POINT POSITIONING ACCURACY OF AIRBORNE LIDAR SYSTEMS: A RIGOROUS ANALYSIS

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1 In: Stilla U et al (Eds) PIA07. Intenational Achives of Photogammety, Remote Sensing and Spatial Infomation Sciences, 36 (3/W49B) POINT POSITIONING ACCURACY OF AIRBORNE LIDAR SYSTES: A RIGOROUS ANALYSIS Noa Csanyi ay¹, ² *, Chales K. Toth² Depatment of Civil and Envionmental Engineeing and Geodetic Science¹ Cente fo apping² The Ohio State Univesity 1216 Kinnea Road, Columbus OH , noa@cfm.ohio-state.edu Commission I, WG I/2 KEY WORDS: Aibone LiDAR, Accuacy Analysis, Eo Popagation, Eo Souces, ulti-senso Systems, GPS/ ABSTRACT: This pape povides a compehensive analysis of the achievable point positioning accuacy fo aibone LiDAR systems consideing all the majo potential eo souces. Using the law of eo popagation, igoous analytical deivation of eo fomulas was pefomed to obtain a eliable assessment of the achievable point positioning accuacy. Fo pactical use, based on the deived analytical fomulas, accuacy figues and tables have been developed. In this pape select example figues illustate the achievable point positioning accuacies of state-of-the-at LiDAR systems fo vaious accuacies of the paametes that influence the LiDAR point positioning accuacy. The analytical deivation-based accuacy plots can also be used as a tool fo choosing the ight system o system configuation fo a desied mapping accuacy, and to help the flight planning, i.e. selecting optimal flight paametes fo a given system to achieve the desied point positioning accuacy. In addition, the developed eo fomulas can also facilitate the analysis of the effects of individual eo souces on point positioning accuacy, although due to the size limitations of the pape this analysis is not included in this pape. 1. INTRODUCTION LiDAR systems ae complex, multi-senso systems consisting of at least thee sensos, the GPS and navigation sensos, and the lase scanne system. Consequently, pope system calibation, including individual senso calibation, inte-senso calibation, and time synchonization between system components is cucial in achieving the equied mapping accuacy. Futhemoe, besides eos in the calibation paametes, thee ae seveal othe eo souces that can degade the accuacy of the deived gound coodinates, such as, fo example, eos in the navigation solution (position and attitude eos), ange measuement eos, etc. In addition, the effect of the vaious eos is influenced by the vaious flight paametes (flying height, flying speed, etc.), teain chaacteistics, and system settings, and accodingly, the dependency of point positioning accuacy on the vaious eo souces is vey complex. As a consequence, the eliable accuacy assessment and pefomance validation of the deived mapping poducts is a vey challenging task. ost publications, discussing the effects of diffeent eo souces on the point positioning accuacy of LiDAR systems typically focus on a single o a few eo souces and do not discuss the combined effect of all eo souces. Baltsavias (1999), fo example, povides an oveview of basic elations and eo fomulas concening aibone lase scanning. Schenk (2001) povides a summay of the majo eo souces fo aibone lase scannes and eo fomulas focusing on the effect of some systematic eos on point positioning. A numbe of papes empiically evaluates the achieved accuacy of specific mapping pojects, nomally using gound contol as efeence (Latypov, 2002; Hodgson and Besnahan, 2004; Hodgson et. al., 2005; Peng and Shih, 2006). LiDAR vendos povide specifications on the appoximate accuacy that can be expected fom thei systems. These values, howeve, ae mostly valid unde specific cicumstances (fo specific flying height, GPS baseline, etc.), and only conside a few eo souces, and consequently ae fequently eithe too optimistic o too pessimistic. Futhemoe, some of the vendos do not clealy state what eo souces ae consideed when they povide the accuacy specifications, which makes it difficult and, in some cases, nealy impossible to compae the achievable accuacies of diffeent