On Error Sources and Mathematical Models for use with GPS

4.9 Lecture Notes For C.I.S. GPS Seminars (Calgary 7 May; Edmonton 28 May; Fredericton, 15 June, 1985) On Error Sources and Mathematical Models for use with GPS Demitris Delikaraoglou Geodetic Survey and Canada Surveys and Mapping Ottawa, Ontario 2. CPS OBSERVABlES The realization of the vast potential of GPS to positionIng very precisely points In regional areas has led to many techniques that Ｎｮ｡ｾｬ･ the geodetic GPS users to use the GPS signals for extending geodetic control and monitoring crustal movements. One of these approaches usually ,.ef.red to as the ,.econstructed crrled phase Is particularly suItable for using the GPS sl8nals for geodetic purposes. The explOration of this concept has been letlvely pursued by CGS. In this section, In order" to place the overall concept In context, .e shall review briefly the structure of the GPS signals and give an overview of the conventional code Vis-i-vis carried phase measurement pl"'ocedures. For complete details, the reader should refer to e.g. Spilker (1978), Bossler- et al. (1980), Hul (1982), Wells et ale (1980). 2.1 Overview of the GPS Signal Characteristics The GPS signals are characterized by a numbel" of components al I of which are based on the fundamental frequency f In :0: 10.23 MHz of the satellite atomic clocks. particular, the satellites emit signals at two-carrier frequencies (to pel"mlt the meesurement of the Ionospheric effect): LI et WI = 154 x 10.23 MHz = 1575.42 MHz L2 et W2 = 120 x 10.23 MHz :0: 1227.60 MHz. These carriers have wavelengths of 19.05 and 24.45 em respectively. Each of these two cerrlers has severe I kinds of modulation superimposed upon It. completed structure of the first slgnel LI as described e.g. In Spilker (1978) Is: The 2 Where BCt) denotes noise, usually modeled as a sum of phase noise ｾＨｴＩ drift 6wlCt) )( and oscillator t. The eosine wave of the carrier Is modulated by a pseudorandom sequence Plct) (+1 or -1) functions known as the P-code CP for "precise" or "protected"). of step It Is a sequence generated by an 211 gar Ithm that repeats Itsel f every 267 days IMllllken and Zoller, 197B).Each seteilite Is assigned a different 7-day segnent of this 267-day period, and generates the P-code sequence associated with Its asslgnnent segment. At each Saturday/Sunday midnight of Universal Time CUT), these sequences are reset to the start of the assigned 7-day segments. The frequency of the P-code Is 10.23 MHz, that Is each 154th carrier wave of II sees a new modulating pulse. The wavelength of the P-code Is 29.31 m. The sine wave of the carrier Is modulated by a unique sequence Glct) of step called the CIA-code (CIA for "coarse" or "clear access"). GICt) Is generated functions In each seteilite by a Gold's algorithm (Gold, 1967) that repeats Itself every mi I II second. During 1 ms a string of 1023 step functions Is generated at a frequency ,of 1.023 MHz. ( Thus every 154Qth carrier wave of II sees a new modulating pulse giving the CIA-code a wavelength of 293.1 m. Clearly a failure to identify the proper millisecond of the carrier arrival creates an error In the range determination of k )( 300 km, where k Is an Integer. Both the cosine and sine waves are, low-frequency stream of data 0, ct) In addition to the two codes. modulated by a containing Information on the satellite position Cephemerls). the satel lite clock correction, Its state of health. information about the other satellites of the system. and the "hand-over word" CHOWl designed to assist the Initiated user to use the P-code (van Olerendonck et 211., 197B]. DICtl Is a sequence of +1 and -1 step functions with a frequency of 50 bits a second. The sine and cosine waves of the carrier have ｴｮ･ｾｦ ｬ､ amp I Itudes ｾＬａｰＮ wave, carrying the CIA-code. Is 3 to 6 cfl stronger than the cosine wave. CIA-code should be received with significantly greater slgnal-to-nolse ratio. The sine Thus the Currently, the second sIgnal L2 consIsts of only the cosine wave of the carrier, I.e. L2(t) • Bp PI (t) cos (/U2t + elt) I that Is, It normally cerrles only the P-eode. civilian users to the P-code wIll It Is likely however that eccess of be denied once the operational constellatIon of satellItes Is In place around' 1988. 2.2 Code vs CarrIer Phase Measurements In standard GPS navigation receIvers, the Incoming signals are cross-c:orrelated wIth similar reference sIgnals generated by the receIver. The latter consIst, In the case of P-code, of slnusolds which are phase modulated by either O· or 180· accordIng to the The CIA modulations are In quadrature, that Is they receiver code generator output. are represented by 90· or 270· phase values. sIgnal and the IncomIng satellite signal required to align Is 15 Is detrmlned by fInding the time shift (correlate) the replIca of the receIver generated code with the Incoming GPS code. light The tIme delay between the reference The tIme delay thus determined when III.Iltlplled by the speed of measure of the slant range to the satellite often called pseudorange. Pseudorenges change due to variations In the ｳ ｡ ｴ ･ ｬ Ｑ ｴ ｾ ｲ ･ ｣ ｬ ｶ ･ ｲ propogatlon delay and are biased by the time offset between the satellIte and receIver clocks. the resolution of pseudorange measurements depends on the accuracy with whIch the Incoming and replica codes can be aligned. few nanoseconds are poss I b Ie. range resolutIon. Clearly Pseudorange measurement resolutions of 15 Hence the measurements are sensitIve to meter-level The dIfference In pseudorange for the Ll and L2 channels Is used to estlmete the fIrst-order IonospherIc contribution to the Ll renge. Users of the P-code slgnels can determIne theIr position wIth respect to the GPS reference system WGS-72 <I.e. the reference system of the GPS orbits) end the offset of their receiver clock from GPS system time by combining the P-code pseudorenge InformatIon for four or more setel lites end correcting for IonospherIc end troposherlc effects on the time deley. Alternetlvely cerrler slgnels. continuously receivers. I.e. to code meesurements geodetIc users of GPS cen make use of the GPS CarrIer phase meesurements Integrated· Doppler count wIth GPS meesurements are essentIally sImilar to mede by TRANSIT setell ite To cerry out this mode of operetlon, the cerrler hes to be reconstructed, get rid of the superimposed P- end C/A- codes by predicting them within the 4 rece Ivel", _tch Ing them through cross-corre let Ion end subtrect i ng them from the rece Ived s Igne I. t \Il1e-de Ieyed The dIf ference of the Incom Ing sete I II te 5 Igne I end. the proper Iy rece Ivel" reference 5Igne I Is then essent I21' Iy 21 sine weve beet frequency. The 180· phese sh Ifts, In th Iseese, cencel out except for the low frequency det 21 IlIOdu Ieted on the cerrler end whose presence Is countered by 21 specie I (Costes) loop thet neutrellzes theIr effect. Aft... removIng the dete lll8ssege, 21 low frequency slgnel Is genereted In the receiver whIch Is phese-Iocked to the beet frequency to Improve the slgnel-te-nolse retlo. It Is this 51gnel which Is often celled the reconstructed cerrler, even though It Is et 21 lower frequency then the orIginal cerrler. 21 Clearly If the trenSl:llltter remelns et fIxed dlstence from the receiver, the phese of the IncomIng slgnel remaIns constent. VIewed from thIs perspective, It Is evIdent thet Its phese chenges wIth the chenge of the renge between the seteillte end receIver. The phese chenges by one cycle whenever the sate I IIte-te-recelver dlstence chenges by one wevelength of the cerrlerfrequency. Th Is type of meesurement Ieeds potent 1211 Iy to the IIlOst precl se Informet Ion ebout the seteillte-te-recelver renges one can obtain. however Is one of emblgulty: The problem with utilIzing this potent I 211 It Is very difficult to locete eccuretely the cycle of the cerrler whose phase Is beIng meesured. phese chenges meesured to en eccurecy of With the present evelleble Instrumentetlon, 21 frectlon of 21 cycle ere possible. cerrler phase meesurements ere sensitive to sub-centimeter renge chenges. Hance However the success of echlevlng this level of posItIonIng eccurecy from cerrler phese meesurements hInges on the cepeblilty to resolve the forementloned emblgulty. We shell dwell on epproeches for resolving thIs emblgulty on the GPS s!gnals In section 3. GPS Error Sources The eccurecy of posItIons obtelned by GPS Is dependent on two generel Inf luences: the errors effecting the the meesurements themselves end the geometric strength of satel lite confIguration being observed. Generelly, the errors thet Influence the GPS measurements fell Into three cetegories those essocleted wIth the setel lites, wIth slgnel propegatlon, end with the receiver. Errors from eech of these sources wll I probebly heve complIcated spectre I properties, 5 . . ." and there will be.correlatlons between some of these errors. However at this stage In GPS development, the error IllOdeis ere 11_1'ted to the siliple epproech of predicting typical standard deviations ofuneorreletedequlvalent renge errors frOlll -.ach error source. Satellite errors consist of errors In the ephemeris (the seteilite Is not where the GPS data message tells us It Is),' and errors In the clock (the seteilite clock perfectly synchronized to "GPS system time"). Is not These seteilite errors are uneorrelated between satel lites, they affect eocIe and carrier phase ..asur_nts equally, end they depend on the nllllber and the fOeation of the tracking stations providing data for orbit determination, the orbital force lIIOdei algorlthlll used, and the sittell Ite constellation geometry (Fell, 1980]. Detailed descriptions of GPS seteilite errors can be found In Schalbly (1976), and Schalbly and Harkins (979). Propagation errors consist of the Ionospheric refraction error, tropospheric refraction error, and multlpath error. COde measurements are subject to Ionospheric group delay, and carrier phase measurements to Ionospheric phase delay. Ionospheric range effect may satellite near horizon) zenith). can be to vary At GPS frequencies, the fran more than 50 m (sunspot maximum, midday, less than a metre (sunspot minimum, night, sate I lite at Since Ionospheric refraction Is frequency dependent, Ll and L2 measurements ､･ｲ｡ｾ｣ to est Imate the ef feet; however, no I se on each of these measurements will propagate Into errors In the Ionospheric refraction estlmete. two It has been shown that the standard deviation of this dual-frequency estimate Is three times the standard CIA-code devl atl on of measurements, the measurement no I se JMart In, thIs Is Important since noise 1978). on 'n the CIA-code case of pseudorange me8surements prevents their use In effectively determining the Ionospheric refraction r8nge effect using the dU81 frequency technique. Also, at the present time only the P-code Is transmitted on the L2 frequency, so th8t dual frequency C/A-eode me8surements are not poss I b I e. mode I s of the For CIA-code measurements the a I ter natl ve I s to use Independent loncsphere. Pred I ctl ons of I onospherl c refract Ion range errors from present day Ionospheric models are 'unllkely to be more thon 75 percent effective, so that for an Ionospheric effect of 50 m, residual errors of up to 15 m may remain. 6 Troposherlc refraction delays vary from about 2 m with theseteilite et zenith, to about 25 m with the satellite at 5- elevation. weather .esurements ｡ｮｾ This delay must be modeled from surfece . . vertical profiles of refractivity must . be assumed to be known. A scale bias In the tropospheric refraction model (typically 4 percent of the tote I tropospheric deley), results In a contribution of • to the tote I GPS range error. Multlpath effects depend lIIelnly on the geometry of the satellite, antenne characteristics and surrounding reflecting sur feces .end are probably a very Irregular function of time. For code meesurements, there Is an additional dependence on the code wavelength so thet the CIA-code effect Is larger than the P-code effect. Indlcetlve rms error velues taken from Mertln (978) end Hul (982) Some for the verlous error Influences on the GPS code end carrier phese l118asurements ere given In Tebles 2.1 end 2.2. 