

Random Errors ◽

Laser Scanners ◽

Engineering Geodesy ◽

Continuous Curvature ◽

Cubic Polynomials

Spline approximation, using both values y i and x i as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the special case of equally weighted and uncorrelated observations, the resulting error vectors are orthogonal to the graph of the spline function and hence can be utilized for deformation monitoring purposes. Based on a functional model that uses cubic polynomials and constraints for continuity, smoothness and continuous curvature, the case of spline approximation with both the values y i and x i as observations is considered. In this case, some of the columns of the functional matrix contain observations and are thus subject to random errors. In the literature on mathematics and statistics this case is known as an errors-in-variables (EIV) model for which a so-called “total least squares” (TLS) solution can be computed. If weights for the observations and additional constraints for the unknowns are introduced, a “constrained weighted total least squares” (CWTLS) problem is obtained. In this contribution, it is shown that the solution for this problem can be obtained from a rigorous solution of an iteratively linearized Gauss-Helmert (GH) model. The advantage of this model is that it does not impose any restrictions on the form of the functional relationship between the involved quantities. Furthermore, dispersion matrices can be introduced without limitations, even the consideration of singular ones is possible. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving CWTLS problems. Using a numerical example it is demonstrated how the GH model can be applied to obtain a spline approximation with orthogonal error vectors. The error vectors are compared with those derived from two least squares (LS) approaches.

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Producing Follower Motions Through Their Digitized Cam Contours

Journal of Computing and Information Science in Engineering ◽

10.1115/1.1505852 ◽

2002 ◽

Vol 2 (2) ◽

pp. 98-105 ◽

Cited By ~ 3

Author(s):

Der-Min Tsay ◽

Meng-Hung Huang ◽

Hui-Chun Ho

Keyword(s):

Least Squares ◽

Contact Point ◽

Spline Approximation ◽

B Spline ◽

Theoretical Design ◽

Analytical Results ◽

Surface Data ◽

Cam Follower ◽

Data Points ◽

Computing Procedure

A computing procedure that can be used to generate follower motions with a higher degree of accuracy by using their digitized 2-D cam surfaces is presented. Based on the geometric relationships at the contact point between a planar cam and its follower, the reverse functions that can be solved to find the follower motion of the cam-follower mechanism are established. To approximate the digitized cam surfaces and to create their derivatives required in determining the follower motions, a least squares periodic B-spline approximation is also introduced. In comparison with analytical results obtained from a theoretical design, the procedure is verified for its accuracy and reliability. Furthermore, by using the cam surface data points taken from a CMM, a practical example is given to show the effectiveness and usefulness of the proposed techniques.

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EXPRESS: Baseline Correction Based on a Search Algorithm from Artificial Intelligence

Applied Spectroscopy ◽

10.1177/0003702820977512 ◽

2020 ◽

pp. 000370282097751

Author(s):

Xin Wang ◽

Xia Chen

Keyword(s):

Artificial Intelligence ◽

Objective Function ◽

Least Squares ◽

Least Squares Method ◽

Search Algorithm ◽

Absolute Error ◽

Search Problem ◽

Baseline Correction ◽

Current Spectrum ◽

Data Points

Many spectra have a polynomial-like baseline. Iterative polynomial fitting (IPF) is one of the most popular methods for baseline correction of these spectra. However, the baseline estimated by IPF may have substantially error when the spectrum contains significantly strong peaks or have strong peaks located at the endpoints. First, IPF uses temporary baseline estimated from the current spectrum to identify peak data points. If the current spectrum contains strong peaks, then the temporary baseline substantially deviates from the true baseline. Some good baseline data points of the spectrum might be mistakenly identified as peak data points and are artificially re-assigned with a low value. Second, if a strong peak is located at the endpoint of the spectrum, then the endpoint region of the estimated baseline might have significant error due to overfitting. This study proposes a search algorithm-based baseline correction method (SA) that aims to compress sample the raw spectrum to a dataset with small number of data points and then convert the peak removal process into solving a search problem in artificial intelligence (AI) to minimize an objective function by deleting peak data points. First, the raw spectrum is smoothened out by the moving average method to reduce noise and then divided into dozens of unequally spaced sections on the basis of Chebyshev nodes. Finally, the minimal points of each section are collected to form a dataset for peak removal through search algorithm. SA selects the mean absolute error (MAE) as the objective function because of its sensitivity to overfitting and rapid calculation. The baseline correction performance of SA is compared with those of three baseline correction methods: Lieber and MahadevanâJansen method, adaptive iteratively reweighted penalized least squares method, and improved asymmetric least squares method. Simulated and real FTIR and Raman spectra with polynomial-like baselines are employed in the experiments. Results show that for these spectra, the baseline estimated by SA has fewer error than those by the three other methods.

