Very often people traveled back to Norway for extended periods of time and therefore giving birth to children there over time, also very often with a large number of family members still in Norway having mothers, sisters and Aunts around was very helpfull.
Regarding Surnames: It was not required in Norway for an individual to have a surname that was passed from generation to generation until 1924. Therefore, many rural individuals did not have true surnames until that time. Also women would not customarilly taking their husbands name thus keeping the name they known by in the community they lived or took offically.
When indiviudals entered the USA they needed a surname and not fully understanding the concept took several different approches, either as noted before taking a farm name, or a patronymic name. If a woman was traveling alone with children she would give here name as she knew it perhaps Anna Cathrine Hasse (a geonymic name), and therefore by default any children would be marked down in US Customs books as Fredrik (mothers lastname), since the individual taking the data probably never asked "what are the children's full name" or the mother responded "Fredrik" and the census taker or Customs person added "Hasse" not the patronymic name that the father was called Hans or his geonymic surname.
Therefore, Technically most rural Norwegians in the 1700's and 1800's never truely had a "surname" just a patronymic name and a geonymic name (i.e. farm name) these only became true surnames either when then moved to the city (or the USA) and came to need one -or- when the Norwegian law required them to take one.
EDIT: Also geonymic names were based on where you currently lived and if the man moved from one farm (Nyhous) to the farm of his wife (Hasse) he would take her geonymic name and become Hans Olson Hasse when before he was Hans Olson Nyhous.