HOIZONAL POSIION OPIMAL SOLUION DEEMINAION FO HE SAELLIE LASE ANGING SLOPE MODEL Yu Wng,* Yu Ai b Yu Hu b enli Wng b Xi n Surveying nd Mpping Insiue, No. 1 Middle Yn od, Xi n, Chin, 710054-640677@qq.com b Aerors Inc., Xi n, No. 1 Middle Yn od, Xi n, Chin, 710054-51149@qq.com, huyu@spcesr.com.cn,rlwngsdkd@16.com Commission I, WG I/ KEY WODS: Lser rnging; Fresnel diffrcion; Slope model; Echo signl; Horizonl posiion ABSAC: According o he Gussin-fi lser echo model nd he errin slope model, he regulr men vlue heorem nd he sympoic principle of he medin poin of he double inegrl men vlue heorem re used o derive he opiml soluion for he horizonl posiion of single-mode lser echo. hrough simulion experimens, he horizonl posiion resuls of he echo signl pek from vrious errin slopes re nlyzed. When ignoring he effec of he mosphere nd he surfce roughness of he rge, considering he geomeric posiion of he Gussin single-mode echo signl pek o be he cener of he lser spo is highly ccure. However, s he ccurcy significnly decreses when he slope is greer hn 6, mking he rnge of he pek vlue of he single-mode echo d (for slope of less hn 6 ) o be he rnge of he geomericl cener of he lser spo cn obin higher degree of ccurcy. 1. INODUCION Lser rnging echnology is new echnology which hs been widely used in erh sciences nd plnery explorion. Sellie lser rnging echnology involves he use of lser limeer inslled on he sellie, o mesure he disnce from he sellie o he ground, nd obin he ground elevion direcly or help n opicl cmer o chieve high-precision 3D posiioning (Wng, 014(); Wng, 014(b); Wng, 013). As resul of he fligh heigh nd he influence of lser energy, he lser bem dimeer is generlly greer hn 30 m on he ground, so i is impossible o obin dense poin cloud. he spo size lso deermines he upper limi of he plne resoluion. herefore, sellie-borne lser limeers re minly used in he mesuremen of high-precision 3D conrol poins, or ground conrol poin nework djusmen. For boh hree-dimensionl conrol poin mesuremen nd disnce mesuremen, he core problem is he mesuremen of he wo-dimensionl coordines of cerin number of poins, nd he corresponding rnge from he sellie o he ground. However, he use of lser echo d o clcule he ground poin locion is ypicl ill-posed problem h involves solving he hree-dimensionl posiion coordines hrough wodimensionl known condiions. As shown in Figure 1, during he cul mesuremen of he lser limeer, he received echo signl is n exension nd superposiion of he echo signl in he ime domin nd he spil domin of ech objec in he lser spo. Processing of he echo signl cn only deermine he heigh disribuion of he lser spo re nd cnno obin he cul rnge vlue of ech poin in he lser fooprin (Li, 007). Mos scholrs regrd he rnge vlue of he pek of he echo d s he rnge of he geomeric cener of he lser spo. However, his ssumpion is no rigorous, becuse he lser spo shows Gussin disribuion in he spce nd ime domins (Grner, 198), nd he energy is collecion of muliple energies wihin he rge s reflecion. If here re mny reflecors on he ground, nd he heigh difference of he ground objec is greer hn he resoluion of he lser rnging, he echo signl will show muli-pek shpe, so using he pek poin s he spo cener locion will resul in lrge error. Even if he ground is fl, he reflecion energy pek will be derived from muliple spos wihin he scope, so we cnno simply regrd he rnge of he pek s he rnge of he geomeric cener. In his pper, heoreicl model for he lser echo signl is firs nlyzed. Simulion experimens hen prove h when ignoring he effec of he surfce roughness of he rge, for slope of less hn 6, regrding he rnge of he echo d energy pek P (x, y) s he rnge of he geomeric cener G (x, y) is highly ccure; however, when he slope is greer hn 6, his ssumpion will resul in lrge error. Figure 1. Inversion of he rnge from echo d.. HEOEICAL ANALYSIS OF HE LASE ANGING OPIMAL SOLUION he lser echo is he superposiion of severl individul echo wves cused by he rge s reflecion from differen disnces. Becuse of he complexiy of he cul objec, he echo is muli-peked, nd i cn be decomposed ino single-pek shpe for nlysis. According o Fresnel diffrcion heory (Li,007) nd Grdner s heory (Grdner 198), he lser * Corresponding uhor doi:10.5194/isprsrchives-xli-b1-17-016 17
