Go get you a piece of indoor/outdoor carpet from your favorite home cheapo. A piece about 5' by 10' won't cost much and can serve as a landing and feet wipe to catch most of the mud. It is used all the time RVers. Keep your scooper handy as some dogs may prefer it over the mud and if so then get a bigger piece. When it gets dirty, just hose it down on a nice day to clean it.
I suppose the obvious would be to pour a concrete patio also with a rug on it.