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Getting started with Least Squares
Kent McMillan replied 8 years, 5 months ago 28 Members · 50 Replies
The engineering company I work for has very little respect for surveying, surveyors, or accurate data in general. It is a level of bizarre that can only exist in a company owned by someone who was born with enough money to never have to worry about security. My choice is between no adjustment and Compass Rule. I choose the former as I do not like the idea of a greater share of the error being massaged into my longer EDM shots. If I used a chain, I would use the CR.
I would debate that you “dump all the cumulative error on the last point” – this is true if you have systematic biases that have not been accounted for, but random error is not necessarily cumulative, it is compensating as well. My understanding of least squares is that is an excellent means of distributing the error in a system of measurements once systematic biases and blunders have been accounted for. Least squares is most effective when you have over determined systems – in other words more observations than unknowns. This means multiple independent redundant observations. It seems to me that in many instances compass rule adjustment is more appropriate for traverse data.
By this do you mean that since (say in a three point traverse), 3-1-2 is initially measured by back sighting to 3 from 1, then 1-2-3 and 2-3-1 (from-at-to) are then measured, it forms a continuous “chain” of measurements, and therefore, there is no “last point”? I.e. Three angles; Three distances?
Totalsurv, post: 339400, member: 8202 wrote: I am looking to get started using least squares and was wondering about best practice for field procedures etc. Could somebody point me in the right direction for documentation, web links or even books? I would be hoping to use a combination of total station and gps data.
I will be using Survnet in Carlson Survey 2011 Intellicad version. Thanks.
Using Least Squares is really easy, I would say much easier than Excel which drives me crazy sometimes.
The programs aren’t very difficult.
Getting into the bones of the actual math, wellll that’s a different issue.If you have used and calculated compass rule then you should understand the basics and be flying along with it quickly.
The good thing is it allows you lots of options and batch adjustments which are impossible with compass rule.Totalsurv,
This is probably more than you are for but the following is my “journey through least squares adjustment”:
In my opinion Least Squares adjustments (LSA) are a fundamental tool that is so ubiquitous today that it should be in every surveyorÛªs toolkit. Having said that, I also know that least squares adjustments can be properly used without having a fundamental knowledge of the mathematics. I used LSA for several years in network adjustments without knowing the mathematics and, looking back on it, realize I did not screw anything up with my lack of knowledge of the math. Starnet and similar programs make it possible to do this because they have very good manuals explaining how to judiciously use LSA. Mike PotterfieldÛªs old Trimble network adjustment manual is also a great one to use. I think it can be found online free.
I was kinda obsessed by learning LSA and did ultimately work through the mathematical foundations. It took me a good five years in my ÛÏspareÛ time with a full time career, job, kids, PTA meetings etc. Back then about the only book available was ÛÏAnalysis and Adjustment of Survey MeasurementsÛ by Mikhail and Gracie as well as another by Mikhail. I worked through every problem twice and it took me five years. Several times along the way I had to take time out to learn the necessary math before I could go any further. First I needed to leave the book to learn probability. This took about a year as I worked through several college textbooks. Then I took another timeout to take linear algebra at night at a community college. Finally I had to jump out for a while to brush up and add to my knowledge of calculus and later statistics. After each of these breaks I went back to Mikhail & Gracie and worked through a few more chapters. All the time I was doing this I was using LSA nearly every day in my work gradually understanding more and more about it as I went along. Eventually I finished the book working through each problem twice. Later I took about another year to go through a totally different least squares course sponsored by Alfred Leick over the internet. It was very good. I also went through the online GPS positioning series of courses from Leick and they were great. I realize most people arenÛªt as obsessed and donÛªt need to be to competently use LSA, but if you want to, something like the route I took is necessary. There really arenÛªt any shortcuts. You have to have a good solid foundation in calculus (including multivariable calculus) probability and statistics (and know the difference) and linear algebra. Then you can put it all together and work through a good textbook on LSA. The one I mention is good as well as a newer one by Paul R. Wolf ÛÏAdjustment Computations: Statistics and Least Squares in Surveying and GISÛ. It takes a while but is worth it. By the way LSA is also foundational to all the various types of range differencing in all GNSS measurements. So itÛªs not just about network adjustments but really covers a lot more.
Cliff
Paul in PA, post: 339455, member: 236 wrote: Least Squares Adjustment does not make observations better, but it does make them different.
I disagree.
Least squares links various measurements together, instead of a reported bearing or distance being based on a single measurement or the compilation of several measurements, it is based on the combined contribution of all measurements in the network. Over and over, I’ve found that my results are better after LSA than before for this reason.
This has also been borne out in the old long range L1 only GPS experiments Dad did. Each individual 100 mile L1 vector was fairly poor on its own, but when combined with other similarly weak vectors in a well structured network, the combination produced remarkably accurate (much, much better) results.
I often have a network of control points. I have multiple measurements into a control point from different directions. Least squares allows you to use all of your data to compute the best coordinate for that control point.
You assign standards errors to your measurements which is the uncertainty of a given measurement. If I turn an angle several times (with new setups) I would expect to get some variation between measurements. Least Squares will change all the measurements (hopefully) a small amount so that they all match. Hopefully the small amount is less than the standard error then you pass chi square (if I understand that correctly). That is reasonable, adjusting a measurement by an amount which is less than the expected variation if I made that same measurement several times.
LS is an analytical method to distribute error based on actual observations. It is a very useful tool and offers a great deal of flexibility. We just completed a survey that is a little less than 19,000 acres in six contiguous tracts. The field work was performed by a combination of 5 crews from two different companies. We had crews running clockwise, counter clockwise and making interior ties. All the traverses and GPS observations were networked and exceeded the ALTA Positional Standard.