9 2.4 Absolute vs Reletlve GPS Positioning Using GPS. both point end ･ｶｬｴｾ In· the seteilite positions. unc:er"teli'ltles ､･ｳｾｬ｢･､Ｎ deleys Just . "bso Iute" . ･｣ｵｾｬｳ positioning ･ｾ trensmlsslon _dl um likely be Obtelned with Ie. 1979; Senus end Hili. 1981). of the geodetic end geodynemlc COlnJnlcty, seteilite end slmultene.ously ｴｾ･｣ｫｬｮｧ to ｉｭｰｾｯｶ･ end point positioning will to use GPS In e different I e I .xIe. slgnels. of the . positioning ｾ ･ ｱ ｵ ｬ ｾ ｟ ｮ ｴ ｳ ｮ･｣ ｳ ･ｾｹ ｢･ｨ ｶｬｯｵｾ clock not .uch better then ebout 50 em ｾ･､ｮａ｛ . ｾｳｴ ｈｯｷ･ｶ ｾＮ｢･｣ ｵｳ･ ponlble. GPS. the ｾ･ｬ ｴ ｶ･ ｮｯｬｴ･ｧ ｰ ｾ ｳｾｯ ･ ｳｯｵｾ｣･ｳ ･ｾ Hence the ｰｾｯ｣･ｳ ･｣ ｵｾ･｣ｹ ＮＬ ｾｯｲ Of the For It will be effect I ng the GPS ､･ｴ ｬ ｾｲｯ｣ between stetlons of teklng edventege of this correletlon cen leed to usuelly en order of II\Itgnltude ｉｾｲｯｶ･ ｮ ｴ ｉｮﾷｴｨ･ｾ ｳｵｬｴ Ｎ One cen envlsege severe I levels of ､ｬｦ ･ｾ ｮｴｬ･ I tis ｮ･｣ ｳ ･ｾｹ on Iy thet the different I e I need not ｴｾ･｣ｫ the same setel I Ites. bleses. end ･ｾ ｦｯｾ｣･､ ｮｯｬｴ･ｧ ｰ ｾ ｳｲｯｾ ･ to seme sequence. ｴｾ･｣ｫ I ｶ･ｾｳ ｴｾ･｣ｫ At the lowest level. GPS et the seme time end In this elSe. the effect of GPS system timing ｨｬｧｨ･ｾＮ One level the the same set of GPS seteilites. elthough not In this elSe, the effect of Ｎｰｨ･ｭ ｾｬｳ ･ｾ ｯｲｳ between In the end specific seteilite At the highest level, the the effects of seteilite end ｾ･｣ ｬｶ･ｾｳ ･ｾ ｮｯｬｴ･ｧ ｰｯｾ forced to In which elSe the error The choice of which dlfferentlel level to use depends on mexlmum. ｾ･｣ ｬｶ･ｾｳ ｮ･｣ ｳ ･ｾｬ ｹ the seme wevefront from the seme seteilite slmulteneously, correletlon ､･ｳｬｾ ｾ･｣ would be correleted. timing bleses would be correleted. ｴｾ･｣Ｚｫ GPS operetlons. ｴｾ･､ Ｍｯｦ ｳ sources Is e between eccurecy end equipment end processing ｳｯｰｨｬｳｾ ｣･ｴｬｯｮＮ As the dlstence between dlfferentlel receivers Increeses, the correletion between effects decreeses. Although theoretlcelly the effective ｾ･ｮｧ･ ･ｲｾｯ of dlfferentlel GPS cen be.of the order of thousends of km due to the high GPS orbits, In prectlce, the optimel effective ｾ･ｮｧ･ will conditions effecting be somewhet ｾ･ｦｲ･｣ｴｬｯｮＮ limited by the correletion dlstence of etmospheric this Is typlcelly ebouT 200 km. 10 3. YARIATIONS OF DIFFERENTIAL GPS OBSERVABLES ( Ther"e are four types of lIlSasurements of GPS sIgna Is wh 1ch have been suggested for dIfferentIal use: dIfferentIal pseudorange, dIfferentIal Integrated Doppler frequency, dIfferential carrier phase and Interfer"ometrlc tIme delay. GPS geodetic receivers How actually accomplish these .asurements Is outsIde the scope of this report. The Interested reader should refer to ••• In this connection, It Is suffIcient to say that although all these observables are Instrumented differently, Instantaneous derivatives. ranges It can be shown that they are between satellites and ground al I functions of stat ions and their the time These quantIties In turn reflect the relatIve geometry of the ground stations and the satellItes and It Is exactly thIs geometry which controls the precision with which relative positIons can be derived (Yanlcek et al., 1983J. Using the code or carrier GPS measurements for relative positioning usually Involves taking differences between measurements, since In this way the effect of errors <cf. section 2.3) which are COIIIllOn to the measurements being differenced are removed or l. greatly reduced. GPS measurements can satel lItes and across time (Paradis and be differenced Wei Is, 1984J. across receivers, across Although many different combinations are possible, the present convention for GPS measurement differencing Is to perform It In the above order, I.e. fIrst across receivers, then across satellites and lastly across time. This differencing results In the following fundamental differential GPS observables: a) a single difference observable (across receIvers) Is the Instantaneous difference In code or phase of a received signal measured by two receivers simultaneously observing the same satellIte; b) a double difference obserYable (across receivers and satellItes) Is then obtained by differencing two single difference observatIons between t.o satell ites, where the ground stations are the same for both single differences; c) a triple difference observable (across receivers, satel lites and time) change In a double dIfference observation from one epoch to the next. Is the sir (t,) . , 1 G PS SATELLITE TRACKING BY DI FFERENTIAL OBSERVATIONS 12 ( Single differences remove or greatly reduce the effects of errors associated with the seteilites, I.e. seteilite clock errors and to a greet extent (depending on base.llne lengths which are short ｣ｾ｡ｲ･､ to the 20,000 kin GPS seteilite altitudes) seteilite ephemeris and atnospherlc refraction errors, double differences the effects of errors associated with any station clock misalignment (which Is a common contributor to both single differences), stetlon clock and triple differences the effects of· residual errors and In the case of seteillte and carrier phase lll8asurements of the phase emblgulty errors. Figure 3.1 gives a sImple conceptual description of the GPS measurement differencIng. A total of eIght range measurements (derived either through code or carrier phase tracking) are shown: from which one can form four single difference observations ( = and 1n turn, two double difference observations k k IP2(tI) - PI(tI») and finally one triple difference observation Ｋ［ｾ･Ｎ In practice, many receivers, satellites and +hree epochs would be Involved. 