Our later instruments are built with internal adjusters, but the internal adjuster is a capricious and none too reliable mechanism. Its reliability depends largely on the assumption that such influences as time, temperature, vibration, shock, etc., which may tend to produce changes in the main optics, will nevertheless have no effect on the very similar but smaller adjuster optics; an assumption which is not invariably correct, since on occasion the internal adjuster will introduce more error than it will take out!
Calibration has been proposed, using another ship for a target, measuring the sextant angle she subtends when abeam, and from this and her known length computing the true distance. The objection to this is that the sextant, while sufficiently accurate for navigational purposes, can hardly handle angular differences of a second or less, which is what the range finder has to do, and hence is of limited value for establishing true distances in this manner.
The following plan is based on the fact that the range finder itself is nothing but an angle-measuring instrument of high accuracy, which may be used to measure the angle subtended by a base line much longer than itself, and thus in effect multiply its own effective base length and obtain a corresponding increase in accuracy.
This is not unlike the procedure when a destroyer checks her short-base range finder against the big ones of a battleship, but the proposed method goes further than this. When we check little one against big one we merely reduce the unknown error of the former to that of the latter, and the only advantage is the fair assumption that the larger instrument will have less error.
In the following method the entire error is found (in theory) and there should be no residual error. There is nothing magical about it. We simply use the same range finder, first in the normal manner to measure the angle which its own base length subtends at some unknown distance, and second, the angle subtended by a larger known base line at the same unknown distance, and between the two form an equation which may be solved to obtain the true distance and hence the error.
But before proceeding further it is necessary to review the theory of range finders.
When a range finder of base length b is trained on a target at true range R, the light rays from the target to the end pentaprisms make with each other an angle q, whose tangent is b/R or, using circular measure for greater convenience and accuracy, q — b/RX206262 seconds. For every range R there is subtended a corresponding parallactic angle q of definite amount.
The rays after entering the instrument would continue at this angle q with each other, but are refracted into parallelism by the compensator when we turn the measuring knob and make coincidence. There is therefore a certain position of the compensator for every range R and angle q, and the scale attached to the compensator is marked at each position with the range corresponding to the angle compensated. Thus, when a 9-foot range finder shows a true range of 9,000 yards on its scale, it has compensated or measured an angle q — 3/9000 X 206262, or 68.7 seconds.
If, however, the instrument read 9,300 on a true range of 9,000, it has in effect recorded an angle of 66.5 seconds; and the linear error of 300 yards is the result of an angular error of 2.2 seconds, introduced through some cause into its optical parts. Should this angular error persist as a constant, it will cause at any other range an error proportional to the square of the range. For example, at 3,000 yards true, the error would be 33 yards, and the scale reading should be 3,033.
The proposed method of calibration rests on the assumption, substantially true, that the linear errors at all ranges do result from a constant angular error in the instrument; constant, at least, until some other change occurs. We may now proceed with the proof of the method.
In Fig. 1 assume a range finder FG of base length b, having a constant angular error D; also two similar targets T and t, forming a known base line B, at the unknown true distance R.
Training the instrument on T, we make coincidence between its two halves in the usual manner. Now if the range finder had no error we would thus measure the angle q correctly; and the correct range R would appear on the range scale. But if there is a constant angular error D, the range scale will show some other value, R', as though an angle q—D had actually been measured, and the target had been at L instead of at T. Record this observed range R'.
Now turn the measuring knob and range scale towards the lower ranges until the two separate targets, T and t, are brought into coincidence. With no error we would thus measure the angle F, subtended by the base line B, plus the base length b (since the lines of sight from T and t to the end pentaprisms now cross each other). Their point of intersection will be at some distance, r, which should appear on the range scale if there were no angular error.
But with the constant error D, the range recorded will have another value, S, as though an angle F—D had been measured, with the targets at T' and t' instead of at T and t, and at a range R" instead of R. Record this observed value, S.
It is here necessary to note and remember that the figure is of necessity much exaggerated. In actuality, the angles q and F are so small that the distances FT, Gt, GN, GT, GM, and GH are substantially equal to each other and to R. Also the angular error D and the resulting range errors TL and tt' are sufficiently small so that the triangles LTM and TFG may be considered similar, as also the triangles t'tN and GHt. The error in our method due to errors in these assumptions will be inappreciable unless the range error, TL, is abnormally big, in which case the obvious procedure is to make an approximate correction of the instrument first, and then a refined correction.
From the similar triangles above noted we get: TM/TL=FG/FT=b/R, or TM= TLXb/R. Also tN/tt'=tH/HG= (B+b)/R, or tN=tt'X(B + b)/R.
But TM equals tN (since they subtend the same angle at the same distance R).
Therefore: TLXb/R=tt'X(B + b)/R, or TL/tt'= (B+b)/b.
Substituting for TL and tt' their equivalents, R' — R, and R" — R, we have: (R'— R) / (R" — R) = (B + b)/b or b(R’ — R) = (B + b) (R" — R).
