I'm reading about the derivation of shallow water equations, and I have a hard time understanding why $$ w(\eta) = \frac{D\eta}{Dt}$$ where $w$ is the vertical velocity, $\eta$ is the (top or bottom) vertical boundary of the layer and $D/Dt$ is the material derivative.
In Atmospheric and Oceanic Fluid Dynamics (2nd ed. 2017, chapter 3.1.2), Vallis gives the following explanation:
At the top the vertical velocity is the material derivative of the position of a particular fluid element. But the position of the fluid at the top is just $\eta$, and therefore $$ w(\eta) = \frac{D\eta}{Dt}$$
Why is the vertical velocity at the top the material derivative?
Edit: Added a schematic illustration of the shallow layer. As you see, $\eta$ is not assumed to be constant.