Introduction. digital elevation models; interpolation; geostatistic; GIS. Received 21 August 2007; Revised 26 May 2008; Accepted 14 June 2008

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1 EARTH SURFACE PROCESSES AND LANDFORMS Earth Surf. Process. Landforms 34, (2009) Copyright 2009 John Wiley & Sons, Ltd. Published online 16 January 2009 in Wiley InterScience ( A comparision of interpolation methods Chichester, EARTH The Earth 9999 ESP1731 Research Copyright John 2006Journal Wiley Science Surface Surf. SURFACE Articles Process. & UK of 2006 Sons, Processes the PROCESSES John British Ltd. Landforms Wiley Geomorphological Landforms AND & Sons, LANDFORMS Ltd. Research Group for producing digital elevation models at the field scale Interpolation methods for producing digital elevation models Saffet Erdogan Afyon Kocatepe University, Faculty of Engineering, Department of Geodesy and Photogrammetry, Ahmet Necdet Sezer Campus, Gazligol Road, Afyonkarahisar, Turkey Received 21 August 2007; Revised 26 May 2008; Accepted 14 June 2008 * Correspondence to: Saffet Erdogan, Afyon Kocatepe University, Faculty of Engineering, Department of Geodesy and Photogrammetry, Ahmet Necdet Sezer Campus, Gazligol Road, Afyonkarahisar, Turkey. serdogan@aku.edu.tr ABSTRACT: Digital elevation models have been used in many applications since they came into use in the late 1950s. It is an essential tool for applications that are concerned with the Earth s surface such as hydrology, geology, cartography, geomorphology, engineering applications, landscape architecture and so on. However, there are some differences in assessing the accuracy of digital elevation models for specific applications. Different applications require different levels of accuracy from digital elevation models. In this study, the magnitudes and spatial patterning of elevation errors were therefore examined, using different interpolation methods. Measurements were performed with theodolite and levelling. Previous research has demonstrated the effects of interpolation methods and the nature of errors in digital elevation models obtained with indirect survey methods for small-scale areas. The purpose of this study was therefore to investigate the size and spatial patterning of errors in digital elevation models obtained with direct survey methods for large-scale areas, comparing Inverse Distance Weighting, Radial Basis Functions and Kriging interpolation methods to generate digital elevation models. The study is important because it shows how the accuracy of the digital elevation model is related to data density and the interpolation algorithm used. Cross validation, split-sample and jack-knifing validation methods were used to evaluate the errors. Global and local spatial auto-correlation indices were then used to examine the error clustering. Finally, slope and curvature parameters of the area were modelled depending on the error residuals using ordinary least regression analyses. In this case, the best results were obtained using the thin plate spline algorithm. Copyright 2009 John Wiley & Sons, Ltd. KEYWORDS: digital elevation models; interpolation; geostatistic; GIS Introduction The shape of the Earth s surface, with valleys, plains, hills and so on, is usually referred to as topography. The representation of topography has always been an important and complex topic for surveyors and cartographers because topography varies continuously over space and has to be flattened to a two-dimensional map (Hu, 1995). Digital elevation models (DEMs) have been used as a tool to represent the Earth s surface in many applications, such as hydrological modelling, precision agriculture, civil engineering, large-scale mapping and telecommunications. A DEM is a numerical data file that contains the elevation of the topography over a specified area, usually at a fixed grid interval over the surface of the Earth. DEMs can be generated using different methods that depend on collection procedures and techniques. Photogrammetric methods, satellite-based techniques and field surveying are direct methods of DEM production. Meanwhile, DEMs are sometimes generated by digitizing existing topographic maps. However, any DEM derived from digitized topographic maps is an approximation of an approximated real world (Carter, 1988). Since many applications rely on DEMs, the quality of DEMs and information about the spatial structure of errors within DEMs are particularly important. The quality of a DEM is a result of individual factors. These generally can be grouped into three classes: (i) accuracy, density and distribution of the source data; (ii) the interpolation process; and (iii) characteristics of the surface (Gong et al., 2000). The first two factors are clearly errors whereas the third should be considered a matter of uncertainty (Fisher and Tate, 2006). The accuracy of source data varies with techniques such as: map digitization; active airborne sensors including interferometric synthetic aperture radar and airborne laser scanning techniques; photogrammetric methods; and field surveying. The density of data changes with different sampling intervals and is also one of the factors affecting accuracy (Chaplot et al., 2006; Weng, 2006). Another factor related to the source data is distribution, which may be

