This paper presents a new approach to extracting a low-dimensional i-vector from a speech segment to represent acoustic information irrelevant to phonetic classification. Compared with the traditional i-vector approach, a full factor analysis model with a residual term is used. New procedures for hyperparameter estimation and i-vector extraction are derived and presented. The proposed i-vector approach is applied to acoustic sniffing for irrelevant variability normalization based acoustic model training in large vocabulary continuous speech recognition. Its effectiveness is confirmed by experimental results on Switchboard-1 conversational telephone speech transcription task.