Adaptive persistence processing on elastic images in an ultrasound system is disclosed. An ultrasound data acquisition unit sequentially acquires a plurality of ultrasound data frames based on ultrasound echo signals reflected from a target object by repeatedly applying and releasing stress to/from the target object. A processing unit calculates inter-frame displacements of the frames between the ultrasound data frames to form elastic images. The processing unit adaptively performs persistence processing on the elastic images in consideration of application or release of the stress to/from the target object.