A synthetic aperture gives a sonar an along-track resolution of half the length of one transducer element, at every range. Range resolution is limited by a different quantity, the bandwidth of the transmitted pulse, and it also holds at any distance.
The along-track resolution of a real aperture is set by geometry, the length of the aperture against the wavelength, and it degrades in proportion to range. Transmitter power and receiver quality do not change it. Synthetic aperture processing exists to remove that dependence on range.
This page develops the result in seven steps, from a single transducer to a synthetic aperture. The numbers throughout come from the array ApertureLab simulates: elements 33 mm long, driven at 300 kHz, where sound in water has a wavelength of 5 mm. The closing figure runs a coarser version of the same array, because it forms its image in the browser, and it states which numbers it changed.
None of the argument is specific to sound. Radar arrived at it first, and synthetic aperture radar (SAR) is the same construction carried out with electromagnetic waves; every step below holds for either. This page works in sound, where the technique is called synthetic aperture sonar (SAS), because the numbers describe a sonar. The last section sets out where the two modalities differ.
Step 1 of 7
An element is a single transducer: the smallest piece of a sonar that turns an electrical signal into sound, and turns a returning echo back into a voltage. It acts as a microphone and a loudspeaker in one part, tuned to a narrow band and coupled to water instead of air.
It has a physical size. The element used throughout this page is 33 mm long. The sonar drives it at 300 kHz, and sound travels at roughly 1500 m/s in seawater, so one wavelength is 5 mm.
Those two lengths, the size of the element and the wavelength, determine almost every result that follows. Every array on this page, real or synthetic, is built from elements like this one.
Step 2 of 7
Sound leaving an element spreads into a cone, and the angular width of that cone is set by the ratio of wavelength to element size:
For a target in the water, the arrival time of the echo measures its range, and a wideband pulse separates two objects a few centimeters apart in range at any distance. Range resolution does not degrade with distance.
The echo carries no comparable information about direction; it establishes only that a scatterer lies somewhere inside the cone. Two objects at the same range, anywhere across that cone, return echoes that arrive at the same instant and add together, so the element records them as one measurement.
Because the cone widens with distance, the ambiguity grows in proportion to range:
A single element therefore produces an image that is resolved in range and smeared along-track, and the along-track smear grows with range. This asymmetry is the problem that synthetic aperture processing addresses.
Beamwidth conventions
Beamwidth can be quoted as the half-power width, the first-null width, or the nominal λ/D. They differ by factors near one. This page uses λ/D throughout so the numbers stay comparable from step to step.
Step 3 of 7
Consider a second element placed a distance d from the first, with the two received signals added together.
For a wave arriving straight ahead, the crests reach both elements at the same moment. The two voltages are in step, and the sum is twice either one.
For a wave arriving off to the side, one element is farther along the incoming wavefront than the other. The extra distance the wave has to cover is d sin θ. When that extra distance is half a wavelength, the two voltages are exactly opposite and the sum is zero. The pair has no response in that direction.
The figure below sweeps the arrival angle.
Neither element changes during the sweep. Each keeps the directivity its own size gives it, and neither carries any information about the direction in which the pair is steered. The whole response curve comes from the phase relationship between two separated measurements.
Step 2 gave a single element a beam 8.6 degrees wide, so a transducer does have directivity of its own. It acquires that directivity by the same mechanism. An element's face is a continuous aperture, and its λ/D beam is the same interference sum carried out across its own width, over a continuum of points instead of two. An element of size D and an array of length L therefore have the same pattern, with D replaced by L.
Directivity is therefore a property of the aperture. The element is the smallest aperture available, and its size is fixed when the transducer is made. A narrower beam requires a longer aperture, and the remaining steps examine how long an aperture can be made.
Step 4 of 7
Adding elements along the same line introduces more cancellation directions and narrows the one direction in which every contribution still adds. For N elements spaced d apart, the beam narrows to roughly λ/L, where L = N·d is the total length of the array.
