[Loran Fundamentals]

[References]

International Loran Association, "Enhanced Loran (eLoran) – Definition document", 2007.

United States Coast Guard Navigation Center

[Fundamentals]

PNT(positioning, navigation, and timing)

eLoran:

- Internationally-standardized PNT

- Takes advantage of the 21st century technology

- Addition of a data channel on the transmitting signal: application-specific corrections(differential Loran messages), warnings(early skywave warnings), signal integrity information, greatly enhances the accuracy over the conventional Loran

- Real-time differential corrections: reference stations that detect tiny variations in the eLoran signal, provides information regarding signal integrity.

Enhanced navigation (e-Navigation): an exceptionally reliable input of position, navigation, and time data. use the combination of GNSS and eLoran

Application (meeting the accuracy)

- Availability, integrity, and continuity of performance requirements

Accuracy (for maritime harbor entrance and approach): 0.004-0.01nm(8-20m)

Availability (for aviation non-precision approach): 0.999-0.9999

Integrity (for aviation non-precision approach): 0.999999(1x10^-7)

Continuity (for aviation non-precision approach): 0.999999(1x10^-7): 0.999-0.9999 over 150 seconds

- Aviation non-precision instrument approaches

- Maritime harbor entrance and approach maneuvers in low-visibility conditions

- Land-mobile navigation

- Location-based services

- Precise time and frequency dissemination: telecommunications, internet communications

- Complement to GNSS(global navigational satellite systems)

System structure: control centers, transmitting stations, monitoring sites

Synchronization of transmission:

- Synchronized to UTC(coordinated universal time)

- Dual-mode receiver: when synchronized to a common time source, eLoran and satellite signals

Why Loran?

- Loran-C: 460-m accuracy

- GPS: started in 1980s. the vulnerability of GPS to disruption by intentional or unintentional interference.

Propagation: travel of the surface of the earth. groundwaves

Propagation delay: for accuracy improvement and to recover precise time (within 50ns). depend on the electrical conductivity of the ground. measured and corrected. User and system monitor receivers will store and employ these signal propagation corrections to maximize the accuracy.

Data channel in eLoran signal:

- Almanac of Loran transmitting and differential monitor sites

- Absolute time based on the UTC, leap-second offsets between eLoran system time and UTC

- Warnings of anomalous radio propagation conditions including early skywaves and singal failure

- Messages that allow users to authenticate the eLoran transmisstions; official-use only messages

- Differential Loran corrections

- Differential GNSS corrections

Transmitting stations: modern solid-state transmitter and control technology. UPS. The time and frequency control systems of the transmitter are designed for eLoran operation and they apply phase corrections in a continuous manner. Multiple cesium clocks.

Users' Equipment:

- All-in-view mode: acquire and track the signals of many Loran stations and use them all to make the most accurate and reliable position and timing measurements.

- MOPS (minimum operational performance standards)

Aviation

- RNP 0.3 (required navigation performance)

Horizontal accuracy: 0.3nm(556m) including human factors. 307m 95% of the time. Signal propagation corrections are published for each

airport and applied by the user receiver in real-time during each phase of operation.

Availability: 0.999-0.9999

Integrity: 10^-7 per hour. probability of providing hazardous misleading information.

Continuity: 0.999-0.9999 over 150 seconds

Maritime

- Shipping industry' strong growth. Sea-lanes are becoming more crowded. Electronic navigation is necessary in the crowded maritime environment.

- e-Navigation services: GNSS. GNSS alone cannot be guaranteed to meet the availability and reliability required. The combination of GNSS and eLoran providing a single combined output data stream (Redundancy by eLoran). "2020 The Vision" - General Lighthouse Authorities of the United Kingdom and Ireland, Oct. 2004.

- IMO (international maritime organization). navigation performance requirements for WWRNS(world wide radionavigation system) for harbor entrances, harbor approaches, costal waters with a high volume traffic and/or a significant degree of risk.

Accuracy: 10m (95%)

Signal availabilty: 0.998 over 2 years

Time to alarm: 10s

Service reliability: 0.9997 over 3 hours

Land mobile application of the eLoran: signal penetration capabilities. urban canyons.

[eLoran signal penetration]

- Steel shipping containers, refrigerated vehicles, storage warehouses

- The development of systems that track items either of high-value or whose safe and timely delivery must be guaranteed. Tracking of hazardous cargoes.

- Performance standards: not required. normally assessed and optimized for user specific applications.

