Least Squares instead of Compass Rule?

[PLS 8176]

First, my understanding of a least squares adjustment is that it is a 'weighted means' adjustment in that it will more heavily apply the adjustment at a point where an error may have more likely occured. In contrast, a compass rule adjustment evenly distributes the error of closure throughout the entire traverse, regardless of how the traverse was modeled (short backsight long foresight gets the same proportional adjustment that a long backsight short foresight would get). It seems obvious that the least squares adjustment would be preferred.

Recently, I overheard an opinion that least squares can only be used when you have taken measurements between all of your traverse points, thus creating a 'triangulation' model for your least squares adjustment. In this case if you traverse around a block littered with houses, a least squares adjustment could not be applied. I have also read that least squares can be applied to any traverse, whether you have tied all of your traverse points together or not.

I want to apply the least squares adjustment to any traverse (using software and looking at Star*Net) that may need adjusting, but have a concern as to which thought is correct. I tend to lean more towards what I read than what I heard, lol.

Can anyone provide some input with some documentation (web link or good book title will work) to support their thoughts? If your response is going to be "Compass Rule is good enough", please explain.

Thanks!!

[dmi]

For a least squares adjustment to work properly you need to collect more than just fs-bs measurements. You need at least three measurement to each point you occupy (more is better). You can use starnet(the only program I have experience with) to run an adjustment on a closed loop traverse. There would not be much difference in the results than a compass rule adjustment.

[RAM]

Least squares will take in account reduntant measurement. Therefore each station needs more than one measurement. Things can be by GPS observation together with conv. traverse and/or sideshots to adjoining points. Three measurements is prefered but can be done with 2. Unless there are extra measurements, a least squares adj. and a compass adj, will yield nearly the same position for a station. Least squares has many practical uses, ie Allen Frank uses it for GLO corner search (See articles in POB). It is another tool in the tool box to used for the correct task.

[Greg Sebourn]

StarNet works well for several applications. I use it to recreate a record traverse from old maps and notes as well as to create search coordinates. You can use it for leveling and you can use it where you only have one distance and angle so long as you understand how it weights the amount of adjustment. As cool of a tool as it is, the user must be smarter than the software to use it correctly... ;^)

[RobCT]

Well that cuts me out Greg...

Rob

[Debbie Naves]

Greg, are you using StarNet to adjust boundaries that may have been created on a "record data" map? If so, do you think it's a better solution than a compass or other adjustment? Why? For this question, my example has a six-course line with each end fixed in place by other establishment methods. Thanks for your input.

[Greg Sebourn]

Debbie,
It depends...

Sometimes it works fine on record boundaries but not always. If I am working on creating a record boundary for a description and there is no field data, the compass rule usually works just fine.

If I am going to be retracing someone's field data, then I almost exclusively use StarNet.

Something to keep in mind is that StarNet doesn't adjust or correct blunders- unless you force it to adjust it. It will, however, help you identify the locations of blunders so that you can determine what the blunder actually entails and make the correction.

[Debbie Naves]

The adjusted framework is already set in this case. I just need to add the six-course line between two points of control on the framework. This line was created by an old parcel map, without benefit of a survey, so an adjustment is needed. I suspect it was laid out 35+ yrs ago on a aerial topo, and scaled to get the line data for that old parcel map. Now it is being established on the ground from the old map info.

I've thought the best establishment would be a compass adjustment, and am curious about what others think of the least squares method for this use. Thanks again.

[Greg Sebourn]

It sounds more like a grant boundary adjustment is in order, wherein the link traverse is scaled and rotated to join accepted control. I might run it through StarNet just to see what the analysis looks like. It might work just fine or there might be a problem with a course or angle.

[Debbie Naves]

A good thought for sure, thanks.

[goodgps]

Oh Debbie . . .
did you say "good" thought ??

Well I'm not very smart, but my take on least squares is that it really works great as a tool to analyize strength of structure as well as error detection, for a geometric figure.
Compass rule just takes the ending error and distributes it proportioantly throughout the traverse assuming the error of each observation to be consistent.

