Why do we observe ripples in Quantum corrals and Scanning Tunneling Microscope (STM), if the wavefunction isn't physically realizable? I recently landed upon an image of Quantum corral and observed ripples around the atoms. Such pattern of ripples were also observed in a movie by IBM - A Boy and his Atom. Upon some research, I found out that these ripples represent the wave properties of matter and particularly the electrons. However, Quantum Mechanics regard the wavefunction as completely abstract and non-physical. My question is Why do we still observe those ripples?

 A:  The role of the wavefunction 
The phrase "completely abstract and non-physical" seems to be a distortion of the statement that the wavefunction itself is not an observable in quantum theory's technical sense. That doesn't mean that the wavefunction is irrelevant! On the contrary, if we want to get accurate predictions out of the theory, then we need to give it accurate information about how the physical system was prepared. That's a minimal description of the role of the wavefunction in quantum theory, and that minimal description applies no matter which "interpretation" we use. (Here, I mean the wavefunction of the whole system, which usually consists of more than a single particle.) Predictions depend on what observable is being measured and on how the system was prepared. The wavefunction is how we tell quantum theory how the system was prepared.
 Why the image shows ripples 
The answer has two parts: what causes the ripples, and why the act of observation doesn't destroy the ripples.
First consider what causes the ripples. An electron can have a wavefunction that is mostly concentrated near the surface of the material but that can still be spread out in the directions tangent to the surface. (Here, I mean the wavefunction of a single electron, treating the rest of the material as a fixed background.) If extra atoms are attached to the surface, such as the ring of atoms shown in the lower-right picture in the question, they affect the shape of a surface-electron's wavefunction. The ring of atoms can "contain" the electron, resulting in a standing wave, as reviewed in detail in cond-mat/0211607. This is analogous to how the wavefunction of an electron in an atom has the form of a kind of standing wave around the nucleus, as illustrated in https://en.wikipedia.org/wiki/Atomic_orbital.
To understand why the act of observation doesn't destroy the ripples, we need to understand something about the process that was used to make these images. I'm assuming that they were made using a scanning tunneling microscope (STM), which works by mechanically scanning a very fine conducting tip over the surface, within a handful of angstroms of the surface. A slight voltage is applied between the tip and the surface, as illustrated in https://en.wikipedia.org/wiki/Scanning_tunneling_microscope, causing a slight current to flow from one to the other. The current flows thanks to the quantum tunneling effect, which allows an electron to cross the few-angstrom gap between the tip and the surface. The distance of the tip from the surface is adjusted to keep the current constant.
The ripples in the image match the shape of the ripples that a single-electron wavefunction would have, but the image is actually showing how close the tip must be to the surface, at each location on the surface, to maintain a constant current. The ripples match what they would be in the single-electron wavefunction because the current flows via tunneling, and an electron's tendency to make that jump is affected by the same factors that determine what the wavefunction of a single surface electron would be in the absence of the applied voltage. We can roughly think of the tunneling phenomenon itself as a modification of the surface-electron wavefunction due to the presence of the conducting tip: if we were to measure the position of an electron, we would find it inside the tip with some probability and on the surface with some other probability.
Maybe the key idea is that the process isn't making measurements of just one electron. A precise measurement of the position of a surface electron would destroy the ripples, unless an electron were put back into the standing-wave pattern before each subsequent position-measurement. In the STM measurement, the electrons are being replaced (a current is flowing), and the STM is not making a precise position-measurement anyway. On the contrary, the electrons in the current must have relatively (but not strictly) well-defined energy instead, so that the standing-wave pattern has a well-defined number of ripples, analogous to an atomic orbital.
