You might agree that the earth's natural noise comes from below, but you might not be willing to agree that it is an impulse function. Of course you are correct. What saves the day is that we depend only upon the autocorrelation of the natural noises. In other words, the theorem is equally true if the autocorrelation of the natural noises incident from below is an impulse function.
The potential nonimpulsiveness of the autocorrelation of the random noise excitation identifies a practical problem which leads to processing design issues.
Expressed back in the framework of our boring lecture room, if all the noise makers are tuned into the key of C, our final images are certain to resonate at that frequency. Luckily, natural noises do seem to have a fairly rich spectrum of frequencies and directions of arrival. Luckily also in imaging theory (and practice), when something is missing, it doesn't ruin the image; it merely degrades it. Imaging theory and practice are not so fragile as inversion. Thus missing frequencies and directions of illumination should not wholly frustrate our activity, but naturally they would limit us accordingly.
Another side of the same problem is more serious. When too much light is coming from one direction we need to shield ourselves from the glare. There is an analog in seismology. We have the practical problem that a great deal of the natural noises come in the form of ground roll. We cannot escape this problem without ``shielding our eyes'' with adequate spatial filtering. This is why simple 2-D arrays seem to us to be inadequate. With Steve Cole's PhD dissertation, we did set out a lot of geophones (4056), and they did suppress a lot of surface noises, enough so that we could see the natural noises arriving from great distances.
I have always felt the great thing about seismology is that it really works. Hundreds and thousands of wiggles and eventually it has often come to make good sense. Our experiments are truly repeatable. In time we really do learn more. Years ago with 2-D seismology we had ``noises'' but now with 3-D seismology, we realize those noises were the geology, now beautifully imaged. Our field has not always treated everyone kindly, but in our field, the gap between theory and practice has always been a good place to put your eyes to see the future.