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In this study, for feature extraction of seismic- and nonlinear trend terms of the nonlinear and nonstationary high-rate Global Navigation Satellite System (GNSS) seismic displacements, we present an adaptive complete ensemble empirical mode decomposition (CEEMD) method based on correlation coefficient. We also introduce an improved CEEMD (ICEEMD) based on multiway principal component analysis (MPCA), hereafter, ICEEMD-MPCA, denoising method for boosting the high-rate GNSS seismic displacements with/without nonlinear trend. The results show that the proposed method is suitable for extracting features of nonlinear and nonstationary high-rate GNSS seismic displacements. The ICEEMD-MPCA denoising method is able to obviously eliminate white noise at high frequencies, remove systematic errors, and effectively preserve the earthquake wave signals. The denoising method reflects that high-rate GNSS has millimeter-level accuracy in the vertical component and submillimeter-level accuracy in the horizontal component. To verify the performance characteristics of the feature extraction and denoising method, the focal mechanism inversion for the 2010 El Mayor–Cucapah earthquake is estimated using the high-rate GNSS data (5 Hz) and collocated strong motion data (50 and 200 Hz). The focal mechanism of the earthquake estimated using the denoised high-rate GNSS with/without nonlinear trend are more consistent with the teleseismic estimates than those estimated using the raw high-rate GNSS and strong motion data. For the GNSS observations, the inversion results are more affected by errors in low-frequency, which accumulate over time and display their signatures as trend in coseismic displacements. It can be concluded that with the adaptive CEEMD method and the ICEEMD-MPCA denoising method, high-rate GNSS at near-field stations only or combined with GNSS/strong-motion records can be applied to yield better information regarding the earthquake focal mechanism inversion, the lower bounds of earthquake magnitudes, and earthquake rupture parameters for small to moderate earthquakes.

We propose a fundamental theorem for eco-environmental surface modelling (FTEEM) in order to apply it into the fields of ecology and environmental science more easily after the fundamental theorem for Earth’s surface system modeling (FTESM). The Beijing-Tianjin-Hebei (BTH) region is taken as a case area to conduct empirical studies of algorithms for spatial upscaling, spatial downscaling, spatial interpolation, data fusion and model-data assimilation, which are based on high accuracy surface modelling (HASM), corresponding with corollaries of FTEEM. The case studies demonstrate how eco-environmental surface modelling is substantially improved when both extrinsic and intrinsic information are used along with an appropriate method of HASM. Compared with classic algorithms, the HASM-based algorithm for spatial upscaling reduced the root-mean-square error of the BTH elevation surface by 9 m. The HASM-based algorithm for spatial downscaling reduced the relative error of future scenarios of annual mean temperature by 16%. The HASM-based algorithm for spatial interpolation reduced the relative error of change trend of annual mean precipitation by 0.2%. The HASM-based algorithm for data fusion reduced the relative error of change trend of annual mean temperature by 70%. The HASM-based algorithm for model-data assimilation reduced the relative error of carbon stocks by 40%. We propose five theoretical challenges and three application problems of HASM that need to be addressed to improve FTEEM.

Airborne light detection and ranging (LiDAR) technology has become the mainstream data source in geosciences and environmental sciences. Point cloud filtering is a prerequisite for almost all LiDAR-based applications. However, it is challenging to select a suitable filtering algorithm for handling high-density point clouds over complex landscapes. Therefore, to determine an appropriate filter on a specific environment, this paper comparatively assessed the performance of five representative filtering algorithms on six study sites with different terrain characteristics, where three plots are located in urban areas and three in forest areas. The representative filtering methods include simple morphological filter (SMRF), multiresolution hierarchical filter (MHF), slope-based filter (SBF), progressive TIN densification (PTD) and segmentation-based filter (SegBF). Results demonstrate that SMRF performs the best in urban areas, and compared to MHF, SBF, PTD and SegBF, the total error of SMRF is reduced by 1.38%, 48.21%, 48.25% and 31.03%, respectively. MHF outperforms the others in forest areas, and compared to SMRF, SBF, PTD and SegBF, the total error of MHF is reduced by 1.98%, 35.87%, 45.11% and 9.42%, respectively. Moreover, both SMRF and MHF keep a good balance between type I and II errors, which makes the produced DEMs much similar to the references. Overall, SMRF and MHF are recommended for urban and forest areas, respectively, and MHF averagely performs slightly better than SMRF on all areas with respect to kappa coefficient.

