The Effect of UNBabc Mapping Function Modification To GPS Tropospheric Delay

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1 Mlysn Journl of Mthemtcl Scences 2(1): 59-71(2008) The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely 1 Hmzh Skdn, 1 Mohd Rzm Au Bkr, 2 Adul Rshd Mohmed Shrff, 3 Mohd Slm Md Noorn, 4 Ad. Nsr Mtor, 5 Azhr Mohmed, 6 Asml Ahmd 1 Dept. of Mthemtcs, Fculty of Scence, Unverst Putr Mlys 2 Fculty of Engneerng, Unverst Putr Mlys 3 Fculty of Scence nd Technology, Unverst Kengsn Mlys, 4 Fculty of Cvl Engneerng, Unverst Teknolog Petrons 5 Deprtment of Survey nd Mppng Mlys, Kul Lumpur 6 Fculty of Informton Technology nd Computer, Unverst Teknkl Mlys Melk. ABSTRACT Tropospherc dely refers to the refrcton of the Glol Postonng System (GPS) sgnl s t psses through the neutrl tmosphere from the stellte to the erth, whch cuses longer dstnce trveled y the sgnl. The zenth tropospherc dely cn e mplfed y mppng functon, especlly for less thn 5 degrees elevton ngle. Mny mppng functons hve een estlshed however the mppng functons gve lrge vlue when the elevton ngles less thn 5 degrees. A modfcton of UNBc mppng functon hs een proposed. The modfed UNBc mppng functon model shows sgnfcnt reducton of mppng functon scle fctor. As the coeffcent of the zenth tropospherc dely, the vlue of mppng functon wll ffect totl tropospherc dely. The modfcton UNBc mppng functon hs mproved the tropospherc dely up to 19.1 percent t two degree elevton ngles. Keywords: Tropospherc, mppng functon, zenth, modfcton INTRODUCTION The ssue of tmospherc dely (error) s extensvely nvestgted to mnmze the postonng error due to tropospherc nd onospherc dely. Tropospherc dely refers to the refrcton of the Glol Postonng System (GPS) sgnl s t psses through the neutrl tmosphere from the stellte to the erth. The effect cuses the dstnce trveled y the sgnl to e longer thn the ctul geometrc dstnce etween stellte nd recever. Tropospherc dely cn e dvded nto hydrosttc (dry) dely nd non hydrosttc (wet) dely s shown n Fgure 1.

2 Hmzh Skdn et l. Atmospherc dely. Tropospherc dely Ionospherc dely Zenth Hydrosttc dely (ZHD) x mppng functon Zenth wet dely (ZWD) x mppng functon Fgure 1: Atmospherc sgnl dely structure At zenth drecton, tropospherc dely contrutes out 2.5m (Ahn, 2005). Hydrosttc (dry) dely contrutes 90% nd wet dely contrutes out 10% of the tropospherc dely (Leck, 1995). Ths hydrosttc component hs smooth, slowly tme-vryng chrcterstc due to ts dependence on vrtons n surfce r pressure (wether cells). So ths prt cn e modeled nd removed wth n ccurcy of few mllmeters or etter usng surfce model (ncludng pressure, temperture nd humdty). It does not therefore crete much of prolem s fr s ts effect on GPS sgnls (El Rny, 2002). Although wet dely s much smller thn the hydrosttc component ut the uncertntes n wet tropospherc dely modelng do plce gret urden on hgh precson GPS pplctons. Referrng to Fgure 2, the tropospherc dely s the shortest n zenth drecton (when the stellte t P) nd ecomes lrger wth the ncrese of zenth ngle (t Q). P Q ZD z TD Fgure 2: Illustrton for olquty fctor (mppng functon) etween zenth nd slnt drecton Projecton of zenth pth delys nto slnt drecton s performed y pplcton of mppng functon or olquty fctor, m(z) tht s defned s (Schuler, 2001): 60 Mlysn Journl of Mthemtcl Scences

