Thin plate spline interpolation

Keller, Wolfgang; Borkowski, Andrzej

doi:10.1007/s00190-019-01240-2

Cited by 63 publications

(33 citation statements)

References 39 publications

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“…In general, when there are two consecutive interferograms with low quality, it may be considered that the common image has high perturbations due to atmospheric effects. To reduce the signal delay, different approaches may be performed, but the most preferable approach is by applying a weather model based on the information from global atmospheric models [40], models generated from GNSS observations [41], models that integrate both the weather forecast and GNSS tropospheric delay estimations [42], or spectrometer observations by systems like the Medium Resolution Imaging Spectrometer (MERIS) on-board the Envisat satellite or the Moderate Resolution Imaging Spectroradiometer (MODIS) on-board the Terra and Aqua satellites [43]. The aggregation to lower frequency observations, in line with the mining extraction plan, results in removing some of the lower coherence data, but highly tropospheric varying conditions still affect DInSAR deformations.…”

Section: Discussionmentioning

confidence: 99%

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Mining Deformation Life Cycle in the Light of InSAR and Deformation Models

et al. 2019

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The Sentinel-1 constellation provides an effective new radar instrument with a short revisit time of six days for the monitoring of intensive mining surface deformations. Our goal is to investigate in detail and to bring new comprehension of the mine life cycle. The dynamics of mining, especially in the case of horizontally evolving longwall technology, exhibit rapid surface changes. We use the classical approach of differential radar interferometry (DInSAR) with short temporal baselines (six days), which results in deformation maps with a low decorrelation between the satellite images. For the same time intervals, we compare the radar results with prediction models based on the Knothe–Budryk theory for mining subsidence. The validation of the results with ground levelling measurements reveals a high level of resemblance of the DInSAR subsidence maps (−0.04 m bias with respect to the levelling). On the other hand, aside from the explicable exaggeration, the location of the subsidence trough needs improvement in the forecasted deformations (0.2 km shift in location, a deformation velocity four times higher than in DInSAR). In addition, a time lag between DInSAR (compatible with extraction) and prediction is revealed. The model improvement can be achieved by including the DInSAR results in the elaboration of the model parameters.

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Section: Discussionmentioning

confidence: 99%

“…The resulting displacement was still represented in the LOS to the satellite. To compute the north-south (d N ), east-west (d E ), and vertical (d U ) components of the deformation vector in the LOS (d LOS ), the following well-known relation (e.g., [42]) was used:…”

Section: Three-dimensional Decompositionmentioning

confidence: 99%

Mining Deformation Life Cycle in the Light of InSAR and Deformation Models

et al. 2019

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“…There are many applications to the spline functions, including gravity data interpolation, satellite position interpolation between observation points, and TEC interpolation between the discrete measurements in the Ionosphere (Keller & Borkowski, 2019). We show how the results obtained in the previous sections can be applied to globally interpolate gravity data.…”

Section: Application In Gravity Data Interpolation In Geodesy: Interpmentioning

confidence: 88%

“…In the spherical domain, the spline interpolation has been investigated in a number of different studies including , Freeden (1981), Keller and Borkowski (2019), Sloan and Womersley (2002), Baramidze et al (2006), and Wahba (1981Wahba ( , 1990. In , spherical splines and zonal kernels are defined in the context of Sobolev and reproducing kernel in interpolating the GRACE satellites' positions in their ground-track repeat orbit and the Total Electron Content (TEC) in the ionosphere are presented.…”

Section: Introductionmentioning

confidence: 99%

Spheroidal Spline Interpolation and Its Application in Geodesy

Shahvandi

Chegini

Safari

et al. 2020

Geodesy and Cartography

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The aim of this paper is to study the spline interpolation problem in spheroidal geometry. We follow the minimization of the norm of the iterated Beltrami-Laplace and consecutive iterated Helmholtz operators for all functions belonging to an appropriate Hilbert space defined on the spheroid. By exploiting surface Green’s functions, reproducing kernels for discrete Dirichlet and Neumann conditions are constructed in the spheroidal geometry. According to a complete system of surface spheroidal harmonics, generalized Green’s functions are also defined. Based on the minimization problem and corresponding reproducing kernel, spline interpolant which minimizes the desired norm and satisfies the given discrete conditions is defined on the spheroidal surface. The application of the results in Geodesy is explained in the gravity data interpolation over the globe.

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“…A variety of the radial basis functions (RBFs) may be adopted, such as the Gaussian [13], multiquadric (MQ) [14], inverse multiquadric (IMQ) [15], and thin plate spline (TPS) functions [16]. Among them, the multiquadric function may obtain more accurate solutions than other RBF interpolations [8] and is often utilized as the RBF interpolation for dealing with PDEs.…”

Section: Introductionmentioning

confidence: 99%

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Liu

Xiao

et al. 2020

Symmetry

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In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

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Thin plate spline interpolation

Cited by 63 publications

References 39 publications

Mining Deformation Life Cycle in the Light of InSAR and Deformation Models

Mining Deformation Life Cycle in the Light of InSAR and Deformation Models

Spheroidal Spline Interpolation and Its Application in Geodesy

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

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