Airborne light detection and ranging (lidar) is becoming a widely adopted technique for capturing elevation data, which are mainly used for creating digital terrain models (DTMs). However, the large size of lidar datasets poses a serious computational challenge to the promising radial basis function (RBF) interpolation method. In this work, to reduce the huge computational cost and improve the interpolation accuracy, random Fourier features are first introduced to approximate the Gaussian kernel of RBFs in feature space, then a random features-based weighted RBF interpolation method is developed. Based on randomized Fourier features, the nonlinear kernel-based training and evaluation of the RBF method is transformed into simple linear operations in feature space, and with the help of weighted ridge regression, the negative effect of the non-Gaussian distribution of lidar datasets on DTM production is reduced. In other words, the combination of randomized Fourier features and weighted ridge regression improves the efficiency and accuracy of the RBF interpolation method. Experiments on simulated datasets indicate that the proposed method performs better than the classical or random features-based RBF method for dealing with non-Gaussian distributed samples, with the former being slightly less accurate than the iterative RBF method due to the low-dimensional random features. However, the computational cost of the new method is much lower compared with the classical or iterative RBFs. Interpolation of airborne lidar-derived points demonstrates that the new method has a computational cost similar to the inverse distance weighting and triangulated irregular network (TIN) approaches, and is significantly faster than the ordinary kriging (OK) or thin plate spline (TPS) methods. Quantitatively, for interpolation of 644,433 points, the proposed method is approximately 833 and 21 times faster than OK and TPS, respectively. Moreover, the new method avoids the surface discontinuity artifacts presented by the OK, TPS, and TIN methods.

Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM) inherently suffers from various errors. Many previous works employed Geoscience Laser Altimeter System onboard the Ice, Cloud, and land Elevation Satellite (ICESat/GLAS) data to assess and enhance SRTM DEM accuracy. Nevertheless, data coregistration between the two datasets was commonly neglected in their studies. In this paper, an automated and simple three dimensional (3D) coregistration method (3CM) was introduced to align the 3-arc-second SRTM (SRTM3) DEM and ICESat/GLAS data over Jiangxi province, China. Then, accuracy evaluation of the SRTM3 DEM using ICESat/GLAS data with and without data coregistration was performed on different classes of terrain factors and different land uses, with the purpose of evaluating the importance of data coregistration. Results show that after data coregistration, the root mean square error (RMSE) and mean bias of the SRTM3 DEM are reduced by 14.4% and 97.1%, respectively. Without data coregistration, terrain aspects with a sine-like shape are strongly related to SRTM3 DEM errors; nevertheless, this relationship disappears after data coregistration. Among the six land uses, SRTM3 DEM produces the lowest accuracy in forest areas. Finally, by incorporating land uses, terrain factors and ICESat/GLAS data into the correction models, the SRTM3 DEM was enhanced using multiple linear regression (MLR), back propagation neural network (BPNN), generalized regression NN (GRNN), and random forest (RF), respectively. Results exhibit that the four enhancement models with data coregistration obviously outperform themselves without the coregistration. Among the four models, RF produces the best result, and its RMSE is about 3.1%, 2.7% and 11.3% lower than those of MLR, BPNN, and GRNN, respectively. Moreover, 146 Global Navigation Satellite System (GNSS) points over Ganzhou city of Jiangxi province were used to assess the accuracy of the RF-derived SRTM3 DEM. It is found that the DEM quality is improved and has a similar error magnitude to that relative to the ICESat/GLASS data.

Surface modeling with very large data sets is challenging. An efficient method for modeling massive data sets using the high accuracy surface modeling method (HASM) is proposed, and HASM_Big is developed to handle very large data sets. A large data set is defined here as a large spatial domain with high resolution leading to a linear equation with matrix dimensions of hundreds of thousands. An augmented system approach is employed to solve the equality-constrained least squares problem (LSE) produced in HASM_Big, and a block row action method is applied to solve the corresponding very large matrix equations. A matrix partitioning method is used to avoid information redundancy among each block and thereby accelerate the model. Experiments including numerical tests and real-world applications are used to compare the performances of HASM_Big with its previous version, HASM. Results show that the memory storage and computing speed of HASM_Big are better than those of HASM. It is found that the computational cost of HASM_Big is linearly scalable, even with massive data sets. In conclusion, HASM_Big provides a powerful tool for surface modeling, especially when there are millions or more computing grid cells.

