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.

scholarly journals On the uncertainty of height anomaly differences predicted by least-squares collocation

2020 ◽  
Vol 10 (1) ◽  
pp. 53-61
Author(s):  
E. Mysen
Keyword(s):  
Least Squares ◽  
Spatial Separation ◽  
Height Anomaly ◽  
Surface Model ◽  
Least Squares Collocation ◽  
Normal Heights ◽  
Gravimetric Quasigeoid ◽  
The Difference ◽  
Grid Interpolation ◽  
Gps Observations

AbstractA network of pointwise available height anomalies, derived from levelling and GPS observations, can be densified by adjusting a gravimetric quasigeoid using least-squares collocation. The resulting type of Corrector Surface Model (CSM) is applied by Norwegian surveyors to convert ellipsoidal heights to normal heights expressed in the official height system NN2000. In this work, the uncertainty related to the use of a CSM to predict differences in height anomaly was sought. As previously, the application of variograms to determine the local statistical properties of the adopted collocation model led to predictions that were consistent with their computed uncertainties. For the purpose of predicting height anomaly differences, the effect of collocation was seen to be moderate in general for the small spatial separations considered (< 10 km). However, the relative impact of collocation could be appreciable, and increasing with distance, near the network. At last, it was argued that conservative uncertainties of height anomaly differences may be obtained by rescaling output of a grid interpolation by \sqrt \Delta, where Δ is the spatial separation of the two locations for which the difference is sought.

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.

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 THE UNIFIED LEVELLING NETWORK OF SARAWAK AND ITS ADJUSTMENT

2016 ◽  
Vol XLII-4/W1 ◽  
pp. 271-275
Author(s):  
Z. A. M. Som ◽  
A. M. Yazid ◽  
T. K. Ming ◽  
N. M. Yazid
Keyword(s):  
Least Squares ◽  
Computational Procedure ◽  
Reference Network ◽  
Major Improvement ◽  
The Past ◽  
Least Squares Adjustment ◽  
The Difference ◽  
Parametric Adjustment ◽  
First Time ◽  
Good Agreement

The height reference network of Sarawak has seen major improvement in over the past two decades. The most significant improvement was the establishment of extended precise leveling network of which is now able to connect all three major datum points at Pulau Lakei, Original and Bintulu. Datum by following the major accessible routes across Sarawak. This means the leveling network in Sarawak has now been inter-connected and unified. By having such a unified network leads to the possibility of having a common single least squares adjustment been performed for the first time. The least squares adjustment of this unified levelling network was attempted in order to compute the height of all Bench Marks established in the entire levelling network. The adjustment was done by using MoreFix levelling adjustment package developed at FGHT UTM. The computational procedure adopted is linear parametric adjustment by minimum constraint. Since Sarawak has three separate datums therefore three separate adjustments were implemented by utilizing datum at Pulau Lakei, Original Miri and Bintulu Datum respectively. Results of the MoreFix unified adjustment agreed very well with adjustment repeated using Starnet. Further the results were compared with solution given by Jupem and they are in good agreement as well. The difference in height analysed were within 10mm for the case of minimum constraint at Pulau Lakei datum and with much better agreement in the case of Original Miri Datum.

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>

Vertical Gravity Gradient Modeling by 3-D Least Squares Collocation and its impact on quasigeoid-geoid separation term

2020 ◽  
Author(s):  
Gonca Ahi ◽  
Yunus Aytaç Akdoğan ◽  
Hasan Yıldız
Keyword(s):  
Least Squares ◽  
Gravity Gradient ◽  
Mountainous Areas ◽  
Least Squares Collocation ◽  
Western Turkey ◽  
Vertical Gravity Gradient ◽  
Gravity Observation ◽  
Vertical Gravity ◽  
Geoid Undulations ◽  
Gradient Modeling

&lt;p&gt;For the quasi-geoid determination by 3-D Least Squares Collocation (LSC) in the context of Molodensky&amp;#8217;s approach, there is no need to measured or modelled vertical gravity gradient (VGG) as the 3-D LSC takes the varying heights of the gravity observation points into account. However, the use of measured or modelled VGG instead of the thereotical value is expected to improve the quasigeoid-geoid separation term particularly in mountainous areas. The VGG measurements are found to be different from the theoretical value in the range of - % 25 and + % 39 in western Turkey. Previously there has been no study using modelled VGGs for gravimetric geoid modelling in Turkey. VGGs are modelled by 3-D Least Squares Collocation (LSC) in remove-restore approach and validated by terrestrial VGG measurements in western Turkey. The effect of using modelled VGG instead of the theoretical one in quasigeoid-to-geoid separation term is found to be significant. The quasi-geoid computed by 3-D LSC in western Turkey is converted to geoids using theoretical or modelled VGG values and compared with GPS/levelling geoid-undulations.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;

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