John Halleck's Least Squares Adjustment of Conditions:


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[*** This page currently has severe notation problems. ***] Inverse of A is sometimes A' and sometimes \A. (Yech... multiple sources of text.)

Introduction to Least Squares for Cave Surveying

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Overview of adjustment by derived conditions

This adjustment method is roughly to use the least squares methods to come up with the corrections that would have to be applied to each involved loop. The actual adjustment of the loop is then done as a proportional distribuition using these correction factors. Because that operation is comparitively easy, and conceptionally obvious, the overall adjustment can be much faster than a (straight forward) adjustment of the full survey. Since many programs already do enough analysis to remove dead ends and dendretic sections the setup can be very straight forward without much additional work.

Adjustment by conditions

The problem given above was:

                P1  P2  P3
set origin   [   1   0   0 ] [ P1 ]     [ SO ]
shot 1       [  -1   1   0 ] [ P2 ]  =  [ S1 ]
shot 2       [  -1   1   0 ] [ P3 ]     [ S2 ]
shot 3       [  -1   0   1 ]            [ S3 ]
shot 4       [   0  -1   1 ]            [ S4 ]

This has one row for each shot. In a network with 1000's of shots, this gives very large (but very sparse) matrix to work with.

There is another method of adjustment that has only one equation per loop. In the adjustment by observations, we basicly decided how much we should weight each traverse for the right global result. In adjustment of conditions, we will decide how much of each loop fragment goes into the final adjustment.

In this example, there are actually three loops, but only two are needed to cover all shots. For this method, every loop must have at least one shot not in another loop, and every shot involved in a loop must be in a loop listed.

We will work the example several ways to match several choices of basic loop.

SETUP: Let's consider first the trivial loop from P1 to P2 described by shots S1 and S2. Let us also consider the loop from P1 to P3 to P2 to P1 described by shots S3, S4, S1. For each shot we need to notice whether or not we are going in the original direction of the shot, or opposite that direction. Setup:

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© Copyright 2000 by John Halleck, All Rights Reserved.
This snapshot was last modified on January 24th, 2001