ANGULAR DEFLECTION WITH A THEODOLITE
At Fort Bragg not long after I started finishing the first sets of .50 caliber tables I very interested in solving another problem that we were having. That is solving the range determination problem to a super fine level. Long before I was in the Special Forces, I was in the conventional army as a combat engineer. A few times, we were on road construction projects in some of the major training areas in Europe. I remembered that there were a couple of engineering officers that had used a mechanical theodolite to lay out the road sites. One of these officers explained a little about the capabilities of these instruments and how they resolve and measure angles to a very fine degree, much better than standard engineer transits.
Triangulation was taught at SOTIC when I first attended the school. Somewhere while investigating range determining possibilities and trying to refine the methods of triangulation, I remembered theodolites and their capability. I called one of the surveying companies in Fayetteville, North Carolina and arranged some time with the owner. A trip downtown and some time with the owner of the store confirmed that I could indeed measure the size of an object in seconds or arc or minutes of angle. Remember earlier in the chapter when we covered Mil Relation and how we discussed the MIL dot reticle? There is a potential for huge error at long ranges because of the low power of the telescopic sights and the strong possibility of the operator messing up the MIL value for a target.
We then talked about using the telescopic sight elevation drum for determining the angular deflection of a target. The single minute of angle or the Ό MOA capability of measuring the target height made for a huge jump in the precision capability in finding the range to the target. Triangulation using a compass and a baseline expanded our range finding capability but this method when using a magnetic compass has the potential for huge error when using a small baseline. The large baseline option meant a lot of moving around for a sniper team, which isnt good site security for the team.
Now we are going into the world of theodolite range finding. This is by far the most state of the art, precise method of determining a range to a target. A theodolite is a precision instrument that costs about $4000.00. That may sound like a lot of money, but consider what the theodolite can do. Here are some of the advantages of the theodolite.
§ Replaces two instruments with one: the spotting scope and a laser range finder or other range finding gear.
§ Ultra high resolution and high power (24X typically) optics make for an excellent spotting scope as well as a range finding instrument.
§ Cost effective. In replacing two instruments with one, you save maintenance money and the theodolite is cheaper than military grade laser range finders.
§ System is totally passive. There is no laser to shoot downrange and risk detection.
§ Night vision systems can be easily adapted to the objective lens assembly or the ocular lens assemblies. A laser pointer can be slaved to the theodolite for designating targets at night when necessary.
§ System can be used on a target that is stationary or an object close to the target that is very small. This depends on the resolving power of the theodolite. A common system available in the military or outside is a 6 theodolite (6 explained in a little bit).
§ This method can also be used like the baseline and compass method by moving the theodolite from Point A to Point B and then measuring Angle B instead of Angle C.
We have used all kinds of units of measure for anything for windage holdoffs, mil relation, and holdovers high and low (MILS). We have also used another unit of measure that is used for elevation and windage (Minutes of Angle). Theodolite range finding uses another unit of measure that is finer than MILS or Minutes. These are seconds of arc. A second of arc is 1/60th of a Minute of Angle. That is a might small angle. Theodolites can read angles to 1 of arc. The more common ones read to 6 of arch. The following units of measure sub-tend as follows:
1 MIL @ 1000 meters = 39.37 Inches
1 Minute of Angle @ 1000 meters = 11.38 Inches
1 Second of Arc @ 1000 meters = .18967 inches
The first theodolite that I was able to obtain for testing I actually borrowed from the Civil Engineering section responsible for property maintenance at Fort Bragg, NC. Thanks guys for the use of that instrument. When I took that instrument back to my place of work it immediately took it out and mounted it on the tripod. I had not had any formal training in the use of this instrument. After playing with the knobs and reading the book that came with it, I had a rough idea how it worked. I knew that the theory was sound because of MIL relation theory and using the Leupold & Stevens M1A scopes to measure the height of a target in Ό MOA clicks.
The first that I did was work out some of the problems on paper and determine the degree of accuracy that I could expect using this method. The theodolite that I borrowed was a Wild T-6 with a 6 resolution capability (meaning it could give you a reading from the optical micrometer in 6 increments). Here are some of the results of that initial paper testing.
