A Technique for Determining the Carrier Phase Differences ...

A Technique for Determining the Carrier PhaseDifferences between Independent GPS

Receivers during Scintillation

Shan Mohiuddin, Todd E. Humphreys, and Mark L. PsiakiCornell University

BIOGRAPHYShan Mohiuddin is a graduate student in Mechanical andAerospace Engineering. He received a B.S. in AerospaceEngineering from Virginia Tech in 2003 and has worked inthe Flight Dynamics Analysis Branch at NASA/GoddardSpace Flight Center. His research focuses on navigationalgorithms for spacecraft, estimation theory, and GNSS re-ceiver technology.

Todd E. Humphreys is a post-doctoral researcher in Me-chanical and Aerospace Engineering. He received a B.S.and M.S. in Electrical and Computer Engineering from UtahState University and a Ph.D. in Mechanical and AerospaceEngineering from Cornell University. His research inter-ests are in estimation and filtering, spacecraft attitude de-termination, GNSS technology, and GNSS-based study ofthe ionosphere and neutral atmosphere.

Mark L. Psiaki is a Professor of Mechanical and AerospaceEngineering. He received a B.A. in Physics and M.A. andPh.D. degrees in Mechanical and Aerospace Engineeringfrom Princeton University. His research interests are in theareas of estimation and filtering, spacecraft attitude and or-bit determination, and GNSS technology and application.

ABSTRACTA method for recovering the carrier phase differences be-tween pairs of independent GPS receivers has been devel-oped and demonstrated in truth-model simulations. Thiseffort is in support of a project that intends to image the dis-turbed ionosphere with diffraction tomography techniquesusing GPS measurements from large arrays of receivers.Carrier phase differential GPS techniques, common in sur-veying and relative navigation, are employed to determinethe phase relationships between the receivers in the imag-

ing array. Strategies for estimating the absolute carrier phasedisturbances at each receiver are discussed. Simulation re-sults demonstrate that the system can rapidly detect the on-set of scintillation, identify one non-scintillating referencesignal, and recover the carrier phase differences accurate to0.1 cycles.

INTRODUCTIONGPS radio waves experience several changes as they passthrough the ionosphere. A quiet ionosphere introduces thewell-known code delay and phase advance effects. Theleft half of Fig. 1 depicts the scenario where a GPS signalpasses through a quiet portion of the ionosphere to a re-ceiver situated on the ground. If the electron density profilechanges slowly across this portion of the ionosphere, thesignal may be assumed to have a single ionospheric piercepoint. Thus the signal is altered only by the number offree electrons encountered along its direct path from thesatellite to the receiver. This quantity, defined as the totalnumber of electrons in a 1-m2 cross section column that isoriented along the signal’s path, is called the Total ElectronContent (TEC). Such a slowly varying electron density pro-file is represented in the left panel of Fig. 2. In this quietionosphere model, which will be referred to in this paperas the bulk-quiescent ionosphere model, the received sig-nal is characterized by an amplitude that is independent ofT EC(x) and a phase that is a function of the electron den-sity at the signal’s pierce point T EC(x1).1

When electron density irregularities are present, theeffect on the signals is more complicated. This situation isdepicted in the right half of Fig. 1 where the signal is re-

1The horizontal dimension x is aligned perpendicular to the local mag-netic field at the ionospheric pierce point. This orientation takes advantageof the fact that ionospheric irregularities, when present, tend to align alongthe local magnetic field lines. Hence, variations in TEC exist primarilyalong only one horizontal direction.

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> 100 m

Electron DensityIrregularities

Ionosphere

GPS Receivers

Observation Plane

Figure 1. The undisturbed and disturbed ionosphere and the scin-tillation imaging array.

xx1

TEC(x)

ObservationPlane

Quiescent Ionosphere Disturbed Ionosphere

TEC(x)

x

Figure 2. The bulk-quiescent ionosphere mode (left) and thewave propagation model (right).

fracted and diffracted by the irregularities. The signal notonly experiences code delay and phase advance due to thefree electrons along the propagation path but also experi-ences constructive and destructive interference as the dis-turbed wavefront propagates down to the receiver. In thiscase, the bulk-quiescent ionosphere model is inadequate todescribe the signal disturbances; a more complex model,based on the physics of wave propagation through irregu-lar media, is required. Such a model is depicted in the rightpanel of Fig. 2, where the transmitted wavefront encountersa TEC profile that varies significantly along the x direc-tion. The uneven lines below the modeled ionosphere rep-resent phase variations in the propagating wavefront. Thereceived signal’s amplitude and phase are now dependenton the electron density profile along the x axis. The sig-

nal at the observation plane exhibits deep power fades andcarrier phase disturbances referred to respectively as am-plitude and phase scintillation [1]. As an illustraton of theeffect of scintillation, Fig. 3 shows a time history of the re-ceived carrier-to-noise ratio and carrier phase disturbancesfor a scintillating signal.

0 5 10 15 2025

30

35

40

45

50

C/N

0, dB

−Hz

0 5 10 15 20−1

0

1

2

Time, s

δ φ,

cyc

les

Figure 3. A time history of the received carrier-to-noise ratio(top) and the scintillation-induced phase disturbances(bottom).

