I've made a couple of aborted attempts at ranting at the Brookings Institution's Amtrak report published on Friday. Instead, I'll just post some graphs I made from their data which seemed interesting.
On Amtrak routes, ridership per train appears to scale with the log of distance. Ridership continues to rise as route length increases, though the rate of increase drops off as routes get longer. Shorter routes tend to have higher frequency of service and more closely-spaced stations. The longest routes also go through some of the least-populous areas of the country (western states). Of course, the sparseness of population doesn't explain why the Empire Builder is the most-ridden long-distance route. The fact that it's basically two (or even three) trains in one might, though.
First, I'll toss in my map of Amtrak station ridership across the country, since I haven't posted about it yet (I still have to work on describing it better at some point):
Now here is ridership per train (annual ridership divided by trains per year) versus distance (log scale), with the size of the circle showing riders per train-mile. Circles are colored according to frequency of service.
This graph plots riders per train-mile versus distance (log scale on both axes), with riders per train as the circle size. Also colored according to frequency of service.
There's a common statement in rail punditry that 400 miles is the limit for viable service, and it's often stated as though ridership suddenly drops off a cliff at that point. I suspect the 400-mile distance really has more to do with political and geographical boundaries. Within the U.S., individual states are often 200 to 400 miles across. In Europe, the Mediterranean Sea is about 400 miles away from Paris.
Anyway, these graphs also indirectly show that ridership scales more or less linearly with frequency of service -- add a second train, and you should expect roughly twice the ridership. So one question worth asking is whether operational costs scale linearly with frequency of service or not.
I'll note that I mostly used frequency of service from Brookings' report. Frequency and route length both become a bit fuzzy since some corridors will only have trains run for part of the distance. I've also noted in the past that Amtrak counts "corridor" trains as Northeast Regionals for the distance they run in the NEC. So the Regional's numbers are probably inflated, while corridor train numbers are deflated (such as the Vermonter). I haven't tried to compensate for that in these graphs. My only change to Brookings data was to tweak the frequencies for the thrice-weekly Sunset Limited and Cardinal and the quad-weekly Hoosier State.