Preliminaries, Planning, & General Rules

1. Preliminaries, Planning, & General Rules Detailed planning: • Essential before performing actual survey work, large or small. Common mistake for inexperienced people: • Arrive on site completely unprepared, not knowing what to do and/or how to do it

2. Often ends up costing much extra time & effort, due to idling of crew (while trying to come to an agreement), inappropriate methods used, and unnecessary mistakes made. Time spent on careful planning = time saved

3. Tips on preliminaries & planning: Study existing maps &/or aerial photos of area if available. Relatively few surveys are performed in unmapped areas, & it is not cheating but common sense to utilize all available information. Distances & angles to be measured: estimated on a scaled map beforehand Rough height differences: obtained using contour lines / photogrammetric techniques

4. These estimates can also be obtained (e.g. by pacing or a compass) during reconnaissance visit, to be discussed next. Pay a reconnaissance visit to site, if at all possible Look for suitable locations for survey stations forming control framework & mark them if possible, noting possible lines of sight & potential obstructions (e.g. clouds or vehicles) to them.

5. Which detail features to include in the survey, what instruments & techniques to use, how to divide labor, & what part of work to carry out on which day, etc.: should plan at this stage, in a way that optimizes manpower & equipment use. Every survey should be designed in such a way that it is impossible for errors to go undetected

6. Introducing redundancies into scheme (e.g. measure all three angles in a triangle, or both diagonals of a quadrilateral) Cannot be achieved by merely repeating the same measurement many times redundancies: not only guard against blunders, but also allow overall statistical treatment of data to improve accuracy

7. Design & produce well-organized booking forms or electronic spreadsheets for recording data, & include entries for redundant measurements & geometric checks to remind the observer. Before leaving base for the actual survey: Be completely familiar with all instruments to be used Make sure they are properly calibrated by qualified personnel such as instruments’ manufacturers

8. Read instruction manuals if available, & become familiar with instruments through practice before the survey. During actual measurements: compare with previously estimated values &/or geometric constraints to detect gross errors right away. Although instruments well adjusted, make measurements as though the equipment is not well adjusted, using techniques so that instrument maladjustment errors will be significantly reduced even if present (techniques: later chapters)

9. Be absolutely sure that no measurement is missed: very time-consuming & expensive to go back & repeat any measurement(s) after a survey is completed, if possible at all. Above all: Safety－No. 1 concern during fieldwork: Hostile environments due to landscape, traffic, animals or people at site Take extreme caution to avoid injuries or damage to instruments.

10. Classification of errors; accuracy & precision Surveying－ measurement science All measurements contain errors • Identify sources of error • Devise methods of dealing with them

11. Three main kinds of errors in surveying: Blunders (or mistakes)–“gross errors” e.g. misreading by a whole meter or a degree; due to carelessness or lack of attention. Often detected by proper checks Team members: check on each other for mistakes Common source of careless mistakes: copying numbers from one place to another.

12. SystematicErrors– Due to some persistent cause, generally in the instrument, but sometimes in a habit of the observer. Thermal expansion of tapes & collimation errors in a level Can be reduced by better technique, but not by averaging readings Do not obey “cancel out” laws of probability

13. All distances measured with an inaccurate tape: same percentage error with same sign, however many times they are measured Only remedy: calibrate the tape more carefully Most serious sort of error; techniques of surveying & instrument calibration are mainly directed against it.

14. Random Errors – Without mistakes & systematic errors: still remain small random errors Due to causes beyond control of observer (e.g. temperature & wind) A matter of chance & subject to laws of probability Magnitude: depends on precision of instrument & observer’s skills

15. Cannot be corrected + & - errors equally probable small errors more frequent than large ones very large errors do not occur  normal distribution to describe errors as random variables Taking average of many readings will help in reducing this type of error Rigorous statistical methods: Least squares adjustment (LSA): in each relevant chapter with reference to particular survey figures

16. Accuracy ≡ closeness of measurement to true value Precision≡agreement between repeated measurements of same quantity If systematic errors dominate results in spite of greatest precautions taken, close agreement among data set may only indicate a consistent instrument & steady observer, i.e. high precision, not necessarily accurate result.

