You will find as you progress in your research that name spellings were highly mutable up until about the 1900s, and some even past that. In my family spellings went from Smailes to Smails to Smail within about three generations. Sometimes the spelling changed from one census to the next. Not as radical a change as yours, but you can imagine in an era where people were often illiterate that the sound of the name could well be interpreted in many ways.
I have come across many instances of people signing official documents with their "mark" - i.e., a scrawled X - quite certain if the head of the household who could only sign with an X, when asked to spell his family name to the registrar of births or the census taker, may well have shrugged his shoulders, leaving the interpretation of how Doughty was spelled up to the person writing it down. Even if they spoke the same language, there could be several ways to spell it.
Speakers of foreign languages immigrating to the new world faced a whole other set of obstacles when it come to spelling out their famiy name, not the least of which was the cultural bias that wanted to help the newcomers be more accepted by "anglicizing" their names.