3 The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely TD m( z) = (1) ZD where : TD totl tropospherc dely (slnt neutrl dely), ZD totl zenth dely, ε - elevton ngle (from ground to stellte). z - zenth ngle (z = 90 -ε ) From Fgure 2 ove, ZD TD = cos z (2) By susttuton nto (2), 1 m( z) = = sec z (3) cos z Unfortuntely, ths secnt model s only n pproxmton ssumng plnr surfce of the erth nd not tkng the curvture of the erth nto ccount (Sstmonen, 1972). The tropospherc delys (TD) t rtrry elevton ngles cn e expressed n terms of the zenth delys (ZD) nd mppng functons m(z) s gven elow: TD = ZD mppng functon TD = ZD mppng functon (4) However, ZD cn e seprted nto two components such s hydrosttc component (zenth hydrosttc dely, ZHD) nd wet component (zenth wet dely, ZWD). TD = ( ZHD + ZWD) m( z) (5) In some cses, the mppng functons for wet nd hydrosttc components re dfferent. Ths representton llows the use of seprte mppng functons for the hydrosttc nd wet dely components (Schuler, 2001): Mlysn Journl of Mthemtcl Scences 61

4 Hmzh Skdn et l. TD = ZHD m ( ε ) + ZWD m ( ε ) (6) where : h w ZHD s zenth hydrosttc dely (m) ZWD s zenth wet dely (m) m (ε) s the hydrosttc mppng functon (no unt) h m (ε) s the wet mppng functon (no unt) w The ojectves of the study re to modfy the model of UNBc mppng functon nd lso to nvestgte ts effectveness y clcultng the tropospherc dely to get the mprovement for hydrosttc dely model nd lso non hydrosttc dely model sed on the orgnl UNBc mppng functon. UNBc MAPPING FUNCTION UNBc s tropospherc mppng functon estlshed y Jmng Guo from the Unversty of New Brunswck (Guo, 2003) usng ry trcng dely vlues, whch hs 3-terms contnued frcton form s shown n equton [7] elow: m c ( ε ) = (7) c where = h or nh ε - elevton ngle (90 z) mh - hydrosttc mppng functon mnh - non hydrosttc mppng functon 62 Mlysn Journl of Mthemtcl Scences

5 The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely From seres of nlyses, prmeter s senstve to the orthometrc heght, nd lttude, φ of the stton wheres prmeters nd could e represented y constnts. The lest-squres estmted prmeters for the hydrosttc functon re: h = ( = h c = h cos φ)/1000 nd for the non-hydrosttc component: nh = ( cos φ)/1000 = nh c = nh In ths study, modfcton of UNBc mppng functon hs een crred out y only focusng on the denomntor of equton [7], wherey the thrd sne functon term to e chnged wth cosne functon whle the rest of the mppng functon remn unchnged. The clculton of reducton percentge wll show the mprovement of the modfed mppng functon. Both current nd modfed mppng functons wll e multpled y zenth tropospherc dely vlue to get the hydrosttc nd non hydrosttc components dely. The summton of the hydrosttc nd non hydrosttc components wll gve the totl tropospherc dely n meter for oth the current nd modfed tropospherc dely. As the coeffcent of the zenth tropospherc dely, the mppng functon vlues should e smller n order to get smller tropospherc dely. The scle fctor vlue t 90 degrees elevton ngle should e unty. FORMATION OF MODIFIED UNBc MAPPING FUNCTION The thrd sne functon hs een selected to e chnged for oth components due to ts smller mppng functon vlue compred to the orgnl mppng functons, s gven n equton [7]. The sne, sne nd cosne comnton of mppng functon s now clled modfed UNBc(h) Mlysn Journl of Mthemtcl Scences 63

6 Hmzh Skdn et l. mppng functon for hydrosttc component nd modfed UNBc(nh) mppng functon for non hydrosttc component, s gven n equton [8] elow: m c = cosε + c ( ε) (8) wherey ll the other terms remn unchnged s gven n equton (7). MATHEMATICAL JUSTIFICATION FOR CHANGING FROM SINE FUNCTION TO COSINE FUNCTION The UNBc mppng functon models for oth hydrosttc nd non hydrosttc re gven s follows: It s true tht π cos ε > sn ε for 0<ε < 4 cos ε + c > c when c s constnt, < when s constnt, cosε + c c <, cosε + c c cosε + c > c when s constnt, >, cosε + c c 64 Mlysn Journl of Mthemtcl Scences

