Research on the Eloran Differential Timing Method

sensors

Letter Research on the eLoran Diﬀerential Timing Method

Yun Li 1,2,*, Yu Hua 1, Baorong Yan 1 and Wei Guo 1

1 National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China; [email protected] (Y.H.); [email protected] (B.Y.); [email protected] (W.G.) 2 Physics Department, University of the Chinese Academy of Sciences, Beijing 100049, China * Correspondence: [email protected]; Tel.: +86-1868-291-1626

Received: 14 September 2020; Accepted: 12 November 2020; Published: 14 November 2020

Abstract: An enhanced long-range navigation (eLoran) system was selected as the backup of Global Navigation Satellite Systems (GNSS), and experts and scholars are committed to improving the accuracy of the eLoran system such that its accuracy is close to the GNSS system. A differential method called eLoran differential timing technology is applied to the eLoran system, which has been used in maritime applications of eLoran. In this study, an application of eLoran differential timing technology in a terrestrial medium is carried out. Based on the eLoran timing service error, the correlation of the timing service error is analyzed in theory quantitatively to obtain the range of the difference station in the ground. The results show that to satisfy the timing accuracy of 100 ns, the action range of eLoran difference station on the land needs to be less than 55 km. Therefore, the eLoran differential method is proposed, and in the difference station, the theoretical calculation is combined with the measurement of the signal delay to obtain the difference information, which is sent to the users to adjust the prediction delay and improve the eLoran timing precision. The experiment was carried out in the Guan Zhong Plain, and the timing error of the user decreased from 394.7287 ns (pre-difference) to 19.5890 ns (post-difference). The proposed method is found to effectively enhance the timing precision of the eLoran system within the scope of action.

Keywords: eLoran; diﬀerence; timing service; error; correlation

1. Introduction After delaying the instalment of enhanced long-range navigation (eLoran) stations for a certain period of time, the United States announced, on 4 December 2018, that eLoran stations will be used as a backup for Global Navigation Satellite Systems (GNSS). Moreover, simultaneously, a high-precision terrestrial time service system is being developed in China. The effective radiated power of eLoran is in the range of 100 to 1000 kW, and it is nearly impossible to jam over large areas with such high power levels. This is the reason why eLoran is used as an alternative to the GNSS system. Unfortunately, eLoran is less accurate than GNSS. Several studies have been conducted to improve the precision of the eLoran system to match that of the GNSS system. Many experts and scholars focus on the prediction of signal propagation delay, especially the additional secondary factor (ASF) under special topographic conditions [1–4]. By using the ASF grid, the accuracy of the eLoran system can be improved [5–8]. To develop the ASF grid, surveys have been conducted for the ASF values. However, conducting investigation and measurement of ASF for the entire area is expensive. Moreover, ASF depends on the ground conductivity along the propagation path of the signal, and any change in conductivity due to weather and season will affect ASF. To supply the real-time variation of the ASF, the eLoran differential timing method has been proposed, especially for maritime applications, in which numerous experiment and data analysis are performed [5,6].

Sensors 2020, 20, 6518; doi:10.3390/s20226518 www.mdpi.com/journal/sensors Sensors 2020, 20, x FOR PEER REVIEW 2 of 14 Sensors 2020, 20, 6518 2 of 14 In this study, the eLoran differential timing service is analyzed theoretically and this method is mainlyIn this used study, in terrestrial the eLoran media. diff erentialThe correlation timing serviceof eLoran is analyzedtiming service theoretically error is analyzed and this methodin theory isquantitatively, mainly used in and terrestrial the action media. range of The difference correlation station of eLoran on the ground timing is service calculated error with is analyzed an accuracy in theoryof 100 quantitatively, ns. Based on the and correlation the action of range the errors, of difference the eLoran station differential on the ground timing is calculatedservice is proposed. with an accuracyThen an of experiment 100 ns. Based is carried on the correlationout to test ofand the verify errors, the the eLoran eLoran differential differential timing timing method service isby proposed.conducting Then a survey an experiment in the Guan is carriedZhong outPlain to of test the and Shanxi verify province. the eLoran differential timing method by conducting a survey in the GuanZhong Plain of the Shanxi province. 2. eLoran Signal and Propagation Delay 2. eLoran Signal and Propagation Delay 2.1. eLoran Signal 2.1. eLoran Signal eLoran is a low frequency (100 kHz) terrestrial navigation and timing system. The transmitters are eLoranorganiz ised a lowinto frequencychains, with (100 one kHz) master terrestrial station navigation and several and secondary timing system. stations The in transmitters each chain. areThe organizedmaster station into chains, transmits with nine one masterpulses stationand second and severalary station secondarys transmit stations eight in pulses each chain. at a time The. Figure master 1 stationillustrates transmits the shape nine pulsesof the andsignal secondary, and the stationsred dot transmitrepresents eight standard pulses z atero a time.-crossing Figure (SZC1 illustrates), which is thethe shape point of at thewhich signal, the andsignal the is redtracked dot represents by an eLoran standard receiver zero-crossing and it is use (SZC),d to calculate which istime the-of point-arrival. at which the signal is tracked by an eLoran receiver and it is used to calculate time-of-arrival.

