The native language of a trajectory is the range versus time curve
Most shooters and ballistic engineers think of bullet behavior as a function of range: how much velocity is left at 1000 yards, how many mils it drops. Ken Oehler points out that treating behavior as a function of time actually carries more information. A single range versus time curve fully describes range, velocity and deceleration. The initial slope at the origin (the first derivative) is muzzle velocity, the slope at any point on the curve is the velocity at that instant, and the second derivative of the curve is the deceleration caused by drag.
That curve is the native language of Oehler System 88. The system uses three skyscreens near the muzzle to measure the slope of the curve at the origin, which is muzzle velocity, and sensors set up at one or more downrange distances to measure the time at which the bullet reaches them. With the origin, the initial slope and one measured range and time point at long range, all that remains is to draw a continuous curve that satisfies those three conditions.
Why a ballistic coefficient cannot be measured near the muzzle alone
The traditional approach first assumes the bullet belongs to some drag function family, usually G1 or G7, then relates the deceleration of the real bullet to that of the mythical standard projectile with a single constant, the ballistic coefficient. Starting from muzzle velocity, you integrate the implied deceleration over time with the assumed ballistic coefficient to get velocity against time, integrate velocity over time to get range against time, and finally convert the result into the table of remaining velocity and time of flight at each range that shooters are used to.
The problem is that the ballistic coefficient is often quoted and rarely understood; for most people it stops at bigger is better. The industry has several techniques for measuring it, and every one of them is difficult and error prone. That is the pit extended range truing avoids. Instead of measuring the deceleration of the bullet at one particular velocity, System 88 considers the total effect the ballistic coefficient accumulates over the whole flight to the target, and uses that accumulated quantity to suppress error. The reasoning for why the traditional path does not work is set out in full in the companion article "Extended Range Truing: Why Measured Data Beats a Better Drag Function" Extended Range Truing: Why Measured Data Beats a Better Drag Function.
Successive approximation: turning extrapolation into interpolation
Extended range truing comes down to a single action. Once System 88 has measured muzzle velocity and the actual time of flight to a distant target, the chosen existing drag function and the conventional procedure are used to predict the time of flight out to that measured range, and successive approximation then adjusts the ballistic coefficient until the predicted time of flight agrees with the observed time of flight.
Plot the result as a range versus time curve and you have one complete curve that satisfies the initial slope and really does pass through the measured time against range point. The conventional procedure has been demoted. It is no longer a tool for extrapolating from the muzzle all the way out to a distant target, only for interpolating between two known points. The curve may not match true behavior exactly at intermediate ranges, but it has been forced into alignment at the far point, and the far end is where error accumulates and hurts most.
Worked example: reproducing a G7 bullet with stepped G1
Oehler uses a purely numerical example to show what stepped ballistic coefficients can do. Take a bullet with a C7 (G7 ballistic coefficient) of 0.250 at a muzzle velocity of 2800 fps in a standard atmosphere. The tables show it entering the transonic region at about 1100 yards. Deliberately backing away from that awkward sonic transition, take 900 yards: G7 gives a time of flight of 1.367 seconds, and a G1 procedure with C1 of 0.495 gives exactly the same 900 yard time of flight. Plot time of flight against range for both and the two curves overlap almost perfectly out to about 1000 yards. If you only need predictions to 1000 yards, you can stop here.
Past 1000 yards the difference becomes obvious, and C1 has to be stepped. At 1350 yards the predicted velocity has decayed to about 1000 fps, and the C7 time of flight at that range is 2.554 seconds. Holding C1 at 0.495 down to 1410 fps and then stepping down to 0.364 matches the 1350 yard time of flight. The two curves now stay together out to about 1400 yards. One step has extended effective range from 1000 yards to nearly 1400 yards.
Correct elevation beyond a mile needs one more step. G7 gives a time of flight of 4.730 seconds at 2000 yards. Holding C1 at 0.364 down to 970 fps and then using C1 = 0.693 in the lower velocity band matches the 2000 yard time of flight. Time of flight now agrees all the way out to 2000 yards and 840 fps; below 840 fps the ballistic coefficient reverts to its initial value.
What the method actually delivers
Theorists may enjoy time of flight curves, but working shooters care about zero elevation correction. Converted to zero correction, the example above differs by at most 0.03 mil inside 1000 yards, stays below 0.1 mil inside 1500 yards, and stays below 0.25 mil inside 2000 yards. In other words, a set of stepped G1 ballistic coefficients really can model a G7 bullet to practical accuracy.
The conclusion generalizes. If stepped G1 can model G7, it can model any bullet. If your solver supports stepped G7 and the bullet itself is closer to G7, the steps needed are smaller and the result is better. Conversely, if the chosen drag function already fits the bullet, the stepped values come out nearly constant. The size of the steps is itself an indicator of how well the drag function fits. When the inputs are measured muzzle velocity and measured time of flight, what you get is a functional description of the bullet under test.
One caution. Stepped ballistic coefficients have been around for years and were often treated as a mild, harmless device, but every step is an abrupt jump. The size of the jump depends on how well the assumed drag function matches the real behavior of the bullet under test, and for G1 against G7 the difference through the transonic region is considerable. On the usual plot of drag coefficient against velocity, G7 has a sharp sonic peak while G1 is smooth and rounded, and a one step jump in ballistic coefficient can be understood as an equivalent step in the drag coefficient curve. Stepped G1 and ordinary G1 are identical up to the first velocity break point, and at that break point the ballistic coefficient drops sharply, which means drag rises sharply. The peak of that step plays a role similar to the sonic peak of G7, forcing the two curves back into alignment after the sonic transition. The curves look different and really are different, but fortunately the predictions they give are essentially the same.
Boundaries in software and in test planning
What if you have no muzzle velocity and time of flight data for the range in question? That is exactly why Oehler built System 88: it measures those numbers with your own gun on your own range. The System 88 software automatically computes the ballistic coefficient that matches the first time of flight. If your downrange sensor sits at a distance where remaining velocity is about Mach 1.2 to 1.1, the resulting ballistic coefficient and muzzle velocity support accurate elevation estimates down to about Mach 1.0, and the reliable prediction range of a stepped ballistic coefficient extends no further than the distance at which you actually measured a time of flight. To cover the whole transonic region with elevation estimates, you have to test at a distance where the bullet has already passed through that difficult region.
The optional Extended Range Truing program for System 88 automates the process for up to four different ranges; installed with SetupTruing.exe, it leaves an icon on the desktop. As of the original paper (July 2016), the program completed the calculation and displayed the stepped ballistic coefficients on screen, but all input had to be typed in by hand, output was shown on screen only, and there was no printed report and no detailed manual. The program expects a first test at a range equivalent to Mach 1.2; extending range through the sonic region calls for additional tests at about Mach 0.9 and beyond. The program handles the different atmospheric conditions, muzzle velocities and distances between one test and the next.
Oehler also mentions one constraint in the field. Setting up several targets in a single test would of course be ideal, but in the example above the midpoint of the trajectory is about 100 feet (about 30 meters) high, and putting a sensor at that height needs a very long pole. That is a physical limit to settle before you plan the range layout.