PDHonline Course L175J (6 PDH) GPS Surveying. Instructor: Jan Van Sickle, P.L.S. PDH Online PDH Center

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1 PDHonline Corse L175J (6 PDH GPS Srveying Instrctor: Jan Van Sickle, P.L.S. 01 PDH Online PDH Center 57 Meadow Estates Drive Fairfax, VA Phone & Fax: An Approved Contining Edcation Provider

2 1 Modle 3 Point Positioning In point positioning, the positions of the satellites are available from the data in their broadcast ephemerides. The satellite clock offset and the ionospheric corrections are also available to a GPS receiver once it has locked on. That is once it has correlated its code with the codes it is receiving. Then it can immediately read the Navigation messages of all for or more satellites it is tracking. Bt the receiver mst assme that all these corrections in the Navigation message are absoltely correct, which is a bit of an exaggeration. In fact, there are errors in these corrections. And along with the timing errors between the receiver and satellite clocks, receiver noise, and any mltipath, they still contaminate the position that reslts. (See modle for a more complete discssion of these biases. Nevertheless this kind of positioning is the flfillment of the original idea behind GPS. It relies on a coded psedorange measrement and can be sed for virtally instantaneos positioning. Psedoranges from the receiver to three GPS satellites provide enogh data to solve three Cartesian coordinates, x, y, and z of the receiver. The forth psedorange to the forth satellite provides the information for the soltion of the receiver's clock offset, dt, as illstrated in the eqations below and Figre 3.1.

3 + c( dt Z +( +(Y X ( = p + c( dt Z + ( + (Y X ( = p + c( dt Z +( + (Y X ( = p + c( dt Z + ( +(Y X ( = p z 4 y 4 x 4 4 z 3 y 3 x 3 3 z y x z 1 y 1 x 1 1

4 3 The ability to achieve so mch redndancy in the measrement of the receiver=s clock error, is one reason the moderate stability of qartz crystal clock technology is entirely adeqate as a receiver oscillator. A niqe soltion is fond here becase the nmber of nknowns is not greater than the nmber of observations, obviosly observing more satellites makes the soltion even better. Bt when the receiver tracks at least for satellites simltaneosly; these for eqations can be solved simltaneosly for every epoch of the observation. An epoch in GPS is a very short period of observation time, and is generally jst a small part of a longer measrement. However, theoretically there is enogh information in any single epoch to solve these eqations. Relative Positioning When two or more receivers are available relative positioning is possible and the accracy of GPS positioning improves. Relative positioning can attain higher accracy than point positioning becase of the extensive correlation between observations taken to the same satellites at the same time from separate stations. In other words, two receivers operating

5 simltaneosly, collecting signals from the same satellites will record errors, bt very similar 4 errors. This is especially tre when yo consider that the distances between sch stations on the earth are very short compared with the 0,000km altitde of the GPS satellites. Relative positioning generally relies on carrier phase ranging instead of code psedoranges. Carrier Phase Carrier phase is the observable at the center of high accracy srveying applications of GPS. It is also known as the carrier beat phase observable. It depends on the carrier waves themselves, the nmodlated L1 and L, rather than their P and C/A codes. There is an interesting approximation that illstrates the potential of carrier phase ranging, The One Percent Rle of Thmb. Assme for the moment most of the errors in the receiver and satellite clocks, receiver noise, ionospheric delay, broadcast ephermeris errors or mltipath are gone. Under sch a circmstance yo cold expect psedoranges to be accrate within abot one percent of the chipping rate of the code sed, whether it is the P code or the C/A code. In practice, positions derived from these codes are rather less reliable than that, of corse, becase they mst rely on the Navigation message corrections for the most part. Bt consider this a P code chip occrs every of a microsecond. So by sing this rle, a P code based measrement cold have a maximm precision of abot 1 percent of abot

