FS Module 5: Survey Computations

What is the closure ratio of a traverse?

Closure ratio = linear misclosure / total traverse perimeter, expressed as 1:N. Linear misclosure = sqrt(lat_error^2 + dep_error^2). Example: if misclosure is 0.05 ft and perimeter is 1500 ft, ratio = 0.05/1500 = 1:30,000. Typical standards: 1:5,000 (rural), 1:10,000 (urban boundary), 1:15,000 (first-order), 1:50,000+ (geodetic).

How do you compute an inverse (from coordinates)?

Given points (X1,Y1) and (X2,Y2): Delta-X = X2 - X1 (departure) Delta-Y = Y2 - Y1 (latitude) Distance = sqrt(Delta-X^2 + Delta-Y^2) Azimuth = arctan(Delta-X / Delta-Y), adjusted for quadrant. This is the fundamental COGO inverse problem.

How does error propagate for the sum/difference of independent measurements?

For Z = A + B or Z = A - B (where A and B are independent): sigma_Z = sqrt(sigma_A^2 + sigma_B^2) Variances add whether the quantities are added or subtracted. This is why a differential level loop accumulates error proportional to the square root of the number of setups.

How does error propagate for a product or quotient?

For Z = A x B or Z = A/B: (sigma_Z / Z)^2 = (sigma_A / A)^2 + (sigma_B / B)^2 Relative (fractional) errors add in quadrature. Example: if distance has 0.1% error and a scale factor has 0.05% error, the combined relative error is sqrt(0.1^2 + 0.05^2) = 0.112%.

What is the degree of curve (arc definition)?

The central angle subtended by a 100-foot arc along the curve. R = 5729.578 / D. Example: D = 5 degrees means R = 1145.92 ft. The arc definition is standard for highway design. (The chord definition, used for railroad curves, defines D as the angle for a 100-ft chord.)

How do you compute the area of a circular segment?

Area = (R^2 / 2)(theta - sin(theta)), where theta is the central angle in radians and R is the radius. A circular segment is the area between a chord and the arc it subtends. Common in right-of-way and easement calculations along curves.

How do you compute latitude and departure for a traverse course?

Latitude = distance x cos(bearing/azimuth) [N/S component] Departure = distance x sin(bearing/azimuth) [E/W component] North latitudes and East departures are positive; South latitudes and West departures are negative.

What is the compass (Bowditch) rule for traverse adjustment?

Distributes the linear misclosure proportionally based on course length: Correction in latitude = -(total lat error) x (course length / total traverse length) Correction in departure = -(total dep error) x (course length / total traverse length) Most commonly used method for adjusting traverses with angle and distance measurements of similar precision.

How do you compute area using the coordinate method?

2A = sum of (Xi x Yi+1 - Xi+1 x Yi) for all points in order around the parcel. Alternatively: 2A = sum(Xi(Yi+1 - Yi-1)) or equivalently sum(Yi(Xi-1 - Xi+1)). The sign of the result indicates direction of traverse (positive = counterclockwise). Take the absolute value and divide by 2 for the area.

What is the DMD (Double Meridian Distance) method for area?

DMD of first course = departure of first course. DMD of next course = DMD of previous course + departure of previous course + departure of current course. 2A = sum of (latitude x DMD) for each course. Area = |sum| / 2. Historically common; coordinate method is now more widely used.

State the horizontal curve formulas involving R, D, L, T, E, and M.

R = 5729.578 / D (for arc definition, D in degrees) T = R x tan(I/2) [tangent length] L = R x I (in radians) [arc length], or L = 100 x I/D E = R x (sec(I/2) - 1) [external distance] M = R x (1 - cos(I/2)) [middle ordinate] LC = 2R x sin(I/2) [long chord] Where I = deflection (intersection) angle, D = degree of curve.

What is the formula for a vertical curve (parabolic)?

y = (g2 - g1) / (2L) x x^2 + g1 x x + elevation of BVC Where g1 = entering grade (%), g2 = leaving grade (%), L = curve length, x = distance from BVC. High/low point at x = -g1 x L / (g2 - g1) from BVC. BVC = PVI - L/2; EVC = PVI + L/2.

How do you convert a bearing to an azimuth?

NE quadrant (N __ E): Azimuth = bearing angle SE quadrant (S __ E): Azimuth = 180 - bearing angle SW quadrant (S __ W): Azimuth = 180 + bearing angle NW quadrant (N __ W): Azimuth = 360 - bearing angle Azimuths are measured clockwise from north (0-360 degrees).

What is the law of cosines?

For a triangle with sides a, b, c and opposite angles A, B, C: c^2 = a^2 + b^2 - 2ab x cos(C) Used when you know two sides and the included angle (SAS), or three sides (SSS). Essential for COGO computations, inverse calculations, and triangle solutions.

What is the law of sines?

a / sin(A) = b / sin(B) = c / sin(C) Used when you know two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA - beware of the ambiguous case). Common in triangle solutions, resection, and intersection problems.

What is the transit rule for traverse adjustment?

Distributes misclosure proportionally based on the absolute value of latitude (for lat correction) and departure (for dep correction): Lat correction = -(total lat error) x |course lat| / sum of |all lats| Dep correction = -(total dep error) x |course dep| / sum of |all deps| Used when angles are more precise than distances.

What is a radial (stake-out) computation?

Given a known point and the coordinates of a target point, compute the azimuth and distance from the known point to the target. This is an inverse computation used to guide instrument setups for construction staking. The instrument is set up on the known point, oriented to a reference direction, and turned to the computed azimuth at the computed distance.

What are the basic principles of least squares adjustment?

Least squares minimizes the sum of the squares of the weighted residuals (v'Pv = minimum). It produces the most probable values when measurements contain only random errors. Requires more observations than unknowns (redundancy). Outputs include adjusted values, residuals, and statistical measures of precision (standard deviations, error ellipses).