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Concept: Multivariate interpolation


Many spatial interpolation methods perform well for gentle terrains when producing spatially continuous surfaces based on ground point data. However, few interpolation methods perform satisfactorily for complex terrains. Our objective in the present study was to analyze the suitability of several popular interpolation methods for complex terrains and propose an optimal method. A data set of 153 soil water profiles (1 m) from the semiarid hilly gully Loess Plateau of China was used, generated under a wide range of land use types, vegetation types and topographic positions. Four spatial interpolation methods, including ordinary kriging, inverse distance weighting, linear regression and regression kriging were used for modeling, randomly partitioning the data set into 2/3 for model fit and 1/3 for independent testing. The performance of each method was assessed quantitatively in terms of mean-absolute-percentage-error, root-mean-square-error, and goodness-of-prediction statistic. The results showed that the prediction accuracy differed significantly between each method in complex terrain. The ordinary kriging and inverse distance weighted methods performed poorly due to the poor spatial autocorrelation of soil moisture at small catchment scale with complex terrain, where the environmental impact factors were discontinuous in space. The linear regression model was much more suitable to the complex terrain than the former two distance-based methods, but the predicted soil moisture changed too sharply near the boundary of the land use types and junction of the sunny (southern) and shady (northern) slopes, which was inconsistent with reality because soil moisture should change gradually in short distance due to its mobility in soil. The most optimal interpolation method in this study for the complex terrain was the hybrid regression kriging, which produced a detailed, reasonable prediction map with better accuracy and prediction effectiveness.

Concepts: Regression analysis, Linear regression, Statistics, Forecasting, Interpolation, Landform, Kriging, Multivariate interpolation


Based on the geo-statistical theory and ArcGIS geo-statistical module, datas of 30 groundwater level observation wells were used to estimate the decline of groundwater level in Beijing piedmont. Seven different interpolation methods (inverse distance weighted interpolation, global polynomial interpolation, local polynomial interpolation, tension spline interpolation, ordinary Kriging interpolation, simple Kriging interpolation and universal Kriging interpolation) were used for interpolating groundwater level between 2001 and 2013. Cross-validation, absolute error and coefficient of determination (R(2)) was applied to evaluate the accuracy of different methods. The result shows that simple Kriging method gave the best fit. The analysis of spatial and temporal variability suggest that the nugget effects from 2001 to 2013 were increasing, which means the spatial correlation weakened gradually under the influence of human activities. The spatial variability in the middle areas of the alluvial-proluvial fan is relatively higher than area in top and bottom. Since the changes of the land use, groundwater level also has a temporal variation, the average decline rate of groundwater level between 2007 and 2013 increases compared with 2001-2006. Urban development and population growth cause over-exploitation of residential and industrial areas. The decline rate of the groundwater level in residential, industrial and river areas is relatively high, while the decreasing of farmland area and development of water-saving irrigation reduce the quantity of water using by agriculture and decline rate of groundwater level in agricultural area is not significant.

Concepts: Regression analysis, Agriculture, Numerical analysis, Interpolation, Polynomial interpolation, Kriging, Spline, Multivariate interpolation


Recent improvements in cardiac computed tomography (CCT) allow for whole-heart functional studies to be acquired at low radiation dose (<2mSv) and high-temporal resolution (<100ms) in a single heart beat. Although the extraction of regional functional information from these images is of great clinical interest, there is a paucity of research into the quantification of regional function from CCT, contrasting with the large body of work in echocardiography and cardiac MR. Here we present the Simultaneous Subdivision Surface Registration (SiSSR) method: a fast, semi-automated image analysis pipeline for quantifying regional function from contrast-enhanced CCT. For each of thirteen adult male canines, we construct an anatomical reference mesh representing the left ventricular (LV) endocardium, obviating the need for a template mesh to be manually sculpted and initialized. We treat this generated mesh as a Loop subdivision surface, and adapt a technique previously described in the context of 3-D echocardiography to register these surfaces to the endocardium efficiently across all cardiac frames simultaneously. Although previous work performs the registration at a single resolution, we observe that subdivision surfaces naturally suggest a multiresolution approach, leading to faster convergence and avoiding local minima. We additionally make two notable changes to the cost function of the optimization, explicitly encouraging plausible biological motion and high mesh quality. Finally, we calculate an accepted functional metric for CCT from the registered surfaces, and compare our results to an alternate state-of-the-art CCT method.