3.1 Differential GPS ModellIng: A Simplified Overview For a pair of ground stations Paand P s observing simultaneously the same satellite, the mathematical model for differential ranging may be written as: or If formulated In terms of the Intersection vector +1 U lR ｾ｡･ + '{ [Vanlcek et al., 19841 ae where, (3.3) 14 +1 + + R with rand R ri receiver stations . S denoting the geocentric Q and . S respectively; position vectors of the +1 e Q +1 • e S satellite and the being the corresponding I along each user-to-satel lite direction and llpCJB Is the difference In unit vectors the slant ranges to the satellite. In the case of P- and C/A-oodemaasurements provided by receivers like the TI-4100. a differential range observable I llpCJB can be derived by differencing two simultaneous pseudoranges to the same satellite as measured at two different stations, I.e. -I • PQ -, . I I I - c6T Q + ( OP ) Ion + (Op ) Q trop CJB I I I I - c6T B + (oPQ a) Ion + ( oPCJB) trop P + eM a P a The I PQ + cM observation equation for such an observable can be written as -I 6paB .. (3.5) where I (OP_ a ) ....., Ion 1 ( OP_ a ) ...., trop .. .. 1 (CPa) trop I - (OP ) a trop (3.6b) 15 ere differential terms for Ionospheric and troposherlc effects and 6T as·• Jg (3.7) S - 4Ta Is a differential correetlon term for the receiver" clock errors. Similarly for receivers like the Mecrometer or the TI-4100 which are able to measure the phase difference of the carrier signal between· Its Internal oscillator and that of the satellite oscl I letor, the phase observable can be model led as, I.e. .1 • a II: II: I where 6f ｾｉ (t I) _ t (T ) a a (f I + !J.f I ) • t - (f + IF> • T a a (3.8a) I f • (t I +6t)-f· (T + ·6T ) a and ｾ a denote satellite and receiver frequency offsets respectively from the a nomInal carrier frequency f. UsIng the relatIonship between the transmit and receive times I t' + 6t l Ｋｾ c where PI Ic Is the propagation delay of the signal, equation (3.8) Is reduced to a f II: - I -c- • Pa • (3.10) In reality the measured phase measurement at some epoch t Is (3.1 Il t6 . I I.e. ｎｾ It consists of a fractional phase part Fr • a correct but unknown a Integer count from an Inlttal epoch tl to epoch t, plus an unknown and arbitrary Integer count ｎｾ at the Initial epoch tl. Clearly If It were not for the station's unknown I I a a clock phase mIsalignment at epoch tl, Fr (t) + N (t) would be a slantrenge (In units of cycles). The Instantaneous phase observable at a single station as described above Is essentially what the TI-4100 provides, whereas the Macrometer V-tOOO provides the difference between such observables taken at two stations at the same epoch. That Is, for differential positioning the Interferometric phase observation can be eXpressed as ｴｾ＼ Ｉ = I l ta<t) - t a (t) . - cf = ( -- f c I (Pe<t) pi (t) ) a 1 top (t) as or t&e<t) = te<t) - ＮｾＨｴＩ I - NJas<t 1) where Nae<tl) Is an unknown number of ful I cycles of This Is usually referred to as the "ambiguity term". during the ｯ｢ｳ･ｲｾ｡ｴｬｯｮ However, differential phase at epoch tl· If neither receiver loses lock session, there Is only one such ambiguity per satellite. If there are breeks In the date, two or more di fferent emb Igu it les per sete I I Ite may resu It. 17 3.2 01 fferentlal GPS Modelling: In developing the The Detailed Observation ｾ ､ ･ ｬ ｳ ｳｬｾ ｬｦ ･､ observation IiIodels Just described, we have tacitly assumed that' the clocks of all the participating receivers In an observing session are perfectly synchronized with respect to 8 chosen time standard such 8s UTC or "GPS time". In practice, one may ol?serve even when clock synchronization Is not practical or Impractical. Futhennore, often synchronIzation .,.rors . y not even be Identified until the data Is processed. Hence, one should be able to accommodate such situations In the data processing stage by accounting for the contribution of such errors Into the model, I.e. by some addItional parameterization In the adjustment leading to the followIng more cOll'Plete development. Slng'e Difference Model Consider a pair of ground stations P and P which are programmed a B to observe slmulte- neously at a sequence of epochs TI (5.14) T1 + (I - 1) • t:.T .. where AT Is the scheduled measurement Interval, and T denotes UTC or GPS time scale. 'In practice the clocks have some errors with respect to the chosen time standard scale. so that measurements In the two stations are actually taken at epochs TaJ = ... + 'I lIT ' ' 'al .' and = respectively, where Alai and AT B1 are the errors of the clocks at the I-th epoch. The transml t times of a sate I lite signa I received In Pa and P Bat Tal and TBI respect ivel y can be expressed as 18 ( k k Pa(t (T al),T aJ) k • Tal - t <Tat) c k k • t <T 81) where Ｌｾ k k PS(t <T 81 ), T 81 ) T 81 - ｉｓｾ 0.16a) . - 6'.al k 0.16b) - 6'.81 c are the combined Ionospheric and troposherlc effects along the range to satellite k from P and P at the I-th epoch. a s slant For the geometric line of sight path length travelled by the signal, the Taylor series approximations 0.17&) ｃｾＮＱＷ｢Ｉ can be used to account for the receiver clock errors. Equation <3.15) to (3.17) can now be substituted Into the simplified model which ｃｾＮＸＩ then becomes k k z • = .pIT al k 4J <T - l Pa( t - (T aI ) , T ai ) c f al k k kk - -- • P c OAk ai a (t <T al k -6Ak l-t<T -N aI a al) a ),T al - f k (3.18b) k fA , - t tT a"l) k - N.... .... ｃｾＮＱＸ｣Ｉ O.18d) 19 k •• f (tl J + -l fkk fk 6,l.k -t(t)-f a I aI k fk ok k • AT aJ - -c- Pa(t (tl J,tl ) - C"" Pa(t (t. J.t. ./Sral k • AT aI k (3.188) - Na Then collecting all similar ter:ms, the detailed observation model of an Instantaneous phase measurement at station P a can be expressed as . (3.19) and sImi larly for the second stetlon P observing "simultaneously" 6 k - Equlltlons f k k N6 and C3.20) 0-. 61 - <:5.19) (}.20) can be combIned to give the Interferometric (single difference) phase observation model as tas · t6 - "(a _ [t (T J _ ｾ . 