Understanding the least-squares adjustment

This topic applies to ArcEditor and ArcInfo only.

A single observation (bearing and distance) from an existing survey point can be used to compute the coordinates for a new survey point. However, relying on a single observation is risky, since there is no way to tell whether the measurement is correct. A second measurement from the same or another existing survey point will confirm, or check, the coordinates defined by the first measurement. Generally, the more measurements fixing the coordinates of a survey point, the more reliable the coordinates. These additional measurements are called redundant measurements.

Weighted average

All measurements contain some degree of error. Therefore, each measurement will compute slightly different coordinates for the same survey point. For practical reasons, there should be one coordinate location for a survey point. A single, best estimate coordinate can be derived by computing a weighted average of the additional or redundant measurements, with each weight defined by the measurement accuracy.

Computing a weighted average

Although the weighted average approach works for a single point, it is not sufficient to compute the coordinates for multiple points in a network such as the parcel fabric. A more advanced method is needed to account for the numerous possible measurement paths between the points. The techniques and algorithms in a least-squares adjustment provide the most rigorous and widely accepted solution for processing a network of measurements and points.

Multiple points in a network

A least-squares adjustment is a mathematical procedure based on the theory of probability that derives the statistically most likely coordinate location of points defined by multiple measurements in a network. In mathematical terms, a least-squares adjustment defines a best-fit solution for weighted measurements by finding a minimum for the sum of the squares of the measurement residuals. A measurement residual is the amount needed to correct a measurement for it to fit into the best-fit solution found by the least-squares adjustment.

Using a least-squares adjustment to adjust a parcel fabric

In the parcel fabric, the least-squares adjustment uses all the measurement data together with control points to estimate the most probable coordinate for every point in the network. This description of the least-squares adjustment can be more easily understood by considering one traverse path between two control points in the fabric network. Fabric points P1 and P5 should be coincident with their corresponding control points CP1 and CP2. The least-squares adjustment adjusts the misclose error between P1 and CP1 as well as P5 and CP2 through the remaining points P2, P3, and P4 such that P1 and P5 become coincident with their control points. The coordinates of P2, P3, and P4 are adjusted into the best-fit solution and the lines are recalculated from the adjusted points. In the parcel fabric, accuracy on parcel lines acts as a weighting system in the least-squares adjustment. Lines with higher weights will adjust less than lines with lower weights. The higher the accuracy, the higher the weight on a parcel line. In the graphic below, the line between P2 and P3 has a high accuracy and thus a high weight. In the least-squares adjustment, line P2–P3 received proportionally less of an adjustment than the other lines in the traverse path.

Least-squares adjustment with control

The residual differences between the original lines and the lines computed from the adjusted coordinates reveal how well the parcel lines fit among themselves and with the control points. A large residual suggests a problem with the parcel line itself or nearby parcel lines, since the original value required a significant change to fit into the best-fit solution.