# What are the fundamental limitations of echolocation in Earth's atmosphere?

I'm writing a cyberpunk novel where one of the characters has an augmentation that allows them to echolocate.

The idea is that there is a device on the center of his forehead (is there a more optimal location?) that produces extremely low-wavelength wavelets of sound at a very high rate. The wavelets are ultrasonic, so he has to have some kind of ear-implant that transfers the information to his brain. Since human brains aren't adapted to processing sound with such extreme precision, he also has to have a brain implant that assists with the necessary computations.

My question is about how powerful this ability could get, assuming there are no restrictions on how fast the implants work.

According to wikipedia, bats can echolocate with a frequency >200 kHz. Let's assume a frequency of 200 kHz. $$\lambda = \frac{343 \;m/s}{200,000 \;Hz} = 1.7 \;mm$$ is the wavelength of the sound, right? Does that mean that anything wider than 1.7 millimeters could hypothetically be detected? Obviously, this can't be true at large enough distances, so I don't think I'm approaching the problem correctly.

What is the upper bound on the frequency that could be used? What is the relationship between resolution and distance? How do both of these things change if the echolocator is moving at high speeds?

• You want hard science? A 2-second google search turned up dozens of academic articles about echolocation. Were none of those articles able to explain what you're looking for? The hard-science tag is basically asking for that level of an answer. It would be polite to be sure that answer doesn't already exist. – JBH Dec 27 '18 at 23:20
• Wavelength can be short or long, but never high or low. And soundwaves are more usually described in terms of frequency, not wavelength, mainly because (1) wavelength depends strongly on the medium of propagation and (2) human ears care about frequency and not about wavelength. For example, many people know that the A above middle C is at 440 Hz, but very few would associate that note with a wavelength of 779.5 mm. – AlexP Dec 28 '18 at 0:14
• "Obviously, this can't be true at large enough distances": not at all obvious. The detection limit comes from the relationship between emitted power, target reflectivity and size, and sensor sensitivity. Remember that the reflected power received back is inversely proportional with the fourth power of the distance. – AlexP Dec 28 '18 at 0:17
• Echolocation in normal, unaugmented humans is a reality. It does take some training to accomplish. This is used by vision-impaired persons, but sighted people can do it too. The simply click their tongue. Echolocation by augmentation could be potentially much more powerful & effective. – a4android Dec 28 '18 at 1:24
• @JBH This shows that bats can detect insects which are .5-1m away, and that they emit 10-20 pulses per second. This talks about how dolphins are adept at detecting small differences in size and shape. Most of the google scholar results focus on specific instances of echolocation, not the possibilities. "Echolocation in Bats and Dolphins" by Thomas et al focuses on the physics and signal processing involved, but I'm not sure the preview will do me any good. – Eben Cowley Dec 28 '18 at 2:02

## Resolution

The 1.7 mm wavelength describes your best resolution for identifying details.

In eyesight, the brain is engaging in a pattern of scanning the same object at multiple angles (see saccade). The brain merges the data interferometrically (much like a hologram). The result is that you can "see" smaller details than your best angular resolution.

So, some math :

$$K$$ is the smallest gap size that can be recognized. It's actually a different number than your wavelength, based on the density of receiver cells and overlap between layers of those receiver cells. You'll have to use some imagination to determine what $$K$$ is. For some additional information, natural rod cells are about 100 microns in size.

Angular size of any object, $$\delta \approx {d \over D}$$

Where $$d$$ is the size of the detail (in any units) and $$D$$ is the distance from the observer to that detail (in the same set of units).

If you imagine this kind of acoustic sight as a video screen, the number of "pixels" of information $$n$$ received by the observer watching an object of size $$d$$, $$D$$ far away is:

$$n = ({\delta \over {2K}})$$

Some research has shown that lighting conditions can provide up to a 10x gain in the number of "pixels" of data received.