SZCSZC

FigureFigure 1. 1.The The enhanced enhanced long-range long-range (eLoran) (eLoran pulse) pulse shape. shape The red. The dot red represents dot represents standard standard zero-crossing zero- (SZC),crossing which (SZC), is thewhich point is the at point which at the which signal the issignal tracked is tracked by an by eLoran an eLoran receiver receiver and and it is it usedis used to to calculatecalculate time-of-arrival. time-of-arrival. 2.2. eLoran Propagation Delay 2.2. eLoran Propagation Delay The eLoran signal is transmitted radially from the transmitter to the receiver, and travels parallel The eLoran signal is transmitted radially from the transmitter to the receiver, and travels parallel to the surface of the earth. During this propagation, the signal does not travel at the speed of light, to the surface of the earth. During this propagation, the signal does not travel at the speed of light, but it is slowed by the atmosphere and the surface of the earth. The time taken for the eLoran signal to but it is slowed by the atmosphere and the surface of the earth. The time taken for the eLoran signal reach the receiver is called propagation time (Tp), which is shown in the following equation to reach the receiver is called propagation time (푇푝), which is shown in the following equation = + Tp푇푝 =PFPF +ASF,ASF, (1)(1) wherewhere if theif the signal signal propagates propagates in the in inﬁnitethe infinite air medium, air medium, the time the taken time to taken reach to the reach receiving the antennareceiving fromantenna the transmitting from the transmitting antenna is calledantenna the is primary called factorthe primary (PF). Owing factor to(PF the). dielectricOwing to properties the dielectric of theproperties earth surface, of the the earth signal surface, will travel the signal relatively will slowlytravel asrelatively compared slowly to that as incompared the atmosphere. to that in This the delayatmosphere. is termed Th asisan delay ASF. is termed as an ASF.

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The refractive index of the atmosphere 푛푠 implies that the speed of the signal is a fraction slower than Thethe speed refractive of light index in ofvacuum. the atmosphere The PF is nrelateds implies to the that distance the speed S, ofand the this signal is expressed is a fraction as follows slower than(Equation the speed (2)) [9,10] of light. in vacuum. The PF is related to the distance S, and this is expressed as follows (Equation (2)) [9,10]. R∫ 푛푠 푑푆 PF = nsdS (2) PF = 퐶 (2) C where 퐶 is the speed of light in a vacuum and is equal to 0.299792458 km/μs, S is the distance of where C is the speed of light in a vacuum and is equal to 0.299792458 km/µs, S is the distance of signal signal propagation, and 푛푠 is the atmospheric refraction index of the ground, which varies with propagation, and n is the atmospheric refraction index of the ground, which varies with temperature temperature humiditys and air pressure. humidity and air pressure. The earth surface medium (seawater and land) delay signal transmission, and the time taken for The earth surface medium (seawater and land) delay signal transmission, and the time taken the eLoran signal in the procedure is ASF. With the decrease in the electrical conductivity of the for the eLoran signal in the procedure is ASF. With the decrease in the electrical conductivity of the surface, a high proportion of the signal will penetrate the ground and the wave-front will propagate surface, a high proportion of the signal will penetrate the ground and the wave-front will propagate more slowly. The calculation formula of ASF is shown in Equation (3) [9,10]. more slowly. The calculation formula of ASF is shown in Equation (3) [9,10]. 106 ASF = 푎푟푔 푊 (푓, 푑, 휎, 휀), (3) 10휔6 ASF = arg W( f , d, σ, ε), (3) where 휔 is the angular velocity in rad/s, 푊ω is the signal attenuation function, which is related to wheresignal ωfrequencyis the angular 푓, the velocity distance in of rad signal/s, W propagationis the signal attenuation푑, the electrical function, conductivit whichy is 휎 related and dielectric to signal frequencyconstant 휀 f ,of the signal distance propagation of signal path propagation, and 푎푟푔d,푊 the is electrical the phase conductivity of the 푊 inσ andrad. dielectric constant ε of signal propagation path, and arg W is the phase of the W in rad. 3. Principle of eLoran Timing Service and Error Analysis 3. Principle of eLoran Timing Service and Error Analysis 3.1. Principle of eLoran Timing Service 3.1. Principle of eLoran Timing Service The transmitter broadcasts the timing signal, which propagated in the channel and arrived at the receiver.The transmitter In the receiver, broadcasts the the signal timing is processed signal, which and propagated demodulated in the to channelobtain the and one arrived-pulse at-p theer- receiver.second (1PPS In the), receiver,and then the the signal receiver is processed local 1PPS and signal demodulated is synchronized to obtain thewith one-pulse-per-second the received 1PPS. (1PPS),Specifically, and thenthe g theroup receiver time pulse local (GTP) 1PPS signal signal received is synchronized by the eLoran with the receiver received is compared 1PPS. Speciﬁcally, with the thelocal group 1PPS timeof the pulse receiver, (GTP) and signal then receivedthe local by1PPS the signal eLoran is receiveradjusted isaccording compared to withthe offset the local Δ휏 1PPS[11–13]. of theThis receiver, principle and is thenillustrated the local in Fig 1PPSure signal 2. is adjusted according to the oﬀset ∆τ [11–13]. This principle is illustrated in Figure2.

1PPS (local) 1PPS (UTC) TOT TOR Output GTP t

t0 tp tr

 t0+tp+tr

Figure 2. PPrinciplerinciple of eLoran autonomous timing.timing.

After a adjustingdjusting the the local local 1PPS 1PPS pulse pulse of of the the receiver receiver to to be be ahead ahead of of 1PPS 1PPS (UTC), (UTC), the the local local 1PPS 1PPS of ofthe the receiver receiver is isused used asas the the opening opening signal signal of of the the counter, counter, and and the the GTP GTP signal demodulated by the receiver is used as the closing signal. Subsequently, Subsequently, the phase difference diﬀerence N is measured between the thesese two pulses as shown below (Equation(Equation (4)).(4)).