The beamwidth is set by the length of the array. Two arrays of the same overall length have nearly the same beamwidth whether they are built from 16 elements or 64. The additional elements reduce ambiguity and raise the signal-to-noise ratio, but they do not narrow the beam. Halving the beam requires doubling the length.
The receive array in these simulations is 36 elements at 33.33 mm, 1.2 m end to end. It narrows the beam from 8.6 degrees to a quarter of a degree, and the footprint at 50 m from 7.5 m to 21 cm. That footprint is still far coarser than the resolution set as the goal in step 5.
Step 5 of 7
Take as the goal an along-track resolution about the size of the element itself, 17 mm, at 100 m. Rearranging δ = Rλ/L gives the array length required:
That is thirty meters of rigid array, held straight to a small fraction of 5 mm and towed through moving water behind a vehicle two meters long.
The requirement follows from the geometry rather than from the materials available, and it sets the practical limit on the length of a real aperture.
Step 6 of 7
In step 3, two elements a distance d apart were two measurements of the same wavefield from two separated points. Nothing in that argument required the two measurements to happen at the same instant, or to come from two different pieces of hardware.
The vehicle already moves in a straight line, along the direction in which the array has to run. One element at position 1 during ping 1, and the same element at position 2 during ping 2, are two measurements from two separated points, which is all that a two-element array provides.
The array can therefore be synthesized from the motion of the vehicle. The sonar transmits a ping, moves, transmits again, and records every complex echo. Afterward, in software, the returns from all the positions are summed with the phase each one carried, exactly as an array sums across its elements. The aperture is synthesized in time instead of being built as hardware, and its length is set by the distance traveled.
The available length follows from the beam. A point at range R stays inside the element's 8.6 degree cone for Rλ/D of travel: 7.5 m of aperture at 50 m range and 15 m at 100 m. The available synthetic aperture grows in proportion to range, which is the same proportionality that degraded the resolution in step 2.
The role of the 36 receivers
Since the synthetic aperture sets the resolution, the 36 receivers of the real sonar serve the sampling rate. The synthetic array has to be sampled about every D/2 of travel or it aliases, and a single element would force a ping rate that limits the survey speed of the vehicle. Recording 36 receivers per ping lays down 36 samples of the synthetic aperture at once, so the vehicle can move farther between pings and the coverage rate rises.
Step 7 of 7
The two results combine directly. The aperture available at range R is Lsyn = Rλ/D. The resolution of an aperture that long is δ = λR / (2·Lsyn), where the factor of two comes from the round trip. The phase of a synthetic array turns twice as fast as that of a real one, because the path is out and back. Substituting:
Along-track resolution becomes half the length of one element, at 10 m and at 100 m alike, and it does not depend on wavelength either. A shorter element gives a finer image, which reverses the behavior of a real aperture, where a longer aperture gives the finer image. The same round-trip factor is why 15 m of travel replaces the 30 m of towed array from step 5.
The result depends on the sum being coherent. Summing hundreds of pings works only if the phase of each one is correct, which requires the position of the sonar at every ping to a small fraction of 5 mm. Underwater there is no GPS (Global Positioning System), and an inertial system drifts well past a millimeter over the tens of seconds an aperture takes to collect. The figure below shows the effect of that error on a vertical scale in decibels (dB), a ratio measure on which every 20 dB is a tenfold drop in amplitude.
At the end of the slider, or with discard the phase selected, the point response becomes a flat line. With the phase discarded, 7.5 m of travel and several hundred pings give no along-track resolution. The target lies somewhere in a 7.5 m smear, which is where a single element left it in step 2. The resolution of a synthetic aperture therefore comes entirely from the phase of the recorded echoes.
The track is therefore recovered from the echoes themselves. Successive pings put different receivers at the same physical point in the water, and those redundant measurements observe the same patch of seafloor; the delay between them measures how the vehicle moved. This technique is micronavigation. Once the track is known, the summation is carried out against the true geometry instead of an assumed straight line, which is what time-domain back-projection does.
