Principles and Practice of Surveying Exam (PS) Geodesy, Datums, and Coordinate Systems Syllabus

Distinguish conformal, equal-area, and equidistant projections.

Conformal: preserves angles/local shape (scale same in all directions at a point) — used for surveying. Equal-area: preserves area but distorts shape. Equidistant: preserves true distance along certain lines. No projection is both conformal and equal-area.

Name the two conformal projections used in the U.S. State Plane Coordinate System and when each is chosen.

Lambert Conformal Conic — for zones that extend mostly east-west. Transverse Mercator — for zones that extend mostly north-south. (Oblique Mercator is used for the Alaska panhandle.)

What is a developable surface? Give the surfaces used for the common projections.

A developable surface can be unrolled flat without distortion: a cone (Lambert Conformal Conic), a cylinder (Mercator / Transverse Mercator), or a plane (azimuthal). The ellipsoid is projected onto these and then unrolled.

Define scale factor $k$ in a projection.

The scale factor $k$ is the ratio of projected (grid) distance to true (ellipsoidal) distance at a point: $$k = \frac{\text{grid distance}}{\text{geodetic distance}}$$ On a secant projection $k<1$ between the standard lines and $k>1$ outside them.

Why are SPCS and UTM designed with a maximum scale distortion of about 1 part in 10,000?

By making the projection secant (cutting the ellipsoid) and limiting zone width, the scale factor is kept between roughly $0.9999$ and $1.0001$. This 1:10,000 limit keeps grid distances close enough to ground distances for many practical survey purposes within a zone.

How is a Lambert Conformal Conic zone made secant, and where is scale exact?

The cone intersects the ellipsoid along two standard parallels. Scale is exactly true ($k=1$) on those two standard parallels, less than 1 between them, and greater than 1 outside them.

In a Transverse Mercator (and UTM) zone, where is scale true and where minimized?

The cylinder is secant along two lines parallel to the central meridian. Scale is exactly 1 on those two lines; it is at a minimum ($k_0$) on the central meridian and increases east/west away from it.

State the UTM zone width, central scale factor, and number of zones.

UTM divides the Earth into 60 zones each $6^{\circ}$ wide in longitude. The scale factor on the central meridian is $k_0 = 0.9996$. Zones are numbered 1–60 eastward starting at $180^{\circ}$W.

What false easting and false northing are assigned in UTM?

False easting is $500{,}000$ m applied to each zone's central meridian (keeping all eastings positive). False northing is $0$ m in the Northern Hemisphere and $10{,}000{,}000$ m in the Southern Hemisphere (keeping northings positive).

What is the purpose of false easting/northing in coordinate systems generally?

To shift the grid origin so all coordinates within the zone are positive and to make eastings clearly distinguishable from northings, avoiding negative values and reducing blunders.

Define the combined factor (grid factor) used to convert ground distance to grid distance.

The combined factor is the product of the scale factor and the elevation (sea-level/ellipsoid) factor: $$CF = k \cdot EF$$ Grid distance $=$ ground distance $\times CF$.

Give the elevation factor (sea-level reduction factor) formula.

$$EF = \frac{R}{R + h}$$ where $R$ is the mean radius of the Earth (or ellipsoid) and $h$ is the height of the observation above the ellipsoid. It reduces a ground distance to the ellipsoid.

How do you convert a ground distance to a grid distance?

Multiply the ground (horizontal) distance by the combined factor: $$D_{grid} = D_{ground} \times k \times \frac{R}{R+h}$$ This first reduces to the ellipsoid (elevation factor) then to the grid (scale factor).

How do you convert a grid distance back to a ground distance?

Divide the grid distance by the combined factor: $$D_{ground} = \frac{D_{grid}}{k \cdot EF} = \frac{D_{grid}}{CF}$$

A line at elevation $h=1600$ m has $k=0.99996$. Using $R=6{,}372{,}000$ m, what is the combined factor?

Elevation factor $EF = \frac{6{,}372{,}000}{6{,}372{,}000 + 1600} = 0.99974897$. Combined factor $CF = 0.99996 \times 0.99974897 \approx 0.99970899$.

