Fundamentals of GIS & Spatial Analysis

Geographic Information Systems (GIS) as a computational framework for storing, querying, analyzing, and visualizing spatially referenced data; The Earth's shape problem: geoids, ellipsoids, and datums (WGS84, NAD83) as the reference surfaces underpinning all coordinate systems; Geographic coordinate systems (latitude, longitude) vs. projected coordinate systems (UTM, Web Mercator): the cartographic projection problem as a mathematical mapping from a curved surface to a flat plane with unavoidable distortion tradeoffs (area, shape, distance, direction); Vector data model: points, polylines, and polygons as geometric primitives with associated attribute tables — the shapefile and GeoJSON as standard serialization formats; Raster data model: georeferenced grids of cell values as the representation of continuous spatial phenomena (elevation, temperature, land cover); Spatial resolution, extent, and the modifiable areal unit problem (MAUP) as fundamental data quality and scale considerations.

The spatial indexing problem: why standard B-tree indexes fail for two-dimensional range queries; R-trees as the canonical spatial index: minimum bounding rectangles, node splitting strategies, and the query algorithm for range and nearest-neighbor searches; Quadtrees as recursive spatial partitioning: region quadtrees for raster data and point quadtrees for vector data; Geohash and S2 cell hierarchies as space-filling curve encodings that map 2D spatial proximity to 1D string prefixes for use in standard key-value stores; Spatial joins: point-in-polygon, line intersection, and buffer overlap as the fundamental binary spatial predicates; Topological relationships formalized: the DE-9IM (Dimensionally Extended 9-Intersection Model) as the mathematical framework defining contains, intersects, touches, crosses, and disjoint; PostGIS as a spatial extension to PostgreSQL: geometry types, spatial indexes (GiST), and ST_ function families as a production spatial query engine.

Affine transformations: translation, rotation, scaling, and shearing as the 3 3 homogeneous matrix operations applied to georeferencing raster imagery; Georeferencing: establishing the correspondence between pixel coordinates and geographic coordinates using ground control points (GCPs) and least-squares polynomial fitting; Datum transformations: Helmert (7-parameter similarity) transformation for converting between geodetic reference frames; Raster resampling methods: nearest-neighbor, bilinear interpolation, and cubic convolution as tradeoffs between accuracy and computational cost when reprojecting grids; Vector reprojection: applying coordinate transformation functions point-by-point to convert geometries between coordinate reference systems; Error propagation in spatial data: positional accuracy, attribute accuracy, and lineage as the components of data quality metadata standards (ISO 19157).

Buffer analysis, Voronoi diagrams, and Delaunay triangulation as the foundational proximity computation primitives; Network analysis on road graphs: shortest path (Dijkstra, A*), service area computation (isochrones), and the traveling salesman problem as spatial optimization; Digital Elevation Models (DEMs): slope, aspect, hillshade, curvature, and viewshed analysis as raster-based terrain derivatives computed via finite-difference kernels; Interpolation of scattered point data to continuous surfaces: IDW (Inverse Distance Weighting), spline interpolation, and Kriging as a geostatistical interpolator that provides uncertainty estimates alongside predictions; Map algebra: local, focal, zonal, and global operations as a raster processing algebra analogous to array broadcasting; Spatial autocorrelation: Moran's I statistic as the formal test of whether nearby locations have more similar values than expected by chance, and its connection to the First Law of Geography.

The electromagnetic spectrum as the data source of satellite remote sensing: optical (multispectral, hyperspectral), SAR (synthetic aperture radar), and LiDAR as complementary sensing modalities; Radiometric calibration: converting raw digital numbers to at-sensor radiance and top-of-atmosphere reflectance as the preprocessing pipeline for optical imagery; Spectral indices as engineered features: NDVI (vegetation), NDWI (water), NDBI (built-up area) as normalized band ratios that isolate physical phenomena; Image classification: supervised (maximum likelihood, SVM) and unsupervised (k-means, ISODATA) approaches to assigning land cover labels to image pixels; Accuracy assessment: confusion matrix, overall accuracy, producer's and user's accuracy, and the Kappa coefficient as the validation framework for classified maps; Cloud computing for remote sensing: Google Earth Engine as a planetary-scale geospatial analysis platform where computation moves to the data rather than the data to the computation.