Prediction of Deviations of the Vertical Using Heterogeneous Data

1976 ◽  
Vol 30 (2) ◽  
pp. 97-108
Author(s):  
Gérard Lachapelle
Keyword(s):  
Least Squares ◽  
Gravity Anomalies ◽  
Heterogeneous Data ◽  
Numerical Results ◽  
Geodetic Data ◽  
West Germany ◽  
Mountainous Areas ◽  
Least Squares Collocation ◽  
Dynamic Information ◽  
Geopotential Coefficients

A method for estimating deviations of the vertical from a combination of topographic-isostatic deviations of the vertical and dynamic information in the form of geopotential coefficients is presented. The method is especially well suited for large areas, either continental or oceanic, where no geodetic measurements, such as gravity anomalies or deviations of the vertical, are available. It is ideally applicable in mountainous areas and along coastlines where the deviations depend greatly on the topography. Numerical results using topographic-isostatic data calculated in Canada, Switzerland and West Germany are presented. Furthermore, if geodetic data such as observed deviations of the vertical and gravity anomalies are available in the area considered, they can be combined with existing estimated deviations by using least squares collocation to achieve a greater accuracy.

Astrogravimetric Leveling by Least Squares Collocation

1977 ◽  
Vol 31 (2) ◽  
pp. 133-150
Author(s):  
Gérard Lachapelle
Keyword(s):  
Least Squares ◽  
Fundamental Equation ◽  
Gravity Anomalies ◽  
Least Squares Collocation ◽  
Empirical Formulas ◽  
Covariance Functions ◽  
Rugged Topography ◽  
The Difference ◽  
Geoid Undulations ◽  
Good Agreement

The application of least squares collocation to astrogravimetric leveling is described and the fundamental equation of astrogravimetric leveling in least squares collocation is derived. This method is easier and quicker to use than classical astrogravimetric leveling and is therefore well suited for extensive application. Six different covariance functions for use in flat and rugged topography areas are presented and comparisons are made between the accuracy estimates obtained when using these covariance functions and those obtained when using Bomford’s empirical formulas for astrogeodetic leveling; results are in very good agreement. Two astrogeodetic profiles located in Canada were used for this purpose. Finally, comparisons are made between the accuracy estimates obtained for the difference of geoid undulations calculated by astrogravimetric leveling when using the covariance functions mentioned and different configurations of observed deviations of the vertical and gravity anomalies.

Numerical solutions of the GEW equation using MLS collocation method

2017 ◽  
Vol 28 (01) ◽  
pp. 1750011
Author(s):  
Ayşe Gül Kaplan ◽  
Yılmaz Dereli
Keyword(s):  
Least Squares ◽  
Collocation Method ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Numerical Results ◽  
Moving Least Squares ◽  
Test Problems ◽  
Least Squares Collocation ◽  
Energy And Momentum

In this paper, the generalized equal width wave (GEW) equation is solved by using moving least squares collocation (MLSC) method. To test the accuracy of the method some numerical experiments are presented. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution was known [Formula: see text], [Formula: see text] error norms and pointwise rates of convergence were calculated. Also mass, energy and momentum invariants were calculated for every test problems. Obtained numerical results are compared with some earlier works. It is seen that the method is very efficient and reliable due to obtained numerical results are very satisfactorily. Stability analysis of difference equation was done by applying the moving least squares collocation method for GEW equation.

A priori noise and regularization in least squares collocation of gravity anomalies

2013 ◽  
Vol 62 (2) ◽  
pp. 199-216 ◽  
Author(s):  
Wojciech Jarmołowski
Keyword(s):  
Least Squares ◽  
Spatial Resolution ◽  
Gravity Data ◽  
A Priori ◽  
Gravity Anomalies ◽  
Geopotential Model ◽  
Large Set ◽  
Least Squares Collocation ◽  
Gravity Field Model ◽  
Two Parameters

Abstract The paper describes the estimation of covariance parameters in least squares collocation (LSC) by the cross-validation (CV) technique called leave-one-out (LOO). Two parameters of Gauss-Markov third order model (GM3) are estimated together with a priori noise standard deviation, which contributes significantly to the covariance matrix composed of the signal and noise. Numerical tests are performed using large set of Bouguer gravity anomalies located in the central part of the U.S. Around 103 000 gravity stations are available in the selected area. This dataset, together with regular grids generated from EGM2008 geopotential model, give an opportunity to work with various spatial resolutions of the data and heterogeneous variances of the signal and noise. This plays a crucial role in the numerical investigations, because the spatial resolution of the gravity data determines the number of gravity details that we may observe and model. This establishes a relation between the spatial resolution of the data and the resolution of the gravity field model. This relation is inspected in the article and compared to the regularization problem occurring frequently in data modeling.

scholarly journals Estimation of gravity noise variance and signal covariance parameters in least squares collocation with considering data resolution

2016 ◽  
Vol 59 (1) ◽  
Author(s):  
Wojciech Jarmołowski
Keyword(s):  
Least Squares ◽  
Likelihood Function ◽  
A Priori ◽  
Gravity Anomalies ◽  
Signal Spectrum ◽  
Noise Variance ◽  
Covariance Model ◽  
Least Squares Collocation ◽  
Uncorrelated Noise ◽  
Data Resolution

<p>The article describes an implementation of the negative log-likelihood function in the determination of uncorrelated noise standard deviation together with the parameters of spherical signal covariance model in least squares collocation (LSC) of gravity anomalies. The correctness and effectiveness of restricted maximum likelihood (REML) estimates are fully validated by leave-one-out validation (LOO). These two complementary methods give an opportunity to inspect the parametrization of the signal and uncorrelated noise in details and can provide some guidance related to the estimation of individual parameters. The study provides the practical proof that noise variance is related with the data resolution, which is often neglected and the information on a priori noise variance is based on the measurement error. The data have been downloaded from U.S. terrestrial gravity database and resampled to enable an analysis with four different horizontal resolutions. These data are intentionally the same, as in the previous study of the same author, with the application of the planar covariance model. The aim is to compare the results from two different covariance models, which have different covariance approximation at larger distances. The most interesting outputs from this study confirm previous observations on the relations of the data resolution, a priori noise variance, signal spectrum and LSC accuracy.</p>

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