N = Δ휏 + (푡0 + 푡푝 + 푡푟) (4) N = ∆τ + (t0 + tp + tr) (4) where 푡0 is the transmission delay of the system obtained from the demodulated message, 푡푔 is propagation delay calculated with the position of transmitter and receiver, and 푡푟 is the receiver

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Sensors 2020, 20, x FOR PEER REVIEW 4 of 14 where t0 is the transmission delay of the system obtained from the demodulated message, tg is delay, which is calibrated in advanced. Then, the offset of local 1PPS signal Δ휏 is obtained using propagation delay calculated with the position of transmitter and receiver, and tr is the receiver delay, Equationwhich is calibrated (5). in advanced. Then, the oﬀset of local 1PPS signal ∆τ is obtained using Equation (5).

Δ휏 = N - (푡0 + 푡푝 + 푡푟), (5) ∆τ= N (t + tp + tr), (5) − 0 The errors of N, 푡0, 푡푝, 푡푟 are transferred to Δ휏. The time difference N is measured by the inner counterThe of errors the re ofceiver N, t,0 ,andtp, t ranare error transferred is introduced to ∆τ .in The the timeprocedure diﬀerence. 푡0 can N is be measured calibrated by accurately, the inner andcounter the error of the after receiver, calibration and an is error approximately is introduced 30 inns. the 푡푝 procedure.is calculatedt0 usingcan be Equations calibrated (1) accurately,–(3), and theand deviation the error afterfrom calibrationthe true value is approximately is approximately 30 ns. 500tp ns.is calculated푡푟 is measured using with Equations an error (1)–(3), less andthan the 20 ns.deviation The majority fromthe of the true timing value error is approximately is due to the error 500 ns. in t푡r0is, 푡푝 measured, 푡푟 which withwill be an addressed error less thanin a future 20 ns. studyThe majority. of the timing error is due to the error in t0, tp, tr which will be addressed in a future study.

3.2. E Errorrror A Analysisnalysis of eLoran Timing Service From the the source source to to the the destination, destination, a signal a signal from from a source a source passes passes through through three threesystems systems to reach to thereach destination. the destination. These three These systems three systems are the transmitter, are the transmitter, propagation propagation channel, channel, and receiver. and receiver.During thisDuring process this process,, the entire the entirepropagation propagation delay delay is composed is composed of transmission of transmission delay, delay, signal signal propagation propagation delay and signal receiving delay, delay, which are shown in Fig Figureure 33.. TheThe signalsignal transmissiontransmission delay error, signal propagation delay error,error, andand signalsignal receivingreceiving delaydelay errorerror makemake upup thethe eLoraneLoran timingtiming error.error.

Signal transmission Signal propagation Signal reception Synchronization deviation between local clock and UTC (NTSC) Transmitter equipment PF Channel delay + + Receiver delay Phase center deviation of transmitting antenna ASF

Figure 3. The entire propagation delay is composed of of transmission delay, delay, signal propagation delay and signal signal receiving receiving delay. delay The. The signal signal propagation propagation consists consists of the primaryof the primary factor (PF) factor and the (PF) additional and the additionalsecondary secondary factor (ASF). factor(ASF). 3.2.1. eLoran Transmission Delay Error 3.2.1. eLoran Transmission Delay Error The eLoran signal transmission error includes the synchronization error between the local time The eLoran signal transmission error includes the synchronization error between the local time in a broadcast station and UTC (NTSC), the transmitter channel delay error and the antenna phase in a broadcast station and UTC (NTSC), the transmitter channel delay error and the antenna phase center error. center error. (1) SSynchronizationynchronization error error between between the the local local time time in in the the station station and and UTC (NTSC) The broadcasting system of eLoran is equipped with an independent atomic clock which is used as the timetime referencereference and and keeps keeps synchronous synchronous with with UTC UTC (NTSC) (NTSC) through through a certain a certain communication communication link. link.The synchronizationThe synchronization error error is generally is generally less than less 5than ns. 5 ns. (2) TTransmitterransmitter channel channel delay delay error

The transmitter channel delay error compri comprisesses signal coding delay delay error error and delay error in the signal modulation modulation,, which can be controlled to be within 30 ns.ns.

(3) EErrorrror of of antenna antenna phase phase center center

In theory, theory, the phase center of the antenna should be consistent with its its geometric center, center, but the change in the signal carrier frequency can can offset oﬀset the phase center, which leads to the uncertainty of the antenna phase center. In In the the 100 kHz carrier frequency band, the phase center fluctuation ﬂuctuation of the antenna is less than 5 ns. The estimated estimated eLoran eLoran transmission transmission delay delay error error is is calculated calculated using using the the following following equation equation 2 2 2 휏 = √5 2+ 30 2+ 5 2 = 30.4138 ns. τ00 = √5 + 30 + 5 = 30.4138 ns.

