Delta-v to overcome atmospheric and gravity drag at 13km?

[Treva31]

Wikipedia says that:

Atmospheric and gravity drag associated with [space] launch typically adds 1.3 to 1.8 km/s to the launch vehicle delta-v required to reach normal LEO orbital velocity of around 7.8 km/s (28,080 km/h).

Does anyone know, or know how to calculate/estimate/simulate the delta-v required to overcome atmospheric and gravity drag (excluding the 7.8 km/s for orbital velocity) if you were launching from 13km altitude? ie a commercial jetliner.

I'm guessing it would be a lot less since its already only around 0.1 atm pressure up there.
I also suspect its very difficult to work out.

 

[mfb]

You can simulate a trajectory numerically, and optimize launch angle, acceleration profile and so on for the reduced atmospheric pressure and some rocket model. Takes a lot of time.
To get a lower estimate on the saved delta-v, you can take a sea-level lauch profile, estimate air drag along its flight path and reduce this accordingly.

Many countries have access to mountains with a height of at least 4 km, but most launch sites are close to sea level: access via ships or highways and a safe landing zone for potential debris are more important than some kilometers of air.

There is more than 1/10 sea-level pressure at a height of 13 km.

 

[Treva31]

You can simulate a trajectory numerically, and optimize launch angle, acceleration profile and so on for the reduced atmospheric pressure and some rocket model. Takes a lot of time.
To get a lower estimate on the saved delta-v, you can take a sea-level lauch profile, estimate air drag along its flight path and reduce this accordingly.

Many countries have access to mountains with a height of at least 4 km, but most launch sites are close to sea level: access via ships or highways and a safe landing zone for potential debris are more important than some kilometers of air.

There is more than 1/10 sea-level pressure at a height of 13 km.

Yea that's probably close enough, thanks :)
I'll post my workings here when I've done it.