6 I J _ f k ( &. k _ &. k ) _ Nk I 6' al a6 (t ) a (3.21e) 20 0: • c - It (T ) - B I tQ(T)-t(T) p I a I t (T )] - a I l(6l81 <3.21 b) '" • = = ( In view of (3.22), equetlon (3.21) reduces to ok P a ｾｔ at ] whIch Is the detailed model upon which subsequent developments will be besed. In our processIng software et GSC, the modelling of the errors essocleted with the etomlc clock sceles of the GPS receivers Is besed ･ｳ ｮｴｬｾＱ e.g. by Devldson et 211., (1983) and Del Ikereoglou (1984). a) lyon the epproech described Under the essumptions that the frequency of the receiver clock Is sUbject only to two kinds of verletlons with time - 21 frequency drift which (sllner In time, end rendom fluctuetlons; b) the frequency drift rete Is constent; c) eny rendom fluctuetions cen be modelled es white noise process hevlng 21 constllr,t standllrd deviation. 21 the clock errcr model we use Is of the form where .. (t) • (3. 24e) F Is the tIme error et synchronlzetlon, al If (3.24b) .. -F- Is the frectlonel frequency offset, end (3.24C> Is the frenctlon frequency drift (egelng) rete of the .clock, end . -'- rto )( ( t) F ;(t) dt represents rendom fluctuetlons In time. . At the 'present time, we heve Ignored errors other then modelled here (e.g. F mey not be constent, or F(t) mey not be whIte noise). The model however could be extended to explicitly Include these effects, If we ere sure of theIr form. Otherwise It may well be preferable to ettach some "age" to the model Itself, end to represent Its reliability es a functIon of Its age through some kind of weighting scheme. In generel, k Apaa In equetlon (3.23) Is determined from broedcest ephemeris dete whiCh k k k One however may develop P ' P (and hence lIP In mey assume to be precisely known. a s aa terms of orbital elements which can be edJusted elong with the stetlon coordinetes, clock errcr parameters and any other nuisance paremeters (e.g. ambiguity terms, etc.). We shell ｾ･ｬ on this problem In section 5. 22 ( 3.2.2 The Double Difference Model As already by differencing two simultaneous single ･ｾｬ｡ ｮ･､Ｌ Ｔｬｦ ･ｲ ｮ｣･ｯ｢ｳ･ｲｾ｡ｴｬｯｮｳ between two satellites, where the ground stations are the seme for both single differences, one may form a double difference observable. The corresponding observation equatlo model follows directly from the form of equation (3.23), I.e. k - f 16T SI -tsr all k k - '# 1 fA Sf l' \. - 1- + 1" I fJ. 1 rr. - - J fk ok k p + (f - f ) I 6T , cae J J (6A Sf ­ &I al ) ­ a al k k k f (&'S'I - OAa'l) - 'k N"' . D.25b) where k N 0.26) In equation ('.25) nClllllnally. the satellIte frequencies fj. fk (for all GPS satellites) are the same to within a fraction of a Hz. This will be especially the case with the operational satellites which will be equipped with high quality c.shlll and rubidium oscillators. Hence the terms (fJ - fk)MaJ and (fJ - fk)AT SI tend to cancel out. so that equation ('.25) reduces to where • '.2.} f ( . 1575.42 MHz or 1227.6 MHz) The Triple Difference Model Lets recall that In the triple difference approach the change In a double difference observlrt Ion from one epoch to the next Is used as the bas I c observab I e. The corresponding observation equation then follows directly frCllll two equations of the form (}.27), I.e. 24 (3.28 ) Note thet I'll the error terms wh Ich dropped out from the doub Ie differences ere 1'1 so absent fran the triple differences. In edd It Ion the setel II te dependent amb I gu Itv terms ere dropped out 1'1 together on· th Is approech. At Integer It turns out. furthermore triple differences allow en eesler autometlc editing of existing CVcle slips. As It Is the cese with the double difference observetlons. observetlons ere methemetlcellv correleted. triple dIfference Clearlv when properlv accounting for the correleted neture of the double or triple difference dete one should achieve the some results es theuncorreleted single differences. Remondl (1984) hI'S formuleted general elgorlthms for exploring the correleted nature of the double or triple difference deta, ｾ pointing out et the seme time thet excellent results are obtlned even when such I' A ( correlation Is Ignored. 25 4. DATA PROCESSING CONSIDERATIONS The tesk of processing GPS data of eny of the types Just desalbed lIllIy naturally be divided Into two distinct steps: (a) e preprocessing step. end (b) a paremeter est Imat Ion step. The goal of the processing Is'ulnly the validation of date quality. edIting of the data for such conditions as valid tncklng IlOde. poor­ signal strength. gross Ll ­ L2 nnge d I ffer-ences. t 11118 teg errors. etc.. removal and the detect Ion and. If poss I b Ie. the of date breaks within the observation series pertlanlng to each satellite. RespectIvely. the main goal of the peremeter estimation step consists of the estimation of the stetlon coordinates (or coordinate dltterences between. fixed and unknown statIons>. clock synchronization parameters or any other -nuisance- parameters such as orbit errOr' paremeters. IIIIblgulty terlDS. etc. In section 4.1 we shall dwell mainly on the preprocessing aspect of GPS data. Most of the discussion on the edIting aspects pertains to data collected using the TI­4100 GPS receiver. However. the discussion that follows on the detection of cycle slips on the carrier phase date Is also applicable to the treatment of Mecrometer ¥­1000 data. Section 4.2 deals with the parameter estimatIon part of the processing. 26 ( 4.1 • , Majority-Voting The process of extracting geodetlcally InterestIng Inforllllttion 1rOlll the stream of recorded r.., data from a T1-4100 Is well described In great detaIl by Texas Instrlllents (1982) and It Is therefore beyond the scope of thIs report. The procedures currently .