Doppler positioningwith GPS

TAMPERE UNIVERSITY OF TECHNOLOGY Department of Electrical Engineering Antti Lehtinen Doppler Positioning with GPS Master of Science Thesis Subject approved by the Department Council on 17.10.2001 Examiners: Prof Keijo Ruohonen (TUT) M.Sc Paula Syrj¨ arinne (Nokia Mobile Phones) Preface This Master of Science thesis is the result of my research on Doppler positioning with the GPS The thesis has been written at the Tampere University of Technology between the autumn of 2001 and February of 2002 I began my work on the GPS field at the Institute of Mathematics in November 2000 My very first task was to a literature survey and find out if Doppler measurements can be used to positioning with the GPS It turned out that the field was unexplored, which gave me a possibility to develop new algorithms and theory I would like to thank professor Keijo Ruohonen for his mathematical assistance and other help he has given throughout the work In addition, I express my gratitude to Paula Syrj¨arinne, who has provided me with GPS related advice, measurement data and excellent remarks on the contents of the thesis Nokia Mobile Phones deserves my thanks for funding the project Moreover, many Nokia employees have given me valuable feedback at the workshops I would also like to thank the personnel of the Institute of Mathematics, especially Niilo Sirola, who has helped me a lot by proofreading the thesis, giving me new ideas and sharing his Matlab functions Furthermore, I am indebted to all the people who have given me the joy of living that helped me to complete the thesis Especially Marjo Matikainen deserves thanks for her support during the work Tampere, 26th February 2002 Antti Lehtinen Iidesranta A 33100 Tampere Tel 040-7291388 i Contents Abstract v Tiivistelm¨ a vi Abbreviations and Acronyms viii Symbols ix Introduction The Doppler Effect 2.1 Physical Background and Applications 2.2 History of Doppler Navigation The Global Positioning System 3.1 History 3.2 The Basic Concepts 3.2.1 Reference Coordinate Systems 3.2.2 Range Positioning 3.2.3 GPS Signal Structure 3.2.4 Pseudorange and Delta Range 10 Pseudorange Positioning 11 3.3.1 Problem Formulation 11 3.3.2 Solution Procedure 12 3.3 ii 3.4 Positioning Error Analysis 13 3.5 Limitations of the GPS 14 Doppler Positioning with the GPS 16 4.1 Need for Doppler Positioning 16 4.2 Comparison between GPS and Transit 17 4.3 The Governing Equations 18 4.3.1 The Doppler Shift Equation 18 4.3.2 The Clock Drift Error 20 4.3.3 Mathematical Problem Formulation 21 Symbolic Methods 23 4.4.1 Solving the Receiver Position 23 4.4.2 Existence and Uniqueness Considerations 23 Standard Numerical Methods 24 4.5.1 Problem Formulation 24 4.5.2 The Newton Method 25 4.5.3 The Gauss-Newton Method 26 4.5.4 Calculating the Derivative Matrix 27 Problem Specific Iterative Methods 30 4.4 4.5 4.6 Positioning Performance 5.1 5.2 31 Measurement Noise 31 5.1.1 Ionospheric Effects 31 5.1.2 Receiver Noise 32 5.1.3 Unknown User Velocity 32 5.1.4 Biased Time Estimate 34 Positioning Accuracy 35 5.2.1 Effect of Measurement Errors on Positioning Solution 35 5.2.2 Doppler Dilution of Precision 36 iii 5.2.3 5.3 Applications of the Doppler Dilution of Precision Concept Case of Normally Distributed Measurement Noise Numerical Results 39 40 43 6.1 Measurement Data 43 6.2 Properties of the Measurement Noise 44 6.2.1 Hypothesis of the Properties of the Measurement Errors 44 6.2.2 Numerical Testing of Normality 45 6.2.3 Numerical Testing of Correlation 45 6.2.4 Magnitude of Errors 47 6.2.5 Conclusions 48 Positioning Performance with Real Data 48 6.3.1 Positioning Accuracy 48 6.3.2 Convergence 50 Positioning Performance with Simulated Data 50 6.4.1 Positioning Accuracy 51 6.4.2 The Distribution of DPDOP Values 51 6.3 6.4 Conclusions 53 iv Abstract TAMPERE UNIVERSITY OF TECHNOLOGY Department of Electrical Engineering Institute of Mathematics Lehtinen, Antti: Doppler Positioning with GPS Master of Science thesis, 58 pages Examiners: Prof Keijo Ruohonen and M.Sc Paula Syrj¨arinne Funding: Nokia Mobile Phones March 2002 Satellite positioning is expected to have hundreds of millions of users in the near future There is a huge commercial potential in personal positioning The standard positioning system nowadays is the Global Positioning System (GPS) The GPS is very accurate in good conditions In indoors and urban areas, however, the standard GPS positioning usually fails This decreases suitability of the GPS for personal positioning The objective of this thesis was to examine a new GPS