Compass rule seems to be more "ground related" whereas least squares can be set up spacially.
It all fun math to play with, just don't set up over a swing tie by misteak. ;-}

[Debbie Naves]

Yes, I said good thought... as I've stated - in this case... all field work is done and adjusted, so at this point there are no "observations" to deal with. The establishment is purely a six-course line that was created on an old parcel map, but never monumented. This survey will monument it for the very first time. The two ends of the six-course line have been established from the adjusted field work already. So how can you apply a "weighted mean" to such an establishment?

[dmi]

The why I see it(but I am always willing to learn more) the question that needs to be addressed is how one would establish error estimates for the six record legs between the held end points. It is easier to think about if you start from a known value and traverse to a know value. It is much easier to come up with accurate assumptions about one's theoritical error estimates (based upon, equipement, proceedures etc)and compare them to those that actually result from the survey work performed.But how it it that we do this with no basis to built accurate assumptions from. the error estimate must be valid otherwise the adjustment is not worth very much. Grasping at straws one could start with 20" per angle and 01' per 100'....and then trial and error... but probably not the right tool i n this case. Each type of adjustment is going to put some type of distortion into the mix. Perhaps it would be best to look at what happens using an irregular boundary adjustment, grant boundary ,etc depending upon the magintude and direction of error.

[Greg Sebourn]

Debbie,
As part of understanding the record data, I would use your end points of where the link traverse connects with the rest of the survey as fixed in StarNet. Then, using the record data and the historical accuracy/precision of the instruments used when the record data was compiled, I would run the link through StarNet and see how well it joins your already-determined lines. Once this is complete, StarNet will tell you just how much it had to adjust and where in order to get it to work. I'm not saying that I would run with what it tells me but I would give it consideration.

I would still tend to lean towards a Grant Boundary Adjustment only because I can confidently say in court that it is the method described by Brown, Wattles, and several other boundary text books.

[dmi]

Greg,
Okay, I see what your plan is for your apriori estimates,but the aposteriori assumptions are weak in the scenario as described. We have only on set of measurement data to use for comparision,"record data from the map". In my humble opinion, insufficient data would yield a partically useless adjustment.
The difference is not likely to differ from compass adjustment so why bother? Now if one acutally ran the line from a known point and closed upon a known point (controling for boundary establishment purposes),with sufficeint data collected to support a least squares adjustment, then you would be on to something.

There is a legal issue to consider and that would be that all the record lines are considered as having the same weight and have to be treated the same in the retracement,unless an error is dicovered and the error would be placed where it occured. There are other factors that would come into play with respect to the subject alignment. But mainly for our purpose, lets exclude all the what ifs and just focus on the least squares issues with its pros and cons.

So absent evidence that there was a particular error, least squares is going to put more error into some legs of the traverse than others and then we would not be treating all the lines equally(or proportionally) and that might make using least squares an indefensible solution.

Just some of my thoughts

[Steve Martin]

I have heard of a court case in San Diego where a Surveyor of a rather expensive lot was called into court to testify why his retracement of a boundary was 1' different from previous surveys. All he could say was that his CAD guy put it into a Least Squares adjustment and he really could not explain what the Least Squares adjustment did or why the line was different. This is all just hearsay and I do not know the citation to the court case, but it illustrates the difficulty defending a least squares adjustment procedure.

Personally I have used a least squares adjustment successfully on several projects beginning about 15 years ago. The particular application was retracing a centerline of old roads. In order to make curves tangent and avoid the grant boundary adjustment issue of holding some angles record, but at the point where each segment begins and ends having a big non-tangency, I entered record data for long segments of roadway into an adjustment and came up with a better retracement in my opinion, based upon the particular facts of these retracements. However, this is not something to let any CAD guy loose on. A small change in the weighting scheme and change your answers significantly. An even 1 foot bust in the chainage used to create the record map, if not identified and corrected for could smear the least squares retracement to a place where the road never was.

[Greg Sebourn]

Dane,
Well said. I think we are on the same page as far as what our end solution would be, Grant Boundary Adjustment where the error distribution is proportional. I would use, and often do, StarNet to locate potential errors in record data once you have located evidence of the traverse line's most likely location. It can be problematic and it does not replace common sense but it can expedite where a problem might be residing. Once the error is located, you would have to determine the best way to deal with it.

