John_T,
I'm afraid that this is rather brief, in transit at home between work, and the thread had slipped to page 2 in my absence.
Yes, I did indeed simplify some points for convenience, the most pertinent of these simplifications being -
(1) The aircraft is field limited on a Balanced Field, i.e. the 1st/2nd segments began right at the end of the TODR, and TODR = TODA = TORA
(2) The quoted airfield Elevation is for the highest point on the runway, i.e. the beginning of the Down-Sloping Runway, or the end of the (same) Up-sloping runway.
(3) The effects of 1st segment have been ignored, as being relatively constant in all cases.
In my work (and I'm sure yours too) it's necessary to make very exact obstacle analysis for RTOW production where Effective Operational Lengths (EOLs) will be frequently used in evaluating a "Balanced" Field / 1st Segment / 2nd Segment limit, i.e. the Field Limit and the Obstacle Limit are equal. In these cases, the TODR will be often well less than the TODR. The gradient to the obstacle, then, will be from the point at the end of the TODR, and, on a Sloping runway, this will always be less than the highest elevation of the runway.
In the example that I quoted (a 1% Down Sloping runway of 2000M TODR, with end elevations of 1000 feet and 934.4 ft), the aircraft must climb from 934.4 ft to the 1100ft AMSL sample obstacle in 1000M, a gradient of 5.05%. If, however the aircraft only required, say, 1500M of the 2000M available, the end of the TODR will be at an elevation of 950.8 ft and be 1500M from the obstacle, a gradient of 3.03%, hardly a 'poofteenth' of a difference! Similar arguments hold good for the Up-Sloping Takeoff direction.
It would have been better if I had said "obstacles are referenced to the lowest portion of the TODR" rather than "obstacles are referenced to the lowest portion of the runway". The latter is correct if the aircraft is truly Field limited, the former applies to everyday performance number crunching.
Where the problem lies is when producing "back-up" General Takeoff Charts in the event that the RTOW is invalidated, and I'm sure that this is a part of your work too. The pilot cannot be expected to do an incremental creep along the runway to calculate the Weight / TODR, the elevation at that point, and the relative vertical and horizontal distance to the obstacle to calculate the 1st/2nd segment limits. It then becomes necessary to quote the worst case in instructing the pilot to compare the overall lowest runway elevation against the obstacle, for safety and more often than not, conservatism.
A lot of pre-surveyed data (e.g. STODs in the Australian system) don't require knowledge of obstacle or runway elevations to do the trigonometric calculations, but when 'raw' data is used from obstacle sources such as survey charts etc., elevation AMSL is the original data to be referenced to Runway Elevation, or the lowest point of it. A very wise P/E once suggested that I use obstacle polygons created from multiple STODs (his name slips my mind
), and, if exact elevation from the point on the runway is used, the obstacle horizontal and vertical scatter comes down to less than a foot vertically, and within a metre or 2 horizontally. If simplified trig is used, not considering the runway elevation at each 'STOD point' the scatter can become quite large.
As a footnote, I use the highest Airport Elevation to calculate the Pressure Height / Density Height portion of the calculation (naturally), but the lowest point of the TOD to compare to the obstacle elevation, irrespective of whether the runway be Up-Sloping or Down-Sloping.. As a further footnote, I use the lowest elevation (as described earlier) against the HIGHEST obstacle in the Takeoff Area to find the Delta to be factored in calculating the MAA, and then add it to the HIGHEST Airfield Elevation.
That's still not the complete story, but hopefully a degree of clarification in where I was coming from in the over-simplified earlier response. I'm only applying principals that my earlier teachers taught me,
P_T, G_S, and.....
J_T.
Regards,
Old Smokey