These two requirements set the structure of a synthetic aperture processing chain. The geometry supplies a range-independent resolution, and the chain realizes it only after the motion of the platform has been measured to a fraction of a wavelength.
Demonstration
The preceding steps established an aperture synthesized from motion, a sum that works only if the phase is correct, and a tolerance on how well the geometry has to be known. The final figure combines all three.
The figure below runs a time-domain back-projection in the browser, the same algorithm ApertureLab's beamformer uses. The sonar is the array from step 4 instead of a single element: thirty-six receive elements with one transmitter at the array center. Its dimensions are chosen so that a browser can compute the image. The elements are 50 mm long and the carrier is 150 kHz, so along-track resolution is D / 2 = 25 mm instead of 17 mm. The 10 mm wavelength widens the element footprint at 50 m to 10.05 m. A full aperture therefore forms in twenty pings instead of several hundred.
Each ping measures thirty-six phase centers, one per receive element, each at the midpoint between the transmitter and that element and so on a 25 mm pitch, half an element length. The vehicle then advances 34 of those spacings, 0.85 m, before the next ping, which leaves exactly two phase centers shared between consecutive pings. Those two are one place in the water measured twice, and micronavigation reads the vehicle's motion from the delay between the two measurements. The fidelity page shows the measured version of that redundancy on simulated data. In this figure each shared phase center is counted once, split between the two pings that made it. Summing both would put a ripple on the aperture weighting and a pair of ghosts on either side of every bright target.
A 6.4 m square patch of seabed at 50 m ground range holds a cylinder, a disc and a scatter of rocks on rippled sand. The vehicle flies 5 m above the bottom, a tenth of the range, so the grazing angle at the patch is 5.7 degrees. Each shadow is that object's own silhouette stretched down-range by that angle. The raw data are simulated once, as 20 pings by 36 channels over a 17 m track, with the exact two-way path from transmitter to scatterer to element for every one of the 720 channel-pings. The simulation includes a Hann-windowed pulse compressed to 25 mm in range, receiver noise, and 10,000 independent scatterers per square meter of seabed. Each slider movement forms the image again from those data.
The two sliders produce failures that look different.
Shortening the aperture blurs the image along-track only. The cylinder stretches into a streak while its extent in range stays as sharp as before, because range resolution comes from bandwidth and does not depend on the distance traveled. This is the asymmetry from step 2, still visible at the end of the chain.
A sound speed error moves the scene before it blurs it. The assumed speed converts echo time into range, so an error slides the whole image in range, by half a meter per percent at 50 m. The along-track focus degrades more slowly, because a common scale factor on every range nearly maps each focusing hyperbola onto that of a target at the scaled range; only the residual curvature mismatch defocuses. Measured on this simulation, a 0.3% error slides the scene six pixels in range and broadens the along-track response from 27.4 mm to 31.8 mm, a change that is difficult to see. At 1% the slide is 20 pixels and the response has nearly quadrupled, to 101 mm. Unlike at short range, 10 m of aperture against 50 m of range leaves about one cycle of quadratic phase error at 1%, so the tolerance is a few tenths of a percent.
This ordering is the practical risk. A navigation error produces a visibly defocused image, whereas a small sound speed error produces a well-focused scene with incorrect geometry. The slider also stays within the variation the sea produces. Sound speed in the ocean runs from roughly 1450 to 1550 m/s, so an assumed 1500 m/s can be 3% wrong, twice the end of this slider. For this reason a sound speed profile is measured rather than assumed.
The figure above images a flat seabed. The surface of focus covers the effect of vehicle motion and a wrong sound speed on objects that stand above the seabed. It also explains why the surface a beamformer images onto behaves like the plane of focus of a lens, using images from ApertureLab's simulator and beamformer inside a 3D scene.
The back-projection above sums complex baseband samples, each multiplied by the carrier phase of its delay. Why the echoes are stored that way, and what happens to the image when that phase is left out or applied with the wrong sign, is the subject of Why baseband?
Comparison
Every step above is independent of the medium. A platform moves and measures a wavefield from a sequence of positions, and the phase relationships between those positions are assembled into an aperture. Synthetic aperture radar is the same construction and was developed first, and the mathematics of image formation is largely shared.
