Precise Computation of Position Accuracy in GNSS Systems

1 P RECI SE C O M PU T A T I O N O F P O SITIO N A CC URA C Y I N GNSS S YST EMS by Jua n Pa blo Bo y e ro Ga rrido ( jpbo ye ro @gm v . e s ) A BSTRA CT Ac cura cy a nd Av aila bility co m puta tio ns f o r a GNSS Sys t e m – or combina tio n o f Syste m s – thro ugh Se rvice V olu m e Sim ula tion s tak e co nside ra ble t im e . The re fo re , t he c o m puta tio n o f the accura c y in 2D a nd 3D a re o fte n sim plif ie d by an a ppr o x im ate so lution . The dra w ba ck is tha t such sim plif ic a t io ns ca n le a d to a cc ura cy re sults t ha t a re t o o c o nse rv a tive (up to 25% in the 2 D c a se a nd up to 43% in t he 3D ca se , fo r a 95% co nfide nce le ve l), which in turn tra nsla tes into pe ss im istic Sys tem A va ilab ilit y . This a rtic le pre se nts a wa y to co m p ute t he e x a c t a cc ura cy , fo r a ny co nfid e nce l e ve l, f o r the on e , two o r thre e dim e nsio na l c a se s, thro ugh the de riva t io n o f c o rre spo nding f a ct o rs. Us ing the fa c to rs int r o duce d he re , a llo ws ge tt ing a cc ura te re sults s w if tl y . The ge ne ric m a t he m a t ica l so lutio n t o co m pute the a c cura c y is p re se nted . Th e a ppro x i m a te a nd pre cise co m puta t io ns a re de scribe d. The n, the e x a c t fa ct o rs tha t sho uld be a pplie d to o bta in t he a c cura c y a t a t yp ic a l co nfide nce l e ve l ( 95%) a re de rive d fo r the t hr e e ca se s. 1. INTROD UCTION In a S a tellite Na viga t io n Syste m , the po sitio n e rro rs in the us e r do m a in ca n be c o m pute d f o r the o ne , tw o o r thre e dim e nsio na l ca se s, a nd the Av a il a bility o f the s ys te m is t he n e va l ua te d a s the pe rce nta ge o f tim e dur ing whic h t he a c cura c y is be t te r tha n the spe cifie d va lue . The s a m e is va l id f o r the v e lo c ity e rro rs. In Se rvice V o lum e Sim ula ti o ns , p o sitio n e rro rs in the us e r do m a in a t ea ch e po c h a re usu a lly m o de le d in e ve ry c o m po ne nt a s Ga us sia n ra ndo m va ria ble s. This de rive s, in turn, fro m the m o de l ing o f t he sa te llit e s ra nging a c cura c y e rro rs a s inde pe nde nt Ga uss ia n ra ndo m v a ria ble s. Sta rting fro m the c o m po ne nts o f the po sitio n do m a in e rro r a t a give n lo ca t io n a nd insta nt, the o ne , tw o o r t hr e e dim e nsio na l e rro rs ca n be co m pute d a t the s pe c ifie d c o nf ide nce lev e l. Fo r insta nce, so m e tim es the a c cura cy i n 3D is give n i n t e rm s o f S EP (Sp he ric a l E rr o r P ro ba ble ), whic h re pre se nts t he ra dius o f t he sph e re c o nta ining t he e rro r w ith 50% pr o ba bilit y . Ot he r tim e s it is m o re co nve nie nt to w o rk with dif fe re nt c o nfide nce le ve ls – 95 % is a no the r typica l va l ue . (se e [ 1] fo r a list o f comm on ly use d m e a sure s a n d the ir de finitio ns). In pra c tice, t he sim ula t io ns run t o c o m pute t he a cc ura cy a nd Av a ila bility fo r a GNSS Syste m – o r co m bina t io n o f S y ste m s - ta ke co nside ra ble tim e . Th a t is why the computa ti o n o f the a cc ura cy in 2D a nd 3D a re of t e n sim plifie d to re duce the c o m puta tio na l lo ad . The dra w ba ck is t ha t such sim plif icatio ns c a n le a d t o a c cura cy re sults t ha t a re t o o con se rva tive an d co nse que ntly im ply pe ssim is ti c re sults in t e rm s o f Syste m Ava ila bility . In t ur n, w o rkin g w ith pe ssimistic A va ilab ilit y re sults c a n ha ve a re le va nt im pa ct w he n de signing a Sys t e m , since im pro ve m e nt o f Ava ila bility r e quire s e ithe r to tra de o f f o t he r Sys te m pe rfo rm a nce pa ra m ete rs, o r to re- dim e nsio n so m e S y ste m co m po ne nts (su ch a s the gro und se gm e nt). 