Fundamentals of Surveying Exam (FS) Field Data Acquisition and Reduction Syllabus

In a level circuit, how do you compute the elevation of a turning point and verify the work arithmetically?

$\text{New elevation} = \text{starting elevation} + \sum BS - \sum FS$. Arithmetic check: the sum of all backsights minus the sum of all foresights equals the net change in elevation from start to finish.

What is a 'turning point' (TP) in differential leveling?

A temporary, stable point on which both a foresight (from the old instrument setup) and a backsight (from the new setup) are taken, allowing the level to be advanced while carrying the elevation forward.

Give the formula for trigonometric leveling elevation difference between an instrument and a point.

$$\Delta \text{Elev} = S \sin \alpha + h_i - h_r$$ where $S$ is slope distance, $\alpha$ the vertical angle above horizontal, $h_i$ the instrument height, and $h_r$ the height of the reflector/target (rod reading).

What combined correction for earth curvature and atmospheric refraction is applied in long-sight trigonometric/differential leveling?

$$C = 0.0206\,K^{2}\ \text{(m, with } K \text{ in km)} \quad\text{or}\quad C = 0.0239\,F^{2}\ \text{(ft, with } F \text{ in thousands of feet)}$$ This combined curvature-and-refraction correction is added to the observed elevation of the distant point (it is roughly $\tfrac{6}{7}$ curvature less $\tfrac{1}{7}$ refraction).

How is balancing backsight and foresight distances used to reduce leveling error?

Keeping backsight and foresight distances approximately equal at each setup cancels systematic errors from earth curvature, atmospheric refraction, and instrument collimation, because these errors are proportional to sight length and affect both readings equally.

How is the allowable misclosure of a level loop typically specified, and what is the correction procedure?

Allowable misclosure is often $C = m\sqrt{K}$ (e.g. $12\ \text{mm}\sqrt{K}$ in km for ordinary work). The loop should close back on its starting elevation; the misclosure is distributed in proportion to the number of setups or the distance leveled, with sign opposite to the error.

Define a 'closed (loop) traverse' versus a 'closed connecting traverse' versus an 'open traverse.'

A closed loop traverse begins and ends at the same point. A closed connecting (link) traverse begins and ends on different points of known coordinates. An open traverse begins at a known point but ends at an unknown point, providing no mathematical check on closure.

What is the geometric check on the interior angles of a closed polygon traverse with $n$ sides?

$$\sum \text{interior angles} = (n - 2)\,180^{\circ}$$ For exterior angles the sum is $(n+2)\,180^{\circ}$. Any difference from the theoretical value is the angular misclosure.

How is angular misclosure in a traverse typically distributed?

It is divided equally among all measured angles (each angle receives $\frac{\text{misclosure}}{n}$ with opposite sign), unless certain angles were measured under poorer conditions, in which case larger corrections may be assigned to them.

Define latitude and departure of a traverse course in terms of its length and azimuth/bearing.

$$\text{Latitude} = L \cos(\text{azimuth}) \qquad \text{Departure} = L \sin(\text{azimuth})$$ Latitude is the north-south component (north +, south −) and departure is the east-west component (east +, west −).

What does it mean for a closed traverse to balance, in terms of latitudes and departures?

For a geometrically closed loop the algebraic sums must theoretically be zero: $$\sum \text{Latitudes} = 0 \qquad \sum \text{Departures} = 0$$ Nonzero sums are the latitude misclosure and departure misclosure.

Give the formula for the linear misclosure of a traverse.

$$\text{Linear misclosure} = \sqrt{(\sum \text{Lat})^{2} + (\sum \text{Dep})^{2}}$$ where $\sum\text{Lat}$ and $\sum\text{Dep}$ are the misclosures in latitude and departure.

Define 'relative precision' (precision ratio) of a traverse.

$$\text{Relative precision} = \frac{\text{linear misclosure}}{\text{total traverse perimeter}}$$ expressed as a fraction with numerator 1, e.g. $\frac{1}{10{,}000}$. Smaller fractions (larger denominators) indicate higher precision.

Describe the Compass (Bowditch) Rule for traverse adjustment.

It distributes misclosure to each course in proportion to that course's length: $$C_{\text{lat},i} = -(\sum\text{Lat})\frac{L_i}{\sum L} \qquad C_{\text{dep},i} = -(\sum\text{Dep})\frac{L_i}{\sum L}$$ It assumes angles and distances are measured with equal precision.

Describe the Transit Rule for traverse adjustment and when it is preferred.

The Transit Rule distributes misclosure in proportion to the magnitude of each course's latitude or departure: $$C_{\text{lat},i} = -(\sum\text{Lat})\frac{|\text{Lat}_i|}{\sum|\text{Lat}|}$$ (similarly for departures). It is preferred when angular measurements are more precise than the linear measurements.

