Date: 201812
Local time of day: 0601-1200
Location: DAL.Airport, TX
Flight conditions: VMC
Light: Daylight
Aircraft: Medium Transport
Operator: Corporate
Operating under: Part 91
Mission: Passenger
Flight phase: Climb
Airspace: Class B DAL
ATC / advisory: TRACON D10
Reporter organization: Corporate
Reporter function: Captain; Pilot Not Flying
Reporter qualification: Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Second reporter function: First Officer; Pilot Flying
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Flight Crew
When detected: In-flight
Result: Flight Crew Became Reoriented; Flight Crew Returned To Clearance
Contributing factors: Aircraft
Primary problem: Aircraft

Narrative: Pax trip [from] DAL. DAL Ground changed our SID from KKITY4 departure to the WORTH1 departure off Runway 31L. SIC changed the routing in the FMS. PIC then did the dual verification by reading the fixes from the FMS and SIC verified from the Departure Plate that the read back was correct. What we did not catch is the FMS legs page routing does not reflect that the Departure Plate has a climb and RIGHT turn to a heading of 013. This is NOT reflected in the FMS when the procedure is loaded. Verifying all the fixes in the departure is insufficient to catch this. We did expect to takeoff on [Runway] 31L and turn right. SIC Pilot Flying took off in ROLL mode with LNV armed. Upon reaching 400 AGL; the flight director prompted a left turn and we both were surprised and said to each other ' I thought we were supposed to turn right?'.  Who is right? Us or the Flight Director with the programmed SID? It took a minute to decide we were right and we were correcting our heading when ATC issued a heading. Later; ATC asked us what we were navigating toward. He said we caught it in time but was curious because this happens on this departure a lot.I told him we had an automation guidance issue. I had not yet figured out the difference in the Departure Plate and the FMS legs page for this departure.We should have caught the discrepancy between the Departure Plate and the FMS but I believe this can happen again if people are not aware of this issue.I suggest Operations issue a notice to make people aware of this trap and make Jeppesen or whoever issues the database for the FMS aware.


Second narrative: Passenger arrived early and I called clearance delivery for our clearance; which had not been available previously. We received a SID which was different from the one filed by the company; so we performed a dual response confirmation of the revised SID routing and altitude; checking each fix along the route. This step later became a source of our problem. Following the subsequent engine start; Ground Control announced an ATIS change which closed the adjacent runway and necessitated a cross-airport taxi to a parallel runway. We were relieved to see that the departure course was unaffected by the runway change. When Tower cleared us for takeoff; we were simply advised of traffic approaching on final approach and that we were cleared for takeoff. I reviewed the initial takeoff actions; stating that we would climb to 2;700 feet; then turn right to a heading of 013 degrees and climb to 5;000 feet and re-checked that LNAV and VNAV were armed. The departure was normal until 400 feet; when I announced 'LNAV Captured. At that point the steering bars commanded a left turn to the first fix. Despite the fact that I had just reviewed the initial action as a right turn; I was immediately struck with the conflict of the precise steering command of the FMS; versus my intended turn. My mind raced to determine if I had missed an initial turn to the left; to be followed by the planned turn to the right at 2;700 feet. Was I right; or was the FMS right? We had; after all; confirmed the accuracy of the FMS programming! With no time to spare; I 'trusted my instruments'; against my instincts; and began a left turn. Simultaneously; I voiced my confusion to the Captain (Pilot Monitoring); saying 'Why is it telling us to turn left??' After a couple of seconds; I regained the sense that I was probably correct and began a reversal of the turn; back to the right. At that point; Departure Control assigned a heading roughly halfway between my current heading and my target heading. From that point onward; the departure was uneventful; with the controller assigning another heading and clearance to proceed direct to a fix along the SID routing. Later; however he did remark that he had seen us initially turn left after takeoff; and wondered why we had done so. He said that he sees a lot of pilots turn left after takeoff on that particular RNAV departure; and that he simply wondered why we pilots were doing so. We explained what had happened and he thanked us for the info; advising us that no harm had been done. The rest of the flight was uneventful.At cruise altitude; we analyzed what had happened and discovered a two-part problem: First; the FMS database does not contain the initial heading for the Worth 1 departure. Second; during our expeditious dual verification of the SID routing; we confirmed that every fix and altitude in the FMS was correct; but we did not confirm what was absent; namely the initial heading displayed in the upper right-hand corner of the departure plate. Although I correctly briefed this turn as I took the runway; I had failed to ensure that it had appeared in the FMS legs page. So I think that I built a trap for myself by thinking; 'The FMS is correctly programmed' (and therefore trustworthy). So when the showdown came between a trustworthy FMS and my own review of the initial actions before takeoff; I blinked; thinking that humans do err; after all. It took a couple of seconds to process these inputs; and to resolve which would be primary.In the future; I shall perform dual verification with an eye toward what might not be read from the FMS; as well as what is found there.


Synopsis: Corporate aircraft flight crew reported temporary disorientation on departure due to initial heading not being in the database of the FMS.