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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.

In order to improve the possibility of correct interpretation of some geophysical phenomena such seismology, it is important to reduce systematic errors such as common mode error (CME) and multipath errors, and to improve the accuracy of high-rate (1–5 Hz) displacements. This paper focuses on accuracy enhancement and noise assessment of high-rate Global Navigation Satellite System (GNSS) position time series of regional network of GNSS stations. Complete ensemble empirical mode decomposition-based multiscale multiway principal component analysis (C-MSMPCA) denoising method is introduced to denoise the high-rate (1–5 Hz) GNSS position time series, whose performance was compared with those of modified sidereal filtering (MSF), stacking filtering, PCA and the recently developed multiscale multiway PCA (MSMPCA). It turns out that C-MSMPCA can significantly eliminate the high-frequency random noise and remove the low-frequency CME, and other systematic errors such as multipath errors. Furthermore, for denoising high-rate GNSS time series, C-MSMPCA is more precise than the other denoising methods. Irrespective of sampling rates and data processing strategies, C-MSMPCA can respectively reach the accuracy of submillimeter- and millimeter-levels on the horizontal and vertical components of high-rate GNSS displacements. Spectral analysis on the 1 Hz coordinate time series from the Sichuan GNSS Continuous Observing Network of China suggests that Gauss-Markov process is an adequate noise model for the noise characteristics of the filtered GNSS data using MSMPCA and C-MSMPCA. The CME magnitudes of MSMPCA are 2.61, 2.55, and 6.72 mm (east, north, and vertical), and those of C-MSMPCA are 2.69, 2.80, and 7.80 mm, respectively.

For many geoscience applications, the quality of digital elevation models (DEMs) is the first and foremost requirement. DEM quality generally depends on the interpolation procedure, especially in forested areas with complex terrain. However, classical interpolation methods do not always preserve terrain features, owing to their isotropic and local estimation nature. Therefore, this paper presents a feature-preserving interpolation method, where a structure tensor is integrated into the radial basis function to ensure its anisotropy. This method alleviates the influence of points located on different surface patches. The performance of the proposed method was compared with those of classical interpolation methods, including the inverse distance weighting (IDW), ordinary kriging (OK), topo to raster (ANUDEM), and natural neighbour (NN) methods. The performance of these methods was tested on benchmark data provided by the International Society for Photogrammetry and Remote Sensing (ISPRS) Commission, and on one private dataset. Both datasets were collected via the airborne light detection and ranging (LiDAR) technique. The results from both datasets demonstrate that from a quantitative perspective, the proposed method produces more accurate DEMs than the classical interpolation methods. Specifically, in terms of root mean square error (RMSE), the proposed method is at least 15.8% and 7.6% more accurate when applied to the ISPRS and private data, respectively. Moreover, the proposed method produces more visually appealing surfaces with a good trade-off between terrain feature retention and noise removal.

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.

Due to the penetration ability of airborne light detection and ranging (lidar) into tree crowns, data pits commonly appear in lidar-derived canopy height models (CHMs). They have a seriously negative effect on the quality of tree detection and subsequent biophysical measurements. To construct a pit-free CHM, an algorithm based on robust locally weighted regression and robust z-score was presented to remove data pits. The significant advantage of the new algorithm is parameter-free, which makes it efficient and robust for practical applications. A numerical test and a real-world example were respectively employed to assess the performance of our method for CHM construction, and its results were compared with those of three classical methods including natural neighbor interpolation of the highest point method, mean and median filters. The numerical test demonstrates that our algorithm is more accurate than the other methods for generating pit-free CHMs under the presence of data pits. The real-world example shows that compared with the classical methods, our method has a better ability of data pit removal. Moreover, our method performs better than the other methods for deriving plot-level maximum tree height from CHMs. In a word, the new method shows high potential for pit-free CHM construction.

To suppress the influence of outliers on function estimation, we propose a least absolute deviation (LAD)-based robust support vector regression (SVR). Furthermore, an efficient algorithm based on the split-Bregman iteration is introduced to solve the optimization problem of the proposed algorithm. Both artificial and benchmark datasets are employed to compare the performance of the proposed algorithm with those of least squares SVR (LS-SVR), and two weighted versions of LS-SVR with the weight functions of Hampel and Logistic, respectively. Experiments demonstrate the superiority of the proposed algorithm.

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.