Light detection and ranging (lidar)-derived elevation data are commonly subjected to outliers due to the boundaries of occlusions, physical imperfections of sensors, and surface reflectance. Outliers have a serious negative effect on the accuracy of digital elevation models (DEMs). To decrease the impact of outliers on DEM construction, we propose a robust interpolation algorithm of multiquadric (MQ) based on a regularized least absolute deviation (LAD) technique. The objective function of the proposed method includes a regularization-based smoothing term and an LAD-based fitting term, respectively, used to smooth noisy samples and resist the influence of outliers. To solve the objective function of the proposed method, we develop a simple scheme based on the split-Bregman iteration algorithm. Results from simulated data sets indicate that when sample points are noisy or contaminated by outliers, the proposed method is more accurate than the classical MQ and two recently developed robust algorithms of MQ for surface modeling. Real-world examples of interpolating 1 private and 11 publicly available airborne lidar-derived data sets demonstrate that the proposed method averagely produces better results than two promising interpolation methods including regularized spline with tension (RST) and gridded data-based robust thin plate spline (RTPS). Specifically, the image of RTPS is too smooth to retain terrain details. Although RST can keep subtle terrain features, it is distorted by some misclassified object points (i.e., pseudooutliers). The proposed method obtains a good tradeoff between resisting the effect of outliers and preserving terrain features. Overall, the proposed method can be considered as an alternative for interpolating lidar-derived data sets potentially including outliers.

Airborne light detection and ranging (LiDAR) datasets with a large volume pose a great challenge to the traditional interpolation methods for the production of digital terrain models (DTMs). Thus, a fast, global interpolation method based on thin plate spline (TPS) is proposed in this paper. In the methodology, a weighted version of finite difference TPS is first developed to deal with the problem of missing data in the grid-based surface construction. Then, the interpolation matrix of the weighted TPS is deduced and found to be largely sparse. Furthermore, the values and positions of each nonzero element in the matrix are analytically determined. Finally, to make full use of the sparseness of the interpolation matrix, the linear system is solved with an iterative manner. These make the new method not only fast, but also require less random-access memory. Tests on six simulated datasets indicate that compared to recently developed discrete cosine transformation (DCT)-based TPS, the proposed method has a higher speed and accuracy, lower memory requirement, and less sensitivity to the smoothing parameter. Real-world examples on 10 public and 1 private dataset demonstrate that compared to the DCT-based TPS and the locally weighted interpolation methods, such as linear, natural neighbor (NN), inverse distance weighting (IDW), and ordinary kriging (OK), the proposed method produces visually good surfaces, which overcome the problems of peak-cutting, coarseness, and discontinuity of the aforementioned interpolators. More importantly, the proposed method has a similar performance to the simple interpolation methods (e.g., IDW and NN) with respect to computing time and memory cost, and significantly outperforms OK. Overall, the proposed method with low memory requirement and computing cost offers great potential for the derivation of DTMs from large-scale LiDAR datasets.

A high accuracy surface modeling method (HASM) has been developed to provide a solution to many surface modeling problems such as DEM construction, surface estimation and spatial prediction. Although HASM is able to model surfaces with a higher accuracy, its low computing speed limits its popularity in constructing large scale surfaces. Hence, the research described in this article aims to improve the computing efficiency of HASM with a graphic processor unit (GPU) accelerated multi-grid method (HASM-GMG). HASM-GMG was tested with two types of surfaces: a Gauss synthetic surface and a real-world example. Results indicate that HASM-GMG can gain significant speedups compared with CPU-based HASM without acceleration on GPU. Moreover, both the accuracy and speed of HASM-GMG are superior to the classical interpolation methods including Kriging, Spline and IDW.