7 The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely A A < cosε + c c when A s constnt If A = 1 +, c so tht equton (6) < equton (7) s gven elow: c cosε + c < c c s lwys true for 0<ε < 4 π Equton (8) cn gve smller vlue thn equton (7) for 0<ε < 4 π. So, equton (8) hs een used s modfed UNBc mppng functon for oth hydrosttc nd non hydrosttc models. Equton (8) cn gve smller vlue thn equton (7) for 0<ε < 4 π. So, equton (8) hs een used s modfed UNBc mppng functon for oth hydrosttc nd non hydrosttc models. Mlysn Journl of Mthemtcl Scences 65

8 Hmzh Skdn et l. COMPARISON BETWEEN THE UNBc MAPPING FUNCTION AND MODIFIED UNBc MAPPING FUNCTION To compre the effectveness etween the UNBc nd modfed UNBc mppng functon, the scle fctor vlues wll e clculted to show the reducton for oth mppng functons s descred elow. Hydrosttc component Modfed UNBc(h) mppng functon s gven n equton [8] s compred wth UNBc(h) mppng functon s gven n equton [7] s shown n Fgure 3 elow: Mppng functon scle fctor Elevton ngle, E (degree) UNBc(h) Modfed UNBc(h) Fgure 3: Comprson etween the UNBc(h) nd the modfed UNBc(h) comnton From Fgure 3, the modfed UNBc(h) grph shows tht t the elevton ngles of less thn 5 degrees, ts scle fctors re much smller. For elevton ngles more thn 5 degrees, ts scle fctors re qute smlr to the orgnl UNBc(h) model. However, t elevton ngle of 90 degree, ts scle fctor s unty. Non hydrosttc component Modfed UNBc(nh) mppng functon s gven n equton [8] s compred wth UNBc(nh) mppng functon s gven n equton [7] s shown n Fgure 4 elow: 66 Mlysn Journl of Mthemtcl Scences

9 The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely Mppng functon scle fctor Elevton ngle, E (degree) UNBc(nh) Modfed UNBc(nh) Fgure 4: Comprson etween the UNBc(nh) nd modfed UNBc(nh) From Fgure 4, the modfed UNBc(nh) shows tht t the elevton ngles of less thn 5 degrees, ts scle fctors re much smller. For elevton ngle more thn 4 degree, ts scle fctors re qute the sme wth UNBc(nh) model. However, t the elevton ngle of 90 degrees, ts scle fctor s unty. IMPROVEMENT OF TROPOSPHERIC DELAY (TD) The mprovement of tropospherc dely cn e shown y comprng the tropospherc dely vlue. The clculton for tropospherc dely wll use equton (1) s gven prevously. The vlues for ZHD = 2.31m nd ZWD = 0.25m cn e clculted from Sstmonen Tropospherc dely model (Sstmonen, 1972). Tropospherc dely usng UNBc mppng functon The tropospherc dely clculton for oth hydrosttc nd non hydrosttc components cn e shown n Tle 1 elow: Mlysn Journl of Mthemtcl Scences 67

10 Hmzh Skdn et l. TABLE 1 : Tropospherc dely (TD) usng UNBc mppng functon Elev. ngle, E ( o ) A = UNBc(h) x ZHD B = UNBc(h) x ZWD TD=A+B (meter) Tle 1 shows tht the elevton ngle strts from 2 degree untl 90 degree. At 90 degree, the tropospherc dely gve the stndrd vlues (2.31m for hydrosttc nd 0.25m for non hydrosttc) due to the vlue of mppng functon for oth components t 90 degree s unty, 1. Tropospherc dely usng modfed UNBc mppng functon The results of clculton of tropospherc dely for UNBc(h) for oth components cn e shown n Tle 2 elow: TABLE 2 : Tropospherc dely (TD) usng modfed UNBc(nh) mppng functon Elev. ngle, E ( o ) C = modfed UNBc(nh) x ZHD D = modfed UNBc(nh) x ZWD TD = C + D (meter) Tle 2 shows tht the elevton ngle strts from 2 degree untl 90 degree. At 90 degree, the tropospherc dely gve the stndrd vlues (2.31m 68 Mlysn Journl of Mthemtcl Scences

