7.4 GIS Spatial Data Models and Topology

5.1 GIS Spatial Data Models & Topology

Geographic Information Systems (GIS) provide the technical framework for capturing, managing, analyzing, and visualizing spatially referenced land information. For professional surveyors sitting for the SURCON examination, mastering spatial data abstraction, topological modeling, and coordinate transformation mechanics is critical. GIS translates complex real-world physical boundaries, terrain elevations, and ownership rights into structured digital formats.

Spatial Data Models: Vector vs. Raster

Spatial data in GIS is conceptualized through two fundamental data abstractions: the Vector Data Model and the Raster Data Model.

The Vector Data Model

The vector data model represents spatial features as discrete geometric objects defined by explicit coordinate pairs $(x, y)$ or $(x, y, z)$ within a specified Coordinate Reference System (CRS). It models spatial reality using three primary primitives:

1. Points (0D): Single coordinate pairs representing zero-dimensional features where area and length are zero (e.g., trigonometric stations, boundary beacons, soil sampling points).
2. Lines / Polyline (1D): Ordered sequences of connected vertices representing one-dimensional linear features with length but no area (e.g., centerlines of roads, property boundary segments, rivers, pipelines).
3. Polygons (2D): Enclosed, multi-sided planar shapes defined by closed loops of vertices where the starting vertex equals the ending vertex. Polygons represent two-dimensional features with measurable area and perimeter (e.g., cadastral survey land parcels, administrative LGA boundaries, water bodies).

Vector data structures can be stored as un-topological spaghetti structures (where each feature is stored independently as an isolated set of coordinates without shared geometry) or as topological vector structures (where adjacent features share common nodes and boundary arcs explicitly).

The Raster Data Model

The raster data model divides geographical space into a continuous, regular tessellation of square grid cells (pixels) arranged in rows and columns. Each cell holds a single numeric attribute value representing a physical variable or land cover class across that cell's area.

Key raster parameter definitions include:

• Spatial Resolution: Defined by the dimensions of a single cell on the ground (e.g., $0.5\text{ m} \times 0.5\text{ m}$ high-resolution aerial imagery vs. $30\text{ m} \times 30\text{ m}$ Landsat satellite imagery).
• Georeferencing & Affine Transformation: Aligning un-projected raster images to a projected coordinate system using Ground Control Points (GCPs) and first-order polynomial affine transformations: $x' = A x + B y + C$ $y' = D x + E y + F$ where $(x, y)$ are raster image pixel coordinates and $(x', y')$ are projected ground map coordinates.
• Continuous vs. Discrete Rasters: Continuous rasters represent phenomena without sharp boundaries (e.g., Digital Elevation Models - DEMs, temperature gradients), whereas discrete rasters represent categorical themes (e.g., land use / land cover classifications).

Vector vs. Raster Paradigm Comparison

Spatial Topology and Planar Enforcements

In GIS, topology refers to the mathematical study of spatial properties and geometric relationships that remain completely unchanged (invariant) under continuous transformations such as stretching, scaling, rotation, or projection. Unlike non-topological geometry, topology explicitly defines how points, lines, and polygons connect, share boundaries, and enclose spaces.

The Three Fundamental Topological Relationships

1. Connectivity (Arc-Node Topology): Line segments (arcs) meet at shared endpoints called nodes. Each arc possesses a directional flow defined by a From-Node (starting junction) and a To-Node (ending junction). This structure allows network tracing (e.g., utility pipelines, road traffic routing) and prevents dangling lines.
2. Containment (Polygon-Arc Topology): An enclosed area (polygon) is defined by an ordered loop of arcs that form its outer perimeter and optional inner loops representing holes or islands. The GIS identifies which arcs enclose a specific polygon identifier, ensuring accurate area calculations.
3. Adjacency (Left-Right Polygon Topology): Every arc maintains explicit references to the polygon lying on its left side and the polygon lying on its right side along its direction of digitizing. This prevents boundary duplication, ensuring adjacent survey parcels share a single boundary line without gaps or overlapping areas.