6 5 a tenth of a microsecond, or one nanosecond. One nanosecond mltiplied by the speed of light is approximately 30 centimeters, 1 percent of the length of a single Pcode chip. The C/A code based psedorange is 10 times less precise. Its chipping rate is 10 times slower. A C/A code psedorange wold have a maximm resoltion of abot 3 meters, that is 1 percent of the length of a single C/A code chip. Ok, now take a look at this same 1 percent rle of thmb applied to the carrier phase observable. Instead of a code chip as the basis of the measrement, it is the wavelength of the carrier. How long is one wavelength of the GPS carriers? The length of a single wavelength of each carrier can be calclated sing the same formla as was sed previosly in Modle 1. c λ = a f Where: λ = the length of each complete wavelength in meters; c a = the speed of light corrected for atmospheric effects; f = the freqency in hertz. The L MHz carrier transmitted by GPS satellites has a wavelength of approximately 19cm.

7 6 c λ = f x 10 mps λ = x 10 Hz λ = 0.19m a The L17.60 MHz freqency carrier transmitted by GPS satellites has a wavelength of approximately 4cm c λ = f x 10 mps λ = x 10 Hz λ = 0.4m a Therefore, sing either carrier, the carrier phase measrement resolved to 1 percent of the 4cm or the 19cm wavelength. The accracy wold be abot mm. Qite an improvement, especially considering that in relative positioning atmospheric errors can be greatly redced by the correlation mention earlier and satellite clock and receiver clock errors can be eliminated by differencing, more abot that in jst a moment. However, first there is something called the cycle ambigity problem that needs to be mentioned. We ve seen before with the EDM, except it is a little more difficlt to solve in GPS An Illstration of the Cycle Ambigity Problem

8 7 While carrier phase, or carrier beat phase as it is sometimes called, is different than the psedorange, the basis of the measrements have some similarities. For example, the fondation of a psedorange measrement is the correlation of the codes received from a GPS satellite with replicas of those codes generated within the receiver. The fondation of the carrier phase measrement is the combination of the nmodlated carrier itself received from a GPS satellite with a replica of that carrier generated within the receiver. It s like a distance measrement by an EDM. As mentioned earlier an EDM sends a carrier wave to the reflector, and generates an identical internal reference. When the external beam retrns from the reflector, it is compared with the reference wave. The difference in phase between the two reveals the fractional part of the measrement, even thogh the nmber of complete cycles between the EDM and the reflector may not be immediately apparent. That is ntil modlated carriers of longer wavelengths are sed (See modle 1 for a more complete discssion of this process. Likewise, it is the phase difference between the incoming signal and the internal reference generated inside a GPS receiver that reveals the fractional part of the carrier phase measrement in GPS. The incoming signal is from a satellite rather than a reflector of corse, bt like an EDM measrement, the internal reference derived from the receiver s oscillator is sed for comparison. And jst like an EDM measrement it is the difference in phase between the two reveals the fractional part of the range measrement. And as the 1 percent rle of thmb illstrates this comparison is capable of a mch more accrate measrement than psedoranges.

9 8 Bt there is a problem. Again, jst like the EDM measrement the nmber of complete cycles is not immediately known. In other words, when the receiver locks onto the carrier it cannot know how many fll wavelengths stand between the satellite and itself at that instant and withot that information it isn t possible to know the fll distance, of corse. There is one helpfl fact abot the cycle ambigity thogh. It has to be an integer. The fll cycles between the receiver and the satellite at lock on mst be a whole nmber. So it is also called the integer cycle ambigity. The sitation is a lot like an nofficial techniqe sed by some nineteenth centry contract srveyors on the Great Plains. The procedre can be sed as a rogh illstration of the cycle ambigity problem in GPS. It was known as the bggy wheel method of chaining. Some of the lines of the pblic land system that crossed open prairies were originally srveyed by loading a wagon with stones or stakes and tying a cloth to a spoke of the wheel. One man drove the team, another kept the wagon on line with a compass and a third conted the revoltions of the flagged wheel to measre the distance. When there had been enogh trns of the improvised odometer to measre half a mile they set a stone or stake to mark the corner and then rolled on, conting their way to the next corner.