3.2.2. eLoran Propagation Delay Error

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3.2.2. eLoran Propagation Delay Error The true value of signal propagation delay is shown in Equation (6). However, it is diﬃcult to obtain the true value of the ns, σ and ε. In eLoran timing service, the calculation is generally used to obtain the signal propagation delay. In the calculation, the value of ns0 is 1.000315 in the international standard atmosphere, and the quantization value ns0, σ0 and ε0 replace the true ns, σ and ε. This is the main reason for the delay error of signal transmission. For Equation (7), the deviation from Equation (6) is the propagation delay error shown in Equation (8), where ∆PF represents the error of PF and ∆ASF is the error of ASF. At present, the error of the eLoran signal propagation delay is better than 500 ns. R nsdS 106 T = PF + ASF = + arg W ( f , d, σ, ε) (6) C ω

6 Sn0s 10 T0 = PFp + ASFp = + arg W ( f , d, σ0, ε0) (7) C ω

∆T = T T0 − = (PF + ASF) (PFp + ASFp) − = (PF PFp)(ASF ASFp) − − = ∆PF + ∆ASF R " # nsdS Sn 106 106 = s + arg W( f , d, σ, ε) arg W( f , d, σ, ε) . (8) C − C ω − ω

3.2.3. eLoran Receiving Delay Error The receiving delay is the time from the moment when the signal propagates to the receiving antenna to the moment when the receiver outputs 1PPS [14–16]. The receiver delay error includes the thermal noise of the receiver and the measurement error introduced during receiver delay measurement [17,18]. After calibrating the receiver’s delay, the receiving delay error can be controlled to be within 20 ns. All the source of errors in eLoran timing is listed in Table1.

Table 1. Source of errors in eLoran timing.

Error Source Magnitude/ns Notes Synchronization error between eLoran time and UTC(NTSC), transmission delay error √52 + 302 + 52 transmitter channel delay error and error of antenna phase center The deviation between the propagation delay error 500 calculation value and the true receiving delay error 20 The delay error of eLoran receiver

4. Correlation Analysis of eLoran Timing Error The differential method used in eLoran timing service is similar to the GPS local difference technology. According to the GPS differential technology, the error correlations between the difference station and the user in an area lay the foundation of the difference technology. The timing error correlation between the difference station and the user in an area is analyzed sequentially. In Figure4, A is the difference station and B represents users. In the two paths OA and OB, weather, geology, and geography are similar, which is the precondition of the difference method. Sensors 2020, 20, 6518 6 of 14 Sensors 2020, 20, x FOR PEER REVIEW 6 of 14

O A

FigureFigure 4. 4. DDiagramiagram of of difference diﬀerence station station and and surrounding surrounding users users..

4.1.4.1. C Correlationorrelation of of eLoran eLoran Transmitting Transmitting Delay Delay Error Error TransmissionTransmission delay delay error error is is one one of of the the key key errors errors that that need need to to be be corrected. corrected. D Diifferenfferencece stations stations andand nearby nearby u userssers receive receive signals signals from from the the same same transmitting transmitting station. station. R Regardlessegardless of of the the location location of of the the differencedifference station station and and the the user, the influenceinfluence ofofthe the transmission transmission error error on on the the di differencefference station station and and the theuser user is theis the same. same. If theIf the error error is is 10 10 ns, ns it, it will will induce induce 10 10 ns ns timing timing error error and and 3 3 m m pseudo-range pseudo-range error error to tousers users and and the the di differencefference station station [19 [19].]. 4.2. Correlation of eLoran Propagation Delay Error 4.2. Correlation of eLoran Propagation Delay Error The propagation delay of eLoran signal comprises PF and ASF. Although these two factors are The propagation delay of eLoran signal comprises PF and ASF. Although these two factors are indistinguishable in propagation delay measurement, their physical properties vary considerably. indistinguishable in propagation delay measurement, their physical properties vary considerably. Therefore, the error correlation of PF and the error correlation of ASF is discussed separately. Therefore, the error correlation of PF and the error correlation of ASF is discussed separately. 4.2.1. The error Correlation of PF 4.2.1. The error Correlation of PF eM = PFM PF0M is the PF error of difference station and eU = PFU PF0U is the PF error of the 푒 = PF −−PF′ 푒 = PF −−PF′ users,푀 and theM differenceM is the between PF error the of twodifference is analyzed station using and Equation푈 푈 (9). First,푈 is the assuming PF error that of the the users, and the difference between the two is analyzed using Equation (9). First, assuming that the difference between the distance of two signal propagation path is ∆d, and the amount of ∆ns is not difference between the distance of two signal propagation path is Δ푑, and the amount of Δ푛푠 is not more than the change in ns within one year in the area, such that ∆ns is not greater than 0.000060 [20,21], more than the change in 푛푠 within one year in the area, such that Δ푛푠 is not greater than 0.000060 then the difference between the primary delay error caused by ns is expressed as follows. [20,21], then the difference between the primary delay error caused by 푛푠 is expressed as follows.

eU eM = (PFU PF0U) (PFM PF0M) |푒|푈 −−푒푀||= |(PF푈 −−PF′푈) −−(PFM −−PF′M )|

푑푈dU× 푛푠ns 푑푈dU× 푛n′푠0s 푑푀dM× 푛n푠 s 푑d푀M× 푛n′푠0s ==|( ( × − × ) )− (( × − × )|) 퐶C − 퐶C − 퐶C − 퐶C

푑 푈dU× Δ∆푛n푠 s 푑d푀M× Δ∆푛n푠 s ==| × − × | 퐶C − 퐶C

Δ∆푛n푠s ==|푑d푈U− 푑d푀M| × | − | × 퐶C 00.000060.000060 ≤ Δ∆푑d× . (9) (9) ≤ ×00.299792458.299792458

4.2.2. Error Correlation of ASF 4.2.2. Error Correlation of ASF eM = ASFM ASF0M is the ASF error of difference station and eU = ASFU ASF0U is the ASF 푒푀 = ASFM − −ASF′M is the ASF error of difference station and 푒푈 = ASF푈 −−ASF′푈 is the ASF error of the user, and the difference between the two is analyzed using Equation (10). ASFM is the ASF error of the user, and the difference between the two is analyzed using Equation (10). ASFM is the with true σ and ε for the difference station, and the ASF0M is the ASF with the quantization value σ0 ASF with true 휎 and 휀 for the difference station, and the ASF′M is the ASF with the quantization value 휎′ and 휀′ for the difference station. ASFU is the ASF with the true 휎 and 휀, and the 퐴푆퐹′U is ASF with the quantization value 휎′ and 휀′ for the user.