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Generalised DOPs with Consideration of the Influence Function of Signal-in-Space Errors

Journal of Navigation ◽

10.1017/s0373463311000415 ◽

2011 ◽

Vol 64 (S1) ◽

pp. S3-S18 ◽

Cited By ~ 20

Author(s):

Yuanxi Yang ◽

Jinlong Li ◽

Junyi Xu ◽

Jing Tang

Keyword(s):

Least Squares ◽

Stochastic Models ◽

Influence Function ◽

A Priori ◽

Functional Model ◽

Global Navigation Satellite Systems ◽

Parameter Estimates ◽

Model Parameters ◽

Integrated Navigation ◽

Satellite Systems

Integrated navigation using multiple Global Navigation Satellite Systems (GNSS) is beneficial to increase the number of observable satellites, alleviate the effects of systematic errors and improve the accuracy of positioning, navigation and timing (PNT). When multiple constellations and multiple frequency measurements are employed, the functional and stochastic models as well as the estimation principle for PNT may be different. Therefore, the commonly used definition of “dilution of precision (DOP)” based on the least squares (LS) estimation and unified functional and stochastic models will be not applicable anymore. In this paper, three types of generalised DOPs are defined. The first type of generalised DOP is based on the error influence function (IF) of pseudo-ranges that reflects the geometry strength of the measurements, error magnitude and the estimation risk criteria. When the least squares estimation is used, the first type of generalised DOP is identical to the one commonly used. In order to define the first type of generalised DOP, an IF of signal–in-space (SIS) errors on the parameter estimates of PNT is derived. The second type of generalised DOP is defined based on the functional model with additional systematic parameters induced by the compatibility and interoperability problems among different GNSS systems. The third type of generalised DOP is defined based on Bayesian estimation in which the a priori information of the model parameters is taken into account. This is suitable for evaluating the precision of kinematic positioning or navigation. Different types of generalised DOPs are suitable for different PNT scenarios and an example for the calculation of these DOPs for multi-GNSS systems including GPS, GLONASS, Compass and Galileo is given. New observation equations of Compass and GLONASS that may contain additional parameters for interoperability are specifically investigated. It shows that if the interoperability of multi-GNSS is not fulfilled, the increased number of satellites will not significantly reduce the generalised DOP value. Furthermore, the outlying measurements will not change the original DOP, but will change the first type of generalised DOP which includes a robust error IF. A priori information of the model parameters will also reduce the DOP.

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ETH–GM21: A new gravimetric geoid model of Ethiopia developed using the least-squares collocation method

Journal of African Earth Sciences ◽

10.1016/j.jafrearsci.2021.104313 ◽

2021 ◽

pp. 104313

Author(s):

Yalew Ephrem ◽

Walyeldeen Godah ◽

Malgorzata Szelachowska ◽

Robert Tenzer

Keyword(s):

Least Squares ◽

Collocation Method ◽

Gravimetric Geoid ◽

Least Squares Collocation ◽

Geoid Model

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Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

16th Design Automation Conference: Volume 1 — Computer Aided and Computational Design ◽

10.1115/detc1990-0032 ◽

1990 ◽

Author(s):

Joanna M. Brown ◽

Malcolm I. G. Bloor ◽

M. Susan Bloor ◽

Michael J. Wilson

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Spline Approximation ◽

The Finite Element Method ◽

B Spline ◽

B Splines ◽

Spline Surface ◽

Spline Surfaces ◽

Spline Basis ◽

Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

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NURBS Surface Reconstruction From a Large Set of Image and World Data Points

19th Design Automation Conference: Volume 2 — Design Optimization; Geometric Modeling and Tolerance Analysis; Mechanism Synthesis and Analysis; Decomposition and Design Optimization ◽

10.1115/detc1993-0373 ◽

1993 ◽

Author(s):

Vassilios E. Theodoracatos ◽

Vasudeva Bobba

Keyword(s):