2 This ar tic le pre se nt s a wa y to c ompute t he e x a c t a cc ura cy fo r a ny con fide nce le ve l f or t he o ne , tw o o r thre e dim e nsio na l cas e s, thro ugh the de riva ti o n o f co rre sp o nding f a c to rs. Us ing the fa c to rs int r o duce d he re - a s loo k up ta ble fo r i ns t a nc e – a llo w s ge tt ing a c cura te re sults s w if t ly . Fi rst, the ge ne ric mathe m a tic a l so lutio n to c o m pute t he a cc ura cy is pre se nte d. The n, t he e x a c t fa c to rs tha t sho uld be ap plie d to o bta in the a c cura c y a t a give n co nf ide nc e le ve l (95%) ar e de rive d f o r the ca se s: m on o dim e nsio na l (single va lue ), bidim e nsio na l (c urv e ) a nd t hr e e dim e nsio na l (s urf a ce ). It i s fa ir to re c a ll t ha t the ge ne ric m a the m a tic a l so luti o n is als o pr e se nte d in [2]. For insta nce, the 2D ca se i s t re a t e d w ith gre a t de tail in Anne x D o f tha t re fe re nce. The p e rs pe ct iv e o f fe re d h e re is , ho w e ve r , dif fe re nt. In [2] t he de ve lop m e nt o f t he ge ne ric m athe m a tic a l so luti o n is o rie nte d in such a w a y tha t t he fina l inte gra l to o bta in a spe c if ic fa c to r ca n be so lve d thro ugh inte rpo la ti o n in co rre spo nding ta ble s. T a king a dva ntag e o f the c o m puta ti o na l po w e r n o w a va ila ble , t he de ve lo pm e nt m a de be lo w is o rie nte d to w a rds e x pre ss io ns whic h a re la t e r o n quickly so lve d thro ugh num e rical inte gra t io n. The i de a b e hind is to a llo w f o r t he fa st c o m puta tio n o f the who le ra nge o f e x a ct fa ct o rs ne e de d, f a ct o rs whic h c a n t he n be dire ct ly a pplie d t o GNSS Sys te m s s im ula t io ns, a nd which ca n a lso he lp in qua ntify ing the ina cc ura cies a ss o ci a te d t o t he sim plifica t io ns in usu a l sim ula tio ns. This is m or e e vide nt fo r t he 3D cas e, w he re ad ditio na l de ve lo pm e nt ha s be e n do ne he re , l e a ding to t he co m puta tio n o f the who le ra nge o f signif ic a nt fa c to rs (s urf a c e ), fro m whic h r e le va nt co ncl us i o ns h a ve a l s o be e n dra wn. 2. M ONODIM ENSION A L CAS E The m o no dim e nsio na l is the us ua l c a se fo r t he c o m puta tio n o f t he V e rtic a l a c cura cy . The po sition do m a in e rro r in the ve rtic a l dim e nsio n is de scribe d by a Ga uss i a n ra ndo m va ria ble wit h z e ro m e a n a nd sta nda rd de via t io n  . The sta nda rd de via t io n co m e s dir e ct ly fro m the third e le m e nt in the dia go na l o f the na viga t io n so lutio n cov a ria nce m a trix . He nce , the pr o ba bilit y de ns it y functio n o f the v e rtic a l er ro r ( x ) is: ) 2 e x p ( 2 1 ) ( 2 2    x x f   Eq. 1 As sh o w n in the ne x t figu re , the a c cura cy re l a ted t o a ce rta in con fide nce lev e l is the v a lue ( e ) fo r w hich the pro ba bility p(e) e qua l s t ha t le ve l.         e e dx x f e x e P e p ) ( ) ( ) ( Eq. 2 3 Figure 1: Pro babi li ty densi ty funct io n and acc uracy fo r a gi ven co nfi dence level in the mono dimensi ona l cas e The e rro r fun c tio n ca n the n be use d to co m pute the e x act fa c to r fo r the de sir e d c o nfid e nce lev e l : ) 2 ( ) ( ) (  e e rf dx x f e p e e      Eq. 3 Fo r e x a m ple , fo r a V e rtic a l a c cura c y w ith 95% c o nfide nce le ve l fo r , p=0.95  e =1.96  , the fa ct o r sh o uld be 1.96 a pplie d. 3. BIDIM ENSIONAL CASE The bidim e nsio na l is the usu a l cas e fo r t he computa ti o n o f the Ho rizo nta l a c cura c y . T he init ia l po int i s no w t he c o va ria nc e m a trix o f the po sitio n do m a in e rro rs in the ho rizo nta l pla ne , tha t is, the uppe r - le ft 2x 2 sub m a trix o f the na viga tio n so lut io n c o va ria nce m a t r ix . 