Meteorological Influence on eLoran Accuracy

Stringent accuracy requirements need to be met for eLoran deployment in marine navigation and harbour entrance and approach. A good accuracy model is therefore required to predict the positioning accuracy at the user’s receiver locations. Accuracy depends on the variations of additional secondary factors (ASFs) and the primary factor delay. The changes in the air refractive index caused variations in the primary factor (PF) delay of the eLoran signal, and current eLoran accuracy models do not take this into account. This paper proposes an improved empirical accuracy model that considers the contributions of changes in the refractive index of the air, often classified as a short term effect. The changes in weather parameters such as atmospheric pressure and temperature increase the time of arrival variance. The developed accuracy model is used to predict the eLoran positioning error in the European maritime region. The results show that the short term ASF variations significantly contribute to the positioning error and must be included in the accuracy model. The results also demonstrate that a 20 m accuracy or better would be achieved in the North Sea, while a 10 m accuracy would be achievable at the SOLAS ports if eLoran was reintroduced in Europe. Nevertheless, the repeatable accuracy around the Irish sea exceeds 80 m and does not meet marine navigation requirements compared to GPS. Coverage can be enhanced by including at least two eLoran transmitters in Ireland.


I. INTRODUCTION
The United States Global Positioning System (GPS) has for decades become an essential part of society. It has been established as the primary source of position, navigation and timing (PNT) information for land, air and sea. While it has superior qualities that make it more appealing than its rivals, the strength of GPS signals, especially at the Earth surface, have been documented to be extremely low [1]. This Achilles heel is present in all Global Navigation Satellite Systems (GNSS) and poses economic, safety and environmental risks associated with over-reliance on a single satellite navigation system. This view was included in the report [33] prepared for the United States Department of Transport. A Low Frequency (LF) terrestrial system, a redundant system with different failure modalities called LOng RAnge Navigation (LORAN) system, was chosen as a suitable backup. It was pointed out that Loran, in its form, cannot meet the stringent standards set by the governing bodies. In 2004, U.S. Federal Aviation Administration (FAA) issued a report confirming eLoran as a suitable backup for GNSS systems. eLoran is designed to serve as a stand-alone and back up to GPS [1]. In 2006, The U.S. Department of Homeland Security and the Department of Transportation funded the Institute for Defense Analyses to form an Independent Assessment Team (IAT) to review the need for an eLoran system as a backup to GPS. The IAT unanimously recommended an upgrade of all Loran systems and that eLoran must be a national GPS backup for 20 years [31]. All eLoran applications must meet the standard set by the IMO [18]. The General Light House Authorities of the United Kingdom and Ireland (GLA) was awarded a 15-year contract [34], but sadly, at the end of 2015, the European eLoran transmitters were shut down following the decision by the nations hosting the transmitters to cease to fund eLoran transmission. Nevertheless, eLoran deployment in Korea is still ongoing. The general eLoran system operations are described extensively in [1]- [4]. Several types of eLoran stated in the eLoran definition document [6] are aviation nonprecision instrument Approaches (NPA), maritime Harbour Entrance and Approach (HEA) manoeuvres, land-mobile vehicle navigation, and location-based services, each with different performance specifications for the four system parameters. These parameters are accuracy, availability, integrity, and continuity. The definition of these parameters are stated in [6], [7], [11].
This work focuses on eLoran for maritime Harbour Entrance and Approach (HEA) manoeuvres where repeatable accuracy is critical. Factors affecting eLoran performance during HEA manoeuvres must be identified and their contributions quantified. These factors are often categorised into long and short term effects and are listed in [23]. The long term effects include seasonal changes in ground conductivity due to rain soaking into the earth, passing weather fronts as well as freeze and thaw of the terrain over which the signals propagate [14]. These lead to signal delay called additional secondary factor (ASF) that varies over time in a stochastic manner.
The short terms effects are mostly due to changes in the meteorological parameters such as temperature, atmospheric pressure, humidity, soil moisture content, and salinity [25]. Even though weather changes affect the primary factor (PF) delay, this PF delay change is normally considered a change in ASF by the eLoran receiver. ASF variations are monitored using a differential Loran (dLoran) set up described in [6], [9]- [11], [14], [20], [22]. By design, system analysts assume that the temporal ASF variations monitored by the dLoran system are the same as those at the user's receiver location. The manifestation of these hazards on accuracy can be mitigated at only port locations but remain prominent elsewhere and are unmitigated due to limited dLoran resources. It is important to note that temporal ASF variations translate to the temporal variation of the position solution. Therefore monitoring and mitigating these variations improves the stability of the position solution.

