Least-squares Collocation Method Research Articles

Exploring oversampling in RBF least-squares collocation method of lines for surface diffusion

Exploring oversampling in RBF least-squares collocation method of lines for surface diffusion

Two alternative techniques for fitting the gravimetric geoid for Egypt

In this paper, two proposed geoid fitting techniques for Egypt's gravimetrically determined geoid and Global Positioning System GPS/levelling-derived geoid are introduced. First, any errors in the available GPS stations are ruled out. These methods rely on the absolute geoid difference, which is the gravimetric geoid height minus the geoid obtained through comparing GPS and levelling. The suggested geoid fitting techniques use an optimization algorithm scheme to choose the minimum number of the best-suited GPS stations to be used for fitting the gravimetric geoid. The least-squares collocation method is used to determine each GPS point's impact on the remaining GPS points. The GPS stations with the least impact on the other points are used for external validations, till an acceptable limit of the influence of the GPS points on the remaining ones (when the lowest standard deviation of the differences between gravimetric and geometric geoids is achieved with minimum average or standard deviation is larger than the accuracy of GPS observations, or standard deviation is larger than the target accuracy for geoid; or until a maximum of 30% of the GPS/levellings are excluded). This method operates in a way that automatically selects the fewest GPS stations that are most suitable for usage in the geoid fitting procedure. The geoid quality is then checked externally using the remaining GPS stations. A kriging trend surface is then taken out of the absolute geoid difference to complete the fitting. The proposed geoid fitting techniques are compared with that using polynomial regression of different degrees. The results proved that the proposed techniques give extremely better results. The findings of this study affirm that the adoption of the proposed techniques yields a geoid with an external accuracy of approximately 19 cm for Egypt.

The Weighted Least-Squares Collocation Method for Elastic Wave Obstacle Scattering Problems

Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is [Formula: see text] for Sobolev kernel with kernel smoothness [Formula: see text]. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles.

Adaptive vertical-deformation field estimation and current tectonic deformation significance analysis in Shanxi rift valley

The Shanxi rift valley is a continental rift system that is characterized by mantle uplift, crustal thinning, and tectonic deformation. A continuous, vertical crustal-deformation field can elucidate the constraints for understanding the mechanisms currently driving the deformation of the rift. Accordingly, we acquired and processed data from 250 continuous and mobile GPS stations located in the Shanxi rift valley and its surrounding region. Accounting for the influence of fault activity, we established the vertical crustal-deformation field of the Shanxi rift valley as a continuous function of space, using an adaptive least-squares collocation method. The main research findings are stated as follows: 1) the adaptive least-squares model yielded a reliable interpolation prediction results with adequate robustness, even for relatively sparse actual observation data. 2) The current general deformation pattern of the Shanxi rift valley exhibits an uplift of the mountainous regions on both sides and subsidence of the central basin. The average uplift rate of the mountain area is 2–3 mm/a, and the subsidence rate of the basin is not uniform and is positively correlated with the spatial distribution of the Cenozoic sediment thickness. However, in certain areas, a high subsidence rate of 10–30 mm/a is associated with human activities such as groundwater exploitation. 3) In summary, the current vertical crustal deformation occurring in the Shanxi rift valley correlates with the pattern over a time scale of millions of years. Overall, it is controlled by regional geological structure pattern, and is influenced by nonstructural factors in the shallow crust, exhibiting both complex and orderly characteristics in its spatial distribution.

On performance of vertical gravity gradient determined from CryoSat-2 altimeter data over Arabian Sea