VALUE OF UNITS OF MEASURE AT 10,000 METERS
§ 1 MIL @ 10,000 meters = 393 inches
§ 1 Minute of Angle @ 10,000 meters = 113.8 inches
§ 1 Second of Arc @ 10,000 meters = 1.883 inches
§ 6 Seconds of Arc @ 10,000 meters = 11.30 inches
RESOLVING POWER OF DIFFERENT INSTRUMENTS USED FOR TRIANGULATION
§ The standard military compass can with great s**** and practice (on a sighting board) can resolve to .5 degrees with visual interpolation (usually a second person taking the reading).
§ The M2 compass that reads in MILS, can be resolve a 2 MILS reading off the compass.
§ The average theodolite can resolve to 6 of arc. Models are available that resolve typically to 1 Minute of angle, 20, 10, 6 and 1 second of arc.
EXAMPLES OF ERROR AT DIFFERENT RANGES WITH 3 DIFFERENT INSTRUMENTS
§ Using a compass: True range = 1572 meters. Baseline = 40 meters. A half degree error in compass reading may result in a or + 427 meter error.
§ Using an M2 compass. True range = 1629 meters. Baseline = 40 meters. A 2 MIL error in compass reading may result in a or + 120 meter error.
§ Using a 6 Theodolite. True range = 1712 meters. Baseline = 40 meters. An error of 18 seconds of arc (3 times the capability of the sight) = 6.4 meter error.
Each theodolite operates the same way. To go into the operation of each knob on each type of theodolite would add many more pages to this book. A major advantage of the theodolite is that it reads both horizontal and vertical angles. This not only gives the operator the vertical or horizontal angular deflection of the target, but it also gives the up or down angle to that target. These instruments have two types of scales that you take your reading from. The first is a vernier scale. These take some tricky reading experience from the operator. If you have ever worked the scale on the iron sight of the M-24, this is that system on steroids. In fact, its probably not workable under field sniper conditions.
The best type of scale is the optical scale. There is another small scope next to the main tube. Inside of this scope is nothing but a scale reading. There are many types of scale readings. Some of these are listed below:
There are two methods for determining the range using a theodolite.
§ METHOD #1. Determining range to a target not at the target site. This method is used when the target is expected in the target area, but has not yet arrived. With this method, the operator must move the theodolite. This lowers the precision of the range determination. Using a large baseline when possible makes up for a lot of this error. Baseline recommendation is 30 meters.
§ METHOD #2. Determining the range to a target using the size of the target as a representative baseline. This method takes advantage of the theodolites resolving power. Even though you are using a smaller target (3 to 15 meters tall or wide) as compared to method #1, which uses a wide baseline of 30 meters or more.
PROCEDURE FOR METHOD #1 / Angle B Measurement
1. The HTI team approaches the final firing point and the team leader designates the point (Point A) for the 1st position of the theodolite. The operator sets in the theodolite and levels the system. The optical micrometer was previously zeroed out at the ORP. You cannot zero out the vertical scale. He immediately releases the theodolite head and slews the scope to the designated target point (Angle AC). The micrometer will read 0 Deg. 0 0. At this point, the operator can take the vertical angle reading to the target if there is any up or down angle. He writes this down in his notes. The observer will need this data for his Slant Angle corrections. He will attach a small fishing weight plumb under the instrument and sink a nail head in the ground.
2. The operator engages the optical micrometer and slews the head right or left until he gets a deflection of 90 Deg. 0 0. He is now looking at Point B and this establishes the azimuth for the baseline (Angle AB). Another operator will have measured the pre-determined baseline length. For the purposes of this example, this is 30 meters. The operator has a small fiberglass wand (tape two vertical levels to this wand to insure it is straight up an down) in his hand that is used to mark the place where he will set a nail in the ground to designate Point B.
3. The theodolite operator will slew the head back to Point C (the target spot) and he should get a reading of 0 Deg. 0 0 on the scale. He will then slew the head again to Point B and this checks his position, his measurement and the precision of Point B. At this point, the triangle is laid in. Points A, B and C is established with a baseline of 30 meters and a 90-degree angle at Point A to B and C.
