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1 Important considerations on thee application of IDW interpolation method Younes Fadakarr Alghalandis ١, Peyman Afzal ٢ Abstractt The utilization of the Inverse Distance Weighting (IDW) interpolationn method has been discussed in this paper for achieving more reliable estimation results. While it is a very common estimation method in a variety of disciplines including mining, environmental, geographical and soil studies, there is s a considerablee lack of discussion around effective use off its critical parameters such as the value of exponent and the estimation error in related literature. Somee experimental tests were conductedd using MATLAB to investigate the responsee of the IDW method regarding variation of the exponent. Comparison of several ٢D resulting graphs verified that the widelyy used value of two for the exponent could be a sufficient default for general applications even in higher dimensions. It was also demonstrated a value higher than ٣٠ could be advised a for generatingg a specific map i.e. Voronoii tessellation diagram. The later is used for spatial connectivity analysiss of sample points in ٢D and ٣D. This result revealed a new form of spatial dependency, which could be addressed in the IDW method. Furthermore, a concept of fuzziness was introduced regarding the variation of the exponent. Application of thiss concept is able to quantify the uncertainty around the t estimation boundaries. Using an exhaustive reference data set the accuracy of the IDW estimation was investigated. As a result, the IDW produced a larger variance of estimationn error when compared to Kriging. More generally, the IDW method provided more erratic estimation results. However, in contrast, it was demonstrated that the IDW overestimates areas with lower values and underestimates areas withh higher values while it is reversed for Kriging estimation. Such an a outcome would be of interest in critical evaluations e.g. risk analysis, too which the IDW is a more conservative method. Also, on the concept of histogram smoothing, it i was shown that the IDW trends toward lower frequencies while Kriging presents a smoother fit to the entire histogram. This can be also interpreted to more erratic estimate using the IDW rather Kriging. Such analyses highlighted the importance of understandingg the uncertainty involved in using the IDW interpolation methodd that can have implications to more reliablee exploration targeting, mine evaluations etc. Keywords: IDW Estimation Uncertainty, Effect of the Exponent, Histogramm Smoothing, Tessellation Risk Analysis, Voronoi ١. Introduction As an initial part of many research and industrial projects, e.g. geographical and geochemical investigations, sampling plays an important role to indentify the trend and the variation of desired d variables such as elevation or elementall distribution. It is favorable to have high-resolution sampling inn order to increase the accuracy of variable estimation; however, implementation of such a densee sampling program is not viable due to economic and technical constraints. With fewer samples, the estimation can be achieved if the distribution is not completely random; moreover, if it i follows thee law of continuity and/orr shows spatial correlation which is known for elemental concentration and it is natural for surface structure [١]. For example, ١٦٠,٠٠٠ samples are required to cover an area of ٤٠٠ square meters with a cell size of one square meter, whilee ١,٠٠٠ samples can be ١ PhD Student, Geosciencess & Mining, WH Bryan Mining & Geology Research Centre, University of Queensland, Brisbane, Australia ٢ Mining Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Irann

