What a drag function actually does
The complete exterior ballistic behavior of a shot can be described by a single curve of distance against time. The slope at the start of that curve is the muzzle velocity, the slope at any point is the velocity at that moment, and the second derivative of the curve represents drag, which is deceleration as a function of time. Put another way, once you have enough points on that curve, what the projectile is doing has already been recorded.
The problem is that in practice you cannot measure the curve continuously; you can only place targets at a few positions. The role of legacy drag functions such as G1, G2 and G7 is to interpolate between those measured points. Each supplies a predefined shape of drag against velocity, which a ballistic coefficient (BC) then scales to your bullet.
Oehler System 88 measures exactly those points on the curve. What it produces is an effective ballistic coefficient: the value that makes the solved long-range time of flight equal the time of flight measured over the same distance. Oehler observes that as long as that equal-time-of-flight condition holds, any reasonable drag function paired with its own appropriate ballistic coefficient predicts intermediate trajectory parameters, drop and wind deflection, that agree very closely between the muzzle and the test target, almost exactly.
It follows that when you collect distance and time data points, the most valuable places to collect them are where the character of the drag starts to change appreciably.
Past the test target, you are betting on the drag function
Once you go beyond the distance of the measured target, how well the prediction agrees with reality depends entirely on the fit between the assumed drag function and the real behavior of this bullet. If the bullet flies like the theoretical G7 standard projectile, G7 will predict correctly at long range and G1 will drift off. If it flies like the theoretical G1 standard projectile, the situation reverses.
What if neither G1 nor G7 fits? The traditional answer is to use a different ballistic coefficient in each velocity band. The stepped ballistic coefficients Sierra has published for years are that same concept, so the idea is not new.
The real difficulty has always been left unstated: how do you measure which ballistic coefficient applies to one particular bullet, fired from one particular rifle, at one particular velocity level? The choice of drag function is an educated guess at best, and the ballistic coefficient itself shifts with parameters such as barrel twist rate, some of which have not even been identified yet. When the ballistic coefficient was measured inside 300 yards, the problem only gets worse.
Three velocity bands: the drag world the projectile actually passes through
Describe one bullet with three common drag functions and adjust each ballistic coefficient so that all three produce the same drag at a muzzle velocity of 2600 fps, and a key feature appears: above roughly Mach 1.2 (about 1350 fps), all three drag curves are relatively flat. Most long-range shooting happens inside that velocity range.
Within this band the common drag functions resemble one another, and can generally be approximated by taking the drag coefficient as inversely proportional to the square root of velocity, so any of them will give an accurate, usable result once paired with a suitable ballistic coefficient. That segment, from the muzzle out to where velocity has decayed to about Mach 1.2 or 1350 fps, is mainly what System 88 is there to characterize with an accurate ballistic coefficient.
The transonic band from Mach 1.2 down to Mach 0.9 (1350 fps down to 1000 fps) is another matter entirely: here the common drag functions differ markedly, as the way the curves cross this stretch makes plain. Oehler is candid about it. Ballisticians have argued over this band for years, he does not pretend to understand it, and he does not try to measure drag between 1350 and 1000 fps, because in practice there is no need to. What is needed is a measurement of the cumulative effect of drag after the bullet has passed through the transition, which becomes a second reliable data point on the distance-time curve.
Below 1000 fps the drag coefficient becomes well behaved again, approaching the constant drag coefficient Newton proposed centuries ago. A third data point taken here yields a third ballistic coefficient, so that prediction and experiment also agree below 1000 fps.
The extension: stitching the bands together with time of flight
What makes this procedure distinctive is the use of cumulative time of flight, collected at measured distances close to the ballistic coefficient step points, as the data points. Behavior between the steps is still predicted by the existing drag functions, subject to one crucial condition: the prediction for the preceding segment must end at a specific point on the distance-time curve, and the prediction for the following segment must begin at that same point. The whole curve is therefore stitched together segment by segment rather than solved piece by independent piece.
The concrete steps are: (1) choose a drag function and use System 88 to measure the ballistic coefficient from the muzzle out to where velocity is about 1350 fps; this set of parameters applies from the muzzle to slightly beyond the measured distance and is normalized to the standard atmosphere. (2) Place a subsonic target at the distance corresponding to about 1000 fps, measure the ballistic coefficient and record the time of flight; in Ballistic Explorer, set the step point at 1350 fps, keep the previously measured value for the upper segment, then use successive approximations to adjust only the ballistic coefficient used below 1350 fps until the solved time of flight matches the measured one, at which point the prediction is valid to somewhat below 1000 fps. (3) Move the target further out to where velocity is well below 1000 fps, keep the first two ballistic coefficients, add a second step point at 1000 fps and solve for a third ballistic coefficient, which extends the prediction well into the subsonic region.