Distinguish geodetic, astronomic, and grid azimuths.

Geodetic azimuth: measured from geodetic (ellipsoidal) north, referenced to the ellipsoid normal. Astronomic azimuth: measured from astronomic north, referenced to the true gravity vertical (plumb line / celestial pole). Grid azimuth: measured from grid north (parallel to the projection central meridian) on the plane coordinate grid.

Give the Laplace equation relating astronomic and geodetic azimuth.

$$\alpha_{geodetic} = \alpha_{astronomic} - \eta \tan\phi$$ where $\eta$ is the east-west component of the deflection of the vertical and $\phi$ is the latitude. This Laplace correction converts an observed astronomic azimuth to a geodetic azimuth.

What is convergence (meridian convergence) $\gamma$ in a projection?

Convergence is the angle between grid north and geodetic (true) north at a point, caused by meridians converging toward the pole while grid lines stay parallel to the central meridian. It is zero on the central meridian and increases away from it.

Give the relationship between geodetic azimuth and grid azimuth using convergence and the t-T correction.

$$t = \alpha - \gamma + (t-T)$$ where $t$ is grid azimuth, $\alpha$ is geodetic azimuth, $\gamma$ is convergence, and $(t-T)$ is the arc-to-chord correction. Equivalently, grid azimuth $=$ geodetic azimuth $-$ convergence $+$ arc-to-chord.

Approximate the convergence angle on a Transverse Mercator/UTM projection.

$$\gamma \approx \Delta\lambda \sin\phi$$ where $\Delta\lambda$ is the longitude difference from the central meridian and $\phi$ is the latitude. (Higher-order terms exist but this is the leading approximation.)

What is the arc-to-chord (t-T) correction and why is it needed?

On the projection, the image of a geodesic is slightly curved, but a measured/plotted line is a straight chord. The (t-T) correction is the small angle between the chord direction ($t$) and the curved projected geodesic direction ($T$). It is needed for precise direction/azimuth reductions, growing with line length and distance from the central line.

What is a local (assumed) datum / assumed coordinate system, and when is it used?

It is an arbitrary, self-contained coordinate system with an assumed origin coordinate and an assumed reference direction (often magnetic or arbitrary north), not tied to a national datum. Used for small isolated surveys (e.g., a single boundary or construction site) where connection to a geodetic datum is unnecessary.

State the basic principle of GNSS positioning (trilateration by ranging).

A receiver determines its position by measuring distances (ranges) to four or more satellites whose positions are known. With four unknowns — three coordinates $(X,Y,Z)$ plus the receiver clock bias $t$ — at least four simultaneous ranges are required to solve the position.

Name and contrast the two fundamental GNSS observables.

Pseudorange (code) observable: derived from code correlation; unambiguous but noisy (meter-level). Carrier-phase observable: measures phase of the carrier wave; very precise (mm-level) but contains an unknown integer cycle ambiguity $N$ that must be resolved.

Write the simplified pseudorange observation equation.

$$P = \rho + c(dt_r - dt_s) + I + T + \varepsilon$$ where $\rho$ is the true geometric range, $c$ the speed of light, $dt_r$ and $dt_s$ the receiver and satellite clock errors, $I$ ionospheric delay, $T$ tropospheric delay, and $\varepsilon$ noise/multipath.

What is the integer ambiguity in carrier-phase GNSS?

The carrier-phase measurement gives the fractional phase plus an unknown whole number of carrier wavelengths between satellite and receiver at lock-on. That unknown integer $N$ must be resolved ('fixed') to achieve centimeter-level positioning.

List major GNSS error sources.

Satellite orbit (ephemeris) errors, satellite clock errors, ionospheric delay, tropospheric delay, multipath, receiver noise, and antenna phase-center variation. Ionospheric delay is frequency-dependent and can be largely removed with dual-frequency observations.

What is DOP and what does a low value indicate?

