After a traverse has been run in the field it will be necessary to adjust that traverse. Some data collector software will allow this to be done in the field. Many surveyors, however, prefer to perform this phase of the survey in the office where they have access to a desktop computer with a large screen. Offices usually have air conditioning and heat so this is sometimes an added incentive to leave the field in favor of the office.

All traverses will contain some error, no matter how carefully the work was performed. Errors can fall into a number of categories. One category consists of blunders or mistakes. These can be caused by writing wrong numbers into a field book. Numbers are sometimes transposed or misread. For example 199.98 might mistakenly be written as 198.99. Sometimes measurements are called out over a walkie-talkie and misunderstood. Although relatively uncommon, the memory of a data collector could become corrupted, or a bad cable connection between a data collector and the instrument might cause a corrupted data transfer. Data collectors have substantially reduced blunders compared to the days when every measurement had to be written in a field book. Normal traverse adjustment procedures will not correct blunders.

Other types of errors are systematic errors. This type of error is not random but occurs as a result of some uniform process such as an instrument that is not adjusted properly, a measuring tape that is defective or an instrument person who consistently sights a target incorrectly. Systematic errors are generally not corrected by traverse adjustment.

Random errors can result from a lack of precision in making measurements or in setting up tripods or tribrachs. These errors are usually small and can result in positive or negative errors so that they might cancel each other out. For example, a tripod is set up 0.01′ to the right of point 2. On the next setup the tripod is set up 0.02′ to the left of point 2. Because surveyors work outdoors, sometimes in inclement weather and on difficult terrain, random errors are very common in land surveying. We adjust our traverses and measurements primarily to reduce the effects of random errors.

There are several methods which can be used to adjust a traverse. One is the Compass Rule . This method assumes that both the angles and distances were measured with similar precision. The compass rule works well when modern instruments, such as total stations, are used, so the method is widely favored for traverse adjustment.

The Transit Rule is another method. This method assumes that angles were measured with greater precision than distances. The Crandall Method is another option where the angles are adjusted first and held fixed while the distances are adjusted using a least squares method. It is a relatively time consuming method.

Another method of adjustment is the Least Squares method of adjustment. This is a more difficult and time consuming method, and before computers it was used primarily for large and complex control traverses where highly precise results were needed. The advent of computers makes the least squares method more commonly used than it was in years past. Most land surveyors now have software that will perform least squares adjustments, and because it is likely to provide a more refined adjustment it is the preferred method for surveys in which a high order of precision is desired. The least squares adjustment uses the theory of probability to determine the statistically most probable coordinate location for each point in a network. The adjustment provides a statistic best-fit for each point. A least squares adjustment has the additional advantage of providing statistics which tells the surveyor something about the degree of confidence of the calculated position of each point. The surveyor can then accept or reject the adjustment.

A final option is to adjust the traverse based on the surveyor’s knowledge of likely sources of error. In this method the error would be placed only in those angles and distances known or believed to have errors. The remaining angles and distances would not be adjusted.

Whatever method is used, the goal is to end up with a traverse that closes perfectly—at least on paper. Because of its long history of use by boundary surveyors and its relative simplicity, our examples in this book will use the compass rule.

A.5.1 Calculating the Error of Closure Consider the traverse shown in Fig.

A.28 . This figure contains the raw data which was measured in the field. Because the following is just an example to illustrate the adjustment of latitudes and departures, the angles were not adjusted. However, if this were a real traverse, it would be best to adjust the angles prior to adjusting the traverse. In this exercise all of the angles have already been converted to bearings.

The data has been compiled in Table

A.6 to show the latitudes and departures and the coordinates.

Table A.6 Traverse raw data table

Notice that the closing point is numbered Point 8, not point 1. Although the survey began at point 1 and returned to point 1, small errors in the measurements have caused the ending point to be in a slightly different location than the point of beginning. In order to avoid confusion between the beginning and end points it is usual to assign these points different numbers. Notice that the beginning and ending coordinates are slightly different. This difference represents the error of closure.

The error of closure can be seen by comparing the north and south latitudes (−0.038) and the east and west latitudes (−0.086). This same error is obtained by taking the difference between the starting and ending coordinates. This makes perfect sense because the coordinates are calculated using the latitudes and departures. Comparing the two gives a good check of the math.