1.1 A PPROXIM A TE CO MPUT A T ION IN THE BIDI MENSI ONAL CAS E T wo m ain line s a re us ua l ly fo ll o w e d fo r t he c o m puta tio n in t he 2D cas e . The sim ple st o ne c o nsis ts of con side ring o nly the e le m e nts in the dia go na l o f the m a t r ix . The Ho rizo nta l a cc ur a cy can be dire ct ly c o m pute d as t he squ a re ro ot the sum o f the tw o e le m e nts in the dia go na l. This is the sa m e o pe ra tio n pe rfo rm e d in the co m puta t io n o f t he HD OP , tha t i s , a ssu m ing UERE e qua l t o 1 m e ter fo r a ll s a tellite s, the Ho rizo nta l a c cura cy w o uld be e qua l to HD OP . A fa ct o r ca n the n be a pplie d to c o nve rt such v a lue into the va l ue cor re spo nd ing to the de sire d le ve l o f co nfid e nce . Sp e c if ic a lly , fo r a 95% co nf ide nce le ve l the f a ct o r ha bit ua lly a pplie d is 1.96. 4 The se con d li ne is i n turn ba se d o n the e i ge nv a lue s o f the 2x 2 submatrix . By c o m puting t he e i ge nv a lue s, the va ria nces o f the e rro rs in o rtho go na l d ire c tio ns a re fo und . Mo re o ve r , the l a rge st e i ge nv a lue co rre spo nds to the va ria nce o f the e rro r a l o ng the wo rst dire c tio n in the pla ne . The situa tion is de pic te d in Figu re 2 , w he re (x ´,y´) re pre se nt the dim e nsio ns t o w hic h t he co va ria nce m atrix is re fe rre d, while the dim e nsio ns (x ,y) re pre se nt t ho se re la tive to the e ige nv a lue s. Figure 2: E rro r el lip se in the ho rizo nt al pla ne The s qua re d e rro r in t he ho rizo ntal pla ne ca n the n be o btain e d a s 2 2 2 2 2 y x e   Eq. 4 W he re x a nd y a re Ga uss ia n ra ndo m va ria ble s w ith ze ro m e a n an d sta nda rd de via tion s  x a nd  y re spe ct ive ly . The y de te rm ine the e rro r e llipse , a s sho w n in Figure 2. At t his sta ge , in o rde r to sim plify the co m puta tio n o f the a cc ur a cy a t a give n le ve l o f c o nfide nce , an a ppro x im a tio n is no rm a ll y do ne . The ide a is t o a ss um e tha t bo th va ria nce s o f the e rro r in the tw o o rtho go na l dire c tio ns a re e qua l t o t he lar ge st o n e (e .g.  =  x ). W ith this c o nse rva tive s im plifica tion , the sq ua re d e rro r be c o m e s a Chi-Squa re d ra ndo m v a ria ble w ith 2 de gre e s o f fr e e do m (se e [3], p187) . The Chi-Squa re d d istributio n is a va ilab le in m o st m a the m a tic a l s o ftw a re to o ls. Alte rna tive ly , it ca n a lso be fo und in t a bula t e d fo rm . The re fo re , w ith the de scribe d a ppr o x im a tio n it is quite e a sy to ob tain the fa ct o r to c o nve rt t he va lue  x into t ha t c o rre spo nding to the de sir e d le ve l of co nfid e nce . S pe c ifica ll y , f o r a 95% co nfide nce le ve l the f a ct o r is 2.447. Thus , the us ua l metho d t o co m pute the Ho rizo nta l a c cura c y ba se d o n eig e nva lue s c o nsis ts of 1) co m puting t he e ige nv a lue s o f the 2x 2 s ubmatrix o f the na viga tio n so lutio n co va ria nce m a t rix ; 2) re tainin g the m a x i m um eig e nva lue , whic h c o rre spo nds t o t he va ria nce o f t he e rro r a lo ng t he w o rst 5 dire ct io n in the pla ne (  ); a nd 3) m ultiplying the sta nda rd de via tion by t he fa ct o r c o rre spo nding t o t he de sire d c o nfid e nce l e ve l ( e .g. 2.447  f o r 95%). The co m puta t io n is de picted in Figure 3 be lo w . Figure 3: Facto r to obt ain the erro r at a given co nfidenc e lev el fro m the l argest varia nce in a two dimensi onal Gaussi an dist ributi on 1.2 PRECISE COM PUT A TION IN TH E BIDIMENSIO NA L CAS E In o rde r to find the e x a ct v a lue link e d t o a giv e n c o nfide nce le ve l, we will s ta rt fro m the e ige nva lu e s o f the 2x 2 sub m a trix o f the na viga tio n so luti o n c o va ria nce m a trix . Since t he e ige nva lue s a re t he va ria nce s o f tw o inde pe nde nt Gau ss ia n distribu t io ns wit h ze ro m ea n, w e ca n fo rm the jo int dis tributio n j(x, y) a s: ) 2 e x p ( 2 1 ) 2 e x p ( 2 1 ) ( ) ( ) , ( 2 2 2 2 y y x x y x y x y f x f y x j             Eq. 5 The distribu t io n is sk e tc he d in Figu re 4. The l o w e r pa rt o f the figure sho ws incre a sing e llipse e rro rs , in line w ith the on e sh o w n in Figure 2 a bo