I've never used interpolation in any way before, so I wanted to think up an example where it might apply.
Let's say I have the vertical advection of vertical lapse rates in temperature, $\frac{\partial T}{\partial z}$, by vertical velocity, $\frac{dz}{dt}$: $$f(x,y,z) = \frac{dz}{dt}\frac{\partial T(x,y,z)}{\partial z}$$
Using the relationships: $\frac{\partial P}{\partial z} = -\rho g$, $\frac{dz}{dt} = w$ and $\omega = -\rho g w$ and taking advantage of chain rule in multivariate calculus we can convert the differential operator: $$\frac{dz}{dt}\frac{\partial}{\partial z} = \frac{dz}{dt}\frac{\partial}{\partial P} \frac{\partial P}{\partial z} = w(-\rho g \frac{\partial}{\partial P}) = \omega \frac{\partial}{\partial P}$$
So now we have vertical advection along pressure, $P = P(x,y,z)$, as a vertical coordinate: $$f(x,y,P(x,y,z)) = \omega \frac{\partial T(x,y,P(x,y,z))}{\partial P}$$. Now let's say I have data all along 3 surfaces of constant height at z =0m, 1m, and 2m. If I wanted to compute $f(x,y,P(x,y,z))$ along a surface of constant pressure at say 1000mb which is around 1m, with 1001mb at 0 m, and 999mb at 2m, we would run into an issue due to pressure varying with height. So let's say we have 999mb and 1001mb perfectly mapped to 0m and 2m, respectively, but at 1000mb, it varies between 0.98m and 1.02m, but we don't actually have data physically existing at those levels, would it be appropriate to interpolate data at 1m onto the surface and compute?