Journal of Geographc Informaton System, 207, 9, 354-37 http://www.scrp.org/journal/jgs ISSN Onlne: 25-969 ISSN Prnt: 25-950 Accuracy Assessment and Comparatve Analyss of IDW, Splne and Krgng n Spatal Interpolaton of Landform (Topography): An Expermental Study Maduako Nnamd Ikechukwu *, Eljah Ebnne, Ufot Idorenyn, Ndukwu Ike Raphael Department of Geonformatcs and Surveyng, Unversty of Ngera, Nsukka, Ngera How to cte ths paper: Ikechukwu, M.N., Ebnne, E., Idorenyn, U. and Raphael, N.I. (207) Accuracy Assessment and Comparatve Analyss of IDW, Splne and Krgng n Spatal Interpolaton of Landform (Topography): An Expermental Study. Journal of Geographc Informaton System, 9, 354-37. https://do.org/0.4236/jgs.207.93022 Receved: March 28, 207 Accepted: June 20, 207 Publshed: June 23, 207 Copyrght 207 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). http://creatvecommons.org/lcenses/by/4.0/ Open Access Abstract It s practcally mpossble and unnecessary to obtan spatal-temporal nformaton of any gven contnuous phenomenon at every pont wthn a gven geographc area. The most practcal approach has always been to obtan nformaton about the phenomenon as n many sample ponts as possble wthn the gven geographc area and estmate the values of the unobserved ponts from the values of the observed ponts through spatal nterpolaton. However, t s mportant that users understand that dfferent nterpolaton methods have ther strength and weaknesses on dfferent datasets. It s not correct to generalze that a gven nterpolaton method (e.g. Krgng, Inverse Dstance Weghtng (IDW), Splne etc.) does better than the other wthout takng nto cognzance, the type and nature of the dataset and phenomenon nvolved. In ths paper, we theoretcally, mathematcally and expermentally evaluate the performance of Krgng, IDW and Splne nterpolaton methods respectvely n estmatng unobserved elevaton values and modelng landform. Ths paper undertakes a comparatve analyss based on the predcton mean error, predcton root mean square error and cross valdaton outputs of these nterpolaton methods. Expermental results for each of the method on both based and normalzed data show that Splne provded a better and more accurate nterpolaton wthn the sample space than the IDW and Krgng methods. The choce of an nterpolaton method should be phenomenon and data set structure dependent. Keywords Spatal Interpolaton, IDW, Krgng, Splne and Modelng Elevaton DOI: 0.4236/jgs.207.93022 June 23, 207
. Introducton Interpolaton ams at fndng the values of a functon f ( x ) for an x between x, x,, x at whch the values of f ( x ) are gven. The dfferent x values 0 gven values n ( ), ( ),, ( ) f = f x f = f x f = f x () 0 0 n n can be obtaned from a mathematcal functon or from an emprcal functon modelled from observatons or experments []. Spatal nterpolaton therefore ams at estmatng values of a spatal phenomenon or functon (temperature, elevaton, etc.) at unobserved/estmated ponts, gven values of the phenomenon at observed/estmated ponts. Spatal nterpolaton has contnued to be an mportant tool for estmatng contnuous spatal envronmental varables for effectve decson makng. Many modelng tools ncludng Geographc Informaton System offer the earth and envronmental scentst the ablty to carry out spatal nterpolaton routnely to generate useful spatal contnuous data for all knds of analyss [2]. Interpolaton becomes very useful and essental n scenaros where, the resoluton, orentaton, or cell sze of a dscretzed surface vares from what s needed. It s also employed when contnuous surface s represented by a data model dfferent from what s desred, and when data spread does not cover an area of nterest totally [3]. Spatal nterpolaton methods gve a means of predctng values of an envronmental parameter at unmeasured locaton usng data from pont measurements wthn the sample space [4]. In an deal stuaton, a fnte set of nputs establshes varatons n an envronmental parameter and they exactly conform to an establshed physcal law. If a relatonshp s establshed, the values of the desred parameter can be correctly estmated. The relatonshp between target varable and the physcal envronment cannot be modeled exactly because of ts complexty [5]. Ths s due to a lack of suffcent knowledge of: (a) the complete lst of nputs nto the model (b) the relatonshp needed to determne the output from these nputs and (c) the mportance of the random part of the system. Estmatng a model