positioning method based on Doppler measurements Doppler positioning with the GPS has generally been overlooked because it provides worse accuracy than the standard method However, the Doppler measurements can be obtained even in very bad conditions Thus, when combined with cellular network assistance, the method may prove valuable In this thesis, a new positioning algorithm is developed The algorithm needs only the Doppler measurements from at least four GPS satellites, the GPS satellite orbital parameters and the current time A general theory for estimating errors in GPS Doppler positioning is also developed Numerical results are compared with the measurements and the accuracy of the algorithm is assessed v Tiivistelm¨ a TAMPEREEN TEKNILLINEN KORKEAKOULU S¨ahk¨otekniikan osasto Matematiikan laitos Lehtinen, Antti: Doppler-paikannus GPS:ll¨a Diplomity¨o, 58 sivua Tarkastajat: Prof Keijo Ruohonen ja DI Paula Syrj¨arinne Rahoitus: Nokia Mobile Phones Maaliskuu 2002 Satelliittipaikannuksella uskotaan l¨ahitulevaisuudessa olevan satoja miljoonia k¨aytt¨aji¨a Henkil¨okohtainen paikannus tarjoaa suuria kaupallisia mahdollisuuksia T¨arkein paikannusj¨arjestelm¨a on nyky¨aa¨n Global Positioning System (GPS) Hyviss¨a olosuhteissa GPS on eritt¨ain tarkka Ongelmia aiheutuu kuitenkin siit¨a, ett¨a perinteinen GPS-paikannusmenetelm¨a ei usein toimi sis¨atiloissa tai kaupunkialueilla T¨am¨a luonnollisesti heikent¨a¨a GPS:n soveltuvuutta henkil¨okohtaiseen paikannukseen T¨am¨an diplomity¨on tarkoituksena oli tutkia satelliittisignaalien kantoaalloista teht¨avien Doppler-mittausten k¨aytt¨omahdollisuutta GPS:ss¨a Doppler-ilmi¨o¨on perustuvaa paikannusta on menestyksekk¨a¨asti k¨aytetty esimerkiksi GPS:n edelt¨aj¨ass¨a, Transit-j¨arjestelm¨ass¨a GPS:n yhteydess¨a Doppler-mittausten k¨aytt¨o¨a paikan selvitt¨amiseen ei kuitenkaan ole k¨aytetty, sill¨a paikannustarkkuuden on oletettu olevan huonompi kuin perinteisell¨a vale-et¨aisyyksiin perustuvalla menetelm¨all¨a Vale-et¨aisyyksien selvitt¨amiseksi GPS-vastaanottimen on kuitenkin kyett¨av¨a demoduloimaan satelliittien l¨ahett¨am¨a navigointisanoma Sen sijaan Dopplermittauksia pystyt¨a¨an useimmiten tekem¨a¨an vaikka signaali olisikin niin v¨a¨aristynyt ettei navigointisanomaa saada T¨ass¨a diplomity¨oss¨a esitet¨a¨an uusi GPS-paikannusmenetelm¨a, jolla voidaan ratkaista vastaanottimen sijainti mittaamalla v¨ahint¨a¨an nelj¨an GPS-satelliitin kantoaaltojen Doppler-siirtym¨at yht¨aaikaisesti Menetelm¨ass¨a ei tarvita navigointisignaalia, joten paikannus voidaan suorittaa my¨os huonoissa signaaliolosuhteisvi sa Satelliittien ratatiedot ja aikatieto pit¨a¨a kuitenkin olla saatavilla Mik¨ali navigointisanomaa ei saada, pit¨a¨a vastaanottimen siis olla yhteydess¨a esimerkiksi matkapuhelintukiasemaan Ty¨ossa kehitet¨aa¨n my¨os yleinen teoriapohja Doppler-paikannuksen virhetarkastelulle Teoria on t¨aysin uusi, mutta perustuu samoihin k¨asitteisiin, joita k¨aytet¨a¨an perinteisess¨a GPS-paikannuksessa Teorian mukaan paikannusvirhett¨a estimoitaessa tarkastelu voidaan eriytt¨a¨a kahteen toisistaan riippumattomaan osaan: mittausvirheiden arvioimiseen ja satelliittigeometrian vaikutuksen laskemiseen Ty¨oss¨a esitet¨aa¨n my¨os paikannusvirheteorian k¨ayttomahdollisuuksia esimerkiksi numeerisen laskennan kevent¨amiseksi Lis¨aksi arvioidaan Doppler–mittauksiin vaikuttavia virhetekij¨oit¨a sek¨a niiden suuruuksia Diplomity¨on numeerisessa osassa uuden paikannusmenetelm¨an toimivuutta kokeillaan todellisen mittausaineiston avulla Paikallaanolevalle vastaanottimelle jolla on tarkka aikatieto saadaan paikannusvirheen arvioksi 100–200 metri¨a Virheen havaitaan kuitenkin riippuvan hyvin voimakkaasti satelliittien sijainnista sek¨a saatavilla olevien satelliittisignaalien m¨a¨ar¨ast¨a Huonoimmissa tapauksissa menetelm¨a osoittautuu antavan paikkaestimaatteja, jotka ovat miljoonien kilometrien p¨a¨ass¨a todellisesta vastaanottimen sijainnista Hyv¨an¨a puolena on se, ett¨a ty¨oss¨a kehitetyn virhetarkasteluteorian avulla kyseiset tilanteet kyet¨aa¨n havaitsemaan Teoria osoittautui toimivaksi my¨os numeeristen simulointien valossa T¨ass¨a diplomity¨oss¨a esitetty paikannusmenetelm¨a voisi toimia varmistusmenetelm¨an¨a perinteisen GPS-paikannuksen ep¨aonnistuessa Paikannusvirheen havaittiin kuitenkin olevan suurempi kuin perinteisell¨a menetelm¨all¨a Virhe voi olla hyvinkin suuri silloin kun saatavilla on vain v¨ah¨an satelliittisignaaleja Lis¨aksi teoreettisesti on odotettavissa, ett¨a vastaanottimen liike aiheuttaa hyvin merkitt¨av¨an paikannusestimaatin heikkenemisen vii