2 INTERPOLATION METHODS FOR PRODUCING DIGITAL ELEVATION MODELS 367 regular, random, progressive, selective, or composite, and so on. Characteristics of surfaces such as flat, hilly or mountainous are the second main factor affecting the accuracy of DEMs (Chaplot et al., 2006). The last factor is the interpolation algorithms that are used to generate DEMs. Many interpolation algorithms exist but there is no definite rule to indicate which algorithm is most suitable for a particular surface. A number of previous studies have examined the effects of interpolation methods based on applications in a range of disciplines (Desmet, 1997; Zimmerman et al., 1999; Priyakant et al., 2003; Hofierka et al., 2005; Robinson and Metternicht, 2005; Fencik and Vajsablova, 2006; Yilmaz, 2007). However, quite a few studies, examining the accuracy of interpolation techniques in comparison of sample density and landform types, have been made. Nonetheless, many users need high-resolution DEMs of small areas for specific applications such as landslide prone and hydrological risk areas. Therefore, field surveying with a theodolite and levelling was used to generate high quality DEMs of the whole topographic surface in this study. Four commonly used spatial interpolation algorithms (inverse distance weighting (IDW), ordinary Kriging (OK), multiquadratic radial basis function (MQ), and thin plate spline (TPS)) were examined and compared for the level of model errors. The main objectives of this study were: (1) to evaluate the effects of (a) the density of raw data and (b) interpolation techniques on the accuracy of DEM generation for a rocky hill; and (2) to examine methods for quantifying the uncertainty of DEMs in this region using spatial measures. By quantifying the amount and distribution of the error introduced by sample interval and interpolation, some statistical expressions of accuracy such as mean error and root mean square error were calculated using different validation techniques. Global and local spatial autocorrelation indices were also employed to quantify the spatial pattern of the uncertainty. Meanwhile the relationship between DEM error and morphometric characteristics of the hill, such as slope and curvature, were examined using the correlation coefficients of linear ordinary least square (OLS) regression analyses. Interpolation Interpolation is a topic of interest to many disciplines including mathematics, earth science, geography and engineering because measurements can be time-consuming, expensive and laborious in many environmental applications. Interpolation is a procedure used to predict values at a location for which there is no recorded observation. It can also be defined as the procedure of estimating the values of properties at unsampled sites within the area covered using existing point observations (Algarni and Hassan, 2001). Interpolation methods can be classified in many ways including local/global, exact/ approximate and deterministic/geostatistical methods. Global interpolators determine a single function that is mapped across the whole region, whereas local interpolators apply an algorithm repeatedly to a small portion of the total set of points. Exact interpolators honour the data points on which the interpolation is based, whereas approximate interpolators are used when there is some uncertainty about the given surface values. Nevertheless, deterministic/geostatistical methods are the most widely used. Deterministic interpolation methods are used to create surfaces from measured points based on either the degree of similarity (e.g. IDW) or the degree of smoothing (e.g. radial basis functions (RBF)). Geostatistical interpolation methods are based on statistics and are used for more advanced prediction surface modelling, which includes error or uncertainty of prediction (Gong et al., 2000). There are many routines available for interpolation and these have been widely tested and documented over the years (Isaaks and Srivastava, 1989; Desmet, 1997; Smith et al., 2005). Although there are many variants, four interpolation methods are widely used and popular in GIS software, and these four were examined. Each of the methods produces different height values across characteristics of the surface. An IDW algorithm determines cell values using a linearly weighted combination of a set of sample points. All points or a specified number of points within a specified radius are used to predict the output value for each unsampled location (Burrough and McDonell, 1988). The general formula of IDW is where Z(s 0 ) is the value predicted for location s 0, N is the number of measured sample points surrounding the prediction location, λ i are the weights assigned to each measured point, and Z(s i ) is the observed value at the location s i. The weights are a function of inverse distance. The formula determining the weights is The power parameter p in the IDW is the significance of the surrounding points upon the interpolated value (Priyakant et al., 2003). When the distance (d) between the measured location and the prediction location increases, the weight that the measured point has on the prediction decreases (Burrough and McDonell, 1988). So a higher power results in less influence from distant points. The RBF methods are a series of exact interpolation algorithms that a surface must go through in each measured sample location. There