systems fom diffeent vendos. Fo example, some LiDAR vendos specify the achievable point accuacy consideing the GPS eos, while othes do not include this eo in thei accuacy specifications. In conclusion, no geneally accepted, compehensive and eliable accuacy assessment tool exists to help with flight o poject planning in ode to achieve the desied accuacy of the final poduct of LiDAR systems. This pape is intended to fill the void by poviding a compehensive accuacy assessment tool fo aibone LiDAR systems that consides all the majo potential eo souces, and consequently a eliable assessment of the achievable point positioning accuacy can be obtained. The fist section of the pape povides an oveview of the analytical deivation of the eo fomulas. In the second section, example plots of the achievable point positioning accuacies of diffeent systems fo vaious accuacies of the paametes that influence the point positioning accuacy ae shown and analyzed. * Coesponding autho. 107

2 PIA07 - Photogammetic Image Analysis --- unich, Gemany, Septembe 19-21, RIGOROUS ERROR PROPAGATION To eliably detemine the achievable point positioning accuacy of aibone LiDAR systems, all the majo potential eo souces that influence the point positioning accuacy have to be consideed in the analytical deivations. The andom eos listed in Table 1 wee consideed duing the analytical deivations. The 2 nd column of Table 1 also shows the symbols fo the espective standad deviation values as used in this pape; the lase beam divegence (γ) itself is obviously not an eo tem, howeve, its effect, the finite footpint size epesents an additional andom eo souce. Eo souce Navigation solution eos Position eos Attitude angle eos Eos in the detemined boesight misalignment angles Range measuement eo Used symbol σ X, σ Y, σ Z σω, σφ, σκ σω b, σφ b, σκ b σ Scan angle eo Lase beam divegence σ β Table 1. Consideed eo souces The eo fomulas fo point positioning accuacy wee deived based on the LiDAR equation (Eq.1.) via igoous eo popagation, see also Figue 1. = + R ( R + b ) (1) whee, L L, R R L 3D coodinates of object point in the mapping fame Time dependent 3D coodinates in the mapping fame, povided by GPS/ (efes to the oigin of the body fame) Time dependent otation matix between the body and mapping fame, measued by Boesight matix between the lase fame and body fame L 3D object coodinates in lase fame b Boesight offset vecto Using the law of eo popagation based on the covaiance matix of the eo souces listed in Table 1, the covaiance matix of the 3D LiDAR positions was deived as descibed in Eq. 2. All analytical deivations wee implemented in ATLAB envionment; due to size limitations of this pape, the deived fomulas ae not shown hee. The effect of lase beam divegence was consideed sepaately fom the eo popagation (but late combined) since the hoizontal eo due to the footpint size can be chaacteized by unifom distibution instead of the nomal distibution. Futhemoe, the boesight offset component was assumed to be eo fee since its effect is negligible, as compaed to the effects of othe eos. γ Figue 1. LiDAR system components and definitions. T C LiDAR = ACA (2) whee C LiDAR [3x3] C [11x11] A [3x11] Covaiance matix of the LiDAR point coodinates Covaiance matix of the position, attitude angles, boesight angles, measued ange, and the scan angle Jacobean matix containing the patial deivatives of X,Y, Z LiDAR coodinates with espect to the diffeent andom vaiables in the LiDAR equation The following assumptions wee made duing the analytical deivations: Flat teain was assumed, sloped teain will cause additional eos in the vetical coodinates as compaed to the eo popagation esults; howeve, this effect can easily be consideed and accounted fo sepaately. The eo fomulas wee developed consideing that the scanning is pefomed in a vetical plane pependicula to the flight diection. The ange measuement accuacy is assumed to be independent of the flying height and scan angle, which fo ealie systems was not exactly the case, since longe ange due to highe flying height and lage scan angle