Abbreviations and Acronyms C/A cm DOD DDDOP DDOP DHDOP DGDOP DGPS DOP DPDOP DVDOP ECEF ECI GDOP GPS Hz m Matlab MHz PRN P(Y) s SA coarse/acquisition code centimetre U.S Department of Defence Doppler drift dilution of precision Doppler dilution of precision Doppler horizontal dilution of precision Doppler geometrical dilution of precision differential GPS dilution of precision Doppler position dilution of precision Doppler vertical dilution of precision Earth-centred Earth-fixed coordinate system Earth-centred inertial coordinate system geometrical dilution of precision Global Positioning System hertz metre Matrix Laboratory, a mathematical software megahertz pseudorandom noise precision code second selective availability viii Symbols × • ∂ α f n t ti wu x ρ˙ ρi ρ˙ i d fi ru x ν1 , ν ω ω θ ρ˙ ˆ ρ ρˆ˙ ρi ρ˙ i ρˆi ρ˙ T ρT i ρ˙ T i σ vector cross product dot product, scalar product euclidian norm, 2-norm, length of a vector partial derivative angle between satellite velocity and satellite-to-user vector frequency measurement error caused by receiver clock drift vector n estimate correction in an iterative algorithm receiver time estimate bias signal propagation time from the ith satellite error in user velocity vector estimate vector x estimate correction in an iterative algorithm machine precision delta range vector measurement noise measurement noise of pseudorange from the ith satellite measurement noise of delta range from the ith satellite clock drift equivalent error satellite i frequency measurement error user position error positioning error vector degrees of freedom transmitted frequency received frequency angle between relative velocity and wave propagation direction delta range measurement vector expected pseudorange vector expected delta range vector pseudorange measurement from the ith satellite delta range measurement from the ith satellite expected pseudorange from the ith satellite true delta range vector true pseudorange from the ith satellite true delta range from the ith satellite standard deviation ix 6.2 Properties of the Measurement Noise In Section 5.1, we introduced the main sources of errors in the GPS Doppler positioning These included the ionospheric effects, the receiver noise, the effect of unknown user velocity and the effect of biased time estimate In the current chapter, we will only consider a stationary receiver with a time estimate of high quality Since the user velocity and the time are accurately known, they are not additional error sources here It should be borne in mind that the positioning errors might be several orders of magnitude greater for a moving receiver Similarly, the results would be significantly weaker, if the current time estimate were biased 6.2.1 Hypothesis of the Properties of the Measurement Errors In this subsection, we will state the hypothesis we have of the properties of the measurement data We will also discuss the methods with which the hypothesis can be validated From the equations 4.13 and 4.14 we can see the nature of the measurements Our hypothesis is that the measurement data can be modelled with the following system of equations: ρ˙ i (t) = v i (t) • where the vector ρ˙ = [ ρ˙1 ρ˙2 its components are uncorrelated r i (t) − r u (t) + d(t) + r i (t) − r u (t) T ρ˙ n ] ρ˙ i (t) (6.1) has multivariate normal distribution and The difficulty of the testing of the hypothesis 6.1 is the unknown clock drift term d(t) The problematic term can be eliminated by considering the differences ij (t) = ρ˙ i (t) − ρ˙ j (t) ij = ρ˙ i − v i • ri − ru ri − ru − ρ˙ j − v j • rj − ru rj − ru (6.2) where the argument t for time has been left out for notational simplicity The right hand side of the equation 6.2 can be calculated from delta range measurements and the current ephemeris In the measurement data there were satellites that were visible during the whole period of 1274 seconds Since the receiver reported the measurement data at Hz frequency, there were 1274 delta range measurements from each of the satellites The sequences ij were formed for all the different combinations i, j such that i < j and j ≤ 44 6.2.2 Numerical Testing of Normality The normality of the differences ij was tested with the Lilliefors test which is implemented in the Matlab 6.1 Statistics Toolbox [Conover] The level of significance used in the test was % In three cases out of the 21 sequences the test rejected the hypothesis of normally distributed data In the figure 6.1 the difference probabilities for one of the sequences ij are plotted Difference probabilities compared