The study of ionospheric disturbances and their ef-fects on GPS signals is of both practical and scientific inter-est. For GPS users, especially those who use carrier phasetechniques for precise relative positioning, the disturbancescaused by scintillation pose a significant problem. Dur-ing quiet periods, ionospheric effects may be removed us-ing models, dual-frequency measurements (if available), ordifferencing techniques. The latter approach, often used insurveying and relative navigation applications, relies on thesingle pierce point and smooth TEC profile assumptions tocancel out the effects of the ionosphere over short base-lines. These assumptions break down if the GPS signalsare scintillating. In this case, precise positioning may beimpossible.

For ionospheric scientists, the altered GPS signals of-fer insight into the structure and dynamics of the disturbedionosphere. A project is underway at Cornell Universitythat intends to use arrays of tens, hundreds, or even thou-sands of inexpensive GPS receivers spread out over tensof kilometers to image the disturbed ionosphere throughdiffraction tomography. This project will used techniquessimilar to those discussed in Ref. [2] to reconstruct TECprofiles based on terrestrial GPS measurements and on wavepropagation models. Figure 1 shows a schematic of smallsegment of the envisioned scintillation imaging array.

Signal amplitude is relatively easy to measure with

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an array of calibrated receivers, but it only represents partof the received signal. The other part, the signal’s phase,is more difficult to measure because a proper measurementwould require the array to be phase-synchronized. The en-visioned array, however, is too large for the receivers tobe connected to a common phase reference. Thus, eachreceiver must use its own independent clock to make itsphase measurements. This introduces phase biases into themeasurements because the individual clock errors are in-separable from the phase measurements.

The question becomes this: What phase informationcan be gained from a non-phase-synchronized array? Oneanswer, rather oddly, comes from the fields of surveyingand relative navigation for which these signal disturbancesrepresent a nuisance. The answer is that phase differencesbetween pairs of independent receivers may be recoveredusing carrier phase differential GPS (CDGPS) techniques—the techniques that surveyors and navigators use to pro-duce precise relative position estimates. Under certain cir-cumstances, these phase relationships may be used as sci-ence data in diffraction tomography. This paper presentsa method that provides this previously unexploited carrierphase data and describes the circumstances under whichthese data may be exploited to image the ionosphere. Untilnow, the proposed imaging techniques only considered am-plitude data. By providing a more complete description ofthe disturbed wavefront at the observation plane, the pro-posed method will improve the quality of the diffractiontomography estimates. The paper describes a technique forrecovering carrier phase differences between pairs of re-ceivers, demonstrates the algorithm in truth-model simula-tions, and illustrates the circumstances in which these datamay be useful in ionospheric tomography.

The remainder of this paper is divided into five ma-jor sections. The first section describes carrier phase tech-niques for relative navigation and how they may be used toproduce phase data from a scintillation imaging array. Thesecond section discusses how these data may contribute tothe tomography effort. The third section develops the al-gorithm for determining the phase differences between re-ceivers. The fourth section describes the simulations andpresents the results. The fifth section offers conclusions.

RELATIVE NAVIGATION AND SCINTILLATIONRelative navigation using CDGPS is a method by which therelative position of a pair of receivers is determined usingaccurate but biased carrier phase measurements. Centimeter-level accuracies, or better, are possible over short baselinedistances of less than 10 km [3]. Figure 4 depicts a relativenavigation scenario in which the relative position vector xbetween receivers A and B is estimated using carrier phase

x BA

j i

Figure 4. The relative navigation scenario.

measurements from pairs of GPS satellites. Only one pair, iand j, is included in the figure. Following the model devel-oped in Ref. [4], the carrier phase measurement at receiverA from GPS satellite j may be written as follows:

λφjA = ρ

jA +c(δtA−δt j)+λ(γ0A −ψ

j0)+ I j

A +T jA +ν

jA (1)

where λ is the nominal GPS carrier wavelength, φjA is the

measured carrier phase in cycles, ρjA is the line-of-sight

range from satellite j to receiver A, c is the speed of lightin a vacuum, δtA is the receiver clock error for receiver A,δt j is the satellite clock error for satellite j, γ0A is the real-valued carrier phase bias associated with receiver A, ψ

j0 is

the real-valued carrier phase bias associated with satellitej, I j

A is the bulk-quiescent ionosphere phase advance ex-pressed as an equivalent distance, T j

A is the troposphere de-lay expressed as an equivalent distance, and ν

jA is a noise

term that models thermal noise and multipath.To eliminate terms that are common to a particular

GPS satellite, measurements from the two receivers are dif-ferenced. A single-difference operator, defined as ∆(∗) j

AB =(∗) j

B−(∗) jA, is used in the following single-differenced mea-

surement equation:

λ∆(φ) jAB = ∆(ρ) j

AB + c(δtB−δtA)

+λ(γ0B − γ0A)+∆(ν) jAB (2)

In the bulk-quiescent ionosphere model, the single differ-enced atmospheric terms may safely be neglected over shortbaselines. Those terms have been dropped from Eq. (2).The clock terms and bias terms associated with satellite jcancel out. A single-differenced receiver clock term c(δtB−δtA) and a single-differenced bias term λ(γ0B−γ0A) remain.A second differencing of the measurements, this time be-

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tween satellites, leads to further cancellation. A double-difference operator, defined as ∇∆(∗) ji

AB = ∆(∗)iAB−∆(∗) j

AB,is used in the following double-differenced measurementequation:

λ∇∆(φ) jiAB = ∇∆(ρ) ji

AB +λN jiAB +∇∆(ν) ji

AB (3)

Now the receiver clock terms have cancelled out, a key re-sult for using CDGPS techniques to recover carrier phasedifferences in a non-phase-synchroized array. Additionally,the double-differenced bias term N ji

AB = (γi0B−γ

j0B

)−(γi0A−

γj0A

) is now guaranteed to be an integer and is referred to asthe integer ambiguity. Double-differenced measurementsare repeated for all commonly visible satellite pairs, andthose measurements are used to resolve the integer ambi-guities and to compute the relative position vector.