The physical constants, however, differ by five orders of magnitude. Sound in water travels 200,000 times slower than light, and almost every difference below follows from that ratio.
| Airborne SAR, Ka-band | SAS, 300 kHz | |
|---|---|---|
| Propagation speed | 300,000 km/s | 1.5 km/s |
| Wavelength | 18 mm | 5 mm |
| Fractional bandwidth | 3.6 % | 20 % |
| Working range | 10 km | 100 m |
| Along-track resolution | 150 mm | 17 mm |
| Round trip to max range | 67 µs | 133 ms |
| Platform travel in that time | 0.4 λ | 53 λ |
| Stop-and-hop error, beam edge | 0.01 λ | 4 λ |
| Pulse rate ceiling from range | 15 kHz | 7.5 Hz |
| Position knowledge | GPS and inertial, centimeters | no GPS; estimated from the echoes |
| Propagation speed knowledge | exact for practical purposes | 1450 to 1550 m/s; must be measured |
| Polarization | horizontal and vertical transmit/receive pairs (HH, HV, VH, VV); a primary channel | none; pressure is a scalar field |
A radar pulse out to 10 km and back takes 67 microseconds, during which a fast aircraft covers under a centimeter. A sonar pulse out to 100 m and back takes 133 milliseconds, during which the vehicle covers 27 cm, more than fifty wavelengths. Transmission and reception therefore occur at different positions.
SAR processing normally treats the platform as stationary for the duration of each pulse. That approximation, called stop-and-hop, costs a radar about a hundredth of a wavelength of path error at the edge of its beam and costs this sonar about four wavelengths. A SAS processor therefore cannot use it, and it models the platform in continuous motion throughout.
A sonar cannot transmit again until the previous echo has returned, so at 100 m range the ping rate cannot exceed 7.5 per second. Step 6 required the synthetic aperture to be sampled every D/2 of travel. Together these limit a single-element sonar to a survey speed of 12 cm/s. With 36 receivers laying down 36 samples of the aperture per ping, the limit rises to 4.5 m/s; this is the purpose of the receive array, and it bounds the speed of a survey vehicle. The pulse rate limit of a radar lies in the kilohertz, thousands of times above what its aperture sampling requires.
A SAR platform carries satellite positioning, which fixes its track to a few centimeters; autofocus then removes the residual error, which at an 18 mm wavelength still exceeds the tolerance derived in step 7. A submerged vehicle has no such fix, and an inertial system drifts past a millimeter long before an aperture is complete. The track has to be reconstructed from the echoes themselves, so micronavigation is a required stage of a SAS processing chain rather than an optional refinement. It is the largest single difference between the two processing chains.
The propagation speed of radio waves can be treated as exact. Sound speed in the sea runs from roughly 1450 to 1550 m/s and varies with temperature, salinity, and depth along the path. An error of a tenth of one percent places a target at 100 m 10 cm from its true position, an error of twenty wavelengths, so the sound speed profile must be measured or estimated. Pressure in water is also a scalar field, so sonar has no polarization channel, which radar uses as a primary channel. Acoustic absorption rises steeply with frequency and couples resolution and range more tightly than in radar, so finer resolution is obtained at the cost of range.
Both are coherent, complex-valued imagers that synthesize an aperture out of motion. Both produce single-look complex imagery with fully developed speckle and the same underlying statistics. Both are focused by the same mathematics, whether time-domain back-projection or a wavenumber-domain method. Both show layover, shadow, and range-dependent geometry, and in both an error in platform position appears as an error in phase.
This shared structure makes the SAR literature a natural starting point for a SAS problem, and it suggests that a model trained on one coherent modality is a credible starting point for the other. The differences listed above concern engineering parameters and leave the physics of coherent imaging unchanged.
ApertureLab simulates this chain forward, from a scene file to raw complex time series, and focuses the result into an image by time-domain back-projection. The two pages below describe the forward model and the processing chain.