Support vector machine for regression (SVR) is an efficient tool for solving function estimation problem. However, it is sensitive to outliers due to its unbounded loss function. In order to reduce the effect of outliers, we propose a robust SVR with a trimmed Huber loss function (SVRT) in this paper. Synthetic and benchmark datasets were, respectively, employed to comparatively assess the performance of SVRT, and its results were compared with those of SVR, least squares SVR (LS-SVR) and a weighted LS-SVR. The numerical test shows that when training samples are subject to errors with a normal distribution, SVRT is slightly less accurate than SVR and LS-SVR, yet more accurate than the weighted LS-SVR. However, when training samples are contaminated by outliers, SVRT has a better performance than the other methods. Furthermore, SVRT is faster than the weighted LS-SVR. Simulating eight benchmark datasets shows that SVRT is averagely more accurate than the other methods when sample points are contaminated by outliers. In conclusion, SVRT can be considered as an alternative robust method for simulating contaminated sample points.

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...

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.

Ground observation is able to obtain highly accurate data with high temporal resolution at observation points, but these observation points are too sparsely to satisfy the application requirements at regional scale. Satellite remote sensing can frequently supply spatially continuous information on earth surface, which is impossible from ground-based investigations, but remote sensing description is not able to directly obtain process parameters. In fact, in terms of fundamental theorem of surfaces, a surface is uniquely defined by the first fundamental coefficients, about the details of the surface observed when we stay on the surface, and the second fundamental coefficients, the change of the surface observed from outside the surface. A method for high accuracy surface modeling (HASM) has been developed initiatively to find solutions for error problem and slow-speed problem of earth surface modeling since 1986. HASM takes global approximate information (e.g., remote sensing images or model simulation results) as its driving field and local accurate information (e.g., ground observation data and/or sampling data) as its optimum control constraints. Its output satisfies the iteration stopping criterion which is determined by application requirement for accuracy. This paper reviews problems to be solved in every development stage and applications of HASM.

A new greedy-based multiquadric method (MQ-G) has been developed to perform LiDAR-derived ground data reduction by selecting a certain amount of significant terrain points from the raw dataset to keep the accuracy of the constructed DEMs as high as possible, while maximally retaining terrain features. In the process of MQ-G, the significant terrain points were selected with an iterative process. First, the points with the maximum and minimum elevations were selected as the initial significant points. Next, a smoothing MQ was employed to perform an interpolation with the selected critical points. Then, the importance of all candidate points was assessed by interpolation error (i.e. the absolute difference between the interpolated and actual elevations). Lastly, the most significant point in the current iteration was selected and used for point selection in the next iteration. The process was repeated until the number of selected points reached a pre-set level or no point was found to have the interpolation error exceeding a user-specified accuracy tolerance. In order to avoid the huge computing cost, a new technique was presented to quickly solve the systems of MQ equations in the global interpolation process, and then the global MQ was replaced with the local one when a certain amount of critical points were selected. Four study sites with different morphologies (i.e. flat, undulating, hilly and mountainous terrains) were respectively employed to comparatively analyze the performances of MQ-G and the classical data selection methods including maximum z-tolerance (Max-Z) and the random method for reducing LiDAR-derived ground datasets. Results show that irrespective of the number of selected critical points and terrain characteristics, MQ-G is always more accurate than the other methods for DEM construction. Moreover, MQ-G has a better ability of preserving terrain feature lines, especially for the undulating and hilly terrains.

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.

Data pits commonly appear in lidar-derived canopy height models (CHMs) owing to the penetration ability of airborne light detection and ranging (lidar) into tree crowns. They have a seriously negative effect on the quality of tree detection and subsequent biophysical measurements. In this study, we propose an algorithm based on robust locally weighted regression and robust z-scores for the construction of a pit-free CHM. A significant advantage of the new algorithm is that it is parameter free, which makes it efficient and robust for practical applications. Simulated and airborne lidar-derived data sets are employed to assess the performance of the new method for CHM construction, and its results are compared to those of three classical methods, namely the natural neighbor (NN) interpolation of the highest point method (HPM), mean filter, and median filter. The results from the simulated data set demonstrate that our algorithm is more accurate compared to the three classical methods for generating pit-free CHMs in the presence of data pits. CHM construction using the lidar-derived data set shows that, compared to the classical methods, the new method has a better ability to remove data pits as well as preserving the edges, shapes, and structures of canopy gaps and crowns. Moreover, the proposed method performs better compared to the classical methods in deriving plot-level maximum tree heights from CHMs. Thus, the new method shows high potential for pit-free CHM construction.

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.

We presented a multiresolution hierarchical classification (MHC) algorithm for differentiating ground from non-ground LiDAR point cloud based on point residuals from the interpolated raster surface. MHC includes three levels of hierarchy, with the simultaneous increase of cell resolution and residual threshold from the low to the high level of the hierarchy. At each level, the surface is iteratively interpolated towards the ground using thin plate spline (TPS) until no ground points are classified, and the classified ground points are used to update the surface in the next iteration. 15 groups of benchmark dataset, provided by the International Society for Photogrammetry and Remote Sensing (ISPRS) commission, were used to compare the performance of MHC with those of the 17 other publicized filtering methods. Results indicated that MHC with the average total error and average Cohen’s kappa coefficient of 4.11% and 86.27% performs better than all other filtering methods.

