Date: 201504
Local time of day: 1801-2400
Location: SFO.Airport, US
Flight conditions: VMC
Light: Dusk
Aircraft: Medium Transport; Low Wing; 2 Turboprop Eng
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Descent
Airspace: Class A ZOA
ATC / advisory: Center ZOA
Reporter organization: Air Carrier
Reporter function: Captain; Pilot Not Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Human factors: Communication Breakdown; Confusion
Second reporter function: First Officer; Pilot Flying
Anomaly: ATC Issue All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance
Detector: Person Air Traffic Control
When detected: In-flight
Result: Air Traffic Control Issued Advisory / Alert; Flight Crew Requested ATC Assistance / Clarification
Contributing factors: Procedure
Primary problem: Procedure

Narrative: While flying the SERFR ONE RNAV ARRIVAL; Oakland Enroute Controller cleared us direct to NRRLI intersection for the arrival Runway 28 transition and pilot's discretion descend to Flight Level 200. The First Officer (Flying Pilot) selected FL200 in the altitude alerter and selected direct to the fix in the FMS. He confirmed the selections with me and executed the selection. After a few minutes we began a 1000 FPM descent. A few moments later a new controller came on the frequency and cleared us to descend via the SERFR ONE RNAV ARRIVAL. At this time the First Officer selected the DIR INTC button on the FMS to confirm the descent rates required to meet the next crossing restrictions on the arrival past NRRLI intersection.  The First Officer then asked me to see if s/he needed the speed restrictions on the arrival. I asked the controller and received 'standby.'  After a minute [the controller] said comply with the published speeds and again said descend via the arrival. After passing NRRLI intersection (published altitude At or Above FL200) the First Officer selected 15;000 (next fix is WWAVS between FL190 & 15;000) in the altitude alerter and we continued the descent. While descending through FL186 the controller queried who had cleared us to descend below FL200. I replied you cleared us to descend via the arrival. S/he then said 'the previous controller had only cleared us to descend to FL200. There was a pause and s/he instructed us to contact NorCal Approach. We never heard from either controller 'descend via the arrival except maintain FL200'. At no time was there a loss of separation with another aircraft.I believe s/he thought by clearing us to descend via the arrival (after we had been told 'Pilots Discretion' to FL200) would only cause us to meet the restriction of FL200 at NRRLI intersection. The clearance was issued the same way that we have become accustom to throughout the country. One center controller will instruct pilot's discretion to a certain altitude and then a new controller will clear us to descend via the arrival. Because of this we had no reason to question the clearance. Proper phraseology was used; however; the order in which the clearance was given caused confusion for the controller. We are taught that a descend via clearance overrides the previous hard altitude and I believe s/he thought the initial hard altitude would remain the clearance limit.As these RNAV arrivals are implemented at major airports there seems to be a fairly steep learning curve for controllers and pilots as to what is expected at each new airport. This should not be the case. We as pilots fly them like we have been trained for all other RNAV arrivals throughout the country. One minor issue that might help alleviate this problem is who issues the descend via clearance. At some locations the enroute controllers issue the clearance. While at others the approach controllers issue the descend via clearance. If this was consistent throughout the country it might be a minor key to avoid the problem during the transition from one facility to the next. The controllers for each specific location seem to think we are going to do a hybrid between the old traditional arrivals and the new arrivals. I believe this creates a major safety issue.


Second narrative: [Report narrative contained no additional information.]


Synopsis: Air Carrier flight crew descended below a previously assigned altitude when given a 'descend via' clearance by a subsequent Controller. Reporter noted confusion on the part of the Controller as to whether or not the descend via clearance cancelled the previously assigned altitude.