least squares collocation
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2022 ◽  
Vol 9 ◽  
Author(s):  
Hamad Al-Ajami ◽  
Ahmed Zaki ◽  
Mostafa Rabah ◽  
Mohamed El-Ashquer

A new gravimetric geoid model, the KW-FLGM2021, is developed for Kuwait in this study. This new geoid model is driven by a combination of the XGM2019e-combined global geopotential model (GGM), terrestrial gravity, and the SRTM 3 global digital elevation model with a spatial resolution of three arc seconds. The KW-FLGM2021 has been computed by using the technique of Least Squares Collocation (LSC) with Remove-Compute-Restore (RCR) procedure. To evaluate the external accuracy of the KW-FLGM2021 gravimetric geoid model, GPS/leveling data were used. As a result of this evaluation, the residual of geoid heights obtained from the KW-FLGM2021 geoid model is 2.2 cm. The KW-FLGM2021 is possible to be recommended as the first accurate geoid model for Kuwait.


2021 ◽  
Author(s):  
◽  
Rachelle Winefield

<p>Each gravity observation technique has different parameters and contributes to different pieces of the gravity spectrum. This means that no one gravity dataset is able to model the Earth’s gravity field completely and the best gravity map is one derived from many sources. Therefore, one of the challenges in gravity field modelling is combining multiple types of heterogeneous gravity datasets.  The aim of this study is to determine the optimal method to produce a single gravity map of the Canterbury case study area, for the purposes of use in geoid modelling.  This objective is realised through the identification and application of a four-step integration process: purpose, data, combination and assessment. This includes the evaluation of three integration methods: natural neighbour, ordinary kriging and least squares collocation.  As geoid modelling requires the combination of gravity datasets collected at various altitudes, it is beneficial to be able to combine the dataset using an integration method which operates in a three-dimensional space. Of the three integration methods assessed, least squares collocation is the only integration method which is able to perform this type of reduction.  The resulting product is a Bouguer anomaly map of the Canterbury case study area, which combines satellite altimetry, terrestrial, ship-borne, airborne, and satellite gravimetry using least squares collocation.</p>


2021 ◽  
Author(s):  
◽  
Rachelle Winefield

<p>Each gravity observation technique has different parameters and contributes to different pieces of the gravity spectrum. This means that no one gravity dataset is able to model the Earth’s gravity field completely and the best gravity map is one derived from many sources. Therefore, one of the challenges in gravity field modelling is combining multiple types of heterogeneous gravity datasets.  The aim of this study is to determine the optimal method to produce a single gravity map of the Canterbury case study area, for the purposes of use in geoid modelling.  This objective is realised through the identification and application of a four-step integration process: purpose, data, combination and assessment. This includes the evaluation of three integration methods: natural neighbour, ordinary kriging and least squares collocation.  As geoid modelling requires the combination of gravity datasets collected at various altitudes, it is beneficial to be able to combine the dataset using an integration method which operates in a three-dimensional space. Of the three integration methods assessed, least squares collocation is the only integration method which is able to perform this type of reduction.  The resulting product is a Bouguer anomaly map of the Canterbury case study area, which combines satellite altimetry, terrestrial, ship-borne, airborne, and satellite gravimetry using least squares collocation.</p>


Author(s):  
В.А. Беляев

Исследованы возможности численного метода коллокации и наименьших квадратов (КНК) на примерах кусочно-полиномиального решения задачи Дирихле для уравнений Пуассона и типа диффузии-конвекции с особенностями в виде больших градиентов и разрыва решения на границах раздела двух подобластей. Предложены и реализованы новые hp-варианты метода КНК, основанные на присоединении внутри области малых и/или вытянутых нерегулярных ячеек, отсекаемых криволинейной границей раздела от исходных прямоугольных ячеек сетки, к соседним самостоятельным ячейкам. Выписываются с учетом особенности условия согласования между собой кусков решения в ячейках, примыкающих с разных сторон к границе раздела. Проведено сравнение результатов, полученных методом КНК и другими высокоточными методами. Показаны преимущества и достоинства метода КНК. Для ускорения итерационного процесса применены современные алгоритмы и методы: предобуславливание; свойства локальной системы координат в методе КНК; ускорение, основанное на подпространствах Крылова; операция продолжения на многосеточном комплексе; распараллеливание. Исследовано влияние этих способов на количество итераций и время расчетов при аппроксимации полиномами различных степеней. The capabilities of the numerical least-squares collocation (LSC) method of the piecewise polynomial solution of the Dirichlet problem for the Poisson and diffusion-convection equations are investigated. Examples of problems with singularities such as large gradients and discontinuity of the solution at interfaces between two subdomains are considered. New hp-versions of the LSC method based on the merging of small and/or elongated irregular cells to neighboring independent cells inside the domain are proposed and implemented. They cut off by a curvilinear interface from the original rectangular grid cells. Taking into account the problem singularity the matching conditions between the pieces of the solution in cells adjacent from different sides to the interface are written out. The results obtained by the LSC method are compared with other high-accuracy methods. Advantages of the LSC method are shown. For acceleration of an iterative process modern algorithms and methods are applied: preconditioning, properties of the local coordinate system in the LSC method, Krylov subspaces; prolongation operation on a multigrid complex; parallelization. The influence of these methods on iteration numbers and computation time at approximation by polynomials of various degrees is investigated.