rnsmission equion shown in Eqs. (1) (4) cn be derived. he common diffuse-reflecion rge-objec model cn be bsrced ino hree cegories: sloped errin, ldder-shped errin, nd vegeion errin. he echo of he ldder-shped errin nd vegeion errin is muli-peked, nd he sloped errin is single-pek. When ignoring he influence of surfce roughness, he model cn be expressed s shown in Eq. (5). he definiion of he symbols used in Eqs. (1) (5) is shown in ble 1. (1) () (3) (4) (5) P r ( Power of he echo pulse signl A Are of he receiving elescope Amospheric enuion coefficien E Energy of he emied pulse z rck heigh of he rnge finder rnsmince of he opicl sysem r MS pulse widh Ndir ngle (bem scnning ngle) Lengh of he bem xis (ρ) efleciviy of rge in he spo ρ ( x 1, y1) rge cross-secion coordine vecor Lser wvelengh c Velociy of ligh in vcuum h (ρ) rge surfce profile in he scn coordine sysem Bem divergence he cener of he spo where energy is 1 S rge inclinion ngle prllel o he direcion of he // e fligh ph S rge inclinion ngle vericl o he direcion of he ( x 1, y1) he coordines of he ground objec in he plne XY fligh ph ble 1. Definiion of he symbols used in Eqs. (1) (5). h he ime is m when P r ( is he pek poin, hen r ( m ) 0 nd Eq. (6) cn be obined from Eq. (1): P, We cn see from Eq. (4) h is funcion of wo independen vribles ( x 1, y1). herefore, in he form of Eq. (6), we define: (7) (6) h Slope errin Puing he slope errin formul (Eq. (5)) ino he definiion of (Eq. (4)), we obin: (8) (9) Figure. he slope model nd is echo signl. he power of n echo signl is similr o Gussin funcion of n independen vrible. he funcion is coninuous nd hs z hmx z hmin pek vlue in he ime inervl [, ] c cos( ) c cos( ) (where h mx represens he heigh of he highes poin of he slope of he lser fooprin, nd h min represens he heigh of he lowes poin of he slope of he lser fooprin. Assuming Becuse ime hs no connecion wih he inegrl vrible, in he double inegrl of P () r, cn be regrded s consn, we regrd g( x1, nd f( x, y, 1 1 s binry funcions of independen vribles x 1 nd y 1. herefore: 1) A ny given momen, f( x, y, 1 1 nd g( x1, cn be regrded s coninuous funcions of nd ; ) As g( x1, is similr o Gussin funcion, γ(z)nd β(ρ, φ) re consn nd greer hn zero. We define D {( u, v ) 0 x x,0 y y} xy 1 1, nd when( x 1, y 1 ) Dxy g( x, y, 0 1 1, f( x, y, 1 1 nd g( x1, sisfy he condiion of he regulr men vlue heorem (Fn, 1978), nd conclusion cn be obined by he heorem direcly: Conclusion 1: les one poin exiss, mking: (10) doi:10.5194/isprsrchives-xli-b1-17-016 18
his mens h les one poin exiss h cn mke he firs order derivive equion of he lser rdr equion bsed on he slope model ( Eq. (10)) is esblished. By furher nlysis of Eq. (10), becuse g x, y, 0, ( 1 1 f ( x, y, m) x y hen (,, ) 0 0 g x 0 1 y1 d x d 1 y, so if P r ( m ) 0,hen 0. Puing 1 Eq. (9) ino Eq. (7), we obin: obined. Figure 3 shows he echo signl pek posiioning error disribuion when he vericl nd horizonl slopes re 16. he Z-xis is he error beween he rnge obined by he echo signl pek vlue nd he disnce from he sellie o he grid poin. king 60 poins s he opiml soluion for he minimum error, he opiml soluion of he echo pek vlue is disribued in srigh line he cener of he spo. (11) If limi nd exis, hen when α=1, When β=1, hen Furhermore, from he heorem bou he regulr men vlue poin (Ci, 005; Guo, 011), i cn be found h he regulr men vlue poin ( x, y ) sisfies he following formuls: (1) (13) By combining he expressions of (Eq. (7)) nd Eq. (10), we cn obin: (14) Furhermore, when from (Eq. (1)) nd Eq. (13), ge. Conclusion : When he lser re of he echo spo grdully reduces, he plne coordine corresponding o he pek vlue of he single-pek lser echo is close o he cener of he spo, x y h is x, y. herefore, in he cul mesuremen process, under he ssumpion h cerin moun of error is permied, he posiion of he pek vlue of he single-pek lser echo cn be considered o be he geomeric cener of he spo. 3 SIMULAION EXPEIMENS he prmeers of he simulion experimens were se ccording o he Geoscience Lser Alimeer Sysem (GLAS) sysem prmeers (Brenner, 000), which re shown in he ble : Prmeer z /km /nm /ns /mrd A /m Figure 600 1064 1 0.09 0.638 Prmeer E /mj (Degree) Figure 0.50 0.50 100 ble. Prmeer seings of he simulion experimens Figure 3. Pek posiion error disribuion when he slope model is 16. o nlyze he opiml soluions of he echo pek vlue of differen ypes of slope model, S // nd S were chnged from 0 o 30, nd he fl posiion error wih 47 ypicl slopes ws simuled, king 60 poins s he opiml soluion for he minimum error. Figure 4 shows he disribuion of he opiml soluion of he echo pek vlue for he differen slopes. Figure 4() is he resul when S // nd S chnge he sme ime, wih chnge inervl