We can calculate the bearing and distance of the error by inversing between Point 8 and Point 1. We are already familiar with how to perform an inverse using Eq.

A.9 :

$$ \tan A = \frac{Departure}{Latitude} $$

Substituting:

$$ \tan A = \frac{ - 0.086}{ - 0.038} = 2.2632 = 66^\circ\; 09^{\prime } \;41^{\prime{\prime }} $$

As a check, if we look at the coordinates for Point 8 and Point 1 we see that both the north and east coordinates for Point 1 are larger than for Point 8. This means that Point 1 must be northeast of Point 8, so the bearing from Point 8 to Point 1 must be pointing northeast. We also notice that the error of the departure is greater than the error of the latitude so the bearing must be greater than 45° (Fig.

A.29 ).

Fig. A.29 Error of closure close-up

Next we calculate the distance between the points using Eq.

A.10 :

$$ distance = \sqrt {latitude^{2} + departure^{2} } $$

Substituting:

$$ distance = \sqrt {0.038^{2} + 0.086^{2} } = 0.094^{\prime } $$

A.5.2 Ratio of the Error of Closure The error of closure is usually expressed as a ratio of the error to the total distance traversed. The ratio is always expressed with the numerator as 1 so the ratio will be 1 foot in X feet.

$$ Ratio \frac{error\; of\; closure}{total \;distance \;traversed} $$

So, in our example the error is 0.094 ft. and the total length of the traverse is 600.07 ft. so, the ratio of Error is:

$$ Ratio \frac{0.094}{600.07} = 0.000157 $$

Dividing both the numerator and denominator by the error gives the ratio:

$$ \frac{1\,foot}{ 6 , 3 8 3 {\,{\text{ft.}}}} $$

Hint: after dividing the error by the distance traversed, you can use the 1/x (inverse) key on your calculator to do the division. Sometimes the ratio is expressed to the nearest 100 units so in our case the ratio would be 1 foot in 6,400 ft.

A.5.3 Adjusting the Traverse The next step is to adjust the traverse. Because the compass rule is one of the most widely used rule for adjusting traverses, we will use it here. We have already calculated the errors in latitude and departure. We will now adjust each latitude and departure so that our traverse closes perfectly.

The compass rule distributes the error to latitudes and departures in proportion to the length of the line. The following equation shows how this is done:

$$ Latitude\; Correction\; Line\; X1 - X2 = \frac{Latitude \;Error*Distance}{Total \;Traverse \;Length} $$

(A.11)

Using the line from Point 1 to Point 2 as an example:

$$ Latitude\; Correction \;Line \;1 - 2 = \frac{ - 0.038*161.20}{600.07} = - 0.010 $$

The same procedure is used for departures.

$$ Departure \;Correction \;Line\; X1 - X2 = \frac{Departure \;Error*Distance}{Total\; Traverse\; Length} $$

(A.12)

The traverse total length is the cumulative length of all traverse lines. In our example the total traverse length is 600.07′.

The adjusted traverse for our example is shown in Table

A.7 . In order to save space the latitudes and departures have been reduced to a single column each. North and east values are positive and south and west values are negative.

Table A.7 Adjusted traverse

In Table A.7 , the Lat. Correction and Dep. Correction columns show the corrections to be made to each latitude and departure in accordance with Eqs. A.11 and A.12 . The column of corrections should be added to be sure that the sum is equal to the total correction. For example, the total correction for the departures is −0.038. The sum of the latitude correction column is also −0.038.

After all of the corrections have been calculated, they are applied to the latitudes and departures to create balanced latitudes and departures. Using this procedure, the columns for balanced latitude and balanced departure are completed. The balanced latitudes and balanced departure columns must be added together to ensure that the totals are zero. In other words, for the traverse to close properly the sum of the positive latitudes must equal the sum of the negative latitudes. The same applies to the departures.

After all of the balanced latitudes and departures have been calculated, they are applied to the coordinates. The beginning and end coordinates must be the same. In our example coordinates for Points 8 and 9 are the same. We now have adjusted coordinates for our traverse.

The bearings and distances shown in Table

A.7 are no longer correct because the coordinates have changed slightly. We need to calculate new bearings and distances for the traverse lines. This is easily accomplished by inversing between the adjusted coordinates. Our adjusted traverse with the adjusted bearings and distances is shown in Fig.

A.30 .

Fig. A.30 Adjusted traverse bearings and distances