2 Where Z i is the neighbor sample point s value,, d i is its distance from the estimation point, n is number of neighbor samples and Z * is an estimation valuee for the estimation point. As shown inn the above formula, f a weighted average value is calculated somehow closer sample points have more weighting on the estimated average. However, for estimation point located too close to a sample position, the IDW result would bee the same as sample value. The value of two for the exponent used for distance weighting (d ٢ i ) is discussed in the following section. ٢.١. The Effects of the Exponent There is no reason to choose the number two as weighting order in the IDWW interpolation method [٧, ٣]. Many researchers have argued it from a mathematical and physical point of o view [٧]. In this section, it is comprehensively explained using the MATLABB [٨] to analyze the effect of variation of the exponent on the estimation results (Fig-١). adequate to preserve key information relating to variable estimation. In general, the sampling strategy involves random positioning to avoid systematic errors [٢]. Generalization of the sampling results to the whole area of study helps to find the total variation using an estimation method. Such estimation provides an overview of the area or a particular point value in geographical, geological, geochemical, spatial studies or can be used to assess grade value and tonnage of deposits in miningg projects. An example off estimation methods the polygonal estimation divides the whole areaa of study too polygonal sub-domains around each sample. Mathematical methods, however, provide a greaterr confidence in estimation using complex solutions [٣]. A most common and traditional method of mathematical interpolationss is the Inverse Distance Weighting W (IDW) method [٣, ٤, ٥], in which an estimated value is calculated based on the nearby sample s values weighted by their distances from the estimation point [٦]. A geostatistical method called Kriging can also be used to calculate weights based on the samples spatial variation. Kriging is famed for its ability to investigate the spatial behavior in geological and mining projects and to provide an estimation error map [٣, ٤], while the IDW is commonly preferred for its simple implementationn [٣] and low-cost and fast running process. These methods are the mostt common interpolation methods in many diverse research areas for example, estimation of the elevation and contouring in geographical and survey studies, investigationn of the trend and the variation v of chemical elements in geochemical investigations, image manipulation and processing in computer vision [٣]. Despite this popularity, there are a few works on comparison between the interpolation methods especially on the effects of parameters on their application e.g. [٣, ٤, ٥]. This paper presents new outcomes of some experimental tests on thee accuracy of the IDW estimation method and proposes new aspects for application of the IDW method such as fuzzy estimation and handling the uncertainty associatedd with the estimation results. ٢. Inverse Distance Weighting Method The IDW method is considered as a straightforward and non-computationally intensivee method [٣], in which, each point of estimation in ١D, ٢D, ٣D (common) or N dimensional space is computed using its nearby sample values weighted by their distances from the estimation point. The general formula for the IDW is: (١) ٠ ٠ ١ ٢ ١ ١ ٢

3 Fig-١. The effects of the variation of the weighting order on the IDW interpolation results; sample points s are on integers ٠ to ٩ in X axis; note the shape and the curvature of interpolation curve in this ١D example. The exponent value equal to zero depicts a simplee average of all data, whichh is shown as the dark blue horizontal line with spikes on the sample locations. Thesee spikes verifies that the IDW I method is an exact estimation e method [ ٣, ٤, ٥] whichh means the estimation e value for all known locations is always the e same as the associated sample value. Equation (١) verifiess this effect when you consider its second line condition. By increasing the exponent, the interpolated points have a betterr estimation regarding nearby sample points. Increasing the exponent above that of two, shows that the weight of closely spaced points dramatically increases. This trend further (value of ten in Fig-١) deforms the estimation curve to a stepping form in which each sample point is located in the middle of the step and a covers half the space between neighbor points. In other words, as the exponent increases the estimation of the IDW interpolation method is becoming more similar to thee Voronoi tessellation [٩] estimation as shown on Fig-٢. Too produce such a map the number n of neighbors to estimate each point was ١٠, while the value of the exponent wass ٣٠. A simple k-nearest Neighbor N algorithm (k=١٠) was coded to choose a k nearest neighbor samples as conditioning data to participate in the IDW estimation of each e point (pixel). That is, the value of two for the exponent in thee IDW interpolation method can be mentioned as itss optimum value (Fig-١), which then the estimation can be placed between two other forms: (١) a total average of all sample points and (٢) a polygonal estimation (Voronoii tessellation, see Fig-٢). The T later is widely used for f spatial connection (correlation) of sample points e.g. [١٠] and ArcGIS. The first form of interpolation is the highest generalization of estimation inn which the whole area of study has only one estimation value (the average value of all samples). On the other hand, h the second form, Voronoi cells, provides individually estimated polygons based on one closest sample value. Therefore,, the IDW interpolation method presents an estimation methodology between these two forms. Byy understanding these circumstances and regarding the subject under estimation, one can choose an appropriate exponent e for the IDW. Therefore, for example, an initial and reliable interpolation can be produced using thee number of two for the weighting exponent. The variation of the exponent, moreover, introduces a fuzzy approach for interpolation. It I is worth mentioning here that the graphical presentation in this case is more important tool to evaluate the structural variation of the interpolation rather than t a blind statistical deviation value. As A shown in Fig-١ the variation of the curve (interpolation values) between two conditioning points is mostly symmetric around the straight line between them meaning that statistically the overall deviation is almost zero. However, the interpolation results are considerably different.