ve . Fro m t his pe rspe ct ive , t he a ppro x im atio n o f a ssu m ing tha t bo t h va ria nce s o f the e rro r in t he tw o o rtho go na l dire c tio ns a re e qua l to the la rge st, can be se e n a s a su bstitution of t he a ct ua l jo int distributio n by a no the r o ne ha ving a m or e spr e a d pro ba bility de ns it y fu nct io n. Thro ugh c o rre spo nding do uble inte gra l, t he tw o dim e nsio na l join t distributio n a llo ws fo r the de riva t io n o f the pro ba bility o f the e rro rs a sso cia t e d to a spe c ific re gio n, p :      y x dy dx y x j p ) , ( Eq. 6 6 Pro viding tha t we a re inte re ste d in t he e rro r (e ) t ha t is no t s urpa ss e d wi th the give n le ve l o f con fide nce , the inte gra l in E q. 6 m ust swe e p the re gio n giv e n by a c ircle o f ra dius e . I n po la r co o rdina t e s, this re gio n is de fine d by  [0,2  ) a nd r  [0,e) , a nd the inte gra l is e x pre ss e d a s (a pply ing  c os r x  a nd  si n r y  ):          2 0 0 ) sin , c os ( ) ( e rd dr r r j e p Eq. 7 Figure 4: Joi nt di stri buti o n o f tw o independent Gaussi an rando m vari ables with zero m ean 7 In the pa rtic ula r ca se s whe re the sta nda rd de via t io n is the sa m e f o r bo t h dis tributio ns - m athe m a tic a lly  x =  y =  - the i nte gra l be co m e s            2 0 0 2 2 2 ) 2 e x p ( 2 1 ) ( e rd dr r e p Eq. 8      e rdr r 0 2 2 2 ) 2 e x p ( 1   ) 2 e x p ( 1 ) 2 e x p ( 2 2 0 2 2   e r e          This a ga in c o rre spo nds to the Chi-Squa re d distributio n w ith 2 de gre e s o f f re e do m , f ro m w hich e ca n be a na lytic a lly de rive d a s: ) 1 ln ( 2 ) 1 l n( 2 2 p p e         Eq. 9 Fo r 95 % co nf ide nce , p=0 .95  e =2.447  , the s a m e re sult pre se nte d a bo ve . In a ge ne ra l c a se , ho w e ve r , the integ ra l to be so lve d is                         2 0 0 2 2 2 2 2 ) sin c os ( 2 e x p 2 1 ) ( e y x y x rd dr r e p Eq. 10                          2 0 0 2 2 2 2 2 ) ) sin c os ( 2 e x p ( 2 1 e y x y x d dr r r                                                   2 0 0 2 2 2 2 2 2 2 2 2 ) ) s i n c os ( 2 e x p s i n c os 1 ( 2 1 d r e y x y x y x                                                       2 0 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 ) sin c os ( 2 e x p sin c os 1 2 1 sin c os 1 2 1 d e d y x y x y x y x y x The fir st te rm in the pr e vio us e x pre ssio n be co m e s e qua l to 1. One wa y t o se e this is by t a king a dva ntag e o f the fa c t tha t p(e) is a jo int distribu ti o n f uncti o n, a nd the re fo re , m ust be no rm a l iz e d to 1. M a the m a t ica lly this mea ns tha t is p →1 whe n e →  . Since , o n the o t he r ha nd, p(e) t e nds p re c ise ly to the e x pre ssio n in t he firs t te rm :                     2 0 2 2 2 2 sin c os 1 2 1 ) ( d e p y x y x Eq. 11 8 W e ca n thus con ve rt Eq. 10 into :                                    2 0 2 2 2 2 2 2 2 2 2 ) sin c os ( 2 e x p sin c os 1 2 1 1 ) ( d e e p y x y x y x Eq. 12 Le t ´ s no w intro duce a pa ra m e te r v = 2 2 / y x   , whic h is t he ra tio be t we e n the la rge st a nd s m a lle st va ria nc e s. Th is pa ra m e te r a llo w s w riting the e qua ti o n a s a f uncti o n o f jus t e/  x a nd v itse lf:                        2 0 2 2 2 2 2 2 ) sin (c os 2 e x p sin co s 1 2 ) ( 1 d v e v v e p x Eq. 13 Re c a ll tha t bo th  x (the la rge st sta nda rd de via ti o n) a nd v a re k no w n a s so o n a s the eig e nva lue s o f the co va ria nce m a t r ix a re a va ilab le , tha t is, fro m the be ginning of this p ro ble m . Fo r a give n v a lue o f p , the re so lutio n o f Eq. 13 a llows ob tain ing the fa c to r e/  x a s a f unct io n o f v . Fo r insta nce , f o r v =1 (which is the cas e whe n  x =  y ) a nd p=95, the fa c to r o bta ine d t hr o ugh this f o rm ula is, a s e x pe c te d, 2.447. Fo r t he re sults pre se nte d he re , t he fo rm ula ha s be e n so lve d num e rically by m ea ns o f Ma tlab (o w n co