Question 1 of 12: Pre-analysis (Design Simulation) of a Geodetic Control Network

Question 1: Pre-analysis (Design Simulation) of a Geodetic Control Network (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

i) Purpose. Pre-analysis (also called network design or simulation) is the process of predicting the quality of a control network before any observations are made. The aim is to guarantee, on paper, that the finished network will meet its specified precision, reliability and economy criteria — so that the surveyor commits field resources only to a configuration already known to work. It answers the questions “where should I put stations, what should I observe, and to what precision, so that the coordinates come out good enough?” It is the surveyor’s equivalent of a structural engineer sizing a member before it is built.

iii) Information required. Pre-analysis needs: (a) approximate coordinates of all stations (from a map, a plan, or a handheld GNSS reconnaissance) — these fix the geometry; (b) the proposed observation plan — which directions, angles, distances, height differences or GNSS baselines will be measured; (c) the a-priori standard deviations of each observation type, taken from the instruments’ specifications (e.g. EDM σ = ±(a + b ppm), angular pointing σ); (d) the datum definition (which stations or quantities are held fixed or constrained); and (e) the target quality criteria — the required precision (error-ellipse size), reliability, and any accuracy standard the network must satisfy (e.g. an FGCS order or a project tolerance).

ii) Step-by-step procedure. The classical simulation loop is:

1. Establish approximate coordinates and the observation scheme. Plot the proposed stations and draw every intended observation (the “sight lines”) onto the network sketch.
2. Form the design (Jacobian) matrix. Linearise each observation equation about the approximate coordinates to build the coefficient matrix $\mathbf{A}$; the coordinates themselves are not needed as values — only the geometry enters $\mathbf{A}$.
3. Assign the weight matrix. Build $\mathbf{P}=\mathbf{Q}_{\ell}^{-1}$ from the a-priori observation variances (the instrument specifications).
4. Compute the predicted covariance of the coordinates. Because no residuals exist yet, the covariance follows purely from geometry and weights: $$\mathbf{Q}_{xx}=\left(\mathbf{A}^{\mathsf T}\mathbf{P}\,\mathbf{A}\right)^{-1}$$ Scaling by the a-priori variance factor $\sigma_0^2$ gives $\boldsymbol{\Sigma}_{xx}=\sigma_0^2\,\mathbf{Q}_{xx}$.
5. Extract the quality measures. From $\boldsymbol{\Sigma}_{xx}$ compute the point error ellipses (semi-axes and orientation from the eigenvalues of each $2\times2$ block), relative ellipses between key points, and reliability numbers (redundancy numbers $r_i$ from the diagonal of $\mathbf{R}=\mathbf{I}-\mathbf{A}(\mathbf{A}^{\mathsf T}\mathbf{P}\mathbf{A})^{-1}\mathbf{A}^{\mathsf T}\mathbf{P}$).
6. Compare against the criteria and iterate. If a predicted ellipse is too large or a redundancy number too small, modify the design and repeat until all criteria are met at least cost.

iv) What can be altered to adjust the scheme. The levers are the quantities that enter $\mathbf{Q}_{xx}=(\mathbf{A}^{\mathsf T}\mathbf{P}\mathbf{A})^{-1}$, classically grouped into the (Grafarend) orders of design: the datum — which points or parameters are held fixed or minimally constrained (Zero-Order Design); the configuration — station positions and the observation plan, i.e. adding, removing or re-routing sights, which changes the rows of $\mathbf{A}$ (First-Order Design); the weights — more precise instruments, more sets or repetitions, or different observation types, which change $\mathbf{P}$ (Second-Order Design); and densification/improvement of an existing network by adding new points and observations to it (Third-Order Design). In practice the cheapest lever is usually adding a few well-oriented observations or strengthening the weakest sights; moving stations is more costly, and buying precision (Second-Order Design) is the last resort because weights cannot economically cure a geometric weakness.