Digital elevation model (DEM) source data are subject to both horizontal and vertical errors owing to improper instrument operation, physical limitations of sensors, and bad weather conditions. These factors may bring a negative effect on some DEM-based applications requiring low levels of positional errors. Although classical smoothing interpolation methods have the ability to handle vertical errors, they are prone to omit horizontal errors. Based on the statistical concept of the total least squares method, a total error-based multiquadric (MQ-T) method is proposed in this paper to reduce the effects of both horizontal and vertical errors in the context of DEM construction. In nature, the classical multiquadric (MQ) method is a vertical error regression procedure, whereas MQ-T is an orthogonal error regression model. Two examples, including a numerical test and a real-world example, are employed in a comparative performance analysis of MQ-T for surface modeling of DEMs. The numerical test indicates that...

The classification of airborne laser scanning ALS point clouds is a necessary step in the process of digital elevation model DEM construction. Motivated by the surface interpolation filtering scheme, a multi-resolution hierarchical classification MHC method was developed to perform classification in our previous research. However, similar to almost all filtering methods, MHC suffers from the problem of misclassifying off-terrain and ground points in complex landscapes with steep slopes and large discontinuities. To improve the classification accuracy in the above landscapes, the original MHC is improved in this article by adopting a region growing-based segmentation method to maximally capture the initial ground seeds of MHC. Furthermore, considering the high sensitivity of principle component analysis PCA to outliers, a robust PCA approach based on least trimmed squares LTS is proposed to estimate point normals. Fifteen groups of data sets provided by the International Society for Photogrammetry and Remote Sensing ISPRS Working Group III/3 are utilized to compare the performances of the improved and original MHCs. Results indicate that the improved MHC is more accurate than the original MHC, irrespective of accuracy measures. In addition, we also compare the results of our method with three new filtering methods developed in the years 2013 and 2014. It is found that regardless of landscapes i.e. urban and rural samples, the improved MHC is averagely more accurate than the three filtering methods in terms of type II error, and irrespective of accuracy measures i.e. types I, II and total errors, our method outperforms the three filtering methods for the rural samples. In short, the robust segmentation-based MHC demonstrates great potential to ALS-derived DEM construction in rural areas.

To overcome the huge volume problem of light detection and ranging (LiDAR) data for the derivation of digital terrain models (DTMs), a least squares compactly supported radial basis function (CSRBF) interpolation method is proposed in this paper. The proposed method has a limited support radius and fewer RBF centers than the sample points, selected by a newly developed surface variation-based algorithm. Those make the linear system of the proposed method not only much sparser but also efficiently solvable. Tests on a synthetic dataset demonstrate that the proposed method is comparable to the smoothing RBF, and far superior to the exact RBF. Moreover, the first is much faster than the others. The proposed method with the RBF centers selected by the surface variation-based algorithm obviously outperforms that with the random selection of equal number. Real-world examples on one private and ten public datasets show that the surfaces of simple interpolation methods including inverse distance weighting, natural neighbor, linear and bicubic suffer from the problems of roughness, peak-cutting, discontinuity and subtle terrain feature loss, respectively. By contrast, the proposed method produces visually appealing results, keeping a good tradeoff between noise removal and terrain feature preservation. Additionally, the new method compares favorably with ordinary kriging (OK) for the generation of high-resolution DTMs in terms of interpolation accuracy, yet the former is much more robust to spatial resolution variation and terrain characteristics than the latter. More importantly, our method is about 4 times faster than OK. In conclusion, the proposed method has high potential for the interpolation of a large LiDAR dataset, especially when both interpolation accuracy and computational cost are taken into account.

Light detection and ranging (LiDAR) technique is currently one of the most important tools for collecting elevation points with a high density in the context of digital elevation model (DEM) construction. However, the high density data always leads to serious time and memory consumption problems in data processing. In this paper, we have developed a thin plate spline (TPS)-based feature-preserving (TPS-F) method for LiDAR-derived ground data reduction by selecting a certain amount of significant terrain points and by extracting geomorphological features from the raw dataset to maintain the accuracy of constructed DEMs as high as possible, while maximally keeping terrain features. We employed four study sites with different topographies (i.e., flat, undulating, hilly and mountainous terrains) to analyze the performance of TPS-F for LiDAR data reduction in the context of DEM construction. These results were compared with those of the TPS-based algorithm without features (TPS-W) and two classical data selection methods including maximum z-tolerance (Max-Z) and the random method. Results show that irrespective of terrain characteristic, the two versions of TPS-based approaches (i.e., TPS-F and TPS-W) are always more accurate than the classical methods in terms of error range and root means square error. Moreover, in terms of streamline matching rate (SMR), TPS-F has a better ability of preserving geomorphological features, especially for the mountainous terrain. For example, the average SMR of TPS-F is 89.2% in the mountainous area, while those of TPS-W, max-Z and the random method are 56.6%, 34.7% and 35.3%, respectively.