11 The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely for hydrosttc nd 0.25m for non hydrosttc) due to the vlue of mppng functon for oth components t 90 degree s unty, 1. Improvement percentge for Tropospherc Dely (TD) The mprovement of tropospherc dely cn e shown y comprng etween modfed UNBc mppng functon nd UNBc mppng functon. As coeffcent to the zenth dely, the mprovement n mppng functon wll drectly ffect the totl tropospherc dely. Current TD s clculted usng UNBc mppng functon nd New TD s clculted usng modfed UNBc mppng functon. The mprovement of the tropospherc dely cn e shown n Tle 3 nd Fgure 5 elow: Tle 3 : Percentge mprovement for the tropospherc dely (TD) Elevton ngle, E Current TD (meter) New TD (meter) Improvement (%) Tropospherc dely,td (m) Elevton ngle, E (degree) Current TD (meter) New TD (meter) Fgure 5 : Comprson for the current nd the new tropospherc dely (TD) Mlysn Journl of Mthemtcl Scences 69

12 Hmzh Skdn et l. Tle 3 shows tht the elevton ngle strts from 2 degree untl 90 degree. For the elevton ngles of more thn 5 degree, the dfferent tropospherc dely etween oth models s too smll nd not sgnfcnt. Fgure 5 shows tht the dfference etween the current tropospherc dely nd new tropospherc dely cn e seen when the elevton ngle s less thn 5 degree. CONCLUSION For the elevton ngles less thn 5 degrees, the reducton percentge of the tropospherc dely s very ovous, especlly for two degrees elevton ngles. On the other hnd, the reducton shows the sgnfcnt mprovement of the mppng functon, ether for hydrosttc or non hydrosttc mppng functons. As the mplfer effect (coeffcent) to the zenth tropospherc dely, the mprovement cn drectly reduce the totl tropospherc dely of the GPS sgnl. Modfed UNBc mppng functon contrutes 19.1 percent to the tropospherc dely mprovement percentge t two degree elevton ngle. The results show tht the modfed UNBc mppng functon s sutle choce to e n lterntve mppng functon, for replcng the orgnl UNBc mppng functon for oth components, hydrosttc nd lso non hydrosttc. REFERENCES Ahn, Y.W Anlyss of NGS CORS Network for GPS RTK Performnce Usng Externl NOAA Tropospherc Correctons Integrted wth Multple Reference Stton Approch, M.Sc. Theses, Unversty of Clgry, Cnd. El-Rny, A. 2002, Introducton To GPS: Glol Postonng System. Artech House mole communcton seres, Boston, London. Guo, J nd Lngley, R.B A New Tropospherc Propgton Dely Mppng Functon For Elevton Angles Down To 2. Proceedng of ION GPS/GNSS 2003, 16 th Interntonl Techncl Meetng of the Stellte Dvson of The Insttute of Nvgton, Portlnd, OR, 9 12 Sept., pp Mlysn Journl of Mthemtcl Scences

13 The Effect of UNBc Mppng Functon Modfcton To GPS Tropospherc Dely Leck, Alfred GPS Stellte Surveyng, 2 nd Edton, John Wley & sons, 1994, USA. Sstmonen, J Atmospherc correcton for troposphere nd strtosphere n rdo rngng f stelltes, Geophyscl monogrph, 15, Amercn Geophyscl Unon, Wshngton, D. C., USA, pp , Schuler, T On Ground-Bsed GPS Tropospherc Dely Estmton, PhD Thess, Unversty of Munch Mlysn Journl of Mthemtcl Scences 71

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