to normal distribution 0.999 0.99 Probability 0.95 0.75 0.50 0.25 0.05 0.01 Difference sequence Normal distribution 0.001 −0.02 −0.015 −0.01 −0.005 0.005 0.01 0.015 0.02 Differences (m/s) Figure 6.1: Normal Probability Plot The Lilliefors test rejected the hypothesis that the sequence whose probabilities are plotted in Figure 6.1 would be normally distributed The reason, as can be seen, are the large errors There seem to be some outliers that not fit to the normal distribution However, this applies only to a small fraction of the data Furthermore, the vast majority of the sequences could be from the normal distribution, according to the Lilliefors test We can conclude that the sequences ij are not perfectly normally distributed, but the normal distribution is still a good approximation for the data 6.2.3 Numerical Testing of Correlation Next, we will consider the correlation between the errors of the delta range measurements from different satellites Again, the clock drift term prevents us from testing the correlation between the delta range measurement errors ρ˙i However, we are able to measure the correlations between the difference sequences ij All the different combinations { ij , kl } were formed such that i < j, i < k < l and l ≤ This resulted in 105 different combinations of {i, j, k, l} The correlations between the sequences were calculated The maximum absolute value 45 of correlation between any two difference sequences ij and histogram 6.2 shows the distribution of the correlations kl was % The Difference sequence correlations 20 Count 15 10 −0.06 −0.04 −0.02 0.02 0.04 Correlation Figure 6.2: Histogram of difference sequence correlations Fisher’s distribution and F statistics Probability 0.8 0.6 0.4 0.2 Theoretical Fisher’s distribution Cumulative F values of correlations 0 F statistic Figure 6.3: Correlation significance test plot The small correlation values imply that the difference sequences are in reality uncorrelated The significance of the correlations is investigated calculating the F statistics for the correlations [Walpole & Myers, p 349] In Figure 6.3 the F statistics of the difference sequence correlations are plotted and compared to the Fisher’s distribution with degrees of freedom ν1 = and ν2 = 1272 The figure 6.3 shows that the F statistics could well be from the Fisher’s distribution This implies that the correlations between the difference sequences ij are not a statistically significant part of the total variance in the sequences Thus, it can be said that the difference sequences ij not correlate 46 6.2.4 Magnitude of Errors The actual error in the delta range measurements could not be determined due to the unknown clock drift error However, the clock drift error was estimated as an average of the differences between the measured delta ranges and expected received frequencies at each instant of time: n ˆ = d(t) ρ˙ i (t) − v i (t) • i=1 r i (t) − r u (t) r i (t) − r u (t) (6.3) This estimate is unbiased and logical consequence of the equation 4.14 The estimates for the measurement noise components ρ˙i were calculated with the help of the equations 6.1 and 6.3 In Figure 6.4, the measurement noises of Measurement noise of two satellites Magnitude (m/s) 0.01 0.005 −0.005 −0.01 Satellite Satellite −0.015 500 1000 time (s) Figure 6.4: Measurement noise two satellites are plotted As one can see from the figure, the average error level contributes more to the error than does the time dependent noise This holds in every measurement analysed so far The reason for this may well be that the measurement period has been quite short, about 20 minutes This is so short a time that for example the ionospheric effects remain about the same during the measurement Since the ionosphere is the biggest source of errors, the nonzero average of the measurement noise during a short period is what could have been expected The standard errors for all the seven measurement sequences were calculated using the estimate 6.3 The standard errors were in the range between 0.003 and 0.006 m/s 47 6.2.5 Conclusions In Subsection 6.2.2 it was shown that the difference sequences ij can be modelled as normally distributed This assumption is valid with relatively high accuracy In Subsection 6.2.3 it was illustrated that the difference sequences ij are uncorrelated In Subsection 6.2.4 we analysed the magnitude of the measurement noise and found out that its order of magnitude in this measurers was about 0.5 cm/s All our numerical testing was done with the difference sequences ij = ρ˙i − ρ˙j because the