2021 ◽  
Vol 183 (22) ◽  
pp. 46-50
Author(s):  
Oyedepo Taiye ◽  
Ayinde Muhammed Abdullahi ◽  
Adenipekun Adewale Emmanuel ◽  
Ajileye Ganiyu

2021 ◽  
Vol 95 (8) ◽  
Author(s):  
P. Zingerle ◽  
R. Pail ◽  
M. Willberg ◽  
M. Scheinert

AbstractWe present a partition-enhanced least-squares collocation (PE-LSC) which comprises several modifications to the classical LSC method. It is our goal to circumvent various problems of the practical application of LSC. While these investigations are focused on the modeling of the exterior gravity field the elaborated methods can also be used in other applications. One of the main drawbacks and current limitations of LSC is its high computational cost which grows cubically with the number of observation points. A common way to mitigate this problem is to tile the target area into sub-regions and solve each tile individually. This procedure assumes a certain locality of the LSC kernel functions which is generally not given and, therefore, results in fringe effects. To avoid this, it is proposed to localize the LSC kernels such that locality is preserved, and the estimated variances are not notably increased in comparison with the classical LSC method. Using global covariance models involves the calculation of a large number of Legendre polynomials which is usually a time-consuming task. Hence, to accelerate the creation of the covariance matrices, as an intermediate step we pre-calculate the covariance function on a two-dimensional grid of isotropic coordinates. Based on this grid, and under the assumption that the covariances are sufficiently smooth, the final covariance matrices are then obtained by a simple and fast interpolation algorithm. Applying the generalized multi-variate chain rule, also cross-covariance matrices among arbitrary linear spherical harmonic functionals can be obtained by this technique. Together with some further minor alterations these modifications are implemented in the PE-LSC method. The new PE-LSC is tested using selected data sets in Antarctica where altogether more than 800,000 observations are available for processing. In this case, PE-LSC yields a speed-up of computation time by a factor of about 55 (i.e., the computation needs only hours instead of weeks) in comparison with the classical unpartitioned LSC. Likewise, the memory requirement is reduced by a factor of about 360 (i.e., allocating memory in the order of GB instead of TB).


Author(s):  
Michael Hanke ◽  
Roswitha März

AbstractIn the two parts of the present note we discuss questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. We provide in Part 1 a robust selection of basis functions and collocation points to design the discrete problem whereas we analyze the discrete least-squares problem and substantiate a procedure for its numerical solution in Part 2.


Author(s):  
Michael Hanke ◽  
Roswitha März

AbstractIn the two parts of the present note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. In the present Part 1, we provide a robust selection of basis functions and collocation points to design the discrete problem. We substantiate a procedure for its numerical solution later in Part 2. Additionally, in Part 1, a number of new error estimates are proven that support some of the design decisions.


2021 ◽  
Vol 9 ◽  
Author(s):  
Zhijie Wei ◽  
Jinyun Guo ◽  
Chengcheng Zhu ◽  
Jiajia Yuan ◽  
Xiaotao Chang ◽  
...  

For the first time, HY-2A/GM-derived gravity anomalies determined with the least-squares collocation method and ship-borne bathymetry released from the National Centers for Environmental Information (NCEI) are used to predict bathymetry with the gravity-geologic method (GGM) over three test areas located in the South China Sea (105–122°E, 2–26°N). The iterative method is used to determine density contrasts (1.4, 1.5, and 1.6 g/cm3) between seawater and ocean bottom topography, improving the accuracy of GGM bathymetry. The results show that GGM bathymetry is the closest to ship-borne bathymetry at check points, followed by SRTM15+V2.0 model and GEBCO 2020 model. It is found that in a certain range, the relative accuracy of GGM bathymetry tends to improve with the increase of depth. Different geological structures affect the accuracy of GGM bathymetry. In addition, the influences of gravity anomalies and data processing method on GGM bathymetry are analyzed. Our assessment result suggests that GGM can be widely applied to bathymetry prediction and that HY-2A/GM-derived gravity data are feasible with good results in calculating ocean depth.


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