[eLoran jamming]

Very difficult because one has to use large antenna structures with a very high peak power

[Relative navigation, Loran-C]

Definition: This is achieved by achieved by subtracting the Loran Time Differences measured at the reference point from the received Time Differences, thus reducing by common-mode cancellation the effect of variations in signal propagation velocity.

Application: aircraft final approach guidance.

Accuracy-limiting factors: GDOP, reference datum accuracy, the response of the receiver's tracking loops to noise and vehicle accelerations, by the local deformations in the shape of the Lines of Position grid.

Achievable accuracy: 100m or less (2d rms)(Loran-C)

[Loran-C system accuracy]

transmitted signal, signal propagation, signal reception, interference or errors from natural and man-made electromagnetic noise, skywave contamination, GDOP, other Loran-C signals, communication information superimposed on the navigation signal, and coordinate conversion.

[DOP, dilution of precision]

PDOP(position DOP); GDOP(geometric DOP); PDOP of less than 6 is useful; DOP multiplied by measurement and other input errors gives the error in position.

[TOA positioning]

P = pseudorange = range + range error; c = vacuum speed of light; dT = receiver clock offsets from GPS Time; dt = satellite clock offsets from GPS Time; d_ion = ionospheric propagation delay; d_trop = trophospheric (air + vapor) delay; e = measurement noise plus unmodeled effects such as multipath

[2DRMS]

95%

[Loran-C coverage modeling]

http://www.colorado.edu/geography/gcraft/notes/gisloran/gisloran.html

[Trends]

S. Basker, "The General Lighthouse Authorities' Loran programme and current status in Europe", Proc. Int. Loran Assoc. 2006 Conv. Tech. Symp., Groton, CT, Oct. 2006.

[DOP]

Earth circumference: 40,076.16 km (at the equator), 40,008.00 km (between N and S)

Earth diameter: 12,756.1 km (at the equator), 12,713.5 km (between N and S) → average value 12,734.8 km → Earth radius = 6367.4 km (10 cm is 1.57´10^-5 time the Earth radius!)

Spherical surface to flat surface conversion: mapping, (latitude, longitude) = (L_a, L_n) → (θ, φ) →

Distance between two points =

Unit vector from to :

unit_vector_from_Rx_to_Tx.tif \

1) TOA

$A = \begin{bmatrix} \frac {(x_1- x)} {R_1} & \frac {(y_1-y)} {R_1} & \frac {(z_1-z)} {R_1} & -1 \\ \frac {(x_2- x)} {R_2} & \frac {(y_2-y)} {R_2} & \frac {(z_2-z)} {R_2} & -1 \\ \frac {(x_3- x)} {R_3} & \frac {(y_3-y)} {R_3} & \frac {(z_3-z)} {R_3} & -1 \\ \frac {(x_4- x)} {R_4} & \frac {(y_4-y)} {R_4} & \frac {(z_4-z)} {R_4} & -1 \end{bmatrix}$

$Q = \left (A^T A \right )^{-1}$

$Q = \begin{bmatrix} d_x^2 & d_{xy}^2 & d_{xz}^2 & d_{xt}^2 \\ d_{xy}^2 & d_{y}^2 & d_{yz}^2 & d_{yt}^2 \\ d_{xz}^2 & d_{yz}^2 & d_{z}^2 & d_{zt}^2 \\ d_{xt}^2 & d_{yt}^2 & d_{zt}^2 & d_{t}^2 \end{bmatrix}$

$\begin{align} PDOP &= \sqrt{d_x^2 + d_y^2 + d_z^2}\\ TDOP &= \sqrt{d_{t}^2}\\ GDOP &= \sqrt{PDOP^2 + TDOP^2}\\ \end{align}$

2) TDOA

Closed form inversion formula for a 3 by 3 matrix

$\mathbf{A}=\left[\mathbf{x_0},\;\mathbf{x_1},\;\mathbf{x_2}\right]$ expressed in three column vectors

$\mathbf{A}^{-1}=\frac{1}{\det(\mathbf A)}\begin{bmatrix} {(\mathbf{x_1}\times\mathbf{x_2})}^{T} \\ {(\mathbf{x_2}\times\mathbf{x_0})}^{T} \\ {(\mathbf{x_0}\times\mathbf{x_1})}^{T} \\ \end{bmatrix}.$

$\det(\mathbf{A})=\mathbf{x_0}\cdot(\mathbf{x_1}\times\mathbf{x_2}).$

Closed form inversion formula for a 2 by 2 matrix

[Loran-C]

http://www.loran-history.info/Loran-C/Loran-C.htm