In this specific case, I still wouldn't be too likely to run with the StarNet calculated position, especially because I couldn't testify as to how it works…well I could a little but I wouldn't feel comfortable with doing it.

Steve,
I once was a "CAD guy", metaphorically speaking. Indeed a little blunder or ignorance on the StarNet user's part can lead to a completely bogus location for a line or point. I have used StarNet to retrace old alignments that I have located some evidence to tie into my survey. This allows me to see if there was a bad deflection angle, transposed numbers for distance, or other errors. The software won't come right out and tell

[Debbie Naves]

Greg, Dane and All - This is a fabulous conversation, exactly what I hoped for. I just do not believe that the least squares solution is applicable to this case. I originally said that a compass rule adjustment applies here for exactly the point that Dane last stated: the record lines must be considered as having the same weight and must be treated the same in the retracement. I have not changed my mind about a compass adjustment, but a grant boundary adjustment may deserve more consideration. Bottom line: A defendable solution is needed, as supported by the notables mentioned.

[McGee]

Retracement is the underlying premise of this discussion. The intent of a retracement is to find oneself in the place and location of the surveyor we are retracing (to follow in the footsteps). Likewise the intent of surveying a boundary as described in a deed is to re-establish the location of the lines that represents the intent of the grantor and grantee. If the intent of a deed is to follow a survey, then the Grant Boundary Solution is usually the appropriate solution lacking extrinsic evidence.

To retrace a survey, one must calibrate their measurements for orientation and scale to the original survey. The process for a Grant Boundary Solution (described in the 1974 Manual of Instructions for the Survey of Public Lands, Section 5-44) is first, to rotate the record traverse to match the basis of bearings of the survey to be retraced by holding the monuments at the two end points, and then proportion the record distances to fit the found monuments.

The reason for proportioning is to reproduce the length of the original surveyors tape so as to follow their footsteps on the ground. It is good to know that you know your distance measuring instrument is accurate and precise compared to the Bureau of Standards; however, many of the old surveys did not have the benefit of knowing the precise standard. Therefore, we must respect their "standard" if we are to find ourselves in the same place as they were. So, by recovering the basis of bearing and the length of their tape or chain, there is a higher probability we will find ourselves in their footsteps, blunders notwithstanding. If they never actually surveyed the line between the monuments then with this solution there is the higher probability we will find ourselves where they would have been, had they ran the line.


Least Squares Adjustment is method for dealing with residuals in a system represented by a math model. Residuals are the differences between the measured values and the adjusted values. Residuals only occur when we have redundancy. Redundancy is when you have more than one independent observation of a measurement. Redundancy can be more measurements than necessary to define the system, i.e. the addition of the diagonals of a quadrilateral. Given sufficient redundancy in a system, least squares will put the adjustments necessary to make the system fit the math model, in the place where they most likely occurred. How does this work. It is a bit of calculus. The solution is in the adjustments (residuals) to the measurements. The least squares solution is unique, in that for a given set of measurements, the sum of the residuals squared is a minimum and no other solution will satisfy this condition. For example, a traverse around a rectangular lot represents a minimally determined system if only three of the courses are measured to determine the coordinates of the four corners. Applying least squares changes nothing. If the fourth course is measured then there is some redundancy because of the one extra bearing and distance. Least squares is still not very useful for distributing errors or finding a blunder with this minimal redundancy and the results of an adjustment may not meaningful. However, measure the two diagonals and add them to the solution and least squares becomes an amazing tool in the surveyors arsenal. Why, because now there are four pathways for which 12 measurements can be simultaneously compared and analyzed. This is called an over determined system and least squares will put errors and blunder where they belong.

In summary, the application of least squares to field surveys with sufficient redundancy is very useful tool that helps the surveyor isolate blunders , distribute random errors where the most likely belong and assess the precision of their surveys. Applying least squares to boundary retracement would be the exception and one would have to have a good explanation for the judge as to why it is a better method than applying linear solutions to find the footsteps.