There is nothing sacred about the numbers 1350 fps and 1000 fps; they are representative choices, and the number of step points can be increased. Tracer drag, for instance, usually changes once the tracer composition burns out, and adding a ballistic coefficient step at that point is reasonable. Most ballistic solvers ask for step points in fps; under standard atmospheric conditions an fps value converts directly to the corresponding Mach number.
The same procedure works with a custom drag function, or with measured long-range drop data in place of time of flight. Conversely, a common drag function can be used to build a stepped ballistic coefficient approximation of a custom drag function, including drag functions derived from Doppler radar such as those Lapua publishes, which lets software that supports only the common drag functions make practical use of that data.
Worked example: chasing a G7 bullet with G1 and G6
Oehler demonstrates with a common load, a Federal .308 Winchester with a 175 grain Sierra bullet. Rather than live-fire data from System 88, he treats the G7 prediction as truth and then tries to approach it with G1 and G6. Ballistic Explorer handles three traces at once: Trace 1 with G7 (the input, the truth), Trace 2 with G1 and Trace 3 with G6, everything else equal.
Trace 1 shows the bullet at 1376 fps at 800 yards, so 800 yards is taken as the first test distance. The ballistic coefficients of Traces 2 and 3 are adjusted until their time of flight at 800 yards matches Trace 1: G1 is trimmed from 0.483 to 0.477 and G6 is changed from 0.269 to 0.286. After that one step, the holdover of all three traces agrees within 0.1 mil from the muzzle out past 1000 yards.
The second step takes 1200 yards, corresponding to about 1000 fps, where Trace 1 has a time of flight of 2.3157 seconds. With the step point added, only the ballistic coefficient used below 1350 fps is adjusted: G1 moves sharply from 0.477 to 0.326 and G6 from 0.286 to 0.277. G1 needing a change that large is no surprise, since the three drag functions differ enormously through the transonic; but after the adjustment the holdover still agrees within 0.1 mil out to 1400 yards.
The third step pushes the target to about 1800 yards, keeps the first two coefficients and adjusts the third until time of flight is 4.278 seconds. In the end all three traces agree closely in time of flight at 800, 1200 and 1800 yards, and the largest difference in holdover is 0.12 mil at 1800 yards with G1 and only 0.05 mil with G6.
The point most worth remembering is this: these stepped ballistic coefficients move as much as they do because G1 and G6 are being used to describe a bullet that in fact obeys G7. Had G7 been used to fit the input data from the start, the ballistic coefficient computed in every velocity band would have stayed at the input value of 0.242. In other words, how far the stepped ballistic coefficients drift from the initial value is itself a quantitative measure of how badly the assumed drag function fits the real bullet. The less the stepped values move, the better the drag function was chosen and the more stable and accurate the result. If the first assumed drag function does not work well, pick another and redo the analysis.
What you have to measure to build this
The measurement requirements are explicit, and counterintuitive: where you put the target is not critical, and exactly how many yards Mach 1.2 or Mach 0.9 falls at in the prediction is not a critical parameter either. What is critical is that the distance from the muzzle to the target be measured precisely. Beyond that you need accurate atmospheric conditions, muzzle velocity and time of flight for every shot.
Compared with working backwards from actual drop at long range, this approach removes the need to estimate long-range drop, and with it the errors from aiming and from shot-to-shot velocity dispersion; System 88 also lets you build a statistically meaningful sample faster. The experimental work can be done on unprepared ground with portable instruments, and the real constraint is finding a test range long enough for remaining velocity to fall well into the subsonic region.
The procedure can be stopped at any step. Step one is simply the ordinary use of System 88, while steps two and three extend the useful range of System 88 (or of other measurement methods) further out. When a bullet has no published drag data at all, this also provides a way to characterize it yourself. The software required is supplied with System 88 and Ballistic Explorer.
Two limitations have to be stated honestly. Oehler notes in the paper that the procedure had not at that time been verified by actual firing, and that the demonstration used simulated data generated from G7; and the step-by-step solving of the ballistic coefficients by successive approximation is tedious enough that he considered the step ought to be automated in software.