4. The theodolite operator will lock the head of the instrument, pick up and move his instrument to Point B. The man with the wand will return to Point A and place his wand vertically on the nail at Point A. The theodolite operator will set his instrument up over the nail using the optical sight on the theodolite for plumbing the sight in. This is also done at the same time the sight is leveled. This takes a bit of practice but like surveying students can attest, with practice it can be done in less than a minute.
5. The system leveled and plumbed directly over Point B. The operator with the micrometer locked, slews the theodolite around until he is directly lined up with the center of the wand. Ensuring that the wand is vertical, he makes final adjustments to the location of the theodolite. At this point, the scale is at 0 Deg. 0 0. He is pointing his sight directly at Point A.
6. He engages the optical micrometer and unlocks the head. He slews the theodolite over and up or down so that the reticle is aiming directly at Point C (target). He looks into the optical micrometer and takes a reading. His direct reading is 88 Deg 38 0. This is written down in his notes. He will have to subtract this from 90 degrees later.
7. As a final check he will slew his theodolite back to the Point A. He should get a 0 Deg. 0 0 reading. This tells him that the original Point A is good and his taking of the reading was solid. There is another way of making your reading more precise.
8. While all of this angle measurement is going on, another member of the team is taking initial meteorological and environmental conditions. He takes a barometric pressure reading and air temperature reading. This is especially important when the FFP is located away from the ORP by a good distance.
9. Repeating Measurement. After the operator has slewed his sight from Point A to Point C and recorded that deflection (he converts this reading to a decimal figure using the calculator 88 Deg. 38 0 becomes 88.63333 Degrees), he disengages his optical micrometer. This allows him to move the sight head without changing his scale reading. He slews the head back to Point A again. He re-engages the micrometer and slews his sight over to Point C. This takes another measurement that is added to the first one.
You must remember that when adding angles, they must be converted to decimal format. This is easy on a scientific calculator. This repeating measurement can be taken as many times as necessary. The acid test is the averaging. The scale in the sight should look like this now. This is on the second repeating measurement. The scale shows 177 Deg. 14 40 for two measurements.
§ Your 3 readings are
88 Deg 38 0 (88.63333)
88 Deg 38 0 (88.63333)
88 Deg 36 0 (88.60000)
§ Total of all readings = 265.86666 or 265 Deg 52 0.
§ Divide the decimal figure by 3 for an Angle B average = 88.62222 or 88 Deg. 37 20 average.
§ Subtract 88.62222 from 90 for the Angle C measurement = 1.37778 or 1 Deg. 22 40.
10. Once the team has determined that the measurements are true and acceptable and they breakdown the equipment leaving only a nail at Point A for a gun position reference. They have gathered their Angle B and Angle C measurements and have their slant angle to the target. They are now ready to begin their calculations to determine the range to the target.
This is a depiction of the intersection through the lens of the theodolite. The optical micrometer is depicted to the right of the road main observation sight. These sights usually are in the 30X range. Because of the high quality of the lenses they make excellent spotting scopes and are quite capable of reading long-range trace. One of their shortfalls is light transmission under conditions of limited visibility.
PROCEDURES FOR METHOD #2 / ANGLE C MEASUREMENT
In this situation, the team is going to use the target itself as a representation of the baseline. Instead of the baseline being at the guns position (30 meters), the size of the target (vertical of horizontal) becomes the baseline for the triangulation formula. This is the preferred method for a couple of reasons.
§ Since the target is in place, you know exactly where it is positioned and wont have to possibly make minor range adjustments. In Method #1, you may have to adjust your range based on the actual location of the target in relation to where you shot the range to (i.e. the road intersection).
§ Much less possible error when taking the angle deflections because the theodolite will remain in one place (Point A) and not have to be moved to Point B to take a deflection for Angle B.
§ Less movement in the FFP area. This is a tactical problem associated with moving in your FFP enough to possibly compromise the team to security patrols that may be operating in the area. The team finishes its angle determinations and returns to the objective rally point and in the meantime an enemy patrol finds evidence of the teams activities at that FFP.
§ You will get a more accurate angle to target deflection for determining the uphill or downhill slant range to target. Another critical factor when using an angular deflection method is that you are looking uphill or downhill at a target, that angle to the target affects the apparent height of the target. You must correct for this angle to target. More about that later.