A sequential adjustment (SA) method is employed to decrease the computational cost of high-accuracy surface modeling (HASM), and the SA of HASM (HASM-SA) is being developed. A mathematical surface was used to comparatively analyze the computing speed of SA and the classical iterative solvers provided by MATLAB 7.7 for solving the system of linear equations of HASM. Results indicate that SA is much faster than the classical iterative solvers. The computing time of HASM-SA is determined by not only the total number of grid cells but also the number of sampling points in the computational domain. A real-world example of surface modeling of digital elevation models (DEMs) with various resolutions shows that HASM-SA is averagely more accurate and much faster than the commonly used interpolation methods, such as inverse distance weighting (IDW), kriging, and three versions of spline, namely regularized spline (RSpline), thin-plate spline (TPS), and ANUDEM in terms of root mean square error (RMSE), mean absolute...

To smooth noises inherent in uniformly sampled dataset, the smoothness of high accuracy surface modeling HASM was explored, and a smoothing method of HASM HASM-SM was developed based on a penalized least squares method. The optimal smoothing parameter of HASM-SM was automatically obtained by means of the generalized cross-validation GCV method. For an efficient smoothing computation, discrete cosine transform was employed to solve the system of HASM-SM and to estimate the minimum GCV score, simultaneously. Two examples including a numerical test and a real-world example were employed to compare the smoothing ability of HASM-SM with that of GCV thin plate smoothing spline TPS and kriging. The numerical test indicated that the minimum GCV HASM-SM is averagely more accurate than TPS and kriging for noisy surface smoothing. The real-world example of smoothing a lidar-derived Digital Elevation Model DEM showed that HASM-SM has an obvious smoothing effect, which is on a par with TPS. In conclusion, HASM-SM provides an efficient tool for filtering noises in grid-based surfaces like remote sensing–derived images and DEMs.

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.

A new method based on orthogonal least square OLS of multiquadric algorithm MOLS is proposed for digital elevation model DEM generalization by the retrieval of critical points from a grid-based DEM to construct a triangulated irregular network surface. The focus is on the method for accurately obtaining the critical points, which maximally retain the important terrain feature lines. The grid-based DEM to be generalized is first approximated in terms of multiquadric method, and then the significances of the grids are assessed based on OLS method with the merit that each selected grid point gives the maximal increment to the explained variance of the desired output. We used six study sites with different terrain complexities to comparatively analyze the generalization accuracies of MOLS and the classical methods including very important method and point-additive method Latticetin under different numbers of retrieved significant points. The results indicate that MOLS averagely performs better than the classical methods for the original DEM generalization in terms of root mean square error. The analytical results also show that MOLS has the better ability in maintaining the feature lines inherent in the original DEM than the classical methods in terms of streamline matching rate. The perfect performance of the newly proposed method can be attributed to the high accuracy of multiquadric method for surface approximation and the effectiveness of OLS for point significance assessment.

In order to avoid the ill-conditioning problem of thin plate spline (TPS), the orthogonal least squares (OLS) method was introduced, and a modified OLS (MOLS) was developed. The MOLS of TPS (TPS-M) can not only select significant points, termed knots, from large and dense sampling data sets, but also easily compute the weights of the knots in terms of back-substitution. For interpolating large sampling points, we developed a local TPS-M, where some neighbor sampling points around the point being estimated are selected for computation. Numerical tests indicate that irrespective of sampling noise level, the average performance of TPS-M can advantage with smoothing TPS. Under the same simulation accuracy, the computational time of TPS-M decreases with the increase of the number of sampling points. The smooth fitting results on lidar-derived noise data indicate that TPS-M has an obvious smoothing effect, which is on par with smoothing TPS. The example of constructing a series of large scale DEMs, located in Shandong province, China, was employed to comparatively analyze the estimation accuracies of the two versions of TPS and the classical interpolation methods including inverse distance weighting (IDW), ordinary kriging (OK) and universal kriging with the second-order drift function (UK). Results show that regardless of sampling interval and spatial resolution, TPS-M is more accurate than the classical interpolation methods, except for the smoothing TPS at the finest sampling interval of 20m, and the two versions of kriging at the spatial resolution of 15m. In conclusion, TPS-M, which avoids the ill-conditioning problem, is considered as a robust 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.