Simplifying: bR'—bR = BR"—BR + bR" —bR, whence
R = (BR"+bR"-bR')/B or (R"(B+b)-bR')/B
From the similar triangles in the figure, R"/S=(B + b)/b, or R"=[S(B + b)]/b; Substituting this value of R" in the preceding equation, we get
R = S(B+b)2 /bB – (bR') / B
which gives us the true range R, in terms of the known quantities B and b, and the quantities obtained by observation, S and R'.
The expressions (B + b)2/bB and b/B are both constants for any one type of range finder and any particular base line between targets; hence they may be designated by the constants K and k respectively, whence the formula becomes R=SK—R'k.
These constants may be computed beforehand; and when the scale readings R' and S have been observed, it is then only necessary to apply these constants K and k to find the true range R.
In Fig. 1 and the foregoing demonstration the true range R was assumed to be less than the observed range R', and the calculated range R"; that is the instrument read high. We arrive at the same formula, however, if R is greater than R' and R", with the instrument reading low.
The first step in the practical procedure is to decide on the two objects on the target ship corresponding to T and t of the base line. These must be such as will afford good coincidence, since poorly defined or dissimilar objects will give inaccurate results. Corresponding sides of two smoke stacks may be used, but not the forward side of one and the after side of another; nor should a mast and a stack or any such combination be used.
The exact horizontal distance should be measured accurately before the calibration is undertaken. Ships habitually together at sea should be supplied with all such practicable base line distances on each other.
The maximum length of base line which is practicable depends on the minimum range reading for which the range scale is graduated and the maximum range at which good coincidence can be made. The relation is given by the following formula: R/S= (B + b)/b, where R is the range, S the lowest graduation on the range scale, and B and b the base line and base length.
For example, the forward sides of Nos. 3 and 4 stacks on a destroyer gave a base line B of approximately 4.583 yards, whereas Nos. 2 and 4 gave 12.875 yards. The 1.5-meter (1.640 yard) range finder reads down to 700 yards. Substituting in the above formula it was found that to use Nos. 3 and 4 would require a working range of 2,660 yards or over, while for Nos. 3 and 4 a range of 6,200 or over would be required. Inside these ranges the targets could not be brought into coincidence.
The larger base line is best, provided good coincidence can be made at the necessary range. Before starting to calibrate, the foregoing calculations should be made to insure that the proposed base line can be used without having to open to a prohibitive range.
The calibrating ship, assuring herself of the exact course and speed of the target ship, opens out to the calculated range for calibration, or without calculation, until the base line targets T and t, may be brought into coincidence with each other.
When steadied on the base course and with target ship bearing ninety degrees relative, the observations are started, coincidence on T alone and on T and t together being alternated as rapidly as possible until a set of at least ten each has been obtained. The ship may then open out to longer range and repeat the process, using either the same base length or a longer one if available. The form shown in Fig. 2 is suggested for the work. In the example shown the errors obtained at 2,697 yards and at 5,878 yards obey the “square of the range” law about as closely as could be expected. Errors in observation will always prevent absolutely consistent results.
Fig. 2 shows the solution for the true range R, and range error R'—R, but for adjusting the instrument on the spot, the following shorter method may be used:
From preceding formulas we know that R"=[S(B + b)]/b. Compute the value of (B+b)/b. Take a few observations of R' and S, multiply the mean of the latter by our value of (B + b)/b to obtain R", and then move the adjuster so as to increase or decrease the range readings accordingly as R" is greater or less than R'. When R' as observed averages up the same as R", computed from 5) then both of them are equal to the true range, R, and the range finder is in correct adjustment.
Any error in the length of the base line B, causes a proportionate error in the value of the true range R, obtained.
If the target ship is not exactly abeam or exactly on the base course, there will be a foreshortening of the base line, producing errors of the following percentages:
Error in course or bearing . .2° 5° 10°
Value of calculated range R 99.9% 99.6% 98.5%
With careful ship handling, smooth weather, and using the mean of several observations, this error should not be of any importance.
The value of calibration by this method will vary with the conditions attending it.
With poorly defined targets, doubtful base line, or with considerable motion of the ship, it will be of little value; but under favorable conditions it will be of more than fair accuracy, and at the very least will serve as a check to tell us whether our range finder has gone very far wrong.
In port where no calibration ranges have been established, a possible application of it would be to erect on shore two conspicuous battens to form the base line B, and use them in the manner above indicated. If the line of the targets were not perpendicular to the line of sight from our anchorage to the targets, we could very easily use its component projected on a line perpendicular to the line of sight.
All the general instructions for accuracy in the use of range finders apply especially when calibration is being done.
In particular it should be remembered that an instrument adjusted at long range will be accurately adjusted for short-range work also, as any residual error decreases as the square of the range; whereas if the adjustment is made at short range only, any residual error will increase as the square and will assume big proportions at long range where the greatest accuracy is required.