Scattered Data Interpolation Using Cubic Trigonometric Bézier

This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with C1 continuity everywhere.We derive the C1 condition on each adjacent triangle.On each triangular patch,we employ convex combination method between three local schemes.The final interpolant with the rational corrected scheme is suitable for regular and irregular scattered data sets.We tested the proposed scheme with 36,65,and 100 data points for some well-known test functions.The scheme is also applied to interpolate the data for the electric potential.We compared the performance between our proposed method and existing scattered data interpolation schemes such as Powell–Sabin(PS)and Clough–Tocher(CT)by measuring the maximum error,root mean square error(RMSE)and coefficient of determination(R^(2)).From the results obtained,our proposed method is competent with cubic Bézier,cubic Ball,PS and CT triangles splitting schemes to interpolate scattered data surface.This is very significant since PS and CT requires that each triangle be splitting into several micro triangles.

Research on Interpolation Method for Missing Electricity Consumption Data

Missing value is one of the main factors that cause dirty data.Without high-quality data,there will be no reliable analysis results and precise decision-making.Therefore,the data warehouse needs to integrate high-quality data consistently.In the power system,the electricity consumption data of some large users cannot be normally collected resulting in missing data,which affects the calculation of power supply and eventually leads to a large error in the daily power line loss rate.For the problem of missing electricity consumption data,this study proposes a group method of data handling(GMDH)based data interpolation method in distribution power networks and applies it in the analysis of actually collected electricity data.First,the dependent and independent variables are defined from the original data,and the upper and lower limits of missing values are determined according to prior knowledge or existing data information.All missing data are randomly interpolated within the upper and lower limits.Then,the GMDH network is established to obtain the optimal complexity model,which is used to predict the missing data to replace the last imputed electricity consumption data.At last,this process is implemented iteratively until the missing values do not change.Under a relatively small noise level(α=0.25),the proposed approach achieves a maximum error of no more than 0.605%.Experimental findings demonstrate the efficacy and feasibility of the proposed approach,which realizes the transformation from incomplete data to complete data.Also,this proposed data interpolation approach provides a strong basis for the electricity theft diagnosis and metering fault analysis of electricity enterprises.

Interpolation and approximation for data living on manifold surfaces

Meshed surfaces are ubiquitous in digital geometry processing and computer graphics. The set of attributes associated with each vertex such as the vertex locations, curvature, temperature, pressure or saliency, can be recognized as data living on mani- fold surfaces. So interpolation and approximation for these data are of general interest. This paper presents two approaches for mani- fold data interpolation and approximation through the properties of Laplace-Beltrami operator （Laplace operator defined on a mani- fold surface）. The first one is to use Laplace operator minimizing the membrane energy of a scalar function defined on a manifold. The second one is to use bi-Laplace operator minimizing the thin plate energy of a scalar function defined on a manifold. These two approaches can process data living on high genus meshed surfaces. The approach based on Laplace operator is more suitable for manifold data approximation and can be applied manifold data smoothing, while the one based on bi-Laplace operator is more suit- able for manifold data interpolation and can be applied image extremal envelope computation. All the application examples demon- strate that our procedures are robust and efficient.

Spatio-temporal variability of surface chlorophyll a in the Yellow Sea and the East China Sea based on reconstructions of satellite data of 2001-2020

Chlorophyll-a(Chl-a)concentration is a primary indicator for marine environmental monitoring.The spatio-temporal variations of sea surface Chl-a concentration in the Yellow Sea(YS)and the East China Sea(ECS)in 2001-2020 were investigated by reconstructing the MODIS Level 3 products with the data interpolation empirical orthogonal function(DINEOF)method.The reconstructed results by interpolating the combined MODIS daily+8-day datasets were found better than those merely by interpolating daily or 8-day data.Chl-a concentration in the YS and the ECS reached its maximum in spring,with blooms occurring,decreased in summer and autumn,and increased in late autumn and early winter.By performing empirical orthogonal function(EOF)decomposition of the reconstructed data fields and correlation analysis with several potential environmental factors,we found that the sea surface temperature(SST)plays a significant role in the seasonal variation of Chl a,especially during spring and summer.The increase of SST in spring and the upper-layer nutrients mixed up during the last winter might favor the occurrence of spring blooms.The high sea surface temperature(SST)throughout the summer would strengthen the vertical stratification and prevent nutrients supply from deep water,resulting in low surface Chl-a concentrations.The sea surface Chl-a concentration in the YS was found decreased significantly from 2012 to 2020,which was possibly related to the Pacific Decadal Oscillation(PDO).

