The Ingredients of Dupire's Formula

In last week's post Yet another differentiation trap, I recapitulated Dupire's formula

We start with a (syntehetic) noisy Black Scholes volatility surface
It starts with a value of 35% at the front left edge (strike 50 percent of spot, maturity 1 month) and drops to a value of 25% in the upper right (strike 150 percent of spot, maturity 5 years). An additional volatiltiy noise (uniformly distributed between 0 and 0.1%) is added. This leads to the following call Black Scholes call values

 We now apply numerical differentiation (50 grid points in each direction) to the different components of Dupire and obtain:

Noisy dC/dK
Noisy dC/dT
But the second derivative explodes

Noisy d2C/dK2

This is obviously the source of severe problems with naively applying Dupire's formula.