usng feld measurement of the parameter of nterest then becomes the only way [6]. For a sample of a target varable Z, denoted as z( s), z( s2),, z( s n ) (where s = ( x, y ) s the locaton context, x, y are the coordnates n geographcal space and n s the number of observatons), the geographcal doman of nterest can be denoted as A. Consderaton s gven to only samples z( s n ) that realzes a process Z = { Z( s), s A } [6]. Accordng to Mtas and Mtasova [7], the choce of sutable nterpolaton methods for dfferent phenomena and dataset presents many dffcultes. The complexty of the modeled felds, dverse spatal data sampled non-optmally, presence of sgnfcant dscontnutes, and noses are common challenges. In addton, datasets obtaned from dverse sources wth dfferent accuraces are usually 3 6 very large ( N 0-0 ). Relable nterpolaton methods sutable for GIS applcatons should meet some necessary requrement such as accuracy and pre- 355
dctve power, dmensonal formulaton, drect estmaton of dervatves, applcablty to large datasets, 3 + computatonal effcency, and ease of use. Currently, t s dffcult to fnd a method that fulfls all the above-mentoned requrements for a wde range of georeferenced data. Therefore, the rght choce of the most adequate method wth approprate parameters for applcatons s paramount. Dfferent methods produce dfferent spatal representatons n dfferent datasets; also, n-depth knowledge of the phenomenon n queston s necessary n evaluatng whch of the nterpolaton methods produces results closest to realty. The use of an unsutable method or napproprate parameters can result n a dstorted model of spatal dstrbuton, leadng to potentally wrong decsons based on msleadng spatal nformaton. A wrong nterpolaton results becomes very crtcal when the estmates are nputs for smulatons, as small error or dstorton can cause models to produce false spatal patterns [7]. Whle external factors e.g. data densty, spatal dstrbuton of sample data, surface type, sample sze and samplng desgn, etc. [4] may affect the performance of an nterpolaton method, an understandng of the mathematcal formulaton of these methods wll provde some nformaton on ther sutablty for terran modelng. Ths paper attempts to examne the accuracy of spatal nterpolaton methods n modelng landform (topography) n relaton to ther mathematcal formulaton. The expermental study of ths work employs an area comprsng a slope and a plan as landform-adaptablty test area and focuses on the comparatve analyss of three commonly used nterpolaton methods of Krgng, Splne, and Inverse Dstance Weghtng, IDW. The followng secton summarzes the theoretcal and mathematcal bass of dfferent known nterpolaton methods ncludng the three nterpolaton methods n queston. Secton 3 ntroduces the accuracy analyss methods used n ths paper whle Secton 4 presents the expermental analyss. Secton 5 dscusses the results and Secton 6 concludes. 2. Mathematcal and Theoretcal Concept of IDW, Splne and Krgng Dfferent spatal nterpolaton methods have been developed n dfferent doman for dfferent applcatons. Accordng to [6], many standard lnear models are specal cases of a more general predcton model. Tobler s frst Law of Geography, everythng s related to everythng else, but near thngs are more related than dstant thngs [8] forms the general prncple of many nterpolaton methods. Some of the methods are only sutable for contnuous data felds that assume normal dstrbuton of the dataset. Spatal Interpolaton methods could be classfed nto two major groups: a) Mechancal/determnstc/non-geostatstcal methods; these nclude among other methods, Inverse Dstance Weghtng (IDW) and Splnes. b) Lnear statstcal/stochastc/geostatstcal methods; whch nclude Krgng among others [4], [6]. 356
2.. Inverse Dstance Weghtng (IDW) Ths method assumes that the value at an unknown locaton can be approxmated as a weghted average of values at ponts wthn a certan cut-off dstance, or from a gven number of the closest ponts (typcally 0 to 30). Weghts are usually nversely proportonal to a power of dstance [9] whch, at an un-sampled locaton s, leads to an estmator as contaned n Equaton (2) below. n ( ) wz ( s) F s = = = m = m j= ( ) z s p s s s s where p s a parameter (typcally = 2) [7]. IDW s a method that s easy to use and readly avalable; t frequently does not produce the local shape mpled by data and produces local extrema at the data ponts [7]. Some modfcatons have gven rse to a class of