Q. What is least squares? - A.O. St. Louis, Mo.

A. Bohlke: Least squares refers to a mathematical procedure that distributes the total error among observed measurements. Each measurement may receive an equal amount of the total error. Alternatively, the measurement may be weighted thereby receiving some portion of the error that relates to the quality of the measurement. Most GPS users rely on a least squares adjustment for distributing the error in their measurements throughout a network of observations.

Dietsch: Least squares is a mathematical procedure that utilizes redundant observations to produce a "Most Probable Value" for some unknown quantity or quantities by "minimizing the sum of the squares of the residuals." Least squares has many applications in math, science and engineering. In surveying and mapping, least squares is normally used to perform network or traverse adjustments or to estimate the transformation parameters that "Best Fit" a set of observations. Least squares is a flexible tool that can be adapted to any math model and which allows the user to evaluate the mathematical and statistical integrity of the measurements in question.

Hurley: The ultimate aim when using least squares is to adjust a set of observations in such a way that the sum of the squares of the residuals get minimized. This is achieved by taking a set of observations and creating a model, applying a mathematical algorithm to that model and then using statistics to evaluate the adjusted data. The end result will yield the most probable value, a value based on the set of observations that will tend toward the truth. Surveyors use least squares methods to verify the integrity of their measurements, and to "fit" their observations onto known positions.

Many different adjustment techniques exist today. Most of them are unique to certain applications. Least squares is an adjustment technique that is more general and a systematic procedure that can be used for all situations. Remember it is not an adjustment technique unique to GPS. It is commonly used in geodesy, photogrammetry, surveying and many other areas. Unfortunately, the details and mathematics of least squares are not easily explained in a couple of paragraphs.

Townsend: Least squares is a criteria often used to indicate convergence of a mathematical problem where there are more observations than unknown parameters. Hence, the solution is over determined causing an imperfect fit of observations to parameters. The least squares criteria dictates that the solution has converged when the sum of the squares of the observation residuals is minimized.

van Diggelen/Martin: It is common knowledge to all surveyors that every measurement, whether collected using conventional surveying equipment or GPS equipment, contains error. These errors can be classified as random errors (inescapable errors due to the precision of the equipment being used) and blunders (mistakes in the measurement). It is extremely important that blunders are detected and removed from measured data, otherwise, the results will be unreliable.

Performing a least squares adjustment has been proven as the best method to detect blunders in survey measurements. Blunder detection is accomplished through statistical analysis of the measured data. These statistical tools help point to data suspected to contain a blunder. Once problem data is discovered, it can be removed from the data set.

A second advantage of a least squares adjustment is the analysis of precision of the adjusted measurement data. Additional statistical tools present in most least squares adjustment software will enable the user to determine the level of precision attained in the survey.

Although detection of blunders and survey precision can be performed through other means, using least squares will give users the best tools to ensure that final adjusted measurements are devoid of blunders, instilling confidence in the reliability and precision of their surveys.

An alternative answer to the question may be: Least squares is the fewest number of engineers required to change a light bulb.

About the participants:

John C. Bohlke serves as GPS technical product manager for Sokkia Corp. in Overland park, Kan. Chris Dietsch is a product test engineer for Trimble Navigation Ltd. in Sunnyvale, Calif. Andrew Hurley is a GPS product specialist at Leica Inc. in Englewood, Colo. Bill Martin is a marketing manager, Survey, at Ashtech Inc. in Sunnyvale, Calif. Bryan Townsend is a GPS specialist with NovAtel Communications Ltd. in Alberta Canada. Dr. Frank van Diggelen is a marketing manager, OEM and Navigation at Ashtech Inc. in Sunnyvale, Calif.