Using acoustics to measure the composition, density, and hardness of layers

Question

I am an engineering student at Alfred University and some fellow classmates and I are doing a competition, but one of the tasks we need to complete is autonomously measuring the composition, density, and hardness of layers of overburden. I was thinking that using acoustics would be possible to accomplish these things. What follows is exactly what we have to do:

"2.Create a digital core that contains the following information: The number of overburden layers in their test station A sequence of the layers in order from softness to hardest The thickness of each layer The digital core should result from information garnered using system telemetry and not via placing a ruler down the hole. Teams may not touch the layers to determine hardness."

The reading has to be continuous. For example another way we could do this is measure the resistance on our auger as it tries to dig a hole. This would provide a continuous reading, but might not be accurate because our frame holding the auger may not be solid.

The question is "can we use sound waves to measure the composition, density, and hardness of each layer?" The total overburden depth will be between 0.5 and 0.8 m. I don't know if this is a large enough volume to measure all of these things using sound waves.

Answer 1

The answer to your question is 'yes': you can probe a medium of arbitrary complexity using sound waves, and relate the propagating wave to the physical properties encountered on its path: the stiffness (Hooke's law), density (Newton's law), and you can locate the layers by figuring out where the stiffness and density change. If the competition only requires you to submit a hypothetical method, such a seismological method certainly fits the bill.

Unfortunately, when it comes to practically implementing such a method; I advice against it. You'll find too many obstacles on your path that require expert input and money. For example:

• You require a very well controlled source: you need to know exactly the source function (the wavelet) that ends up in your medium; which is distorted by the actuator's frequency response, and which is distorted by the usually weak coupling between your source and the medium.
• You require a high-resolution receiver as well, to record the signal that propagated within the medium, which is similarly not trivial to create.
• After you have solved these (difficult!) problems, you have to turn your recording at the surface into a quantitative 2-D/3-D physical model of the Earth, i.e. a model that tells you the velocity of the medium and the thickness of each layer. This requires extensive data-processing and typically requires many numerical wave simulations; only then, when your numerical simulation predicts data that resembles the data that you recorded, you may assume that you may have found the approximate physical state of the medium. This technique is called Full-Waveform Inversion, and is the only one I'm aware of that will provide the answer to all your questions.

The problems in the last point are due to the fact that surface recordings provide a massively underdetermined problem. Say, as an example that you drop a weight onto the medium, and then it takes 100ms before you hear the first echo come from the medium. Then you infer that the wave traveled back and forth to depth with a certain velocity. This gives an equation: $100\ \text{ms} \approx \frac{2h}{V}$ that doesn't reveal what the depth $h$ actually is or what the velocity $V$ actually is! The problem is thus poorly constrained by the data, and you require a lot of data to get a good understanding of the subsurface. Then, furthermore, your waves propagate in an elastic medium which excites many types of waves: compressional waves, shear waves, surface waves, $\dots$, which all travel at different speeds and are sensitive to different physical properties. You require some good data-processing to isolate only the wave that you are interested in!

As said in the comments, the size of the medium is no limitation, you simply scale up the wave's frequencies to the ultra-sound region! It is generally thought that you can locate objects in the Earths subsurface when they are separated by a distance $d\geq\lambda/2$, where $\lambda=\frac{V}{f}$ is the wavelength, with $V$ the velocity of the medium and $f$ the wave's frequency. The average speed of a compressional wave in rocks is about 5km/s, thus you require a frequency of at least $f=\frac{5\,000\ \text{m/s}}{2d\ \text{m}}$ to find your object of size $d$. If you resolution requires separation up to 1cm, you require a wave with a frequency of at least 80 $\text{kHz}$: well into the ultra-sound territory!

I'm sorry to give such a demotivating answer, but seismology is simply too complex to be a viable project for anything but a fully-fledged multi-year commercial or PhD project. My suggestion would be to build a variation to a Cone Penetration Test, which measures the resistance to pushing a cone into the ground (which is directly related to the hardness of the rock!) at a fixed speed. Whenever the resistance changes you are encountering a new layer, such that the size and number of layers is easily inferred. That would satisfy all the requirements, I think. Then again, I have no expertise with the Cone Penetration Test, and perhaps it is not as easy as I think...