Remote-sensing-derived elevation data sets often suffer from noise and outliers due to various reasons, such as the physical limitations of sensors, multiple reflectance, occlusions and low contrast of texture. Outliers generally have a seriously negative effect on DEM construction. Some interpolation methods like ordinary kriging (OK) are capable of smoothing noise inherent in sample points, but are sensitive to outliers. In this paper, a robust algorithm of multiquadric method (MQ) based on an Improved Huber loss function (MQ-IH) has been developed to decrease the impact of outliers on DEM construction. Theoretically, the improved Huber loss function is null for outliers, quadratic for small errors, and linear for others. Simulated data sets drawn from a mathematical surface with different error distributions were employed to analyze the robustness of MQ-IH. Results indicate that MQ-IH obtains a good balance between efficiency and robustness. Namely, the performance of MQ-IH is comparative to those of the classical MQ and MQ based on the Classical Huber loss function (MQ-CH) when sample points follow a normal distribution, and the former outperforms the latter two when sample points are subject to outliers. For example, for the Cauchy error distribution with the location parameter of 0 and scale parameter of 1, the root mean square errors (RMSEs) of MQ-CH and the classical MQ are 0.3916 and 1.4591, respectively, whereas that of MQ-IH is 0.3698. The performance of MQ-IH is further evaluated by qualitative and quantitative analysis through a real-world example of DEM construction with the stereo-images-derived elevation points. Results demonstrate that compared with the classical interpolation methods, including natural neighbor (NN), OK and ANUDEM (a program that calculates regular grid digital elevation models (DEMs) with sensible shape and drainage structure from arbitrarily large topographic data sets), and two versions of MQ, including the classical MQ and MQ-CH, MQ-IH has a better ability of maximally reducing the impact of outliers, while faithfully preserving terrain features. Theoretically, MQ-IH is not a promising interpolation method, and some side effects can be found from its simulation results. For example, the contours of MQ-IH are coarser than ANUDEM in some locations of the real-world study site, and its hill shade may not strictly agree with the real-world surface at rough terrain. Furthermore, the computing cost of MQ-IH is much bigger than that of the classical interpolation methods. However, compared with the classical interpolation methods, MQ-IH has significant potential for interpolating remote-sensing-derived elevation data sets.

The inverse-distance weighting (IDW) method is considered as one of the most popular deterministic methods and is widely applied to a variety of areas because of its low computational cost and easy implementation. In this paper, we show that the classical IDW is essentially a zeroth-order local kernel regression method with an inverse distance weight function. Thus, it suffers from various shortcomings, such as the boundary bias. Considering the advantages of the local polynomial modeling technique in statistics, the classical IDW was generalized into a higher-order regression by the Taylor expansion and then computed by means of a weighted least-squares method. Surface modeling of rainfall fields in China indicated that the generalized IDWs with the first- and second-orders are more accurate than the classical IDW in terms of root mean square error (RMSE). The example of digital elevation model construction with a group of sample points showed that the two generalized IDWs have better RMSE and mean error than the classical IDW. Furthermore, the second-order IDW has a better performance than the ordinary kriging in terms of RMSE. A theoretical analysis demonstrated that the gradient-plus-inverse distance squared method presented by Nalder and Wein (Agric For Meteorol 92(4): 211-225, 1998) is a first-order form of the generalized IDW expanded on spatial coordinates and elevation. In a word, the generalized IDW can incorporate multiple covariates, which can better explain the interpolation procedure and might improve its accuracy.