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and ε0 for the diﬀerence station. ASFU is the ASF with the true σ and ε, and the ASF0U is ASF with the quantization value σ0 and ε0 for the user.

eU eM = (ASFU ASF0U) (ASFM ASF0M) | − | − − −

= (ASFU ASFM) (ASF0U ASF0M) − − −

= [F(dU, σ, ε) F(dM, σ, ε)] [F(dU, σ0, ε0) F(dM, σ0, ε0)] − − −

= ∆ASF ∆ASF0 . (10) − According to radio wave propagation theory, the calculation of ASF is quite complex, as shown in Equation (3). However, if Equation (3) is used, it is very inconvenient to study the error correlation of ASF in Equation (10). The relationship between ASF and distance d can be simplified by using another algorithm for convenience engineering calculation, but the residual error must meet the conditions of the military standard long-wave ground wave propagation [5]. When the signal propagation distance is greater than 100 km, the ASF and distance d are almost linear, and the relationship between the SF and distance is fitted with a linear polynomial ASF = a d + b [22]. × First, using Equation (3), the ASF is calculated as 0 < d 2000 and the relation between ASF and d ≤ is obtained on a typical propagation path. Second, the relationship between ASF and distance d is fitted by the least square algorithm using the calculated data. Finally, if the fitting residual error meets the conditions of the military standard, the fitting is termed reasonable. The simulation result is listed in Table1, and the limit error set by the military standard is presented in the last column. The fitting on five types of paths are sufficient to satisfy the condition, but the fitting residual error of seawater and average land exceed the military standard, in Table2.

Table 2. The ﬁtting analysis of secondary delay (100 km < d).

Maximum in Military Ground Type Mean (µs) Standard (µs) Maximum (µs) Standard (µs) Seawater 4.9830 10 16 0.0304 0.0915 0.02 − × − Land with good conduction 1.2401 10 15 0.0156 0.0334 0.05 − × − Wet ground 9.9587 10 16 0.0061 0.0991 0.1 − × − Average Land 1.3052 10 15 0.0237 0.0251 0.2 − × − Dry land 2.5275 10 15 0.0722 0.0763 0.2 − × − Dryer Ground 3.3674 10 15 0.0900 0.1065 0.2 − × − Driest Ground 3.1111 10 15 0.0659 0.1257 0.2 − × −

In Figure5, the fitting curve of the sea surface is shown in the top and that of the average land is on the bottom. The red line represents the result of the calculation and the blue line represents the fitting result. The maximum residual error appears near 100 km. With the increase in distance, the fitting results also improve. The distance between the difference station and the eLoran broadcast station is generally greater than 100 km. Therefore, for the sea surface and the average land path, it is also reasonable to fit the relationship between ASF and d with a polynomial SF = a d + b, when the × distance is more than 100 km. SensorsSensors2020 2020, ,20 20,, 6518 x FOR PEER REVIEW 88 of of 1414

Figure 5. Fitting curve of seawater and average land. Figure 5. Fitting curve of seawater and average land. For the typical signal propagation path, the value of the electrical conductivity has a certain range.For For the example, typical in signal the third propagation column of path, Table the3, it value can be of seen the thatelectrical the electrical conductivity conductivity has a ofcertain the seawaterrange. For is betweenexample 3, in and the 7, third which column are the minimumof Table 3, and it can the b maximume seen that values, the electrical respectively. conductivity When the of maximumthe seawater value is between of σ is considered, 3 and 7, which the corresponding are the minimum fitting and parameters the maximum are a andvalues,b and respectively when the. 휎 푎 푏 minimumWhen the value maximum of σ is value considered, of is the considered corresponding, the fittingcorresponding parameters fitting are a 0parametersand b0. The are variation and in ASFand duewhen to uncertaintythe minimum in σvalueuncertainty of 휎is fromconsidered, seasonal the and corresponding weather effects fitting is smaller parameters than the are variation 푎′ and in푏′. ASF The brought variation from in ASF maximum due to to uncertainty the minimum in 휎 of uncertaintyσ. from seasonal and weather effects is smaller than the variation in ASF brought from maximum to the minimum of 휎. Table 3. Additional secondary factor (ASF) fitting coefficient of the typical path. Table 3. Additional secondary factor (ASF) fitting coefficient of the typical path. Equivalent Earth Ground Type ε σ Fitting Coefficients EquivalentRadius Coe Earthfficient Ground Type 휺 흈 Fitting Coefficients 7 Radius Coefficient a 0.0020 b 0.1996 Seawater 70 1.14 − 7 3 a a0 0.00200.0021 b b0 −0.19960.1878 Seawater 70 1.14 − 3 0.055 a′ a 0.00210.0028 b′ b −0.18780.0292 Land with good conduction 40 1.13 Land with good 0.055 0.017 a a0 0.00280.0035 b b0 0.02920.2318 40 1.13 conduction 0.017 0.017 a′ a 0.00350.0035 b′ b 0.23180.2284 Wet ground 30 1.11 0.0170.0055 a a0 0.00350.0046 b b0 0.22840.5795 Wet ground 30 1.11 0.00550.0055 a′ a 0.00460.0047 b′ b 0.57950.5730 Average Land 22 1.08 0.00550.0017 a a0 0.00470.0062 b b0 0.57301.2559 Average Land 22 1.08 0.00170.0017 a′ a 0.00620.0063 b′ b 1.25591.2535 Dry land 15 1.06 0.00170.00055 a a0 0.00630.0067 b b0 1.25352.4676 Dry land 15 1.06 0.000550.00055 a′ a 0.00670.0070 b′ b 2.46762.5488 Dryer Ground 7 1.05 0.000550.00017 a a0 0.00700.0055 b b0 2.54883.6182 Dryer Ground 7 1.05 0.000170.00017 a′ a 0.00550.0055 b′ b 3.61823.8882 Driest Ground 3 1.06 0.000170.000055 a a0 0.00550.0049 b b0 3.88823.732 Driest Ground 3 1.06 0.000055 a′ 0.0049 b′ 3.732 Based on the simplified relationship between ASF and distance d, the ASF error comparison between user and differential station is calculated using Equation (11). Including the PF and the

