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Over the past decades, plenty of filtering algorithms have been presented to distinguish ground and non-ground points from airborne LiDAR point clouds. However, with the existing methods, it is difficult to derive satisfactory filtering results on rugged terrains with dense vegetation due to the low-level penetration ability of laser pulses. Therefore, a multi-level interpolation-based filter is developed in this paper. The novelty of the algorithm lies within its usage of multi-scale morphological operations and robust z-score to correctly select ground seeds as more as possible, and a terrain-adaptive residual threshold to adapt to various terrain characteristics. Rural samples provided by International Society for Photogrammetry and Remote Sensing (ISPRS) were employed to assess the performance of the proposed method and its results were compared with 15 filtering algorithms developed in recent 5 years (2015-2019). Results show that the proposed method with the optimized parameters produces the best accuracy with the average total error and kappa coefficient of 1.89% and 87.88%, respectively. We further filtered high-density point clouds in six forested areas with different vegetation covers and terrain slopes. Results demonstrate that the proposed algorithm is more accurate than the well-known filtering methods including morphological-based filter, progressive TIN densification filter (PTD), improved PTD and cloth simulation filter, with the average total error decreased by 26.2%, 19.9%, 3.8% and 40.4%, respectively. Moreover, the DEMs of the proposed method have lower average root mean square errors than the four classical filters. Therefore, the proposed method can be considered as an effective ground filtering algorithm for airborne LiDAR point clouds in forested areas.

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.

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.

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.

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.

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.

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.

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 have found that a spatial interpolation of mean annual temperature (MAT) in China can be accomplished using a global ordinary least squares regression model since the relationship between temperature and its environmental determinants is constant. Therefore the estimation of MAT does not very across space and thus exhibits spatial stationarity. The interpolation of mean annual precipitation (MAP), however, is more complex and changes spatially as a function of topographic variation. Therefore, MAP shows spatial non-stationarity and must be estimated with a geographically weighted regression. A statistical transfer function (STF) of MATwas formulated using minimized residuals output from a high accuracy and high speed method for surface modeling (HASM) with an ordinary least squares (OLS) linear equation that uses latitude and elevation as independent variables, abbreviated as HASM- OLS. The STF of MAP under a BOX-COX transformation is derived as a combination of minimized residuals output by HASM with a geographically weighted regression (GWR) using latitude, longitude, elevation, impact coefficient of aspect and sky view factor as independent variables, abbreviated as HASM-GWR-BC. In terms of HASM-OLS and HASM-GWR-BC

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.

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.

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.

In addition to classical methods, namely kriging, Inverse Distance Weighting (IDW) and splines, which have been frequently used for interpolating the spatial patterns of soil properties, a relatively more accurate surface modelling technique is being developed in recent years, namely high accuracy surface modelling (HASM). It has been used in the numerical tests, DEM construction and the interpolation of climate and ecosystem changes. In this paper, HASM was applied to interpolate soil pH for assessing its feasibility of soil property interpolation in a red soil region of Jiangxi Province, China. Soil pH was measured on 150 samples of topsoil (0-20 cm) for the interpolation and comparing the performance of HASM, kriging. IDW and splines. The mean errors (MEs) of interpolations indicate little bias of interpolation for soil pH by the four techniques. HASM has less mean absolute error (MAE) and root mean square error (RMSE) than kriging, IDW and splines. HASM is still the most accurate one when we use the mean rank and the standard deviation of the ranks to avoid the outlier effects in assessing the prediction performance of the four methods. Therefore, HASM can be considered as an alternative and accurate method for interpolating soil properties. Further researches of HASM are needed to combine HASM with ancillary variables to improve the interpolation performance and develop a user-friendly algorithm that can be implemented in a GIS package. (C) 2009 Elsevier B.V. All rights reserved.

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.