Fundamentals of Surveying Exam (FS) Geospatial Systems and Coordinate Geometry Syllabus

Why are a minimum of four satellites required for a GNSS position fix?

There are four unknowns: three position coordinates ($X, Y, Z$) and the receiver clock bias $\Delta t$. Because the receiver clock is not synchronized to GPS time, each measured range is a 'pseudorange' containing clock error, so a fourth satellite equation is needed to solve for the clock offset along with position.

Define a pseudorange and write its basic observation equation.

A pseudorange is the apparent satellite-to-receiver distance from signal travel time, biased by the unsynchronized receiver clock: $$P = \rho + c\,\Delta t + \varepsilon$$ where $\rho$ is the true geometric range, $c$ the speed of light, $\Delta t$ the receiver clock bias, and $\varepsilon$ combined other errors.

Contrast code-phase (pseudorange) positioning with carrier-phase positioning in terms of precision.

Code-phase positioning uses the C/A or P code with precision at the meter level (autonomous positioning). Carrier-phase positioning measures fractions of the carrier wavelength (about $19$ cm for L1) and, after resolving integer ambiguities, achieves millimeter-to-centimeter precision used for geodetic surveying.

What is the integer ambiguity in carrier-phase GNSS, and why must it be resolved?

The receiver can measure the fractional carrier phase precisely but cannot directly count the unknown whole number of full wavelengths between satellite and receiver at lock-on. This unknown integer is the integer (cycle) ambiguity. Resolving (fixing) it to the correct integer converts the precise phase measurement into a precise range, enabling centimeter accuracy.

Describe static GNSS surveying and its typical application.

In static surveying, two or more receivers occupy fixed stations and log carrier-phase data simultaneously for an extended period (typically 30 minutes to several hours). Long observation allows reliable ambiguity resolution and produces the highest accuracy, used for control network/geodetic work and long baselines.

What is Real-Time Kinematic (RTK) GNSS, and what accuracy does it provide?

RTK uses a stationary base receiver on a known point transmitting carrier-phase corrections in real time to a roving receiver, which resolves ambiguities on the fly. It delivers centimeter-level accuracy (typically $1$–$2$ cm horizontal) instantly, used for stakeout and topographic surveys, with range limited (about $10$–$20$ km) by base-rover separation.

What is a Real-Time Network (RTN) / Networked RTK using a CORS infrastructure?

An RTN uses a network of permanent Continuously Operating Reference Stations (CORS) to model atmospheric and orbital errors over a region and stream corrections (e.g., VRS, FKP, MAC) to rovers. It provides RTK-level accuracy without the surveyor needing to set up a private base, over a much larger coverage area.

Differentiate absolute (autonomous) positioning, differential positioning (DGPS), and PPP.

Absolute/autonomous: single receiver, meter-level. Differential (DGPS): corrections from a base on a known point remove common errors, sub-meter to meter (code) or cm (carrier). PPP (Precise Point Positioning): single receiver using precise satellite orbit/clock products achieves cm-dm accuracy globally without a local base, but requires a long convergence time.

List the major GNSS error sources affecting GPS measurements.

Satellite clock errors, satellite orbit (ephemeris) errors, ionospheric delay, tropospheric delay, multipath, receiver clock error, receiver noise, and antenna phase-center variation. Selective Availability (now disabled) was a former intentional error.

How does the ionosphere affect GNSS signals, and how is the delay largely eliminated?

The ionosphere is dispersive: it delays the code and advances the carrier by an amount proportional to total electron content and inversely proportional to frequency squared ($\propto 1/f^{2}$). Because the effect is frequency dependent, using dual-frequency observations (L1 and L2) lets the receiver form a combination that nearly eliminates the first-order ionospheric delay.

Why is the tropospheric delay harder to remove than the ionospheric delay, and how is it handled?

The troposphere is non-dispersive at GNSS frequencies, so dual-frequency combinations cannot cancel it. It depends on temperature, pressure, and humidity (wet and dry components). It is handled with mapping-function models (e.g., Saastamoinen, Hopfield) and by estimating residual zenith delay as a parameter; differencing also cancels it over short baselines.

What is multipath error and how can it be mitigated?

Multipath occurs when the signal reaches the antenna via reflections (from ground, water, buildings) in addition to the direct path, corrupting the range. Mitigation: choke-ring or ground-plane antennas, avoiding reflective surfaces, longer observation times, antenna placement away from obstructions, and advanced receiver correlator designs.

Define DOP (Dilution of Precision) and name its common components.

DOP is a dimensionless number describing how satellite geometry amplifies measurement error into position error: $$\sigma_{position} = DOP \times \sigma_{measurement}$$ Components: GDOP (geometric), PDOP (position 3D), HDOP (horizontal), VDOP (vertical), and TDOP (time). Lower DOP (good spread of satellites) is better; PDOP under about $4$ is desirable.

Why does poor satellite geometry (high DOP) degrade a GNSS fix even with good ranging?

When satellites are clustered close together in the sky, the intersecting spheres meet at shallow angles, producing an elongated, poorly defined error volume. A wide, evenly distributed spread of satellites makes the spheres intersect near right angles, tightening the solution and lowering DOP.

What is a CORS station and its role in GNSS network design and datum realization?