meant weake signal esponse, and consequently less accuate ange measuement. Howeve, accoding to LiDAR vendos, fo the state-of-the-at systems, the ange measuement accuacy does not noticeably degade fo longe anges. Duing the deivations the vaious eos mentioned above wee consideed to be uncoelated with each othe, howeve, any coelation between vaiables could easily be consideed by changing the covaiance matix to nondiagonal. Using the deived eo fomulas, based on the accuacy of the paametes listed in Table 1, the achievable point positioning accuacy can be computed fo any given LiDAR system opeated at diffeent flying heights, etc. 108

3 In: Stilla U et al (Eds) PIA07. Intenational Achives of Photogammety, Remote Sensing and Spatial Infomation Sciences, 36 (3/W49B) 3. EXAPLES OF ACHIAVABLE LIDAR POINT POSITIONING ACCURACY In the following examples, select accuacy plots ae shown. The figues wee geneated using typical accuacy values fo the vaious eo souces that epesent the state-of-the-at LiDAR systems. Fo the flight paametes, such as flying height, and the maximum scan angle values o in some figues value anges that ae fequently used in LiDAR mapping, wee consideed. The accuacy of the navigation paametes, the position and attitude angle accuacies used in these examples ae chosen based on the post-pocessed accuacy specifications of the Applanix POS/AV T systems. Since the examples ae intended to illustate the pefomance that can be expected fom state-ofthe-at systems, only the accuacies povided by the high-end systems, such as POS/AV T , ( wee consideed fo the examples. Consequently, fo the geneation of the following accuacy plots, the accuacy anges listed in Table 2 wee used. Fo the accuacy of the boesight misalignment angles, typical achievable standad deviations of the calibated boesight misalignment angles wee consideed (Buman, 2001; Skaloud and Lichti, 2006). Table 2 contains the values that wee used in the geneated examples. The ange measuement was assumed to have a 1 cm standad deviation (1σ), this value is based on the system specifications of state-ofthe-at LiDAR systems. It should be emphasized that ou example plots assume had sufaces; the anging accuacy used in ou computations is obviously not valid in vegetated aeas. The accuacy of scan angle measuement is typically not addessed in the liteatue o in the system specifications povided by LiDAR vendos. In the examples below, a quantization eo with º standad deviation (1σ) was assumed, which was mentioned as a valid value fo the Riegel LQ-280 system in (Campbell, et. al, 2003). The lase beam divegence (γ) of 0.3 mad was consideed based on LiDAR system specifications of moden LiDAR systems. Paamete Value (1σ) 5-15 cm, σ X, σ Y σ Z cm (1.5*σ X, σ X =σ Y )) σω, σφ σκ (2*σω, σω=σφ) σω b, σφ b 10 σκ b 30 σ 1 cm σ β º Table 2. Standad deviation values of paametes assumed fo the illustated examples Fo the sake of simplicity, the plots below illustate the accuacy of the LiDAR point coodinates in a local ight-handed coodinate system that has its X-axis aligned in the flight diection, Y-axis points to the scan diection, and Z up. Futhemoe, fo the geneation of these plots all thee aicaft attitude angles (oll, pitch, and heading) wee assumed to be zeo fo the same eason, but any othe value could be used in the deived accuacy fomulas. In all plots shown in the pape the vetical (Z) accuacy is shown in ed, the accuacy in the scan diection (Y) is maked with geen colo, and in the flying diection (X) it is shown in blue. 3.1 The Effect of Attitude Angle Eos Figue 2 and 3 illustate the effect of attitude angle eos on the achievable point positioning accuacy as a function of scan angle fo 600 m and 1500m flying height, espectively. To bette show the effects of the attitude angle eos, in these figues all othe vaiables ae consideed to be eo fee. The Sigma omega phi -axis of these figues show the σω, σφ values, the σκ value was taken accoding to the atio shown in Table 2. As the figues show, the attitude angle eos have stonge effect on the hoizontal positions than on the