The spelling of the name "Rogerson" can be explained using the International Phonetic Alphabet (IPA) transcription. The first syllable "Ro" is pronounced as /rəʊ/, with a schwa sound followed by the long 'o' sound. The second syllable "ger" is pronounced as /dʒər/, with a soft 'g' and the schwa sound. Finally, the third syllable "son" is pronounced as /sʌn/, with a short 'u' sound and the 'n' sound at the end. Therefore, the correct spelling of the name "Rogerson" is justified by the sound and phonetic transcription.