are several types of RBF including thin plate spline, spline with tension, completely regularized spline, multiquadratic function and inverse multiquadratic spline. Multiquadratic function is considered by many to be the best (Yang et al., 2004). The RBFs are conceptually similar to fitting a rubber membrane through the measured sample values while minimizing the total curvature of the surface. The selected RBF algorithm determines how the rubber membrane will fit between the values (Burrough and McDonell, 1988). Because an RBF method is always an exact interpolator, it can be introduced as a smoothing factor to all the methods in an attempt to produce a smoother surface (Yang et al., 2004). This method is a linear combination of the different basis functions where φ(r) is a radial basis function, r = s i s 0 is euclidean distance between the prediction location s 0, and s i w i : i = 1, 2,..., n are weights to be estimated. In MQ function and in TPS N Z( s ) = λ Z( s ) 0 λ i n = i= 1 (1) (2) (3) φ(r) = (r 2 + σ 2 ) 1/2 (4) φ(r) = (σ.r) 2 ln (σ.r) (5) Here, σ is the optimal smoothing parameter, which is calculated by minimizing the root mean square errors using cross validation (Bishop, 1995). i p di0 N p di0 i= 1 z( s0) = wiφ( si s0) + w i= 1 i n+ 1

3 368 EARTH SURFACE PROCESSES AND LANDFORMS The other interpolation method, Kriging, is a powerful geostatistical method which depends on mathematical and statistical models for optimal spatial prediction. In classic statistics it is assumed that observations are independent and that there is no correlation between observations. However, in geostatistical methods, information on spatial locations allows distances between observations to be computed and autocorrelation to be modelled as a function of distance (Burrough and McDonell, 1988). Kriging uses the semivariogram, which measures the average degree of dissimilarity between unsampled values and nearby values, to define the weights that determine the contribution of each data point to the prediction of new values at unsampled locations (Krivoruchko and Gotway, 2004). The important step in Kriging is adjustment of the experimental model to the appropriate type of variogram model. There are many models and each has its own basic structure, which is the function of the distance among data (Fencik and Vajsablova, 2006). There are several types of Kriging, such as simple, ordinary, universal, indicator, disjunctive and probability Kriging, of which simple, ordinary and universal are linear predictors. The difference between these methods is in the assumptions about the mean value of the variable under study (Krivoruchko and Gotway, 2004). The general Kriging model is based on a constant mean μ for the data and random errors ε(s) with spatial dependence. Z(s) = μ(s) + ε(s) (6) where Z(s) is the variable of interest, μ(s) is the deterministic trend and ε(s) is the random, autocorrelated errors. Variations on this formula form the basis of all the different types of Kriging. Ordinar Kriging assumes a constant unknown mean and estimates mean in the searching neighbourhood, whereas simple Kriging assumes a constant known mean. Thus these two methods model a spatial surface as deviations from a constant mean where the deviations are spatially correlated. approximately locations were sampled at regular 2, 4, 10 and 20 m intervals, giving a mean density of 2500, 625, 100 and 25 points ha 1. These points were determined with the coordinates of X, Y, Z using a level (Pentax AL-320 with a precision of ±0 8 cm) and electronic distance meter (Zeiss Elta RS 45 with a precision of ±1 cm). The coordinates were calculated using the National Horizontal and Vertical Control Network and input to the database using ArcGIS 9.2 software. Methodology Many software packages require that original data (survey data) are interpolated onto a regular grid for visualization and analysis of the elevation model (Robinson and Metternicht, 2005). ArcGIS, which is an integrated collection of GIS software products for building complete organizations, was used in this study. The Geostatistical Analyst module of ArcGIS 9.2 is a set of models and tools developed for geostatistical analyses. This study was evaluated using Geostatistical Analyst in two steps. The first step was examining geostatistical data with exploratory data analysis tools for dependency, stationarity and distribution of input data. If data are independent, it makes little sense to analyse them; if data are not stationary, they need to be made so, usually by data detrending and transformation. Geostatistical analyses work best when data are Gaussian; if not they need to be made close to the Gaussian distribution (Krivoruchko, 2005). In the second step, after receiving information on dependency, stationarity and distribution of the data, four interpolation methods were performed on this regularly sampled data. All the interpolation methods parameters were optimized using cross validation because cross validation is the technique most commonly used as an exploratory procedure to find the most suitable model among a number of models (Davis, 1987; Smith et al., 2005). The parameters used in the evaluations are shown in Table I. Study Area and Topographic Characterization To compare the effects of interpolation process and data density on DEM accuracy, a rocky hill near the campus of Afyon Kocatepe University, Turkey, was chosen as a study area (Figure 1). The study area is approximately m 2 with an average elevation of 1005 m above sea level. In this area Evaluation of DEM Accuracy Error predictions may provide important information about the deficiencies of a method, so may be an important input when using and comparing methods for particular applications (Smith et al., 2003). The accuracy of interpolation methods can be evaluated from different aspects. The most straightforward is Figure 1. Study area. This figure is available in colour online at

