Multibeam Swath and Refraction

What is modelled

One ping is a fan of narrow beams leaving a transducer 2.2 m under the surface. Each beam bends as the sound speed changes with depth: sound speed rises with depth in a uniform gradient and changes by a step across a thermocline. The seafloor is a plane with an adjustable slope. The time of flight of each beam is recorded and the foot point is reconstructed either by following the bent ray (correct) or by pretending the water has a single sound speed (orange curve).

Equations

• Sound speed c(d) = c0 + g (d - 2) + step / (1 + e-(d - zt)/2.2) with c₀ = 1492 m/s.
• Snell's law for layered water: the horizontal slowness p = sin θ / c is the same everywhere on a ray.
• Ray geometry dx/dz = p c / √(1 - p²c²), travel time t = ∫ ds / c.
• In a constant gradient g the exact ray is a circular arc, x = [√(1 - p²c0²) - √(1 - p²c²)] / (g p).
• Swath width in uniform water 2 d tan θmax.

Checks run in the page

• Straight beams in uniform water against the geometric foot point.
• Conservation of sin θ / c along a traced ray.
• Nadir time from the traced ray against an independent integral of dz / c.
• The traced ray in a constant gradient against the exact circular-arc formula.

Honest simplifications

The seafloor is a plane, there is no ship roll, pitch or heave, no beam footprint, no bottom detection error and no absorption or noise. Real systems measure beam angles with array processing and also correct for the sound speed at the transducer. The ping rate limit shown is only the time for the longest echo to return.

• Choose Uniform water: the orange and green curves coincide and the swath is 2 tan(angle) times the depth.
• Choose Strong thermocline: outer beams bend away from the vertical and the straight-ray assumption puts the seafloor in the wrong place.
• Tilt the seafloor and watch the swath become asymmetric.