Concepts: Optimization, 3D computer graphics, Maxima and minima, Multivariate interpolation, Subdivision surface, Catmull–Clark subdivision surface, Loop subdivision surface


Aerosol is an important component of the atmosphere that affects the environment, climate, and human health. Remote sensing is an efficient observation method for monitoring global aerosol distribution and changes over time. The daily Moderate Resolution Imaging Spectroradiometer (MODIS) level-2 aerosol optical depth (AOD) (Collection 6) product (10 km resolution) is often used to study climate change and air pollution. However, the product is prone to yielding large amounts of data gaps due to the unfeasibility of retrieving reliable estimates under cloudy conditions, and these data gaps inevitably affect the results and analysis of the product’s application. In this study, a geostatistical data interpolation framework based on the spatiotemporal kriging method was implemented to interpolate satellite AOD products in Beijing, China. Compared to the ordinary kriging method for filling data gaps, the spatiotemporal interpolation not only utilizes spatial autocorrelation but also considers the temporal and spatiotemporal autocorrelations between different locations. In the study region, the completeness of the spatiotemporal-interpolated AOD product reaches 67.73%, which is significantly superior to the completeness of the original MODIS product (14.27%) and that of the spatial kriging-interpolated AOD product (33.3%). The cross-validation results show that the mean absolute error of the spatiotemporal kriging results (0.07) is lower than that of the ordinary kriging (0.09).

Concepts: Regression analysis, Climate, Linear interpolation, Atmosphere, Numerical analysis, Interpolation, Kriging, Multivariate interpolation


Spatial interpolation method is the basis of soil heavy metal pollution assessment and remediation. The existing evaluation index for interpolation accuracy did not combine with actual situation. The selection of interpolation methods needs to be based on specific research purposes and research object characteristics. In this paper, As pollution in soils of Beijing was taken as an example. The prediction accuracy of ordinary kriging (OK) and inverse distance weighted (IDW) were evaluated based on the cross validation results and spatial distribution characteristics of influencing factors. The results showed that, under the condition of specific spatial correlation, the cross validation results of OK and IDW for every soil point and the prediction accuracy of spatial distribution trend are similar. But the prediction accuracy of OK for the maximum and minimum is less than IDW, while the number of high pollution areas identified by OK are less than IDW. It is difficult to identify the high pollution areas fully by OK, which shows that the smoothing effect of OK is obvious. In addition, with increasing of the spatial correlation of As concentration, the cross validation error of OK and IDW decreases, and the high pollution area identified by OK is approaching the result of IDW, which can identify the high pollution areas more comprehensively. However, because the semivariogram constructed by OK interpolation method is more subjective and requires larger number of soil samples, IDW is more suitable for spatial prediction of heavy metal pollution in soils.

Concepts: Regression analysis, Evaluation, Maxima and minima, Heavy metal music, Interpolation, Kriging, Multivariate interpolation, Geostatistics


A comparative study was made of three methods of interpolation - inverse distance weighting (IDW), spline and ordinary kriging - after optimization of their characteristic parameters. These interpolation methods were used to represent the electric field levels for three emission frequencies (774kHz, 900kHz, and 1107kHz) and for the electrical stimulation quotient, QE, characteristic of complex electromagnetic environments. Measurements were made with a spectrum analyser in a village in the vicinity of medium-wave radio broadcasting antennas. The accuracy of the models was quantified by comparing their predictions with levels measured at the control points not used to generate the models. The results showed that optimizing the characteristic parameters of each interpolation method allows any of them to be used. However, the best results in terms of the regression coefficient between each model’s predictions and the actual control point field measurements were for the IDW method.

Concepts: Regression analysis, Mathematics, Radio, Curve fitting, Interpolation, Kriging, Multivariate interpolation, AM broadcasting


The appropriate spatial interpolation methods must be selected to analyze the spatial distributions of Potentially Toxic Elements (PTEs), which is a precondition for evaluating PTE pollution. The accuracy and effect of different spatial interpolation methods, which include inverse distance weighting interpolation (IDW) (power = 1, 2, 3), radial basis function interpolation (RBF) (basis function: thin-plate spline (TPS), spline with tension (ST), completely regularized spline (CRS), multiquadric (MQ) and inverse multiquadric (IMQ)) and ordinary kriging interpolation (OK) (semivariogram model: spherical, exponential, gaussian and linear), were compared using 166 unevenly distributed soil PTE samples (As, Pb, Cu and Zn) in the Suxian District, Chenzhou City, Hunan Province as the study subject. The reasons for the accuracy differences of the interpolation methods and the uncertainties of the interpolation results are discussed, then several suggestions for improving the interpolation accuracy are proposed, and the direction of pollution control is determined. The results of this study are as follows: (i) RBF-ST and OK (exponential) are the optimal interpolation methods for As and Cu, and the optimal interpolation method for Pb and Zn is RBF-IMQ. (ii) The interpolation uncertainty is positively correlated with the PTE concentration, and higher uncertainties are primarily distributed around mines, which is related to the strong spatial variability of PTE concentrations caused by human interference. (iii) The interpolation accuracy can be improved by increasing the sample size around the mines, introducing auxiliary variables in the case of incomplete sampling and adopting the partition prediction method. (iv) It is necessary to strengthen the prevention and control of As and Pb pollution, particularly in the central and northern areas. The results of this study can provide an effective reference for the optimization of interpolation methods and parameters for unevenly distributed soil PTE data in mining areas.

