# Error Propagation in Successive Least Square Adjustment

I have a certain problem in surveying in which I'm trying to do some error analysis. I'll layout the problem first.

• My current goal is to evaluate the variance-covariance matrix for the position of a point T on the ground.
• Now to first calculate the position of T, I observe this point from 4 other points (S1, S2, S3, S4).
• The observation is done in terms of (Ri,θi,φi) and is converted to (xi,yi,zi) position of T.
• Now using simple least square adjustment I calculate the adjusted position of (X,Y,Z).
• I need to find out the net variance-covariance matrix of (X,Y,Z) provided I know the following information

Available information

• Var-Covar matrix of the positions of S1, S2, S3, S4.
• Var-Covar matrix of the observations (Ri,θi,φi) that is converted into the matrix for (xi,yi,zi)
• Var-Covar matrix of (X,Y,Z) obtained through the co-factor matrix in least square adjustment procedure - this matrix is only a by-product of the adjustment - it assumes no uncertainty in S1, S2, S3, S4.

Question

I need to know how can I propagate the errors in the positions of S1, S2, S3, S4 into the errors of point T.

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Here are the general ideas:

1) Treat errors coming from different sources as independent with mean 0.

Moral: When two errors add together, the covariance matrices merely add up. The least square procedure in your case merely returns the average of 4 observations, so the final matrix $A=\frac 1{16}\sum_i A_i$ and each $A_i$ is the sum of the covariance matrix of $S_i$ plus the covariance matrix of the vector going from $S_i$ to $T$.

2) When having non-linear dependences and small errors, linearize near the observed values.

You have the conversion from polar to Cartesian here. I'm not 100% sure what notation you use for polar, but let's say the conversion is $$x=R\sin\theta\cos\phi, y=R\sin\theta\sin\phi, z=R\cos\theta$$
Suppose you want to compute some covariance for the coordinates of the Cartesian direction vector, say $E[dx dz]$ where $dx$ is the error in $x$, etc. Linearizing yields $$dx=(\sin\theta\cos\phi)dR+(R\cos\theta\cos\phi)d\theta-(R\cos\theta\sin\phi)d\phi, \\ dz=(\cos\theta)dR-(R\sin\theta)d\theta$$ Now multiply out and open parentheses. You will get various expressions of the kind $$F(R,\theta,\phi) dR d\phi$$ (or (dR)^2, or...). Read the expectations $E[dR d\phi]$ from the polar covariance matrix you have, plug them in, and do the arithmetic.

That's all it takes.

P.S. This is done under the assumption that S_i were independently measured. If they themselves were obtained by measurements from the same point, their errors are not completely independent any more and the picture gets more involved. Let me know if this effect is of importance in your setting.

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