now with typos corrected...Re: Fwd: estimating gap errors

[WarrenZ]

Now with typos corrected!
------
Error estimation algorithm using survex

1. Measure H_old and V_old between the survey stations of
interest (say, station_1 and station_2), using aven.

2. Add the following to the survex file

*begin
*sd dx dy <delta_H>
*sd dz <delta_V>
*data cartesian from to northing easting altitude
station_1 station_2 0 0 0
*end

The effect here is to put a leg of zero length and known error
between the two stations of interest, and the aim is to measure
how the survey re-adjusts in response.  Replace <deltaH>  and
<deltaV>  by numerical estimates of the error (try starting
with 1.0 m for example).

3. Re-solve the survey and measure H_new and V_new.

4. Then

  sd_H = delta_H * sqrt(H_old / H_new - 1)

is an estimate of the standard deviation for the horizontal gap,

  sd_V = delta_V * sqrt(V_old / V_new - 1)

is likewise for the vertical gap.

5. If you find that H_new is very close to H_old, try decreasing
delta_H.  If you find H_new is very small, try increasing delta_H.
The optimum to aim for is H_new approximately half H_old.
Same for vertical error.

-oOo-

The physics analogy, if it is useful, is to measuring the effective
resistance between two points in a circuit by measuring the open-loop
potential drop and the closed-loop current.

-oOo-

I know sd_H and sd_V aren't the full story - one would in general
expect a variance-covariance matrix - and of course it all assumes
there are no gross errors!

Proof: Let sigma_1 be the variance for the unperturbed network, sigma_2
be the variance for the added leg, and sigma be the variance for the
perturbed network.  Let dx_1 be the station separation for the
unperturbed network, dx_2 be the station separation for the added
leg, and dx the station separation for the perturbed network (just
considering Eastings for example).

By the parallel addition rule one has

dx / sigma = dx_1 / sigma_1 + dx_2 / sigma_2

1 / sigma = 1 / sigma_1 + 1 / sigma_2

Now, we know all the dx's, in particular dx_2 = 0 is imposed.  We also
know sigma_2 because we assigned it.  We don't know sigma or sigma_1.
Eliminating sigma between these two equations and solving for
sigma_1 gives

sigma_1 = sigma_2 (dx_1 / dx - 1)

Taking the square root and filling in the known values gives the above
equation (for Eastings, ditto for Northings and height differences).

Patrick