An object of the invention is to estimate the quantity of N2O produced in an ammonia oxidation catalyst in a case where the ammonia oxidation catalyst is provided in an exhaust passage of an internal combustion engine. In a case where an ammonia oxidation catalyst having the function of oxidizing ammonia is provided in an exhaust passage of an internal combustion engine, the quantity of N2O produced in the ammonia oxidation catalyst is estimated based on the difference between a measurement value of an upstream NOx sensor provided in the exhaust passage upstream of the ammonia oxidation catalyst and a measurement value of a downstream NOx sensor provided in the exhaust passage downstream of the ammonia oxidation catalyst.