multvarate blended IDW surfaces and volumes [9]. The assumpton for IDW s that measured ponts closer to the unknown pont are more lke t than those that are further away n ther values. The weght s gven as: p d λ = (3) n = p d where d s the dstance between x 0 and x, p s a power parameter, and n s the number of measured ponts used for the estmaton. The man factor affectng the accuracy of IDW s the value of the power parameter [0]. Weghts dmnsh as the dstance ncreases, especally when the value of the power parameter ncreases, so nearby samples have a heaver weght and have more nfluence on the estmaton, and the resultant spatal nterpolaton s local [0]. The choce of power parameter and neghborhood sze s arbtrary []. The most popular choce of p s 2 and the resultng method s often called nverse square dstance or nverse dstance squared (IDS). The power parameter can also be chosen based on error measurement (e.g., mnmum mean absolute error), resultng n optmal IDW [2]. The smoothness of the estmated surface vares drectly wth the power parameter, and t s found that the estmated results become less satsfactory when p s and 2 compared wth when ps 4 [2]. IDW s referred to as movng average when ps zero [3], lnear nterpolaton when ps and weghted movng average when ps not equal to [3]. 2.2. Splne Splnes belong to a group of nterpolators called Radal Bass Functons (RBF). Methods n ths group nclude Thn-Plate Splne (TPS), Regularzed Splne wth Tenson, and Inverse Mult-Quadratc Splne [4]. These models use mathematcal functons to connect the sampled data ponts. They produce contnuous elevaton and grade surfaces whle lmtng the bendng of the surface produced to a p j (2) 357
mnmum. RBF models are best employed n smooth surfaces for whch the avalable sample data sze s large as ther performance s less than optmum for surfaces wth apprecable varatons spannng short ranges.rbf does not force estmates to mantan the range of the sampled data n these models lke n IDW [4]. Splne functons are the mathematcal equvalents of the flexble ruler cartographers used, called splnes, to ft smooth curves through several fxed ponts. It s a pecewse polynomal consstng of several sectons, each of whch s ftted to a small number of ponts n such a way that each of the sectons jon up at ponts referred to as break ponts. Ths has the advantage of accommodatng local adjustments, f there s a change n the data value at a pont, and s preferable to a smple polynomal nterpolaton because more parameters can be defned, ncludng the amount of smoothng [3]. Splnes are normally ftted usng low order polynomals (.e. second or thrd order) constraned to jon up. They may be two-dmensonal (e.g. when smoothng a contour lne) or three dmensonal (when modelng a surface). The smoothng splne functon also assumes the presence of a measurement error n the data that needs to be smoothed locally [3]. Among the many versons and modfcatons of splne nterpolators, the most wdely used technque s the thn-plate splnes [5] as well as the regularzed splne wth tenson and smoothng [7]. 2.2.. Regularzed Splne wth Tenson For regularzed splne wth tenson and smoothng, the predcton s gven by: z( s ) = a + n w R( v ) (4) 0 = where a s a constant and R(v ) s the radal bass functon gven by: R ( v ) = E ( v ) + In ( v ) + C E (5) and v = ϕ h (6) where ( ) [ ] 2 E v s the exponental ntegral functon, C E = 0.57725 s the Euler constant, ϕ s the generalzed tenson parameter and h 0 s the separaton between the new and nterpolaton pont. The coeffcents a and w are obtaned by solvng the system, n = 0 2 w = 0 (7) ( ) δ ϖ ϖ ( ) n = j 0 a + w R v + = z s ; j =,, n. (8) whereϖ0 ϖ are postve weghtng factors for a smoothng parameter at each locaton [7]. The tenson parameter ϕ determnes the dstance over whch the gven ponts nfluence the resultng surface, whle the smoothng parameter controls the vertcal devaton of the surface from the sample locatons. The use of an approprate combnaton of tenson and smoothng produces a surface that correctly fts the emprcal knowledge about the expected varaton [7]. 2.2.2. Thn Plate Splne Wahba and Wendelberger [6] formulated thn plate splnes (TPS), prevously 358