of 1. Figure 4(b) is he resul when he S // direcion chnge inervl is nd he S direcion is fixed 15. I cn be seen from Figure 4 h in ech kind of slope, he disribuion of he opiml soluion is srigh line, nd ll he srigh lines cross hrough he cener of he lser spo. (): Chnge in boh he horizonl nd vericl direcions he plne coordine rnge on he ground ws se s D xy {( x, y) 0 x 30,0 y 30}, (pproximely he size of he lser spo, nd he spo re fooprin ws divided ino 60 60 grid. Experimen 1: Using he echo signl, he corresponding rnge of he pek of he echo wve cn be obined. When he ground model nd sellie coordines re known, he disnce from he sellie o he ground grid locion cn be clculed. he wo ses of disnce vlues re mched in he rnge of cerin error, nd he corresponding muliple soluions ( x, y ) cn be doi:10.5194/isprsrchives-xli-b1-17-016 19
increses slighly, bu he vriion is very smll nd he overll error is less hn cm. However, he error increses o 10 cm bove 7. (b): Chnge in he horizonl direcion, wih he vericl direcion fixed Figure 4. Opiml disribuion of he echo signl pek vlue in differen slope grdiens. Experimen : In order o prove h he bes posiion of he lser echo signl pek vlue for he slope model is he geomericl cener of he spo, 4 poins round he cener of he spo were rndomly seleced o compre wih he wo norms, nd he resuls re shown in ble 3 nd ble 4. Y X 6 m 1 m 15 m 18 m 4 m 6 m.8384.91 1.970 3.7781.405 1 m 4.9189.8083 1.6447 0.9793 1.8476 15 m 3.3780 1.4 0.040 0.9775 3.196 18 m.4091 0.66 1.81.1108 4.54 4 m.0051 3.1309 4.1701 4.9763 7.0076 ble 3. Horizonl nd vericl chnge he sme ime, 5 poins of he wo norms. X Y 6 m 1 m 15 m 18 m 4 m 6 m 8.7006 6.0453 4.498 3.395 0.8043 1 m 5.8037 3.150 1.6101 0.5489.1946 15 m 4.1154 1.4690 0.560 1.314 3.8745 18 m.9116 0.3366 1.378.46 5.0791 4 m 0.880.6673 4.15 5.318 7.9743 ble 4. Horizonl chnge, wih he vericl direcion fixed, 5 poins of he wo norms As cn be seen from he bles, he furher wy from he cener poin, he higher he ccurcy, nd he lrger he vlue of he wo norms, he worse he posiion ccurcy. herefore, seing he rnge of he slope echo pek P (x, y) o be he rnge of he spo cener cn obin higher precision hn seing i o ny oher vlue. Experimen 3: In order o compre he influence of differen slope grdiens on he locion error of he spo cener, he locion error of he spo cener of slope ws compred, where he slope grdien ws chnged from 0-30 1 inervls (see Figure 5). I cn be seen from Figure 5 h when he slope chnges from 0 o 6, he rnge error of he spo cener Figure 5. he spo cener locion error for differen slopes. herefore, when selecing he rnge of he lser echo signl pek s he pproxime rnge of he cener of he spo, o ensure high ccurcy, i is bes o selec errin slope grdien of less hn 6, in boh he prllel nd vericl direcions o he fligh rck. 4 CONCLUSION In his pper, ccording o he sellie lser limeer echo signls nd he slope model, hrough heoreicl nlysis, i is shown h by reducing he spo size, he energy cener grdully pproches he geomeric cener of he spo. Using he GLAS sellie prmeers, he simulion resuls for slope d vrying from 0-30 show h he opiml soluions of he horizonl posiion corresponding o he echo signl pek ll cross hrough he spo cener. his indices h he posiion of he cener poin of he spo of he slope model cn be used s he posiion of he echo pek, nd higher precision cn be ensured wih slope of less hn 6. In he cul mesuremen process, he rnge of he cener poin of single-pek echo wve (for slope of less hn 6 ) cn be used s he rnge of he geomeric cener of he spo G (x, y). In his pper, he effec of errin roughness ws no considered, nd he simulion experimens were crried ou for slopes of less hn 30. Our fuure reserch will consider he influence of errin roughness nd errin slopes bove 30. ACKNOWLEDGEMENS his work ws suppored by he Nionl Nurl Science Foundion of Chin (projec 41571443). EFEENCES Brenner, A., Zwlly, H.J., Benley, C., e l,000. Derivion of rnge nd rnge disribuions from lser pulse wveform nlysis for surfce elevions, roughness, slope, nd vegeion heighs. Geoscience Lser Alimeer Sysem (GLAS) Algorihm heoreicl Bsis Documen. Ci, W., 005. Asympoic Quliy on Poin of Men Vlue for Double Inegrls. Journl of Shnghi Universiy of Elecric Power, (1), pp. 90 9. Fn, Y., 1978. Advnced Mhemics. High Educion Press, Beijing, Chin. Grdner, C.S., 198. rge signures for lser limeers: n nlysis. Applied Opics, 1(3), pp. 448 453. doi:10.5194/isprsrchives-xli-b1-17-016 130
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