multi-frequency gnss robust carrier tracking for

J. Space Weather Space Clim. 2017, 7, A26© J. Vilà-Valls et al., Published by EDP Sciences 2017DOI: 10.1051/swsc/2017020

Available online at:www.swsc-journal.org

RESEARCH ARTICLE

Multi-frequency GNSS robust carrier tracking for ionosphericscintillation mitigation

Jordi Vilà-Valls1,*, Pau Closas2 and James T. Curran3

1 Centre Tecnològic de Telecomunicacions de Catalunya (CTTC/CERCA), Barcelona, Spain2 Department of Electrical and Computer Eng., Northeastern University, Boston, MA 02115, USA3 European Space Agency (ESA), Noordwijk, The Netherlands

Received 11 May 2017 / Accepted 14 August 2017

*Correspon

This is anOp

Abstract – Ionospheric scintillation is the physical phenomena affecting radio waves propagating from thespace through the ionosphere to earth. The signal distortion induced by scintillation can pose a major threatto some GNSS application. Scintillation is one of the more challenging propagation scenarios, particularlyaffecting high-precision GNSS receivers which require high quality carrier phase measurements; and safetycritical applications which have strict accuracy, availability and integrity requirements. Under ionosphericscintillation conditions, GNSS signals are affected by fast amplitude and phase variations, which cancompromise the receiver synchronization. To take into account the underlying correlation among differentfrequency bands, we propose a new multivariate autoregressive model (MAR) for the multi-frequencyionospheric scintillation process. Multi-frequency GNSS observations and the scintillation MAR aremodeled in state-space, allowing independent tracking of both line-of-sight phase variations and complexgain scintillation components. The resulting joint synchronization and scintillation mitigation problem issolved using a robust nonlinear Kalman filter, validated using real multi-frequency scintillation data withencouraging results.

Keywords: GNSS / ionospheric scintillation / multivariate AR modeling / robust tracking / carrier phasesynchronization / adaptive nonlinear Kalman filter

1 Introduction

Global Navigation Satellite Systems (GNSS) is thetechnology of choice for most position-related applicationswhen it is available (Dardari et al., 2015).AGNSS receiver relieson a constellation of satellites to estimate a set of rangemeasuresfrom which to compute its position. The main challenges ofGNSS technology arise when operating in complex, harshpropagation scenarios which are naturally impaired by multi-path, shadowing, high dynamics, or ionospheric scintillation. Inthe last decade, ushered by an ever increasing demand foravailability, accuracy, and reliability, the mitigation of thesechallenges has steered intense research on advanced receiverdesign (Amin et al., 2016).

Among these challenges, in this paper we focus onionospheric scintillation. Because such disturbance is notrelated to the local environment, as in the case of multipath orshadowing, it can degrade receiver performance even underideal open-sky conditions. Ionospheric scintillation is caused

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en Access article distributed under the terms of the Creative CommonsAunrestricted use, distribution, and reproduction in any m

by a disturbance in the portion of the ionosphere throughwhich the GNSS signals propagates, an effect particularlyrelevant in polar and equatorial regions. It can producerapidly varying constructive and destructive interferencebetween multiple scattered signals. This turbulent behaviorcan render the carrier phase difficult to track by the receiver.One of the major challenges of severe scintillation propaga-tion conditions is known as canonical fading, which ischaracterized by a combination of strong fading and rapidphase changes in a simultaneous and random manner (Kintneret al., 2009; Humphreys et al., 2009). From a synchronizationstandpoint, measuring the signal phase during such a fade isvery challenging because rate of change of phase changes is atits highest when the signal amplitude is near its lowest(Humphreys et al., 2005). This may lead the receiver tomomentarily lose carrier synchronization, resulting in eithercycle-slips or loss of lock. Unfortunately, the accuracyimprovement of modern high-precision receivers is based onthe use of carrier phase-based positioning techniques, and soit is extremely sensitive to scintillation perturbations(Jacobsen & Dähnn, 2014; Prikryl et al., 2014; Banville &Langley, 2013, Banville et al., 2010).

ttribution License (http://creativecommons.org/licenses/by/4.0), which permitsedium, provided the original work is properly cited.

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J. Vilà-Valls et al.: J. Space Weather Space Clim. 2017, 7, A26

Traditional synchronization relies on well-known closed-loop DLL/PLL architectures (Kaplan, 2006). However,receiver architectures based on DLL/PLL are not always ableto provide continuous and reliable phase tracking (Humphreyset al., 2005). Most of state-of-the-art (SoTA) techniquespropose methods to robustify DLL/PLL schemes via finetuning and augmentation (Yu et al., 2006; López-Salcedo et al.,2014), either by using adaptive bandwidth schemes (Skoneet al., 2005) or combined architectures (Chiou, 2010; Won,2014). It has already been shown that Kalman filter (KF)-basedsolutions provide better performances and robustness in thesesituations (Humphreys et al., 2005; Macabiau et al., 2012),being nowadays the performance benchmark for the develop-ment of new methodologies (Vilà-Valls et al., 2017b). Thenatural extension is to use adaptive KF-based solutions (Zhanget al., 2010; Won et al., 2012), which aim at sequentially adaptthe filter parameters. However, even these advanced techni-ques do not solve the estimation versus mitigation trade-off(Vilà-Valls et al., 2015b), because the filter is not able todecouple the different phase contributions (i.e., dynamics andscintillation), and so may provide poor scintillation mitigationcapabilities. A scalar tracking approach to overcome theselimitations was proposed in Vilà-Valls et al. (2015a,b), wherethe scintillation physical phenomena is statistically modeledusing autoregressive (AR) processes and embedded in the filterformulation. This approach was shown to outperform theSoTA techniques using synthetic data, being effectively able todiscriminate between dynamic and scintillation contributionsto the estimated carrier-phase. The Cornell Scintillation Model(CSM) (Humphreys et al., 2009, 2010), a popular syntheticsignal simulator that generates scintillation amplitude andphase traces, was used to test the AR-based filteringmethodology. Recent studies of real equatorial and high-latitude data have shown that scintillation characteristics arefrequency-dependent (Carrano et al., 2012; Sokolova et al.,2015; Jiao et al., 2016). While phase disturbances may becorrelated, deep amplitude fades tend to occur at differenttimes on different frequencies, thus motivating the potentialperformance gain of using inter-frequency aiding and multi-frequency architectures.

This paper exploits the statistical relationships betweenscintillation at different frequency bands to propose a multi-frequency carrier tracking method that virtually mitigatesscintillation component. The paper extends our previous workon single-frequency carrier tracking loops under scintillationconditions (Vilà-Valls et al., 2015a,b), for which a state-spaceformulation and a Kalman filter (KF) was considered. Here, anaugmented state-space formulation considering multiplefrequencies is investigated and a tracking method proposedon the top of it. Remarkably, the resulting method is validatedwith real data measurements showing notable performance.

To summarize, in this article, we further explore the ideaand preliminary results in Vilà-Valls et al. (2017a) with thefollowing four main contributions:
– multivariate AR ionospheric amplitude and phase scintil-lation modeling, featuring a joint fitting of the multi-frequency scintillation time series. An optimal model orderselection based on the Bayesian Information Criterion isused. This is discussed in Section 3;
–
the novel multi-frequency joint carrier phase andscintillation state-space model (SSM) formulation, that
of 14

takes into account the potential correlation at differentfrequency bands is introduced in Section 4;

–
the proposed robust filter that performs state estimation onthe aforementioned SSM is discussed in Section 5. Thesolution is based on a nonlinear, adaptive KF formulation;
–
as opposed to previous works where synthetic data wasused, the proposed method is analyzed using realionospheric scintillation data in Section 6.
2 Ionospheric scintillation

Scintillation is caused by local variations in the structure ofthe ionospheric electron density along line-of-sight betweenthe satellite and the receiver. The receiver observes signals thatare refracted and diffracted such that the phase and amplitudeof the signal exhibits large stochastic perturbations. Theseverity of the scintillation is traditionally quantified by twoindices: an amplitude scintillation index, denoted S4; and aphase scintillation index, denoted sf. The indices arecomputed on a per-signal basis and indicate average intensityof the signal variations over the preceding minute. They havebeen used for some decades and as a result there exist richdatabases of historical data for a wide range of observationpoints.

Unfortunately these indices offer no insight into the time-correlatedness of the instantaneous phase and amplitude ofscintillation observed on multiple frequencies. A number ofrecent studies have provided empirical evidence that the phasescintillation exhibits a high degree of correlation betweendifferent GNSS frequencies, including L1, L2 and L5. It hasbeen suggested that this correlation persists except in the caseof strong amplitude scintillation, when deep amplitude fadesoccur (Sokolova et al., 2015; Carrano et al., 2012; Jiao et al.,2016). This implies that even when strong phase scintillation isobserved, which is not uncommon at high latitudes, thesevariations in the carrier will be correlated across frequencies.When a receiver observes both amplitude and phasescintillation, then it has been observed that the phase variationsare correlated depending on the scintillation intensity, butbecome decorrelated once deep amplitude fades begin to occur(Sokolova et al., 2015). These deep amplitude fades, it hasbeen shown, tend to be uncorrelated across frequencies(Carrano et al., 2012).

From a carrier tracking perspective, these two observationscan be exploited to improve and robustify carrier phasetracking. Firstly, the fact that the phase variations are highlycorrelated implies that a centralized, or joint tracking of thecarrier across multiple GNSS frequencies can improvetracking accuracy (Sokolova et al., 2015). Secondly, the factthat deep amplitude fades occur at different times on differentfrequencies, implies that it is highly unlikely that a receiverwill loose lock on multiple frequencies simultaneously,suggesting that it should be possible to assist one frequencywith another (Sokolova et al., 2015; Carrano et al., 2012).These concepts are explored in the work presented here.

The experimental work presented in this manuscript, wasconducted on data obtained from the Joint Research Centre(JRC) Scintillation Repository a collection of data with morethan 10 hours of scintillation events recorded over Hanoi in

2040 2060 2080 2100 2120 2140time (s)

-50

-40

-30

-20

-10

0

10A

mpl

itude

(dB

)

GPS L1GPS L2GPS L5

Fig. 1. An example of 100 s of real equatorial scintillation amplitudedata for a severe scintillation event at three different GPS frequencybands.


March and April, 2015 (Curran et al., 2015b). These datasetswere based on a reconfigurable quad-channel front-end,Fourtune, which had been programmed to collect triple-frequency data, L1, L2 and L5, using 1-bit complex samplingat rates of 5 MHz, 5 MHz, and 30 MHz, respectively (Curranet al., 2015b). The recording system was configured tocontinuously record 50-minute datasets and to post-processeach using an L1 software defined receiver for the purposes ofbasic scintillation detection. Those datasets in which severescintillation was identified were then archived for post-processing, and the others discarded (Curran et al., 2015a).

The post-processing stage employed a multi-frequencyopen-loop software receiver which exploited precise knowl-edge of the receiver location, and the well-disciplinedreference oscillator to generate accurate reference carrierand code local replicas (Curran et al., 2015b). Oncedemodulated to complex baseband, the correlator valuescorresponding to each observed GNSS signal were processedto estimate the phase and amplitude perturbations induced byionospheric activity. Being an open-loop post-processingscheme, a batch estimation of the amplitude and phase waspossible, providing highly accurate and reliable characteriza-tion (Curran et al., 2015b). From these 1 kHz estimatesofcarrier phase and amplitude, traditional measures ofscintillation activity were computed, such as S4 and sf.

Figure 1 shows an example of real equatorial ionosphericscintillation amplitude data for a scintillation event at threedifferent GPS frequency bands on one particular satellite. Thisis an example of strong scintillation, exhibiting a few fadesbelow –30 dB, and many prolonged fades below –10 dB.

3 Multivariate AR scintillation signal model

Ionospheric scintillation can be modeled as a multiplica-tive channel js(t), with the received base-band signal being

xðtÞ ¼ jsðtÞsðtÞ þ wðtÞ; jsðtÞ ¼ rsðtÞejusðtÞ; ð1Þ

where rs (t) and us (t) are the ionospheric scintillation envelopeand phase components, respectively; s(t) is the baseband

o

equivalent of the transmitted signal; and w(t) is a randomnoiseterm including both thermal noise and any other disturbance.The amplitude scintillation strength is typically described bythe scintillation index S4, and is usually considered withinthree main regions (Humphreys et al., 2009): weak, moderateand severe,

S4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðr4s Þ � ðEðr2s ÞÞ2

ðEðr2s ÞÞ2

s;

S4 � 0:3 ðweakÞ0:3 < S4 � 0:6 ðmoderateÞ0:6 < S4 ðsevereÞ

;

8<: ð2Þ

It was shown in Vilà-Valls et al. (2013, 2015a,b) for thesingle-frequency case that both scintillation components canbe well approximated using an autoregressive (AR) process.The scalar AR scintillation model is given by

rs;k ¼Xqi¼1

girs;k�i þ kþ hr;k; hr;k�Nð0; s2hrÞ; ð3Þ

us;k ¼Xpi¼1

bius;k�i þ hu;k; hu;k � Nð0; s2huÞ; ð4Þ

with k a constant value, and the mean of the amplitudescintillation process equal to m ¼ k / ð1�Pq

i¼1 g iÞ; {g i, bi}are the set of AR coefficients, and {s2

hr,s2

hu} the driving noise

variances. In Vilà-Valls et al. (2017a), we considered thisscalar AR approximation to build a multi-frequency SSM,where the model order selection and AR parameters wereobtained from synthetic scintillation data. Using this scalarapproximation in a multi-frequency scenario is a simplisticapproach, because it does not take into account the possiblecorrelations among frequencies, and therefore loses thecharacteristics of the physical phenomena (see Section 2).To overcome the limitations of the scalar AR processes modelin this context, we propose a new multivariate AR (MAR)ionospheric scintillation formulation. Real ionospheric scin-tillation data was used, selecting the model order andparameters.

Definition 1

(MAR(p)) Amultivariate ARmodel of order p is defined as

zk ¼ wþXp

i¼1

Aizk�1 þ ek ; ð5Þ

where p is the model order; zk = [zk,1, ... , zk,d]⊤ is the kth

sample of a d-dimensional time-series; A1 to Ap ∈ Rd�d arethe coefficient matrices of the MAR model; the random vectorterm ek = [ek,1, ... , ek,d]

⊤ is a zero-mean Gaussian randomvariable with covariance matrix Se ∈ Rd�d. ek is assumeduncorrelated in time (i.e., E {ekeℓ} = 0 for k≠ ℓ), but possiblycorrelated among time-series (i.e., Se can containoff-diagonalelements); w∈ Rd is a parameter vector of intercept terms toallow a nonzero mean time-series, which implies an expected

f 14


value of

zk ¼ I�Xp

i¼1

Ai

!�1

w: ð6Þ

The two main MAR modelling challenges are: (i)selectionthe correct model order selection p, and (ii) thetime-series analysis to obtain an estimate of the MAR modelparameters, fw; fAigpi¼1;Seg. Both points are discussed in theremainder of this section.

3.1 MAR model order selection

In the statistical and signal processing literature, theAkaike Information Criterion (AIC) (Akaike, 1974), exten-sions of the AIC such as the corrected AIC (AICc) and thegeneralized information criterion (GIC), and the BayesianInformation Criterion (BIC) (Schwarz, 1978), are widely usedmetrics for model order selection. The latter has become themethod of choice mainly because: i) while the AIC has anonzero overfitting probability, this tends to zero in the BICwhen the sample size grows, and ii) it is an asymptoticapproximation of the optimal maximum a posteriori (MAP)rule (Stoica and Selen, 2004). In this contribution we use theBIC as the model order selection criterion.

In this subsection we define the quantities for a generictime-series composed of N samples, y ∈ RN, and assume u isthe vector parameters of the model with order p, which we aimto select.

Definition 2

(BIC) The Bayesian Information Criterion is defined as(Stoica and Selen, 2004)

BIC ¼ �2 ln ppðy; uÞ þ p lnN ; ð7Þ

with y being the available data, u the parameter vector (whosedimension can vary with the model order) and p corresponds tothe model order. pp (y, u) is the likelihood function whichdepends on the parameter vector u and the model order p, andthe maximum likelihood estimate of the parameters is givenby u ¼ argmaxu ln ppðy; uÞ: The BIC rule selects the order pthat maximizes (7).

Definition 2 corresponds to the BIC when a single time-series is fitted. We are, however, interested in jointly fittingmultiple time-series (from the various frequencies). Thisconcept is generalized in Definition 3.

Definition 3

(Multivariate BIC) The Bayesian Information Criterion forthe multiavariate ARmodel MAR(p) in Definition 1 yields to amodel order estimate according to Neumaier and Schneider(2001).

o

p ¼ argminp

Lp

d� 1� dpþ 1

N

� �lnN

� �; ð8Þ

where Lp ¼ ln det�ðN � dp� 1ÞSeðpÞ

�and SeðpÞ is an

estimate of the noise covariance matrix obtained from the Nsamples.

The estimates of the noise covariance matrix SeðpÞ aretypically obtained for a range of model order values, pmin � p� pmax. An efficient iterative way to compute an estimateof Lp using a QR factorization and a stepwise downdatingstrategy is shown in Neumaier and Schneider (2001).

3.2 MAR model parameters

Several MAR model parameters estimation methods basedon the minimization of the prediction error are available in theliterature. A thorough comparison of the most commonestimators is available in Schlögl (2006). The main conclusionis that when enough samples are available (i.e., N ≥ 400), theperformance of the different methods is similar. In thiscontribution we propose to use the stepwise least squaresestimation method proposed in Neumaier and Schneider(2001), which is available as a MatlabTM package named ARfit(Schneider & Neumaier, 2001). This method first rewrites theMAR model as a regression model, and then obtains theparameters using a least squares method.

3.3 Ionospheric scintillation MAR formulation

The MAR multi-frequency ionospheric scintillation modelconsidering M frequency bands (i.e., B1 to BM) is given by

rs;k ¼Xqi¼1

Girs;k�i þ kþ er;k ; er;k ∼Nð0;SrÞ; ð9Þ

us;k ¼Xqi¼1

Gius;k�i þ eu;k; eu;k ∼Nð0;SuÞ; ð10Þ

where rs;k ¼ ½rB1s;k; ⋯ ; rBM

s;k �T, us;k ¼ ½uB1s;k; ⋯ ; uBM

s;k �T, and theensemble of q þ p þ 3 MAR model parameters is given byfGigqi¼1; k; fDigpi¼1;Sr;Su��

. Notice that each of these matri-ces has dimension M � M, given by the number of frequencybands. It is worth saying that the MARmodel noise covariancematrices, Sr and Su, provide a good indicator of the inter-frequency correlation of the time-series. It has been foundempirically from the estimated noise covariances that whileSr

is almost diagonal (i.e., no correlation in the ionosphericscintillation amplitude), significant values off the maindiagonal appear in Su (i.e., higher inter-frequency correlationin the ionospheric scintillation phase).

A key point to exploit theMAR scintillation approximationwithin a filtering framework is the model order selection. InVilà-Valls et al. (2013, 2015a,b), the selection was doneinspecting the power spectral densities (PSD), and subse-quently in Vilà-Valls et al. (2017a), we used a statistical testbased on the partial autocorrelation functions (PAF). Bothanalyses were done using synthetic data. In this contributionwe use the BIC in Definition 3 and perform the fitting on realmulti-frequency scintillation data.

f 14

500 1000 1500 2000Time (s)

40

45

50

55

C/N

0 (

dBH

z)

500 1000 1500 2000Time (s)

0

0.2

0.4

0.6

0.8

S4

GPS L1 C/AGPS L2 CMGPS L5 Q

1600 1800 2000 2200 2400 2600 2800Time (s)

303540455055

C/N

0 (

dBH

z)

1600 1800 2000 2200 2400 2600 2800Time (s)

0

0.2

0.4

0.6

0.8

1

S4

GPS L1 C/AGPS L2 CMGPS L5 Q

Fig. 2. C/N0 and S4 scintillation index estimation at three GPSfrequency bands for two ionospheric scintillation events recordedover Hanoi in March and April 2015.

0 2 4 6 8 10 12Scintillation Amplitude Model Order q

0

1000

2000

0 2 4 6 8 10 12Scintillation Phase Model Order p

0

500

1000

1500

Fig. 3. Empirical model order probability density functions obtainedfrom real multi-frequency ionospheric scintillation data.


Figure 2 shows theC/N0 (carrier-to-noise density ratio) andamplitude scintillation index estimations, S4, at three differentGPS frequency bands (i.e., L1, L2, and L5). These resultscorrespond to two ionospheric scintillation events recordedover Hanoi in March and April 2015 (cf. Sect. 2). While thefirst event (top figure) corresponds to a moderate to lowscintillation scenario, the second one (bottom figure) is clearlya strong scintillation case. Thus, the real data available (over 3hours of equatorial scintillation) contains both low, moderateand severe scintillation scenarios.

To obtain the model order for both amplitude and phasescintillation processes, we processed over 4 hours of 3-dimensional time-series to compute the BIC metrics in (8).BIC results are obtained from non-overlapped sequences of N= 500 samples (5 s). In contrast to the single-frequency fittingto synthetic data given in Vilà-Valls et al. (2013, 2015a,b), theconclusion from the MAR analysis is that a suitable modelorder to fit the majority of scintillation scenarios is given by q =6 and p = 5, i.e. a MAR(6) and MAR(5) for the scintillationamplitude and phase, respectively. These results are in contrastto the analysis of previous contributions (based on synthetic

o

data), where lowerorders were sufficient. To justify the choiceof the model order, we plot in Figure 3 the histograms of theorder selection obtained from real multi-frequency ionosphericscintillation data.

4 Multi-frequency GNSS and ionosphericscintillation state-space model

4.1 Single frequency GNSS signal model

The generic GNSS baseband transmitted signal is given by

sðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2PxðtÞ

pdðt � tðtÞÞcðt � tðtÞÞejuðtÞ; ð11Þ

with Px (t) the received power, d(t) the navigation message andc(t) the spreading code, t(t) the code delay and u(t) = 2pfd(t)þue(t) the carrier phase, where fd(t) is the carrier Dopplerfrequency shift and ue (t) a carrier phase term including otherphase impairments. After the acquisition stage, the samples atthe output of the correlators are (Spilker, 1996)

yk ¼ AkdkRðDtkÞsin ðpDf d;kTsÞ

pDf d;kTsejð2pDf d;kTsþDukÞ þ nk;

where Ts is the sampling rate, k stands for the discrete time tk =kTs,Ak is the signal amplitude at the output of the correlators, dkis the data bit, R(⋅) is the code autocorrelation functionand Dtk ;Df d;k;Duk

n oare, respectively, the code delay,

Doppler shift and carrier phase errors. Focusing on carrier-tracking, we work under the assumption of perfect timingsynchronization and data wipe-off, resulting in a signal model

yk ¼ akejuk þ nk; nk � Nð0; s2

n;kÞ; ð12Þ

where amplitude ak and phase ukmay include any non-nominalharsh propagation disturbance. For instance, including theionospheric scintillation into the signal model leads to ak =Akrs,k and uk = ud,k þ us,k. The dynamics phase term ud,k refersto the line-of-sight (LOS) phase evolution due to the relativemovement between the satellite and the receiver.

f 14


Standard carrier tracking architectures, being proportionalplus integral controller in the form of a PLL, model ud,k using aTaylor approximation of the time-varying phase evolutionaccording to the expected dynamics. Typically, a 3rd orderrepresentation of the dynamics phase evolution is considered

ud;k ¼ u0 þ 2pðf d;kkTs þ f r;kk2T2

s / 2Þ; ð13Þ

where u0 (rad) is a random phase value, fd,k (Hz) the carrierDoppler frequency, and fr,k (Hz/s) the Doppler frequency rate.

4.2 Multi-frequency GNSS state-space model

In a multi-frequency architecture, a single satellitetransmits simultaneously at different frequencies, thereforeat the receiver side several measurements are available.Although the methodology in this paper is general for anyGNSS signal, we focus on a modern GPS triple band satellite,which transmits on the L1, L2 and L5 frequencies. In this case,the measurements are given by

yL1kyL2kyL5k

264

375 ¼

aL1k e

jðuL1d;k

þuL1s;k

Þ

aL2k e

jðuL2d;k

þuL2s;k

Þ

aL5k e

jðuL5d;k

þuL5s;k

Þ

26664

37775þ

nL1knL2knL5k

264

375; ð14Þ

which may be useful to rewrite in real form as

yL1i;kyL1q;kyL2i;kyL2q;kyL5i;kyL5q;k

26666666664

37777777775

|fflfflffl{zfflfflffl}yk

¼

aL1k

cosðuL1k ÞsinðuL1k Þ

" #

aL2k


" #

aL5k


" #

26666666664

37777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}hðxkÞ

þ

nL1i;knL1q;knL2i;knL2q;knL5i;knL5q;k

26666666664

37777777775

|fflfflfflffl{zfflfflfflffl}nk

; ð15Þ

where for each frequency, yk = yi,kþ iyq,k, uk = ud,kþ us,k and nk= ni,k þ jnq,k.

The different LOS carrier-phase evolutions, whenreferencing all of the dynamics to L1, are given by

uL1d;k

uL2d;k

uL5d;k

264

375 ¼

uL10 þ 2pðf d;kkTs þ f r;kk

2T2s=2Þ

uL20 þ 2pf L2=f L1ðf d;kkTs þ f r;kk

2T2s=2Þ

uL50 þ 2pf L5=f L1ðf d;kkTs þ f r;kk

2T2s=2Þ

264

375; ð16Þ

with f L1 , f L2 and f L5 being the carrier frequencies at thedifferent bands. It is important to notice that the same Dopplerfrequency terms, fd,k and fr,k, appear in both frequencies. Themulti-frequency noise diagonal covariance matrix is

Rk ¼ diagðs2nL1;k

; s2nL1;k

; s2nL2;k

; s2nL2;k

; s2nL5;k

; s2nL5;k

Þ / 2:

In general, xk refers to the state to be tracked and is defined

as xk≐ x⊤d;k ; x⊤r;k; x

⊤u;k

h i⊤with dimension nx = 5 þ 3q þ 3p and

o

xd;k≐ uL1d;k; u

L2d;k ; u

L5d;k; f d;k; f r;k

⊤; ð17Þ

xr;k≐ r⊤s;k ; r⊤s;k�1; ⋯ r⊤s;k�qþ1

⊤; ð18Þ

xu;k≐ u⊤s;k; u⊤s;k�1; ⋯ u⊤s;k�pþ1

⊤: ð19Þ

The LOS dynamic carrier-phase state evolution

xd;k ¼

1 0 0 2pTs pT2s

0 1 0 2pdL2Ts pdL2T2s

0 0 1 2pdL6Ts pdL5T2s

0 0 0 1 Ts

0 0 0 0 1

0BBBB@

1CCCCA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Fd

xd;k�1 þ vd;k ; ð20Þ

where dL2 ¼ f L2 / f L1 , dL5 ¼ f L5 / f L1 and vd;k � Nð0;Qd;kÞstands for possible uncertainties or mismatches on thedynamic model. The covariance Qd,k is usually a priori fixedaccording to the problem under consideration.

From the MAR scintillation model in (9), the multi-frequency scintillation amplitude state evolution is

xr;k ¼G1 ⋯ Gq�1 Gq

IM⋱

IM

0BB@

1CCA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Fr

xr;k�1 þ kr þ vr;k; ð21Þ

where the elements not specified are 0, IM refers to the identitymatrix of dimension M � M, vr;k � Nð0;Qr;kÞ with blockdiagonal covariance matrix Qr,k= blkdiag (Sr, 0M(q-1)�M(q-1))and kr is a null vector except for the firstM values given by k.Equivalently, we define the scintillation phase state evolutionfrom (10) as

xr;k ¼D1 ⋯ Dq�1 Dq

IM⋱

IM

0BB@

1CCA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Fu

xr;k�1 þ vu;k ; ð22Þ

with vu;k � Nð0;Qu;kÞ and block diagonal covariance matrixQu,k = blkdiag (Su, 0M(p-1)�M(p-1)). Finally, the complete stateevolution equation reads,

xk ¼ diagðFd;Fr;FuÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Fk

xk�1 þ kT þ vk ; ð23Þ

where the Gaussian process noise has a block diagonalcovariance matrix Qk = blkdiag(Qd,k, Qr,k, Qu,k). Equations(15) and (23) define the new SSM formulation. Notice that incontrast to the SSM in Vilà-Valls et al. (2017a), in this case the

f 14

Fig. 4. Block diagram of the new multi-frequency nonlinear EKF-based architecture.

Table 1. Real equatorial multi-frequency ionospheric scintillation data characterization.

Event Low Scint Moderate Scint Severe Scint

# 1 t= 1500–2500s t = 100–1500s

# 2 t= 100–1800s# 3 t= 100–1700s t= 1700–2800s# 4 t = 700–2000s t= 100–700, 2000–2500s# 5 t= 100–2900s# 6 t = 700–950s, 1300–1700s t= 100–700, 950–1300, 1700–2900s

The vector differential operator is defined as ∇ ¼ ∂∂x1 ; . . . ;

∂∂xN

h i.


possible correlation between scintillation amplitude and phaseat different frequency bands is accounted for.

5 New robust filtering methodologyIn this article we elaborate on the previous KF-based

architecture proposed in Vilà-Valls et al. (2017a), and extendedin Section 4, where the scintillation physical phenomena ismathematically modeled and embedded into the state-spaceformulation. A nonlinear extended KF (EKF) formulationallows to directly operate with the received signal samples,avoiding the problems associated to the use of a phasediscriminator, that is, nonlinearities, loss of Gaussianity andpossible saturation at lowC /N0.Using the SSMin (15) and (23),and the MAR formulation in Section 3.3, recall that the mainparameters of the filter are

FkðCÞ ¼ diagðFd;FrðfGig6i¼1Þ;FuðfDigpi¼1Þ;

Qk ¼ blkdiagðQd;k ;Qr;kðSrÞ;Qu;kðSuÞÞ;

Rk ¼ diagðs2nL1;k

; s2nL1;k

; s2nL2;k

; s2nL2;k

; s2nL5;k

; s2nL5;k

Þ / 2;

whichmainlydependon the amplitudeMAR(6) andphaseMAR(5) setofparametersobtainedfromfitting to realmulti-frequencydata, C ¼ ½fG ig6i¼1; kT ; fDig5i¼1;Sr;Su�. The complete state

o

xk (nx = 38) to be tracked is defined in equation (25), Qd,k is apriori fixed by the user from the expected dynamics, and themeasurementnoisevariances (i.e.,s2

nL1;k, s2

nL2;kands2

nL5;k) canbe

sequentially adjusted from the nominalC /N0 estimators at eachfrequency band, being available at the receiver.

The main idea behind the EKF is to linearize the nonlinearmeasurement function around the predicted state estimate, andthen use the standard linear KF equations (Anderson andMoore1979). In this case, the measurement function is given by

hkðxkÞ ¼

AL1k r

L1s;kcosðuL1d;k þ u

L1s;kÞ

AL1k r

L1s;ksinðuL1d;k þ u

L1s;kÞ

AL2k r


L2s;kÞ

AL2k r


L2s;kÞ

AL5k r


L5s;kÞ

AL5k r


L5s;kÞ

26666666664

37777777775; ð24Þ

and the linearized (Jacobian) measurement matrix to be used inthe KF measurement update step is1

~Hk ¼ ∇hkðxkjk�1Þ ¼ ~Hd;k; 06� 2; ~Hr;k ; 06� 15; ~Hu;k ; 06� 12
1
f 14

Table 2. Root mean square phase tracking error considering real scintillation data at different GPS frequency bands for the low scintillationevents #3 (t = 200–700s) and #2 (t = 200–700s).

Event #3 RMSEt(L1) RMSEt(L2) RMSEt(L5)

PLL 0.0804 0.0772 0.0767

KF 0.0465 0.0419 0.0392AKF 0.0465 0.0421 0.0392AKF-AR 0.0451 0.0408 0.0373AEKF-AR 0.0116 0.0122 0.0116MAR-EKF 0.0093 0.0092 0.0069


PLL 0.1476 0.0812 0.0766



with the different matrices given in equations (26)–(28). Notethat the matrix is evaluated at the predicted state, xk ¼ xkjk�1,with u

Ljkjk�1≐u

Ljd;kjk�1 þ u

Ljs;kjk�1 for each frequency.2

xk≐ uL1d;k ; u

L2d;k; u

L5d;k; f d;k ; f r;k; r

L1s;k; r

L2s;k; r

L5s;k ; . . . ;

hrL1s;k�5; r

L2s;k�5; r

L5s;k�5; u

L1s;k ; u

L2s;k ; u

L5s;k; . . . ; u

L1s;k�4; u

L2s;k�4; u

L5s;k�4�⊤

ð25Þsee equation (26) below

~Hu;k ¼

�AL1k r

L1s;kjk�1sinðuL1kjk�1Þ

AL1k r

L1s;kjk�1cosðuL1kjk�1Þ

0 �AL2k r

L2s;kj

0 AL2k r

L2s;kjk�

0

0

266666666664

~Hd;k ¼

�AL1k r


AL1k r


0 �AL2k r

L2s;kj

0 AL2k r

L2s;kjk�

0

0

266666666664

2k|k-1 refers to the predicted estimate at time k using measurements up

to time k - 1, and k|k refers to the estimated value at time k usingmeasurements up to time k.

o

~Hr;k ¼

AL1k cosðuL1kjk�1Þ 0 0

AL1k sinðuL1kjk�1Þ 0 0

0 AL2k cosðuL2kjk�1Þ 0

0 AL2k sinðuL2kjk�1Þ 0

0 0 AL5k cosðuL5kjk�1Þ

0 0 AL5k sinðuL5kjk�1Þ

266666666664

377777777775;

ð27Þsee equation (28) below

0 0

0 0

k�1sinðuL2kjk�1Þ 0

1cosðuL2kjk�1Þ 0

0 �AL5k r


0 AL5k r


377777777775; ð28Þ

0 0

0 0

k�1sinðuL2kjk�1Þ 0

1cosðuL2kjk�1Þ 0

0 �AL5k r


0 AL5k r


377777777775; ð26Þ

f 14

500 1000 1500 2000 2500Time (s)

0

0.2

0.4

0.6

0.8

S4

20150326 160024 PRN06

GPS L1C/AGPS L2CMGPS L5Q

0.6

20150402 130756 PRN06


For the sake of completeness, the EKF formulation of thegeneral carrier-phase tracking problem is sketched inAlgorithm 1. Notice that the Kalman gain, Kk, is computedfrom the possibly time-varying system noise matrices, Qk andRk, which implies that the filter is adapting its behavior to theactual working conditions and has an implicit adaptivebandwidth, in contrast to the traditional PLL with constantcoefficients (i.e., fixed bandwidth) (Vilà-Valls et al. 2017b).

Algorithm 1 Carrier-phase tracking EKF formulation

200 400 600 800 1000 1200 1400 1600 1800Time (s)

0

0.2

0.4

S4

500 1000 1500 2000 2500Time (s)

00.20.40.60.8

1

S4

20150402 130756 PRN30

500 1000 1500 2000 2500Time (s)

00.20.40.60.8

1

S4

20150402 151046 PRN06

GPS L1C/AGPS L2CMGPS L5Q

500 1000 1500 2000 2500Time (s)

0

0.2

S4

20150408 125245 PRN01

500 1000 1500 2000 2500Time (s)

00.20.40.60.8

1

S4

20150408 125245 PRN30

Fig. 5. Estimated scintillation index S4 for the 6 sets of real multi-frequency equatorial scintillation data, i.e., #1 (top) to #6 (bottom).

In practice, we compute an estimate of the MARparameters using a subset of the real multi-frequencyionospheric scintillation data available. To obtain meaningfulresults, in the simulations we use a small sample set to obtainthe model parameters and process the rest of the scintillationtime-series to assess the filter performance. In other words, wetest the robustness of the method by decoupling the modelfitting from the state tracking. Finally, the block diagram of thenew multi-frequency nonlinear EKF-based architecture forefficient scintillation mitigation is sketched in Figure 4. Itincludes both the state estimation procedure and the update ofthe prediction/estimation error covariance matrices.

6 Results and discussion

The performance of the proposed method (named MAR-EKF in the simulations) is analyzed using the real equatorialmulti-frequency ionospheric scintillation data described inSection 2. We processed 6 sets of data (i.e., over 4 hours ofdata), each corresponding to a different scintillation event,containing low, moderate and severe ionospheric scintillationpropagation conditions (see Eq. (2)). The different events aredescribed in Table 1, with the corresponding estimation of thescintillation index S4 shown in Figure 5. From the state vectorin (25), the parameters of interest are the phase componentsrelated to the dynamics of the receiver, i.e., obtaining a goodestimate of these parameters directly implies good ionosphericscintillation mitigation capabilities. Therefore, this is the finalgoal of the new receiver architecture. Note that in someapplicationsit may be interesting to obtain an estimate of thescintillation amplitude and phase components, rs,k and us,k .

The results obtained with the MAR-EKF are compared tothe current state-of-the-art techniques, and the latest contri-butions using scalar AR scintillation model approximations:
– 3rd order PLL with equivalent bandwidth Bw = 5 Hz.
of

–

14

Standard discriminator-based KF-based tracking.
– Adaptive discriminator-based KF (AKF), adjusting themeasurement noise variance from the C/N0 estimate.
–
The scalar AKF-AR presented in Vilà-Valls et al. (2015b),only tracking thescintillation phase, and using a C/N0
estimate to tune the measurement noise variance at theoutput of the discriminator.

–
The scalar AEKF-AR presented in Vilà-Valls et al. (2015a),tracking both scintillation amplitude and phase, treatingeach frequency separately.

200 300 400 500 600Time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Est

imat

ion

erro

r fo

r d,

L5(r

ad)

AKF-ARAEKF-ARMAR-EKF

Fig. 6. Steady-state instantaneous dynamics phase estimation errorfor the moderate scintillation event #1 (t = 200–600s) at L5.

Table 3. Root mean square phase tracking error considering realscintillation data at different GPS frequency bands for the moderatescintillation events #1 (t = 500–1000s) and #4 (t = 1300–1800s).


PLL 0.0886 0.1304 0.1490



PLL 0.0988 0.1267 0.1299



We consider a Doppler profile given by an initial randomphase in [-p, p], initial Doppler fd,0 = 50 Hz and ratefr,0 = 100Hz/s, and a relatively low nominal C /N0 = 30 dB–Hz. The signal is sampled at Ts = 10 ms. The measure ofperformance is the root mean square error (RMSE) on thecarrier-phase of interest, i.e., uL1d;k , u

L2d;k and uL5d;k . We define the

RMSE (radians) over time for each frequency Lj as

RMSEtðLjÞ ¼ 1

Nt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNt

i¼1

uLjd;i � u

Ljd;i

� �2vuut ; ð29Þ

which is obtained from long real data sets of Nt samples.The proposed methodology is characterized via an in depth

analysis, considering the steady-state performance for severalrepresentative scenarios: (i) a low scintillation scenario whichis used as an architecture validation, (ii) scintillation mitigationcapabilities in challenging moderate and severe scintillationscenarios, (iii) robustness to modeling mismatch, where wetake into account possible scintillation under and overestima-tion, and (iv) the estimation performance for the ionosphericscintillation amplitude and phase components.

6.1 Low scintillation scenario: architecture validation

In general, it is held that low ionospheric scintillationconditions are given for S4 < 0.4. From Figure 5, we easilyidentify these conditions in the scintillation events #1 (t = 1500– 2500s), #2 (t = 100 – 1800s), #3 (t = 100 – 1700s) and #5 (t =100 – 1900s). A desired behavior of any tracking methoddesigned to mitigate moderate and severe scintillationconditions is that it performs at least as good as the standardtracking techniques when low or no scintillation is present.Therefore, analyzing the performance of the proposed MAR-EKF in such scenarios can be understood as a representativearchitecture validation procedure.

We show the RMSEt (Lj) in radians, obtained for twodifferent low scintillation scenarios (500s of data), in Table 2.In this case, we can see that the performance obtained with the

two EKF-based solutions, i.e., AEKF-AR and MAR-EKF, issimilar but slightly better with theMAR-EKF. Moreover, thesemethods provide a lower RMSE than the one given by the PLLand the three discriminator-based KF solutions, i.e., KF, AKFand AKF-AR. Therefore, we have two main conclusions: i) theproposed architecture is a valid approach even if noscintillation is present in the signal, that is, the filter is robustand including the scintillation components into the state to betracked does not induce an error on the estimation of thedynamics related phase, and ii) it may be preferable to avoiddiscriminator-based architectures and operate directly with thereceived signal samples. Moreover, in this case we can includethe scintillation amplitude into the state formulation, thusbeing aware of possible fading and amplitude variations.

6.2 Assessing the scintillation mitigation capabilitiesin moderate and severe scintillation scenarios

The main goal of the proposed architecture is to effectivelycounteract the undesired ionospheric effects in moderate andsevere scintillation scenarios, without sacrificing performancein nominal scenarios (cf. Section 6.1). In this section, weanalyze the performance of the MAR-EKF under harshpropagation conditions as per Table 1.

6.2.1 Moderate scintillation

For the sake of completeness, we plot an example of thesteady-state instantaneous dynamics phase estimation error forthe moderate scintillation event #1 (t= 200–600s) at L5,obtained with the AKF-AR, AEKF-AR and the MAR-EKF.Notice that we do not plot the results for the other methodsbecause their performance is significantly poorer (see Table 3for RMSE results). In view of the results, the benefits of thenew multi-frequency approach with respect to the other filtersusing scalar AR approximations are apparent.

To obtain statistically more representative results, we showthe RMSEt (Lj), obtained for events #1 (t = 500 – 1000s) and #4

of 14

1800 1900 2000 2100 2200Time (s)

0

0.5

1

1.5

2

2.5

3

3.5

Est

imat

ion

erro

r fo

r d,

L2 (

rad)

AKF-ARAEKF-ARMAR-EKF

Fig. 7. Steady-state instantaneous dynamics phase estimationerror forthe severe scintillation event #6 (t= 1800–2200s) at L2.

2140 2145 2150 2155 2160 2165 2170Time (s)

-2

-1

0

1

2

Pha

se

s

2140 2145 2150 2155 2160 2165 2170Time (s)

10-1

100

Am

plitu

de

s

Real Scintillation DataMAR-EKF Estimate

Fig. 8. Scintillation amplitude and phase estimation for the severescintillation event #3 at L5.

Table 4. Root mean square phase tracking error considering realscintillation data at different GPS frequency bands for the strongscintillation events #3 (t = 2000–2500s), #4 (t = 2000–2500s) and #6(t = 1700–2200s).


PLL 0.5400 0.6568 0.6928



PLL 0.1527 0.2324 0.2330



PLL 0.7083 0.8431 0.8805



(t = 1300 – 1800s), in Table 3. These results verify theappropriate behavior of the new architecture, indicating that itoutperforms the other methods under moderate scintillationconditions. Again, there is a significant performance improve-ment attained when avoiding the use of a discriminator.Specifically, the performance obtained with the AEKF-AR andMAR-EKF is improved with respect to the othermethods. This

is due to the Gaussianity assumption,which is does not holdwhen working with a discriminator under fading conditions(Vilà-Valls et al. 2017b).

6.2.2 Severe scintillation

Finally, we assess the performance of the MAR-EKF inchallenging, severe scintillation conditions. Figure 7 shows anexample of the steady-state instantaneous dynamics phaseestimation error for the severe scintillation event #6 (t= 1800–2200s) at L2, obtained with the AKF-AR, AEKF-AR and theMAR-EKF. The best estimation performance is given by theMAR-EKF.

The RMSEt (Lj), obtained for events #3 (t = 2000 – 2500s),#4 (t = 2000 – 2500s) and #6 (t = 1700 – 2200s), is shown inTable 4. These results confirm that the new architecture issuperior to the rest of tracking methods also under severescintillation, providing a significant performance improvementwith respect to standard approaches and the scalar EKF-basedsolution in Vilà-Valls et al. (2015a), which justifies the interestand importance of considering a multi-frequency approach.Overall, the MAR-EKF appears to be relatively robust againsta wide range scintillation propagation conditions.

6.3 Robustness to modeling mismatch

To fully characterize the proposed methodology, weanalyze the impact of an incorrect MAR model parametersestimation on the filterperformance, that is, fitting thescintillation amplitude MAR(6) and phase MAR(5) to a setof real data with a different scintillation behavior than theactual signal being processed. Two cases are considered: i)underestimation of the scintillation conditions, which impliesthat while the filter deals with a severe scintillation scenario,the MAR model parameters are fitted to moderate scintillationdata; and ii) overestimation of the scintillation conditions,which refers to themoderate scintillation case with MARmodel parameters fitted to severe scintillation data.

of 14

0 500 1000 1500 2000 2500 3000Time (s)

-8

-6

-4

-2

0

2

4

Pha

se e

rror

(cy

cles

)

PLLStd KFAKFAKF-ARAEKF-ARMAR-EKF

Fig. 9. Phaseerror for event #6, GPS L2.

Table 5. Root mean square phase tracking error considering realscintillation data under modeling mismatch for the severe scintillationevent #6 (t = 1700–2200s) and the moderate scintillation event #6 (t =1300–1700s).

Event #6 Severe RMSEt(L1) RMSEt(L2) RMSEt(L5)

MAR-EKF under 0.3922 0.3196 0.3248

MAR-EKF correct 0.2344 0.2087 0.2007

Event #6 Moderate RMSEt (L1) RMSEt (L2) RMSEt (L5)

MAR-EKF over 0.1118 0.1128 0.1341

MAR-EKF correct 0.0894 0.0701 0.0681

Table 6. Root mean square scintillation components trackingerrorconsidering real scintillation data for the moderate scintillation event#1 (t = 500–1000s) and the severe scintillation event #3 (t = 2000–2500s).

Event #1 RMSEtðrs;L1Þ RMSEtðrs;L2Þ RMSEtðrs;L5ÞAEKF-AR 0.0587 0.1047 0.0769

MAR-EKF 0.0605 0.1034 0.0740

Event #1 RMSEtðus;L1Þ RMSEtðus;L2Þ RMSEtðus;L5ÞAKF-AR 0.0800 0.1776 0.1698

AEKF-AR 0.0581 0.1035 0.0965MAR-EKF 0.0456 0.0764 0.0551

Event #3 RMSEtðrs;L1Þ RMSEtðrs;L2Þ RMSEtðrs;L5ÞAEKF-AR 0.0969 0.0933 0.0987

MAR-EKF 0.0784 0.0790 0.0771

Event #3 RMSEtðus;L1Þ RMSEtðus;L2Þ RMSEtðus;L5ÞAKF-AR 0.5044 0.4431 0.5202

AEKF-AR 0.3421 0.2916 0.2702MAR-EKF 0.1968 0.1945 0.1916

Table 7. Number of cycle slips for the real ionospheric scintillationevent #6.

PLL KF AKF AKF-AR AEKF-AR MAR-EKF

Event #6 L1 1 0 1 1 0 0

Event #6 L2 5 2 16 2 0 0Event #6 L5 8 6 12 1 1 0


The RMSEt (Lj), obtained for the severe scintillation event#6 (t = 1700 – 2200s) considering an underestimated fitting,and for the moderate scintillation event #6 (t = 1300 – 1700s)with an overestimated fitting, is shown in Table 5. It isinteresting to note that the performance degradation whenhaving a model mismatch is reasonably bounded. In practice, itis always desirable to overestimate, rather than underestimate,the scintillation intensity.

6.4 Scintillation amplitude and phase estimation

In this section we analyze the estimation performance forthe scintillation amplitude and phase components, rs,k and us,k .We define the corresponding RMSE over time for eachfrequency Lj as

RMSEtðrs;LjÞ ¼1

Nt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNt

i¼1

ðrLjs;i � rLjs;iÞ2

vuut ; ð30Þ

RMSEtðus;LjÞ ¼1

Nt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNt

i¼1

ðuLjs;i � uLjs;iÞ2

vuut ; ð31Þ

which are obtained from Nt data samples. Notice that standardmethods track the complete phase of the signal, therefore wedo not have an estimate of the scintillation components. Theresults obtained with theMAR-EKF are compared to the scalarAKF-AR (i.e, only scintillation phase) and AEKF-AR.

First we plot an example of the scintillation amplitude andphase estimates delivered by the MAR-EKF, for the severescintillation event #3 at L5, in Figure 8. We can observe aremarkable recovery of scintillation components by the MAR-EKF filter. Both RMSEtðrs;LjÞ and RMSEtðus;LjÞ are shown inTable 6. Again, we confirm that the MAR-EKF provides thebest estimation performance for both scintillation amplitudeand phase components. The performance gain with respect tothe other methods using scalar AR approximations is clear forthe severe scintillation scenario. In the moderate scintillationcase, the performance obtained with the AEKF-AR and theMAR-EKF are equivalent.

of 14


6.5 Cycle slip performance analysis

To conclude the characterization of the new methodology,we assess its robustness to cycle slips, which is particularlynecessary in carrier-based positioning techniques such as real-time-kinematic (RTK) and precise-point-positioning (PPP)that rely on the integrity of carrier phase measurements.

In terms of cycle slips, the most challenging scenario is thesevere scintillation event #6. We can see in Figure 5 that thisevent has the highest values for the scintillation index S4 . Weshow the phase error for the different methods in Figure 9.While some cycle slips appear with the discriminator-basedPLL, KF, AKF and AKF-AR, the methods including bothscintillation components into the SSM formulation, AEKF-AR, MAR-EKF, are more robust (i.e., no cycle slips). Theseresults confirm the enhanced filtering performance of the newapproach. Notice that for this particular scenario, the numberof cycle slips with the AKF is higher than with the standardPLL and KF. This is because the filter is not able to cope withthe strong fadings during severe scintillation due to the fastchanges on the estimated C/N0, which controls the noisevariance at the output of the discriminator. This is also the casewith the discriminator-based AKF-AR, which does not takeinto account the scintillation amplitude. The number of cycleslips for event #6 and the three GPS frequency bands is shownin Table 7, which again confirm the superiority in terms ofrobustness of the new MAR-EKF.

7 Conclusions

Carrier-phase measurements can be severely degraded dueto ionospheric scintillation. This paper proposes a new trackingmethodology that is able to extract the desired phasecomponent, used for positioning, from the scintillationcomponent. A key point is first to model both ionosphericscintillation components, amplitude and phase, using an ARmodel formulation, and then include them into the state to betracked by the filter. This allows to decouple the phasecontribution due to the LOS dynamics from the phaseperturbations induced by the ionosphere. Particularly, we wantto exploit the advantages of multi-frequency architectures,because the scintillation characteristics are frequency-depen-dent. To take into account possible correlations amongfrequencies we must resort to multivariateAR models. Anew state-space model is introduced and a Kalman-type stateestimator operating on the outputs of the correlatorsimplemented. The new method is validated using realscintillation data recorded in Hanoi, March and April 2015.It has been shown that the new filter outperforms state-of-the-art solutions in the tested scenarios, both in terms of meansquare error and robustness to cycle slips. This approachprovides a solution to the estimation/mitigation challengewhen dealing with scintillation corrupted measurements.

One of the main challenges in the design of the proposedsolution is the estimation of the multivariate AR modelparameters. While in the analysis conducted in this article theseparameters have been obtained by fitting the desired model to asmall part of the real time-series available (i.e., then processing adifferent part of the data), in real-life conditions, where thescintillation conditions evolve over time, we should consider aniterative procedure to adapt the filter to the actual working

conditions. Another important point is how to fix the LOS phasedynamics covariance matrix, which depends on the expecteddynamics. To fully characterize the new method, it would bedesirable to have a larger set of data, recorded at different pointsover the earth, at different latitudes and containing severalevents. Even if the set of data available is representative ofdifferent scintillationconditions, itwouldbe interesting toobtaina statistically more representative analysis.

Acknowledgments. This work has been supported by theSpanish Ministry of Economy and Competitiveness throughproject TEC2015-69868-C2-2-R (ADVENTURE) and by theGovernment of Catalonia under Grant 2014–SGR–1567. Theeditor thanks two anonymous referees for their assistance inevaluating this paper.

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