Equations (1) through (3) assume a quiet ionosphere.Consider the situation where the signals encounter irregu-larities in the ionosphere. The measured phase becomes

λφjA = ρ

jA + c(δtA−δt j)+λ(γ0A −ψ

j0)

+λδφjA + I j

A +T jA +ν

jA (4)

where the new term λδφjA is the product of the nominal

wavelength and phase disturbance due to scintillation. Af-ter double differencing, the measurements become

λ∇∆(φ) jiAB = ∇∆(ρ) ji

AB +λN jiAB

+λ∇∆(δφ) jiAB +∇∆(ν) ji

AB (5)

Notice that the same terms that cancelled out of Eq. (1)have canceled out here, but a double-differenced phase dis-turbance term remains.

For the moment, consider the terms in Eq. (5) fromthe perspective of trying to recover information about thedisturbed ionosphere. Suppose the double-differenced rangeand the integer ambiguities are known. What remains arethe measured double-differenced phase, the double-differencedphase due to scintillation, and a noise term. Rearrangingthe terms in Eq. (5) with the measured term, the knownterms, and the noise term on the right, the following CDGPSmeasurement residual can be formed:

λ∇∆(δφ) jiAB = λ∇∆(φ) ji

AB−∇∆(ρ) jiAB−λN ji

AB

−∇∆(ν) jiAB (6)

This measurement residual represents a combination of thesingle-differenced phase disturbance from satellite i and thesingle-differenced phase disturbance from satellite j:

λ∇∆(δφ) jiAB = λ[(δφ

iB−δφ

iA)− (δφ

jB−δφ

jA)] (7)

Given only this information, it is impossible to determinewhether the ionospheric irregularities that caused the phasedisturbances were encountered on the path from satellite ior satellite j, or both.

Suppose, however, that one of the signals is not scin-tillating, as depicted in the top drawing of Fig. 5. Thenthat signal’s contribution to the double-differenced phasedisturbance may be neglected, leaving only the phase dif-ference between the receivers caused by disturbances onthe signal from satellite i:

∆(δφ)iAB = λ[(δφ

iB−δφ

iA)−���

���:0

(δφjB−δφ

jA)] (8)

In the authors’ experience with scintillation monitoring, theassumption that at least one visible signal experiences neg-ligible scintillation is valid in nearly all situations. Consid-ering measurement residuals from pairs of receivers through-out the scintillation imaging array, a map of the phase dif-ferences can be constructed, as shown in bottom drawingof Fig. 5. The next section considers how these data maybe used in diffraction tomography.

xBA

j i

A

D C

B

CDGPS equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + Ij

A + T jA + νj

A

λ∆φjAB = λ(φj

B − φjA)

= ∆(ρ)jAB + c(δtB − δtA) + λ(γ0B − γ0A) + ∆(ν)j

AB

λ∇∆(φ)jiAB+λ(∆(φ)i

AB − ∆(φ)jAB)

= ∇∆(ρ)jiAB + λ[(γi

0B− γj

0B) − (γi

0A− γj

0A)

︸ ︷︷ ︸Nji

AB

] + ∇∆(ν)jiAB

Scintillation equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + λδφj

A + IjA + T j

A + ν̃jA

λ∇∆(φ)jiAB = ∇∆(ρ)ji

AB + λN jiAB + λ∇∆(δφ)ji

AB + ∇∆(ν)jiAB

λ∇∆(δφ)jiAB = λ[(δφi

B − δφiA) − (δφj

B − δφjA)]

∆(δφ)iAB

∆(δφ)iAC

∆(δφ)iAD

CDGPS block equations

zxk

zNk

zrk

=

Rxxk RxNk

0 RNNk

0 0

[

xk

N

]+

νxk

νNk

νrk

zrk∼ N (0, I)

β = 50

4 × β

1

Figure 5. The relative navigation scenario with one satellite scin-tillating (top) and the phase relationships in the scintil-lation imaging array (bottom).

Before going forward, however, one question remainsto be answered: How are the double-differenced range andthe integer ambiguities determined? The answer is that theyare the products of a calibration stage in which the rela-tive navigation solution is computed during a quiet periodprior to the onset of scintillation. Ionospheric irregularitiesmost often occur near the magnetic equator after local sun-set. Since GPS satellites are typically in view for hours,the undisturbed signals tracked before local sunset are thesame signals that are disturbed at the onset of scintillation,an important and scientifically interesting stage of scintil-

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lation phenomena. Since the integer ambiguities are con-stant, they remain valid, provided there are no cycle slips,for all continuously tracked signals—even after scintilla-tion begins. Furthermore, the receivers in the array arestatic; that is, their relative position vectors are constant.If estimated, these relative position vectors may be used inconjunction with the known GPS ephemerides to calculatethe double-differenced range.

PHASE DIFFERENCES IN DIFFRACTIONTOMOGRAPHYThe phase differences recovered from the CDGPS mea-surement residuals do not contain all the information re-quired to fully describe the phase disturbance of a GPSwavefront across a scintillation imaging array. For that, theabsolute carrier phase disturbance at each receiver wouldhave to be known. Suppose, however, that the absolute car-rier phase disturbance at one receiver could be estimated.That receiver would act as an anchor for the array, allow-ing the absolute carrier phase disturbances at the other re-ceivers to be computed using the measured phase differ-ences. This concept is illustrated in the bottom drawing ofFig. 5, where receiver A acts as the array’s anchor.

Two scenarios can be considered. In the first, eachreceiver in the array is equipped with an inexpensive fre-quency reference, e.g., a temperature compensated crystaloscillator. In this case, the absolute carrier phase distur-bance at receiver A would have to be re-estimated at eachmeasurement step as part of the overall tomography esti-mation problem. For N receivers in the array and M sam-ples, the scintillation imaging array provides M(N−1) newpieces of data to the tomography estimator.

In the second scenario, receiver A is equipped witha very stable oscillator, e.g., an ovenized crystal oscillator.For this case, consider the carrier phase measurement equa-tion for receiver A from satellite i:

λφiA = ρ

iA + c(δtA−δt i)+λ(γ0A −ψ

i0)

+λδφiA + Ii

A +T iA +ν

iA (9)

Assuming that the absolute position of the receiver is knownand that the bulk-quiescent ionosphere term I j

A and tropo-sphere term T j

A maybe modeled and removed, the followingunknowns remain: the GPS satellite ephemeris errors (theposition components of which are assumed to be includedin the line-of-sight range term ρi

A); the receiver clock error,δtA; the real-valued carrier phase measurement biases, γ0A

and ψi0; and the phase disturbance due to scintillation. The

combined effect of the GPS ephemeris errors, the stable re-ceiver clock error, and the measurement biases would benearly constant over relatively short sample periods. The

tomography estimator could use data from the imaging ar-ray to estimate this bias once for a given sample period,resulting in a measurement residuals that contain the timehistory of the absolute carrier phase disturbances at receiverA. In this scenario, the diffraction tomography estimatorgains MN−1 new pieces of data.

It is important to keep in mind that, until now, tomog-raphy estimators have exploited only amplitude data. Theuse of the phase data offered by this paper’s method willimprove the quality of the tomography estimates.

SOLUTION ALGORITHMThe solution algorithm is divided into four steps: calibrat-ing the array using CDGPS techniques, detecting the on-set of scintillation, identifying a non-scintillating referencesignal, and recovering the phase differences between pairsof receivers.

The calibration step is carried out by performing therelative navigation solution using the CDGPS techniquesdescribed in Ref. [5]. The problem formulation in Ref.[5] employs a square-root information (SRI) implementa-tion of a sequential least-squares estimator. The estimatorassembles double-differenced carrier phase measurementsfor the commonly tracked pairs of satellites, considers theambiguity a priori information, and performs the standardsquare-root information factorizations [6]. The followingblock upper-triangular system of equation results: zxk

zNk

zrk

=

Rxxk RxNk

0 RNNk

0 0

[ xkN

]+

νxk

νNk

νrk

(10)

where the vectors zxkand zNk

are the SRI vectors that asso-ciated with the relative position xk and ambiguity N states,respectively. The matrices Rxxk and RNNk are square, upper-trangular SRI matrices through which the states and the SRIvectors are related. RxNk is a dense matrix that relates theambiguity state vector and the SRI vector zxk

. The terms νxk

and νNkare zero-mean, unit-variance, white-noise random

vectors. The vector zrkis the SRI residual, and the vector

νrkis the associated zero-mean, unit-variance, white-noise

random vector. Specialized techniques are used to resolvethe integer ambiguity vector N, and the relative positionvector xk is calculated by back substitution. These esti-mates will be used in subsequent steps to construct carrierphase difference measurement residuals.

An important feature of square-root information dataprocessing is that, in the absence of measurement anoma-lies, the square-root information residual vector zrk

is aGaussian distributed, zero-mean, unit-variance random vec-tor, i.e., zrk

∼N (0, I). Measurement anomalies like carrierphase cycles slips or disturbances due to scintillation make

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0 5 10 15 20 25 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Jrk

Γ(Jrk,N)

false alarmprobability

CDGPS equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + Ij

A + T jA + νj

A

λ∆φjAB = λ(φj

B − φjA)

= ∆(ρ)jAB + c(δtB − δtA) + λ(γ0B − γ0A) + ∆(ν)j

AB

λ∇∆(φ)jiAB+λ(∆(φ)i

AB − ∆(φ)jAB)

= ∇∆(ρ)jiAB + λ[(γi

0B− γj

0B) − (γi

0A− γj

0A)

︸ ︷︷ ︸Nji

AB

] + ∇∆(ν)jiAB

Scintillation equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + λδφj

A + IjA + T j

A + ν̃jA

λ∇∆(φ)jiAB = ∇∆(ρ)ji

AB + λN jiAB + λ∇∆(δφ)ji

AB + ∇∆(ν)jiAB

λ∇∆(δφ)jiAB = λ[(δφi

B − δφiA) − (δφj

B − δφjA)]

∆(δφ)iAB

∆(δφ)iAC

∆(δφ)iAD

CDGPS block equations

zxk

zNk

zrk

=

Rxxk RxNk

0 RNNk

0 0

[

xk

N

]+

νxk

νNk

νrk

zrk∼ N (0, I)

β = 50

4 × β

1

Figure 6. The Chi-squared distribution for the cost variable Jrk .

zrkdeviate from that distribution. This fact is used to de-

velop a threshold test for detecting such anomalies. To thisend, scalar cost is defined as the sum-of-squares of the ele-ments of zrk

:

Jrk =N

∑p=1

(zprk)2 (11)

In this equation, the superscript p indicates the pth elementof the vector zrk

. The sum-of-squares of N Gaussian dis-tributed random variables with zero mean and unit varianceis distributed as a Chi-squared random variable with a meanof N and variance of 2N, i.e., Jrk ∼ χ2

N .For example, in the absence of cycle slips and scin-

tillation, the cost variable Jrk for N = 11 should be drawnfrom the distribution shown in Fig. 6. If an anomaly occurs,the value of Jrk increases so that it appears to be drawn fromthe long tail of the distribution. If a value is far out on thetail, the probability that it is in fact drawn from the distri-bution decreases. An acceptable false alarm probability isspecified as the area under the distribution’s tail beyond thevalue of a test statistic β. (Note: the false alarm probabilityarea in the figure is exaggerated. More realistic areas resultin larger β values.) The cost at each sample is comparedto the test statistic; once exceeded, the hypothesis that ameasurement anomaly has occurred is accepted.

Figure 7 plots the time history of Jrk computed using50 Hz carrier phase data for another example case. Noticehow the cost increases relatively slowly after the scintilla-tion begins. This behavior differs from that of a full cycleslip in which the cost typically jumps two or more ordersof magnitude. This contrast makes distinguishing betweenscintillation and cycle slips easy.

One may argue at this point that a more traditionalapproach for scintillation detection could be used, such as

0 20 40 60 80 1000

50

100

150

200

250

300

Samples, k

J rk

CDGPS equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + Ij

A + T jA + νj

A

λ∆φjAB = λ(φj

B − φjA)

= ∆(ρ)jAB + c(δtB − δtA) + λ(γ0B − γ0A) + ∆(ν)j

AB

λ∇∆(φ)jiAB+λ(∆(φ)i

AB − ∆(φ)jAB)

= ∇∆(ρ)jiAB + λ[(γi

0B− γj

0B) − (γi

0A− γj

0A)

︸ ︷︷ ︸Nji

AB

] + ∇∆(ν)jiAB

Scintillation equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + λδφj

A + IjA + T j

A + ν̃jA

λ∇∆(φ)jiAB = ∇∆(ρ)ji

AB + λN jiAB + λ∇∆(δφ)ji

AB + ∇∆(ν)jiAB

λ∇∆(δφ)jiAB = λ[(δφi

B − δφiA) − (δφj

B − δφjA)]

∆(δφ)iAB

∆(δφ)iAC

∆(δφ)iAD

CDGPS block equations

zxk

zNk

zrk

=

Rxxk RxNk

0 RNNk

0 0

[

xk

N

]+

νxk

νNk

νrk

zrk∼ N (0, I)

β = 50

4 × β

1Figure 7. threshold.

monitoring the commonly used S4 index. That index how-ever, is typically calculated by averaging data over a rel-atively long time interval, e.g., 30-60 s. For a real-timescintillation monitoring system, or for real-time scintilla-tion detection in relative navigation applications, anomaliesmust be detected nearly instantaneously. This paper’s pro-posed threshold test method typically achieves detection afew tenths of a second. This fast detection is illustrated inFig. 7 in terms of 50 Hz samples.

The next step in the solution algorithm is to iden-tify at least one non-scintillating signal. One side effectof rapidly detecting measurement anomalies is that thereis insufficient information in the data history to determinewhich signals are free of scintillation. In order to considermore data, a second, higher threshold is set to define theendpoint of an analysis interval. In the example case, thissecond threshold is set to 4×β, illustrated in Fig. 8. Theinterval begins at the last sample prior to the onset of scin-tillation for which the cost Jrk is less than N, the theoreticalmean for a quiet ionosphere. This interval shown betweenthe endpoints K1 and K2 in Fig. 8.

The analysis of the interval data requires the forma-tion of CDGPS measurement residuals that rely on knowl-edge of the relative position vectors and the integer ambi-guities. Locking in these values completes the calibrationstep. The relative position vector is averaged over an arbi-trary number of samples prior to the beginning of the anal-ysis interval; the integer ambiguities are taken from the lastestimate before the analysis interval. Once these calibra-tion products are saved, the following double-differencedmeasurement residual is formed for all the possible ji-pairsof GPS satellites:

∇∆φjiresk

= λ∇∆(φ) jiAB−∇∆(ρ) ji

AB−λN jiAB−∇∆(ν) ji

AB (12)

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CDGPS equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + Ij

A + T jA + νj

A

λ∆φjAB = λ(φj

B − φjA)

= ∆(ρ)jAB + c(δtB − δtA) + λ(γ0B − γ0A) + ∆(ν)j

AB

λ∇∆(φ)jiAB+λ(∆(φ)i

AB − ∆(φ)jAB)

= ∇∆(ρ)jiAB + λ[(γi

0B− γj

0B) − (γi

0A− γj

0A)

︸ ︷︷ ︸Nji

AB

] + ∇∆(ν)jiAB

Scintillation equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + λδφj

A + IjA + T j

A + ν̃jA

λ∇∆(φ)jiAB = ∇∆(ρ)ji

AB + λN jiAB + λ∇∆(δφ)ji

AB + ∇∆(ν)jiAB

λ∇∆(δφ)jiAB = λ[(δφi

B − δφiA) − (δφj

B − δφjA)]

∆(δφ)iAB

∆(δφ)iAC

∆(δφ)iAD

CDGPS block equations

zxk

zNk

zrk

=

Rxxk RxNk

0 RNNk

0 0

[

xk

N

]+

νxk

νNk

νrk

zrk∼ N (0, I)

β = 50

4 × β

1

CDGPS equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + Ij

A + T jA + νj

A

λ∆φjAB = λ(φj

B − φjA)

= ∆(ρ)jAB + c(δtB − δtA) + λ(γ0B − γ0A) + ∆(ν)j

AB

λ∇∆(φ)jiAB+λ(∆(φ)i

AB − ∆(φ)jAB)

= ∇∆(ρ)jiAB + λ[(γi

0B− γj

0B) − (γi

0A− γj

0A)

︸ ︷︷ ︸Nji

AB

] + ∇∆(ν)jiAB

Scintillation equations:

λφjA = ρj

A + c(δtA − δtj) + λ(γ0A − ψj0) + λδφj

A + IjA + T j

A + ν̃jA

λ∇∆(φ)jiAB = ∇∆(ρ)ji

AB + λN jiAB + λ∇∆(δφ)ji

AB + ∇∆(ν)jiAB

λ∇∆(δφ)jiAB = λ[(δφi

B − δφiA) − (δφj

B − δφjA)]

∆(δφ)iAB

∆(δφ)iAC

∆(δφ)iAD

CDGPS block equations

zxk

zNk

zrk

=

Rxxk RxNk

0 RNNk

0 0

[

xk

N

]+

νxk

νNk

νrk

zrk∼ N (0, I)

β = 50

4 × β

1

analysis interval

0 20 40 60 80 1000

50

100

150

200

250

300

Samples, k

J rk

Jrk =N∑

i=1

(zirk

)2 ∼ χ2N

zjik = λ∇∆(φ)ji

AB − ∇∆(ρ)jiAB − λN ji

AB − ∇∆(ν)jiAB

Jji =K2∑

k=K1

(zjik )2

Jji =∫ t2

t1

[zjimr(t)]

2dt

t1 t2 K1 K2

λ∆(δφ)iAB = λ∇∆(φ)ji

AB − ∇∆(ρ)jiAB − λN ji

AB − ∇∆(ν)jiAB

φ(x, y) A(x, y)

2

Jrk =N∑

i=1

(zirk

)2 ∼ χ2N

zjik = λ∇∆(φ)ji

AB − ∇∆(ρ)jiAB − λN ji

AB − ∇∆(ν)jiAB

Jji =K2∑

k=K1

(zjik )2

Jji =∫ t2

t1

[zjimr(t)]

2dt

t1 t2 K1 K2

λ∆(δφ)iAB = λ∇∆(φ)ji

AB − ∇∆(ρ)jiAB − λN ji

AB − ∇∆(ν)jiAB

φ(x, y) A(x, y)

2

Figure 8. The analysis interval.

An interval cost is defined for each pair by summing thesquares of the measurement residuals over the analysis in-terval:

J jiint =

K2

∑k=K1

(∇∆φjiresk

)2 (13)

Each pair is ranked according to its interval cost. Rely-ing on the previously stated assumption that at least onesignal experiences negligible scintillation, the lowest costpair should contain at least one clean signal. The choicebetween the signals in the lowest cost pair is guided byconsidering the costs of each of those signals paired withall the remaining signals. While this selection method maybe more formally posed in terms of statistical hypothesistesting, this ad hoc approach has proven effective.

Let the non-scintillating satellite be designated by j.Then by substituting the relative position vectors and the in-teger ambiguities determined from the calibration stage, thescintillation-induced phase differences between receivers Aand B on the signal from satellite i can be expressed as thefollowing CDGPS measurement residual equation, whereλ∇∆(φ) ji

AB, ∇∆(ρ) jiAB, and λN ji

AB are known:

λ∆(δφ)iAB = λ∇∆(φ) ji

AB−∇∆(ρ) jiAB

−λN jiAB−∇∆(ν) ji

AB (14)

RESULTS FROM TRUTH-MODEL SIMULATIONSThe solution algorithm proposed in this paper has beendemonstrated in truth-model simulations. Two differentsimulators have been combined for this purpose. The firstis a scintillation simulator that was originally developed fortesting GPS receiver tracking loops. It models amplitudevariations as following a Ricean distribution and the spec-trum of the rapidly varying component of complex-valued

scintillation as following a low-pass second-order Butter-worth filter. The severity of the simulated scintillation iscontrolled by two parameters: the S4 index and a decorre-lation time constant τ0. Figure 9 shows a comparison be-tween empirically-derived scintillation data for which S4 =0.91 and τ0 = 0.5 s on the left and simulated data with thesame parameters and with a nominal carrier-to-noise ratioof 45 dB-Hz on the right. The top plots represent the re-ceived carrier-to-noise ratio time histories; the bottom plotsrepresent time histories of the carrier phase disturbancesdue to scintillation. Notice how the deep power fades inboth the empirical and simulated data are accompanied byrapid, half-cycle phase transitions. This behavior stressesa receiver’s tracking loops and may cause CDGPS algo-rithms to diverge from the correct integer ambiguity esti-mates. For more details on this simulator, refer to Ref. [7].

0 2 4 6 8 10 12 14 16 18 2025

30

35

40

45

50

C/N 0, d

B−Hz

Simulated Data

0 2 4 6 8 10 12 14 16 18 20−5

−4

−3

−2

Time, s

δ φ, c

ycle

s

0 2 4 6 8 10 12 14 16 18 2025

30

35

40

45

50

C/N 0, d

B−Hz

Real Data

0 2 4 6 8 10 12 14 16 18 20−1

0

1

2

Time, s

δ φ, c

ycle

s

Figure 9. A comparison between empirically-derived scintilla-tion data (left) and simulated scintillation data (right).

To generate a scintillation scenario, a typical two-receiver static baseline (100-m) CDGPS scenario is gen-erated. A random number of GPS satellites is chosen to bescintillating. The scintillation start times are chosen ran-domly from a window that begins after a sufficient amountof non-scintillating data has been generated to allow thecalibration stage to resolve the integer ambiguities. The S4

and τ0 parameters are randomly specified for both scintil-lating satellites and non-scintillating satellites. The scin-tillating satellites’ parameters are drawn from a Gaussiandistribution that result in strong scintillation, whereas thenon-scintillating satellites’ parameters are drawn from adistribution that result in weak scintillation. The mean val-ues and standard deviations of these distributions are sum-marized in Table 1. Once the parameters for a particular

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Table 1. Scintillation parameter distributions for simulations.

S4 τ0 (s)

Strongmean 0.9 0.5

st. dev. 0.1 0.1

Weakmean 0.1 30

st. dev. 0.1 10

signal have been specified, scintillation time histories aregenerated at receiver A and receiver B independently. Thisapproach does not take into account any spacial or time cor-relation between the scintillation received at receiver A andthe scintillation received at receiver B. The authors believethis approach is reasonable considering the large separationdistances envisioned. The question of correlation may beaddressed in the future by using a more complicated scin-tillation model based on phase screens.

The carrier phase measurements are modified to in-clude not only the scintillation-induced carrier phase dis-turbances but also an increase in carrier phase measurementerror associated with the instantaneous drops in carrier-to-noise ratio. This phase measurement model, however,omits a potential source of error: those introduced by thephase-lock loop’s inability to accurately track rapid phasechanges. The question of how to design phase-lock loopsthat can track such phase transitions is the subject of ongo-ing research [8, 9]. This paper assumes that the simulatedphase-lock loops exhibit robust phase tracking capability.Figure 10 shows the carrier-to-noise ratio time history fora scintillating signal at one of the receivers during a typi-cal simulation. The scintillation begins about 12 s into thescenario and exhibits the typical deep fading of severe scin-tillation.

Before considering the results from the phase differ-ence recovery algorithm, it is interesting to consider theeffect that severe scintillation has on precise relative navi-gation algorithms. Figure 11 shows the relative position er-ror magnitude time history during an example simulation.Once the integer ambiguities have been resolved, the rel-ative position error magnitude is less than 5 mm. Whenthe first deep power fade and associated phase disturbanceare encountered, the relative navigation estimator divergesfrom the correct integer ambiguity estimates, resulting inan instantaneous jump in error from 5 mm to 14 m. The er-ror eventually reaches 24 m. Errors like these may have se-rious consequences for certain applications—precision air-craft approach and landing, for instance.

When the phase difference recovery algorithm is ap-plied to the simulated data, it detects the onset of scin-

0 2 4 6 8 10 12 14 16 18 2026

28

30

32

34

36

38

40

42

44

46

Time, s

C/N 0 a

t Rec

eive

r A, d

B−Hz

Figure 10. The time history of the received carrier-to-noise ratioat receiver A.

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

Time, s

Erro

r Mag

nitu

de, m

Figure 11. The time history of the relative position error magni-tude during scintillation.

tillation in a few tenths of a second worth of data, cor-rectly identifies a non-scintillating satellite, and providesthe phase relationships between the two receivers for allof the other tracked satellites. Figure 12 shows the timehistory of the true and recovered phase difference betweenthe receivers for the satellite pair that includes the refer-ence satellite, PRN 8, and a scintillating satellite, PRN 17.Although the recovery algorithm detects the onset of scin-tillation quickly, it does not begin reporting phase differ-ences until after the analysis interval. Once reporting, therecovered phase difference tracks the true phase differenceswell, even during the fast phase transitions that are associ-ated with deep power fades.