A. CONTRIBUTIONS
This paper is a continuation of the previous work done by the authors [20] and now adds the contribution of the changes in meteorological parameters on the transmitter ranging errors experienced at the mariner's location. This result in an improved and realistic accuracy prediction model for harbour entrance approaches (HEA). The work in previous studies assumed that a significant contribution to the eLoran accuracy could be attributed to the long-term temporal variations in the times-of-arrival (TOAs) of Loran signals while ignoring the short term effects contributions. The authors in [22] stated that pseudo-range errors experienced at the receiver's position are in the short term caused by the changes in atmospheric parameters along the transmitter-receiver propagation path. The inclusion of the contribution of the atmospheric parameters results in poor repeatable accuracy values in regions where there is no DeLoran.
Therefore, including the contribution of the meteorological parameters in the TOA variance model gives a realistic estimate of the repeatable accuracy performance in a coverage area. Position accuracy is assessed to determine if it meets the performance standard set by the IMO. In summary, the contributions of this research are as follows -The effects of long-term ASF variations are included in the accuracy model -Models the impact of the short term ASF variations and includes this in the accuracy model to depict accuracy performance in areas not designated as reference stations -Applies the long term and short term accuracy models at regions not designated as reference stations. The models in the literature assume that all areas host a reference station. -Overall, the proposed accuracy model is more realistic and represents what happens in an eLoran operated coverage area than the models seen in the literature. This article is divided into five sections. Section II describes the improved accuracy model. The model parameters for the contributions of meteorological parameters on the transmitter's pseudo-range error measured at the receiver's location are derived in section III. Section IV discusses the results, while Section V concludes the article.

II. RELATED WORKS
Loran based coverage prediction using repeatable accuracy started in the early 1990s in Europe. Table 1 summaries the findings of several works that studied the hazards that introduce position errors and repeatable accuracy models.

A. EXISTING ACCURACY MODELS
This section describes existing accuracy models found in the literature. Repeatable accuracy describes how stable the measured position is overtime [19]. It represents the scatter of the user's static position and is determined assuming that the uncertainty in the phase measurements is the same. Authors of [15], [20], [23] described the factors that contribute to phase measurement uncertainty. The atmospheric noise measured at the front-end of the receiver influences the signal to noise ratio (SNR) and the residual number of pulses left after application of CRI mitigation procedures [17]. The eLoran position accuracy follows the procedure described in [5], [7], [17], [20], [29].