The sensitivities of the initial value and the sampling information to the accuracy of a high accuracy surface modeling (HASM) are investigated and the implementations of this new modeling method are modified and enhanced. Based on the fundamental theorem of surface theory, HASM is developed to correct the error produced in geographical information system and ecological modeling process. However, the earlier version of HASM is theoretically incomplete and its initial value must be produced by other surface modeling methods, such as spline, which limit its promotion. In other words, we must use other interpolators to drive HASM. According to the fundamental theorem of surface theory, we modify HASM, namely HASM.MOD, by adding another important nonlinear equation to make it independent of other methods and, at the same time, have a complete and solid theory foundation. Two mathematic surfaces and monthly mean temperature of 1951-2010 are used to validate the effectiveness of the new method. Experiments show that the modified version of HASM is insensitive to the selection of initial value which is particular important for HASM. We analyze the sensitivities of sampling error and sampling ratio to the simulation accuracy of HASM.MOD. It is found that sampling information plays an important role in the simulation accuracy of HASM.MOD. Another feature of the modified version of HASM is that it is theoretically perfect as it considers the third equation of the surface theory which reflects the local warping of the surface. The modified HASM may be useful with a wide range of spatial interpolation as it would no longer rely on other interpolation methods.

Since spatial datasets are subject to sampling errors, a smoothing interpolation method should be employed to remove noise during DEM construction. Although least squares support vector machines (LSSVM) have been widely accepted as a classifier, their effect on smoothing noisy data is almost unknown. In this article, the smoothness of LSSVM was explored, and its effect on smoothing noisy data in DEM construction was tested. In order to improve the ability to deal with large datasets, a local method of LSSVM has been developed, where only the neighboring sampling points around the one to be estimated are used for computation. A numerical test indicated that LSSVM is more accurate than the classical smoothing methods including TPS and kriging, and its error surfaces are more evenly distributed. The real-world example of smoothing noise inherent in lidar-derived DEMs also showed that LSSVM has a positive smoothing effect, which is approximately as accurate as TPS. In short, LSSVM with a high efficiency can be considered as an alternative smoothing method for smoothing noisy data in DEM construction.

Thin plate spline (TPS) has been widely accepted as a method for smooth fitting of noisy data. However, the classical TPS always has an ill-conditioning problem when two sample points are very close. Although the modified orthogonal least squares-based TPS (TPS-M) avoids this ill-conditioning problem, it is not completely immune to over-fitting when sample points are noisy. In this paper, a regularized least squares method of thin plate spline (TPS-RLS) was developed, which adds a weight penalty term to the error criterion of orthogonal least squares (OLS). TPS-RLS combines the advantages of both regularization and OLS, which avoid the over-fitting and the ill-conditioning problems simultaneously. Numerical tests indicate that irrespective of the standard deviation of sampling errors and the number of knots, TPS-RLS is always more accurate than TPS-M for smooth fitting of noisy data, whereas TPS-M would have a serious over-fitting problem if the optimal number of knots were not determined in advance. The real-world example of fitting total station instrument data shows that among the classical interpolation methods including IDW, natural neighbor and ordinary kriging, TPS-RLS has the highest accuracy for a series of DEMs with different resolutions, especially for the coarse one. Surface modeling of DEMs with contour lines demonstrate that TPS-RLS has a better performance than the classical methods in terms of both root mean squared error and relief shaded map appearance.

The multiquadric method (MQ) with high interpolation accuracy has been widely used for interpolating spatial data. However, MQ is an exact interpolation method, which is improper to interpolate noisy sampling data. Although the least squares MQ (LSMQ) has the ability to smooth out sampling errors, it is inherently not robust to outliers due to the least squares criterion in estimating the weights of sampling knots. In order to reduce the impact of outliers on the accuracy of digital elevation models (DEMs), a robust method of MQ (MQ-R) has been developed. MQ-R includes two independent procedures: knot selection and the solution of the system of linear equations. The two independent procedures were respectively achieved by the space-filling design and the least absolute deviation, both of which are very robust to outliers. Gaussian synthetic surface, which is subject to a series of errors with different distributions, was employed to compare the performance of MQ-R with that of LSMQ. Results indicate that LSMQ is seriously affected by outliers, whereas MQ-R performs well in resisting outliers, and can construct satisfactory surfaces even though the data are contaminated by severe outliers. A real-world example of DEM construction was employed to evaluate the robustness of MQ-R, LSMQ, and the classical interpolation methods including inverse distance weighting method, thin plate spline, and ANUDEM. Results showed that compared with the classical methods, MQ-R has the highest accuracy in terms of root mean square error. In conclusion, when sampling data is subject to outliers, MQ-R can be considered as an alternative method for DEM construction.

