35 ACCURACY AND ERRORS IN ASTRO-NAVIGATIONAL COMPUTATIONS
131. ACCURACY IN ASTRO-NAVIGATION
The required degree of accuracy of any navigational result varies with the use to which that result is to be put. If, for example, a navigator wishes to keep his vessel in a channel, then assuming that he knows that his vessel is somewhere near the middle of the channel, the required degree of accuracy of his fixes should be related to the channel width. It would be pointless to try to fix his ship to the nearest cable if the width of the channel were 10 miles. On the other hand, accuracy to the nearest cable would be insufficient if the channel width were only 1 cable.
Safe navigation requires the navigator to fix his vessel and set her courses within certain safe limits. Any combination of errors within these safe limits will not endanger the vessel. The wider the safe limits prescribed by the navigator, the smaller will be the required degree of accuracy of the quantities and processes involved in producing a navigational result.
Since it is seldom possible to take sextant sights from a small vessel in a seaway to an accuracy of better than 1′ or 2′, it might appear to be pointless to worry about decimals of a minute of arc in G.H.A.. or Dec.. This, however, is not the case. All figures should be calculated as accurately as the tables permit for two reasons: –
- Errors due to working only in whole numbers may happen to be all in the same direction as the inevitable error in the sight-taking, thus making it even worse, and.
- If one develops the habit of being meticulous with decimals, one is more likely to be right with units and tens – which do matter.
Perhaps rather surprisingly, the most difficult part of Astro-navigational work is not in remembering what to do (the standard sight form will solve this), nor how to do it. It is to be accurate. Inevitably, a number of figures have to be looked up, written down and added or subtracted. Simple as this appears to be, this is where the majority of errors are made, particularly by beginners who do not fully appreciate the serious consequences which can arise at sea from simple (but nevertheless dangerous) arithmetical errors. It is maddening when a sight makes nonsense and it is found, on checking, to be due to a simple arithmetical error. It is dangerous when a sight appears to make sense but nevertheless contains an undetected arithmetical error.
As we reminded you at the beginning of the Astro-navigation study, it is of no use saying to the coxswain of the lifeboat which rescues you after your vessel has stranded on the rocks But I only made a little mistake. Arithmetical accuracy is essential in Astro-navigation, and the experienced navigator follows the following rules in order to achieve this: –
- Write all figures down – do NOT rely on memory or attempt mental arithmetic at any time.
- Be methodical. Follow a regular sequence of working and DO NOT attempt shortcuts.
- Write all figures neatly, boldly and clearly so that there is no risk of their being misread. Write figures accurately under each other so that adding and subtracting is easy – misaligned figures lead to column dodging.
- CHECK every figure, check every addition, check every subtraction, check every ‘name’ N-or S. E or W). and check every sign (+ or -). TICK every item checked, and do not use the answer until you have ensured that every figure, name, sign, etc. has been ticked as checked. This may sound tedious, but in practice takes only a few moments and could be the means of averting a navigational disaster.
Check all subtractions (the most frequent cause of the error) by adding up backwards, for ex: –
– 51º 55′.0 Check by adding 01º 10.0 to 51º 55.0 to ensure it makes 53º 05.0
= 01º 10′.0
132. Arithmetical Accuracy and Precision
Nautical Astronomy is not an exact science. The measurements made by a navigator when using a sextant or a chronometer, the quantities extracted from the N.A. and from nautical tables, and the computational processes employed when reducing sights, are all liable to error. An intelligent Astro-navigator should aim to understand the nature of the several errors which may influence the degree of accuracy of his observed positions.
Before discussing navigational errors a few remarks on arithmetic and its processes will be relevant. By arithmetic, we mean the mathematics of computation in which numbers are used. It is useful to distinguish between the terms accuracy and precision.
In many, if not all, navigational processes we deal with measurements of quantities which are continuous as opposed to those which are composed of discrete and separate elements. The quantities measured by a navigator when using a sextant or chronometer are called by arithmeticians approximate numbers.
An approximate number is incorrect because of the error which exists between it and its true value. Let us suppose that the length of the rod illustrated in fig. 35-1 is to be measured with each of three rules, labelled in the figure A., B., and C. divided into inches, half inches and quarter inches respectively.
Using ruler A. we find that the rod is 3 inches long to the nearest inch. Using ruler B we find it to be (7 x ½). i.e. 3½ inches to the nearest half-inch, and using ruler C we find the length to be (13 x ¼), i.e. 3¼ inches to the nearest quarter inch.
In each case the length of the rod is expressed in terms of the nearest exact unit of measurement of the ruler. The result, therefore, is accurate to within a half unit of the measurement given. The accuracy of the result using ruler A is within ± ½ inch of the value. That using ruler B is within ± ¼ inch of the true value; and that using ruler C is within ± ⅛ inch of the true value.
The term precision is used to denote the degree of accuracy of an approximate number. The smaller is the degree of accuracy the more precise is the measurement. Precision and accuracy are the concern of the navigator, not only when using navigational instruments, but also when extracting quantities from nautical tables.
The fractional numbers used in most navigational processes are expressed in decimals. The precision of a decimal quantity is indicated by the number of digits to the right of the decimal point. Consider the quantity the magnitude of which is 11.64 precisely. The quantity may be described as 11.6 or 12 to mean respectively that its magnitude lies between 11.55 and 11.65 or between 11.5 and 12.5. We say, therefore, that if the magnitude of the quantity is described as 11.6 the description is more precise than by giving it as 12, etc.