1) INFLUENCE OF GRI AND SNR ON RANGING ERRORS
This section describes the influence of group repetition interval (GRI), Cross rate Interference (CRI) and SNR on ranging errors. The accuracy model described in [5], [7], [16], [17] are employed by assuming that there is differential Loran (dLoran) everywhere in the coverage area. The TOA variance for any transmitted eLoran signal measured at the user's receiver is given by Year Citations [25] The researchers of this work demonstrated the disparity of the meteorological parameters at various points along the propagation path. They argued that using values of meteorological parameters measured at a single point, as is the practice case, is inaccurate, especially at the receiver position to represent the overall meteorological changes along the propagation path. Their proposed method, called generalized regression neural network (GRNN), better predicts the TOA delay. 2021 2 [26] Hargreaves et al. gave a general overview of the capabilities of the coverage software developed for the General Light House Authorities. The paper described the spatial decorrelation contribution to the ranging error at the user as being equal to the 10% of the corrected ASF. The TOA variance model also included the contribution of the skywave ground ratio to the ranging errors. . 2015 1 [7] The authors of this article proposed two things: an eLoran SNR definition and blanking techniques to mitigate the effects of atmospheric noise on eLoran repeatable accuracy over the British Isles. This TOA variance model resulted in an improved eLoran repeatable accuracy compared to the one used in [2] 2011 8 [31] The authors of this work assessed the temporal ASF models for eLoran in aviation application. They also examined the performance of the model bounds on the pseudo ranges and how they translate into the achieved position errors. This work responded to the task given to the Loran Integrity panel assigned by the FAA to determine if eLoran can be a suitable backup candidate to GPS.
2008 11 [19] This paper proposed receiver signal processing techniques for mitigating the effects of carrier wave interference which was prevalent at that time 1993 32 [6] Boyce investigated the impact of atmospheric noise on position accuracy and proposed techniques to mitigate its impact 2007 14 [17] Safar investigated the effects of cross-rate interference inside eLoran receivers and proposed the mitigation techniques to improve repeatable accuracy 2014 12 [12] The authors investigated the accuracy mismatch obtained when using ASF corrections from nearby monitor sites. The goal was to assess if corrections from different nearby sites could be interpolated in order to achieve the stipulated accuracy requirements The authors stated that the ASF exhibits diurnal variation with temperature and rainfall of 30ns and 100ns, respectively. These are equivalent to 9m and 30m.
2017 0 where N a is the number of averaged pulses, b 1 is the transmitter jitter, b 2 represents other unknown sources of error, and γ represents the SNR of each transmitter in view at the user's location. Sherman Lo et al. [16] employed equation (1) in their Loran coverage prediction . Safar et al. [7] proposed that the TOA variance is given by where L m is the inbuilt losses of the receiver under study.
where L b represents the pulses that remain after blanking and GRI is the group repetition interval. Equation (2) can be verified using the maximum likelihood estimation technique.

2) INFLUENCE OF SKYWAVE ON RANGING ERRORS
The authors in [35] studied the skywave influence on ranging errors and considered it as a threat and hazard to the safety and position accuracy of a navigation system. The ranging error introduced is modelled as a function of the skywavegroundwave ratio [26], [29] The work in [36] proposed a high precision method that can be employed in the receiver frontend to separate skywave occurring as early as 35µ s from groundwave under low SNR conditions. While the described models presented a significant leap in progress towards modelling repeatable accuracy, it is essential to note that many authors modelled repeatable accuracy under the assumption that dLoran is deployed everywhere in  the geographical area. This assumption suggests that each grid point in the coverage area is a port or harbour, which is unrealistic and uneconomical. Therefore, this work aims to model realism by using the long and short term ASF variations to dilute the effects of dLoran in regions that are not harbours and ports. The following section describes our proposed model.

III. PROPOSED ACCURACY MODEL
This section starts by summarising the long term ASF variation model we developed in our previous work [20], [29] and then goes on to describe the short term ASF variation model to enhance the accuracy model used for eLoran service volume coverage prediction.