HERMITE—BIRKHOFF INTERPOLATION OF SCATTERED DATA BY RADIAL BASIS FUNCTIONS

For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon's thin-plate splines,Hardy's multiquadrics,and inverse multiquadrics.

Improved interpolation method based on singular spectrum analysis iteration and its application to missing data recovery

A novel interval quartering algorithm （IQA） is proposed to overcome insufficiency of the conventional singular spectrum analysis （SSA） iterative interpolation for selecting parameters including the number of the principal components and the embedding dimension. Based on the improved SSA iterative interpolation, interpolated test and comparative analysis are carried out to the outgoing longwave radiation daily data. The results show that IQA can find globally optimal parameters to the error curve with local oscillation, and has advantage of fast computing speed. The improved interpolation method is effective in the interpolation of missing data.

Complex seismic wavefi eld interpolation based on the Bregman iteration method in the sparse transform domain

In seismic prospecting, fi eld conditions and other factors hamper the recording of the complete seismic wavefi eld; thus, data interpolation is critical in seismic data processing. Especially, in complex conditions, prestack missing data affect the subsequent highprecision data processing workfl ow. Compressive sensing is an effective strategy for seismic data interpolation by optimally representing the complex seismic wavefi eld and using fast and accurate iterative algorithms. The seislet transform is a sparse multiscale transform well suited for representing the seismic wavefield, as it can effectively compress seismic events. Furthermore, the Bregman iterative algorithm is an efficient algorithm for sparse representation in compressive sensing. Seismic data interpolation methods can be developed by combining seismic dynamic prediction, image transform, and compressive sensing. In this study, we link seismic data interpolation and constrained optimization. We selected the OC-seislet sparse transform to represent complex wavefields and used the Bregman iteration method to solve the hybrid norm inverse problem under the compressed sensing framework. In addition, we used an H-curve method to choose the threshold parameter in the Bregman iteration method. Thus, we achieved fast and accurate reconstruction of the seismic wavefi eld. Model and fi eld data tests demonstrate that the Bregman iteration method based on the H-curve norm in the sparse transform domain can effectively reconstruct missing complex wavefi eld data.

Brittleness index predictions from Lower Barnett Shale well-log data applying an optimized data matching algorithm at various sampling densities

The capability of accurately predicting mineralogical brittleness index (BI) from basic suites of well logs is desirable as it provides a useful indicator of the fracability of tight formations.Measuring mineralogical components in rocks is expensive and time consuming.However,the basic well log curves are not well correlated with BI so correlation-based,machine-learning methods are not able to derive highly accurate BI predictions using such data.A correlation-free,optimized data-matching algorithm is configured to predict BI on a supervised basis from well log and core data available from two published wells in the Lower Barnett Shale Formation (Texas).This transparent open box (TOB) algorithm matches data records by calculating the sum of squared errors between their variables and selecting the best matches as those with the minimum squared errors.It then applies optimizers to adjust weights applied to individual variable errors to minimize the root mean square error (RMSE)between calculated and predicted (BI).The prediction accuracy achieved by TOB using just five well logs (Gr,ρb,Ns,Rs,Dt) to predict BI is dependent on the density of data records sampled.At a sampling density of about one sample per 0.5 ft BI is predicted with RMSE~0.056 and R^(2)~0.790.At a sampling density of about one sample per0.1 ft BI is predicted with RMSE~0.008 and R^(2)~0.995.Adding a stratigraphic height index as an additional (sixth)input variable method improves BI prediction accuracy to RMSE~0.003 and R^(2)~0.999 for the two wells with only 1 record in 10,000 yielding a BI prediction error of>±0.1.The model has the potential to be applied in an unsupervised basis to predict BI from basic well log data in surrounding wells lacking mineralogical measurements but with similar lithofacies and burial histories.The method could also be extended to predict elastic rock properties in and seismic attributes from wells and seismic data to improve the precision of brittleness index and fracability mapping spatially.

