7.1 Error Theory and Least Squares
Key Takeaways
- Errors divide into gross errors (blunders), systematic errors and random errors; only random errors are treated by adjustment theory.
- The standard error of the mean of n equally weighted observations is the standard error of a single observation divided by the square root of n.
- For independent quantities the variance of a sum or difference is the sum of the variances, which is why misclosure tolerances scale with the square root of the number of stations.
- Least squares minimises the sum of weighted squared residuals and, for normally distributed random errors, yields the most probable values of the unknowns.
- Weights are inversely proportional to variances, so a distance measured with twice the standard error carries one quarter of the weight.
7.1 Error Theory and Least Squares
Why this sits in a SURCON guide: Paper I of the Pupil Surveyor's cadre is Mathematical and Computational Techniques, and the whole of the Survey Regulations' tolerance structure - 1:3,000, 3 seconds times the square root of N, a 30-second range in three azimuths - is applied error theory. You cannot defend a Folio computation without it.
1. Three kinds of error
| Type | Character | Treatment |
|---|---|---|
| Gross error (blunder) | A mistake: transposed digits, wrong station occupied, misread staff, a tape length omitted | Detect and eliminate. Blunders are not adjusted away. Checks - closed circuits, double chaining, repetition on a different zero - exist to find them |
| Systematic error | Follows a physical law and accumulates with a consistent sign: tape not standard, temperature, sag, slope, collimation, refraction, index error | Model and correct, or eliminate by procedure |
| Random error | Small, unpredictable in sign, following a normal distribution; the irreducible residue after blunders are removed and systematics corrected | Adjust, by least squares |
The Survey Regulations attack all three. Regulation 5(A) requires one face right and one face left, which cancels collimation and trunnion-axis error - systematic. Regulation 4 requires closed circuits and, where closure is impossible, double chaining in opposite directions plus a check azimuth - blunder detection. Regulation 5(C) requires corrections for standard, slope, and where necessary temperature and sag - systematic modelling. What is left is random, and that is what an adjustment distributes.
2. Measures of precision
For $n$ observations $x_i$ of a single quantity with mean $\bar{x}$:
Standard error of a single observation
Standard error of the mean
The second is the reason regulation 5(B) requires three separate sun or star observations for azimuth. Three observations reduce the standard error of the mean by a factor of $\sqrt{3} \approx 1.73$, and - more importantly - three values allow a range check. Two observations that disagree tell you there is a problem but not which one is wrong.
Worked example. Three azimuth observations give 127 deg 14' 20", 127 deg 14' 42" and 127 deg 14' 31". The mean is 127 deg 14' 31" and the range is 22 seconds - inside the regulation 5(B) limit of 30 seconds, so the set is acceptable. Residuals are -11", +11" and 0"; $\sigma = \sqrt{(121+121+0)/2} = 11.0''$, and $\sigma_{\bar{x}} = 11.0/\sqrt{3} = 6.4''$.
3. Propagation of error
For a function $y = f(x_1, x_2, \dots, x_n)$ of independent quantities,
Three consequences carry most of the practical weight:
Sums and differences. For $y = x_1 \pm x_2$, $\sigma_y^2 = \sigma_1^2 + \sigma_2^2$. Variances add, standard errors do not.
A chain of $n$ equal steps. If each of $n$ angles carries standard error $\sigma$, the accumulated error in the final bearing is
That is exactly the form of regulation 5(A)(iii): maximum misclosure of bearings = 3 seconds times the square root of N. The regulation is a tolerance of three seconds per station propagated through $n$ stations, and it caps $N$ at 30 because beyond that the accumulated tolerance ceases to be a useful check.
The same form appears in levelling, where loop misclosure tolerances are conventionally expressed as a constant times $\sqrt{k}$ with $k$ the route length in kilometres - because a levelling line is a chain of independent set-ups.
Products. For $y = x_1x_2$, relative errors combine: $\left(\frac{\sigma_y}{y}\right)^2 = \left(\frac{\sigma_1}{x_1}\right)^2 + \left(\frac{\sigma_2}{x_2}\right)^2$. This is why area accuracy degrades faster than linear accuracy, and why regulation 31(j) reduces the number of decimals stated as parcels grow.
4. Weights
Where observations are of unequal quality, weight is inversely proportional to variance:
where $\sigma_0$ is an arbitrary reference ("the standard error of unit weight"). Doubling the standard error of an observation reduces its weight to one quarter.
Common weighting conventions:
- Levelling - weight inversely proportional to route length (a longer line has more set-ups and more accumulated variance).
- Traverse angles - equal weight where observed identically; otherwise inversely proportional to variance.
- Distances - EDM standard error is typically quoted as $a + b,D$ (a constant plus a proportional part), so weight declines with length but not linearly.
5. Least squares
The principle
Where there are more observations than the minimum needed - redundancy - the observations will not be perfectly consistent. Least squares selects the set of values for the unknowns that minimises the sum of the weighted squared residuals:
For observations whose random errors are normally distributed, this yields the most probable values of the unknowns. That is its justification - not that it is convenient, but that it is the maximum-likelihood estimator under the normal error model that random errors actually follow.
The observation-equation form
Linearise each observation about approximate values of the unknowns:
where $\mathbf{A}$ is the design matrix of partial derivatives, $\mathbf{x}$ the vector of corrections to the approximate unknowns, $\mathbf{l}$ the vector of observed-minus-computed values, and $\mathbf{v}$ the residuals. With weight matrix $\mathbf{W}$, the normal equations are
What comes out besides coordinates
An adjustment that returns only adjusted coordinates has wasted most of its value. The by-products are the quality control:
- Residuals $\mathbf{v}$ - a residual far larger than its expected size flags a probable blunder in that observation.
- Standard error of unit weight, $\sigma_0 = \sqrt{\dfrac{\mathbf{v}^T\mathbf{W}\mathbf{v}}{r}}$, where $r$ is the redundancy (observations minus unknowns). If this comes out far from 1, either the weighting was wrong or a blunder survived.
- The cofactor matrix $(\mathbf{A}^T\mathbf{W}\mathbf{A})^{-1}$ - from which the standard errors of the adjusted coordinates and the error ellipses are computed. The error ellipse tells you the direction in which the network is weak, which no single accuracy figure can.
- Redundancy numbers - how much each observation is checked by the others. An observation with near-zero redundancy is unchecked, and a blunder in it will pass undetected.
Minimum constraint versus over-constrained
A minimally constrained adjustment fixes only what is necessary to define position, orientation and scale, and therefore reveals the internal quality of the observations. An over-constrained adjustment holds several control points fixed and forces the network onto them, absorbing any distortion in the control into the residuals.
Run the minimally constrained adjustment first. If the internal fit is good but the over-constrained fit is poor, the problem is in the control, not in your observations - which, under regulation 6, is exactly the situation in which a full report must be made to the Surveyor-General so that inquiry may be made into the positions of the old marks.
Three azimuth observations of a line give 127 deg 14' 20", 127 deg 14' 42" and 127 deg 14' 31". Do they satisfy regulation 5(B), and what is the standard error of the mean?
Why does the regulation 5(A) tolerance for misclosure of bearings take the form 3 seconds multiplied by the square root of N?
In a weighted least squares adjustment, an observation is found to have a standard error twice that of the others. What weight should it carry relative to them?
A network adjusts well when minimally constrained but poorly when several existing control points are held fixed. What does this indicate, and what does the Survey Regulations require?