Smooth 2D Data with Discontinuous and Artificial Jumps
If I understand you correctly you want to smooth the data (Namely reduce "Noise") yet regular filters would ruin the data on discontinuities.

What you need is an Edge Preserving Filter.
You can try the Bilateral Filter or Anisotropic Filter.
I have an advanced implementation of the Anisotropic Filter - Fast Anisotropic Smoothing of Multi Valued Images Using Curvature Preserving PDE.

I hope it works.