Every program agrees, and none of them hits
The author of an exterior ballistics program can usually explain in careful detail how he treats the drag function, the ballistic coefficient (BC), muzzle velocity and the atmosphere before producing a trajectory. The method may be the old Siacci approach, modern numerical integration, or Pejsa's clever approximation. The math holds up, the implementation is sound, and he has long since compared his results against the other well-known programs. They always agree. In fact almost every modern ballistics program returns the same answer to the same problem.
The trouble starts at the range, where the predictions and the actual impacts do not line up. Dr. Ken Oehler's observation is that this contradiction does not point at the math, it points at the inputs. The programs agree with each other because they are fed the same set of assumptions, and those assumptions are not necessarily true of your rifle and your lot of ammunition.
Three inputs, three kinds of uncertainty
The first input is the drag function. A ballistician will tell you to use G1 because it is the standard, or to use G7 because it fits modern bullets better than the old G1, and then hand you the choice. What he does not tell you is that none of the drag functions on offer really matches your bullet. Some are simply closer than others. Oehler compares it to ordering hunting boots from a catalog that lists whole sizes only and says nothing about width. They may go on your feet, but at best they are a compromise.
The second input is the ballistic coefficient. BC can be read as a grade: how good is my bullet compared with the reference projectile that was used to define this drag function? The number usually comes from the bullet maker or from a third-party measurement, but wherever it comes from, it was most likely measured over a relatively short distance, typically 200 yards, and often from drag measured close to the muzzle. We then expect it to support predictions beyond 1000 yards. Oehler's comparison is using a runner's first 100 yards to predict his time for a mile. If you really want the mile, look up how long his mile took last week instead of extrapolating from his speed over the first 100 yards and how much he lost in the second 100.
The third input is muzzle velocity, and it is the easy one. A modern chronograph makes a respectable muzzle velocity straightforward to obtain, and temperature and barometric pressure can be measured and logged to recover the correct air density. Temperature matters more, and changes faster, than most shooters assume; humidity is negligible.
How large is each error
Start with the choice of drag function. Using Ballistic Explorer, take the expected muzzle velocity, zero the sight at 1000 yards with G1, then compare three drag functions: G1, the century-old standard; G7, the best approximation for modern long-range bullets; and the velocity-squared drag law Newton proposed three centuries ago. The G7 and Newtonian BC values are converted by the usual convention, so that their drag equals that of G1 at muzzle velocity. At 1000 yards the G7 trajectory falls 6 inches below G1, and the Newtonian one sits only 6 inches above G1. Note what is being compared: the spread among three predictions, not the gap between prediction and real bullets. As long as the inputs hold, those paths stay fairly close. The differences grow at higher muzzle velocity and longer range, and especially where the bullet crosses the transonic region.
Next, the uncertainty in BC itself. BC is known to vary with velocity, from bullet to bullet and from rifle to rifle. Bullet to bullet is manufacturing tolerance and is frequently observed below 1%. Rifle to rifle is far more significant. It shows up not only between different twist rates or different muzzle crowns but between two "identical" rifles, and a 5% difference from one lot of ammunition fired in different rifles is not unusual. Oehler's view is that many of the variables cannot be measured or even identified, so taking BC uncertainty as ±5% is a reasonable assumption. Using G1 again, with 3500 fps muzzle velocity and the same sight setting, and comparing BC values of 0.500, 0.475 and 0.525, the 1000-yard trajectories differ by about 6 to 7 inches, which is barely visible.
Last comes muzzle velocity, which the shooter can influence by choosing ammunition. A 100 fps spread is routine for factory hunting ammunition, 50 fps is common for match-grade ammunition, and some bench-rest shooters who focus on long range claim 5 fps for carefully selected handloads. Note that these spreads do not include rifle-to-rifle differences, nor the variation caused by temperature. Estimating with a velocity difference of ±1%, the swing seen at 1000 yards is about the same size as the two items above.
Taken one at a time, 6 or 7 inches looks tolerable. But these errors add. The ballistician is in effect taking a model that may not match your bullet, correcting it with bullet data measured only at short range, pairing it with a questionable muzzle velocity, and predicting what your bullet will do far downrange. Added together, the error can be much larger.
The military answer: Doppler radar
About sixty years ago the military demoted the whole idea of drag functions and ballistic coefficients to approximation and discussion. Military practice now uses Doppler radar data to build a Firing Table for each standard round. That amounts to a custom drag function for every projectile, after which its ballistic coefficient is declared to be 1.000 when used with that table. It works, and it costs an enormous amount of time and money.
There is a turn in this history. During the first half of the twentieth century ballisticians spent most of their effort improving the prediction models, which is why G1 and G2 through G7 and G8 were formalized. They used high-end spark photography to measure deceleration and drag precisely over short distances near the muzzle. By mid-century, once large Doppler radars and their attached computers had become common at the proving grounds, most of that military effort was abandoned.
What makes Doppler radar so good? Its raw output is already a record of velocity (Doppler frequency) against time. It can run for several seconds and track a bullet for miles, covering everything from muzzle velocity down to the low subsonic segment at impact. A computer then converts velocity against time into distance against time and drag against time, and finally into drag against velocity. That drag function fits the shot just observed exactly. There is no guessing which canned drag function to use, and no guessing the exact BC. They use what actually happened on the last shot to predict what will happen on the next.