4 Fig-٢. The IDW ٢D interpolation map (colorful cells) produced by increasing the weighting order (w) too ٣٠. The resulting map is completely matched with the Voronoi tessellation diagram (black lines) ٣. Estimation using Kriging Method Kriging is a geostatistical method, which is widely used for spatial interpolation [٣]. The basics of this method are similar to the previous method,, weighted estimation using nearby sample values [١١]. In this method the effect of neighboring point grades (values) on an estimation point are appliedd as weights, which are calculated by probing of the spatial variation of data points. Technically speaking based on the calculation of variograms of data and then fitting models, the parameters (range of effectiveness, sill and nugget effect) are extracted. Then using the variogram model parameters, a matrixx of covariance/variogram is used for extraction of weighting values. λ ; ١ λ ١ ١, ١ ١, : : : λ, ١, Where m is the average of all data, λ is matrix of the weighting values, Z is value of neighbor points, n is the number of neighbor points, c(.) is a covariance/variogram value for pairs, and Z * is the estimation value of Kriging. The above equation is well-known as simple Kriging (SK) [١١]. Note there are some assumptions to make the results of Kriging statistically valid e.g. stationary for mean value. These assumptions are accepted to simplify s the system of matrices to be solvable, although they are not always valid in real cases [٧, ١١]. Nevertheless, Kriging is widely accepted because it quantifies the error involved, which means a Kriging map has a confidence value of correctness accompanying the estimation. A variety of ١ ١, ٠ :, ٠

5 Kriging methods exist, such as indicator, ordinary, universal and disjunctive, which creates more opportunities for its use in other studies [١١]. However, for the purpose of this paper SK was w used. ٤. Application of Interpolation Methods Interpolation methods are widely used in several sciences and industries. This is evident from the variety of available computation packages utilizing the IDW and Kriging methods. Because of thee simplicity off the IDW method, it has been widely used as an initiall standard for spatial interpolation procedures in geographic information science [١٢, ١٣] and has been implemented in many GIS software packages [٣]. Furthermore, there are many common software packages such as ArcGIS, Surfer, Datamine, ILWIS etc belonging to different disciplines including civil, surveying, mining, geography and remote sensing that have specific tools for interpolation including the IDW and Kriging. These methods are also common in the other applied sciences such as statistics, computer visualization and image processing [١٤, ١٥]. As described before, the results of IDW interpolation method are fully bounded to lowest and highest values within used data set, while the Kriging providess estimation without this constraint. Therefore, they should be used in consistency to the purpose of the estimation. For example, in geochemical mapping any result out of the detection limits considered as invalid estimation. Since in geochemical explorations the data boundaries are used to definee the proper thresholds to separate between anomaly and background values, it is critical that the estimatess to be limited to these constrains c avoiding deformation of thee anomaly map and misleading the exploration project. Therefore, for this case, the IDW method would be recommended as a reliable estimator. ٥. Case Study and Comparisons For this study, a sample dataset was selected fromm the data package of Surfer software [٦]. Then a dense ٢D grid of data ( cell size ١x١m) was interpolated usingg linear Kriging as the reference valuess grid. Since the dense dataset is used once as a reference dataset to draww randomly a much smaller dataset (thee ratio less than ١/١٥٩), the method of interpolation exceptionally is not critical. The linear Kriging was used here because of its smoothing effect, which helps to preserve the structural trend of data well. Fig-٣ shows the referencee grid as a colored map and ١٠٠٠ drawn sample locations as s small black dots. Using the Surfer mapping package the t IDW interpolation with a value of two for the exponent and Kriging were applied on all ١٠٠٠ samples to produce two interpolated maps as shown in Fig-٤. F Then the residual maps [١١, ٧] were generated by subtraction of real data from both IDW and Kriging results. Fig-٥ shows the resulting maps of the IDW and Kriging interpolation as colored background. As shown on Fig-٥, it can be e concluded that for this case study, the IDW interpolation method presents error areas that are larger in size and scale than Kriging, which presents less erratic values, and smaller error areas. This result has been statistically presented in Fig-٦ as a histogram graph of both maps. The graph shows that the real-idw residual map has a very wide distribution function while the real-kriging residual map shows a very narrow histogram. As a statistical result, thee deviation (or variance) of residuals for the Kriging method is much smaller than the IDW, which explains a higher average accuracy of the Krigingg rather IDW..