In order to improve the computational speed of the method of surface modeling based on the theorem of surfaces (SMTS), a modified Gauss-Seidel (GS) method (MGS) was introduced and a MGS of SMTS (SMTS-MGS) has been developed. Numerical tests show that SMTS-MGS is more than twice as fast as SMTS-GS and much faster than the classical iterative methods provided by MATLAB 7.0. The computing time of SMTS-MGS is proportional to the first power of the total number of grid cells in the computational domain, whereas the classical SMTS computing time is proportional to the third power of the total number of grid cells. A real-world example of constructing a series of DEMs of Dongzhi tableland was employed to comparatively analyze the simulation accuracies of the two versions of SMTS including SMTS-MGS and the classical SMTS, and the well parameterized classical interpolation methods including the inverse distance weighted technique (IDW), kriging, thin plate spline (TPS), regularized TPS, TPS with tension, and ANUDEM Version 4.6.3, at spatial resolutions of 5, 12, 20, and 25m. The real-world example demonstrates that SMTS-MGS with the same accuracy as SMTS is approximately as accurate as the third-order TPS and much better than other classical interpolation methods at almost all spatial resolutions, except for ANUDEM at a spatial resolution of 5m. On average, SMTS-MGS provides the best results with a minimum of computing time.

High accuracy surface modeling method (HASM) has the capacity of constructing a surface with a higher accuracy than classical methods. However, surface construction with HASM requires a large amount of computing time due to the intensive computing arising from solving the large linear systems. An effective linear system solving method is essential to cutting down the computing time so that HASM could be applied in constructing large scale surface with high resolution. This paper aims to study the performances of seven linear system solution methods, including HASM-GS, HASM-MGS, HASM-DSPM, HASM-CG, HASM-PCG, HASM-GCG and HASM-GPCG, thus providing a guideline for choosing the right method for HASM under different situations of computing environment. These seven methods were tested against a real-world DEM surface in the Boyang lake basin, which owns a complex topography. Three performance metrics were used, including convergence rates, computing time and consumed memory size. We found that to attain the same accuracy HASM-DSPM method consumed the least memory, HASM-GCG and HASM-GPCG needed less computing time than other methods and HASM-PCG had the fastest convergent rate. This study could provide valuable suggestions for HASM users to make a wise choice of linear system solving method for different surface construction applications.

We have presented a new method for generating grid-based digital elevation models (DEMs) based on Coons patch. A series of Gaussian synthetic surfaces with various surface complexities were employed to comparatively analyze the simulation accuracies of Coons patch and the classical interpolation methods, such as IDW, Natural neighbor, SPLINE and KRIGING. Results indicate that irrespective of terrain complexity, Coons patch is much more accurate than the classical interpolation methods, and the accuracy of Coons patch is about 327.30, 15.19, 3.66 and 40.07 times as high as the accuracies of IDW, Natural neighbor, SPLINE and KRIGING on average. A real-world test area was used to test the effect of sampling density on the simulation accuracy. The results indicate that Coons patch has a higher accuracy than the classical interpolation methods regardless of sampling density. In a word, Coons patch as an accurate interpolation method can be used for grid-based DEM construction.

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.

The concept and the computation of terrain representation error (ETR) are investigated and total DEM error is presented as an accuracy index for DEM evaluation at a global level. A promising method of surface modelling based on the theorem of surfaces (SMTS) has been developed. A numerical test and a real-world example are employed to comparatively analyze the simulation accuracy of SMTS and the classical interpolation methods, including IDW, SPLINE and KRIGING performed in ARCGIS 9.1 in terms of sampling and interpolation errors and of total DEM error. The numerical test shows that SMTS is much more accurate than the classical interpolation methods and ETR has a worse influence on the accuracy of SMTS than those of the classical interpolation methods. In a real-world example, DEMs are constructed with SMTS as well as the three classical interpolation methods. The results indicate that, although SMTS is more accurate than the classical interpolation methods, a real-world test indicates that there is a large accuracy loss. Total DEM error composed of, not only sampling and interpolation errors, but also ETRs can be considered as a good accuracy measure for DEM evaluation at a global level. SMTS is an alternative method for DEM construction.

A fast and robust interpolation filter based on finite difference TPS has been proposed in this paper. The proposed method employs discrete cosine transform to efficiently solve the linear system of TPS equations in case of gridded data, and by a pre-defined weight function with respect to simulation residuals to reduce the effect of outliers and misclassified non-ground points on the accuracy of reference ground surface construction. Fifteen groups of benchmark datasets, provided by the International Society for Photogrammetry and Remote Sensing (ISPRS) commission, were employed to compare the performance of the proposed method with that of the multi-resolution hierarchical classification method (MHC). Results indicate that with respect to kappa coefficient and total error, the proposed method is averagely more accurate than MHC. Specifically, the proposed method is 1.03 and 1.32 times as accurate as MHC in terms of kappa coefficient and total errors. More importantly, the proposed method is averagely more than 8 times faster than MHC. In comparison with some recently developed methods, the proposed algorithm also achieves a good performance.