4 INTERPOLATION METHODS FOR PRODUCING DIGITAL ELEVATION MODELS 369 Table I. Method parameters Methods Intervals Power Model Major range Partial sill Lag size Number of lags Number of neighbours Number of divisions OK 2 Spherical OK 4 Spherical OK 10 Spherical OK 20 Spherical TPS 2 0 Circular TPS 4 0 Circular TPS 10 0 Circular TPS 20 0 Circular IDW Circular IDW Circular IDW Circular IDW Circular MQ 2 0 Circular MQ 4 0 Circular MQ Circular MQ Circular to predict some error indices such as mean error, mean absolute error, root mean square error, and so on, that characterize the interpolation accuracy via different validation techniques. There are a number of validation methods. Several authors recommend cross validation for the evaluation of the accuracy of interpolation methods (Kravchenko and Bullock, 1999; Webster and Oliver, 2001; Smith et al., 2003). The cross-validation method involves using all the raw data for comparison. The main advantage of this method is a clearly defined and user independent formulation that can be implemented on the surface. The method is more reliable for surfaces with a sufficient number of representative input points (Hu, 1995). The most common form of cross validation is the leave one technique. This technique involves omitting one point before the interpolation process; performing the interpolation then predicts the value of the omitted point and the difference between the predicted and actual values of the omitted point is then calculated. This process is repeated for all samples. Another validation method used in this study was the split-sample method. This method can be used to assess the stability of the interpolation algorithm (Declercq, 1996; Smith et al., 2005). In this method some raw data are omitted, interpolation is performed, and the difference between the predicted and measured values of the omitted values is calculated. This difference is used as a measure of the stability of the interpolation algorithm (Declerq, 1996; Smith et al., 2005). Another method is the use of an independent set of sample data that is never used in the interpolation process (Desmet, 1997; Robinson and Metternicht, 2005). For each point the deviation between the actual and predicted values is calculated, and accuracy is tested according to these values. Mean, minimum, maximum, mean absolute, root mean square errors, and so on, are the statistical means that are usually employed to evaluate the overall performance of interpolation methods. The measure most widely used as a single aspatial global statistic is the root mean square error (RMSE), which measures the dispersion of the frequency distribution of deviations between the original points and interpolated points. The main attraction of RMSE lies in its straightforward concept and easy computation (Weng, 2006), mathematically expressed as: RMSE = 1 N { zx ( x } 2 i) z( i) i= 1 N (7) where; (x i ) is the predicted value, z(x i ) is the observed value, and N is the number of values. The RMSE expresses the degree to which the interpolated values differ from the measured values, and is based on the assumption that errors are random with a mean of zero and normally distributed around the true value (Desmet, 1997). In a number of studies, the mean error has not been found to equal zero and therefore some researchers have recommended the use of mean absolute error and standard deviation indices (Desmet, 1997; Fisher and Tate, 2006). Meanwhile it must be noted that these descriptive statistics are single summary indices and assume uniform values for entire DEM surfaces. This assumption is not always true, and many authors have suggested that the distribution of errors will show some form of spatial pattern (Fisher and Tate, 2006; Weng, 2006). An important way to examine the distribution of error is to create prediction error maps. These maps have the advantage of clearly indicating where serious errors occur. Another method is the usage of global (Morans I, Getis-Ord General G) and local spatial autocorrelation (LISA, Getis Ord Gi*) measures to examine the extent of error clustering. Moran s I is a global measure of spatial autocorrelation which is produced by standardizing the spatial autocovariance by the variance of the errors. The range of possible values of Moran s I is taken as 1 to 1, since positive values indicate spatial clustering of similar error values and negative values indicate clustering of dissimilar error values. Moran s I indicates clustering of high or low error values, but it cannot distinguish between these situations. The General G statistic is usually used to understand clustering of high or low error values. A large or larger than expected value for the G statistic means that high error values are found together, while conversely a low value for the G statistic means low error values are found together (Ord and Getis, 1995). These global spatial data analyses show error clustering but they do not show where the clusters are. To investigate the spatial variation as well as the spatial associations it is possible to calculate local versions of Moran s I and the General G statistic. The LISA (local indicators of spatial association: Anselin s formed I value, 1995) were used to detect local pockets of dependence that may not show up when using global spatial autocorrelation methods. Lastly, the relationship between DEM error and morphometric characteristics (slope and curvature) of the hill was examined via linear ordinary least square regression analysis. For each DEM, correlation