Yes it could have been done without LS. If you want to do something different, that is your right. As for me and my operation, we are going to use LS.
- There is no better way to combine GPS Vectors, terrestrial data, and levelling data than least squares.
- There is no better way to weigh and utilize redundant data than least squares.
- Adjustments of any sort are for dealing with small random errors, not for papering over blunders.
- Least squares is a great way to detect and correct blunders.
- For most day to day jobs, if your data is correct, running it through an LS adjustment takes a few minutes.
- If your data isn’t correct the time spent in LS detecting and correcting blunders is time well spent.
- You don’t need to know calculus or matrix algebra to run a least squares program. It is enough that the programmer knew it.
- You do need some conceptual knowledge of statistics to analyze LS results.
Paul in PA said: Ô
Least Squares Adjustment does not make observations better, but it does make them different.I partially disagree. I think any adjustment only “improves” measurements to the extent that the stochastic and mathematical model of the adjustment reflect where the errors are coming from. Another way of putting that is to say that the geometry of the network has to be solid (well-conditioned), and the a priori weighing scheme has to be realistic. In other words, the internal redundancy gives one more overall “measurement” that is applied on top of the internal redundancy already achieved by doing things like repeating angles and distances etc. However there is usually another factor involved that often degrades the actual observations. This is the error already inherent in the positional values of the control points that the measurements are tied to. Least Squares can only spread this error into the actual observations of the survey, and this can seriously degrade original observations especially with contemporary high accuracy equipment that most surveyors use today. It is not unusual to have internal survey accuracies that are better than the accuracy of the control points that are tied into. This was especially noticeable in the past when tying GPS surveys to NAD27 values, but still occurs in other types of surveys today.
For those of you that do not use LS, how do you know (I mean really know) if you are meeting RPA?
Are you performing an independent survey of higher order on your own survey to determine if your survey meets the standards?
Paul in PA, post: 339455, member: 236 wrote: I thought I had made it clear, I do not adjust my everyday field work. If it falls within my expectation of erros, I accept it for what it is. Honest measurements with error.
Quote me on his, “Least Squares Adjustment does not make observations better, but it does make them different.
The measurements are not closer to the truth, mathematical perfection is not what surveying is about.”Least squares is much more useful in checking your product than it is in being your product.
What do I prove by adjusting my field angle by 1″ or a distance by 0.01′, when I use a 5″ gun and a bipod on a found pipe that is deformed or a rebar with a not so flat top?
Paul in PA
Paul, least squares used correctly does make your observations better. As you go around a traverse your true position gets further and further from your measured position. Provided that there is no blunder, When you add in redundancy and process it with least squares you get an answer that will be closer to the true value. If you do have a blunder, least squares is great way to locate it.
If you don’t adjust your field measurments how do you close your maps?
“…I accept it for what it is. Honest measurements with error…”
Without least squares you use only the measurements and ignore the error. With it, you account for the the errors. I ask, which is more honest?JBrinkworth, post: 339622, member: 6179 wrote: For those of you that do not use LS, how do you know (I mean really know) if you are meeting RPA?
Are you performing an independent survey of higher order on your own survey to determine if your survey meets the standards?
I notice that no one has answered this question….
..and if you want to generate true state plane coordinates, no?
There are 2 reasons for not using least squares:
1. The cost of the program.
2. StubbornessIf someone claims some other reason they are in denial.
3. Unwilling to learn something new.
Agree, the issue (assuming no blunders) is not the small (hopefully) errors in each line, it is the propagation of those errors throughout the traverse. If ìÄAZ n = ö?ìÄAZ 12 + ìÄë± 22 + ìÄë± 32 +…. ìÄë± n2, then ignoring the propagation through a traverse is a folly in my opinion.
RCliffWilkie, post: 339612, member: 10285 wrote: Paul in PA said: Ô
Least Squares Adjustment does not make observations better, but it does make them different.I partially disagree. I think any adjustment only “improves” measurements to the extent that the stochastic and mathematical model of the adjustment reflect where the errors are coming from. Another way of putting that is to say that the geometry of the network has to be solid (well-conditioned), and the a priori weighing scheme has to be realistic. In other words, the internal redundancy gives one more overall “measurement” that is applied on top of the internal redundancy already achieved by doing things like repeating angles and distances etc. However there is usually another factor involved that often degrades the actual observations. This is the error already inherent in the positional values of the control points that the measurements are tied to. Least Squares can only spread this error into the actual observations of the survey, and this can seriously degrade original observations especially with contemporary high accuracy equipment that most surveyors use today. It is not unusual to have internal survey accuracies that are better than the accuracy of the control points that are tied into. This was especially noticeable in the past when tying GPS surveys to NAD27 values, but still occurs in other types of surveys today.
Interesting you should mention this because one of the things that I like about Move3 (LSA software) is the report of the quality of your controls upon performing a “Free” (unconstrained or minimally constrained adjustment. You really get to see the slop you’re inheriting from some of these agencies.
JBrinkworth, post: 339622, member: 6179 wrote: For those of you that do not use LS, how do you know (I mean really know) if you are meeting RPA?
Are you performing an independent survey of higher order on your own survey to determine if your survey meets the standards?
“RPA”? I assume you mean Relative Positional Accuracy, which cannot be done as you infer.
Least Squares Adjustment however can give an “estimate” of Relative Positional Precision, but as per Minimum Standard Detail Requirements for ALTA/ACSM Land Title Surveys, 3.E.iii.
“Relative Positional Precision is a measure of how precisely the surveyor is able to monument and report those positions; it is not a substitute for the application of proper boundary law principles. A boundary corner or line may have a small Relative Positional Precision because the survey measurements were precise, yet still be in the wrong position (i.e. inaccurate) if it was established or retraced using faulty or improper application of boundary law principles.”
Paul in PA
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