called Laplacan smoothng splnes, for modelng clmatc data [4]. A basc soluton to the b-harmonc equaton, has the form z( r) = r 2 log r (9) where r s the dstance between sample ponts and un-sampled locatons [7]. The relaton below approxmates the surface wth mnmum bend ( ) = + + + 2 3 ( 0 ) n = f s a a x a y wz s s (0) where the terms a, axay 2, 3 model the lnear porton of the surface defnng a flat plan that best fts all control ponts usng least squares, the last term models the bendng forces due to m sampled ponts, w are control ponts coeffcents and s s 0 s the separaton of sampled pont s and locaton s 0. The unknowns a, axay 2, 3 and w are evaluated usng the relaton L V = w aaa () where L ( ) T K P 2 3 P 0 = T and V s a vector of pont heghts. K s a matrx of the dstance between sampled ponts and P s a matrx of the sampled ponts coordnates. L s obtaned by calculatng the nverse of L or solvng EQ. wth V replaced wth the matrx of the heghts of sampled ponts H padded wth zeros [8]. Once the unknowns are evaluated, one can compute EQ.0 to determne the heghts of unknown ponts.tps computes a smoothng factor by lmtng the Generalzed Cross Valdaton functon, GVC, makng for a comparatvely sturdy model as lmtng the GVC mproves the accuracy of estmatons and s less relant on the accuracy of the model tself. TPS gves a determnaton of spatal accuracy [7]. (2) 2.2.3. Inverse Mult-Quadratc Splne The relaton below gves the nverse mult-quadratc splne functon f = + s s ( s) 2 (3) where s s s the Eucldean dstance between control ponts s and the unknown pont s [5]. The surface s modelled by the functon ( ) ( ) z s = n af s (4) = where the weghts such that a are selected to ensure exact estmatons at each data pont n ( ) ( ) z = z s = a f s, =,, n (5) j= j j and s computed by the relaton z = Fa (6) where z s replaced by a vector of sampled data values, F s a square functon matrx gven by, 359
FIJ = f ( sj ),, j =,, n (7) The estmaton functon generated wth these weghts s smooth and exact at sampled data ponts [9]. Splnes have been wdely seen as hghly sutable for estmaton of densely sampled heghts and clmatc varables [7], [5]. Among ts dsadvantages, the nablty to ntegrate larger amounts of auxlary maps n modelng the determnstc part of change as well as the arbtrary selecton of the smoothng and tenson parameters have been wdely crtczed [7]. Predctons obtaned from splnes therefore are largely dependent on decsons lke the order of polynomal used, number of break ponts, etc. taken by the user. Splnes may also be modeled not to be exact to avod the generaton of excessvely hgh or low values common wth some exact splnes [3]. Unlke the IDW methods, the values predcted by RBFs are not constraned to the range of measured values,.e., predcted values can be above the maxmum or below the mnmum measured value [4]. 2.3. Krgng Krgng, synonymous to geostatstcal nterpolaton, began n the mnng ndustry as a means of betterng ore reserve estmaton n the early 950 s [6]. Mnng engneer D. G. Krge and statstcan H. S. Schel formulated t. After almost a decade, French mathematcan G. Matheron derved the formulas, establshng the entre feld of lnear geostatstcs []. Krgng s founded on a concept of random functons wth the surface or volume assumed one realzaton of a random functon wth a known spatal covarance [7]. Regonalzed varable theory assumes that the spatal varaton of any varable can be expressed as the sum of the followng three components: a) A structural component havng a constant mean or trend. b) A regonalzed varable, whch s the random but spatally correlated component. c) A random but spatally uncorrelated nose or resdual component. d) Mathematcally, for a random varable z at x, the expresson s Z x = m x + ε x + ε (8) ( ) ( ) ( ) where m( x ) s a structural functon modelng the structural component, ε ( x ) s the spatally auto-correlated stochastc resdual from m( x ) (the regonalzed varable), and ε s random nose whch s normally dstrbuted wth an average of zero and a varance 2 σ [3]. Ordnary Krgng Ordnary Krgng (OK) s a standard verson of Krgng where predctons are based on the model, ( ) = µ + ε( ) Z s s (9) where µ s the fxed statonary functon or global average, and ε ( s ) s the stochastc but spatally correlated part of the varaton. Predctons are made as 360