10 9 A GPS receiver is like the man assigned to cont the trns of the wheel. He is spposed to begin his cont from the moment the crew leaves the newly set corner, bt instead, sppose he jmps into the wagon, gets comfortable and takes an nschedled nap. When he wakes p the wagon is on the move. Trying to make p for his laxness, he immediately begins conting. Bt at that moment the wheel is at a half trn, a fractional part of a cycle. He conts the sbseqent half trn and then, back on the job, he intently conts each and every fll revoltion as they come arond. His tally grows as the cycles accmlate, bt he is in troble and he knows it. He cannot tell how far the wagon has traveled; he was asleep for the first part of the trip. He has no way of knowing how far they had come before he woke p and started conting. He is like a GPS receiver that cannot know how far it is from the satellite when it starts conting phase cycles. They can tell it nothing abot how many cycles stood between itself and the satellite when the receiver was locked on and began tracking. Those nknown cycles are the cycle ambigity. The 360 cycles in the carrier phase observable are wavelengths λ, not revoltions of a wheel. Bt there is a soltion to the cycle ambigity problem. It is called differencing. Differencing In GPS the word differencing has come to represent several types of simltaneos baseline soltions of combined measrements. Now, obviosly, differencing is only possible sing two or more receivers and relative positioning. And it sally involves carrier phase, or carrier beat phase, measrements. The lines between the pairs of receivers are called vectors or baselines. In these illstrations only two receivers are sed, bt there cold certainly be more

11 and the principles wold be the same. OK, the most freqently sed differencing methods are known as the single difference, doble difference, and triple difference. 10 Single Difference A single difference, also known as a betweenreceivers difference, can refer to the difference in the simltaneos carrier phase measrements from one GPS satellite as measred by two different receivers (Figre 3..

12 Figre 3. 11

13 1 Since the two receivers are both observing the same satellite at the same time the satellite clock error is eliminated in a singledifference. It is a bons that the atmospheric biases and the orbital errors recorded by the two receivers in this soltion are also nearly identical, and can be virtally eliminated too. Unfortnately, there are still two factors in the carrier beat phase observable that are not eliminated by single differencing. The difference between the integer cycle ambigities at each receiver and the difference between the receiver clock errors remain. Doble Difference There is a GPS soltion that will eliminate the receiver clock errors. It involves the addition of what might be called another kind of single difference, also known as a betweensatellites difference. This term refers to the difference in the measrement of signals from two GPS satellites, as measred simltaneosly at a single receiver (Figre 3.3.

14 Figre

15 14 The data available from the betweensatellites difference allow the elimination of the receiver clock error. In this sitation, there can be no difference in the clock since only one receiver is involved in each of the component single differences. And the atmospheric effects on the two satellite signals are again nearly identical as they come into the lone receiver, so the effects of the ionospheric and tropospheric delays are virtally eliminated as well. By sing both the betweenreceivers difference and the betweensatellites difference, a doble difference is created. This combination is virtally free of receiver clock errors and satellite clock errors, bt still there is one stbborn factor in the carrier beat phase observable that is not eliminated. The integer cycle ambigity is still in there. Triple Difference Combining two doble differences creates A third kind of differencing. Each of the doble differences involves two satellites and two receivers. The difference next derived is between two epochs. The triple difference is also known as the receiversatellitetime triple difference (Figre 3.4, the difference of two doble differences of two different epochs.

16 Figre

17 16 In the triple difference two receivers observe the same two satellites dring two consective epochs. This soltion can be sed to qantify the integer cycle ambigity becase if all is as it shold be it is constant over the two observed epochs. Actally, a triple difference is not sfficiently accrate for short baselines. It is sed to find the integer cycle ambigity. Once the cycle ambigity is determined it can be sed with the doble difference soltion to calclate the actal carrier phase measrement. Here s how it works, from the moment of a receiver s lock onto a particlar satellite; there are actally three components to the total carrier phase observable. φ = α + β + N where : φ = total phase α = fractional initial phase β = observed cycle cont N = cycle amigity First is the fractional initial phase, which occrs at the receiver at the first instant of the lockon. The receiver starts tracking the incoming phase from the satellite. The receiver grabs onto the satellite's signal at some fractional part of a phase. It is interesting to note that this fractional part does not change for the dration of the observation and so is called the fractional initial phase. It is symbolized here by α.

