Abstract: The spatial distribution of precipitation is an important aspect of water-related research. The use of different interpolation schemes in the same catchment may cause large differences and deviations from the actual spatial distribution of rainfall. Our study analyzes different methods of spatial rainfall interpolation at annual, daily, and hourly time scales to provide a comprehensive evaluation. An improved regression-based scheme is proposed using principal component regression with residual correction (PCRR) and is compared with inverse distance weighting (IDW) and multiple linear regression (MLR) interpolation methods. In this study, the meso-scale catchment of the Fuhe River in southeastern China was selected as a typical region. Furthermore, a hydrological model HEC-HMS was used to calculate streamflow and to evaluate the impact of rainfall interpolation methods on the results of the hydrological model. Results show that the PCRR method performed better than the other methods tested in the study and can effectively eliminate the interpolation anomalies caused by terrain differences between observation points and surrounding areas. Simulated streamflow showed different characteristics based on the mean, maximum, minimum, and peak flows. The results simulated by PCRR exhibited the lowest streamflow error and highest correlation with measured values at the daily time scale. The application of the PCRR method is found to be promising because it considers multicollinearity among variables.

Abstract: Rainfall data in continuous space provide an essential input for most hydrological and water resources planning studies. Spatial distribution of rainfall is usually estimated using ground-based point rainfall data from sparsely positioned rain-gauge stations in a rain-gauge network. Kriging has become a widely used interpolation method to estimate the spatial distribution of climate variables including rainfall. The objective of this study is to evaluate three geostatistical (ordinary kriging (OK), ordinary cokriging (OCK), kriging with an external drift (KED)) and two deterministic (inverse distance weighting, radial basis function) interpolation methods for enhanced spatial interpolation of monthly rainfall in the Middle Yarra River catchment and the Ovens River catchment in Victoria, Australia. Historical rainfall records from existing rain-gauge stations of the catchments during 1980-2012 period are used for the analysis. A digital elevation model of each catchment is used as the supplementary information in addition to rainfall for the OCK and KED methods. The prediction performance of the adopted interpolation methods is assessed through cross-validation. Results indicate that the geostatistical methods outperform the deterministic methods for spatial interpolation of rainfall. Results also indicate that among the geostatistical methods, the OCK method is found to be the best interpolator for estimating spatial rainfall distribution in both the catchments with the lowest prediction error between the observed and estimated monthly rainfall. Thus, this study demonstrates that the use of elevation as an auxiliary variable in addition to rainfall data in the geostatistical framework can significantly enhance the estimation of rainfall over a catchment.

Abstract: It is practically impossible and unnecessary to obtain spatial-temporal information of any given continuous phenomenon at every point within a given geographic area. The most practical approach has always been to obtain information about the phenomenon as in many sample points as possible within the given geographic area and estimate the values of the unobserved points from the values of the observed points through spatial interpolation. However, it is important that users understand that different interpolation methods have their strength and weaknesses on different datasets. It is not correct to generalize that a given interpolation method (e.g. Kriging, Inverse Distance Weighting (IDW), Spline etc.) does better than the other without taking into cognizance, the type and nature of the dataset and phenomenon involved. In this paper, we theoretically, mathematically and experimentally evaluate the performance of Kriging, IDW and Spline interpolation methods respectively in estimating unobserved elevation values and modeling landform. This paper undertakes a comparative analysis based on the prediction mean error, prediction root mean square error and cross validation outputs of these interpolation methods. Experimental results for each of the method on both biased and normalized data show that Spline provided a better and more accurate interpolation within the sample space than the IDW and Kriging methods. The choice of an interpolation method should be phenomenon and data set structure dependent.