Topological Errors & Automated Cleaning

When converting field cadastral survey data into GIS vector layers, digitizing and measurement errors frequently create topological defects:

• Dangles (Undershoots & Overshoots): Undershoots occur when an arc fails to extend far enough to touch an intersecting boundary node; overshoots occur when an arc extends past an intersecting node. Automated node snapping within a defined cluster tolerance merges nearby vertices.
• Slivers & Gaps: Tiny, spurious polygons formed along shared boundaries due to non-identical digitizing of the same parcel edge. Clean-up routines eliminate slivers by merging them with adjacent larger polygons based on area thresholds.
• Unclosed Polygons & Self-Intersections: Polygons whose initial and terminal vertices fail to match, or lines that cross over themselves forming invalid figure-eight geometries.

Spatial Data Queries, Buffering, and Map Projections

Spatial Analysis Operations

• Attribute Queries: Filtering features using Structured Query Language (SQL) based on non-spatial tabular values (e.g., WHERE Land_Use = 'Commercial' AND Area_SqM > 1000).
• Spatial Queries & Predicates: Selecting features based on location relative to other spatial layers using OGC topological predicates: Intersects, Contains, Within, Touches, Crosses, and Disjoint.
• Vector Buffering: Generating enclosed proximity zones around point, line, or polygon features at a specified distance. Euclidean buffering applies constant distances on flat Cartesian planes, whereas geodesic buffering calculates true distances over the curved surface of the ellipsoid.

Map Projections & Datum Transformations in GIS (QGIS & ArcGIS)

Geodetic coordinates measured on the curved Earth ellipsoid ($\phi, \lambda$) must be mathematically projected onto flat two-dimensional map surfaces $(E, N)$. In Nigeria, cadastral surveying and digital mapping utilize two primary reference systems:

1. Minna Datum (Clarke 1880 Ellipsoid): The traditional national geodetic datum for Nigeria, projected onto Universal Transverse Mercator (UTM) Zones 31N ($3^\circ\text{E}$ central meridian), 32N ($9^\circ\text{E}$ central meridian), and 33N ($15^\circ\text{E}$ central meridian).
2. WGS84 Datum (GRS80 Ellipsoid): The global geodetic datum used by satellite GNSS (GPS). Coordinate values referenced to WGS84 (EPSG:4326) differ from Minna Datum coordinates by up to hundreds of meters on the ground.

To transform coordinates between WGS84 and Minna Datum in GIS software (QGIS or ArcGIS Pro), surveyors must apply Bursa-Wolf 7-Parameter Transformations or Molodensky 3-Parameter Transformations:

• Translation Parameters: $(\Delta X, \Delta Y, \Delta Z)$ accounting for origin center shifts.
• Rotation Parameters: $(R_x, R_y, R_z)$ accounting for axial orientation differences.
• Scale Factor: $(\Delta S)$ accounting for relative ellipsoid scale variations.

# PyQGIS Script: Automated Topological Validation and Datum Transformation
from qgis.core import QgsVectorLayer, QgsCoordinateReferenceSystem, QgsCoordinateTransform, QgsProject

# Load raw cadastral survey shapefile (Minna Datum / UTM Zone 31N)
layer_path = "/data/cadastral_parcels_minna.shp"
src_layer = QgsVectorLayer(layer_path, "Cadastral_Parcels", "ogr")

# Define Source (Minna UTM 31N - EPSG:26331) and Target CRS (WGS84 - EPSG:4326)
src_crs = QgsCoordinateReferenceSystem("EPSG:26331")
tgt_crs = QgsCoordinateReferenceSystem("EPSG:4326")

# Set up Coordinate Transformation engine
transform = QgsCoordinateTransform(src_crs, tgt_crs, QgsProject.instance())

# Iterate features, validate geometry, and calculate transformed area
for feature in src_layer.getFeatures():
    geom = feature.geometry()
    if not geom.isGeosValid():
        print(f"Topological Error detected in Parcel ID: {feature['PARCEL_ID']}")
    else:
        area_ha = geom.area() / 10000.0
        print(f"Parcel {feature['PARCEL_ID']} is Topologically Valid. Area: {area_ha:.4f} Ha")