6 Fig-٣. Reference data grid (coloredd background) with uniform random samples (dots) Fig-٤. IDW and Kriging interpolation i maps (colored background) of all ١٠٠٠ samples (black dots)

7 Fig-٥. Residual maps of IDW and Krigingg interpolation depicted considerable differencess between their results: the mapp of Kriging residuals (errors) shows less but bigger deviation islands. Fig-٦. Histogram of IDW and Krigingg interpolation residuals Fig-٧ shows another comparison between the IDW and Kriging interpolation. It is shown, that with the IDW method, generally, higher grades values have been estimated to be lower than corresponding references values, while lower grade values have been estimated ass higher than the corresponding reference values. These results are reversed for the Kriging interpolation in which higher grades values have h been estimated higher than the corresponding references values, while lower grade values have been estimated lower than the corresponding referencee values. Such results emphasize the importance of consistency of choosing a right estimation method for example in geochemical investigations and resource estimation where both IDW and Kriging are widely used for generating the estimation.

8 Fig-٧. Residual maps of IDW and Kriging interpolations ٥.١. Histogram Smoothing Effect The calculation of the histogram forr all three dataset including reference data, the IDW and Kriging results r has been done by coding in MATLAB. The resulting histograms are shown in Fig-٨ and have been already

9 standardized for better presentation and comparison. As shown in Fig-٨, it can c be observed that Kriging method fairly presents averagee values for all a classes, which is shown as a more smoothed curvee (red colored). On the other hand, the IDW interpolation shows a higherr respect to lower class bounds (blue colored). This effect can be verified according to the previouss section (seee Fig-٧ and its description). Therefore,, Kriging interpolation method presents a more smoothed histogram and is recommended for use where a smoothed distribution shape is required (e.g. normal distribution approximation).. Fig-٨. Histograms of three dataset: reference data, IDW and Krigingg interpolation results ٦. Conclusions The effect of adjusting estimation parameters such as the value of the exponent e in IDW method has been investigated. It was shown the commonly used number of two for the exponent could be seen as suitable default for general estimation purposes. However, it was demonstrated that the variation of thee exponent provides an opportunity to generate different estimations e between a total average of all sample points and a polygonal estimation. The later is well known as Voronoi tessellation, which is widely used for analyzing the t spatial connection (correlation) of sample points. These results develop the application of the IDWW method as ١) a fuzzy estimator and ٢) a spatial connectivity analyzer. Application of a fuzzy IDW estimator r is able to quantify the uncertainty around the estimation boundaries, which is particularly critical inn mine evaluation. Although, the IDW method produced a larger variance of estimation error (more erratic estimation results) when compared to Kriging, however, it was also shown that the IDW overestimates areas with lower values and underestimates areas with higher values while it is completely reversed for Kriging K estimation. Such an outcome also would be of interest in critical evaluations e e. g. risk analysis in mine evaluation, to which the IDW is a more conservative method. The histogram smoothing effect was investigated forr IDW method in comparison to the referencee data and the Kriging results. As a result, the IDW trends toward lower frequencies while Kriging presents a much smoother fit to thee entire histogram. This result verifies that t Kriging generates better overall estimation than IDW. Therefore, Briefly, the IDW method could be recommended forr geochemicall mapping where the data boundaries in critical for threshold-based target separations and also in mine evaluation where a lower risk is appreciated.

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