measurement noise ρ˙i cannot be measured It is difficult to state anything about the nature of the measurement noise sequences ρ˙i For our purposes, we can model the measurement noise as having a multivariate normal distribution without correlation between the different sequences It has to be borne in mind that the measurement noise cannot be modelled as white noise This is because the errors change very slowly with time as was explained in Subsection 6.2.4 6.3 Positioning Performance with Real Data In this section, the positioning results with real measured data are given The positioning was carried out using Matlab The used algorithm was presented in Subsection 4.5.3 Since the algorithm is iterative, we need to investigate both its accuracy and its convergence 6.3.1 Positioning Accuracy The Gauss-Newton algorithm was used to calculate the position estimates independently at each instant of time The initial guess given for the algorithm was the centre of the Earth The iteration was continued until the norm of the √ correction term x fell under , where the machine precision was 2.2e-16 In Figure 6.5, the magnitudes of the positioning errors are demonstrated in the case where seven satellites were used to form a position estimate once a second The standard positioning error is about 105 meters The Doppler position dilution of precision measures were calculated with the equation 5.25 The DPDOP value changed very slightly during the measurements and was about 9700 seconds The good positioning accuracy and low DPDOP value were due to the usage of the high number of seven satellites According to the theory developed in Chapter and using the equation 5.29, we get that the standard error in the delta range measurements is about 1.0 cm/s 48 Histogram of positioning errors with satellites 300 250 Count 200 150 100 50 90 100 110 120 130 Error magnitude (m/s) Figure 6.5: Positioning errors Next, we will illustrate the effect of the satellite geometry on the positioning accuracy The positioning was now performed using only four of the seven visible satellites in the calculations All 35 different geometries involving four satellites were tested The DPDOP values were also calculated for the different geometries In Figure 6.6, the positioning error magnitudes are plotted versus the Doppler position dilution of precision values The positioning error magnitudes and the DPDOP values have a correlation of 94 %, the slope of the fit line being 1.0 cm/s The result implies that the positioning errors and the DPDOP values actually have a linear dependence on each other, as suggested in Chapter In addition, the result implies that the order of magnitude of the delta range errors in the measurements is cm/s Figure 6.7 is the same plot with the logarithmic scale on both axes The purpose of this figure is to show that the linear fit predicts the positioning error reasonably well even with smaller DPDOP values Positioning errors versus DPDOP values Positioning error (m) 3500 3000 2500 2000 1500 1000 500 Data points Linear fit 0 0.5 1.5 DPDOP (s) 2.5 x 10 Figure 6.6: Error dependence on geometry 49 Positioning error (m) Positioning errors versus DPDOP values 10 10 Data points Linear fit 10 10 DPDOP (s) Figure 6.7: Error dependence on geometry, logarithmic plot 6.3.2 Convergence Most of the algorithms in the numerical analysis, including the Gauss-Newton method, are iterative This means that the search for the solution is initiated from an initial guess point, marked by x0 A tremendously important question is whether the algorithm will converge towards the correct solution independent of x0 The answer in most cases is no If x0 is too far from the correct solution, the iteration may get stuck in a local minimum or approach infinity Any general proofs for the area of convergence cannot be given Consequently, the numerical testing of convergence is vital Extensive testing was performed in order to achieve numerical results for the convergence area The results were very optimistic The iteration converged in all the cases where x0 was near the Earth The first divergent case took place only when the initial guess x0 was more than 17 000 kilometres biased from the real receiver location When the bias in x0 was more than 17 000 kilometres, some of the iterations diverged When the bias was more than 23 000 kilometres, all the iterations diverged We can conclude that the algorithm has a very large convergence area The iteration can be initiated from the centre of the Earth without the fear of divergence 6.4 Positioning Performance