de ; se e a lso [4] fo r o nline av aila ble co de to c o m pute p tak i ng as inpu t the co va ria nc e m atrix ). Co nve rs e ly , the f a c to r by w hich t o m ultiply the large st sta nda rd de via t io n (  x ) ca n a lso be de riv e d a s a fu nct io n o f the in ve rse o f the s qua re r o o t pa ra m e ter v p re vio usly de fine d r =1 / √ v (a s we ll wit h 1/ v ). Tha t is, inste a d o f div iding the la rge st va ria nce by the s m a l le st o ne , dividing the s m a lle st sta nda rd de via t io n by the lar ge st o ne . Alt ho ugh this wa y m a y cre a t e so m e pro ble m s w ith the co m puta tio ns whe n the va lue o f the a bscissa te nds to ze ro ( c o m puta tio ns whic h c a n be m a de s e pa ra t e ly w ith the pa ra m e te r v if ne e de d), it ha s the a dva ntag e o f pre se nting a m o re co m pa c t plo t, since the ra nge r =1/ √ v go e s o nly fr o m 0 t o 1. Fo l lo w ing tha t li ne , Figure 5 dis pla ys the f a ct o r fo r the bidim e nsio na l cas e a s a fun c tio n of t he ra t io smalle st/la rge st sta nda rd de via t io n r , sp e cifically fo r the typica l 95% co nfid e nce le ve l (up), a s w e ll a s fo r o t he r pe rc e nta ge s (do w n). The c ur ve fo r the 50% con fide nce le ve l is include d in Figure 5, an d it ca n be c he c k e d tha t i t m a tc he s the curve in Figure 8 fro m [2], whe re the sa m e v a lue s a re c o m pute d using a dif fe re nt m e tho d (ge ne r a l inte gra l is de ve lo pe d in a dif fe re nt m an ne r so t ha t it c a n be so lve d by inte rpo l a tion of re le va nt t a ble s). Fro m Figure 5 it c a n be se e n a s we ll tha t t he e x a ct f a c to r fo llows f o r a ll pe rce nta ge s a fun c tio n w hich incre as e s m o no t o no usly wit h the r a tio s m a lle st/lar ge st sta nda rd de via tio n. 9 Figure 5: Facto r to m ul tipl y t he larges t st d to o bt ain the 2D erro r , as a func tio n of the ratio st dm i n/st dmax, for ty pica l 95% co nfi dence (up) and fo r se veral % (do w n) The e x a c t fa ct o r f o r the typica l 95% co nfide nce le ve l is co m pa re d in the ne x t f igur e a ga inst the fa c to r whic h m ultiplie s t he la rge st sta nda rd de via ti o n in the tw o simplifie d a ppr o a che s de scribe d a bo ve , na m e ly , the a ppro x im ate a ppro a c h w ith e ige nva lue s bu t using a uniqu e single fa ct o r (2.45 ); a nd t he a ppro a c h ba se d o n the squa re ro o t the sum o f the tw o e le m e nts in t he dia go na l o f the 2x 2 su bm a t rix m ultiplie d by 1.96. 10 Figure 6: Co m p ariso n of fa cto r multi plyi ng t he larges t st d to o bta in the 2D err or at 95% co nfidence as a functi o n o f the rati on st dmin/st dm ax It be comes visible in Figure 6 tha t the a ppro x im a te a ppr o a ch o f using a unique single f a c to r (2.45) to co m pute t he Ho rizo nta l a c cura cy c a n le a d to a consid e ra ble o ve re stim a t io n of the e rro r . Fo r insta nc e , t a king a n inte rm e dia t e po int , v =3 ( r =0 .58) tha t a ppr o a ch re pre se nt s a n e rro r incre m e nt o f 18.2% w ith re spe ct to t he e x a c t va lue (2 .45/2.07 = 1.182). The m a x im um o f the o ve re stim a tio n o c curs fo r the c a se w he n v →  ( r → 0) , tha t is , w he n the e rro r in o ne o f the d im e nsio ns is cle a rly do m ina nt, a nd the f acto r fo r p=0.95 t e nds t o 1 .96, t he o ne o btain e d in the m o no dim e nsio na l ca se . In tha t ca se , the o ve re stim atio n is e qua l to (2 . 45 -1.96) /1.96 = 25%. In t he sa m e m a nne r , t he a ppro a ch ba se d on the squa re ro o t t he sum of the tw o e le m e nts in t he dia go na l o f the 2x 2 subm a trix a lso le ad s to ov er e stim a t io n. W ith such a ppro a c h, the m a x im um o f the o ve re stim a t io n o c curs fo r the c a se whe n r →1 – the re is no do m ina nt dim e nsio n – sin c e 1.96  √(2  2 x ) = 2.77  x , w ith ov e re stim a tio n o f (2.77-2.45 )/2.45 = 13%. 