There have been studies done on the lower bound of information (in bits) that the human brain needs to recognize a detail (https://people.csail.mit.edu/torralba/publications/howmanypixels.pdf) :

| #bit (n)= | 4 | 8 | 16 | 32 | 64 | 128 |

| Color | 20% | 40% | 60% | 80% | 90% | 100% |

| Greyscale | 10% | 20% | 50% | 70% | 90% | 100% |

And, speaking of color, if you'd like to have color for ultrasound "vision", the transmitter and and receiver would need to use a few different frequencies spaced some distance apart, but not too far apart.

## Effect of Rayleigh Scattering on Ultrasound "Illumination"

What may be a limiter with ultrasound is Rayleigh scattering, which happens in the acoustic domain just as it does in the optical one.

Much like a flashlight, the "ultrasound" receiver is generating the phonons that are bouncing off the target and back to the receiver.

If you are using omnidirectional illumination, the intensity of those phonons ($$I$$) drops off with the square of the distance $$I = {I_0 \over D^2}$$

For more focused illumination (like a flashlight) $$I = {I_0 \over {D sin \theta}}$$, where $$\theta$$ is the half-angle of the cone the flashlight covers.

Overlooking a number of parameters, Rayleigh scattering adds a $$1 \over {D^2 \lambda^4}$$ term to this effect, where $$\lambda$$ is your ultrasound wavelength. Putting that all together, for omnidirectional illumination, the total equation would be :

$$I = {I_0 \over D^2} - {I_0 \over {D^2 \lambda^4}}$$

(it's actually much more complicated than that - especially for acoustics - the above is a very rough estimate. For details, check out : https://en.wikipedia.org/wiki/Rayleigh_scattering#Cause_of_the_blue_color_of_the_sky and, for acoustics, https://asa.scitation.org/doi/full/10.1121/1.4918298)

## Reflectance and Camouflage

This might be starting to feel ridiculous. If this is a commonly used technology, some people will have studied how to hide from it.

The amount of reflected energy, $$R$$, is equal to the intensity of the phonons "illuminating" it ($$I$$) and some material-specific constant describing the material's ability to absorb or reflect the energy ($$\epsilon$$). Reflectance ($$\epsilon$$) ranges from 0 to 1.

$$R = I \epsilon$$

## Summarizing Illumination : Detection in Relative "Light" and "Dark"

For the receiver to be able to detect an object at range, enough reflected intensity needs to be picked up by the receiver to overcome noise ($$N$$). You'll have to decide for yourself, but in an urban environment where ultrasound is used for many things it may be very noisy.

The equation is :

Signal-to-Noise ratio, $$SNR$$ must be greater receiver gain/sensitivity. $$SNR = {db(R A)\over{db(r^2) db(N)}} = { db(R A) - 2 db(R) - db(N) }$$

db() is a function transforming the intensity values to decibel-meters (dbm)

For some ideas of real-world gains, I found a quick ultrasonic receiver online with a gain of 19 dbm.

## Gains from Pattern Recognition

Other studies have shown gains of up to 40 dbm from observing a signal and noticing patterns that are indicative of a signal.

I apologize for not pre-calculating all of that into a final result. I would recommend, for a set of frequencies, pre-figuring how far a X-watt ultrasonic "lamp" will be able to illuminate in urban or quiet conditions.

## Pratical Example

A character in your cyberpunk settings has tracked a target into the woods and is hunting in the dead of a moonless night by echolocation. You've pre-determined that, in this environment, the ultrasound "lamp" the character is using is only effective out to 300 feet.

Actually, the prey is savvy and has on camo gear designed for ultrasound, and the lamp is only good to 150 feet. However, the hunter is actively searching and you've pre-calculated that increases the effectiveness of the lighting to about 225 feet.

Based on the frequencies chosen, the target's size, and the distance, the cyberpunk is only getting 8 "pixels" of information : there is only a 20% chance of the hunter "seeing" it's quarry. Should have paid for the color upgrade - the chance would have been 40%.

• Wow, excellent answer. And thanks for mentioning the camouflage; while I wasn't intending for echolocation to be a common technology, my antagonists have very advanced technology, so I will probably be working echolocation camo in somehow. – Eben Cowley Dec 29 '18 at 0:21
• Glad to help. Hope you have fun with it! – James McLellan Dec 29 '18 at 14:03