with Simulated Data Testing with the real data is naturally fruitful, but testing with simulated data is also necessary The algorithm can be tested very extensively with simulated data because data collection causes no problems Simulated data can also be 50 used for testing how the algorithm behaves in rare and unexpected situations In addition, the testing is more controlled since the magnitudes of the simulated errors are known Altogether, more than 140 000 simulations were performed with different combinations of different simulated receiver locations, initial guesses, noise magnitudes and numbers of satellites The used satellite geometries were taken from real ephemerides 6.4.1 Positioning Accuracy The positioning accuracy was now tested with the simulated data Multinormally distributed white noise was added to the theoretically calculated delta ranges Thereafter, the Gauss-Newton algorithm was used to compute the position estimate for the simulated receiver In Figure 6.8, the position errors are plotted versus DPDOP × σρ˙ , where σρ˙ is the standard deviation of the added noise The Positioning errors versus DPDOP × σρ˙ Positioning error (m) 2.5 x 10 1.5 0.5 0 0.5 1.5 DPDOP × σρ˙ (m) 2.5 x 10 Figure 6.8: Error dependence on geometry and noise linear fit of the data points in Figure 6.8 gives an index of determination of 0.92 This means that the positioning error can be estimated with a high accuracy as the product of DPDOP and σρ˙ 6.4.2 The Distribution of DPDOP Values We have now seen that the positioning error can be easily estimated as a product of DPDOP and magnitude of delta range measurement noise The measurement noise is hard to analyse and depends on the receiver On the contrary, the DPDOP values are easy to analyse The DPDOP values are easy to compute, as was explained in Chapter In addition, the computation of DPDOP values requires only knowledge of the current ephemeris, time and a reference position but no measurements 51 A big number of DPDOP values were computed using different ephemerides from different days and choosing different satellite combinations The satellites whose elevation angle was less than five degrees were rejected When using all the available satellites, the DPDOP values were most of the time below 10 000 seconds Furthermore, when using the four best satellites, the DPDOP values were most of the time below 13 000 seconds The importance of selecting the right satellites is demonstrated by the fact that using the worst four satellites produced DPDOP values as high as 1010 seconds Figure 6.9 shows the variation of the best and the worst four satellites’ DPDOP values at a single location Note the logarithmic scale The total number of visible satellites was eight DPDOP values during 13 minutes 10 10 DPDOP (s) 10 10 10 Worst satellites Best satellites 10 100 200 300 400 500 600 700 800 time (s) Figure 6.9: DPDOP values with the best and worst satellites The lesson from Figure 6.9 is that positioning with low number of satellites is risky Sometimes the positioning results may be very good, the order of magnitude being 100–200 meters Sometimes, however, the positioning estimate may be millions of kilometres biased This all does not depend on measurement errors but satellite geometry Luckily, calculating the DPDOP values gives the user information whether the positioning geometry is bad or not Still much more testing is required to be able to draw conclusions about the general distribution of the DPDOP values For example, it would be interesting to know if some areas on Earth are more advantageous for Doppler positioning than others In addition, the dependence of the DPDOP value on the reference position is also important 52 Chapter Conclusions This thesis discussed the feasibility of using Doppler positioning with the GPS The predecessor of the GPS system, namely the Transit Doppler positioning system, was introduced It was shown that the standard GPS positioning method is very accurate, but it cannot be used in weak signal conditions Thus, there is need for new positioning techniques A heuristic comparison between the GPS and Transit Doppler positioning showed that the GPS might be at least as accurate as the Transit This fact justifies further analysis The thesis continued by developing a new positioning algorithm, beginning from the governing physical equations The iterative algorithm uses Doppler shifts from at least four simultaneously measured satellite signals The Doppler shifts are transformed into quantities called delta ranges The algorithm does