LONG TERM ASF VARIATION MODEL
This model takes into account the contribution of the long term ASF variations on eLoran transmitter ranging error. The eLoran propagation path is a mixture of seawater and land paths. The receiver-transmitter land path distances are determined using the conductivity data provided by ITU. Fig. 3 shows different ground conductivity segments across Europe represented by numbers ranging from 1 to 9. The number 1 here represents seawater conductivity, while other numbers represent different terrain conductivities. We found that more than 70% of the coverage regions have the same conductivity values. The United Kingdom and Ireland also have more than 70% of similar conductivity values. Because of this, we assumed a direct correlation between the land path distance and ASF variations. This assumption is consistent with the results obtained by the same study conducted in the US [37]. The eLoran temporal ASF data of four eLoran stations Lessay, Anthorn, Soustons and Sylt, was measured at the Harwich, UK reference station from October 2009 to October 2010. We then determined the land path distance from the transmitter to the reference station. We fitted the yearly ASF variations of the transmitters, transmitter powers, and land path distances into a proposed model given by Equation (4).
where D represents the land path distance in km, P is the power transmitted in kW, P 0 is the highest transmission power of all the coverage transmitters, while α and β are derived from the model. The coefficients of the proposed model were found to be α = 0.17 m and β = 0.0106 km −1 . The square of equation (4) is represented by σ 2 LT ASF throughout this paper. We use the error propagation laws to state the TOA variance in the following form where σ 2 N oise , σ 2 LT ASF and σ 2 ST ASF are the TOA variances due atmospheric noise, long term ASF and short term ASF variations respectively. The term σ 2 Other represents the TOA variance due to other error sources and is considered to be negligible in this study.
Substituting equation (3) into (6) gives where K m is a constant equals to L m × 337.5 2 8 × 10 5 . Equation (7) suggests that longer GRIs increases the TOA variance. Depending on the value of the horizontal dilution of precision, the position accuracy may become large. Good position errors is obtained when the receiver selects the lower GRI transmitters in the position solution. The TOA variance, σ 2 T X , in regions not designated as reference stations when only σ 2 N oise and σ 2 LT ASF are considered is given by The pseudorange error is the square root of the TOA variance and is given by This study assumes that ASF corrections remain valid at a 50 km radial distance from the reference station. This assumption was verified using the results from the eLoran trials held at Harwich, United Kingdom. The TOA variance is given by (10) shows the blanking loss, land path and GRI influence on the pseudorange error. The results depicted in fig. 1 suggest that a pseudo-range error increases with increasing GRI and decreases with an increase in the SNR. Fig. 4 indicate that a decreasing SNR and an increase in the land path distance leads to an exponential growth in the pseudorange error.

SHORT TERM ASF VARIATION MODEL
The literature [25], [38]- [41] proposes that the atmospheric refractive index at any grid point is given by  Wenzel et al. [22] proposed that the wet term of the refractive index is insignificant and that it is sufficient to model the dry term only. Equation (11) reduces to η = 1.0 + 77.6 P T × 10 −6 (12) where P is the pressure and T is the temperature. Our short term ASF model uses temperature and pressure information for a yearly period from October 2009 to October 2010. The dataset used was extracted from the European Centre for Medium-Range Weather Forecasting (ECMWF) ERA-VOLUME 4, 2016  . We used a bilinear transformation to interpolate and fit the data into a grid equivalent to the coverage area. Fig. 5 and 6 show pressure and temperature plots over the European eLoran service area. Sea areas experience high temperatures and lower atmospheric pressure compared to other areas. By definition, the signal propagating in a medium with a refractive index of η travels with a speed equal to c/η. The speed of a Loran signal propagating through the air space, PF, is given by where d is the propagation distance, c is the speed of light in  a vacuum, and η is the refractive index of air. The primary factor expressed in terms of refractivity N is given by This can be rewritten as The first part of equation 15 is the propagation delay introduced by the air refractivity, while the second is the propagation time if the medium is vacuum since c is the speed of light in vacuum. The PF variance in square seconds is given by The PF variance in square meters is given by where σ N is the standard deviation of the refractivity of the primary factor delay along the propagation path, and d is the path distance. Equation 17 represents the short ASF variation model (σ 2 ST ASF ) and is used to modify the TOA variance model at regions not designated as harbours or ports.

1) Calculation of σ 2 N from the Weather data
This section describes the steps taken to determine σ N at any grid point.
-Store the daily atmospheric pressure and temperature data for each grid point from October 2009 to October 2010 -Calculate daily Refractivity for each grid point using equation 12. -Calculate the refractivity variance, σ 2 N for each grid point, using the whole year data. Fig. 11 shows standard deviation of refractivity. The results suggest that the standard deviation is higher in Norway compared to the rest of Europe.
2) Calculation of σ 2 N along the propagation path As the signal propagates from the transmitter to the receiver located at a particular grid point, it experiences a variance in the TOA due to changes in the weather parameters. In this study, the steps taken to estimate σ 2 N along the propagation path are as follows -Divide the propagation path into equal segments -Load the yearly σ 2 N refractive variance for the whole grid.
-The total σ 2 N experienced by the transmitted signal to the grid point equals the sum of σ 2 N for each segment. This idea is consistent with the propagation of error laws.
Equation (17) represents the contribution of the weather parameters on the transmitter ranging errors. It models TOA variance introduced by weather changes at grid points that are not harbours and ports. A piecewise function shown by equation (19) presents an improved accuracy model, which includes the contribution of atmospheric noise and long and short term variations. This model shows that the pseudorange error at grid points outside the ports and harbours is worse than when measured at port and harbour locations. The model realism imitates a real eLoran system.

