How deep is deep? Look shallow from the photo, so grinding down and polishing may be the answer. Your local marble and tile or tombstone shop should be able to do it. I had a verde antique marble slab from a former soda fountain repolished and it serves as the hearthstone of a fireplace. Usually they do this sort of thing after hours, the owner turns his back.

I used the stone powder-epoxy mixture once and it wasn't too bad- but I could still see it. It was that slightly gray with darker lines, much like yours- this was a very popular (and probably more abundant) kind at one time.
I replaced a broken top to a side table with new marble, too many fragments missing and mending impossible. Rather expensive, as beveling is so much per inch.