Interactive Geolocation Techniques

Explore how different positioning methods work by dragging emitters and watching measurements update in real-time. Understand AOA, TDOA, FDOA, TOA and combined techniques.

Drag the red emitter to see how measurements change dynamically. All angles, distances, and timing values update in real-time.
AOA

Angle of Arrival

AOA determines location by measuring the angle at which signals arrive at multiple receiving stations. Two bearing lines from different stations intersect at the emitter location. Drag the emitter to see how the bearing angles change and affect positioning accuracy.

Ground-Based Space-Based
45.0deg
315.0deg
141.4km
141.4km
90.0deg
--km²
Emitter (Draggable)
Sensors
Bearing Lines
Error Region
200 km
±2°
θ = atan2(y_emitter - y_sensor, x_emitter - x_sensor)
Location = Intersection of bearing lines from multiple sensors

How It Works

  • Each sensor measures the bearing angle to the emitter
  • Directional antennas or phased arrays determine signal direction
  • Two bearings create two lines that intersect at the target
  • More sensors improve accuracy and resolve ambiguities

Accuracy Factors

  • Intersection angle affects precision (90 is optimal)
  • Angular measurement error amplifies with distance
  • Multipath causes bearing errors
  • Antenna array size limits angular resolution
TOA

Time of Arrival (Trilateration)

TOA measures the absolute propagation time from emitter to receivers. Each receiver defines a range circle centered on itself. The intersection of three circles determines the 2D position. Requires synchronized clocks between emitter and all receivers. GPS extends this to multilateration using 4+ satellites with atomic clocks. Drag the emitter or receivers to see how geometry affects measurements.

Ground-Based GPS/GNSS
141.4km
141.4km
141.4km
0.472ms
0.472ms
0.472ms
--km²
Emitters (Draggable)
Receivers (Draggable)
Range Circles
Error Region
±10 km
E: 1/5
R: 3/5
d = c × t (where t is propagation time)
(x - x_i)² + (y - y_i)² = d_i² for each receiver i

How It Works

  • Emitter transmits signal with known timestamp
  • Receiver measures arrival time (requires clock sync)
  • Range = speed of light × travel time
  • Three ranges define 2D position via trilateration

GPS Multilateration

  • Satellites broadcast precise time via atomic clocks
  • Receiver computes pseudoranges to 4+ satellites
  • 4th measurement solves for receiver clock bias
  • Intersection of range spheres = 3D position + time
GPS

GPS Multilateration (Ranging)

GPS uses multilateration – measuring ranges to multiple satellites with atomic clocks. Each satellite defines a range sphere. The red intersection region shows where all range bands overlap. Start with 2 satellites to see ambiguity, then add more for a unique fix. The 4th satellite resolves receiver clock bias. Drag satellites and receiver to explore geometry effects on PDOP.

Space-Based Navigation
20,200km
20,200km
20,200km
20,200km
2.5
--km²
GPS Receiver (Draggable)
Satellites (Draggable)
Range Circles + Error Bands
Intersection Region (Position Fix)
8 ns
SVs: 2/6
Pseudorange: ρ = c × (t_rx - t_tx) = R + c × δt_rx + ε
Position solution requires ≥4 satellites: 3 for (x,y,z) + 1 for clock bias δt

How GPS Multilateration Works

  • Each satellite transmits its position + atomic clock time
  • Receiver measures signal travel time (pseudorange)
  • Range spheres from 4+ satellites intersect at receiver
  • 4th satellite solves for receiver clock error

Error Sources & PDOP

  • Satellite geometry affects position accuracy (DOP)
  • Wide satellite spread = low PDOP = better accuracy
  • Satellites clustered together = high PDOP = poor fix
  • Ionospheric/atmospheric delays add range errors
TDOA

Time Difference of Arrival

TDOA measures the difference in arrival times between synchronized receivers. Each receiver pair defines a hyperbola of possible locations. The intersection of multiple hyperbolas pinpoints the emitter. Unlike TOA, TDOA does not require emitter clock synchronization – only receivers must be synchronized. Drag the emitter and receivers to explore how geometry affects the hyperbolas and location accuracy.

Ground-Based Space-Based
0.471ms
0.471ms
0.471ms
0.000ms
0.000ms
--km²
Emitter (Draggable)
Receivers (Draggable)
TDOA Hyperbolas
Error Region
2
±15 µs
Ambiguous - add receiver for unique fix
Δt₁₂ = (d₁ - d₂) / c
Hyperbola: |√((x-x₁)² + (y-y₁)²) - √((x-x₂)² + (y-y₂)²)| = c × Δt₁₂

📐 Why TDOA Creates a Hyperbola (with Ambiguity!)

A hyperbola is defined as the set of all points where the difference of distances to two fixed points (foci) is constant.

|d₁ - d₂| = constant
⚠️ Both Branches are Valid!

Points P and P' on opposite branches have the same distance difference. With only 2 receivers, the emitter could be on either side!

This is why TDOA needs 3+ receivers — to resolve which branch the emitter is on.

How It Works

  • Signal arrives at different times at each receiver
  • Time difference defines a hyperbola (constant path difference)
  • Two hyperbolas from 3 receivers intersect at the target
  • Requires precise time synchronization (nanoseconds)

Applications

  • E-911 cellular positioning (OTDOA)
  • Satellite signal geolocation
  • ADS-B aircraft multilateration
  • Gunshot detection systems
FDOA

Frequency Difference of Arrival

FDOA exploits the Doppler shift caused by relative motion between the emitter and moving collectors. Here, two satellites orbit in figure-8 patterns with a phase offset, creating continuously varying relative velocity. As each satellite moves through its orbit, the radial velocity toward the emitter changes, producing different frequency shifts. The difference in these shifts constrains the emitter location to an iso-FDOA curve.