In this picture, the operator is looking over his #1 gun. He has acquired the left side of the missile launcher and his micrometer is set at 0 Deg. 0 0. The vertical micrometer which you cannot set to zero is showing a down angle of 8 Deg. 0 0. 8 degrees is the angle from the gun to the target. The Cosine of 8 degrees is .99027. The operator will multiply his true range in meters against this value to obtain the slant range correction. The procedures for a horizontal angle deflection are as follows:
1. The team performs an area reconnaissance for the final firing position. When the FFP is located the team leader designates the position for this base gun and the theodolite is set up in this position. The instrument is leveled. For this mode, there is no need to plumb and mark the site.
2. The operator slews the theodolite head to zero the optical micrometer. He then disengages the micrometer so that he can move the head without changing the micrometer setting. The Reticle is aligned with the left or right extreme measurable side of the object to be ranged. (Example P. 37). Check to insure that the micrometer is at 0 Deg. 0 0. The operator engages the optical micrometer and slews the theodolite head to the opposite side of the target, inducing a change in the reading on the optical micrometer. In the case here, the deflection is 0 Deg. 20 0. The vertical deflection is 8 Deg. 0 0.
3. At this point, the team has all the data that they need to determine the range to the target. They have the size of the target in meters, the horizontal angle deflection of that target, and they have the angle from gun to the target.
4. Repeating Measurement. A repeating measurement is recommended whenever the tactical conditions permit. A repeating measurement works as follows.
§ After the first measurement, the operator disengages the micrometer and slews the head back to the left side of the target.
§ He re-engages the micrometer and dials the reticle back to the opposite side of the target for a second measurement. The 20 reading should continue to escalate to a 40 or so reading. It may vary 15 or 30 seconds. The micrometer is again disengaged and the head is panned back to the left side of the target.
§ The third measurement. The operator re-engages the micrometer and for the third time, he pans the reticle across the target. This repeating measurement is much more accurate than a single reading. Do this for, as many times, as you have time for or until you are satisfied with your measurements.
§ Remember to convert your angle findings to decimal format before adding them together. YOU MUST DO THIS TO OBTAIN SIN AND COSINE FUNCTIONS ALSO.
§ Your three ANGLE C readings are:
0 Deg 20 0 (0.33333) Range = 1375 Meters
0 Deg 20 30 (.34167) Range = 1342 Meters
0 Deg 21 0 (.35000) Range = 1310 Meters
§ Total of all readings = 1.02500 or 1 Deg. 1 30 for Angle C.
§ Divide the decimal figure by 3 for an Angle C average = .34167 or 0 Deg. 20 30 average.
The Angle C method is much more desirable than Angle B method. There is much less movement around the final firing position and that means less exposure time to a potential enemy position. The team can move forward to obtain the Angle C data with the theodolite and then return to the objective rally point to calculate their data. Another team member should be taking the meteorological conditions while the range finding operations are being conducted.
With practice an operator can take an Angle C reading in about 2 or 3 minutes. The major factor in this time is the time it takes to level the theodolite. It is critical that the theodolite be leveled before taking the measurement. This method works very well for a target that is wide. Trucks, missiles laying in the travel mode, radar vans, boats, aircraft and railcars all work very well for determining Angle C on the horizontal plane. We have covered the two major methods for determining Angular Deflection using a theodolite. Both of these techniques are extremely accurate, but one is much more difficult than the other method.
Determining Angle C on the vertical plane is as fast as doing the horizontal method but it requires a little more calculator use. This is because you cannot zero the vertical angle reading scale on a theodolite. It will always read something between zero and 90 degrees.
PROCEDURES FOR METHOD #2 / ANGLE C MEASUREMENT USING THE VERTICAL PLANE
Using the vertical plane (measuring the height of the target instead of the width) is in some ways desirable to the horizontal plane. Certain types of targets are much taller than they are long, especially missiles in the launch mode. Certain vans that have extendable mast antennas are much taller when that mast is extended than when it is lowered. In may cases such as in the picture below, you can use a full ground to top of the radar measurement because the ground line is clearly defined. This will depend on the operators ability to see clearly defined edges.
Another factor for taking a vertical measurement is the angle of the vehicle to the theodolite position. In the example to the left, the vehicle is on about a 40-degree angle to the gun position. The radar is also on that 40-degree angle. In this case, although the radar is tipped back a little, you will get a more accurate angle deflection taking a vertical measurement that you will with the horizontal measurement. You also save time using the vertical measurement because you do not have to zero out the optical micrometer before taking your 1st deflection. Repeating measurements create a bit of a problem though. Here are the steps for a vertical measurement:
1. After the team leader has fixed the location for his FFP, the theodolite operator locates his theodolite over the base gun position. He levels his instrument and is ready to take the first reading. There is no need to zero the horizontal optical micrometer and you cannot lock or unlock the vertical micrometer. That reading will constantly change as long as your are moving the sight up and down. As with the horizontal method, there is no need to plumb and mark the sight other than to mark it for the base guns position.
2. The operator places the reticle at the bottom of the target. This is the point on the target where he knows he has a solid real world measurement for height in meters and he can clearly see that point through his scope. He notes the reading for the vertical deflection. In this case, the reading is 8 Deg. 15 00. Target height is 11 meters.
3. He will now slew the sight reticle to a point halfway up from the bottom. This is to take the slant angle setting to the target. This will also give him the angle that he needs to correct the target heights visual appearance to him based on the downhill angle to that target. The more severe this angle, the smaller the target is going to appear to him and this must be corrected for. Mid-point reading is 8 Deg. 27 50. In this case, a 11-meter high target will appear to the eye to be 10.88 meters high for measurement. You cant tell this, but you know that target is 11 meters high. You can only slew the sight from the bottom to the top of the target based on what you can see of that target. There is no way to mentally project where the top of that target would be if the angle to the target were 0 degrees. So how do you correct for this misrepresented angle? You must take your 3rd and final reading to correct for this optical error.
4. The sight is now elevated to the top of the target and a reading is taken. The reading is 8 Deg. 40 00. To obtain the deflection, subtract the bottom angle reading from the top angle reading. Remember to convert your angles to decimal format to subtract.
§ 8 Deg. 40 0 (8.66667 Deg) 8 Deg. 15 0 (8.25000) = .41667 or 0 Deg. 25 00
§ Angle C (Vertical) = 0 Deg. 25 00
5. Remember we said that because there is an 8+ degree angle to the target, that target is going to appear smaller to the operator. He can only take a reading on what he sees and cannot project where the top of the target would be on a horizontal plane (0 degree angle to target). Now is the time to correct the vertical deflection for that optical error. Here are the steps to correct the vertical deflection.
§ Total height of target in Deg Min Sec = 0 Deg 25 0
§ Angle to Target = 8 Deg. 27 5
CALCULATOR SEQUENCE
§ Enter > .25
§ Divide by
§ Enter > 8.275
§ COS key
§ =
§ .25263 corrected Angle C (this is not in decimal format, dont change it)
6. Now, how much does this minor correction affect the range to the target and the subsequent MOA elevation setting? Lets find out.
§ Using an uncorrected 0 deg. 25 0 / Range = 1513 meters / 79.75 MOA
§ Using the corrected angle of 25 26.3 / Range = 1487 meters / 77 MOA
§ A difference of 2.75 minutes of angle at 1500 meters = a difference in the strike of the round by 47 on the target. Not too bad, but can you afford it? We havent even corrected this data for slant angle to the target and the other MET and ENV conditions. The errors add up fast.
NOTE: Whenever you are reading the angle to target and you are shooting uphill or downhill, the target is going to appear farther away from you. When you correct the deflection for this angle, your range number should be smaller. If the range number is larger, you multiplied instead of divided. Remember you still must correct the True Range to target for the slant angle.
The two effects are not the same. The first, the optical error due to angle corrects the apparent size of the target for range. The second, correcting the True Range for the Slant Angle, corrects the range for the effects of gravity.
This all sounds rather long and drawn out. The first couple of times it is. As you practice the use of the calculator and learn how to use the different memory capabilities, you will speed up. In the appendix for calculator operations, there are programs that are written for the Hewlett-Packard HP20S scientific calculator. This is the only (as far as I know) calculator that is easily programmed using keystroke programming. Its the best.