17. Fig. 1.3 Center of dartboard: “true value” which observer is trying to aim at

18. Precisions of instruments: standard errors (or standard deviation, “”) Theodolite: “ = 5” in user menu: indicates reliability of observations Weights：attached to observations ~ 1 /2 Weighted observations (next section)

19. x2 + x3 x2 + x3 x1 x1 + x2 x1 + x2 + x3 Least Squares Adjustment of Random Errors • To determine: three lengths x1, x2 & x3 by various direct or indirect measurements Fig. 1.4: Generic problem

20. Data: assume no blunders/ systematic errors More observed than minimum necessary Contain random errors Different sets of data yields different answers

21. Observations carry different weights (i.e. degree of reliability): Table 1.1 Quantity x1 Observed Value 3.0 Weight of Observation 1 x1 + x2 6.1 2 x1 + x2 + x3 11.2 3 x2 + x3 8.1 2

22. Assignment of weights observers’ skills and experience precisions of different instruments used No need to worry about such details here except to work with given weights How to get unique set of “best” values for x1, x2, x3?

23. Principle of least squares (LS) Best estimates: “most probable values” (MPV’s） Difference between observed value (#’s) & corresponding MPV (unknown) ≡residual: Residual = (Observed Value – its MPV) (1.1)

24. LS condition: The most probable values of observed quantities are such that they minimize the total weighted sum of square residuals, SSR =  weight  (residual)2(1.2)

25. LS principle: starting point (could be derived based on other principles, e.g. “maximum likelihood estimators”: probability theory) Practical concern: numerical answers for x1, x2 & x3 from (1.2) Minimization problem, solved by calculus (see Ex. 1.5)

26. Matrix approach: only involves matrix algebra Advantage: readily performed on a spreadsheet. Pretend: all weights = unity (will remove this restriction later) Vector of unknowns, [x1, x2, x3]T ≡x (superscript T = “transpose”)

27. Left-most column in Table 1.1: re-write as matrix product:

28. Table 1.1 rephrased in matrix form: “~” means “have the observed values of”

29. Let A≡aforesaid 43 matrix of 0’s & 1’s (i.e. coefficients on x) (“design matrix”); Vector k≡ observed values  Ax ~ k

30. LS principle: “best”x: must minimize overall “discrepancy” between Ax & k, i.e. (squared) magnitude of residual vector, hence Minimize rTr where r = Ax–k (1.3) Well known problem, solved in linear algebra

31. Take partial derivatives w.r.t. x, followed by matrix algebra: can show: x must satisfy (ATA)x = ATk(1.4) (1.4): “normal equations” Solution: obviously x = (ATA)–1ATk (1.5) (ATA)–1AT≡“left pseudo-inverse” of A (why?)

32. Formal proof of (1.4) & (1.5): Skeel & Keiper (1993) We will simply use this result to solve practical problems Return to original problem: unequal weights Only minor modifications needed: Introduce diagonal “weight matrix”,

33. and “square root of W”, Simple algebra (1.2) is minimization of = (W1/2r)T(W1/2 r) i.e. minimize dot product of “weighted residual vector”r’ = W1/2r with itself

34. In terms of W, A & x: r’ = (W1/2A)x – (W1/2 k) to be minimized Note: Quantities in ( ): all constants independent of x  same problem described in (1.3) except: “A”: nowW1/2A, “k”: nowW1/2 k.

35. (1.5): still valid after making aforementioned replacements x = [(W1/2A)T(W1/2A)]–1(W1/2A)TW1/2 k Recall: (BC)T = CTBT for matrices B & C, (for diagonal matrix) WT = W, (W1/2)T = W1/2, & W1/2 W1/2 = W,  Solution simplifies to x = (ATWA)–1ATWk (1.6)

36. Sub. numerical values for A, k, and W = Diag [1,2,3,2] = [3.055, 3.045, 5.082]T Most probable values: x1 = 3.055, x2 = 3.045, x3 = 5.082.

37. Matrices arise frequently, not only in CIVL 102 (LSA of survey figures), but also in CIVL 337 (matrix structural analysis) CIVL 253 (hydrology: regression analysis) Geotechnical reliability analysis etc. Become efficient in handling matrix computations with some form of software as earlyas possible

38. Ch.2: lots of LSA using matrix approach. spreadsheet for matrix transpose, multiplication & inversion, as well as other useful mathematical tasks for surveying.