ESTIMATION OF INTERPOLATION ERROR IN DEMS USING STATISTICAL METHODS

Transcription

1 ESTIMATION OF INTERPOLATION ERROR IN DEMS USING STATISTICAL METHODS Robet PAQUET, Austalia Key wods: DEM, inteolation eo, ALS, hotogammety, Delaunay tiangulation SUMMARY To wok with DEMs euies a knowledge of thei accuacy, to avoid having an eo lage that the level of details to achieve the task. Such oblems can lead to unlanned costs, esuveys, disutes, fo examle in flood studies (suious wells), defomation detection (tying to detect mining subsidence effects smalle than the eo), stockile volume calculations (lage disceancy in estimation by buye and selle). The eo can be divided into two comonents: the obsevation eo (with its many diffeent asects esulting fom the method of obsevation) and the inteolation eo (whee heights ae estimated between obsevations). The total eo is thus the estimate of the obsevation eo oagated into the inteolation eo. The obsevation eo is often difficult to detemine, esecially whee the senso is mounted on a latfom in motion, such as an aicaft in aibone lida. It is howeve moe eadily estimated fo examle with theodolites and teestial lase scannes. The inteolation eo in a DEM is a function of the oughness of the suface and the oint density of the DEM. It does not deend on the latfom chosen fo the obsevations. This aticle descibes a statistical method to estimate the inteolation eo in a DEM, egadless of which method was used fo the obsevation. The eo is then modelled on the density, using linea egession. The idea is to comae the inteolation eo to the level of eo euied fo the task. The accuacy/density model can be used to lan suveys at the obsevation hase to obtain a level of eo adeuate fo the task. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 1/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

2 ESTIMATION OF INTERPOLATION ERROR IN DEMS USING STATISTICAL METHODS Robet PAQUET, Austalia 1. INTRODUCTION DEMs ae a oduct used widely fo engineeing alications such as wate flow modelling, infastuctue design, gound defomation studies o civil engineeing woks. Eos in the DEMs can be the souce of contact disutes as fo examle the calculation of volumes, esulting in lengthy legal battles. Assessing the eo fo gound infomation whee the oints ae inteolated between the obsevations is a difficult task, as the eo in the obsevations is oagated to the oints, which also have uncetainty due to inteolation. Moeove, some DEMs have thei nodes themselves aleady inteolated fom aw data, esulting in a secondgeneation inteolation. Thid geneation inteolations ae also available: fo examle hotogammetic data may be used to geneate contous, which ae digitised in tun to ceate a DEM, fom height infomation is sought at oints inteolated between the nodes. This aticle demonstates two simle statistical techniues to assess the inteolation associated with a given DEM and to ossibly model the inteolation, to ovide a ediction tool. 2. ESTIMATION OF INTERPOLATION ERROR 2.1 Oveview of the data The data is a DEM geneated diectly fom ALS (aibone lase scanning). The aea coveed is Islington Pak, between the sububs of Tighes Hill, Islington and Mayville in the inne west of the city of Newcastle uon Hunte, NSW. The data was ovided to the autho by AAMHATCH fo eseach uoses. It is a tyical uban ALS suface, whee the obsevations have been cleaed of buildings, tees and infastuctue to ovide gound infomation. The data is a set of 27,729 oints. Figue 1 shows a lot of the data set and an aeial hoto (coutesy of Google Eath). The data set shown has its numbe of oints educed to 6,000 oints fo claity. Note the coodinates system: the data oiginally on a MGA gid is shifted and cented on the oigin. This is to minimise calculation eo oagation due to oeations with lage numbes. The data ovides gound infomation: the dense aeas ae in the steet and in the ak, the moe atchy aea eesents aeas whee building and tee canoy infomation was emoved to ovide gound infomation. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 2/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

3 Figue 1: Data and Aeial Photo of Islington Pak, NSW 2.2 Bootstaing Method Comments about Inteolation The inteolated height of a oint has inceasing uncetainty with inceasing distance fom a measued oint (of known height). In a TIN envionment, the inteolated height of a mak is calculated using the known heights of the vetices of the tiangle enclosing the said mak. The uncetainty is thus at a maximum fo maks whose satial osition is at the geometic cente of the enclosing tiangle. The inteolation eo fo seveal maks in the same tiangle will theefoe have diffeent values deending of the osition of the maks in the tiangle. The otential magnitude of the eo also vaies with the size of the tiangle. The inteolation eo is also a function of the vaiation of elief within the tiangle. Fo examle the height of any oint in a tennis cout o a sots oval is not likely to be influenced by the size of the tiangle, as the gound has no sloe by design. Assessing the mean suae oot of the inteolation eo of a suface is indeed a vey comlex task Bootstaing Bootstaing methods ae basically e-samling methods to obtain comlex statistical chaacteistics of samles which would ove difficult to obtain with taditional analytical methods (Efon and Tibshiani 1998). The ocessing owe of moden esonal comutes is ut to use to e-samle the data to be analysed many times aound in ode to ceate a stochastic model fom which conclusions can be dawn Estimation of inteolation eo To evaluate the inteolation eo at a oint in the oiginal set is a difficult task, as outlined in Section FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 3/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