Concepts: Interpolation, Hunan, Kriging, Multivariate interpolation, Geostatistics, Radial basis function, Chenzhou, Variogram


Non-uniform rational B-spline (NURBS) surface fitting from data points is wildly used in the fields of computer aided design (CAD), medical imaging, cultural relic representation and object-shape detection. Usually, the measured data acquired from coordinate measuring systems is neither gridded nor completely scattered. The distribution of this kind of data is scattered in physical space, but the data points are stored in a way consistent with the order of measurement, so it is named quasi scattered data in this paper. Therefore they can be organized into rows easily but the number of points in each row is random. In order to overcome the difficulty of surface fitting from this kind of data, a new method based on resampling is proposed. It consists of three major steps: (1) NURBS curve fitting for each row, (2) resampling on the fitted curve and (3) surface fitting from the resampled data. Iterative projection optimization scheme is applied in the first and third step to yield advisable parameterization and reduce the time cost of projection. A resampling approach based on parameters, local peaks and contour curvature is proposed to overcome the problems of nodes redundancy and high time consumption in the fitting of this kind of scattered data. Numerical experiments are conducted with both simulation and practical data, and the results show that the proposed method is fast, effective and robust. What’s more, by analyzing the fitting results acquired form data with different degrees of scatterness it can be demonstrated that the error introduced by resampling is negligible and therefore it is feasible.

Concepts: Measurement, Computer-aided design, Systems of measurement, Interpolation, Splines, Multivariate interpolation, Non-uniform rational B-spline


Information about the distribution and abundance of the habitat-forming sessile organisms in marine ecosystems is of great importance for conservation and natural resource managers. Spatial interpolation methodologies can be useful to generate this information from in situ sampling points, especially in circumstances where remote sensing methodologies cannot be applied due to small-scale spatial variability of the natural communities and low light penetration in the water column. Interpolation methods are widely used in environmental sciences; however, published studies using these methodologies in coral reef science are scarce. We compared the accuracy of the two most commonly used interpolation methods in all disciplines, inverse distance weighting (IDW) and ordinary kriging (OK), to predict the distribution and abundance of hard corals, octocorals, macroalgae, sponges and zoantharians and identify hotspots of these habitat-forming organisms using data sampled at three different spatial scales (5, 10 and 20 m) in Madagascar reef, Gulf of Mexico. The deeper sandy environments of the leeward and windward regions of Madagascar reef were dominated by macroalgae and seconded by octocorals. However, the shallow rocky environments of the reef crest had the highest richness of habitat-forming groups of organisms; here, we registered high abundances of octocorals and macroalgae, with sponges, Millepora alcicornis and zoantharians dominating in some patches, creating high levels of habitat heterogeneity. IDW and OK generated similar maps of distribution for all the taxa; however, cross-validation tests showed that IDW outperformed OK in the prediction of their abundances. When the sampling distance was at 20 m, both interpolation techniques performed poorly, but as the sampling was done at shorter distances prediction accuracies increased, especially for IDW. OK had higher mean prediction errors and failed to correctly interpolate the highest abundance values measured in situ, except for macroalgae, whereas IDW had lower mean prediction errors and high correlations between predicted and measured values in all cases when sampling was every 5 m. The accurate spatial interpolations created using IDW allowed us to see the spatial variability of each taxa at a biological and spatial resolution that remote sensing would not have been able to produce. Our study sets the basis for further research projects and conservation management in Madagascar reef and encourages similar studies in the region and other parts of the world where remote sensing technologies are not suitable for use.

Concepts: Scientific method, Regression analysis, Coral, Remote sensing, Interpolation, Kriging, Multivariate interpolation, Bilinear interpolation


Soil moisture data can reflect valuable information on soil properties, terrain features, and drought condition. The current study compared and assessed the performance of different interpolation methods for estimating soil moisture in an area with complex topography in southwest China. The approaches were inverse distance weighting, multifarious forms of kriging, regularized spline with tension, and thin plate spline. The 5-day soil moisture observed at 167 stations and daily temperature recorded at 33 stations during the period of 2010-2014 were used in the current work. Model performance was tested with accuracy indicators of determination coefficient (R (2)), mean absolute percentage error (MAPE), root mean square error (RMSE), relative root mean square error (RRMSE), and modeling efficiency (ME). The results indicated that inverse distance weighting had the best performance with R (2), MAPE, RMSE, RRMSE, and ME of 0.32, 14.37, 13.02%, 0.16, and 0.30, respectively. Based on the best method, a spatial database of soil moisture was developed and used to investigate drought condition over the study area. The results showed that the distribution of drought was characterized by evidently regional difference. Besides, drought mainly occurred in August and September in the 5 years and was prone to happening in the western and central parts rather than in the northeastern and southeastern areas.

Concepts: Standard deviation, Root mean square, Mean squared error, Mean absolute percentage error, Interpolation, Root mean square deviation, Multivariate interpolation, Statistical deviation and dispersion