SUMMARYHigh-precision and high-resolution satellite altimetry data from CryoSat-2 are widely utilized for marine gravity inversion. The vertical gravity gradient is a crucial parameter of the Earth's gravity field. To evaluate the performance of vertical gravity gradient determined from CryoSat-2 altimeter data, the pre-processed along-track sea surface heights (SSHs) are obtained through error correction. The study area focused on the Arabian Sea and its surrounding region, where the along-track geoid was derived by subtracting the mean dynamic topography of the ocean from the along-track SSH of CryoSat-2. The residual along-track geoidal gradients were obtained by adjusting the along-track geoid gradients calculated from CryoSat-2 altimeter data using the remove-restore method. This was done by subtracting the geoid gradients calculated by the gravity field model XGM2019e_2159. After obtaining the residual along-track geoidal gradients, the residual gridded deflections of the vertical (DOV) are calculated using the least-squares collocation (LSC) method. The residual gridded DOV are then used to compute the residual gridded gravity anomaly gradients in the study area using the finite-difference method. After restoring the gravity anomaly gradients computed by the XGM2019e_2159 model, a high-resolution gravity anomaly gradient model with a resolution of 1′ ×1′ is obtained for the Arabian Sea and its surrounding area. To evaluate the accuracy of the gravity anomaly gradient model derived from CryoSat-2, it was compared with the SIO V32.1 gravity anomaly gradient model released by the Scripps Institution of Oceanography. The comparison showed that the root mean square (RMS) of the differences between the two models is 7.69E, demonstrating the high accuracy and precision of the vertical gravity gradient determined from CryoSat-2 altimeter data.

Least-squares collocation: a spherical harmonic representer theorem

SUMMARY The functional analysis of the least-squares collocation (LSC) for gravity potential modelling using m measurements is revisited starting from an explicit spherical harmonic expansion. A spherical harmonic representer theorem (SHRT) is given: the model of the potential is a linear combination of m kernels or covariances. This theorem is independent of the specific forms of the data-fitting loss and the regularizer, showing that it is a stronger result than the LSC theory. The corresponding reproducing kernel Hilbert space is explicitly specified. When the least-squares data-fitting loss and the quadratic regularizer are employed, the SHRT gives exactly the LSC method for variable prediction. The nominal prediction precision assessment of the SHRT and that of the LSC are also explicitly compared; this contributes to the unification of the deterministic and stochastic analyses of the LSC theory.

Fault locking behavior of the Longmenshan Thrust Belt preceding the 2008 Wenchuan earthquake

Fault locking inversion can clarify the occurrence mechanisms of large earthquakes and the seismic hazards of future earthquakes. Due to limited observations in mainland China, fault locking inversions have typically been unstable. To investigate a more suitable fault locking behavior of the Longmenshan Thrust Belt (LTB) preceding the 2008 Wenchuan earthquake, GNSS velocity field data during 1999–2007 and the least-squares collocation method (LSCM) are used in this study. Despite the limited stations, after interpolating the GNSS velocity with the LSCM, uncertainties are reduced to some extent during the inversion. Based on the interpolation data, the inversion shows two locked areas that are consistent with the coseismic slip distribution of the Wenchuan earthquake. The third locked area is located on the southern LTB. A checkerboard scheme is proposed, and historic earthquakes larger than M 3.0 since 1950 are analyzed to recheck our inversion. According to the inverted result, the Wenchuan and Lushan earthquakes are two independent events. Additionally, we propose that future destructive earthquakes may occur on the southern segment of the LTB. The maximum magnitude of a future possible earthquake is Mw 8.2.

Development and Verification of a Simplified hp-Version of the Least-Squares Collocation Method for Irregular Domains

Development and Verification of a Simplified hp-Version of the Least-Squares Collocation Method for Irregular Domains

Verified simulation of the stationary polymer fluid flows in the channel with elliptical cross-section

This work is aimed at the numerical analysis of the stationary non-isothermal flows of an incompressible viscoelastic polymer fluid in the channels with elliptical cross-sections. The description of such flows is done on the basis of the mesoscopic approach, and the resolving equations are derived. For solving them three algorithms, which use different techniques of constructing the approximate solutions, are designed: the least-squares collocation method based on the piecewise polynomial approximations, which lead to the overdetermined systems of linear algebraic equations; the finite element method, which uses weak formulations; and the non-local method without saturation, which operates with the global approximations in the elliptical coordinate system and with the matrix Sylvester equations. The proposed algorithms are verified by solving the test problem with the known analytical solution. Further we use them for the numerical analysis of the polymer fluid flows with its parameters varying in wide ranges. Comparison of the results obtained by the different algorithms shows their high performance and confirms that the solution of the considered non-linear problem exists and that it was computed accurately. The singularities of the obtained stationary solutions are analyzed. Taking them into account within the proposed algorithms enables us to increase the accuracy and the speed of simulations.