4 Figue 2: Inteolation Eo Calculation Conside now the tiangulated gou of 4 oints in Figue 2. The e-samling of this gou may delete the mid-oint and use the thee oute oints in the tiangulation, elacing the thee tiangles with one lage tiangle. Since we know the coodinates of the 4 th oint, the inteolation eo is calculated as the diffeence between the known RL of the midoint and its inteolated RL in the new lage tiangle fomed. To calculate the inteolated RL of the oint, we use the easting and nothing coodinates of the oint, and the euation of the enclosing tiangle, defined by: Ax By Cz D 0 (Euation 1) Whee the coefficients A, B, C and D ae detemined with the coodinates of the vetices P, Q and R of the enclosing tiangle, as: A y ( z D Ax B x ( z C x ( y z ) y ( z By Cz z ) x ( z y ) x ( y z ) y ( z z ) z ) x ( z y ) x ( y z ) y ) (Euation 2) To obtain a stochastic model of the inteolation eos, the oiginal suface of 27,729 oints is teated in the following manne: - At each iteation, 25 oints ae andomly selected fom the oiginal suface (the 25 oints ae thus diffeent fom iteation to iteation). - The suface of 27,704 oints still in the data set is thinned out andomly to hold 26,000 oints. The inteolation eo is then calculated fo the 25 oints. - This oeation descibed in the two items above is eeated 25 times. Recall that fo each of the 25 iteations, the suface of 26,000 oints is diffeent, and the 25 oints ae diffeent. We obtain theefoe a samle of 625 inteolation eo measuements fo a andom density of 26,000 oints, all geneated fom a eal suface of 27,729 oints. - The next 25 iteations ae undetaken in the same manne, with diffeent oints, diffeent sufaces, both icked andomly, but with a suface made of 25,750 oints, that is, a decement of 250 oints fom the evious (and initial) 25 iteations. The oiginal suface has theefoe 25 oints andomly chosen, then is thinned out off 1,954 oints instead of 1,704 oints to geneate a suface of 25,750 oints density. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods FIG Congess 2010 Sydney, Austalia, Ail /11

5 - The next 25 iteations (i.e. Iteations No.51 to No.75) geneate a suface of 25,500 oints, thus anothe decement of 250 oints. - The numbe of decements (of 250 oints) is set to 102. The stochastic model is theefoe constucted with 625 inteolation eos ove 103 diffeent density of the same suface, using only obseved data. The ogamme did the 64,375 calculations in 2,466.4 seconds. DEM Density (No. of Points) Table 1: (Patial) Results of Bootstaing Method No of Calculations Mean of RL (25 ts by 25 Diffeences (m) Iteations) Standad Deviation of means (m) 26, , , , , , , , Note that the numbe of calculations does not always add to 625, as the algoithm: - stos dulication of any of the 25 andom oints icked fo the calculation of inteolation, although it is not likely to haen when the ool suface is lage but note that it haens when the suface is small. - evidently does not etun a calculation when the oint is outside the eimete of the suface and an enclosing tiangle cannot be found. A lot of the model density Vs mean inteolation eo is shown in Figue 3 below. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 5/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

6 FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods FIG Congess 2010 Sydney, Austalia, Ail 2010 Figue 3: Plot Density Vs Mean Inteolation Eo Figue 3 shows that, as exected, the inteolation eo value is deceasing as the density of the DEM is inceasing. The inteolation eo can then be used in conjunction with the othe souces of eo to assess the eo budget of the DEM. A geneic eo model is: RMSE DEM RMSE RMSE (Euation 3) 2 Inteolation 2 EveythingElse whee Eveything Else includes the suvey eos, the euiment and method eos, etc (see (Hodgson and Besnahan 2004) fo a detailed examle). The eo model is an essential decision tool. A dense DEM may not be waanted fo the task, but is vey demanding on comuting time and may euie excessive comute esouces fo the maniulation of the data, in which case the DEM can be thinned out. On the othe hand, the eo model may eveal that the eo is too lage and not fit fo uose. 3. Pediction Model The aim of this exeiment is to ceate a density-inteolation eo model, which could be used to edict one fom the othe. The statistical used is linea egession. In ode to achieve this exeiment and validate it, we use a thinned-out DEM to constuct the model, which is veified with the oiginal dense data. 6/11

