# Error Analysis

All measurements, including ultrasonic measurements, however careful and scientific, are subject to some uncertainties. Error analysis is the study and evaluation of these uncertainties; its two main functions being to allow the practitioner to estimate how large the uncertainties are and to help him or her to reduce them when necessary. Because ultrasonics depends on measurements, evaluation and minimization of uncertainties is crucial.

In science the word "error" does not mean "mistake" or "blunder" but rather the inevitable uncertainty of all measurements. Because they cannot be avoided, errors in this context are not, strictly speaking, "mistakes." At best, they can be made as small as reasonably possible, and their size can be reliably estimated.

To illustrate the inevitable occurrence of uncertainties surrounding attempts at measurement, let us consider a carpenter who must measure the height of a doorway to an X-ray vault in order to install a door. As a first rough measurement, she might simply look at the doorway and estimate that it is 210 cm high. This crude "measurement" is certainly subject to uncertainty. If pressed, the carpenter might express this uncertainty by admitting that the height could be as little as 205 or as much as 215 cm.

If she wanted a more accurate measurement, she would use a tape measure, and she might find that the height is 211.3 cm. This measurement is certainly more precise than her original estimate, but it is obviously still subject to some uncertainty, since it is inconceivable that she could know the height to be exactly 211.3000 rather than 211.3001 cm, for example.

There are many reasons for this remaining uncertainty. Some of these causes of uncertainty could be removed if enough care were taken. For example, one source of uncertainty might be that poor lighting is making it difficult to read the tape; this could be corrected by improved lighting.

On the other hand, some sources of uncertainty are intrinsic to the process of measurement and can never be entirely removed. For instance, let us suppose the carpenter's tape is graduated in half-centimeters. The top of the door will probably not coincide precisely with one of the half-centimeter marks, and if it does not, then the carpenter must estimate just where the top lies between two marks. Even if the top happens to coincide with one of the marks, the mark itself is perhaps a millimeter wide, so she must estimate just where the top lies within the mark. In either case, the carpenter ultimately must estimate where the top of the door lies relative to the markings on her tape, and this necessity causes some uncertainty in her answer.

By buying a better tape with closer and finer markings, the carpenter can reduce her uncertainty, but she cannot eliminate it entirely. If she becomes obsessively determined to find the height of the door with the greatest precision that is technically possible, she could buy an expensive laser interferometer. But even the precision of an interferometer is limited to distances on the order of the wavelength of light (about 0.000005 meters). Although she would now be able to measure the height with fantastic precision, she still would not know the height of the doorway exactly.

Furthermore, as the carpenter strives for greater precision, she will encounter an important problem of principle. She will certainly find that the height is different in different places. Even in one place, she will find that the height varies if the temperature and humidity vary, or even if she accidentally rubs off a thin layer of dirt. In other words, she will find that there is no such thing as one exact height of the doorway. This kind of problem is called a "problem of definition" (the height of the door is not well-defined and plays an important role in many scientific measurements).

Our carpenter's experiences illustrate what is found to be generally true. No physical quantity (a thickness, time between pulse-echoes, a transducer position, etc.) can be measured with complete certainty. With care we may be able to reduce the uncertainties until they are extremely small, but to eliminate them entirely is impossible.

In everyday measurements we do not usually bother to discuss uncertainties. Sometimes the uncertainties are simply not interesting. If we say that the distance between home and school is 3 miles, it does not matter (for most purposes) whether this means "somewhere between 2.5 and 3.5 miles" or "somewhere between 2.99 and 3.01 miles." Often the uncertainties are important, but can be allowed for instinctively and without explicit consideration. When our carpenter comes to fit her door, she must know its height with an uncertainty that is less than 1 mm or so. However, as long as the uncertainty is this small, the door will (for all practical purposes) be a perfect fit, x-rays will not leak out, and her concern with error analysis will come to an end.