If the train had a loudspeaker on the front emitting a pure sinewave of known frequency, louder than anything else in the environment, you could probably just u

If the train had a loudspeaker on the front emitting a pure sinewave of known frequency, louder than anything else in the environment, you could probably just use a frequency counter and the Doppler formula. Given just some microphones picking up whatever sound the train makes on its own, it's not obvious to me that there's a simple solution.