This paper investigates if the color appearance model accurately predicts the lightness of a display whose surface features specular reflection such as a mirror display. Eight observers participate in psychophysical experiments and the results are compared with model prediction value. It is found that the model overestimates the lightness under the average surround condition where there is an intense specular reflection on the display surface. Based on the measurement results, we figured out that the conventional measurement method cannot measure practical specular reflection value. Thus we propose a new measurement method and modify the background parameter to include the specular reflection value. The proposed CIECAM02 considering specular reflection improves the coefficient of variation (CV) value under average surround condition from 20.56 to 12.40 while maintaining CVs under dim and dark surround conditions. In conclusion, we found that people perceive the specular reflection image as a background.