Dilution of Precision (DOP) describes how satellite geometry amplifies measurement error into position error. Low DOP (well-spread satellites) gives good geometry/precision; high DOP (clustered satellites) degrades the solution. Variants include PDOP, HDOP, VDOP, and GDOP.

Define WGS 84 and ITRF and their relationship.

WGS 84 is the geocentric reference frame/ellipsoid used by GPS (maintained by the U.S. DoD). ITRF (International Terrestrial Reference Frame) is the highly precise global frame maintained by the IERS. Recent WGS 84 realizations are aligned to within a few centimeters of contemporaneous ITRF realizations.

Why must an ITRF/WGS 84 position include an epoch?

Because tectonic plates move (centimeters per year), station coordinates are time-dependent. A position is only meaningful with its epoch (e.g., 2010.00) plus a velocity, so coordinates can be propagated to a common date for comparison.

What is OPUS and what does it provide?

OPUS (Online Positioning User Service) is an NGS web tool: a user uploads a static GNSS data file from a single receiver, and OPUS processes it against CORS reference stations to return precise NSRS coordinates (typically in NAD 83 and ITRF) for the occupied point.

What is the CORS network?

The Continuously Operating Reference Stations network — a set of permanent GNSS base stations whose precise coordinates/velocities are published by NGS. Their archived data provide the control framework used by OPUS and for differential/post-processed positioning, giving access to the NSRS.

Contrast forward and inverse geodetic position computations.

Forward (direct) problem: given a starting point's coordinates, a geodetic azimuth, and a distance, compute the coordinates of the second point and the reverse azimuth. Inverse problem: given two points' coordinates, compute the geodetic distance and forward/reverse azimuths between them.

What rigorous algorithms are commonly used to solve the geodetic forward/inverse problem on the ellipsoid?

Vincenty's iterative formulae and, more rigorously, Bowring's or Karney's algorithms solve for geodesic distance and azimuths on the ellipsoid. They account for ellipsoidal eccentricity, unlike simple plane or spherical approximations.

Why can plane (grid) coordinate geometry be used directly in SPCS but not on the raw ellipsoid?

Because SPCS is a conformal projection onto a flat grid, plane trigonometry (COGO) applies directly to grid coordinates. On the ellipsoid you must use geodetic (geodesic) formulas because the surface is curved and azimuths and distances follow geodesics, not straight lines.

What corrections relate an observed (ground) horizontal angle/direction to a grid direction in SPCS?

Two corrections: convergence $\gamma$ (rotates geodetic north to grid north) and the arc-to-chord $(t-T)$ correction (accounts for the curved projected geodesic vs. the straight grid chord). Distances additionally need the combined (grid) factor.

Dual-frequency GNSS receivers are used mainly to remove which error, and why does it work?

They remove (most of) the ionospheric delay. The ionosphere is dispersive — its delay is inversely proportional to the square of the frequency ($\propto 1/f^{2}$) — so comparing two frequencies (e.g., L1 and L2) allows the ionospheric term to be modeled and eliminated by an ion-free linear combination.

What is the difference between relative (differential) and absolute (PPP) GNSS positioning?

Relative/differential positioning uses simultaneous data from a base station (known coordinates) and rover to cancel common errors via differencing, yielding a vector/baseline. Absolute Precise Point Positioning (PPP) uses a single receiver with precise satellite orbit/clock products to obtain a global position without a local base.

Why are orthometric heights, not ellipsoidal heights, used for engineering/hydraulic work?

Water flows according to gravity (equipotential surfaces), which the geoid represents. Orthometric heights are gravity-related, so they correctly indicate which way water flows. Ellipsoidal heights are purely geometric and can decrease in the direction water actually flows, making them unsuitable for drainage.

State the geodetic-to-grid reduction sequence for a measured slope distance in SPCS.

(1) Reduce slope to horizontal (ground) distance; (2) multiply by the elevation factor to reduce to the ellipsoid; (3) multiply by the scale factor to reduce to grid. Steps 2 and 3 combined are the combined/grid factor: $D_{grid} = D_{horiz} \times EF \times k$.