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ASF, the entire propagation delay error of the user and the diﬀerential station is compared using Equation (12).

eU eM = ∆ASF ∆ASF0 | − | −

= a(dU dM) a0(dU dM) − − −

= (a a0)(dU dM) − −

= (a a0) ∆d (11) − × 0.000060 eU eM = ∆d + ∆d (a a0) . (12) | − | × 0.299792458 × −

4.3. Correlation of Receiving Delay Error The delay error of the receiver is primarily the thermal noise of the receiver equipment, which has no correlation [19,23], but we can reduce the error by measuring the relative delay of the receivers [24]. The residual error is less than 5 ns after relative delay measurement and calibration.

4.4. Correlation of the Entire Timing Error After the correction by the difference station, the residual error for the user is shown in Table4. How much is the distance from the user to the difference station, which can meet the timing accuracy of 100 ns. Using Equation (13), the distance is calculated and is listed in Table5. On dryer ground path, the effective range is approximately 55 km, which is the minimum range. To achieve 100 ns precision, the effective range of the difference station on the land should be less than 55 km, in the condition of the distance greater than 100 km from the difference station to the broadcasting station.

Table 4. Residual error after correction.

Error Item Residual Error Post-Diﬀerence/µs The error transmitting delay 0 ∆n The error of propagation delay D s + (a a ) D × C − 0 × The error of receiving delay 0.005 ∆n sum D s + (a a ) D + 0.005 × C − 0 ×

Table 5. Scope of diﬀerence station in a typical path.

The Range of Diﬀerence Ground Type abs(∆a) Station/km Seawater 0.0001 316.6667 Ground with good conduction 0.0007 105.5556 Wet ground 0.0011 73.0769 Average Ground 0.0015 55.8824 Dry Ground 0.0004 158.3333 Dryer Ground 0.0015 55.8824 Driest Ground 0.0006 118.7500

0.000060 eU eM D ( + a a0 ) + 0.005 0.1 | − | ≤ × 0.299792458 − ≤

0.095 D (13) ⇒ ≤ 0.000060 0.299792458 + abs(∆a)

5. The eLoran Difference Timing Method Within 55 km around the difference station, there is a strong correlation with the timing error, and the offset between the difference station and the user timing error is not more than 100 ns. Sensors 2020, 20, x FOR PEER REVIEW 10 of 14

5. The eLoran Difference Timing Method Within 55 km around the difference station, there is a strong correlation with the timing error, Sensorsand the2020 ,offset20, 6518 between the difference station and the user timing error is not more than 10010 of ns. 14 Therefore, we can package the uncertainty in the timing error of the difference station into differential correction and send them to the surrounding users to correct the propagation delay of the users and Therefore,to improve we the can timing package accuracy the uncertainty. For the difference in the timing station, error the of thedifferential difference message station intois calculated differential by correctioncombining and the send measurement them to the and surrounding calculation users of tosignal correct propagation the propagation delay, delay andof the the differential users and tocorrection improve model the timing is generated accuracy. and For sent the to di thefference user. station,For users, the the diff differentialerential message correction is calculated is calculate byd combiningto modify the the measurement prediction value and calculation of propagation of signal and propagation improve delay,the accuracy and the diofff erentialtime service. correction The modelcalculation is generated and prediction and sent of to the the signal user. propagation For users, the delay differential are shown correction using Equation is calculated (7). to modify the predictionIn a region, value a site of propagationis selected as and the improve difference the station accuracy for ofsurrounding time service. users. The In calculation the difference and predictionstation, it ofis important the signal propagationto measure the delay signal are shownpropagation using delay Equation to calculat (7). e the difference message. MeasurementIn a region, devices a site, is such selected as eLoran as the monitoring difference stationreceiver, for time surrounding interval counter, users. In data the acquisition difference station,equipment it is, importantand UTC (NTSC) to measure is used the. signalWithout propagation UTC (NTSC) delay, the tolocal calculate time-synchronized the difference to message. the UTC Measurement(NTSC) is also devices, enough. such as eLoran monitoring receiver, time interval counter, data acquisition equipment,For the and difference UTC (NTSC) station, is used. first, Withoutthe accurate UTC (NTSC),position thecoordinate local time-synchronized of the monitoring to the receiver UTC (NTSC)antenna is is also calibrated enough. accurately to calculate the distance from the transmitter to the receiver. Second, For the difference station, first, the accurate position coordinate of the monitoring receiver antenna the signal propagation delay 푇푝 is calculated using Equation (7). Then, the monitoring receiver isreceive calibrateds the accurately eLoran signal to calculate and demodulate the distance the from 1PPS, the transmitter which is compared to the receiver. with Second, the UTC the (NTSC) signal T propagationthrough the delaytime intervalp is calculated counter to using obtain Equation the time (7). difference Then, the M monitoring. Finally, the receiver time difference receives M the is eLoran signal and demodulate the 1PPS, which is compared with the UTC (NTSC) through the time collected to calculate the 푀 − 푇푝 to form a differential model and is transmitted to the surrounding interval counter to obtain the time difference M. Finally, the time difference M is collected to calculate users in real-time. the M Tp to form a differential model and is transmitted to the surrounding users in real-time. For− the user, with the precise position of the receiver antenna, the circle distance from the user For the user, with the precise position of the receiver antenna, the circle distance from the user to the broadcast station is calculated. The signal propagation delay 푇′푝 is calculated according to to the broadcast station is calculated. The signal propagation delay T is calculated according to Equation (7), then the correction is calculated according to the received0p difference information to Equation (7), then the correction is calculated according to the received difference information to obtain more accurate signal delay and synchronize the 1PPS to UTC (NTSC). The principle of eLoran obtain more accurate signal delay and synchronize the 1PPS to UTC (NTSC). The principle of eLoran differential time service is shown in Figure 6. differential time service is shown in Figure6.