Abstract: Near surface air temperature (NSAT) is a primary descriptor of terrestrial environmental conditions. In recent decades, many efforts have been made to develop various methods for obtaining spatially continuous NSAT from gauge or station observations. This study compared three spatial interpolation (i.e., Kriging, Spline, and Inversion Distance Weighting (IDW)) and two regression analysis (i.e., Multiple Linear Regression (MLR) and Geographically Weighted Regression (GWR)) models for predicting monthly minimum, mean, and maximum NSAT in China, a domain with a large area, complex topography, and highly variable station density. This was conducted for a period of 12 months of 2010. The accuracy of the GWR model is better than the MLR model with an improvement of about 3 °C in the Root Mean Squared Error (RMSE), which indicates that the GWR model is more suitable for predicting monthly NSAT than the MLR model over a large scale. For three spatial interpolation models, the RMSEs of the predicted monthly NSAT are greater in the warmer months, and the mean RMSEs of the predicted monthly mean NSAT for 12 months in 2010 are 1.56 °C for the Kriging model, 1.74 °C for the IDW model, and 2.39 °C for the Spline model, respectively. The GWR model is better than the Kriging model in the warmer months, while the Kriging model is superior to the GWR model in the colder months. The total precision of the GWR model is slightly higher than the Kriging model. The assessment result indicated that the higher standard deviation and the lower mean of NSAT from sample data would be associated with a better performance of predicting monthly NSAT using spatial interpolation models.

Abstract: Land Use and Land Cover (LULC) classification is vital for environmental and ecological applications. Sentinel-2 is a new generation land monitoring satellite with the advantages of novel spectral capabilities, wide coverage and fine spatial and temporal resolutions. The effects of different spatial resolution unification schemes and methods on LULC classification have been scarcely investigated for Sentinel-2. This paper bridged this gap by comparing the differences between upscaling and downscaling as well as different downscaling algorithms from the point of view of LULC classification accuracy. The studied downscaling algorithms include nearest neighbor resampling and five popular pansharpening methods, namely, Gram-Schmidt (GS), nearest neighbor diffusion (NNDiffusion), PANSHARP algorithm proposed by Y. Zhang, wavelet transformation fusion (WTF) and high-pass filter fusion (HPF). Two spatial features, textural metrics derived from Grey-Level-Co-occurrence Matrix (GLCM) and extended attribute profiles (EAPs), are investigated to make up for the shortcoming of pixel-based spectral classification. Random forest (RF) is adopted as the classifier. The experiment was conducted in Xitiaoxi watershed, China. The results demonstrated that downscaling obviously outperforms upscaling in terms of classification accuracy. For downscaling, image sharpening has no obvious advantages than spatial interpolation. Different image sharpening algorithms have distinct effects. Two multiresolution analysis (MRA)-based methods, i.e., WTF and HFP, achieve the best performance. GS achieved a similar accuracy with NNDiffusion and PANSHARP. Compared to image sharpening, the introduction of spatial features, both GLCM and EAPs can greatly improve the classification accuracy for Sentinel-2 imagery. Their effects on overall accuracy are similar but differ significantly to specific classes. In general, using the spectral bands downscaled by nearest neighbor interpolation can meet the requirements of regional LULC applications, and the GLCM and EAPs spatial features can be used to obtain more precise classification maps.

Abstract: Temperature time series with high spatial and temporal resolutions are important for several applications. The new MODIS Land Surface Temperature (LST) collection 6 provides numerous improvements compared to collection 5. However, being remotely sensed data in the thermal range, LST shows gaps in cloud-covered areas. We present a novel method to fully reconstruct MODIS daily LST products for central Europe at 1 km resolution and globally, at 3 arc-min. We combined temporal and spatial interpolation, using emissivity and elevation as covariates for the spatial interpolation. The reconstructed MODIS LST for central Europe was calibrated to air temperature data through linear models that yielded R2 values around 0.8 and RMSE of 0.5 K. This new method proves to scale well for both local and global reconstruction. We show examples for the identification of extreme events to demonstrate the ability of these new LST products to capture and represent spatial and temporal details. A time series of global monthly average, minimum and maximum LST data and long-term averages is freely available for download.

Abstract: Precipitation is a primary climate forcing factor in catchment hydrology, and its spatial distribution is essential for understanding the spatial variability of ecohydrological processes in a catchment. In mountainous areas, meteorological stations are generally too sparse to represent the spatial distribution of precipitation. This study develops a spatial interpolation method that combines meteorological observations and regional climate model (RCM) outputs. The method considers the precipitation–elevation relationship in the mountain region and the topographic effects, especially the mountain blocking effect. Furthermore, using this method, this study produced a 3-km-resolution precipitation dataset from 1960 to 2014 in the middle and upper reaches of the Heihe River basin located on the northern slope of the Qilian Mountains in the northeastern Tibetan Plateau. Cross validation based on the station observations showed that this method is reasonable. The rationality of the interpolated precipit...