5 370 EARTH SURFACE PROCESSES AND LANDFORMS coefficients were calculated for slope and curvature parameters of the terrain. Then a multivariate ordinary least square linear regression model was used to identify relationships between error and slope-curvature characteristics. Results It has been demonstrated that DEM error can vary to a certain degree with different interpolation algorithms and data density (Desmet, 1997; Weng, 2006). The level of this error is important for many specific applications. Therefore, four of the most widely used interpolation algorithms were compared with different data densities, which were determined regularly. DEMs generated from surveyed data with the 2 2 sample interval are shown in Figure 2. The main purpose of this study was to quantify the amount and distribution of error introduced by interpolation methods and data density for this rocky hill. As a first step, interpolation methods and data density were validated by cross validation, split-sample and jack-knifing methods as mentioned above. To compare the accuracy of interpolation methods and data density statistically, mean error, maximum error, minimum error and root mean square error indices were calculated via validation methods. Cross-validation values are shown in Table II. Cross validation is a useful indicator of the general characteristics of interpolation methods, but it cannot be used as a measure for the robustness of algorithms (Smith et al., 2003). Therefore the split-sample method and jack-knifing were used to assess the model s overall accuracy and stability, as measures of the robustness of the methods. The split-sample method was performed using 95, 75 and 50 per cent of the raw data for investigation. The results of the split-sample validation investigation are shown in Table III. Jack-knifing by using an independent set of the sample was performed with 173 irregularly surveyed test points. These test points were compared with the estimates obtained by interpolation algorithms. The results of this validation are shown in Table IV. To measure the precision of methods, RMS is usually used. Small RMS values are required, because it is desirable that the predicted data should be close to the raw data. When we compared the results of the validation methods, the IDW algorithm provided the worst interpolation and produced the greatest overall error of all the methods. The MQ and OK methods had nearly similar results, but TPS was the best interpolator according to accuracy indices produced by validation methods. To compare the differences between methods interpolation predictions by considering the data density, the prediction values were plotted against measured values. The scatter points for the TPS at 2, 4, 10 and 20 m intervals are Figure 2. Digital elevation models generated from surveyed data with 2 2 sample interval with contours. This figure is available in colour online at

6 INTERPOLATION METHODS FOR PRODUCING DIGITAL ELEVATION MODELS 371 Table II. Results of cross-validation with different sample intervals Methods Sample interval Mean error (+) error Minimum ( ) error Root mean square error OK OK OK OK TPS TPS TPS TPS IDW IDW IDW IDW MQ MQ MQ MQ Table III. Results of split-sample validation with different sample intervals Model Percentage raw data Depth error ( ) error Mean RMSE Depth error ( ) error Mean RMSE OK TPS IDW MQ OK TPS IDW MQ OK TPS IDW MQ OK TPS IDW MQ OK TPS IDW MQ OK TPS IDW MQ Table IV. Results of independent data set validation with different sample intervals Methods Sample interval Mean error (+) error ( ) error Root mean square error OK OK OK TPS TPS TPS IDW IDW IDW MQ MQ MQ

7 372 EARTH SURFACE PROCESSES AND LANDFORMS Figure 3. Measured values versus predicted values and correlation coefficients using the thin plate spline method. shown in Figure 3. The spread of the plots increased generally as sample density decreased. For all densities and interpolation methods, linear correlation coefficients were over 0 9. The TPS method had the best correlation coefficients for all densities compared with the other methods ( ). Accuracy indices produced by validation methods assume single uniform error values for the entire region. However, several researchers have identified this assumption as being invalid (Fisher, 1998; Carlisle, 2005). Therefore, it is important to investigate the spatial variation and distribution of errors considering the stationarity. A way to do this is to show the errors using choropleth symbol maps. The spatial distribution of errors was examined by plotting