Difference Component Differential Tp M-TP station t0 tp tr 1PPS(UTC) Transmit 1PPS Receive 1PPS OUTPUT GTP M

1PPS(local) 1PPS(UTC) Transmit 1PPS Receive 1PPS OUTPUT GTP

tp User t0 tr  T p M-TP

N FigureFigure 6. 6.Principle Principleof ofdi differentialfferential timing. timing. 6. Verification of eLoran Differential Timing Method 6. Verification of eLoran Differential Timing Method The experimental research is carried out to verify the result of the difference technology. In the The experimental research is carried out to verify the result of the difference technology. In the experiment, a survey is carried out in the difference station and the users, but with a different purpose. experiment, a survey is carried out in the difference station and the users, but with a different The measurement in difference station is to calculate the difference in correction mode, and the test in purpose. The measurement in difference station is to calculate the difference in correction mode, and user site is to verify the difference technology effect. The offset between the difference result and the the test in user site is to verify the difference technology effect. The offset between the difference measured value of delay is used to evaluate the effect of the timing differential method. The schematic result and the measured value of delay is used to evaluate the effect of the timing differential method. of the test equipment is shown in Figure7. The BD receiver is used as the standard time and frequency The schematic of the test equipment is shown in Figure 7. The BD receiver is used as the standard source, which have been synchronized to the UTC (NTSC) in advance. The time interval counter is time and frequency source, which have been synchronized to the UTC (NTSC) in advance. The time used to measure the signal delay, which is collected in the data acquisition system. interval counter is used to measure the signal delay, which is collected in the data acquisition system.

Sensors 2020 20 Sensors, 2020, 6518, 20, x FOR PEER REVIEW 11 of11 14 of 14 Sensors 2020, 20, x FOR PEER REVIEW 11 of 14

BD eLoran BDReceiver eLoranReceiver Receiver Receiver 1PPS GTP 1PPS GTP

open close open close Time interval Time intervalcounters counters

Data Dataacquisition acquisition system system Figure 7. Measurement equipment connection. Figure 7. MeasurementMeasurement e equipmentquipment c connection.onnection. The measurement is carried out in GuanZhong Plain in Shanxi Province. The area has wet ground,The measurementmeasurement with conductivity is is carried carried σout = out0.01 in GuanZhong inand Guan dielectricZhong Plain constant Plain in Shanxi in ε Shanxi= 30. Province. The Provi basic Then ce.information area The has area wet ofhas ground, test wet points ground,withis conductivity shown with conductivityin Tableσ = 0.01 6. Taking andσ = 0.01 dielectric Wu and Gong dielectric constant (WG) asεconstant= the30. difference The ε = basic 30. The station information basic and information Mei of test Xian points of(MX) test is as shownpoints the user, isin shown Tablethe 6distance. in Taking Table between Wu 6. Taking Gong WG (WG)Wu and G asong MX the ( isWG) di 44.8930ﬀerence as the km stationdifference. The anddistribution station Mei Xian and of (MX) testMei pointsas Xian the (MX) user,is shown theas the distance in user,Figure 8. thebetween distancePu Cheng WG between and (PCH) MX WGis is the 44.8930 and eLoran MX km. isbroadcast 44.8930 The distribution km station.. The distribution of test points of istest shown points in is Figure shown8. in Pu Fig Chengure 8. Pu(PCH) Cheng is the (PCH) eLoran is the broadcast eLoran station.broadcast station. Table 6. The information of the measurement point. Table 6. The information of the measurement point.point. Measure Point Longitude Latitude The Distance/km PF/μs SF/μs TOA MeasureMeasure Point Point Longitude LatitudeLatitude TheThe Distance Distance/km/km PFPF/µs/μs SFSF/µ/μss TOATOA Wu. Gong 108.2200 34.2618 143.2942 478.1286 0.9422 479.0708 WuWu.. MeiGong Gong. Xian 108.2200108.2200107.7348 34.2618 34.261834.3014 143.2942143.2942180.7000 478.1286478.1286602.9402 0.94220.9422 1.0964 479.0708479.0708 604.0366 MeiMei.. Xian Xian 107.7348107.7348 34.301434.3014 180.7000180.7000 602.9402602.9402 1.09641.0964 604.0366604.0366

