I think that is why systems try to hit the missile on the way up before RV separation. The navy has Missile Systems, such as the TBMD section of the Aegis weapons system, that uses US/ship based SM-3 missiles to hit a Ballistic missile on it's ascent. However, this means you have to detect the BM early and launch your missiles from somewhere in the ocean so that you could hit the target around it's apex. They do this by sharing radar data between land and ship based radars, getting the SM-3 close to target, and then using a kinetic warhead to make contact with the BM.
I would think that the patriot missile system uses a similar theory. It probably use a radar to detect the scud launch and then calculate the correct time to deploy a missile for intercept. The patriot would either be flown close to the scud using telemetry based systems on the patriot, or by using mid course/terminal guidance from the patriot launcher. Terminal guidance would fly the missile in front of the scud and "patriot go boom". If the terminal guidance is slightly off, you don't go boom in front of the scud and the scud hits it target.
In some fire control radars, the terminal guidance is constructed of data the radar receives from a missile downlink, where the missile reports it's position to the launcher in order to use a more precise radar on the ground to direct the missiles path towards a target. I think it's more popular to use a combination of missile communications uplink/downlink and combine it with Ground based radar data, to determine where the missile is, where the target is, and where the expected point of missile intercept should be.