-ployed at GSC for this purpose have been discussed by Beck (1985). In essence, following the transfer of collected data from the field cartridge re<:Ol"'dlng devleelnto a 9-tr"ack tape, and the appropriate decoding of the "satellite ephemeris" and "measurements" reports (Ibid), Jl8asurenent reports from each stet Ion. (a) Non-Integer second, time tags. of t Ime teg week rejected, a .aJorlty-votlng process Is applied on the 11'1 Is cons I sts of a date qua Iity check for: Whenever the navigation processor generated GPS time I s not an I nteger second, the who Ie .asurement resu I t I ng I n a loss of up to 4 output records each time. report Is Th I s shou I d only occur while the receiver Is acquiring new satellites and Is calculatIng an Initial navigation solution (which Includes time); c (b) Valid tracking mode. The measurement report Indicates at what stage of the signal acqu I sit Ion process the rece I ver was I n at the t Ime of the Nasurement. The tracking mode parameter Indicates Idle mode, search mode, pull-In mode, frequency locked reading loop, datil phase locked loop, P-code handover, mode. Only when PLL measurements for II reading and phase datil mode Is locked loop Indicated, (PLLl are the satellIte accepted; (c) Measurement QualIty. The TI receIver Itself generates a quality word for both Ll and L2 trackers; a "0" Indicates a valid measurement; other values Indicate things such occurrences as cycle slips, loss of phase lock durIng measurement, etc. Unless the quellty word (for both Ll and L2 data) Is "0" the datil measurement deta Is reJ ected; (d) Signal Strength. Measurement noise Is proportlonlll to the signal strength (Ibid). Unless the slgnlll to noise rlltlo Is grellter thlln or equal to lind 32 for L2 deta, the trllcker meesurement dlltll Is rejected; ｾＵ dB-Hz for Ll dlltll 27 (e) Gr'oss Ranse 0 I fference Errors. To ensure that gross range errors were detected.. the Ll pseudOl"enge and LZ pseudorange ere dlffereneed; If the absolute difference Is great... then 10'. the particular tracker data Is reJected; (f) Time TaS Error. frCllll For an unknown reason (at this time). the Masur_nt Incorrect Integer campaign, It report ｾｯｮ･ ｴｳＮ appeared that have occasionally SOllll!l correct pseudoranges calculated fractional parts but From the data collected, during the Ottawa test the error was always a.ultlple of 3 seconds. Investigation shawed that the error was In the receiver GPS tllll8 tag; this affects both the t l . tag end the Ll and LZ pseudorariges. If the error was 3 seconds, the time tag and pseudoranges wee corrected; If It was greater. the lII8asur...nts were rejected. 28 TAILE 4.1: ..... SNPLE OUTPUT Of FROCIWf RRR4 FROCESSINB FILE IS RE147 ..... EPHEMER IS OUTPlTT FILE IS RE14 7E OBSERVATION OUTPlTT FILE IS RE147A Ｑｌｾｃ Ｑ Ｇｾ 715.00 41 CNOL2 40 roe'll 200.00 BRBW DLLSM 200.00 4.0000 Total I of records read , of rec. rejected for non­I nteger time teg I of obs. reJectedfor poor track Ing mode I of obs. rejected for poor quality vector I of obs. reJ ected for poor sIgna I strength , of obs. rejected for Iarge range d Iff , of rec. corrected for tIme tag errors I of OBSERVATION RECORDS OUTPUT WI DE At«> NARROW AGCW, AGCN CNOL 1, CNOL 2 COB'll, CRBW DLLSM, DLLDM DLLSM, PLL[)o\ DLLDF PLLSM 4.0000 5.6000 PLLDF 4.8000 ACGN 642.50 4559 2:5 2579 41 ,eo 0 285 14525 Automatic Gain Control Ll (dB) ratio on Ll, L2 (dB ｾ Hz) Pre­detection Bend widths ­ code and carrier (Hz) Code delay lock loop SLlll and difference band widths (Hz) Carrier phase lock loop SLlll end difference band widths (Hz) ･ｳｬｯｎｾｴＭｬ｡ｮｧｬｓ Table 4.1 shows a typical outcome of this editing process (form our program RFQRM) on the measurement reports collected on one station during a five hour observIng session. For day 147 a tota I of 4559 measurement reports were recorded. dIstInction between records (reports) and observations. . Note here the A record which Is actually a TI­4100 measurement report may consist of up to four valid observations, one for each satel J Ite. reports. Normally non­Integer time tag problems occur for less than 50 measurement To our best JUdgement, It appears that this problem occurs when measurement reports are output while the TI­processor Is not In "navigation" mode; this often happens during the start of an operational sessIon, but It may well happen for as long as the duration of an entire tracking sessIon. 29 Most of the obs8l"vatlons being rejected due to poor tracking occurs dur I n9 wh I ch less than four sate I I I tes are be I ng tracked at a time. Its present (satellite conf 19uratlon channels) tradc.ed. FInally fractIonal but the when even TI-41oo less receiver than four records satellItes at. periods Th I sis because I n datil for ere four vIsIble trackers end being It should be noted that correctIng t l . tags whIch have correct Incorrect Integer components can selv.ge a consIderable emount of observatIons per session (285 observatIons or 20J of the total observatIon In the exemple of Tab Ie 4.1). 4.1.2 Cycle slip detection throuSh Polynomial fitting Polynomial fitting procedures for cycle slIp detectIon on carrier phase data have been applIed successfully with "semI-raw" Meercmeter date e.g. by Beutler et al., (1984) • . Essentially the fittIng process Is applIed on a date serIes of sIngle difference date for each satellite being tracked. An algebraIc polynomIal of degree r r pet) I z mzO a m (t - to)m (4.1 ) Is used to approximate n single differences observed by two stations on seteillte k k d i (t) where ｾ ＨｴＩ = ｾＨｴＩ ｾＨｴＩ - I z (4.2 ) 1,2, ••• ,n Is the I-th obs8l"ved single difference of phase and ＩｴＨｾ Is Its theo- retical counterpart computed from approximate sate I lite and station coordinates. If there are n breaks on the date of a particular session, one ends up dividing the total observation sessIon In In subintervals so that the dlff8l"ences k d l are approxI- mated by the pIecewIse contInuous function P (t) ! gsl r aO g + Lam (t - to) m, m=l ttl n I.e. an additional bias term Is added to the original polynomial I • n (4.3) for each subInterval 30 Although seemingly straightforward, the foregoing approech tecitiyesSlllle5 that the ( division of the observetlon .esslon onto .ublntervals (I.e. the location of existing data breaks) Is known .