4. THREE DIM ENSIONAL CA SE Fo r the thre e dim e ns io na l c a se it is quite sim ple to e sta blish a pa ra lle lism wi th t he tw o dim e nsio na l ca se , the s tar ting po int be ing t he c o va ria nce m a t rix o f t he po sitio n do m a in e rro rs in the thre e dim e nsio na l spa c e , tha t is, the uppe r -le ft 3x 3 s ubm a trix o f the po sitio n so lution c o va ria nce m atrix . The sa m e re a so ning is va lid fo r the ve locity e rro rs, pro vide d tha t the c o va ria nce m a trix f o r t he ve lo c ity so lutio n is b uilt in the sa m e way as t he o ne fo r the po sition so l utio n, t he o nly dif fe re nce be ing the use o f ra nge ra t e s inste ad o f UERE´s . 11 1.3 A PPROXIM A TE CO MPUT A T ION IN THE THREE DIM ENSIONAL CASE The sa m e two o pti o ns sho wed in the tw o dim e nsio na l ca se a re f o llo w e d in this ca se . The o ne co nside ring just the e le m e nts in t he dia go na l o f the m a trix c o m pute s the a c cura c y a s the squa re ro o t the sum o f the t hr e e e le m e nts (s im ila r t o t he P D OP co m puta t io n). The se c o nd o pti o n is ba se d o n the e ige nva lue s o f the 3x 3 su bm a t r ix . B y computing the e ige nv a lue s, the va ria nces of t he e rro rs in t hr e e o rtho go na l d ire ct io ns a re fo und. The s qua re d e rro r in the ho rizo nta l pla ne c a n be t he n o bt a ine d a s 2 2 2 2 2 2 2 z y x e    Eq. 14 W he re x , y a nd z a re Ga uss ia n ra ndo m va ria ble s wit h z e ro m e an a nd sta nda rd de via tion s  x ,  y a nd  z re spe ct ive ly . The y de te rm ine the e rro r ellips o i d, w he re the lar ge st e ige nva lue cor re spo nds to t he va ria nc e o f the e rro r a lo ng the wo rst dire c tio n in the thre e dim e ns i o na l sp a ce. Afte r do ing t he sa m e sim plif ic a t io n o f a ssu m ing tha t t he va ria nce s o f the e rro rs in the t hr e e o rtho go na l dire ct io ns a re e qua l t o the la rge st o ne , the squa re d e rro r be c o m es a Chi -Squa re d ra ndo m v a ria ble w ith 3 de gre e s o f f re ed o m . Fro m t he Chi-Squa re d distributio n, t he fa c to r c o rre spo nding t o t he de sire d c o nfid e nce lev e l can be o bt a ine d. He nc e , fo r t he 9 5% co nf ide nce le ve l, the fa ct o r fo r t he 3D spa c e be c o m e s 2.795. Co nse que ntly , the fo rm ula a pplie d f o r the er ro r is e= 2.795  (  be ing the la rge st e ige nva lue fro m the 3x 3 cov a ria nce m a trix ). 1.4 PRECISE COM PUT A TION IN TH E THREE DIM ENSI ONA L CA SE In o rde r to de rive the e x a c t a cc ur a cy va lue fo r a give n con fide nce le ve l in t he t hr e e d im e nsio na l ca se , the de riva t io n yie lding to E q. 5 ne e ds to be ex te nde d. W ith the thre e e ige nva lue s, re pre se nting the va ria nc e s o f t hr e e inde pe nde nt Ga ussia n distributio ns w ith ze ro m ea n, we can fo rm t he join t distrib utio n j(x,y , z) as : ) 2 e x p ( 2 1 ) 2 e x p ( 2 1 ) 2 e x p ( 2 1 ) ( ) ( ) ( ) , , ( 2 2 2 2 2 2 z z y y x x z y x z y x z f y f x f z y x j                   Eq. 15 No w it is a triple inte gra l t he o ne w hich so lve s the pro ble m . The pro ba bility p o f the e rro rs as so ciate d to a s pe c ific re gio n is de t e rm ine d by :        z y x dz dy dx z y x j p ) , , ( Eq. 16 T o illus tra te t he an a lo gy wi th t he bid im e nsio na l c a se , w e can wo rk o ut the c o unte rpa rt o f Eq. 8 , na m e ly , t he pa rticula r ca se whe re the s ta nda rd de via tion is the sa m e fo r t he t hre e distribu t io ns,  x =  y =  z =  . The j o int dis tributio n turns int o : ) 2 e x p ( 2 1 ) , , ( 2 2 2 2 3    z y x z y x j             Eq. 17 12 In po la r co o rdina te s (u sing   c os si n r x  ,   si n sin r y  a nd  c os r z  ), the inte gra l ha s to be do ne o ve r the po l a r va ria ble s  [0,  ] ,  [0,2  ) a nd r  [0,e ):                           0 2 0 0 2 2 3 sin ) 2 e x p ( 2 1 ) ( e d r rd dr r e p Eq. 18                 e e dr r r d r dr r 0 2 2 2 3 2 0 0 2 2 2 3 ) 2 e x p ( 2 