I give up. Stoopid maffs.

Please post the question.

Not grasping the true problem as this seems to be a fairly simple trigger-nom-uh-tree problem.

Using an American football expression, tell him to "PUNT".

What are we to compute? Maybe best estimates of the positions of two points and their standard errors? How many measurements are given?

"A problem well stated is a problem half solved."~Charles Kettering

Thanks for the replies guys. I'll paraphrase the question and get back to you. I'm trying to not cross any ethical lines here.

The problem has to be linearized by using derivatives of angles and distances with respect to coordinates. Then you form the matrix equations.  Here are some notes that could help.

https://www.dropbox.com/s/esgmuxk2s0mqio4/LSQ_notes.doc?dl=0

The question itself:

"The following network was observed with both angles and distances. There are two fixed stations P and S.

The following distances have been observed with a standard deviation of +/-0.015m +/- 4ppm

Use the least squares method to determine the adjusted coordinates of Q and R. Note: you will need to iterate the solution using a convergence limit of 0.001."

So, that's the actual question itself, with the actual data he was given. As I said, I know how to do this when directions are given, but not angles.

To be clear, I'm trying to tread carefully here on someone else's behalf. So I'd rather not cross any lines. I'm only looking for the variation of coordinates formula that deals with angles and two unknown points.

Thank guys.

Mick.

@bill93 Thanks Bill. That doesn't address the problem at hand, but it's an interesting read. Anything that suggests we fudge figures to make it work is an interesting read!

Did that document come from Microsurvey?

It's a braced quad question, with two fixed points and two free points. It requires the variation of coordinates method to solve it. My problem is, the question itself gives angles + distances, whereas I learned how to solve it with directions + distances.

My guess would be the professor is making the students work through several different ideas in one problem.  If there are two fixed points with the coordinates listed, as you show in your follow-up post, then there is also a fixed azimuth between them.  Would that not then mean there are azimuths from the fixed line PS (by adding observed angles) such that you would have directions (not just raw angles).

https://www.researchgate.net/publication/254250473_A_review_of_least_squares_theory_applied_to_traverse_adjustment

that document

I put that document together by reading stuff from various sources. I pretty much copied the procedures for getting derivatives for distances, changing to my own notation, and then developed the others.

That was my working paper as I wrote my own general 2-D LS program with some nice features like angles between lines that you don't know the vertex point (including parallels) and specified differences between lengths that are yet undetermined. Once you see how the basics work, such add-ons are easy. What wasn't easy was an attempt at graphics, and I've got a bug (probably memory leak) that crashes it sometimes.

@jon-payne I went down that road first. Finding a way to convert the angles to directions was my first choice, because I know the directions formula.

This would require starting with the azimuth on the P-S and swinging all the angular readings onto it. But the first swinging a second angle onto a first angle means swinging it onto an angle that has a SD that hasn't been solved yet. For example, if you take the azimuth P-S, then calculate the direction P-R by adding the negative angle to it, P-R ends up on the P-S azimuth. Of course. But the direction P-R still has the 6.5" SD attached to it that hasn't been dealt with yet.

If you then decide to swing line P-Q onto line P-R, you are adding a second angular SD (4.5") to a first angular SD (6.5") that hasn't been solved yet. That's fine too, because you can propagate those SDs.

That's all fine, and can (through the process of creating one giant mess) be figured out eventually. Rinse, lather, repeat. Then you end up using this equation:

But the professor doesn't want it done with that equation. He wants this:

Now, with the direction equation, or an angular equation that I convert to a direction equation, you work with a start point and an end point of each line. Then, depending on whether each start or end is fixed or free, you slot those numbers into your A matrix accordingly.

But when working with angles directly, you don't have a start point and end point. You have a left line and a right line. That's the part that is causing the problem, because the formula above is for resections, where your start point in unknown and you are relating your reading to two know points.

Similarly, that equation could be flipped to work for the sections of the quad where you dealing with intersections instead of resections. But what do you when, for example, you are at Q and reading to P, S and R? You are reading to known, known and unknown.

That's where I am stuck.

But the direction P-R still has the 6.5" SD attached to it that hasn't been dealt with yet.

It has been too long since I have done one of these, but shouldn't the SDs be worked in as part of a weight matrix.

The following distances have been observed with a standard deviation of +/-0.015m +/- 4ppm

I think this would also similarly need to be taken into account within a weight matrix.

"but shouldn't the SDs be worked in as part of a weight matrix."

Yes they should, and do. But does the matrix know to propagate as the angles accumulate?

Regardless, it doesn't solve the issue of the requirement to work with angles rather than directions.

I’m just a simple country land surveyor, not a mathematician, Jim!

But the first swinging a second angle onto a first angle means swinging it onto an angle that has a SD that hasn't been solved yet. For example, if you take the azimuth P-S, then calculate the direction P-R by adding the negative angle to it, P-R ends up on the P-S azimuth. Of course. But the direction P-R still has the 6.5" SD attached to it that hasn't been dealt with yet.

Perhaps I am missing something, but if angles are converted to directions, the SD for each direction is independent of which direction everything is referenced to.

Angular SD is just the sum of two direction SDs, the square root of the sum of the squares of each direction SD.

If the foresight and backsight were observed the same manner (2DR, 4DR, etc for both FS and BS) then the a priori SD for each direction is the SD of the angle divided by the square root of 2.

So for the first angle P-Q-R (SD of 4.5") the equivalent direction SD for each direction (both P-Q and P-R) would be ~3.2", because adding two directions that each have that SD results in an angle between those two directions of 4.5" SD. Next angle P-R-S with SD of 6.5" translates to 4.6" SD directions, and so on.

Orienting each direction to a common starting azimuth doesn't change its individual a priori SD.

I really wish I hadn't left Ghilani's Adjustment Computations at work, I am sure the answer is in there somewhere. Maybe I need a second copy for my home office...