Least Squares ◽

Vision System ◽

Physical Object ◽

Free Form ◽

Approximation Technique ◽

Global Coordinate System ◽

Large Set ◽

Affine Invariant ◽

Nurbs Surface ◽

Data Points

Abstract In this paper an approach is presented for the generation of a NURBS (Non-Uniform Rational B-splines) surface from a large set of 3D data points. The main advantage of NURBS surface representation is the ability to analytically describe both, precise quadratic primitives and free-form curves and surfaces. An existing three dimensional laser-based vision system is used to obtain the spatial point coordinates of an object surface with respect to a global coordinate system. The least-squares approximation technique is applied in both the image and world space of the digitized physical object to calculate the homogeneous vector and the control net of the NURBS surface. A new non-uniform knot vectorization process is developed based on five data parametrization techniques including four existing techniques, viz., uniform, chord length, centripetal, and affine invariant angle and a new technique based on surface area developed in this study. Least-squares error distribution and surface interrogation are used to evaluate the quality of surface fairness for a minimum number of NURBS control points.

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Outlier removal method for the reﬁnement of optical measured displacement ﬁeld based on critical factor least squares and subdomain division

Measurement Science and Technology ◽

10.1088/1361-6501/ac476c ◽

2021 ◽

Author(s):

Bo Wang ◽

Chen Sun ◽

Keming Zhang ◽

Jubing Chen

Keyword(s):

Least Squares ◽

Displacement Field ◽

Critical Factor ◽

Least Square ◽

Outlier Removal ◽

Full Field ◽

Removal Method ◽

Strain Estimation ◽

Data Points ◽

Measured Displacement

Abstract As a representative type of outlier, the abnormal data in displacement measurement often inevitably occurred in full-field optical metrology and significantly affected the further evaluation, especially when calculating the strain field by differencing the displacement. In this study, an outlier removal method is proposed which can recognize and remove the abnormal data in optically measured displacement field. A iterative critical factor least squares algorithm (CFLS) is developed which distinguishes the distance between the data points and the least square plane to identify the outliers. A successive boundary point algorithm is proposed to divide the measurement domain to improve the applicability and effectiveness of the CFLS algorithm. The feasibility and precision of the proposed method are discussed in detail through simulations and experiments. Results show that the outliers are reliably recognized and the precision of the strain estimation is highly improved by using these methods.

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Autoregressive processes with infinite variance

Journal of Applied Probability ◽

10.1017/s0021900200105121 ◽

1977 ◽

Vol 14 (02) ◽

pp. 411-415 ◽

Cited By ~ 5

Author(s):

E. J. Hannan ◽

Marek Kanter

Keyword(s):

Least Squares ◽

Autoregressive Model ◽

Infinite Variance ◽

Autoregressive Processes ◽

Stable Law ◽

Least Squares Estimators ◽

Data Points

The least squares estimators β i(N), j = 1, …, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered when the disturbances have a distribution attracted to a stable law of index α < 2. It is shown that N1/δ(β i(N) – β) converges almost surely to zero for any δ > α. Some comments are made on alternative definitions of the βi (N).

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Can One Promote a Delegation of the Bilateral Aid to Multilateral Donors to Improve Foreign Aid Effect on Economic Growth in ECOWAS (Note 1) Countries?

Advances in Politics and Economics ◽

10.22158/ape.v2n1p51 ◽

2019 ◽

Vol 2 (1) ◽

pp. 51

Author(s):

W. Jean Marie Kébré

Keyword(s):

Economic Growth ◽

Panel Data ◽

Least Squares ◽

Foreign Aid ◽

Bilateral Aid ◽

Multilateral Aid ◽

Multilateral Institutions ◽

Negative Effect ◽

High Degree ◽

The Relationship

This article analyzes the relationship between external aid and economic growth in the ECOWAS region, with a focus on bilateral and multilateral aid effects. The key idea behind this analysis is an argument of Svensson (2000) that multilateral aid is more effective than bilateral aid because of the high degree of altruism of bilateral donors. He therefore suggested a delegation of bilateral aid to multilateral institutions. To appreciate his suggestion, this analysis used panel data from the 16 ECOWAS countries from the period 1984 to 2014. The results of the estimates, based on the dynamic least squares estimator (DOLS), show a negative effect of foreign aid on economic growth. This negative effect on economic growth persists when the components of aid are introduced into the model. In addition, results highlight that governance is a channel through which foreign aid affect positively economic growth. In these conditions, bilateral aid is more effective on economic growth than multilateral aid. These results about foreign aid received by ECOWAS countries invalidates Svensson’s (<a title="Svensson, 2000 #5" href="#_ENREF_1">2000</a>) theory. Therefore, a delegation of bilateral aid to multilateral institutions is not relevant because bilateral aid contributes more to economic growth if governance is taken into account.

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