7 3.1.1 Exeiment Fo this exeiment, the oiginal data set of 27,729 oints is thinned out once and andomly to a set of 10,500 oints. The algoithm esented in Section 2 is used to analyse the esulting suface with 95 decements of 100 oints of 25 iteations each, stating fom a suface of density 10,000 oints. The vaiation of the means and standad deviation of the means is shown in Figue 4 below Linea Model Figue 4: Means and SD of Inteolation Eo Vs Density In ode to use linea egession, seveal data tansfomations ae tied and a tansfomation using the log of the density and the log of the means is adoted (Kutne, Nachtsheim et al. 2004). The main eason fo that choice is the staightness of the mean, as shown in Figue 5. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 7/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

8 Figue 5: Tansfomed Data The line is fitted on the means using linea egession techniues. The euation of the line is: Y X (Euation 4) The goodness of the fit of the model and the esidual lot ae shown in Figue 6. Figue 6: Linea Model Fit and Residual Plot FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 8/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

9 Figue 5 shows that the tansfomation adoted staightened the data in such a way that a linea model can be consideed. A visual insection of the esidual lot indicates a slight deatue constancy of the eo vaiance. The lot is of the megahone tye, with an inceasing eo vaiance as the density incease. Desite this, Figue 6 shows an excellent fit of the linea model with well-distibuted esiduals and no aaent tends Pediction et validation To validate the model, which was ceated fom a uniue DEM of 10,500 oints, we simly extend the linea model to density of 26,000 oints and comae it with the data obtained in Section 3. The validation is done though lots in Figue 7. Figue 7: Pediction and Validation The validation of the ediction shows that the model is ove-edicting fo the dense DEMs. This is veified by the esidual lot which shows moe negative esiduals at highe density. The data and the model is tansfomed back and lotted in Figue 8. It confimed that the model is ove-edicting the inteolation eo. The ediction is uite accuate if taking into consideation that the ediction is fo a DEM almost thee times as dense as the DEM used fo modelling. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 9/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

10 Figue 8: Data and Model (un-tansfomed) 4. Conclusions End uses of DEMs aely uey the accuacy of this oduct. Poblems can occu whee the DEM ae used fo examle fo engineeing design o volume calculations and the eo is lage than the euiements of the intended usage. On the othe hand, the DEMs can also be vey lage files euiing abundant comute esouces, in which case it would be beneficial to thin out the DEM befoe maniulation. This aticle descibes simle statistical tools to: - Assess the inteolation eo in the DEM, and - Geneate a Linea Regession model of the Density Vs Inteolation Eo in ode to edict the eo outside the available DEM density. These models ae useful if the estimated eo is lage than the taget eo, and if a eduction of the eo is necessay, fo suvey lanning uoses, although extaolation has to be used with caution. The assessment of the inteolation eo is using a coss validation (o bootstaing) statistical techniue. These techniues ae emiical and heled to obtain statistical infomation difficult to find with theoetical techniues. FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 10/11 FIG Congess 2010 Sydney, Austalia, Ail 2010

11 ACKNOWLEDGMENT I would like to thank AAMHATCH fo oviding the ALS data used in this aticle. REFERENCES Efon, B. and R. J. Tibshiani, 1998, An Intoduction to the Bootsta, Ed. Chaman & Hall/CRC, 436., London. Hodgson, M. E. and P. Besnahan, 2004, "Accuacy of Aibone Lida-Deived Elevation: Emiical Assessment and Eo Budget", Photogammetic Engineeing and Remote Sensing 70(3): Kutne, M. H., C. J. Nachtsheim, et al., 2004, Alied Linea Statistical Models, Ed. McGaw-Hill Iwin, 1396., fifth edt., Boston BIOGRAPHICAL NOTES Robet Pâuet has degees in both civil engineeing and suveying, and a doctoate in hotogammety. His cuent woks and eseach includes analysis of subsidence data. CONTACTS D Robet Pâuet Mine Safety Oeations Industy & Investment NSW PO Box 344 Hunte Region Mail Cente NSW 2310 Austalia Tel Fax obet.auet@industy.nsw.gov.au Web site: htt:// FS 2D Usage of Image Techniues in Suveying Robet Pâuet Estimation of Inteolation Eo using Statistical Methods 11/11 FIG Congess 2010 Sydney, Austalia, Ail 2010