Every image-forming instrument is in focus somewhere. A camera lens brings exactly one distance to a sharp point, and a photographer learns to see that plane of focus even though nothing in the scene marks it. A synthetic aperture sonar has an equivalent, and it matters to anyone who reads seafloor imagery or trains models on it: the beamformer, the processing that turns recorded echoes into an image, draws the seabed on a chosen surface, and anything that stands off that surface is displaced, and is blurred as well whenever the vehicle's path is not straight.
The scene below makes that surface visible. The teal sheet is the surface the beamformer images, and it can be dragged up and down. Beside three objects on the seabed, a wedge, a truncated cone and a cylinder, floats a point target, the single ideal scatterer that engineers use to measure an imaging system, at a height you choose. Every image below was made by ApertureLab's own simulator and beamformer, which forms each pixel by adding every recorded echo at the delay that point would produce; the controls step through the settings those images were computed for.
Point target image
Along-track cut, dB against offset in cm
Range cut, dB against offset in cm
Controls
Point target
8 m × 8 m, along-track up, range to the right (12 to 20 m from the track). × where the point target stands · ○ where it is drawn
This is the whole patch, beamformed onto the same surface with the same settings: sand at 20,000 scatterers per square metre, the wedge, truncated-cone and cylinder meshes of ApertureLab's dataset, and the point target. Surfaces are brightest where they face the track squarely, and each object hides the bed behind it, which is the dark acoustic shadow stretching away from the track.
The images were computed ahead of time for every combination of the controls, 792 in all, from twelve simulations (one for each motion and point-target height) and a beamforming run for each surface height, sound speed and aperture.
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A synthetic aperture is built by listening from many positions along the vehicle's path and adding the echoes so that one chosen point's contributions line up. What makes them line up is the range history: the distance from each position on the track to the point. For a straight, level track, that history depends, apart from a shift along the track, on only one number, the point's closest range to the track. Every point at the same closest range, whether it sits on the seabed, halfway up a mast or in open water, produces the same history.
The set of points sharing a closest range is a cylinder wrapped around the track, drawn in amber around the selected object. A straight-track beamformer cannot tell points on it apart, and so it cannot defocus them either: the point target comes out as sharp 3 m above the seabed as on it. The real array gives up about a decibel of peak off the surface, because its transmitter and receivers sit apart and the vehicle keeps moving while the sound travels, which the simple cylinder picture leaves out; the width of the point does not change.
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The beamformer still has to put each echo somewhere, and it puts it where the cylinder meets the image surface. For an object standing above the surface, that meeting line lies nearer the track than the object itself, so the object is drawn closer in than it stands. This is layover. Raised to 5 m, the point target is drawn about 2.5 m nearer the track than the point it floats above.
Because layover is a known function of height, a system that knows an object's height can move it back; blur cannot be undone that way, and the next section shows where it comes from.
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A real vehicle sways and heaves. The beamformer is given its measured path, so points on the image surface stay sharp: for them, the correction for the path is exact. A point above the surface is seen along a flatter line of sight than the surface point it is drawn on, so the same sideways or vertical movement of the vehicle changes its range by a different amount. That difference is a phase error, and on average it grows with the point's height above the surface.
The size of the error depends on how the motion varies within one synthetic aperture, which here is about 3 m long. A slow swell, whose wavelength is several times the aperture, moves the vehicle almost in a straight line during that time, and a point target 5 m up stays sharp, though it gives up about 3 dB of peak. Chop, whose wavelength is comparable to the aperture, bends the path within it, and the higher the point target floats, the more of its peak it loses: about 2 dB at 1 m and 8 to 9 dB at 3 and 5 m, where its energy spreads along the track over about 11 cm. The along-track cut shows the energy leaving the main lobe.
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Raise the surface to the point target's 5 m and the target comes back into focus under the same chop, because it now lies on the surface whose path correction is exact. The seabed, now 5 m below the surface, blurs instead, and so would a point target left on it, which loses 7 dB and spreads over 12 cm. This is the sonar counterpart of focusing a lens on a subject: one height is sharp, and the sharpness falls off above and below it. The surface can be lowered beneath the bed as well; the seabed then stands above the surface it is drawn on, is drawn nearer the track than it lies, and under motion blurs as a raised point does.
The analogy has a limit worth keeping. A lens has a plane of focus because of its optics; a straight-track synthetic aperture has none, and only acquires one when the vehicle moves off the straight line. Survey systems therefore image onto the best estimate of the seabed they have, and objects that stand well above it are the ones the vehicle's motion defocuses.
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The beamformer converts distance to time with an assumed speed of sound. If the water carries sound at 1500 m/s and the beamformer assumes 1530 m/s, every range is stretched by 2 percent, and the curvature of every range history is predicted wrongly as well. Unlike motion, this error does not depend on height; it defocuses the whole image, and it does so by an amount that grows with range, so no single choice of surface undoes it.
Small errors mostly stretch the geometry and leave the focus nearly intact: at 1510 m/s the point target loses about half a decibel. At 1530 m/s it loses more than 3 dB and doubles in width, and the loss grows quickly beyond that, and the ocean itself carries sound at anywhere from about 1450 to 1550 m/s, which is why the speed of sound is measured on the vehicle rather than assumed.
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The resolution of a synthetic aperture along the track improves with the length of track it integrates over, so using only part of the available aperture coarsens the image in inverse proportion: at 35 percent the point target is more than twice as wide. It also shortens the stretch of motion that has to add coherently: under chop, the point target at 3 m loses about 4 dB of peak with 35 percent of the aperture instead of more than 8 dB with all of it.
A photographer who stops a lens down trades light and sharpness for depth of field, and the trade here is similar: a shorter aperture gives a coarser image that tolerates more height and more motion. A beamformer can make this choice region by region, from what is known about the scene and the vehicle.
How the figures are made
The sonar is the array the rest of this site describes: 36 receivers 30 mm long at 300 kHz with 60 kHz of bandwidth, advancing 34 phase-centre spacings per ping so that consecutive pings share two, on a track 10 m above a flat sand bed. The seabed is simulated with ApertureLab's point-scatterer engine at 20,000 scatterers per square metre, with the objects as meshes and their acoustic shadows cast by the simulator. The vehicle sways and heaves sinusoidally, the heave about half the sway at 1.3 times its frequency, and the beamformer is always given the true path.
Every image is a time-domain back-projection formed on a flat surface at the height the control sets, with the sound speed the control sets, shown through the dataset's display chain with one fixed brightness scale per aperture, so that a blurred point really is dimmer than a focused one. The point target's own response is isolated by simulating each scene a second time without it and subtracting the two images, which is exact because the imaging chain is linear; the point-target panel and the numbers in the readout are measured on that response.
For the steps that lead up to a synthetic aperture, see Why synthetic aperture?; for the full imaging chain, including micronavigation, see Image formation. On a circular track the depth plane becomes a true plane of focus; that case has its own page, The plane of focus in circular SAS.