n T ( 0) ( 0) ( ) 0 z s = w s z s = z (20) OK λ = where λ 0 s the vector of krgng weghts ( w ) and z s the vector of n samples at prmary locatons. Krgng, n a way, s an mprovement of the nverse dstance nterpolaton where the key problem of nverse dstance nterpolaton (the determnaton of how much mportance s gven to each neghbor) s addressed n such a way that the estmated weghts account for the true spatal autocorrelaton structure. The novelty Krgng has n the analyss of pont data s the dervaton and plottng of the sem-varance dfferences between the neghborng values [6]. Assumng statonarty, one can estmate a sem-varogram, γ ( h ), for data z( s ), defned as where h s the dstance between pont x and x 0 [2]. Ths relates to the spatal covarance C( h ) by, γ ( h) = var ( ) ( 0 ) 2 z x z x (2) ( ) ( 0) ( ) γ h = C C h (22) where C ( 0) s the sem-varogram value at nfnty(sll) [7]. The Ordnary Krgng (OK) weghts are evaluated by multplyng the covarances, = C C (23) λ 0 0 where C s the covarance matrx derved for an n x n samples matrx wth one addtonal row and column added to ensure the sum of weghts s equal to one, and C 0 s the vector of the covarances at a new locaton [6]. The covarance at a dstance of zero ( C ( 0) ) s by defnton the mean resdual error [6]. Expermental varograms usually have some characterstc features among whch are: ) Low values of h have small varance wth varance ncreasng n drect proporton to h, levelng off at a certan pont to form the sll. 2) At dstances less than the range (the dstance at whch the varance levels off), ponts closer together are more lkely to have smlar values than ponts further apart whle at dstances greater than the range, ponts have no nfluence upon themselves. The range therefore gves an dea of how large the search radus needs to be for a dstance-weghted nterpolaton. 3) The semvarance when h s zero has a postve value referred to as the nugget and ndcates the amount of non-spatally autocorrelated nose [3]. The semvarance dsplayed n an expermental varogram s modeled by a mathematcal functon dependng on the shape of the expermental varogram. A sphercal model s used when the varogram has a classc shape, an exponental model when the approach to the sll s more gradual. A Gaussan model s used when the nugget s small and the varaton s very smooth, and a lnear model when there s no sll. A varogram contanng a trend that has to be modeled separately s ncreasngly steep wth larger values of h. If the nugget varance s large and the varogram shows no tendency to gradually vansh wth smaller values of h, or the dstance between observatons s larger than the range (.e. 36
sample ponts are too far apart to nfluence one another), then nterpolaton s not reasonable and the best estmate s the overall mean of the observatons. Anose-flled varogram showng no partcular pattern may mean that the observatons are too few. A varogram that dps at dstances greater than the range to create a hole effect shows the sample space may be too small to reflect some long wave-length varaton n the data [3]. The nterpolated surface s constructed usng statstcal condtons of unbasedness and mnmum varance [7]. Three mportant requrements for Ordnary Krgng are: () The trend functon s fxed () The varogram s nvarant n the entre area of nterest () The target varable s (approxmately) normally dstrbuted. These requrements are often not met and consttute a serous dsadvantage of Ordnary Krgng [6]. However, a major beneft of the varous forms of krgng (and other stochastc nterpolaton methods) s that estmates of the model s predcton errors can be calculated, ncorporated n the analyss, and plotted along wth the predcted surface. Such error nformaton s an mportant tool n the spatal decson makng process [4]. 3. Accuracy Assessment Methodology The accuracy evaluaton ndces commonly used nclude, the Mean Error (ME), Mean Absolute Error (MAE), Mean Squared Error (MSE) and Root Mean Squared Error (RMSE). For n observatons, p predcted value, and o observed value these ndces are evaluated usng the expressons lsted below: ME = n ( p o ) n = MAE = n n p o = MSE = n ( p o ) 2 n = RMSE = n ( p o ) n = 2 2 (24) (25) (26) (27) [4] ME s used for determnng the degree of bas n the estmates often referred to as the bas [0]. Snce postve and negatve estmates counteract each other, the resultant ME tends to be lower than the actual error promptng cauton n ts use as an ndcator of accuracy [4]. RMSE provdes a measure of the error sze, but s senstve to outlers as t places a lot of weght on large errors [4]. MSE suffers the same drawbacks as RMSE. Whereas MAE s less senstve to extreme values [20] and ndcates the extent to whch the estmate can be n error [2]. MAE and RMSE are argued to be smlar measures and they gve estmates of the average error, but do not provde nformaton about the relatve sze of the average dfference and the nature of dfferences comprsng them [20]. Of course, cross-valdaton s used together wth these measurements to assess the performance of the nterpolaton methods [4]. In ths paper the ME and RMSE (aval- 362
