You may be surprised to know that it is not at all uncommon for Surname spellings to differ from generation to generation or even from person to person within a family. The 'correct' spelling depends on the individual him or herself. The spelling of a surname may differ for many reasons including the literacy of the individual, transcription error in a birth certificate or other public record, intentional changes made during immigration to fit the destination nation's primary language, or even a fallout with other family members leading to a desire to create a new, independent identity.
In my research, I have come across people who were born with one name on their birth certificate, used a different spelling in their young life, and were buried under yet another spelling! It's common to find public records of the same individual with different surname spellings from census to census, birth to marriage to probate to death records.
From a research point of view, knowing the 'correct' spelling of a surname doesn't always help you because of these inconsistencies. It helps to 'cast a wider net' sometimes. I'm not sure why you asked the question, so I may be off-base with this answer.
Ultimately, the only way to know the 'correct' spelling of a person's surname is to ask the person in question. If they are deceased, there are several schools of thought on what would be 'correct.' Some people maintain that a birth record establishes the 'true' or 'correct' name for a person, others might argue that point.