Random Low Patch⁃rank Method for Interpolation of Regularly Missing Traces

Assuming seismic data in a suitable domain is low rank while missing traces or noises increase the rank of the data matrix,the rank⁃reduced methods have been applied successfully for seismic interpolation and denoising.These rank⁃reduced methods mainly include Cadzow reconstruction that uses eigen decomposition of the Hankel matrix in the f⁃x(frequency⁃spatial)domain,and nuclear⁃norm minimization(NNM)based on rigorous optimization theory on matrix completion(MC).In this paper,a low patch⁃rank MC is proposed with a random⁃overlapped texture⁃patch mapping for interpolation of regularly missing traces in a three⁃dimensional(3D)seismic volume.The random overlap plays a simple but important role to make the low⁃rank method effective for aliased data.It shifts the regular column missing of data matrix to random point missing in the mapped matrix,where the missing data increase the rank thus the classic low⁃rank MC theory works.Unlike the Hankel matrix based rank⁃reduced method,the proposed method does not assume a superposition of linear events,but assumes the data have repeated texture patterns.Such data lead to a low⁃rank matrix after the proposed texture⁃patch mapping.Thus the methods can interpolate the waveforms with varying dips in space.A fast low⁃rank factorization method and an orthogonal rank⁃one matrix pursuit method are applied to solve the presented interpolation model.The former avoids the singular value decomposition(SVD)computation and the latter only needs to compute the large singular values during iterations.The two fast algorithms are suitable for large⁃scale data.Simple averaging realizations of several results from different random⁃overlapped texture⁃patch mappings can further increase the reconstructed signal⁃to⁃noise ratio(SNR).Examples on synthetic data and field data are provided to show successful performance of the presented method.

Triangular Curved Surface Construction of A Series ofArbitrary Disordered Points in Space

Constructing Bernstein-Bezier triangular interpolating curve surface interpolating a series of arbitrary disordered data points is of considerable importance for the design and modeling of surfaces with a variety of continuity information. In this article. a kind of simple and reliable algorithm that can process complex field triangular grid generating is presented, and a group of formulae for determining triangular curved surface with wholly C1 continuity are given. It can process arbitrary non-convex boundary and can be used to construct surfaces inner holes.

Impact of urban expansion on meteorological observation data and overestimation to regional air temperature in China

Since the implementation of the reform and opening up policy in China in the late 1970s, some meteorological stations ＇entered＇ cities passively due to urban expansion. Changes in the surface and built environment around the stations have influenced observa- tions of air temperature. When the observational data from urban stations are applied in the interpolation of national or regional scale air temperature dataset, they could lead to overes- timation of regional air temperature and inaccurate assessment of warming. In this study, the underlying surface surrounding 756 meteorological stations across China was identified based on remote sensing images over a number of time intervals to distinguish the rural sta- tions that ＇entered＇ into cities. Then, after removing the observational data from these stations which have been influenced by urban expansion, a dataset of background air temperatures was generated by interpolating the observational data from the remaining rural stations. The mean urban heat island effect intensity since 1970 was estimated by comparing the original observational records from urban stations with the background air temperature interpolated. The result shows that urban heat island effect does occur due to urban expansion, with a higher intensity in winter than in other seasons. Then the overestimation of regional air tem- perature is evaluated by comparing the two kinds of grid datasets of air temperature which are respectively interpolated by all stations＇ and rural stations＇ observational data. Spatially, the overestimation is relatively higher in eastern China than in the central part of China; however, both areas exhibit a much higher effect than is observed in western China. We concluded that in the last 40 years the mean temperature in China increased by about 1.58℃, of which about 0.01℃ was attributed to urban expansion, with a contribution of up to 0.09℃ in the core areas from the overestimation of air temperature.