A Continuously Operating Reference Station is a permanently mounted GNSS receiver/antenna logging data continuously at a precisely known position. The NOAA/NGS CORS network provides the framework that realizes the national datum (NAD83), supports OPUS solutions, and serves as control for differential and PPP positioning.

What is OPUS and what does it provide to surveyors?

OPUS (Online Positioning User Service), run by NGS, processes a user-submitted static GNSS data file against nearby CORS stations and returns precise published coordinates (NAD83 and ITRF) for the occupied point, typically within centimeters, without the user operating a base of their own.

Distinguish a horizontal datum from a vertical datum, with U.S. examples.

A horizontal datum defines latitude/longitude on a reference ellipsoid (e.g., NAD27 on Clarke 1866; NAD83 on GRS80). A vertical datum defines heights/elevations relative to a reference surface (e.g., NGVD29 based on tidal stations; NAVD88 based on a single tidal benchmark, Father Point/Rimouski, and the geoid).

Differentiate ellipsoidal height ($h$), orthometric height ($H$), and geoid height ($N$), and give the relationship.

Ellipsoidal height $h$ is measured from the ellipsoid (what GNSS provides). Orthometric height $H$ is measured from the geoid (mean sea level, what leveling provides). Geoid height (geoid undulation) $N$ separates the two surfaces. The relationship is: $$h = H + N \quad\Rightarrow\quad H = h - N$$

How does a surveyor obtain accurate orthometric heights from GNSS ellipsoidal heights?

Apply a geoid model (e.g., NGS GEOID18) that supplies the geoid undulation $N$ at the point, then compute $H = h - N$. GNSS gives $h$ directly; without the geoid model the elevations would be ellipsoidal, not the orthometric elevations used for engineering and drainage.

Define a GIS (Geographic Information System) and its core functional capabilities.

A GIS is a computer system for capturing, storing, managing, analyzing, and displaying spatially referenced (geographic) data. Core capabilities: data input/capture, storage and database management, spatial query and analysis (overlay, buffering, proximity), and visualization/output (maps and reports).

Contrast a LIS (Land Information System) with a general GIS.

A LIS is a specialized GIS focused on land parcels and legal/cadastral land records—ownership, boundaries, easements, tax assessment, and land use. It emphasizes parcel-based data and legal accuracy (cadastre), whereas a general GIS handles any geographic theme (environmental, transportation, utilities, demographics).

Compare the vector and raster data models in GIS.

Vector represents features as points, lines, and polygons defined by coordinate geometry—precise boundaries, compact for discrete features, good topology. Raster represents space as a grid of cells (pixels), each with a value—well suited to continuous data (elevation, imagery) and overlay math, but resolution-limited and storage-heavy.

What are the three vector geometry primitives and an example feature each represents?

Point (zero-dimensional, e.g., a survey monument or well), Line/arc (one-dimensional, e.g., a road centerline or stream), and Polygon/area (two-dimensional closed feature, e.g., a parcel or lake). Vertices store coordinates; topology stores connectivity.

Define topology in GIS and give the rules it enforces.

Topology describes the spatial relationships between features—connectivity, adjacency, and containment—independent of exact coordinates. Rules include: polygons must not overlap or have gaps, lines must connect at nodes, and adjacent polygons share boundaries. Topology enables network analysis and ensures data integrity.

List the components of spatial data quality typically reported in metadata.

Positional accuracy (horizontal and vertical), attribute (thematic) accuracy, logical consistency (topological correctness), completeness, lineage (source and processing history), and temporal accuracy (currency/timeliness). These describe how fit a dataset is for a given use.

What is metadata in the GIS context and why is it required?

Metadata is 'data about data'—documentation describing a dataset's content, source, accuracy, projection/datum, scale, creation date, lineage, and contact. It is required so users can assess fitness for use, correctly combine datasets (matching datum/projection), and avoid misusing data of unknown quality; standards include FGDC CSDGM and ISO 19115.

Distinguish accuracy from precision in the context of spatial data.

Accuracy is closeness of a measured or recorded value to the true value (correctness). Precision is the closeness of repeated measurements to one another (repeatability) and the level of detail recorded. Data can be precise but inaccurate (tightly clustered but biased) or accurate but imprecise.

What is a coordinate transformation, and how does it differ from a map projection conversion?

A coordinate transformation mathematically relates coordinates in one reference frame/datum to another (e.g., NAD27 to NAD83, or local to geodetic) by translation, rotation, and scaling. A projection conversion changes how coordinates are mapped onto a plane within the same datum. A datum transformation changes the underlying reference frame itself.

Describe the 2D conformal (Helmert/four-parameter) coordinate transformation and its parameters.

A 2D conformal (similarity) transformation maps one plane system to another preserving shape, using four parameters: a scale factor $s$, a rotation $\theta$, and two translations $T_X, T_Y$: $$X = s(x\cos\theta - y\sin\theta) + T_X$$ $$Y = s(x\sin\theta + y\cos\theta) + T_Y$$ At least two common control points are required to solve it.

What is a seven-parameter (3D Helmert/Bursa-Wolf) datum transformation, and what does each group of parameters represent?

It relates two 3D geodetic datums in ECEF space using seven parameters: three translations ($\Delta X, \Delta Y, \Delta Z$) shifting the origin, three rotations ($\omega, \phi, \kappa$ or $R_X, R_Y, R_Z$) aligning the axes, and one scale factor ($s$). It is the standard rigorous transformation between datums such as NAD83 and ITRF/WGS84.