-prlorl. Such brNks are often not obvious at the time of the observation and can only be detected by a post11 ..Ion close examination of the data. In an effort to . .ke this detections as autoutlc as possible (I.e• • Ithout operator's Intervention), our practical epproech to ttlls probl_ .Ittl the T1-4100 has been to k elCam I ne the rate of change of success I ve d I val ues, I.e. whenever Is greater ｾｉｲｬ｣｡ｬ ｹ choice. addItional than some prescribed threshold (usually 2.7 cycles/sec as determined for IIIOSt our the data .e have processed so far seems to be a setlsfactory ThIs Is an Indication of the occurrence of a cycle slIp In which case an bIas term Is Introduced In the polynomial approximation which In effect adjusts al I subsequent phase data for that subinterval by an Integer number of cycles ( l so that they ere consistent with the observed phase rates. At the end of the process, the comp lete sequency forms a cont I nuous "cyc I e 5 II p free" phase data ser Ies whose dIfferences from epoch to epoch are correct but which as a whole stili has an unknown of f set of an Integer nIIIlber of cyc les. The process descr I bed I n sect I on 4.3 can be used to -determine and correct these offsets or If thIs Is not entirely possIble at least to come up with some good approximatIons for these ambIguIties, which they can be dealt wIth adequately In the flna' estImatIon step •. From our practical experiences with this approach on various data sets collected on different days under different tracking conditions have concluded that the methods work well In detecting large cycle slips. Slips of small magnItUdes (f.e. smaller than those detectab I e by the rate of change of phase chosen thresho I dl may be detected by examIning the residuals resulting from the adjustment for the polynomial coefficients In (4.}). 31 Su ch an· I nspect I on usua I IY resu I ts· In hay I ng to redef I ne the I nt....va I boundar I es, reject possible outll ....s and repeat the process until a satisfactory solution Is found. In the data sets we heve dealt with at GSC the criterion of equation (4.4) was not all of the times adequate for achieving the wphase connection", but th..... were tines we had occasional anomalies phases or rates or boths. Enabling the phase connection algorithm to Identify and bypass these bad data points proved to be real thorny problem which can .aslly I.ad to a complete br.akdown of the process, at least from the practical viewpoint and the ease of applying It to these data. From our Investigations, we led to believe that this situation can probably be caused by Incorrect tracking thresholds set In the receiver during the observation session (cf. Table 4.2). Normally for static operation, "NAY MODE" should be set equal to "0", for low dynamics should be set equal to "1" which also Is tile default value In the TI-4100). When one or the other race I ver uses higher track I ng bandw I dths dur I ng the observ I ng session, one ends up with nolsl.... single difference data for these stations which Is much more difficult to ,handle using the polynomlcal fitting algorithm. 4.1.3 The Ambiguity Resolution Problem In the previous process only cycle 51 Ips we calculated. carrier phase data stili remains. The Initial ambiguity In the Since the TI-4100 receiver can continuously track the carrier phase from point to point over a satellite pass, the ernblgulty problem reduces to recovering two unknown ernblgultles for each satellite - one for Ll and one for L2. The fact that TI-4100 receivers provide cerrler phase measurements on both Ll and L2 It offers a natural choice In using both the Ll and L2slgnals to resolve these cycle ambiguities. Furthermore, what comes handy In the connection Is that P-code group de I ays are aI so prQv I dad on L 1 and L 2. These are much less prec I se than the carrier phases but are unambiguously measured. Hence a. combination of the two types of Information makes a truly synergistic mix. In this section we dIscuss two of our software alternative Implementations of a mixed group delays/carrier phases algorithm for cycle ambiguity resolution. 32 The MARM Algorithm ( In the fIrst approach .. have utilized an elgorlttvn originally proposed by Melbourne (1982) (thus referring. to as the crow ｾｉ｢ｯｵｲｮ･ ｾ｢ｬｧｵｬｴｹ ｾｯｬｵｴｬｯｮ ｾｴｨｯ､Ｉ and utilized by et 211. (1984) with SERIES­X date. Lets recell thet affer differencing the dete between two stetlons P and P • the S a observed single difference In the phase of aslgnel from satellite J Is given by + W aB,L - (L J + as kN e ) f fT L (4.5) L ｾ J where NaS,L denote the cycle ambiguities on Ll and L2. differences of group deley, ( kN phese on Ll e Is a known end L2, constent J LaB Is the true times the (but unknown) (unknown) content of the Ionosphere along the two slgnel peths. Instrumente I are the observed single aB,L and. (and etmospherlc) errors on Ll and L2 thet remeln of the phese dete between stetlons. dlfferentlel dHferentlel *J election represent reslduel ｲｾｴ the differencing SImI lerly, the P­code group deleys on Ll end L2 et the same epoch are given by ｾ J = aB,L where ­J L as,L *' L J a8,L (4.6 ) L­Ll,l2 ere the observed group deleys end *J L aa,L represent reslduel Instrumente I group deleys. ­J *J Cleerly, If It were not for 1tIe error terms LaB,L end L ,L' equetlons (4.5) a6 would be vIewed as four equetlons with four unknowns, nemely would then be solved unIquely. Ｇ｡ ｾ kNe , Nevertheless, even In the presence of Ｎ｟ｾ ｾ｡ｓＬｌｬ end *' ""P, L (4.6 ) end end ＲｌＬｂ｡