What is the most rigorous traverse adjustment method, and on what principle is it based?

The Least Squares method. It adjusts all observations simultaneously to minimize the sum of the weighted squares of the residuals ($\sum w v^{2} = \min$), producing the statistically most probable values and allowing rigorous error analysis.

State the Coordinate Method (Double Meridian Distance, DMD) formula for the area of a closed traverse.

$$\text{Area} = \frac{1}{2}\left|\sum (\text{DMD}_i \times \text{Lat}_i)\right|$$ where the DMD of the first course equals its departure, and each subsequent DMD equals the previous DMD plus the previous departure plus the current departure.

Give the coordinate (shoelace) formula for the area of a closed traverse from station coordinates.

$$\text{Area} = \frac{1}{2}\left|\sum_{i=1}^{n} X_i (Y_{i+1} - Y_{i-1})\right| = \frac{1}{2}\left|\sum (X_i Y_{i+1} - X_{i+1} Y_i)\right|$$ with the points taken in order around the polygon.

What is Double Meridian Distance (DMD) of a traverse course?

The DMD of a course is the sum of the meridian distances of its two endpoints (twice the meridian distance of its midpoint). DMD of the first course = its departure; DMD of any course = previous DMD + previous course's departure + this course's departure.

What is the difference between accuracy and precision in survey measurements?

Accuracy is the closeness of a measurement to its true value (affected by systematic errors and blunders). Precision is the closeness of repeated measurements to one another (degree of refinement/repeatability), reflecting random error scatter.

Classify the three types of errors in surveying measurements.

(1) Blunders/mistakes — gross human errors, must be detected and removed; (2) Systematic errors — follow physical laws, same sign/magnitude under same conditions, can be modeled and corrected (e.g. tape temperature); (3) Random errors — small, equally likely positive or negative, treated statistically.

How does random error in a sum of independent measurements propagate?

For a sum of independent quantities, errors add in quadrature: $$\sigma_{\text{sum}} = \sqrt{\sigma_1^{2} + \sigma_2^{2} + \cdots + \sigma_n^{2}}$$ For $n$ measurements each with error $\sigma$, the total is $\sigma\sqrt{n}$.

What is the error of the mean for $n$ repeated equal-precision observations?

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$ where $\sigma$ is the standard deviation of a single observation. The mean's precision improves with the square root of the number of observations.

What is 'leveling' (plate-bubble) error and how is the effect of an out-of-level vertical axis on horizontal angles minimized?

If the vertical axis is not truly vertical, horizontal angles are in error, and this error is NOT removed by reversing faces. It is minimized by careful leveling of the plate bubble and, for vertical-axis tilt effects on vertical angles, using a compensator or a sensitive vertical-circle index bubble.

What is the purpose of double centering (double sighting) when prolonging a straight line with a theodolite?

To eliminate instrumental errors (collimation and horizontal-axis errors): sight the backsight in direct mode and set a forward point, then plunge to reversed face, re-sight the backsight, and set a second point. The true prolongation is the midpoint between the two forward points.

Why must EDM slope distances be reduced to horizontal AND then often to a grid/ground datum before traverse computation?

Slope-to-horizontal removes the vertical component; further reductions (sea-level/ellipsoid factor and grid scale factor) convert ground horizontal distance to the mapping projection so coordinates computed from latitudes and departures are consistent with the state plane grid.

For trigonometric leveling, what makes reciprocal observations (measuring vertical angles from both ends simultaneously) valuable?

Simultaneous reciprocal vertical-angle observations cancel the effects of earth curvature and atmospheric refraction, because those errors are equal and opposite at the two ends, yielding a more accurate elevation difference over long sights.

In a link (connecting) traverse between two known control points, what replaces the 'sum of latitudes = 0' check?

The computed coordinates carried through the traverse must match the known coordinates of the closing control point. Misclosure equals the difference between computed and known $X$ and $Y$ (departures and latitudes must sum to the known $\Delta X$ and $\Delta Y$ between the endpoints).

What is the typical field procedure ('angle to the right') for measuring traverse angles with a total station?

Backsight the previous station with the horizontal circle zeroed, then turn clockwise to foresight the next station; the displayed angle is the clockwise (right) angle. Observations are usually repeated in direct and reversed face and averaged to cancel instrumental errors.

Give the formula for the area of a tract bounded partly by an irregular boundary using the Trapezoidal Rule with equal offset spacing $d$.

$$A = d\left(\frac{h_0 + h_n}{2} + h_1 + h_2 + \cdots + h_{n-1}\right)$$ where $h_0, h_1, \dots, h_n$ are the perpendicular offsets to the irregular boundary at equal interval $d$.