Those who metal detect and GPS the locations of their relics often find interesting trends in their waypoint locations that need to be analyzed. One such analysis is the determination of the line of fire and shooters location for a series of linearly trending bullets.
Below is an air photo map of some data taken from an 1861 practice area adjacent to a Civil War campsite. The data shows the positions of a suite of .54 caliber bullets, each having a distinctive nose sprue. Initially, it was believed that these bullets were fired from a US Model 1841 "Mississippi Rifle" because of their caliber. However, closer examination showed that the fired bullets of the suite all showed low impact deformation and all had four distinct barrel groove impressions. This ruled out the Model 1841 and confirmed that they were fired from an Austrian Lorenz Model 1854 Rifled Musket. These were the only .54 caliber bullets in several hundred bullets taken from the location.
Problem: Where are they shot from? Where were the shooters standing?
It's not easy determining a line of fire. Shots have a tendency to "fan out" around a target at an arc of about 3 degrees from the shooting position This is particularly true if the shooter is new to firearms. There is also a tendency to "pull" the weapon when it discharges and overshots of the target are often made by inexperienced shooters.
One thing works to the benefit of those doing the analysis: "Drops" are usually located near where the shooting took place. On the Air photo, two drops were found close together at one end of the trend. A third drop (probably a fired bullet with low deformation) was also found about halfway up the trend but this is questionable as it was one of the earliest finds and may have been misidentified. Other than these drops, all of the other bullets of the suite were shot and exhibit some form of low impact with the ground.
What is the best way to find the line of fire when the number of bullet locations is small? Many would simply "eye-ball" a line through the data but this would not be very accurate. Some would try to fit a straight line through the data using a "least-squares" fit. but this could lead to errors as well.
Use the Median
Often when people have a small number of pieces of data (usually less
than 20) and they want to find a middle or typical value, they use the
average (or mean value). This is incorrect. Take the following numbers
where there are a
couple of values (8 and 18) that are "outliers". Let's say
we are measuring the
diameter of the badly deformed bullet shown in the picture to the left and we got
the following measurements:1,2,2,3,3,3,3,4,4,5,8,18
totals 56 with an average of 4.7.
This
is certainly not the best "guess" for the diameter of the bullet. A
better answer for a typical value would be somewhere around 3.
The
median or middle value of the above group is 3. The median has the big
advantage of not being influenced by outlier values like 8 and 18.
Typical
outliers values
are anomalously high or low values that are the result of measurement
errors, or in the case of measuring the dimensions of a bullet,
errors resulting from deformation of the bullet either from
loading, firing, or impact with the ground.
Whenever the number of measurements are small it is always better to use the median rather than the average to get a typical measurement.
If we apply the same thinking to a bullet trend we already know because of the outward fan of the bullet trend from the shooter that there will be outliers to the location data. If we use the "least-squares" method for finding a straight line through the data, the trend line will be "pulled" in the direction of the outliers much like the above example using the average.
Fortunately, there is an easy-to-use method to fit a straight line through a relic field which is not influenced by outliers and uses the median values of the data. It's called the median-median method and is available on Texas Instruments graphing calculators that most
high school students carry.
If you are handy with spreadsheets you can make the calculations there
as well.
An Example Calculation
In the table to the left are the GPS latitude and longitude map coordinates of the 16 bullet locations mentioned in the Introduction:. The coordinates are of the form DD MM.mmm or degrees with decimal minutes which is typical output for the GPS. Usually the length and width of such a bullet trend is under 100 yards in length and width.
The area is so small that the coordinate degrees and whole minutes are all the same so this helps make the calculation easier since we will use only the decimal part of the coordinates.
Notice that we have sorted the values of latitude and corresponding values of longitude. Next , divide the dataset into 3 equal regions. If the number of values cannot be divided by 3 equally then place a smaller number (whatever is left over after the first and third regions are filled) in the middle region. Thus, 16/3 equals 5.3 so region 1 is of size 6 values; region 2 in the middle is size 4 values, and region 3 is of 6 values. Check: 6+4+6 = 16
Next we will calculate the median value of the decimal part of the coordinates for each region. This is shown in the table below.X refers to latitude and Y refers to longitude.
Once the data are divided into the three regions, the median of the X- and median of the Y-values are calculated for each region. The resulting three points for these data are termed the median-median points For Region 1 the values are (750, 574.5); Region 2 are (761.5, 565.5) and Region 3 are (791, 562)
The slope of our best line involves values from Region 1 and 3 and is calculated as the value b:
and the Y-intercept is given by the value a::
This yields the equation for the median-median line: Y(estimated) = a + bX or Y(estimated) = 801.3272 + (-30488)X
If we substitute 2 values for X into the equation we can estimate 2 values for Y that lie on the line. We then plot the estimated points on an air photo (shown below) and draw a line showing the best firing line.
