TDOA Buoy Array 2D: Hyperbolic Wildlife Localization

Buoy Array

Hydrophone buoys 4

Speed of sound in water 1500 m/s

Buoy clock jitter (σ) 0.3 ms

Tagged animal swim speed 1.0×

Show hyperbola branches

Show Gauss-Newton path

Live Localization

Position error (last fix)

— m

Mean error

— m

Fixes solved

GN iterations (last)

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How it works

Each robotic buoy is a synchronized hydrophone receiver at a known position B_i. A tagged animal emits an acoustic ping that spreads outward at the local speed of sound c. Every buoy records its own arrival time t_i. Picking buoy 0 as reference, the range difference to every other buoy is a direct measurement:

r_i = c · (t_i − t_0) = |P − B_i| − |P − B_0|

For a fixed pair of buoys, every point P satisfying a constant range difference r_i lies on one branch of a hyperbola with foci B₀ and B_i — drawn live on the map for every buoy pair. With 3+ buoys (2+ independent hyperbolas) the intersection pins down the animal's position P = (x, y) — this is TDOA multilateration, the same principle behind GPS, Loran-C and cellular E911 localization.

Because r_i(P) is nonlinear in P, the solver here uses Gauss-Newton least squares: starting from the buoy centroid, it linearizes the residuals f_i(P) = (|P−B_i| − |P−B_0|) − r_i with the Jacobian

∂f_i/∂x = (x−B_ix)/|P−B_i| − (x−B_0x)/|P−B_0|
∂f_i/∂y = (y−B_iy)/|P−B_i| − (y−B_0y)/|P−B_0|

and repeatedly solves (JᵀJ)·Δ = −Jᵀf for the correction Δ until it converges (typically 6-10 iterations, traced live in the panel below the map and plotted on the convergence chart). Buoy clock jitter injects Gaussian timing noise into every t_i, which is exactly why real telemetry arrays need tight clock sync — a few milliseconds of jitter at c ≈ 1500 m/s already moves the fix by meters.

Tagged animal (true position) Hydrophone buoy TDOA position estimate Hyperbola branch (buoy pair)

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Gauss-Newton convergence (|Δ| per iteration, last solve)