18 17 Second is the integer nmber of fll cycles of phase that occr from the moment of the lock to the end of the observation. It is symbolized by β, the observed cycle cont. This element is the receiver=s consective conting of the change in fll phase cycles, 1,, 3, 4..., between the receiver and the satellite as the satellite flies over. Of the three terms, β is only nmber that changes that is, if the observation proceeds correctly. Third is the integer cycle ambigity N. It represents the nmber of fll phase cycles between the receiver and the satellite at the first instant of the receiver's lockon. N does not change from the moment of the lock onward, nless that lock is lost. In other words, the total carrier phase observable consists of two vales that do not change dring the observation, the fractional phase α, and the integer cycle ambigity N. Only the observed cycle cont β, changes, nless there is a cycle slip. Cycle Slips In fact, the triple difference makes the detection and elimination of cycle slips relatively easy. A cycle slip is a discontinity in a receiver=s continos phase lock on a satellites signal. The coded psedorange measrement is immne from this difficlty, bt the carrier beat phase is not.

19 18 Relative positioning by carrier phase measrement is the primary vehicle for high accracy GPS srveying. Simltaneos observations, doble differencing in postprocessing, and the sbseqent constrction of networks from GPS baselines are the hallmarks of geodetic and control work in the field. The strengths of these methods generally otweigh their weaknesses, When lock is lost a cycle slip occrs. A power loss, an obstrction, a very low signaltonoise ratio, or any other event that breaks the receiver=s continos reception of the satellite=s signal cases a cycle slip. That is, the receiver loses its place in its cont of the integer nmber of cycles β and, as a reslt, N is completely lost and the receiver has to start over from scratch. There are several methods that may be sed to regain a lost integer phase vale, N. The triple difference is one of the better alternatives in this regard, as stated earlier the triple difference does not depend on the initial integer ambigity, becase it is a constant in time. Therefore, when a large residal does appear in its component doble differences it is very likely cased by a cycle slip. Even better the obstrcted signal can be singledot by isolating all available satellite pairs ntil the problem is fond. This tility in fixing cycle slips is the primary appeal of the triple difference. It can be sed as a preprocessing step to weed ot cycle slips and provide a first position for the receivers. Smmary

20 particlarly where there can be an nobstrcted sky and relatively short baselines and where the length of observation sessions is not severely restricted. 19 However, conditions are not always so ideal. Where obstrctions threaten to prodce cycle slips, coded psedorange measrements may offer an important advantage over carrier phase. Psedorange measrements also may be preferred where accracy reqirements are low and prodction demands are high. Differencing is an ingenios approach to minimizing the effect of errors in carrier phase ranging. It is a techniqe that largely overcomes the impossibility of perfect time synchronization. Doble differencing is the most widely sed formlation. Doble differencing still contains the initial integer ambigities, of corse. And the estimates of the ambigities generated by the initial processing are sally not integers, in other words, some orbital errors; atmospheric errors, etc. remain. Bt with the knowledge that the ambigities oght to be integers, dring sbseqent processing it is possible to force estimates for the ambigities that are in fact integers. When the integers are so fixed, the reslts are known as a fixed soltion, rather than a float soltion. It is the doble differenced carrier phase based fixed soltion that makes the very high accracy possible with GPS. However, in this discssion of errors it is important to remember that mltipath, cycle slips, incorrect instrment heights, and a score of other errors whose effects can be minimized or

21 0 eliminated by good practice are simply not within the prview of differencing at all. The navoidable biases that can be managed by differencing inclding clock, atmospheric, and orbital errors can have their effects drastically redced by the proper selection of baselines, the optimal length of the observation sessions, and several other considerations inclded in the design of a GPS srvey. Bt sch decisions reqire an nderstanding of the sorces of these biases and the conditions that govern their magnitdes. The adage of, "garbage in, garbage ot," is as tre of GPS as any other srveying procedre. The management of errors cannot be relegated to mathematics alone.