the location and magnitude of errors. The resultant plots of errors introduced by the interpolation methods using the data sampled with 20-m intervals are shown in Figure 4. The size of the points represents the magnitude of errors. Red points represent errors larger than ±1 m. It can be shown that errors larger than ±1 m occurred more often in the IDW method. Locations of errors larger than ±1 m were concentrated in similar places in TPS and OK. Similarly when we examined the distribution of errors, it was observed that the errors were largely coincident with the rocky parts of the surface and these errors were larger in the low-density samples. The errors in TPS with 4-m sample intervals are shown in Figure 5. The size of the points shows the magnitude of errors. The black points represent errors larger than ±0 3 m. However, interpretation of the choropleth error maps can be difficult when there is too much data. Another way to observe and analyse the distribution of errors is to create error maps. These maps show where serious and anomalous errors are occurring and clustering. Therefore, prediction error surfaces were created to show the spatial pattern of errors resulting from interpolation algorithms. Comparison of such surfaces can be extremely informative with respect to the occurrence and magnitude of errors in relation to terrain slope-curvature and distribution of input data (Shearer, 1990; Weng, 2006). An error surface created to show the spatial pattern of absolute errors resulting from TPS with a 4-m interval is shown in Figure 6. It is clear from the maps that errors tend to cluster in rocky regions of the study area where slope and curvature change rapidly, especially around the crest of the hill. Another comparison was the test of spatial autocorrelation. All methods with different surfaces had significant values of Moran s I, indicating a considerably high degree of clustering (Table V). Similarly, significant General G statistic values of surfaces with 2, 4 and 10-m intervals were larger than expected, indicating that high error values were found together (hot spots). However, a detailed look at the autocorrelation values revealed that some interpolators did a better job than others. The TPS method produced the least amount of clustering in terms of revealing the systematic errors that resulted from underrepresentation of rocky areas. Although Kriging is the best linear unbiased estimator, the OK method produced reasonably low values of clustering similar to the MQ method. On the other hand, IDW produced the highest values of clustering. Global spatial autocorrelation indices show clustering but they do not show where the clusters are. Therefore, to investigate the spatial variation as well as the spatial associations, it is possible to calculate local versions of Moran s I, for each location (Anselin, 1995). The LISA investigates those clusters of error values with similar values and those clusters of error values with different values. A high value of I means that the point is surrounded by features with similarly high or low error values, whereas a low value of I means that the point is surrounded by features with dissimilar error values. The z score shows the statistical significance of the I value for the distance specified. All interpolation methods generated clusters around the peak of the hill. The TPS method produced the least amount of clustering whereas IDW produced the most. Error surface and LISA results of the TPS algorithm with 4-m intervals are shown in Figure 6. Red points show the clusters of high-high values.

8 INTERPOLATION METHODS FOR PRODUCING DIGITAL ELEVATION MODELS 373 Figure 4. Spatial distribution of errors with 20-m sample intervals. This figure is available in colour online at journal/espl Table V. Global spatial autocorrelation indices of Moran s I and Getis-Ord General G with mean absolute errors of DEMs Interpolation methods Moran s index Expected index z score Observed General G Expected General G z score IDW IDW IDW IDW MQ MQ MQ MQ TPS TPS TPS TPS OK OK OK OK

9 374 EARTH SURFACE PROCESSES AND LANDFORMS Figure 5. Spatial distribution of errors with 4-m sample intervals and location of these errors over the hill. This figure is available in colour online at Finally, relationships between DEM error and the slopecurvature morphometric parameters of the hill were examined for each DEM using correlation coefficients that were calculated for each terrain parameter and elevation error. Nevertheless, it was expected that both terrain parameters acting in combination would influence the spatial variation in DEM error (Carlisle, 2005). Therefore, multivariate ordinary least squares linear regression analysis was used to model any such relationship. Table VI summarizes the values of multivariate ordinary least squares linear regression model correlations with slope and curvature parameters. According to the results, the curvature parameter had a greater effect than slope parameter on residuals. Residual errors from the MQ algorithm showed the strongest correlations with terrain parameters. Correlation values of OK were found to be close to values of MQ. Residual errors from the IDW algorithm had weaker correlations with terrain parameters than the other three methods. Significant correlations indicate that there is a relationship between