On the sensitivity of implementations of a least-squares collocation method for linear higher-index differential-algebraic equations

The present paper continues our investigation of an implementation of a least-squares collocation method for higher-index differential-algebraic equations. In earlier papers, we were able to substantiate the choice of basis functions and collocation points for a robust implementation as well as algorithms for the solution of the discrete system. The present paper is devoted to an analytic estimation of condition numbers for different components of an implementation. We present error estimations, which show the sources for the different errors.

H-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications

h-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications

Interseismic Fault Coupling and Slip Rate Deficit on the Central and Southern Segments of the Tanlu Fault Zone Based on Anhui CORS Measurements

The Tanlu fault zone, extending over 2400 km from South China to Russia, is one of the most conspicuous tectonic elements in eastern Asia. In this study, we processed the Global Positioning System (GPS) measurements of Anhui Continuously Operating Reference System (AHCORS) between January 2013 and June 2018 to derive a high-precision velocity field in the central and southern segments of the Tanlu fault zone. We integrated the AHCORS data with those publicly available for geodetic imaging of the interseismic coupling and slip rate deficit distribution in the central and southern segments of the Tanlu fault zone. This work aims at a better understanding of strain accumulation and future seismic hazard in the Tanlu fault zone. The result indicates lateral variation of coupling distribution along the strike of the Tanlu fault zone. The northern segment of the Tanlu fault zone has a larger slip rate deficit and a deeper locking depth than the southern segment. Then, we analyzed three velocity profiles across the fault. The result suggests that the central and southern segments of the Tanlu fault zone are characterized by right-lateral strike-slip (0.29–0.44 mm/y) with compression components (0.35–0.76 mm/y). Finally, we estimated strain rates using the least-squares collocation method. The result shows that the dilatation rates concentrate in the region where the principal strain rates are very large. The interface of extension and compression is always accompanied by sudden change of direction of principal strain rates. Especially, in the north of Anhui, the dilatation rate is largest, reaching 3.780×10−8/a. Our study suggests that the seismic risk in the northern segment of the Tanlu fault zone remains very high for its strong strain accumulation and the lack of historical large earthquakes.

Comparison of kriging and least-squares collocation – Revisited

Abstract The well-known to physical geodesists method of least-squares collocation and the geostatistical method of kriging probably known to the broader audience are compared. Both methods are rooted in Wiener–Kolmogorov’s (W–K) prediction theory; but, since necessity is the mother of invention, the W–K foundations have been extended to satisfy the needs of particular applications. The paper presents a link or rather an equivalence of the two methods as far as their basic forms are considered (specialization to geodetic boundary-value problems, covariance propagation between functionals and nonlinear geostatistical methods are excluded from this comparison). Only scalar random fields (univariate case) and the assumption of a second-order structure of a random function are considered. Due to the equivalence of their basic formulas, both techniques share the same advantages and disadvantages. The paper also shows the difference as to the predicted values and prediction variances in case of exact and filtered (noise reduction) prediction models. This theoretical comparison of the methods has practical implications because of readily available geostatistical software that, in local as well as global applications, can be used for predictive problems occurring in geodesy and surveying.

PERFORMANCE OF SITE VELOCITY PREDICTION IN SUNDALAND

Abstract. Global Positioning System (GPS) technique has been extensively implemented in determination of crustal deformation globally. With the ability of providing solution up to milimeter (mm) level, this technique has proven to provide a precise estimate of site velocity that represents the actual motion of tectonic plate over a period. Therefore, this study aims to evaluate the site velocity estimation from GPS-derived daily position of station, respective to the global plate motion model and predicted site velocity via Least-Squares Collocation (LSC) method within the tectonically active region of Sundaland. The findings have indicated that stations with precise velocity estimates were consistent with global plate model and predicted velocity, with velocity residuals of 5 mm – 10 mm. However, stations that were severely impacted by continuous earthquake events such as in Sumatra were believed to be induced by the impact with consistently large velocity residuals up to 37 mm. Following the outcomes, this study has provided an insight on the post-seismic decay period plate motion which are induced by continuous tectonic activities respective to modelled plate motion.