able on the software used) are used to evaluate the performances of the IDW, Splne and Krgng nterpolaton methods consdered n the expermental analyss part of ths study. Cross valdaton s performed as further test of predcton accuracy. 4. Expermental Analyss The study area as descrbed n Fgure and Fgure 2 s an expanse of land measurng about twenty-one (2) hectares located n Ikot Ukapon lattude 50'0''N and longtude 7 59'0''E n the North-Eastern part of Akwa Ibom state of Ngera. The area s largely hlly wth a plan coverng about twenty (20%) percent of ts total area. Orthometrc heght data, obtaned from the Offce of the State Surveyor General, consst of four hundred and sxty-two (462) randomly sampled ponts spread across the area collected through feld survey usng a Kolda K9-T seres dfferental GPS. Data so obtaned from the feld samples was n Mcrosoft Excel.xls format. The lowest elevaton of the area s 27.08m whle the hghest elevaton s 98.79m above mean sea level. Prelmnary data exploraton usng ESRI s Arc- GIS (verson 9.3) shows the data s not normally dstrbuted as shown by dstrbuton parameters n Table. Fgure. Map of the studyng area showng the dstrbuton of data ponts wthn the study area. 363
Fgure 2. The contour map of the studyng area. Table. Dstrbuton parameters of the based elevaton data. Count 462 Mnmum value 27.08 Maxmum value 98.79 Mean 7.754 Standard devaton 2.633 Skewness 0.6962 Kurtoss 2.0252 Medan 8.497 st quartle 52.994 3 rd quartle 88.886 4.. IDW Method The followng contour fll surface shown n Fgure 3, s generated for IDW nterpolaton wth power, p of 2, smoothng factor of 0.5and neghborhood sze of 5 for the based data. 4.2. Splne Method Regularzed Splne nterpolaton, mplemented as Radal Bass Functon(RBF), wth order 2 gves the contour fll map n Fgure 4 for power = 2, smoothng factor = 0.5 and neghborhood sze = 5. 4.3. Krgng Krgng works on the assumpton that the data set s normalzed, therefore, we carred out Box-Cox normalzaton on the data before mplementng Krgng nterpolaton. 364
Fgure 3. IDW contour fll map. Fgure 4. The regularzed Splne contour fll map. The data was dvded nto a tranng and test subsets n a rato of 80:20 usng the Geostatstcal Analyst tool n ArcGIS and optmal parameter values (generated by ArcGIS and sets the best possble value for each parameter) used for predctons on the tranng subset. The test subset was then used wth these optmal parameters for valdaton. The data dstrbuton parameters after Box-Cox normalzaton s as shown n Table 2. The contour fll maps generated usng Krgng wth the Gausan model and 365
Table 2. Dstrbuton parameter of data after normalzaton usng Box-Cox wth transformaton. count 462 Mnmum value 26.08 Maxmum value 97.79 mean 70.754 Standard devaton 2.633 skewness 0.6962 kurtoss 2.0252 medan 80.497 st quartle 5.994 3 rd quartle 87.886 Sphercal model respectvely for auto-calculated values for nugget, sll, mdrange, a lag sze of 54.40, and lag number = 2 are as shown Fgure 5(a) and Fgure 5(b). The maps for IDW, Splne and Krgng after the optmal valdaton of the data are shown n Fgures 6(a)-(c). 5. Result and Dscusson From the predcton errors tabulated n Table 3 and Table 4, the level of bas n estmaton s lowest for Krgng and hghest for IDW as ndcated by the respectve MEs. Ths presence of bas s expected, as the heght data was not random as shown n Table. Box-Cox normalzaton of the data dd not result n a normal dstrbuton ether (see Table 2), ths means the data was very based. Ths scenaro s frequently encountered n practce as feld collecton of elevaton data usually focuses on capturng perceved changes n elevaton rather than on randomness. The bas for the valdated data however, s hghest for Krgng and lowest for Splne lkely because of the reducton n the number of samples used for valdaton. A better measure of the error n predcton, the RMSE, s lowest for Splne and hghest for Krgng for both predcton and valdaton shown n Table 3 and Table 4. Ths ndcates an elevaton model that s closer to what s on ground for Splnes. However, outsde the areas where nterpolaton data where obtaned, Splne produces unrelable predctons. It s therefore not sutable for cases where data outsde the captured area s desred (extrapolaton). IDW produces a model that s better than that of Krgng but not as good as the Splne model. Its predctons outsde the captured area are also better than that of Krgng and Splne. Ths does not mply Krgng s not sutable for terran modelng or wll not perform better than both IDW and Splne. Krgng assumes normal dstrbuton of data and models the spatal dstrbuton of a geographcal event as a realzaton of a functon that s random. Its predctons therefore are dependent on the data satsfyng the statstcal crtera of unbasedness and mnmumvarance. Its mathematcal formulaton makes t unsutable for data that 366