3) Description of Repeatable Accuracy Calculation Method
The International Maritime Organisation (IMO) [18] stipulates the position error not exceeded 95% of the time. Repeatable accuracy is given by The flowchart in fig. 10 summarises repeatable accuracy calculation for eLoran . Before starting repeatable accuracy calculation within the geographical area, we load conductivity, ground field strength, skywave and atmospheric noise data into the coverage software written using MATLAB. The geographical area is divided into equal grid points using a specified grid resolution set to 0.1 in latitude and longitude. At the starting location where repeatable accuracy is measured, the 95 percentile atmospheric noise is estimated, and the SNRs of all the transmitters in view are calculated. Once this is done, the algorithm checks if the transmitters meet SNR criteria SN R ≥ −10, skywave-groundwave ratio(SGR) criteria. The SGR criteria is stated in [29]. If three or more transmitters meet the SNR and SGR criteria, position accuracy is the determined using algorithm described in [2], [29]. Repeatable accuracy cannot be determined when less than three transmitters meet coverage criteria because three or more are needed to perform geolocation. After three or more transmitters meet the coverage criteria, the algorithm checks if the resulting position accuracy meets the 10m or less accuracy set by IMO for HEA manoeuvres. The receiver's location is out of coverage if it does not meet IMO accuracy requirements. Otherwise, it is said to be in coverage.

COMBINED LONG TERM AND SHORT ASF VARIATION MODEL
The proposed accuracy model includes the long-term and short term ASF variations under the following assumptions -Similar ground conductivity variations are experienced everywhere in the geographical area. -The long-term ASF variations increase with increasing Land path distances. -The short-term ASF variations are due to the changes in meteorological parameters. -The pseudorange error due to the skywave signal is insignificant. -Carrier wave interference influence on the position error is negligible. -Every port and harbour has a reference station or deploys a dLoran system -eLoran meets the 10 m accuracy requirement set by the IMO at 50 km radial distances from all ports and harbours. -The receiver signal processing module can blank out all CRI contaminated pulses. -The averaging process eliminates transmitter jitter and its effects. -The European Loran System transmitters are still in operation. The reason behind these assumptions is to derive a repeatable accuracy model that can be modified as more data becomes available. Sophisticated receiver signal processing techniques minimize the influence of skywave, carrier wave and Cross Rate Interference. However, they can be modelled and represented by σ 2 Other . Equation (19) determines the TOA variance at any grid point. If the reference station-grid point distance is less or equal to 50 km, the top part of the piece-wise function shown by equation (19) represents the signal TOA variance; otherwise, the lower part, which includes long and short term ASF models represents the signal TOA variance. For distances less or equal to 50km, we assume that dLoran mitigates the long and short term TOA variance hazards.
The TOA variances of the transmitter are used in the position solution to determine position error or repeatable accuracy using the procedure described by fig. 10.