Geostatistical characterization of local DEM error is usually based on the assumption of a stationary variogram model which requires the mean and variance to be finite and constant in the area under investigation. However, in practice this assumption is appropriate only in a restricted spatial location, where the local experimental variograms vary slowly. Therefore, an adaptive method is developed in this article to model non-stationary variograms, for which the estimator and the indicator for characterization of spatial variation are a Voronoi map and the standard deviation of mean values displayed in the Voronoi map, respectively. For the adaptive method, the global domain is divided into different meshes with various sizes according to the variability of local variograms. The adaptive method of non-stationary variogram modeling is applied to simulating error surfaces of a LiDAR derived DEM located in Sichuan province, China. Results indicate that the locally adaptive variogram model is more accurate than the global one for capturing the characterization of spatial variation in DEM errors. The adaptive model can be considered as an alternative approach to modeling non-stationary variograms for DEM error surface simulation.

A new method for filling voids is developed by improving the approach to high-accuracy surface modelling (HASM), which is based on the first fundamental coefficients and the second fundamental coefficients of surfaces. The first fundamental coefficients are used to calculate the lengths of curves, angles of tangent vectors, areas of regions and geodesics on the surface. The second fundamental coefficients reflect the local warping of the surface, namely its deviation from the tangent plane at the point under consideration, which can be observed from outside the Earth. Nine regions with different landform complexities in hilly, plateau and mountainous areas are selected for testing the performance of HASM by comparing those ones of the classic methods such as triangulated irregular network (TIN), inverse distance weighted interpolation (IDW), advanced Spline method (ANUDEM), Spline and Kriging. The results demonstrated that the HASM void filling always has the highest accuracy regardless of the landform complexity, void area and auxiliary data.

A robust estimator based on the M-estimation principle (REMP) has been developed for digital elevation model (DEM) accuracy assessment. Adaptive weights were employed to respond to a broad class of DEM error distributions, and an iterative procedure in terms of REMP starting from robust initial estimates with a high breakdown point was introduced. Original DEMs with the resolution of 2 m were obtained by means of light detection and Ranging (LiDAR) from two study sites. DEM errors in each study site were, respectively, calculated based on 100 checkpoints captured by real-time kinematic (RTK) in terms of stratified random sampling strategy. Each group of DEM errors was, respectively, contaminated by five groups of outliers from different distributions. Thus, ten groups of simulated DEM errors were employed to comparatively assess the estimation accuracies of REMP and the classical estimators. The results indicated that under the non-normal distribution of DEM errors, the classical non-robust estimators are seriously influenced by the non-normality. Some robust estimators such as 10%-trimmed or Winsorized mean and normalized median absolute deviation (MADN) are not very robust to resist the influence of outliers. REMP, slightly affected by the non-normal distribution of DEM errors, is more accurate than the classical estimators. The robust methodology can adapt to the DEMs, especially the ones derived from remote sensing such as LiDAR or digital photogrammetry in non-open terrain.

An adaptive method is employed to speed up computation of high accuracy surface modeling (HASM), for which an error indicator and an error estimator are developed. Root mean-square error (RMSE) is used as the error estimator that is formulated as a function of gully density and grid cell size. The error indicator is developed on the basis of error surfaces for different spatial resolutions, which are interpolated in terms of the absolute errors calculated at sampled points while paying attention to the landform characteristics. The error surfaces indicate the magnitude and distribution of errors in each step of adaptive refinement and make spatial changes to the errors in the simulation process visualized. The adaptive method of high accuracy surface modeling (HASM-AM) is applied to simulating elevation surface of the Dong-Zhi tableland with 27.24 million pixels at a spatial resolution of 10 m 10 m. Test results show that HASM-AM has greatly speeded up computation by avoiding unnecessary calculations and saving memory. In addition, HASM-AM improves simulation accuracy.