The problem is that this route shuts civilian and amateur ballisticians out. A usable Doppler system costs about as much as a fleet of limousines, trained operating crews are hard to find, and a range longer than a mile is needed. For a long time the civilian side was left with canned drag tables plus a ballistic coefficient measured near the muzzle.
The one long-range quantity civilians can measure well: time of flight
The civilian position is this: we can measure muzzle velocity, estimate BC, and then fill in downrange velocity and time of flight by prediction. The only thing downrange we can genuinely attempt to measure is trajectory height, the drop at the point of impact, and trajectory is both very important and very hard to measure consistently. In other words, we have almost nothing with which to check our own predictions.
The answer Oehler's team found is that long-range time of flight (TOF) is the only long-range parameter that can be measured accurately and reliably. Time of flight, taken together with muzzle velocity and a known distance, describes exactly the accumulated effect of drag over the whole flight path. With that measurement you know precisely which ballistic coefficient to pair with your chosen drag function so that the two agree at long range.
Why not derive BC from the velocity difference between two chronographs? Oehler says that does not work in practice. The spacing between the two has to be very large before it produces a meaningful velocity loss, and if the velocity loss is not large compared with the absolute accuracy of the chronographs, the computed BC is worthless. Once the spacing is that large it is also too far to get every shot reliably through the downrange screens. The most accurate BC comes from muzzle velocity plus long-range time of flight.
The muzzle velocity end can be raised a level too. A shooter who measures with his own rifle across the expected temperature range can approach 0.25% rather than the 1% assumed in the earlier example, but that is not an easy standard to reach: chronograph screens must be spaced at least 4 feet apart and ammunition consistency has to be taken very seriously. Thanks in part to some military research, Oehler also found that four points on the time-distance curve are enough to measure muzzle velocity very accurately. Muzzle velocity and long-range time of flight then yield a very accurate ballistic coefficient with any of the common drag functions.
Oehler pays particular attention to the velocity band used in long-range shooting. Muzzle velocity in these applications is almost always between Mach 2 and Mach 3.5, and the shooter wants the bullet to stay above Mach 1 all the way to the target, because odd things happen near the speed of sound. Inside that band he found that the choice of drag function hardly matters, provided BC is derived from muzzle velocity and time of flight to 1000 yards. Return to the original comparison: with the same sight setting, running G1, G7 and Newton on BC values measured from the same 1000-yard time of flight, the gap between the G1 and G7 trajectories shrinks from 6 inches to 1 inch, and even Newton's original formula comes close. Moving the measurement distance from 200 yards out to 1000 yards improved BC measurement accuracy by a factor of 4 to 5.
One objection is that this measures an average BC over a long distance rather than the true BC measured over shorter distances at distinct velocity levels. The objection is entirely correct, but Oehler's answer is that if the chosen drag function really matched the bullet, the measured BC would stay constant across every velocity band anyway. Long time of flight measures the accumulated effect of drag over the whole path, and we simply express that conveniently as one equivalent ballistic coefficient. Turned around, if the BC values measured after changing muzzle velocity still agree with one another, that verifies that your chosen drag function suits your bullet. In other words, measuring BC accurately at very long range matters far more than which drag function you pick; force the analysis to agree far out and the close-range agreement follows. (As an aside, the fact that BC shifts with velocity band is itself a symptom that the assumed drag function does not fit the bullet, and adjusting BC with velocity is only a crutch for bending the drag function into agreement with the test results.)
What it looks like as hardware: System 88
Why have ballisticians not made heavy use of long time of flight before now? Oehler's guess is that many wanted to, but outside of Doppler radar there was no practical instrument for it. His team spent several years developing a measurement system that makes it workable, bringing the cost from the order of a fleet of limousines down to the order of a compact sedan.
What Oehler System 88 provides is exactly that: an accurate measurement of long-range time of flight, the time that is the key to deriving the correct ballistic coefficient. It consists of two or more identical timing units linked by a radio network to a Windows computer. The first unit measures muzzle velocity accurately with proven Skyscreen III screens at a wide spacing. The second unit is usually placed at 1000 yards or beyond as an acoustic target. More important than the impact position, that acoustic target supplies an accurate stop signal for time of flight. Third and fourth units are optional extra target positions, set before or after the main acoustic target, so one shot yields several times at different distances and gives redundant measurements of the ballistic coefficient.
The acoustic target uses Oehler's new target microphones, far more rugged than the earlier versions and well suited to a portable System 88. The terminal target uses a square microphone array with excellent accuracy; intermediate target positions can instead place the four microphones in a straight line for fly-over or fly-by sensing.
One last point deserves an honest hearing. Much about drag behavior in the transonic region remains unknown, and questions about stability (twist rate, spin decay, velocity) and about how the airflow changes as a decelerating bullet crosses the transonic region are still unanswered. Oehler claims no magic answer, but he does stress that the total drag effect of whatever happened can now at least be measured. A bullet that has already fallen to subsonic speed will not trigger the acoustic target, and time of flight is still critical, so the system was designed with that in mind and work continues on alternative target sensors able to provide a stop signal for subsonic bullets.