vetical one, and eos in all thee coodinate diections incease fo highe flying heights. Futhemoe, the effect of attitude angle eos on the coodinate accuacy in the scan diection does not change with the scan angle, while the accuacy in both the flying diection and the vetical coodinate diection degades with inceasing scan angles (towads the sides of the LiDAR stips). This inceasing effect of attitude angle eos in the flying diection is caused by the κ angle eo that has inceasing effect fo lage scan angles, while the accuacy degadation of the vetical coodinates towads the sides of the stip is due to the ω attitude angle eo that also has inceasing effect with lage scan angles. Figue 2. Effect of attitude angle eos on point positioning as a function of scan angle fo H=600 m. Figue 3. Effect of attitude angle eos on point positioning as a function of scan angle fo H=1500 m. 109

4 PIA07 - Photogammetic Image Analysis --- unich, Gemany, Septembe 19-21, The Effect of All Navigation Eos To illustate the effect of the accuacy of all the navigation paametes, Figue 4 and 5 illustate the achievable point positioning accuacy as a function of the accuacy of the aicaft position and attitude angles fo 600 m and 1500m flying height, espectively. To bette show the effects of navigation eos, in these figues all othe vaiables ae consideed to be eo fee. The figues illustate the point positioning accuacies at 10º scan angle, and in ode to also show the effect of aicaft position and the attitude eos sepaately fom each othe; the point positioning accuacies wee computed stating at zeo navigation eos, but the ealistic values ae in the anges shown in Table 2. As Figue 4 illustates, fo lowe flying heights, the accuacy of the aicaft position has a elatively lage effect on the point positioning accuacy as compaed to the angula eos (in paticula, on the vetical coodinates). Note that the highe positioning eos in the vetical coodinates fo lage aicaft position eos in this figue ae caused by the 1.5 atio of σ Z /σ X,Y (which is athe ealistic) used in the computation. This might be a bit supising, since LiDAR vetical accuacy is known to be bette than the hoizontal one, which, as the othe figues show, is nomally tue due to the othe eos that affect the hoizontal position moe (especially the beam divegence and attitude eos). As Figue 5 shows, as the flying height inceases, the attitude angle eos have inceasing effect on the point positioning accuacy, while the effect of aicaft position eos do not incease. 3.3 All Eo Souces Consideed Figue 6 and 7 illustate the point positioning accuacies fo the same cases as Figue 4 and 5, but in these figues all othe eo souces with accuacies listed in Table 2 wee also consideed. The zeo aicaft position eo and zeo attitude eo (which is obviously not a ealistic case) is only intended to show the effect puely of all othe eo souces excluding the navigation eos. As these plots show, these othe eo souces have stonge effect on the hoizontal position than on the vetical, and especially fo highe flying height, the vetical LiDAR point accuacy is indeed bette as compaed to the hoizontal accuacy (except fo some unealistic cases when the accuacy of the aicaft position is much wose than that of the attitude angles). Figue 6. Standad deviation of point positioning fo H=600m at 10º scan angle as a function of navigation eos, all eos consideed. Figue 4. Effect of navigation eos on point positioning fo H=600 m at 10º scan angle. Figue 7. Standad deviation of point positioning fo H=1500m at 10º scan angle as a function of navigation eos, all eos consideed. 3.4 Flying Height and Scan Angle Dependency Figue 5. Effect of navigation eos on point positioning fo H=1500 m at 10º scan angle. Finally, Figue 8 illustates the point positioning accuacies achievable as a function of flying height and scan angle. This plot is intended to show the case of a state-of-the-at LiDAR system including a highly accuate navigation solution. Fo this figue the following standad deviations wee consideed: σ X =σ Y. =5cm, σ Z =7.5cm, σω=σφ=15, σκ=30, the accuacy of the othe paametes wee assumed as shown in Table 2. As 110