In expressing a number which is a multiple of 10, 100, 1000. etc, there is sometimes a doubt as to its precision. If. for example, the tonnage of a ship is described as 2000 tons, the tonnage may be taken to mean between 1500 and 2500; or between 1900 and 2100; or between.1990 and 2010 tons.
A common way of expressing the degree of precision of a numerical quantity is to state the number of significant figures. Significant figures in a number, as the name implies, are those which occupy places which indicate their significance. For example, in the number 32.04 there are four significant figures, for we know that the figures represent three tens, two units, nought tenths, and four hundredths, respectively.
In determining the number of significant figures in a number, caution is necessary in respect of zeros. Zeros interspersed between digits are always significant figures (as in the example above) but zeros written to the right of a digit in a whole number present difficulty in determining which figures are significant. In the above example with tonnages, we could say that the tonnage of the ship is 2000 tons to four, three, or two significant figures, according to whether we wish it to mean 2000 tons precisely; or between 1990 and 2010 tons; or between 1900 and 2100 tons respectively.
It is important to bear in mind that in any arithmetical computation, the result can never be more accurate than the least precise value used. An example will illustrate this. Suppose we wish to add 10, 6.4 and 5.35. From the above remarks 10. as written, may mean between 9.5 and 10.5 ; 6.4 may mean between 6.35 and 6,.45 ; and 5.35 may mean between 5.345 and 5.355. It follows that the result of the addition may lie between (9.5 + 6.35 + 5.345), i.e. 26195, and (10.5 + 6.45 + 5.355), i.e. 22.305. The sum of the three numbers is 21.75 precisely, only if the numbers are 10, 6.4 and 5.35 precisely in each case. If the three numbers are not precise the result 21.75 may give the computer a false indication of accuracy. So much for arithmetical precision and accuracy.
The term error applies to the difference between a correct and a corresponding incorrect value arising from imperfections in the instruments or methods used in obtaining a result. Errors are to be distinguished from what statisticians euphemistically call blunders. Blunders are due largely to carelessness and are commonly called mistakes.
135. Random Errors and Errors in Position Lines
In contrast to systematic errors, random errors are those which cannot be predicted. These include errors which are inherent in practical work. In navigation, random errors are seldom very big. In measuring an altitude with a sextant, the accuracy of the measured result is influenced by several factors which include: indistinct horizon, abnormal refraction, changing H. of. E. due to rolling, pitching, or heaving of the vessel. Moreover, the limit of accuracy with which the sextant can be read may also result in error. All of these are examples of random or chance errors. Random errors are governed by the mathematical laws of probability. The term probability may be defined as the proportional frequency of occasions on which some stated event takes place.
If., for example, a series of Sext. Alt. observations are made, and the observations are liable to random, but not systematic errors, the probability of the result of any given observation being greater or less than the corresponding true value is expressed as 0.5. This follows because the probability or chance of a particular observation producing too high a result is equal to that of its producing too low a result. If a large number of observations affected by a random error were made and the results plotted against the probability of it happening, the result would tend to be a curve known as a Normal or Gaussian Curve of Errors (illustrated in fig. 35-2).
This is a bell-shaped curve symmetrical about an ordinate representing the proportional frequency of observations yielding the correct value. The ordinates of points on the curve to the right of the central ordinate represent the proportionate frequency of observations yielding too high a result, and those of points to the left represent the proportionate frequency of observations yielding too low a result. The curve illustrates that the possibility of a random error occurring falls off as the size of the error increases.
The value of the error at the two ordinates placed symmetrically about the central ordinate, and which limit half the total area under the curve, is called the probable error or 50 per cent error, so that the probable error may be defined as the error such that 50% of the observations will have an error greater, and 50% of the observations will have an error smaller, than the probable error.
For the Astro-navigator, the ideal result of an Astronomical observation is a position line which is drawn on a chart or plotting sheet. Because random errors are to be expected in the process involved in taking and reducing a sight, the ideal result., viz. a true line of position on the chart, is not to be expected.
The practical result of an Astronomical observation is a position band on the chart the width of which may be regarded as being proportional to the probable, that is the 50 percent, error. The band may be considered to be the projection on the chart of part of a ridge the shape of which, at right angles to the band, corresponds to the Gaussian curve of errors, as illustrated in fig. 35-3.
A fix in the simplest case is obtained by crossing two position lines. In the ideal case, in which the position lines are free from error, the required fix is the point of intersection of the two position lines. If, on the other hand, two position lines are affected by random error, the 50 per cent bands of error will intersect to form a diamond.
It may at first be thought that there is a 50 per cent chance of the vessel’s position falling within the diamond formed by the intersection of the 50 per cent bands of error, but this is not the case. Statistical analysis reveals that the 50 per cent probability area is an ellipse which fits into a diamond having dimensions 1¾ times those of the diamond formed by the intersecting 50 per cent position bands. In fig. 35-4. the ellipse of 50 per cent error fits into the diamond WXYZ, the dimensions of which are 1¾ times those of the diamond ABCD.
Students recognising the possibility of errors affecting Astro position lines should at once realise that the concept (common amongst navigators) that information used in computing a vessel’s position is perfectly reliable, is false and unrealistic.
A vessel’s position obtained as a result of systematic evaluation of the information used in deriving it may be described as a probable position, because recognition of probable error has been made. Probable error in a position line is estimated : there is no other practical way of evaluating it. The estimation of probable error in Astro-navigation is related to a man’s skill and experience as a navigator. Skill, in this connection, is related to an understanding of the nature of the errors which may affect navigational observations and processes. Without this understanding, errors cannot be handled systematically and intelligently. The practical treatment of navigational errors will now be discussed.