not need to demodulate the navigation message of the satellite signals, provided that the satellite orbital parameters and the current time are known from other sources Thus, the algorithm is ideally suitable for weak signal conditions, such as indoors and urban areas Details affecting the accuracy of the GPS Doppler positioning were covered It was pointed out that for a stationary receiver with perfect time information, the ionospheric effects are the main error source But when the user is in motion or if the time information is biased, the order of magnitude of the positioning error is expected to rise substantially In addition, a new theory for estimating positioning errors was developed According to the theory, the positioning error analysis can be divided into two separate fields, namely measurement error analysis and satellite geometry analysis The final positioning error estimate can then be achieved by multiplying the two effect together The result is essentially a modification of the dilution of 53 precision concept in standard GPS positioning Furthermore, it was shown that the new theory can be used in satellite selection as well as in error estimation The last part of the thesis concentrated on numerical results Real measurement data was used in the analysis The properties of the measurement errors were first analysed The average error delta range level in the data set was found to be about 0.6 cm/s The result is in line with the literature, which estimates the ionospheric error to be at most 1.6 cm/s The measurements from different satellites were found to be uncorrelated The measurement errors were almost normally distributed, but there was slight tendency for larger errors than the normal distribution fit expected However, it was concluded that only a minor error is done if the measurement errors are modelled with multinormal distribution A Matlab algorithm was used to perform positioning with the newly developed algorithm The algorithm was noticed to behave well and converge from an initial estimate even thousands of kilometres away The positioning results were surprisingly good, the error being on average 105 meters when seven satellites were used Moreover, the positioning errors were found to be well in line with the theory developed for estimating the errors Because of the limited measurement data set, also simulated data was used to test the positioning performance The positioning error estimation theory was further tested and found to be valid Variation of the satellite geometry was also studied It became evident that the satellite geometry affects the positioning error much more than the ionospheric effect, because its variability is many orders of magnitude larger With satellites, for example, the positioning error estimate may be well below 200 meters when the geometry happens to be good However, the worst case scenario with only four available satellites gives positioning estimates that are millions of kilometres biased This thesis has shown that the Doppler positioning is a considerable option for a rough positioning estimate if the standard GPS positioning fails A positioning estimate can be achieved with only four satellites The navigation data need not be demodulated if the satellite positions and velocities are achieved from other sources However, the accuracy of the method with only four satellites is not guaranteed Increasing the number of satellites diminishes the risk of a bad positioning estimate The theory for estimating positioning errors turned out to be extremely useful Using the theory, bad satellite geometries are noticed and the user can be warned about unreliable positioning results The theory can also be used for selecting the most suitable satellites when computational efficiency prevents one from using all the visible satellites 54 The positioning algorithm was developed assuming that the user is stationary or that its velocity vector is known The applicability of the method for mobile devices is significantly diminished by the fact that even a slow receiver motion weakens the positioning performance drastically For example, a velocity of 1.0 m/s is expected to raise the magnitude of positioning errors by two orders of magnitude Thus, for a mobile receiver, a modification for the algorithm is necessary The modified algorithm would solve for the receiver velocity as well as its position The modified algorithm would require at least seven satellites In addition, the accuracy is further weakened 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