ｾ *, L J aB.L' It cen be shown thet equetlons (4.5) end (4.6) solved recursively yIeld est Imetes of the cycle embtgultles on II end l2 given by :s:s J f2 + L l' f2 Ll • aB,Ll ­ｾ ｾ 12 L2 f2 L2 as,L 1 ­J f Ll ｾｊ + 'T aB,Ll 2f aB,L 2 f L 1 L2 f2 _ f2 Ll L2 • -J + - t aB,L 2 (4.7e) as,L 1 - J f 't L1 ｾｊ aB,Ll aB,L2 _I _I ｾ where the terns ON"aB,Ll end ON"aB,L2 result from the Instrllllentel error terlllS taB,L end ｾ *J 'TaB,L end ere given by slmller expressions =: ｾ aB,L 1 f2 + f2 L l ' L2 f2 ­ f2 Ll L2 2\ = aB,L2 f 1 L2 f2 ­ f2 Ll L2 f *J 'T L 1 aB,L 1 *J f 'T Ll aB,L 1 2f 1 'f L L2 f2 _ f2 Ll L2 f2 + f2 L1 L2 f2 _ f2 Ll L2 f f *J 'T L2 aB,L2 *J 'T L2 aB,L2 - ­ t*j aB,Ll t *J aB,L2 For the GPS frequencies of 1575.42 MHz end 1227.6 MHz efter eveluetlng the coefficients In breckets, equetlons (4.7) become 34 (4.9b) ( ｾ｟ｑ with similar expnasslons for and ｬｌＬｾ In practical terms, the success of achieving an ambiguity resolution at the (desired) sub­cycle level hinges on the capability of reducing the effect of the Instrumental errors *J *J latter when associated 'aB,L and "l'ae,L' especially the corruption of the P­code deleys. *J *J given by equetlon (4.9) would be Integers or very nearly so. ｴｾＬｌ and the multlpath Obviously If perfact InstrUlll8ntatlon was to become avellable so thet 'aB,L and tae,L .ere zero, the estimates of 15 hardly the cese since with ｌＬｾＧｬＢ the cycle ambiguities In practice, however, this consist of both rendom and systemetlc perts due to both system noise end reslduel etmospherlc errors not removed by differencing between stet Ions. Crow et 21 I, (1984) me Inta In thet such 11m Itet Ions on th I5 approach cen neverthelss be greetly overcome by long term evereglng leedlng to everage estimates of J aNaa,L with 21 resulting error given by where the Instrumentel cerrler phase error Is considered negligible error In the P­code deleys, C1 ｣ｯｾ｡ｲ･､ to the Is the receIver noise error In the Ll P­code group delay, p the factor (2 eccounts for the dlfferenlng between stetlons, and k denotes the retlon of the L2/L 1 groupd de Iay error due to the nom Ine I dIHerence In the power Ieve I5 of the trensmltted slgnels, and n Is the number of velues used In the everaglng. The Grid Seerch Algorithm The long term aver eg Ing process of the MARM 21 Igor Ithin Just descrl bed Can be nice Iy complemented end/or checked agelnst an alternetlve elgorlthm whIch searches for the Integer cycle emblgulties on purely geometrical principles. of the approech first proposed In Devldson et al. Besed on (1983) 21 ､･ｬ ｾ ｳ version the seerch algorithm essentially storts from the sema equetlons (4.5) end (4.6) or rather the difference of the corresponding equations for Ll end L2. Thet Is, from equetlon (4.5) we get ｾ 35 ｾ as,L2 +Jas,L 1 as,L 1 Jas,L2 + f f .L 1 1 ­J ­J. • -('t ,Ll ­ TaS,L) aS *J . as,L2 *J , *J + (T ,ll­ ToB,L) + as C•• 111> + f L2 fL1 • *J as,Ll • kN. -:r- - ­­:­r L2 • 1 fL1 fL.2 • • *J as,L2 f2 L2 *J as,L 1 f2 Ll c•• 11 b) using equation (•• 6) to get fran the P­code group delays a f Irst­order approximation of the·dlfferentlal Ionospheric election content along the signal paths from the satellite to the two ground stations. Next let us assume that (4.12) ｊｾＬｌ where are some approximate values for the sought Integer eycle ambiguities ..c>.,J . Jo and Y'laS,L are Integer corrections to NaB,L to be determined•. Substituting Jaa,L (4.12) Into (4.11) yields after some straightforward manipulation ｾ &Jas,Ll aB,L2 . + \1 (j ｾ as,L2 ­ JOoB,L2. ｾ ) + (T*J ) 't*J as,L2 aB,L 1 ­ . ­ aB,L 1 + t­t 0 aB,Ll \1 ｾＪｪ TJ aB,L2 . aB,Ll \2 \2 TaB,L 1 + *j as,L 1 f2 Ll + • aB,L2 f2 L2 . q* (4.13a) or Ｈ ｾ f f ) ｾ Ll ｾ as,L2 . q C4.13b) Eech observetlon elong 21 seteilite pess gives us one setup of the kind described by ( The letter can be regarded es equetlon C4.13). reel spece coordinated by exls Is exectly known ｾ｡Ｘ ,Ll CI.e. ｾＺＢ end equel 21 streight line In the two-dlmenslonel 'to f /f L2 In'tercept q Is effec:t.d by errors (I.e• •:kL end setup of equetlon C4.13) does where ｾ｡ｂＬｌ 1 end ｾ｡ｂＬｌＲ • Ll ｔｾＬｌＩＮ 60177), • Beceuse of Integers. solution to 'this problem cen be found 'through 21 grid shown In figure 4.1. 120/150 not generelly heve en exect solutIon, ere· both Nevertheless, ･ｬｾ ｳ ｾ wIth respect 'to 'the ' whose slope _L2 ""6>,Ll whereas Its 'these errors the Ｈｾ Ｌｌ 'the .. o 1 ' ｾＬｌＲＩ best epproxlmate end fest search on an Integer The best solution cen shnply be defined es 'the point On this grid 'that 11es closest to the real streight line defined by equation C4.13). This Is essentially 21 three step procedure whereby: e) for each Integer velue of ｾ｡ｂＬｌｬ £ {-39,.-3B, ••• ­l,O,l, the corresponding In'teger value for &-lJ ••• 37,3B} ｾｯ up, L2 Is found from equation (4.13) as (4.14b) II: aB,L2 where [ b) ] denotes "the Integer ve I ue 0 f"; because of the truncation In (4.14b) the Integer pair be I ow the rea I Ｈｾｯ U1P,Ll Ｇｾ｡ U1P,L2 ) should lie II ne def J ned by equat Ion (4.13), wherees the closest po' nt to the rea I II ne I n the Integer gr I d may lie ebove or be I C1W the I I nee Hence there are two possible solutions given by the Integer paIrs (&t_OL 1' ｾ｟ｯ up, c) up, L ) end 2 (&t..o L1' &(0 L2 + up, up, 1) (4.15 ) the best solution Is then the point given by the Integer pair whIch satisfies the condItion '7 I.e. the closest point to the real line. Its hou I d be noted I n th I. connect Ion that the gIven procecilre can lead to a perIod I c solution .Ith periods 77 (&taa• L1 , ｾ｡ＬｌＲＩ Is and 60 along the a solution to the search for all Integer values of k are also solutions. the approximate ｡ｭ｢ｬｧｕ ｴｬ･ｳｾ Ｎｌ 77A ). Ll • 60). L2 approximatIon of should be ＩｎＺｾ Normelly the UoP,L MARM to this level. approach may be necessary. ｾｮｷｯ ｫ &I and Ll probl_, ｾ axes L2 - that (cSJ_ L·1 + 17K, wp, ｡ｾ Is, If + 6Ok) wp.L2 Hence, to be able to start the process .Ith an algorithm can accuracy provide a of 14.67 m (I.e. good enough If this Is not the case. an IJ'teraltlve InitIal ( ( GEOMETRIC INTERPRETATION OF THEAM8IGU'T\" EOUATION v • c. 10 ｾﾭ t ｾ 10 ｾ :; I Ｎｾ -. .•..... ... .•... c ­J 10 ｾ .... ­ L.­­.J a: : ,..-....! , ｾ : ··. )(J.... I ··· '\ ",-1<"" ·· · " ·· ·· .•••••.••• 1.:: ••ｾＨＭＧ • 'lett, ·· ·· · .,··.... ····. ·· ·· ··· ··· ... ,.· . ••• -t-·· ...• ·· ··· · • ｾ !E-t I .c:e.t "0 - lLl.... •Z ­J ｾ •••••••••• 10 z Q to- a: ｾ ｾ ｾ to- ｾ m lL 0 > ｾＮｦ a:: t- lU ｾ 0 L&J C) "0 ···