DEM errors and terrain characteristics. The curvature parameter gives a good indication of the amount of error while sample density increases. The slope parameter was found to have the greatest values when the data interval was 10 m. Meanwhile a combination of parameters could give a better signal about the amount of error than a single parameter could. Conclusion The magnitude of uncertainty from the interpolation is subject to many factors. The research revealed that the magnitude and distribution of errors in a DEM of the hill were strongly Table VI. Correlation coefficients of regression model results Methods Slope Curvature Combination of slope and Curvature IDW IDW IDW IDW OK OK OK OK TPS TPS TPS TPS MQ MQ MQ MQ related to the varying characteristics of the terrain, sampling density and interpolation algorithm. This study demonstrated that the IDW algorithm produced the greatest overall error. This probably arose because of its inability to model the steep surfaces that are common in hill areas. In the IDW surfaces with 10- and 20-m intervals the bull s eye effect was too strong. As well as introducing error, the stepped appearance of IDW means that it did not produce the most realistic looking representation of the hill areas. The method introduced a little more error than TPS and MQ. Although the

10 INTERPOLATION METHODS FOR PRODUCING DIGITAL ELEVATION MODELS 375 Figure 6. Error surface and local indicators of spatial association (LISA) results of error values. This figure is available in colour online at model parameter choices, such as search radius, sill, range, nugget and minimum/maximum number of data for OK were optimized using cross validation, it should be noted that the results were related to the parameters rather than to OK itself. The disadvantage of OK was the length of the process time for high volume data. However, it is interesting to note that, where the density of data is low, OK seemed to exhibit a larger and more generalized approximation than the DEMs produced by IDW. The TPS method was the most appropriate and effective with MQ. Normally, as shown in Table II, as distances between the points in the raw (elevation) data increased, the RMS values also increased (the relationship was close to linear). With the exception of the IDW method, the other methods showed almost the same RMS values. In OK, while distances between the points increased in the raw data, the smoothness of the surface increased more according to the TPS and MQ methods. Split-sample validation was performed using sample sizes of 95, 75 and 50 per cent of the raw data for all the methods to assess the stability of the methods given a smaller input of raw data. It was found that the OK, TPS and MQ methods were relatively stable. Similar values were obtained for these three methods while the quantity of raw data decreased. Meanwhile, IDW was also found to be stable, but the values obtained for this method were rather larger than for the other three methods as the quantity of raw data decreased. The difference in errors created at varying sample densities was assessed using the split-sample, jack-knifing and crossvalidation methods. It was found that OK, TPS and MQ produced relatively parallel increasing RMS error values. When the spatial distribution of errors was investigated, it was found that all the large errors (±) were clustered around the steep surface of the hill area as shown in Figure 7. Error indices such as RMS, mean absolute error, and so on, alone were insufficient to indicate DEM uncertainty. To understand the DEM uncertainty, spatial distributions of the model residuals from the four modelling algorithms were investigated using error surfaces and the global and local Moran coefficients. Global Moran s I indices indicated that errors were concentrated with all methods and all sample densities. The Getis and Ord General G method showed that high error values were concentrated with DEMs obtained from input data with 2-, 4- and 10-m intervals. Then LISA was utilized to investigate spatial distribution and heterogeneity in model residuals using four interpolation algorithms with ordinary least squares (OLS) as the benchmark. According to OLS regression results, residual errors with interpolation algorithms showed strong correlations both alone and in combination with terrain parameters of slope and curvature. For the study area, highly adjusted regression coefficients changing between 0 4 and 0 9 indicated that the spatial distribution of DEM error could be modelled with OLS regression modelling of terrain parameters with a high degree of success. Meanwhile, OLS regression is a global technique in that a single regression model is created that best fits the whole residual data set over the entire study area (Carlisle, 2006). This study area had a highly variable terrain character and DEM errors have a high spatial autocorrelation that show spatial non-stationarity. Therefore, relationships between residual errors and terrain parameters would not show spatial stationarity. The limitations of OLS regression as a consequence of its assumption of spatial stationary were discussed in another study (Fotheringham et al., 2002). This limitation will be examined in a further study using a geographically weighted regression model.

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