TDOA Solution & Results

Case FA-2026-001 | Muzzle blast source localization via multilateration

SOLUTION

Position

~2.6m south of stage center

Local coords: (-0.5, -2.2)m | Range across PA models: (-0.5 to 0.9, -3.3 to -2.2)m

95% Confidence Ellipse

2.42m × 1.01m

Rotated -151° from east | Area: 7.7m²

Best-Fit RMSE

0.005ms

Distance from Stage

~2.55m (bearing 169°)

PA model spread: <0.7m total

RECEIVER ARRAY & SOLUTION

Local coordinate system: receivers (colored circles), PA speakers (blue triangles), TDOA solution (diamond), 95% confidence ellipse (dashed orange).

SATELLITE VIEW

Geographic positions on satellite imagery. Click any marker for details. The red dashed ellipse shows the 95% confidence region with proper rotation.

ESRI World Imagery. Receiver positions from KML (±0.5m). Solution and confidence ellipse from TDOA multilateration.

TDOA MEASUREMENTS

Reference receiver: R2. Corrected TDOAs (zero PA correction model):

R3 − R2

-1.684ms

R3 closer to source

R4 − R2

+5.025ms

R4 farther from source

R1 − R2

+6.549ms

R1 farthest from source

Receiver	Distance to Source	Residual
R3	~11.9m (closest)	<0.01ms
R2	~13.5m	<0.01ms
R4	~14.9m	<0.01ms
R1	~15.8m (farthest)	<0.01ms

TDOA HYPERBOLA INTERSECTION

Each TDOA defines a hyperbola — the locus of positions consistent with the measured time difference between a receiver pair. With 3 independent TDOAs, the hyperbolas intersect at a unique point: the source location.

Three TDOA hyperbolas intersecting at the solution. Green: R2–R3, Blue: R2–R4, Orange: R2–R1.

UNCERTAINTY QUANTIFICATION

Monte Carlo Simulation (N=10,000)

Each iteration perturbs receiver positions, onset times, and speed of sound within their measured uncertainties, then re-solves the TDOA system. The scatter of solutions defines the confidence region.

Position

±0.5m (1σ)

TOA

±1.0ms (1σ)

Speed of Sound

±0.7 m/s (1σ)

Convergence

100%

10,000 Monte Carlo solutions. Dashed orange: 95% confidence ellipse (2.4 × 1.0m). Inner dashed: 1σ ellipse.

Jackknife Leave-One-Out

Each receiver is removed in turn and the solution re-computed. This tests which receivers most influence the result.

Dropped	Shift	Interpretation
R1	~0.7m	Most influential — only receiver north of stage, provides all N-S constraint
R3	~0.0m	Redundant with R2 at 1.6m separation — acts as consistency check
R4	~2.4m	Largest shift — R4 anchors east-west position, most geometric leverage

Error Budget

Source	Magnitude	Position Impact
Synchronization	±0.021ms	<0.007m (negligible)
Onset picking	±1–2ms	~0.3–0.7m (dominant)
Receiver position	±0.5m	~0.3–0.5m
Speed of sound	±0.7 m/s	~0.2m at 15m range
PA correction	<1ms effective	<0.22m per ms
Combined (95%)	—	2.4 × 1.0m ellipse

CONCLUSIONS

The muzzle blast originated approximately 2.6 meters south of the stage center (bearing 169°), within the audience/crowd area directly in front of the speaker podium. The 95% confidence region is a 2.42 × 1.01 meter ellipse (7.7m²).

This result is robust: it persists across 5 independent PA correction models, survives jackknife analysis with only moderate shifts, and the dominant uncertainties (onset picking and receiver position) are well-characterized.

This analysis locates the sound source. It does not identify the shooter or address any questions beyond the acoustic evidence.