(a) (b) Fgure 5. (a) and (b) depcts the Krgng contour fll map after Box-Cox normalzaton created wth Gaussan model and wth sphercal model respectvely. 367
(a) (b) (c) Fgure 6. (a) Inverse Dstance Weghted (IDW) Predcton Map; (b) Splne; (c) Ordnary Krgng Predcton Map. 368
Table 3. Predcton errors for the three nterpolaton methods at optmal parameters. METHOD IDW SPLINE KRIGING ME 0.589 0.0608 0.0989 RMSE 3.488 2.0 4.374 ME s the mean predcton error and RMSE s the root mean square standardzed error of predcton. Table 4. Predcton errors after valdaton. METHOD IDW SPLINE KRIGING ME 0.396 0.06838 0.492 RMSE 3.76.892 4.02 ME s the mean predcton error and RMSE s the root mean square standardzed error of predcton. s not normally dstrbuted or dffcult to normalze. Splnes on the other hand use a physcal model varyng n accordance to the varaton n the elastc propertes of the estmaton functon. It tends to do well wth modelng physcal phenomena such as terran. IDW uses a lnear combnaton of values at captured event locatons, assgns weghts by an nverse functon of the separaton between the event locaton to be estmated and ponts captured to estmate values of the unknown locaton. Though weghts are specfed arbtrarly, ArcGIS software provdes an optmal weght management functon that assgns a weght that s most sutable for ponts wthn the captured data set. Predctons are nfluenced by ths weght assgnment but are more relable n terms of error than what s obtaned usng Krgng. It s acknowledged that the Krgng does very well wth covarate data such as temperature data, but the data has to be captured as randomly as possble. Ths s often not acheved. A good knowledge of the data used as well as the strengths and weaknesses of the avalable nterpolaton methods s necessary n decdng on a method to use for nterpolaton for a gven purpose. 6. Concluson In ths study, Splne provdes a more accurate model and result for the elevaton data obtaned drectly from feld survey that was not homogenously randomzed and not normalzed. From the nterpolaton result we obtaned, Splne method outsde the data area also reaffrms that predctons by RBFs are not constraned to the range of measured values,.e., predcted values can be above the maxmum or below the mnmum measured value. Tan and Xu [22] concluded from ther experment on terran modelng usng data from a dgtzed map that IDW gave a better model n terms of accuracy than Splne or Krgng. Ths s most lkely due to means from whch the test data were acqured. Ther test data were dgtzed from a contour map and were homogenously dstrbuted. The knowledge of the source of data may therefore be of mportance n the choce of nterpolaton method. References [] Kreyszg, E. (2003) Advanced Engneerng Mathematcs. 8th Edton, John Wley 369
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[9] Smth, T.E. (206) Notebook on Spatal Data Analyss. http://www.seas.upenn.edu/~ese502/#notebook [20] Wllmott, C.J. (982) Some Comments on the Evaluaton of Model Performance. Bulletn Amercan Meteorologcal Socety, 63, 039-33. https://do.org/0.75/520-0477(982)063<309:scoteo>2.0.co;2 [2] Nalder, I.A. and Wen, R.W. (998) Spatal Interpolaton of Clmatc Normals: Test of a New Method n the Canadan Boreal Forest. Agrcultural and Forest Meteorology, 92, 2-225. https://do.org/0.06/s068-923(98)0002-6 [22] Tan, Q. and Xu, X. (204) Comparatve Analyss of Spatal Interpolaton Methods. Sensors and Transducers, 65, 55-63. Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: Acceptng pre-submsson nqures through Emal, Facebook, LnkedIn, Twtter, etc. A wde selecton of journals (nclusve of 9 subjects, more than 200 journals) Provdng 24-hour hgh-qualty servce User-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research work Submt your manuscrpt at: http://papersubmsson.scrp.org/ Or contact jgs@scrp.org 37