4 ) 2 e x p ( 2 2               e dr r r 0 2 2 2 2 2 ) 2 e x p ( 2 4     Applyin g the c ha nge o f va ria ble 2 2 2 /  r t  , w hich im plie s 2 /  rdr dt  , a nd c ha nging the inte gra ti o n lim its a c co rdin gly ,      2 2 2 0 2 / 1 ) e x p ( 2 ) (   e dt t t e p Eq. 19 This inte gra l f a lls into t he c a t e go ry o f t he so calle d inc o m ple te ga m m a fun c tio n 1       x a dt t t x a 0 1 ) ex p ( ) , (  Eq. 20 In t his c a se , a=3/2; 2 2 2 /  e x  . The re fo re , t he pro ble m c a n be so lve d by find ing the va l ue fo r w hich p x 2 ) , 2 / 3 (    Eq. 21 The in c o m ple te ga m m a f unctio n is a s we ll im ple m e nte d in m o st m a the m a t ica l so ftw a re too ls (s uch as M a tlab ), o r c a n o t he rwise be fo und ta bula t e d. Thro ugh t he se m e a ns, it c a n be de rive d tha t fo r p=0.95 , the c o rre spo nding x  3.908, w hich in t ur n lea ds t o e  2.795  , in line with t he ab o ve re sult. In t he ge ne ra l thre e dim e nsio na l ca se , the triple i nte gra l in po la r coo rdin a tes ha s the fo rm :                                                                       0 2 0 0 2 2 2 2 3 sin ) c os sin sin co s sin 2 e x p ( 2 1 1 ) ( e z y x z y x d r rd dr r e p Eq. 22 T a king the dire c tio n of x as t he on e co rre spo n ding to the l a rge st e ige nva lue , tw o pa ra m e ter s m = x y   / a nd n = x z   / ca n be de fin e d, bo t h ra nging fr o m 0 to 1. No t ice tha t the dire c tio n of x is de t e rm ine d by the la rge st e ige nv a lue , an d the dire ct io n s y a nd z fo llo w a trire ct a ngula r trihe dro n a c co rding ly . 1 I n s o m e re fe re nce s (s e e [5], p26 0), t he ter m inco m ple t e ga m m a f unc tio n is a pplie d to the no rm a l iz e d fun c tio n          x a dt t t a a x a x a P 0 1 ) ex p ( ) ( 1 ) ( ) , ( ) , (  13 Applyin g the n the cha nge o f va ria ble x r t  /  , w hich im plie s x dr dt  /  , a nd cha nging t he inte gra tion lim its a c co rdin gly , the int e gra l be c o m e s:                                                        0 2 0 / 0 2 2 2 2 2 3 sin ) c os si n s i n c os sin 2 e x p ( 2 1 1 ) ( x e d d dt t n m t n m e p Eq. 23 It ca n be se e n t ha t the inte gra l de pe nds o n t he t wo pa ra m e ter s m , n , a s well a s o n the fa c to r x e  / . The fa c to r fo r p=0.95 ha s be e n num e rically fo und he re fo r e a ch c o m bina t io n o f m , n , t he o utc o m e be ing the figur e be lo w . Figure 7: Facto r to m ul tipl y t he larges t st d to o bt ain the erro r at 95% conf idenc e in a three di mensi onal Gauss ian di stri buti on Figure 7 sho w s t he fa ct o r by w hich the la rge st std ha s to be m ultiplie d in o rde r to find the ra dius co rre spo nding to the 95 % o f the e rro r in 3D. The fa c to r is pre se nte d as a functio n o f the ra tio s be tw e e n e ach o f the o t he r s tan da rd de via t io ns a nd the la rge st s tan da rd de via t io n. Fro m the Figure , it ca n be o bse rve d tha t the inte gra l is sy m m e t rica l in m , n . This im plie s tha t , o nce t he squa re ro o t o f the lar ge st e ige nva lue is t a ke n t o ge t x  , t he othe r tw o e i ge nv a lue s c a n be inter cha nge a bly se le c te d to o bta in m a nd n . 14 As a c ro ss che ck, w he n the pa ra m ete rs m an d n a re e qua l t o 0, whic h m ea ns t he re is o nly o ne re le va nt dimen sio n, the f a ct o r be c o m es 1.96, in line w ith t he m o no dim e nsio na l ca se . Co nve rs e ly , w he n the t hr e e sta nda rd de via t io ns a re e qua l ( m = n =1), the fa c to r be comes  2.795 ; in line wit h the a bo ve re sult. And whe n o ne of the s tan da rd de via tion s te nds to 0, the surf a c e re duce s t o a curv e w hich m a tc he s the o ne in Figur e 5. In te rm s o f o ve re stim a t io n, Figure 7 m a ke s co m pre he nsib le the dif fe re nce w he n usin g o nly the fa c to r de rive d in t he a ppro x im ate ap pro a ch. Fo r ins ta nce , the o ve re stim a t io