The picture below shows such a line drawn through the locations of the .54 caliber Lorenz bullets locations and the best firing line through the data:
Reference: Walters, E.; Morrell, C.; and Auer, R. (2006) "An Investigation of the Median-Median Method of Linear Regression" Journal of Statistics Education, Vol 14 No. 2
http://www.amstat.org/publications/jse/v14n2/morrell.html
Whenever the number of measurements are small it is always better to use the median rather than the average to get a typical measurement.
If we apply the same thinking to a bullet trend we already know because of the outward fan of the bullet trend from the shooter that there will be outliers to the location data. If we use the "least-squares" method for finding a straight line through the data, the trend line will be "pulled" in the direction of the outliers much like the above example using the average.
Fortunately, there is an easy-to-use method to fit a straight line through a relic field which is not influenced by outliers and uses the median values of the data. It's called the median-median method and is available on Texas Instruments graphing calculators that most
high school students carry.
If you are handy with spreadsheets you can make the calculations there
as well.An Example Calculation
In the table to the left are the GPS latitude and longitude map coordinates of the 16 bullet locations mentioned in the Introduction:. The coordinates are of the form DD MM.mmm or degrees with decimal minutes which is typical output for the GPS. Usually the length and width of such a bullet trend is under 100 yards in length and width.
The area is so small that the coordinate degrees and whole minutes are all the same so this helps make the calculation easier since we will use only the decimal part of the coordinates.
Notice that we have sorted the values of latitude and corresponding values of longitude. Next , divide the dataset into 3 equal regions. If the number of values cannot be divided by 3 equally then place a smaller number (whatever is left over after the first and third regions are filled) in the middle region. Thus, 16/3 equals 5.3 so region 1 is of size 6 values; region 2 in the middle is size 4 values, and region 3 is of 6 values. Check: 6+4+6 = 16
Next we will calculate the median value of the decimal part of the coordinates for each region. This is shown in the table below.X refers to latitude and Y refers to longitude.
| Region 1 | Region 2 | Region 3 | ||||||||
| X | Y | X | Y | X | Y | |||||
| 732 | 573 | 757 | 566 | 778 | 577 | |||||
| 734 | 576 | 760 | 562 | 778 | 568 | |||||
| 749 | 583 | 763 | 570 | 786 | 554 | |||||
| 751 | 571 | 767 | 565 | 796 | 544 | |||||
| 752 | 581 | 797 | 565 | |||||||
| 755 | 571 | 818 | 559 | |||||||
| Median X | Median Y | Median X | Median Y | Median X | Median Y | |||||
| 750 | 574.5 | 761.5 | 565.5 | 791 | 562 | |||||
Once the data are divided into the three regions, the median of the X- and median of the Y-values are calculated for each region. The resulting three points for these data are termed the median-median points For Region 1 the values are (750, 574.5); Region 2 are (761.5, 565.5) and Region 3 are (791, 562)
The slope of our best line involves values from Region 1 and 3 and is calculated as the value b:
b.
= (562 - 574.5) / (791 - 750) = -.30488
and the Y-intercept is given by the value a::
a =
((574.5 + 565.5 + 562) - b(750
+761.5 +791)/3) = 801.3272
This yields the equation for the median-median line: Y(estimated) = a + bX or Y(estimated) = 801.3272 + (-30488)X
If we substitute 2 values for X into the equation we can estimate 2 values for Y that lie on the line. We then plot the estimated points on an air photo (shown below) and draw a line showing the best firing line.
The picture below shows such a line drawn through the locations of the .54 caliber Lorenz bullets locations and the best firing line through the data:
The
above air photo shows the GPS location of 16 - .54 caliber bullets
believed to be fired from the Model 1854 Lorenz rifled musket. It is
believed that the shooter shot from near the location of the drops near
the lower portion of the bullet trend. A best-fit median-median line
though the data points shows the estimated line of fire. The
approximate distance of the trend is 200 yards and more bullets may be
found along the trend to the north.
This method
(median-median) could also be utilized in identifying skirmish lines
and regimental battle lines by using dropped bullets and percussion cap locations.
Acknowledgment: This page was greatly helped by the comments Kim Allen (a.k.a. "SoArk")
Acknowledgment: This page was greatly helped by the comments Kim Allen (a.k.a. "SoArk")
Reference: Walters, E.; Morrell, C.; and Auer, R. (2006) "An Investigation of the Median-Median Method of Linear Regression" Journal of Statistics Education, Vol 14 No. 2
http://www.amstat.org/publications/jse/v14n2/morrell.html