Modelling and Simulation of Deformation and Failure of Reinforced Concrete Beams under Four-Point Bending

A new mathematical model for the four-point bending of reinforced concrete beams is developed and investigated. The model takes into account multi-modulus concrete behavior, nonlinear stress-strain relationships, and damage evolution. An algorithm for a numerical implementation of the model is proposed. The corresponding boundary value problem is solved by the hp-version of the least-squares collocation method in combination with an acceleration of an iterative process based on Krylov subspaces and parallelizing. Special attention is given to the influence of mathematical model parameters on the results of numerical simulation. The results are compared with experimental data and three-dimensional simulation. A satisfactory agreement is shown

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

The determination of an accurate national geoid model remains one of the main challenges of geodetic research projects in Ethiopia. The main aim of this study is to develop a new gravimetric geoid model, named the ETH–GM21, over Ethiopia. This new geoid model is based on a combination of terrestrial and airborne gravity data together with the state-of-the art combined Global Geopotential Model (GGM) and the high-resolution SRTM3 (Shuttle Radar Topography Mission with a spatial resolution of 3 arc-second) global digital elevation model. The ETH–GM21 has been determined using the remove-compute-restore (RCR) procedure and the least–squares collocation method. The accuracy of ETH–GM21 gravimetric geoid model has been evaluated using geometric geoid heights obtained from GNSS (Global Navigation Satellite Systems) and levelling data distributed over the country. On the basis of this evaluation the estimated accuracy of geoid heights from the ETH–GM21 geoid model is at the level of 13 cm. The ETH–GM21 can be recommended as a first relatively accurate vertical reference surface for a definition of heights within the whole Ethiopia.

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations\u2014Part 2: the discrete least-squares problem

In 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.

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations\u2014Part 1: basics and ansatz function choices

In 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.

Evaluating Accuracy of HY-2A/GM-Derived Gravity Data With the Gravity-Geologic Method to Predict Bathymetry

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.

Collocation and FFT-based geoid estimation within the Colorado 1 cm geoid experiment

In the frame of the International Association of Geodesy Joint Working Group 2.2.2 “The 1 cm geoid experiment”, terrestrial and airborne gravity datasets along with GPS/leveling data were made available for the comparison of different geoid modeling methods and techniques in the wider area of Colorado, USA. We discuss the methods and procedures we followed for computing gravimetric quasi-geoid and geoid models and geopotential values from the available datasets. The procedures followed were based on the remove-compute-restore approach using XGM2016 as a reference geopotential model. The higher frequencies of the gravity field were computed via the residual terrain correction, using (a) the CGIAR-CSI SRTM digital elevation model with the classical technique and (b) a spectral one. Least-Squares Collocation was used for the downward continuation of the airborne data and for gridding. Finally, the geoid models were obtained by applying Least-Squares Collocation and spherical FFT-based methods, while the influence of the orthometric height correction on geoid heights was taken into account by employing simple and complete Bouguer reductions. All results were evaluated with available GPS/leveling benchmarks. Moreover, potential values were determined in support of the International Height Reference System/Frame. From the results acquired, a final accuracy of 5–7 cm for the determined geoid models was achieved depending on the adopted method and data combination, without considering the accuracy of the GPS/leveling data used for their evaluation. The contribution of the airborne gravity data was deemed as limited in combination solutions although the airborne only solution provided equal level of accuracy to the terrestrial and the combined ones. Better consistency was obtained on the points of the GSVS17 line, when compared to the GPS/leveling data, where an accuracy of 2.4 cm and 2.8 cm was reached for the FFT and LSC based methods, respectively.