Quadrilateral Traverse Adjustments

A traverse is a survey where you have occupied each station and measured each angle and distance between points. An example of a strong shape often used for studying deformation, such as on a landslide (Baum and others, 1988; Johnson & Baum, 1987) or for from an earthquake (Cruikshank and other, 1996; Johnson and others, 1997), is a braced quadrilateral. This is where all the angles and distances in a quadrilateral are measured repeatedly and averaged for accuracy. The quadrilateral is then measured at some later date and strain calculated from the change in shape of the quadrilateral.

The sum of the interior angles for a quadrilateral should be 360°. Any deviation from that is a measure of error. Also, the triangles that make up the quadrilateral should all obey the laws of sines and cosines - any deviation is a measure of error. Needless to say, either the distance measurements or the angles could be in error. We also have an over-determined system. For example measuring two angles and one side of a triangle is sufficient to construct the whole triangle (although there would be no error control). If we measure all three sides and angles we have much more information than we need, and could possibly construct several different triangles using the data. Probably the best method for solving over-determined systems is the methods of least-squares.

Least Squares Adjustment

An outline of the method of general least squares is given in the following PDF format document (from the G423/523 Course Web pages).

Since we know that the quadrilateral should have no error, and we have convinced ourselves that there is no obvious bad data in out data set, we can adjust the angles and distances so that out shape meets all the geometric criteria. The most complete adjustment methods uses least-squares to adjust angles and distances so that you make the smallest possible changes (Moffitt, and Bouchard, 1992; Wolfe and Ghilani, 1997). One method for adjusting quadrilaterals is given by Smith & Varnes (1987). Copies of FORTRAN programs from Smith & Varnes are available here: Triangle or Quadrilateral. Note that the programs require ISML routines. An Excel workbook is also available for making these adjustments to a quadrilateral.

Coordinate Adjustment

Simpler ways of adjusting traverses are shown in many Geology field methods texts. A sense of error in a traverse can be obtained by calculating coordinates for each point following the path of your traverse. After completing the traverse you should return to the starting coordinates, any difference is a measure of your total error. For a quadrilateral you would perform the following calculations:

Fix one corner of quadrilateral, say the SW corner. This will have coordinates of (0,0,0). Assign the direction from the SW to the SE corners to be "east". Use your data to calculate the coordinate of the SE corner.
From the SE corner, use the angle between lines from the SE to SW corner and SE to NE corner to calculate the coordinates of the NE corner.
From the NE corner, use the angle between lines from the NE to SE corner and NE to NW corner to calculate the coordinates of the NE corner.
From the NW corner, use the angle between lines from the NW to NE corner and NW to SW corner to calculate the coordinates of the SE corner. The difference between this calculated coordinate and (0,0,0) is your error of closure.

This closure error can then be adjusted to zero using either the compass rule or the transit rule. An Excel workbook showing these correction methods is available here.

Compass Rule

In this method the coordinate error is distributed in proportion to the length of traverse lines. The assumption is that the greatest error will come from the longest shots.

Northing adjustment = Length of traverse line to point / Total length of traverse × Northing closure error
Easting adjustment = Length of traverse line to point / Total length of traverse × Easting closure error
Elevation adjustment = Length of traverse line to point / Total length of traverse × Elevation closure error

Transit Rule

In this method, the coordinate error is distributed in proportion to the amount that various coordinates change between points.

Northing adjustment = Change in Northing for traverse line to point / Sum of absolute values of all changes in northing for all traverse lines × Northing closure error
Easting adjustment = Change in Northing for traverse line to point / Sum of absolute values of all changes in northing for all traverse lines × Easting closure error

References

Baum, R. L., Johnson, A. M., and Fleming, R. W., 1988, Measurement of slope deformation using quadrilaterals, United States Geological Survey, Bulletin 1842B, p. 23.

Cruikshank, K. M., Johnson, A. M., Fleming, R. W., and Jones, R., 1996, Winetka deformation zone. Surface expression of coactive slip on a blind fault during the Northridge earthquake sequence: Denver, CO, United States Geological Survey Open-File Report 96-698, p. 70.

Johnson, A. M., and Baum, R. L., 1987, BASIC programs for computing displacement, strains, and tilts from quadrilateral measurements: United States Geological Survey Open-File Report 87-343, Reston, Virginia, p. 19.

Johnson, A. M., Fleming, R. W., and Messerich, J. A., 1997, Growth of a tectonic ridge, United States Geological Survey Open-File Report 97-153, p. 94.

Moffitt, F. H., and Bouchard, H., 1992, Surveying: New York, New York, Harper Collins, 848 p.

Smith & Varnes (1987), Least-squares adjustment of triangles and quadrilaterals in which all angles and distances are observed: Surveying and Mapping, v. 47, no. 2, p. 125-142.

Wolf, P. R., and Ghilani, C. D., 1997, Adjustment computations: Statistics and least squares in surveying and GIS, Wiley Series in Surveying and Boundary Control: New York, John Wiley & Sons, 564 p.