Abstract: Heterogeneous data fusion from disparate geospatial sensors has drawn increasing attention in multimedia. Unfortunately, environmental sensors are usually sparsely and preferentially located, which restricts situation recognition of geographical regions and results in uncertainty in derived inferences. Spatial interpolation is an effective way to solve the problem of data sparsity, which demands the availability of related data sources. However, these data sources are usually in different resolutions, distributions, scales, and densities, which poses a major challenge in data integration. To address this problem, we present a novel spatial interpolation framework to incorporate diverse data sources and model the spatial processes explicitly at multiple resolutions. Spectral analysis is deployed to generate features at multiple spatial resolutions and to improve the interpolation accuracy at unobserved locations. A statistical operator based on the spatial Gaussian process is implemented and integrated into a geospatial situation recognition system, which can analyze heterogeneous spatio-temporal data streams derived from sensors. To verify the effectiveness and efficiency of the proposed framework, this framework is applied to the PM2.5 air pollution application. Experiments conducted in California, USA, demonstrate that the proposed method outperforms state-of-the-art approaches.

Abstract: In the present study, to understand the spatial distribution characteristics of the ETo over India, spatial interpolation was performed on the means of 32 years (1971–2002) monthly data of 131 India Meteorological Department stations uniformly distributed over the country by two methods, namely, inverse distance weighted (IDW) interpolation and kriging. Kriging was found to be better while developing the monthly surfaces during cross-validation. However, in station-wise validation, IDW performed better than kriging in almost all the cases, hence is recommended for spatial interpolation of ETo and its governing meteorological parameters. This study also checked if direct kriging of FAO-56 Penman–Monteith (PM) (Allen et al. in Crop evapotranspiration—guidelines for computing crop water requirements, Irrigation and drainage paper 56, Food and Agriculture Organization of the United Nations (FAO), Rome, 1998) point ETo produced comparable results against ETo estimated with individually kriged weather parameters (indirect kriging). Indirect kriging performed marginally well compared to direct kriging. Point ETo values were extended to areal ETo values by IDW and FAO-56 PM mean ETo maps for India were developed to obtain sufficiently accurate ETo estimates at unknown locations.

Abstract: 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.

Abstract: This paper considers the problem of co-array interpolation for direction-of-arrival (DOA) estimation with sparse nonuniform arrays. By utilizing the much longer difference co-array associated with these arrays, it is possible to perform DOA estimation of more sources than sensors. Since the co-array may contain holes (or missing lags), interpolation algorithms have been proposed to fully utilize the remaining elements of the co-array beyond that captured in the contiguous ULA segment. However, the quality and stability of interpolation performed by such algorithms (especially in presence of modeling errors) have not been analyzed. This paper provides a unified analysis of co-array interpolation algorithms to bound the interpolation error in terms of modeling errors. The results are universal in the sense that they can be applied to analyze any algorithm that utilizes the positive semidefinite (PSD) structure of the interpolated covariance matrix. The general framework is then applied to analyze specific algorithms and simulations are conducted to study their interpolation errors.

Abstract: Accurate assessment of spatial and temporal precipitation is crucial for simulating hydrological processes in basins, but is challenging due to insufficient rain gauges. Our study aims to analyze different precipitation interpolation schemes and their performances in runoff simulation during light and heavy rain periods. In particular, combinations of different interpolation estimates are explored and their performances in runoff simulation are discussed. The study was carried out in the Pengxi River basin of the Three Gorges Basin. Precipitation data from 16 rain gauges were interpolated using the Thiessen Polygon (TP), Inverse Distance Weighted (IDW), and Co-Kriging (CK) methods. Results showed that streamflow predictions employing CK inputs demonstrated the best performance in the whole process, in terms of the Nash–Sutcliffe Coefficient (NSE), the coefficient of determination (R2), and the Root Mean Square Error (RMSE) indices. The TP, IDW, and CK methods showed good performance in the heavy rain period but poor performance in the light rain period compared with the default method (least sophisticated nearest neighbor technique) in Soil and Water Assessment Tool (SWAT). Furthermore, the correlation between the dynamic weight of one method and its performance during runoff simulation followed a parabolic function. The combination of CK and TP achieved a better performance in decreasing the largest and lowest absolute errors compared to any single method, but the IDW method outperformed all methods in terms of the median absolute error. However, it is clear from our findings that interpolation methods should be chosen depending on the amount of precipitation, adaptability of the method, and accuracy of the estimate in different rain periods.

Abstract: 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.

Abstract: In this paper, we propose a new interpolation technique which considers all the neighboring pixels as well as their impact on the reference pixels to provide better quality interpolated image and a new data hiding scheme which embeds the secret data in the interpolated pixels by taking into account the human visual system so that quality of the resultant image is maintained. The proposed interpolation technique is an improvement of the existing neighbor mean interpolation (NMI) technique in such a way that the interpolated image would have more resemblance to the input image. The proposed interpolation technique has less computational cost like NMI as it does not perform much computation during estimation unlike B-Spline, Bilinear Interpolation etc. The proposed data hiding scheme comes into the category of reversible data hiding scheme as the input image can be reconstructed after extraction of the entire secret data at the receiver side. Thus, it reduces the communication cost. Furthermore, the proposed data hiding scheme identifies the smooth and complex regions of the interpolated (or cover) image by dividing the same into blocks. It then embeds more bits into the complex regions of the image so that data hiding capacity as well as the image quality can be enhanced. The experimental results shows that the percentage increment in the PSNR value and capacity of the proposed scheme with respect to Chang et al. method is in the range of 0.26 to 30.60% and 0.87 to 73.82%, respectively. Moreover, the modified NMI yields higher PSNRs than other interpolating methods such as NMI, BI, and ENMI.

Abstract: We propose a new approach for the radial basis function (RBF) interpolation of large scattered data sets. It uses the space subdivision technique into independent cells allowing processing of large data sets with low memory requirements and offering high computation speed, together with the possibility of parallel processing as each cell can be processed independently. The proposed RBF interpolation was tested on both synthetic and real data sets. It proved its simplicity, robustness and the ability to handle large data sets together with significant speed-up. In the case of parallel processing, speed-up was experimentally proved when 2 and 4 threads were used.

Abstract: The general concept of spatial interpolation is first discussed, particularly in the contexts of spatial sampling and the geographical nature of spatial data. Then two major interpolation approaches, deterministic and statistical/geostatistical, are differentiated. The inverse-distance weighting (IDW) method is introduced as a specific method using a spatial averaged weighting scheme based upon the inverse of distance between the sampled locations and the location of predicted value. Parameters involved in specifying the IDW method are highlighted using examples to illustrate their impacts on the interpolation results. Limitations, potential algorithmic problems, and an extension of the IDW method are also discussed. Keywords: distance-decay parameter; first law; kriging; spatial interpolation; spatial sampling; weights

Abstract: The marker-in-cell method is generally considered a flexible and robust method to model the advection of heterogenous non-diffusive properties (i.e., rock type or composition) in geodynamic problems. In this method, Lagrangian points carrying compositional information are advected with the ambient velocity field on an Eulerian grid. However, velocity interpolation from grid points to marker locations is often performed without considering the divergence of the velocity field at the interpolated locations (i.e., non-conservative). Such interpolation schemes can induce non-physical clustering of markers when strong velocity gradients are present (Journal of Computational Physics 166:218–252, 2001) and this may, eventually, result in empty grid cells, a serious numerical violation of the marker-in-cell method. To remedy this at low computational costs, Jenny et al. (Journal of Computational Physics 166:218–252, 2001) and Meyer and Jenny (Proceedings in Applied Mathematics and Mechanics 4:466–467, 2004) proposed a simple, conservative velocity interpolation scheme for 2-D staggered grid, while Wang et al. (Geochemistry, Geophysics, Geosystems 16(6):2015–2023, 2015) extended the formulation to 3-D finite element methods. Here, we adapt this formulation for 3-D staggered grids (correction interpolation) and we report on the quality of various velocity interpolation methods for 2-D and 3-D staggered grids. We test the interpolation schemes in combination with different advection schemes on incompressible Stokes problems with strong velocity gradients, which are discretized using a finite difference method. Our results suggest that a conservative formulation reduces the dispersion and clustering of markers, minimizing the need of unphysical marker control in geodynamic models.