IV. RESULTS AND DISCUSSION
This section presents the repeatable accuracy results obtained using our developed accuracy model. We first draw the reader's attention to the repeatable accuracy plot shown in fig. 2. The results obtained assume that every grid point has dLoran capability. The contours show accuracy results. It can be seen that the East part of the coverage area has better repeatable accuracy than the West. There are no transmitters in the Republic of Ireland; hence accuracy is poor in the West of the coverage area due to bad HDOP. We noted that this model needed to be modified as it is unrealistic and uneconomical to have every grid point having a reference station. Fig. 12 shows the eLoran repeatable accuracy for the chosen geographical area without differential ASFs. The land path model is essentially used to determine repeatable accuracy at all regions that are not harbours or ports in the coverage area. A comparison between fig. 2 and fig. 12 suggest that the accuracy has become poorer in fig. 12 as expected due to the contribution of the land path model. It is important to note that we have added a component to the TOA variance matrix, resulting in poor repeatable accuracy compared to fig. 2. The best position accuracy of 0-10m in fig. 12 is in some parts of the Atlantic ocean. Repeatable accuracy is above 10m in most parts of the British Isles and the Atlantic ocean. The results show that none of the areas in the British Isles meets the IMO accuracy requirements. The next step is to include the results for grid points hosting reference stations. The plot in fig.12 is a good representation of the coverage elsewhere except at the SOLAS ports and harbours. This section discusses results using Long term ASF model and the reference station coverage prediction model. We obtained from the GLAs the coordinates of places used as harbours and ports where TOA variance is only due to atmospheric noise. The next step of our algorithm is to include dLoran accuracy in the harbour and port designated areas. When using a system with dLoran capability, such regions will have accuracy below 10m as the IMO resolution requires.

A. INFLUENCE OF THE LONG TERM VARIATIONS ON RANGING ERRORS
It is reasonable to assume that all harbours and ports have reference stations and meet the IMO repeatable accuracy requirement of 10 m at 50 km radial distances. Various eLoran trials conducted up to 100 km from the Harwich harbour in the United Kingdom verified this assumption. The accuracy at port and harbour regions is factored into the accuracy plot to make it more realistic. Fig. 13 shows the repeatable accuracy modified by the land path model at other regions in the coverage area except at harbours and ports. The dark blue circles represent the accuracy performance at port regions in the coverage area. Fig. 13 demonstrates a much-improved accuracy model which is realistic than the one shown in [2]. This plot shows that IMO accuracy requirements are met at reference station locations and some parts of the Atlantic Ocean. Repeatable accuracy is poor in the west of the British Isles, Irish sea and the Republic of Ireland because of poor transmitter geometry. This model ignored the weather effects on repeatable accuracy. The following section discusses the results of the short term variation in the TOA variance model. FIGURE 13: Shows repeatable accuracy plot after including the long-term ASF variations outside port regions while meeting a 10m IMO requirement at all harbours.

B. INFLUENCE OF THE SHORT TERM VARIATIONS ON RANGING ERRORS
Repeatable accuracy varies temporally and spatially in the coverage area due to changes in the weather parameters. DLoran can only correct the effect of these changes at locations within the harbour's ASF correction spatial decorrelation limits. Beyond this, the position error is unmitigated due to limited dLoran resources. The accuracy plot in fig. 13 is made more realistic by including the effects of the changes in meteorological parameters at all areas except at ports and harbours. The results shown in Fig. 14 show that positioning accuracy in the other areas except SOLAS ports is worse compared to fig. 13. The coverage plot also shows better coverage in the North Sea than in the Irish sea. The results suggest that the IMO accuracy requirement are only met at harbours and ports. The results are consistent with those obtained by the GLAs during eLoran trials. Poor repeatable accuracy in the Irish Sea and some parts of Ireland is due to poor transmitter geometry. Installation of at least two transmitters in Ireland is needed to enhance transmitter geometry. The added transmitters are likely to improve the accuracy plot in the region between the Irish Sea and the British Isles. Fig. 14 portrays a real eLoran system.

V. CONCLUSIONS
This paper has presented a realistic eLoran accuracy model for HEA. The results have demonstrated that the short term ASF variations significantly contribute to the positioning error and must be included in the accuracy model. The results indicate that eLoran can achieve a 20 m accuracy or better in the North Sea, while a 10 m accuracy would be achievable at the SOLAS ports if eLoran was reintroduced in Europe. The results also show that eLoran meets navigational requirements for HEA. Nevertheless, the repeatable accuracy around the Irish sea exceeds 80 m and does not meet marine navigation requirements compared to GPS. Coverage can be enhanced by including at least two eLoran transmitters in Ireland.