5 In: Stilla U et al (Eds) PIA07. Intenational Achives of Photogammety, Remote Sensing and Spatial Infomation Sciences, 36 (3/W49B) the figue illustates, as the flying height inceases, the accuacy of the vetical coodinates do not significantly decease (especially fo smalle scan angles), while the hoizontal point positioning accuacy does. Towads the LiDAR stip edges (with highe scan angles) all thee coodinates show degading accuacy; in the scan diection this degadation is small, while in the flying diection and vetical diection the eos incease moe. The highe degadation in accuacy in the flying diection as compaed to the scan diection that is noticeable in the figue can be explained by the fact that eos in the heading and in the κ boesight angle affect the accuacy in the flying diection, and this effect significantly inceases with highe scan angles. 5. REFERENCES Baltsavias, E.P.,1999. Aibone lase scanning: basic elations and fomulas. ISPRS Jounal of Photogammety & Remote Sensing Vol. 54, pp Buman, H., 2000b.Calibation and Oientation of Aibone Image and Lase Scanne Data Using GPS and, PhD dissetation, Photogammety Repots No. 69, Royal Institute of Technology, Stockholm, 107 p. Campbell, J. L.,. Uijt de Haag, and A. Vadlamani, The Application of LiDAR to Systhetic Vision System Integity, IEEE Digital Avionics Systems Confeence, 2003, DASC 03, Oct. 2003, Vol. 2., pp Hodgson,. E. and P. Besnahan, Accuacy of Aibone LiDAR-Deived Elevation: Empiical Assessment and Eo Budget, Photogammetic Engineeing & Remote Sensing, Vol. 70, No. 3., ach 2004, pp Hodgson,.E., J. Jensen, G. Rabe, J. Tullis, B. A. Davis, G. Thompson, and K. Schuckman, An Evaluation of Lidadeived Elevation and Teain Slope in Leaf-off Conditions, Photogammetic Engineeing & Remote Sensing, Vol. 71, No. 7., July 2005, pp Latypov, D., Estimating elative lida accuacy infomation fom ovelapping flight lines. ISPRS Jounal of Photogammety and Remote Sensing, Volume 56, Issue 4, July 2002, pp Figue 8. Standad deviation of point positioning as a function of flying height and scan angle, all eos consideed. 4. CONCLUSIONS This pape pesented a compehensive analysis of the achievable point positioning accuacy of aibone LiDAR systems using igoous analytical deivations via eo popagation. LiDAR systems in eality ae athe complicated and consequently not all potential eo souces could be accounted fo in this analysis. Howeve, as all the majo eo souces wee consideed, theefoe, we believe that fo well-calibated LiDAR systems the deived fomulas povide a athe ealistic and eliable accuacy assessment when the accuacies of the consideed paametes that influence the point positioning accuacy ae easonably well known. Peng,. H. and T. Y. Shih, Eo Assessment in Two Lida-deived Datasets. Photogammetic Engineeing & Remote Sensing, Vol. 72, No. 8, August 2006, pp Schenk, T., odeling and Analyzing Systematic Eos in Aibone Lase Scannes. Technical Notes in Photogammety, vol. 19. The Ohio State Univesity, Columbus, USA. 46 p. Skaloud, J. and D. Lichti, Rigoous appoach to boe-sight self-calibation in aibone lase scanning. ISPRS Jounal of Photogammety & Remote Sensing Vol. 61, pp Using the deived eo fomulas, based on the accuacy of the navigation solution, the boesight misalignment angles, the anging and scan angle accuacy, and lase beam divegence, the achievable point positioning accuacy can be computed fo any given LiDAR system, opeated at diffeent flying heights, etc. The accuacy plots that wee deived based on the analytical deivations can be used as a tool fo choosing the ight system fo given application equiements, and to help with flight planning to decide on the optimal flying height, and maximum scan angle to achieve the desied point positioning accuacy. Due to the length limitations of this pape, the effects of individual eo souces on point positioning accuacy wee not analyzed hee, but the developed eo fomulas also facilitate that type of analysis. 111

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