Principles, Accuracy, and Calibration of Radio Positioning

Positioning classification
- Based on Lines of Position (LOP) systems:
- Circular systems (based on distances)
- Polar systems
- Hyperbolic systems
- Parameters for geometric quality
- Statistics and Accuracy
- Modeling:
- Circular systems
- Polar systems
- Hyperbolic systems
- Calibration for:
- Circular systems
- Hyperbolic systems

Circular Method (Active):
- Also known as the ‘range/range’ method
- Involves sender and receiver on a vessel
- Coastal stations receive signals and respond, usually by pulse or phase.
- Systems used:
- Synchronised clock on vessel (e.g., atomic clock or GNSS time)
- Determines distances to 2 or 3 coastal stations.
- Results in circular LOPs.
Circular Method (Passive):
- ‘Pseudorange’ Method:
- Represents false circular method, operates as a hyperbolic system.
- Coastal stations send signals, synchronised (e.g., with atomic clocks).
- Options include pulse or continuous wave.
- Reception through:
  - Time difference or Phase difference.
  - Determines distance differences => results in hyperbolic LOPs needing 3 coastal stations to obtain intersection of 2 hyperboles.
Polar Method (Active):
- Uses both distance and bearing.
- Coastal station determines vessel direction using radio-goniometer.
- Interaction involves pulse sent from coastal station to the vessel and back.
- High accuracy achieved due to orthogonal intersections of LOPs, utilizes telemetry for position transmission.
Hyperbolic Method:
- Involves a master and slave(s) system, which are synchronised.
- Continuous waves are sent; phase differences correspond to distance differences, leading to hyperbolic LOPs.

Random Errors:
- Precision is measured by individual and accidental deviations relative to the average value.
- Caused by observation or equipment imperfections.
- Inevitable errors that cannot be corrected even by calibration.
- Characteristic clustering of measured values around average position, where good precision indicates small random errors.
Statistics:
- Boxplot Representation:
- Displays first, second, and third quartiles.
- Defines outlier as distance to box > 1.5 * box height.
- If no outliers, whiskers connect to smallest and largest values.
Standard Deviation (SD):
- More commonly used than CEP for scientific purposes.
- Sensitive to small numbers of large errors.
- SD is calculated by SD = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N-1}}
- For a normal Gaussian distribution, 68% of values lie within 1 standard deviation from the mean, and 95% lie within 2 standard deviations.
Circular Error Probable (CEP):
- Defined as the radius of a circle, containing 50% of all measurements around the best guess for the “correct position.”

Systematic Error:
- Defined as the deviation of the average of measurements from the 'true' position, which is often unknown.
- Accuracy inversely relates to the size of the systematic error.
- The most probable value can be approximated by assuming minimal systematic errors.
Sources of Systematic Errors:
- Non-properly calibrated sensors; should be calibrated using more accurate values or compensated computationally.
- Assumptions or simplifications in mathematical models can introduce errors; utilize precise models for accuracy.

Repeatability:
- Refers to positional variation with repeated measurements using the same instrument over time at the same location.
- Low test-retest reliability causes variability in results.
- Difference between repeatability (variation at the same place and time) and reproducibility (variation at the same location but different instances).
Blunders and Reliability:
- A blunder is defined as a careless mistake that can be easily traced.
- Reliability pertains to the ability to identify blunders, with improved reliability from more data availability.

Essential for checking:
- Installation quality of the positioning chain.
- Coordinate precision of reference beacons.
- Stability of transmission parameters and good radio paths.
Calibration Practices:
- Involves comparing measurements with known references (baseline crossing techniques) to verify accuracy.
- Different methods for circular and hyperbolic systems involve adjustments for delays and biases to achieve accuracy.

Distance Measurement Corrections:
- Assumptions regarding the difference in shortest pathways while incorporating radio path lengths and height variations.
- Address variations in speed of propagation due to atmospheric conditions and adjustments based on height differences.