Space-Based Figure-8 Orbits
+2.45kHz
-1.83kHz
+4.28kHz
7.5km/s
7.5km/s
--km²
Ground Emitter (Draggable)
SAT-1 (Figure-8 Orbit)
SAT-2 (Phase Offset)
Orbit Paths
Iso-FDOA Line
Error Region
7.0 km/s
±50 Hz
f_doppler = f₀ × (v_radial / c)
FDOA = Δf = f₀/c × (v₁·cos(θ₁) - v₂·cos(θ₂))

How It Works

  • Two satellites in same orbital plane at slightly different speeds
  • Each measures different Doppler shift based on geometry
  • FDOA = difference creates iso-Doppler curves on Earth
  • Combined with TDOA for single-pass geolocation

🚂 The Train Whistle Analogy

Imagine two people at different positions hearing the same train whistle:

  • Person closer to the train's path hears a bigger pitch change
  • Person further away hears a smaller pitch change
  • The difference in pitch changes = FDOA
  • FDOA tells you the emitter's angle relative to both receivers
  • Here, the "people" are satellites moving at different speeds!

Key Characteristics

  • LEO satellites ideal due to high orbital velocity (~7 km/s)
  • Speed difference creates changing FDOA over time
  • Accuracy improves with larger velocity differential
  • Works best for continuous wave signals
INTERFERENCE

Satellite Interference Geolocation

Dual-satellite geolocation locates interfering ground emitters using signals received through 2 adjacent GEO satellites at a single ground station. The station measures TDOA (path length difference) and FDOA (Doppler difference from Earth rotation). Drag the interferer and satellites to explore how geometry affects accuracy. This technique is used operationally for interference hunting in satellite communications.

Space-Based Interference Hunting
72,000km
72,000km
0.000ms
+0.00Hz
4.0°
--km²
Interferer (Draggable)
SAT-A (Draggable)
SAT-B (Draggable)
Ground Station
TDOA Hyperbola
Location Region
4°
±10 µs
±30 mHz
Path Difference: Δd = (d_iA + d_AG) - (d_iB + d_BG)
TDOA: Δt = Δd / c
FDOA: Δf = (f₀/c) × ω_earth × Δr × cos(lat)   (Earth rotation effect)

📍 Location Fix: Two Crossing Hyperbolas (with Ambiguity!)

Two hyperbolas intersect at TWO points — creating ambiguity:

TDOA Hyperbola

North-South oriented
(constant path difference)

FDOA Hyperbola

East-West oriented
(Earth rotation Doppler)

⚠️ Resolving Ambiguity

• Prior knowledge of emitter region
• 3rd satellite measurement
• Multiple time samples

Why roughly perpendicular? TDOA depends on longitude; FDOA on latitude — orthogonality gives good geometry!

How It Works

  • Interfering signal reaches both GEO satellites
  • Ground station receives signal via both satellite paths
  • TDOA from path length difference creates a hyperbola
  • FDOA from Earth rotation creates another hyperbola
  • Two hyperbolas intersect at 2 points (ambiguity!)
  • Prior knowledge or 3rd sat resolves true location

🎠 The Merry-Go-Round Analogy for FDOA

Imagine you're on a spinning merry-go-round (Earth rotating) throwing balls at two stationary friends (GEO satellites).

  • As you spin, balls thrown forward (toward your motion) arrive faster
  • Balls thrown backward arrive slower — this is the Doppler effect
  • Each friend sees a different ball speed based on your angle to them
  • The difference in speeds tells them WHERE you are on the ride!
  • Near the equator (fast spin) = big FDOA; near poles (slow spin) = small FDOA

Operational Considerations

  • Requires satellite ephemeris data
  • Adjacent satellites (2-6° separation) work best
  • Higher FDOA sensitivity at lower latitudes
  • Accuracy: typically 10-50 km CEP
  • Used by operators to locate illegal/accidental interference

Technique Comparison

Technique Min. Sensors Clock Sync Requirements Moving Platform Typical Accuracy Best Application
AOA 2 Not Required Optional 0.5 - 5 km Direction finding, DF networks
TOA 3 (2D) / 4 (3D) Required: Emitter + Receivers Optional 1 - 10 m Ranging systems, surveying
Multilateration
(GPS Ranging)		 4+ satellites Atomic clocks on satellites Optional 1 - 10 m GPS/GNSS navigation
TDOA 3 (2D) / 4 (3D) Receivers only (not emitter) Optional 10m - 1 km Cellular E-911, passive geolocation
FDOA 2 (moving) Required Required 1 - 10 km LEO satellite geolocation
Dual-Satellite
Interference Geo		 2 GEO sats + 1 station Required (ground station) Stationary GEO 10 - 50 km Interference hunting, spectrum monitoring
RSS 3+ Not Required Optional 10 - 100 m Indoor WiFi positioning

Clock Sync: TOA vs TDOA

  • TOA: Emitter and all receivers must share synchronized clocks to measure absolute propagation time
  • TDOA: Only receivers need synchronized clocks – measures time difference, eliminating emitter clock dependency
  • TDOA is ideal for passive geolocation of non-cooperative emitters

GPS Multilateration Principle

  • Each satellite broadcasts its position and precise atomic time
  • Receiver measures pseudorange to 4+ satellites
  • 4th satellite solves for receiver clock bias
  • Intersection of range spheres determines 3D position