n is in t he or de r o f 14% fo r c a se s w he re two o f the e rro rs a re sim ila r a nd do m ina nt c o m pa re d to the t hird o ne , tha t is m =1, n =0. (2.8/2.45=1.14) . The o ve re stim atio n o f the a c cura cy e rro r i n 3D c a n be a s high a s 43%, c o rre spo nding to the cas e s whe re the e rro r in o ne dire ct io n is do m ina nt wit h re sp e c t to the o the r tw o ( m = n =0 ; 2.8/1.96=1 .43). In the sa m e wa y , the a ppro a ch ba se d o n the squ a re ro o t the sum o f the t wo e le m e nts in the dia go na l o f the 2x 2 sub m a trix a lso le a ds to o ve re stim a t io n. W ith s uch a ppro a ch fo r t he 95% co nfide nce lev e l c o m puta ti o n in 3D, the m a x im um o f the o ve re stim a tio n o cc urs f or t he cas e whe n m = n =1 – the re i s no do m ina nt dim e nsio n – since 1.96  √(3  2 x ) = 3.395  x , w ith ov e re stim a tio n o f (3.39 5 -2 .795)/2.795 = 21 . 5%. 5. SUMM AR Y This a rtic le pre se nts a wa y to co m p ute t he e x act po sitio n a cc ur a cy fo r GNSS Sys te m s, fo r an y de sire d co nfide nce lev el. The re sults a re va lid a s well f o r ve lo c ity a c cura c y . In pra ct ice , the a cc ur a c y a nd Ava ila bility co m puta t io ns f o r a GNSS Sys tem a re do ne t hr o ugh S e rvice V o lum e Sim ula tion s, w hich ta ke a lon g ti m e , s o t ha t the co m p uta tion of the a c cura cy in 2D a nd 3D a re o fte n sim plif ie d to re duce the co m puta t io na l loa d. It ha s be e n sho w n he re tha t such s im plifica tion s ca n le a d to a c cura cy re sults tha t a re to o co nse rv a tive (up to a 25% in t he 2D c a se a nd up t o a 43% in t he 3D c a se ), a nd co nse que ntly im ply pe ss im istic re sults in te rm s o f Syste m Ava ila bility The ge ne ric m a the m a tic a l so l utio n to compute the a c cura c y fo r the o ne , tw o an d thre e dim e ns io na l ca se s ha s be e n de rive d - t he first t wo ca se s be ing no rm a lly li nk e d to the V er tic a l a nd Ho rizo nta l a c cura c ie s re spe ct iv e ly . Fro m t he ge ne ric mathe m a tic a l s o lutio n, t he ex a c t fa c to rs tha t ca n be a pplie d t o o bta in the accura c y a t a ny de sire d c o nfide nce lev e l m a y be co m pute d. The fa c to rs ha ve be en c o m pute d fo r the thre e cas e s: m o no dim e nsio na l (s ingle va lue ), bidim e ns io na l (curv e ) a nd thre e dim e nsio na l (s urf ace ), f o r a typica l co nfide nce l e ve l (9 5%). By us ing the se fa ct o rs, the a c cura cy – a nd co nse que ntly the Ava ila bility – o f a GNSS Sys t e m ca n be o bt a ine d in a f a st a nd cor re c t m a nne r . Alt ho ugh t he de t a ils o f t he im ple m e nta tion o f t he fa c to rs a re no t e x plicit ly discuss e d in this a rtic le , ta king f o r e x ample t he 2D cas e , w he re t he fa c to rs fo ll o w a curv e , e a sy i m ple m e nta t io ns can be de vis e d in the fo rm o f a lo o k-up t a ble , or in t he fo rm o f a functio n fitting the curve w hich c a n in t ur n be q uickly e va lua t e d. 15 REFERENCES 1. Fra nk va n Digge le n “GPS A ccura c y: Lie s, Da m n Lie s, a nd Sta tistic s” , GPS W o rld, No ve m be r 29, 1998 2. A e ro na utic a l Cha rt a nd Inf o rm a t i o n Ce nte r , USAF , T e chnica l Re po rt No . 96 “Principle s of Erro r The o ry a nd Ca rto gra phic Applica ti o ns” , Fe brua ry 19 62 3. A tha na sio s Pa po ulis, P ro ba bility , Ra ndo m V a ria ble s a nd Sto cha stic P ro ces se s, M cGra w -Hill, Se con d Edit io n. 4. Da vis , T . a nd K le de r , M . “Co nf ide nc e re gio n ra dius.”, M a thw o rks Ce nt r a l Fil e E x cha nge . ( http:// ww w .m athwo rks.co m /m a tlab c e ntra l/ f ilee x c ha nge /10526 -co nfide nce -re gio n- ra dius/co nte nt/ crr . m ), Ma rch 2008. 5. M. Abra m o w itz a nd I. A. Ste gun. Ha ndbo o k of m athe m a t ica l fun c tio ns w ith f o rm ula s, gra phs , a nd m a t he m a tic a l ta ble s. T e nth P rin ti ng , De ce m be r 19 72, w ith c o rre c tio ns