Figure 8. Distribution of test point. Figure 8. Distribution of test point. Figure 8. Distribution of test point. The measurement data in WG and MX is shown in Figure9. From the changing trend of the curve, theThe variation measurement of propagation data in WG delay and in MX the MXis show is coincidentn in Figure with 9. From the WG the overchang time.ing trend In WG, of the curve,The measurement the variation data of propagation in WG and delay MX isin showthe MXn in is Figurecoincident 9. From with the the changWG overing time.trend Inof WG, the the curve,the measurement the variation data of propagation and the theoretical delay in calculation the MX isresults coincident are combined with the WG to compute over time. the In di WG,ﬀerence the information.measurement In the data calculation and the andtheoretical forecasting calculation of the di resultﬀerences are model, combined the least to compute square principle the difference is measurementinformation. data In andthe calculationthe theoretical and calculationforecasting resultof thes differenceare combined model, to computethe least squarethe difference principle is information. In the calculation and forecasting of the difference model, the least square principle is

Sensors 2020, 20, x FOR PEER REVIEW 12 of 14 Sensors 2020, 20, 6518 12 of 14 usedSensors for 20 20the, 20 difference, x FOR PEER station. REVIEW The sliding window is 600 and the forecast time is 300 s, with 1 as12 ofthe 14 fitting order. In other words, the data in 10 min forecast for 5 min. usedused for for the the di ﬀdifferenerencece station. station. The The sliding sliding window window is is 600 600 and and the the forecast forecast time time is 300is 300 s, withs, with 1 as1 as the the ﬁttingfitting order. order. In In other other words, words, the the data data in in 10 10 min minthe forecast delay forecast of for WG for 5 min. 5 min. 510.7

510.6

Delay/μs 510.5

510.4 0123456789 Time/s 104 the delay of MX 635.7

635.6

Delay/μs 635.5

635.4 0123456789 Time/s 104

Figure 9. Delay measurement in Wu. Gong (WG) and Mei. Xian (MX). Figure 9. Delay measurement in Wu. Gong (WG) and Mei. Xian (MX). As shown in FigureFigure 10, 9. Delaythe standard measurement error inis Wu.reduced Gong from (WG) 32.7356 and Mei. ns Xian pre-correction (MX). to 18.9630 ns post-correction.As shown in FigureThe average 10, the of standard all the errorerrors is in reduced 394.7287 from ns 32.7356pre-difference, ns pre-correction and the average to 18.9630 of nsthe As shown in Figure 10, the standard error is reduced from 32.7356 ns pre-correction to 18.9630 ns post-correction.error is 19.5890 The ns averageafter correction. of all the The errors accuracy in 394.7287 is improved ns pre-di byfference, an order and of the magnitude. average of It theis proved error post-correction. The average of all the errors in 394.7287 ns pre-difference, and the average of the isthat 19.5890 the nstiming after service correction. accuracy The accuracy of eLoran is improvedis improved by anby orderthe difference of magnitude. method, It is and proved is within that the the error is 19.5890 ns after correction. The accuracy is improved by an order of magnitude. It is proved timingrange service of the difference accuracy of station. eLoran is improved by the difference method, and is within the range of the diffthaterence the station.timing service accuracy of eLoran is improved by the difference method, and is within the range of the difference station. 500 pre-correction 450

400 Error/ns 350

300 0123456789 Time/s 104

100 post-correction 50

Error/ns 0

-50 0123456789 Time/s 104

Figure 10. Eﬀect of diﬀerence method in MX. Figure 10. Effect of difference method in MX.

7. Conclusions Figure 10. Effect of difference method in MX.

7. Conclusions

Sensors 2020, 20, 6518 13 of 14

7. Conclusions First, eLoran signal and the propagation delay on the path was introduced, and then the eLoran timing error was analyzed. Second, the correlation of eLoran timing error was analyzed quantitatively in theory based on eLoran timing error and it was found that the action range of the difference station on land should not be more than 55 km to achieve 100 ns accuracy. Finally, based on the correlation between user and difference in timing error, an eLoran differential timing technology was proposed. In the difference station, the theoretical calculation method is combined with the measurement method of the signal delay to obtain the offset which is the common error for the difference station and the users. The offset data were then processed to obtain the difference information which is sent to the users so as to adjust the propagation delay and improve the timing precision. To validate the correctness and validity of the proposed method, experimental verification was carried out in Guan. Zhong Plain. In the experiment, the measurement for the users and the difference station are carried out synchronously to verify the effect of differential technology. The timing precision of the users is improved from 394.7287 to 19.5890 ns, which shows that this method can significantly improve the timing accuracy of the system. Although the range of the difference station in the land is smaller than that of the maritime applications, it can effectively enhance the timing precision of the eLoran system within the action scope.

Author Contributions: Y.L. gave the analysis of eLoran timing error, the difference timing method and process the measure data to verify the method. B.Y. contributed to the discussion of the difference timing method. W.G. provided the scheme of actual measurement. Y.H. supervised the work, arranged the architecture and contributed to the writing of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research is funded by the Natural Science Foundation of China for Distinguished Young Scholars, grant number 11803040 and Frontier Science of the Chinese Academy of Sciences, grant number QYZDJ-SSW-JSC034. Conflicts of Interest: The author declares no conflict of interest.

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