Abstract: The focus of this paper is on evaluating different spatial interpolation methods for the construction of radio environment map using field measurements obtained by cyclostationary-based mobile sensors. Mobile sensing devices employing cyclostationary detectors provide lot of advantages compared to the widely used energy detectors, such as robustness to noise uncertainty and ability to distinguish among different primary user signals. However, mobile sensing results are not available at locations between the sensors making it difficult for a secondary user (possibly without a spectrum sensor) to decide whether to use primary user resources at that location. To overcome this, spatial interpolation of test statistics measured at limited number of locations can be carried out to create a channel occupancy map at unmeasured locations between the sensors. For this purpose, different spatial interpolation techniques for the cyclostationary test statistic have been employed in this paper such as inverse distance weighting, ordinary kriging, and universal kriging. The effectiveness of these methods is demonstrated by applying them on extensive real-world field measurement data obtained by mobile-phone-compliant spectrum sensors. The field measurements were carried out using four mobile spectrum sensors measuring eight digital video broadcasting-terrestrial (DVB-T) channels at more than hundred locations encompassing roughly one-third of the area of the city of Espoo in Finland. The accuracy of the spatial interpolation results based on the field measurements is determined using the cross-validation approach with the widely used root mean square error as the metric. Field measurement results indicate that reliable results with spatial coverage can be achieved using kriging for cyclostationary based test statistics. Comparison of spatial interpolation results of cyclostationary test statistics is also carried out with those of energy values obtained during the measurement campaign in the form of received signal strength indicator. Comparison results clearly show the performance improvement and robustness obtained using cyclostationary based detectors instead of energy detectors.

Abstract: Spatial interpolation methods have been applied in many environmental research studies. However, it is still a controversial issue to select an appropriate interpolation method for the conversion of discrete sampling sites into continuous maps. This study aimed at selecting an optimal interpolation method to analyze the spatial pattern of atmospheric N deposition in South China. N deposition was calculated by 259 moss sample data. Four spatial interpolation methods, including inverse distance weighting (IDW), radial basis function (RBF), ordinary kriging (OK), and universal kriging (UK), were utilized for modeling the spatial distribution of N deposition. It is the first time that these methods were applied to analyze N deposition in South China. Validation method was used to evaluate the interpolation precision of the various methods, and the cross-validation method was used to evaluate their interpolation accuracy. Comparison of predicted values with measured values indicated that OK was the optimal method for analyzing the spatial distribution of N deposition in this study; it had the highest precision (mean error (ME) = -0.059, root-mean-square error (RMSE) = 5.240, mean relative error (MRE) = 0.129, mean absolute error (MAE) = 4.007) and the lowest uncertainties (standard deviation (SD) = 5.47, coefficient of variation (CV) = 0.15). RBF produced similar results as good as OK, while the worst performed interpolation method was UK. By using the OK method for analyzing N deposition, this work revealed systematic temporal and spatial variations in atmospheric N deposition in South China.

Abstract: Rapid satellite-to-site visibility determination is of great significance to coverage analysis of satellite constellations as well as onboard mission planning of autonomous spacecraft. This paper presents a novel self-adaptive Hermite interpolation technique for rapid satellite-to-site visibility determination. Piecewise cubic curves are utilized to approximate the waveform of the visibility function versus time. The fourth-order derivative is used to control the approximation error and to optimize the time step for interpolation. The rise and set times are analytically obtained from the roots of cubic polynomials. To further increase the computational speed, an interval shrinking strategy is adopted via investigating the geometric relationship between the ground viewing cone and the orbit trajectory. Simulation results show a 98% decrease in computation time over the brute force method. The method is suitable for all orbital types and analytical orbit propagators.

