azimuth readings with high inclination shots

[Stelios Zacharias]

Let's say I want to survey a deep shaft and we are not happy with the accuracy of azimuth readings when the inclination is in the range of 70-90 degrees (as it will be with the shaft). We cannot measure plumb-lines as the shaft is rigged with re-belays set so as to avoid the one being directly below the other.

In an effort to get better azimuth measurements between two stations, I figured that if we take an azimuth reading from the from station (A) and from the to station (B) to a third point (C) on the other side of the shaft (which is about 10m wide), given that we know the distance and inclination between A and B (call it X meters), this could be used to calculate the real azimuth between the two stations in the case that the high inclination gives an inaccurate reading.

This is a doddle to draw (compass and protractor stuff), but it should also be possible to work it out mathematically - I could have done it while still at high-school, but the use it or lose it maxim has worked against me.

The solution will come if we set up a 2D Cartesian problem as follows:
-two lines intersect at C - we know their gradients (given by converting the azimuths to Cartesian)
-we want to find the gradient of the line (to be reconverted to polar coordinates later) which links the two lines thus drawn at the point where they are only X meters apart. There are going to be two such lines and it should (from context) be easy to figure out which of the two it is.

The cop-out of course is to tell me to go buy a disto whatsit and measure AC and BC, and knowing all distances reduce it to a pure trig question, but I don't have a disto and am not likely to get one.

Questions:

-Has anyone else come across this problem and solved it, ie is there an equation available into which the numbers (two azimuths and one distance) can be inserted to get the third azimuth
-Has anyone solved it another way?
Am I fretting too much about something which will yield too small a benefit in terms of accuracy?

Many thanks for your time,
Stelios

[Dwight Livingston]

Stelios

If I understand your proposal, I believe you don't have enough information to fix a solution. You have the angle of AC-BC and the length of AB. You need one more measure: angle of AB-AC, angle of AB-AC, length of AC, or length of BC, none of which you say you can get. Theoretically you could add another point across the shaft and figure out where the lines cross, but I don't believe it would be very accurate using Suuntos.

How about holding out a plumb line while shooting azimuth - I've done that with good results.

Dwight

[icave]

Sorry if this potential low tech answer doesn't suite, but couldn't you use a protractor, rigged with a short plum line and a cheap laser pointer attached to the protractor. It should be pretty simple to attach the laser pointer to the protractor and attach the plum line to the center axis of the protractor. Then all you'd have to do is line up the laser pointer with your station and read the degree at which the string crosses the protractor.

I know it's pretty low tech, but I would imagine that when rigged properly, you should be accurate to within a degree or so.

[Stelios Zacharias]

Thanks for both of your responses.

i-cave: I am after an azimuth reading - as I read it, what you describe would give inclination readings.

Dwight: I am still going over the thing in my mind (visualising the problem with pieces of meccano in my mind's eye...). You might well be right that another measurement is needed, as there is more than one triangle which can be made with the same angle at the apex and different length sides.

When you say to use the plumbline, would I be right to assume you mean to tie a weight to a string and hang it from the higher of the two stations then take the azimuth from the lower station to the weighted string? If so, this is certainly very doable.

[cavemanjonny]

As described, your problem has two solutions and is therefore ambiguous. If you know roughly what the azimuth should be, you can choose which of the two solutions will work, however.


NOTE: As I created the above image, what I have labelled as azimuths aren't actually azimuths. It won't change the trig, however.

Look at http://www.analyzemath.com/Triangle/SineLaw.html for more info.

[Dwight Livingston]

Stelios

For a plumb line I do mean a string with a weight on it. I also mean hold the weighted string at arms length. Move the string to align with the station and shoot the string near your hand. Yes, that leaves you short a hand to light your compass. I've done it wearing a carbide cap lamp where the flame lit the compass okay. Maybe you need a special gooseneck lamp? I've though about surveying with a very small glow-stick, also.

Perhaps a good solution is a precisely made stick you attach to your compass, one that holds a plumb line out about a foot directly in the line of sight.

Jonny

You show the two possible right triangles. I believe there are also a whole range of triangles that are not right triangles.

[cavemanjonny]

Jonny

You show the two possible right triangles. I believe there are also a whole range of triangles that are not right triangles.

I didn't intend for them to be right triangles, but yeah, you're right. I was wrong. There are an infinite amount of solutions to that triangle.

[ian mckenzie]

Honestly, I'd just measure the depth and inclination, and fudge the compass to the nearest 10 degrees or so. If the shaft is near vertical, it won't make more than a smidge of difference to the final map. You'll find a greater error by being 0.5 degrees out on several long horizontal shots than you will be by being 5 degrees out on a near-vertical shot.

[Dwight Livingston]

When you say to use the plumbline, would I be right to assume you mean to tie a weight to a string and hang it from the higher of the two stations then take the azimuth from the lower station to the weighted string? If so, this is certainly very doable.

And, though that wasn't what I meant, hanging a plumb line from the upper station and shooting from the lower station is a very good solution, better than averaging a crappy front sight and a crappy back sight.

Dwight

[George Dasher]

My first choice would be to take a whole bunch of azimuths (that you consider ok) and then take an average.

My second choice would be to find some way of using a plumb line or setting some in-between stations.

But 90% of the time I can make Option A work.

Remember, your survey doesn't have to follow the rope. It can go all over the place.

[Martin Sluka]

Let's say I want to survey a deep shaft and we are not happy with the accuracy of azimuth readings when the inclination is in the range of 70-90 degrees (as it will be with the shaft). We cannot measure plumb-lines as the shaft is rigged with re-belays set so as to avoid the one being directly below the other.

Check: http://www.cavediggers.com/MartinSlukaCL/index.html

But cheap laser ponter is not a good idea. The beam is not colimated and the laser diode is not fixed. Use colimated laser module instead.

Martin