# 10.1 Introduction

The term ‘adjustment’ suggests that some form of cheating might take place during the process of converting surveying observations into results. This is not necessarily the case. As explained in Chapter 2, surveying observations always have two particular properties: 1 they contain errors (random, systematic and the occasional gross error); 2 the system of observations should always be such that more observations have been taken than would be strictly necessary to obtain the required result. This combination of properties means that no set of surveying observations is ever exactly consistent. If, for instance, the purpose of the observations is to fix the position of a new station, there will be no position which will exactly concur with all the data. All we can do is choose a position for the point which gives the best agreement with the observational data, and say that this is the most likely position of the point. The process of finding this ‘best’ position is called adjustment. It only involves cheating if, in choosing a ‘best’ position for the point, we ignore some of our observations simply because they do not appear to agree well with the other observations. The adjustment of large quantities of observations with several stations whose positions are unknown is a tedious job and is best done by computer. Several programs exist for this purpose and are able to take detailed account of what constitutes the ‘best’ fit of the available data. The process they use is called ‘least-squares adjustment’ and will be described later in this chapter. Often, though, it is adequate to use a simpler method which does not need elaborate computer software; one such method is called a Bowditch adjustment, and a simplified version of this adjustment is described below.

This adjustment method is best suited to a traverse (either in a horizontal plane or in the vertical direction). A basic version of the general method will be described with reference to a simple example. All distances will be assumed to be small, so that the ‘(t−T) corrections’ described in the previous chapter can be ignored. A classic traverse to find the grid1 positions of two new stations is shown in Figure 10.1. The grid positions of stations A, B, E and F are already known, and the positions of stations C and D need to be found.

A scheme of observations to find these positions with some degree of redundancy is also shown in the figure. Angles are measured as shown, at stations B, C, D and E, and the horizontal distances BC, CD and DE are also measured. Note that the angles are measured using the previous station as the reference object; at D, for instance, C is used as the reference object, giving an observed angle CDE greater than 180°. Given these data, it is first possible to calculate the grid bearing of the line BA by simple trigonometry; then, by simply adding the measured angle ABC, the grid bearing of the line BC. Knowing this bearing and the length of the line BC,2 we can calculate a preliminary grid position for C. We can now repeat this process, to calculate preliminary grid positions for D and then for E. If all our readings were totally error-free, the preliminary grid position we would calculate for E would be exactly the same as its known grid position, which we already have as part of our data. 1 Multiplying the horizontal distance calculated by an EDM device by the factor RE/(RE+h), where RE is the radius of the earth (6.381×106 m) and h the height of the instrument above the ellipsoid (see Chapter 11, up to equation 11.6). Note that the ellipsoidal height of the instrument includes the height of the instrument above the geoid (i.e. its benchmark height), plus the height of the geoid above the ellipsoid in the area where the observation is made. Note also that this adjustment produces a negligible effect if h is less than about ±20 m.

2 Multiplying the distance calculated above by the local scale factor, as described in Section 9.3. For all but the largest traverses, this factor will be the same for all the distances in the traverse. Having done this, the easting and northing components of the vector BC can be calculated, using the computed bearing and distance from B to C. Adding the grid coordinates of B to this vector gives the preliminary co-ordinates for C. This process is then repeated to find preliminary co-ordinates for D and E as well. Finally, the co-ordinates which have just been computed for E are compared with its known co-ordinates, and the error or mis-closure is found. As described above, 1/3 of this mis-closure is now subtracted from C, and 2/3 from D, to yield accepted values (i.e. best guesses) for the co-ordinates of these two points.

A sample calculation of a Bowditch adjustment is given in Appendix E. The goal of adjustment is to choose the most likely co-ordinates for points whose positions are unknown, given the available readings. Once these co-ordinates have been chosen, the calculated angles and distances between these points and the fixed points will not all be exactly the same as those which were observed; any difference between each calculated and observed value must be presumed to be the error in the reading. Assuming that these errors have a statistically normal distribution, it can be shown that the most likely co-ordinates for the unknown points are those which yield the smallest value when the squares of all the errors are added together. The process of finding these co-ordinates is therefore known as ‘least-squares adjustment’. The exact process involved in least-squares adjustment4 is best explained by reference to a simple example. Suppose we wish to find the most likely easting and northing co-B, to take three measurements: (1) the distance AC, (2) the distance BC and (3) the angle BAC. Because of errors in these measurements, it will not in general5 be possible to find any position for C which will exactly concur with all the data. The process starts by taking a guessed position for C, and then seeing how this guess should be altered, in the hope of bringing the sum of the squares of the errors to a minimum. Taking C’s guessed co-ordinates to be (CN,CE), we can calculate the distance and bearing from A to this position for C (Figure 10.4): 5 It is always possible that the errors in a set of observations will cancel each other out, to give the impression that they do not exist at all. Adding further redundancy to a set of observations reduces the likelihood of this undesirable possibility.

This calculated distance will in general not be equal to the measured distance, labelled as m1 in Figure 10.4. We define the difference between the calculated value and the measured value6 to be the current error for this reading, i.e.: Now consider the two degrees of freedom we have for altering the position of C, denoted as x1 and x2 in Figure 10.4. In Figure 10.5, we can see that adding the quantity x1 to C’s easting will increase the calculated distance AC by approximately x1 sin α, provided x1 is small compared to dAC. Likewise in Figure 10.6, we can see that adding the quantity x2 to C’s northing will increase AC by approximately x2 cos α, again provided x2 is small. 6 The actual distance measured must first be reduced to the ellipsoid (see Chapter 11) and must then be multiplied by the local grid scale factor before being used in this equation.  which shows how the 3D vector e is a linear function of the 2D vector x (provided the elements in x are small), with A and k containing only constants. Clearly the goal is to set the elements in x so as to make the components in e as small as possible.

However, there is one further complication to be overcome first; whereas e1 and e2 are distances, e3 is an angle, so its value cannot be compared directly with the other two values. Even in the case of the two distance measurements, it may be that one measurement is known to be much more accurate than the other, so that, say, a 1 mm error in measurement 1 is as likely as a 10 mm error in measurement 2. To overcome this problem, weighted errors are used in place of the actual errors, obtained by dividing each error by the estimated standard deviation which might be expected in repeated observations of the same measurement. We define a weighted vector v such that: