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On Least Squares
I normally enjoy reading "Point to Point" by Joel Leininger and agree with what he has to say, or at least admit that he has a good argument. However, his article in the September issue ("Boundaries by the Numbers"), betrays a truly fundamental misunderstanding of the purpose of least squares.
He correctly states, "Most leastsquares analysis software allows the user to `weight’ the various factors to be considered in the solution, so that factors which are more certain receive less adjustment, and vice versa." Then he goes completely off track and adds, "Thus, one could, for instance, weight some of the monuments to be `fixed’ and force all adjustment into the other elements." One could, yes, but this is a complete misuse of least squares!
Least squares is to the compass rule as a total station is to a theodolite: it’s newer, fancier, more versatile, and generally more precise, and has nothing whatsoever to do with anything but measurements.
Leininger errs by thinking that weights should be assigned to monuments. This is utterly wrong. "Fixing" monuments in least squares is the same as "fixing" the measurements between them and what competent surveyor is so arrogant as to assume he or she knows exactly the true distance and bearing between any two points? Compare and contrast to a GPS survey: GPS requires control points to which the rest of the survey is adjusted. These control points are "fixed" just as Leininger suggests monuments should be. However, they are not fixed due to a long history of use, or because they best fit the record, or because a landowner can point to them and say "There’s my corner!" They are fixed because a prior surveyor carefully and precisely measured their position. The accuracy of the published coordinates is higher than that of the subsequent survey. In most resurveys, the new survey is more accurate, and should not be distorted to fit the old.
The proper use of weighting is to inform the software which measurements are more accurate than others. GPS should be weighted higher than conventional traverse work, which should be weighted higher than anything done with compass and tape. Weighting should be determined as soon as a user has a new least squares program and prior to any survey. It may require a few surveys to get the feel of new software, and as some surveys will always be noticeably better or worse than average, it will be necessary to occasionally tweak the estimates a bit to get acceptable closure, but changes should be slight and, in general, weighting should be independent of the findings of any particular survey.
So then "Geometric checks ensure the integrity of the measurements and provide the fuel on which adjustment depends." This is exactly what least squares does: adjusts and checks. And reports back its estimate of the quality of the adjustment based on the checks. And that’s all. After a least squares adjustment is finished, a surveyor may use the COGO routine to check his or her measurements against previous records or determine how far a found monument is from the "true" position. This is part of ranking the evidence which Leininger mentioned, where the measurements may help support or undermine the decision to accept a given monument as the corner. But when it comes to "weighing" a monument, least squares should be as involved as my bathroom scale.
—Edith Forkner, LS (and, for what it’s worth, BS, mathematics)
More On Least Squares
I would like to thank Joel Leininger for the first factual document written to discourage the use of least squares adjustment for boundary retracement. It is a program invented by engineers who have little or no understanding of surveying. It is now used to force-fit a mathematical solution in lieu of actual record-supported retracement based on evidence and seniority. Please continue your efforts to educate the new generation of younger, less experienced, computer savvy, soon-to-be-greenhorns, like their P.E. buddies, known for overuse of available software in lieu of true boundary retracement.
—Paul Ruzenski, LS Huntington, NY
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