# Techniques: Measurement Errors

Version date: 16 October 2016

In the previous sections we have seen that most of the survey techniques that can be used underwater rely on being able to measure distances and depths. All distance and depth measurements have errors. This section explains these errors, why some survey techniques are more accurate than others and why we take check measurements. The ideas in this section are based on statistics so if it gets a bit complicated then skip to the next section and come back here later.

#### Types of Error

There are three types of errors; mistakes, systematic errors and random errors.

- Typical mistakes include reading the wrong numbers from a tape measure, making a measurement with the tape snagged around some ship's structure or reading the wrong values from a form when processing the measurements. Mistakes are sometimes called gross errors or blunders.
- Systematic errors are ones that can be repeated and can be accounted for in processing. If you calibrate a tape measure against a known standard and find that it always measures distances that are too long, the difference is a systematic error and can be removed when the measurements are processed.
- Tape measurement made several times under the same conditions is unlikely to give exactly the same value for each measurement. Judgement of the tape reading will vary as will tension on the tape depending on how hard you pull. If you remove the mistakes and the systematic errors then some variation in the repeated measurements will still be seen, this is called random error.

The procedures we use for making measurements and then processing them to form a site plan should aim to remove mistakes and systematic errors. We cannot remove the effects of random error but we can ensure that they are kept within acceptable limits.

#### Mistakes

There are two ways of getting rid of mistakes, avoid making them or find them during processing. It is usually better to avoid making mistakes at all, finding mistakes in processing means a repeat measurement must be made and having to repeat work is expensive.

If there is no independent check on the position of a point then there may well be a mistake in one of the measurements and the position will be wrong. Finding mistakes that have been made is done using check measurements. To find the position of a point in three dimensions needs three measurements, any extra measurements can be used as a check on the other three. Extra measurements are called redundant measurements.

Making extra redundant measurements takes time so a balance is needed between making the check measurements and minimising on work. The simple rule is that each point should have at least one check measurement for detail work, for positioning the control points at least two should be used.

#### Systematic Errors

Systematic errors in tape and depth measurements can be found by calibrating the tape measures and depth sensors against a known standard.

#### Random Errors

If a number of divers make a tape measurement between the same two points the measurements will be slightly different, even after mistakes and systematic errors have been removed. All measurements show these differences called random errors, only the size of the random error varies.

A random error is one whose value depends on chance and the analysis of random errors based on statistics. If we take our set of tape measurements we can look at a few of its characteristics. Tape and depth measurements are usually normally distributed, this means in practice that:

- Small errors occur frequently, large errors occur less frequently.
- Very large errors are likely to be mistakes
- Positive and negative errors of the same size are just as likely to happen.

For our set of measurements, called a sample, we can calculate an average or mean value and we can also calculate a standard deviation. The standard deviation gives an indication of precision, the smaller the standard deviation the better quality the measurements.

#### Practicalities

We have said that all measurements have random errors and the size of the random error can vary, but how does that affect our underwater survey?Calculating the position of a point is in fact only half of the problem, in surveying it is also important to work out how well you know the position. Giving a value for the quality of the position is similar to saying 'the point is here, plus or minus X metres'. What we need to work out is the size of the plus or minus X value. If we do not do this then the plus or minus number could be small or very large and you would not know, this makes the position of the point meaningless if we are trying to do accurate work.If the measurements used to position a point have small random errors, or are precise, then the position error for a point will also be small. This is the reason why techniques using angles to record positions underwater are used for assessment surveys or for small areas. The precision of a tape measure is much smaller than that of an angle measurement made with a divers compass.

A problem also arises if you make extra, redundant measurements as a check on position. Because of the random errors the measurements will never fit together to give a position at a single point, lots of positions can be calculated using different selections of the measurements in the set. It is not possible to calculate the true position for the point so we must find the most likely or most probable position using some maths.

#### The Least Squares Adjustment

If we go back to our set of tape measurements it happens that the most probable value for the true measurement is the average or mean value. If we take each measurement in turn and compare it with the average value, the difference between them is called the residual.Now if we want to compute the position of a point using lots of measurements, not just the minimum three required, we can use a technique called Least Squares to tell us the most probable position. In short, the most probable position is the one where the sum of the squares of the residuals is the smallest. In that position, you add together the residual for each measurement ignoring the sign and the total will be smaller than that for any other position.Least Squares is also used in setting up control and in Direct Survey Measurement (DSM) to deal with the direct tape measurements between points that are so difficult to plot by hand. All you have to do is to type in the measurements into a computer program and the least squares turns them into positions for points. Least squares produces a single solution no matter how many measurements are used, the type of measurements or how they were collected. The technique also tells you how well the measurements fit together so least squares can be used to help find mistakes.

We may also need to find an answer using measurements with different precision, fortunately least squares can do this as well.

#### Recommendations

- Expect to make mistakes
- Make check measurements
- Be aware of the effects of measurement error
- Use least squares to adjust measurements for accurate work