travertine cleaning

Hello, we have a home the floors of which are almost entirely travertine.  On the travertine covered steps from the garage are very dark dirt (scuff) stains.  What is the best way to clean them, since they do not come clean through normal twice weekly mopping with mild soapy solution (neutral ph)?

 

The travertine has been polished to a very low luster.

 

After cleaning, should the travertine be sealed somehow?  If so, with what product?