Long Range is all about Ballistics. Beyond that, no more than luck can be expected without it.

Background

It’s a well established scientific fact that air temperature influences muzzle velocity, and under some field conditions that variation can be quite significant, in particular for taking a long range shot.

Recent tests showed a rate of change of about 2.5 to 4.0 feet/sec per 1°C (1.8°F) depending on how sensitive the load’s powder is to air temperature.

Just to give a basic perspective, a muzzle velocity variation of +/- 30 feet/sec can introduce a change in the trajectory’s path of about 1.0 MOA at 1000 yards, (+/- 0.5 MOA) and of course, there is uncertainty that must be accounted for.

So, there is enough “statistical significance” to relate muzzle velocity changes to changes in air temperature, since there is enough statistical evidence that there is a variation, not implying that the difference is necessarily large.

It’s important to realize that the focus of this analysis is on the very first shot, from a cold barrel. Then air temperature is the only meaningful and readily available parameter that any shooter can easily take a reading of.

Powder temperature is the real and crucial factor that determines muzzle velocity, as it’s related to the Maximum Average Peak Pressure. From a cold barrel, it’s closely statistically correlated to Air temperature.

Now, the problem is how can we estimate the predicted muzzle velocity (for a given system, comprised of a particular firearm and load) when facing those variations in air temperature…bearing in mind that not all temperature values can be covered during the data collection process.

Mathematical support

In engineering applications, data collected from the field are usually discrete and the physical meanings (relationship among the observed variables) of the data are not always well recognized.

To estimate the outcomes and, eventually, to have a better understanding of the physical phenomenon, a more analytically controllable function that fits the field data is desirable as well as required.

The mathematical process of finding such a fitting function is called “Data Regression”, also known as “Curve Fitting”.

On the other hand, the method of estimating the outcomes in between sampled data points is called “interpolation”, while the method of estimating the outcomes beyond the range covered by the existing data is called “extrapolation”.

Data Regression is a vital part of statistics. It refers to techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one (or more) independent variables.

For our purposes, the independent variable is the air temperature and the corresponding dependent variable is the muzzle velocity. This is the basic relationship we are interested in.

The goal of regression analysis is to determine the values of parameters for a function that cause the function to best fit a set of data observations that you provide by taking measurements of the involved variables.

It’s also of interest to typify the deviation of the dependent variable (muzzle velocity) around the regression function, which can be described by a probability distribution, especially for the case of a “Linear” regression.

Most commonly, regression analysis estimates the conditional expectation of the dependent variable (Muzzle velocity) given the independent variable (air temperature). That is, the “standard value” of the dependent variable when the independent variable(s) are held fixed.

Both the method and procedure presented here can focus on “quantiles” (points taken at regular intervals), or other location parameters of the conditional distribution of the dependent variable given the independent variable. This is a very important aspect to take into consideration.

Regression analysis is widely used for prediction (including forecasting of time-series data). Under controlled circumstances, Regression analysis can be used to infer fundamental relationships between the independent and dependent variables.

The Solution

A large number of both methods and techniques for carrying out regression analysis have been developed during the last 300 years, so we can hardly call this a “new” branch of technology in general terms. However research continues as new challenges are tackled every day requiring novel approaches.

Essentially, a user gathers field data in the form of Known Data Points (KDPs), which are MV/Temp data pairs. And from that data, different methods will try to make a “best fit”. As can be expected, the better a method fits the KDPs, the better and more trustable will be the predicted values.

Common methods such as Linear Regression and ordinary Least Squares Regression can provide fair results, if and only if, the gathered data shows a good response to a linear representation. In other words, it correlates well with a “straight line”.

This linear approach is the most common in use, especially by some ballistics programs that incorporate a way to estimate muzzle velocity based on predefined changes of some independent variable (air temperature is the usual one)

The first problem we find with a linear approach is that it rarely fits the KDP pairs, since a perfect correlation is very difficult to observe in the dataset (field data)

This means that if your log shows that at 65°F your measured MV is 3000 feet/sec, then a linear method will not yield that value at the same air temperature of 65°F.

The Goal

Clearly, we need to define objective criteria to help us in the selection of the best Regression method, and only then we can make our mind which one best fits our field data.

One of the criteria is well understood (as it’s obvious) for everyone dealing with Regression. That is to find the best function that matches as closely as possible our field data. In short, that “correlates well”.

The second criteria is how good the methods are for predicting both interpolated and extrapolated values. Why? Because that’s where regression will show its value and potential to us shooters as a predictive tool

Interpolating is a must since it’s impossible to have a log of all possible intermediate values that are within the limits of our data. Extrapolating is, by the same token, an essential capability, because we need to know what’s going to happen with values of the independent variable (muzzle velocity) when we have no data outside of our log limits.

In order to understand and visualize the strengths and weakness of the used methods, it’s interesting to see how they execute under two clearly different and alternative scenarios.