Abstract: In this article a rational ball cubic B-spline representation based image interpolation scheme is proposed with tension parameter to interpolate two dimensional natural images. Genetic algorithm is used to detect the optimal value of tension parameter, so that sum square error is minimum. Comparison among the proposed interpolation scheme and the existing image interpolation techniques in terms of some state-of-the-art image quality metrics with classical Peak Signal-to-Noise Ratio (PSNR), depicts that suggested ball cubic B-spline form produces better results and is more suitable for problems related to image interpolation.

Abstract: Windstorms result in significant damage and economic loss and are a major recurring threat in many countries Estimating surface-level wind speeds resulting from windstorms is a complicated problem, but geostatistical spatial interpolation methods present a potential solution Maximum sustained and peak gust weather station data from two historic windstorms in Europe were analyzed to predict surface-level wind speed surfaces across a large and topographically varied landscape Disjunctively sampled maximum sustained wind speeds were adjusted to represent equivalent continuously sampled 10-minute wind speeds and missing peak gust station data were estimated by applying a gust factor to the recorded maximum sustained wind speeds Wind surfaces were estimated based on anisotropic and isotropic kriging interpolation methodologies The study found that anisotropic kriging is well-suited for interpolating wind speeds in meso- and macro-scale areas because it accounts for wind direction and trends in win

Abstract: Issues of missing data have become increasingly serious with the rapid increase in usage of traffic sensors. Analyses of the Beijing ring expressway have showed that up to 50% of microwave sensors pose missing values. The imputation of missing traffic data must be urgently solved although a precise solution that cannot be easily achieved due to the significant number of missing portions. In this study, copula-based models are proposed for the spatial interpolation of traffic flow from remote traffic microwave sensors. Most existing interpolation methods only rely on covariance functions to depict spatial correlation and are unsuitable for coping with anomalies due to Gaussian consumption. Copula theory overcomes this issue and provides a connection between the correlation function and the marginal distribution function of traffic flow. To validate copula-based models, a comparison with three kriging methods is conducted. Results indicate that copula-based models outperform kriging methods, especially on roads with irregular traffic patterns. Copula-based models demonstrate significant potential to impute missing data in large-scale transportation networks.

Abstract: Ionospheric delay effect is a critical issue that limits the accuracy of precise Global Navigation Satellite System (GNSS) positioning and navigation for single-frequency users, especially in mid- and low-latitude regions where variations in the ionosphere are larger. Kriging spatial interpolation techniques have been recently introduced to model the spatial correlation and variability of ionosphere, which intrinsically assume that the ionosphere field is stochastically stationary but does not take the random observational errors into account. In this paper, by treating the spatial statistical information on ionosphere as prior knowledge and based on Total Electron Content (TEC) semivariogram analysis, we use Kriging techniques to spatially interpolate TEC values. By assuming that the stochastic models of both the ionospheric signals and measurement errors are only known up to some unknown factors, we propose a new Kriging spatial interpolation method with unknown variance components for both the signals of ionosphere and TEC measurements. Variance component estimation has been integrated with Kriging to reconstruct regional ionospheric delays. The method has been applied to data from the Crustal Movement Observation Network of China (CMONOC) and compared with the ordinary Kriging and polynomial interpolations with spherical cap harmonic functions, polynomial functions and low-degree spherical harmonic functions. The statistics of results indicate that the daily ionospheric variations during the experimental period characterized by the proposed approach have good agreement with the other methods, ranging from 10 to 80 TEC Unit (TECU, 1 TECU = 1 × 1016 electrons/m²) with an overall mean of 28.2 TECU. The proposed method can produce more appropriate estimations whose general TEC level is as smooth as the ordinary Kriging but with a smaller standard deviation around 3 TECU than others. The residual results show that the interpolation precision of the new proposed method is better than the ordinary Kriging and polynomial interpolation by about 1.2 TECU and 0.7 TECU, respectively. The root mean squared error of the proposed new Kriging with variance components is within 1.5 TECU and is smaller than those from other methods under comparison by about 1 TECU. When compared with ionospheric grid points, the mean squared error of the proposed method is within 6 TECU and smaller than Kriging, indicating that the proposed method can produce more accurate ionospheric delays and better estimation accuracy over China regional area.