Figure 13 shows a time history of the errors in therecovered phase differences. After the analysis interval,

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0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8PRN Pair: 8−17

Time, s

∆ (δ

φ),

cyc

les

MeasuredTrue

Figure 12. The time histories of the true and recovered phase dif-ferences.

which is marked by the large errors between 12 s and 13 sin scenario time, the phase difference errors are less thanabout 0.1 cycles and include two components: a rapidlyvarying part and a nearly constant part. The rapidly vary-ing component is caused by the carrier phase measurementnoise at the two receivers. Notice the jump in error whenthe carrier-to-noise ratio drops twice between 14 s and 15 sin scenario time. The nearly constant component is causedby the assumption that the phase disturbance due to scin-tillation for the reference satellite is zero. That satellitedoes in fact experience scintillation—very weak scintilla-tion with a small S4 index and a very long decorrelationtime constant τ0. The effect is a small, slowly varyingphase disturbance that looks like a bias over the severalseconds considered in the example scenario. Other, smallereffects that may contribute to this bias component includeerrors in the relative position vector estimate and errors inthe broadcast GPS satellite ephemerides.

CONCLUSIONSA method has been developed that recovers the phase rela-tionships between pairs of independent GPS receivers op-erating in large scintillation imaging arrays. These phaserelationships contribute to the science data that will be usedto image the disturbed ionosphere with diffraction tomog-raphy techniques. The phase recovery method uses CDGPStechniques to calibrate a non-phase-synchronized array andto recover the phase differences between the receivers. Theonset of scintillation is detected with a threshold test, and amethod for identifying a non-scintillating reference signalhas been developed. The system has been tested in truth-model simulations. The simulations generate two-receiverstatic baseline CDGPS scenarios that include random andsystematic errors. The simulated carrier phase measure-

0 5 10 15 20−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Time, s

∆ (δ

φ)

Err

or,

cyc

les

Figure 13. The time history of the error in the recovered phasedifferences.

ments are altered to include the effects of scintillation andthe associated increase in phase measurement error.

The system rapidly detects the onset of scintillation,typically within a few tenths of a second. It defines an anal-ysis interval, typically about 1 s long, that provides enoughinformation to correctly identify a non-scintillating refer-ence satellite. The recovered phase differences accuratelyrepresent the true phase differences, including tracking thefast half-cycle phase transitions associated with deep powerfades. The errors are typically less than 0.1 cycles and havetwo components: a rapidly varying part that is due to thephase measurement error and a bias part that is due to theerroneous assumption that the reference signal has no scin-tillation effects. That signal does in fact experience veryweak scintillation with a long decorrelation time constant,an effect that would appear to be a bias over the short sam-ple periods considered in the simulations.

The diffraction tomography estimator’s sensitivity tothese errors has yet to be determined. If it is very sensitive,a more sophisticated, model-based approach to recoveringthe carrier phase disturbances may be necessary.

References[1] Yeh, K.C., and Liu, C.-H., “Radio Wave Scintillations

in the Ionosphere,” Proc. of the IEEE, 70, 324-360,1982.

[2] Bernhardt, P.A., Siefring, C.L., Galysh, I.J., Rodilosso,R.F., Doch, D.E., MacDonald, R.L., Wilkens, M.R.,and Landis, G.P., “Ionospheric Applications of theScintillation and Tomography Receivers in Space (CIT-RIS) Used with the DORIS Radio Beacon Network,”Journal of Geodesy, Vol. 80, Nos. 8-11, Nov. 2006, pp.473-485.

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[3] Parkinson, B.W., and Enge, P.K., “Differential GPSand Integrity Monitoring,” in Global Positioning Sys-tem: Theory and Applications, Vol. II, Parkinson, B.W.,and Spilker, J.J. Jr., eds., American Institute of Aero-nautics and Astronautics, Washington, 1996, pp. 3-49.

[4] Psiaki, M.L., and Mohiuddin, S., ”Modeling, Measure-ment, and Simulation of GPS Carrier-Phase for Space-craft Relative Navigation,” Journal of Guidance, Con-trol, and Dynamics, to appear.

[5] Mohiuddin, S., and Psiaki, M.L., “High-AltitudeSpacecraft Relative Navigation Using Carrier-PhaseDifferential Global Positioning System Techniques,”Journal of Guidance, Control, and Dynamics, Vol. 30,No. 5, September-October 2007, pp. 1427-1436.

[6] Bierman, G.J., Factorization Methods for Discrete Se-quential Estimation, Academic Press, New York, 1977.

[7] Humphreys, T.E., Psiaki, M.L, and Kintner, P.M Jr.,“Simulating Ionosphere-Induced Scintillation for Test-ing GPS Receiver Phase Tracking Loops,” to be sub-mitted to IEEE Transactions on Aerospace and Elec-trical Systems, preprints are available upon request.

[8] Psiaki, M.L., Humphreys, T.E., Cerruti, A., Powell,S.P., and Kintner, P.M. Jr., “Tracking L1 C/A and L2CSignals through Ionospheric Scintillations,” to appearin the Proc. of the 2007 ION GNSS Conf., Sept. 25-28,2007, Fort Worth, TX.

[9] Humphreys, T.E., M.L. Psiaki, B.M. Ledvina, and P.M.Kintner Jr., “GPS Carrier Tracking Loop Performancein the Presence of Ionospheric Scintillations,” Proc. ofthe 2005 ION GNSS Conf., Sept. 13-16, 2005, Instituteof Navigation, Long Beach, CA.

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