Series 1: Foundations of Computational Geography

Series Overview

This foundational series reconstructs geography as a quantitative modelling discipline, introducing mathematical concepts through observable spatial phenomena. Rather than treating mathematics as prerequisite knowledge, we develop calculus, differential equations, and computational thinking through geographic problems that provide intuition and motivation.

Pedagogical Philosophy

Mathematics emerges from observation. We begin with patterns visible in landscapes and human systems, then develop mathematical tools to describe and predict these patterns. This inverts the traditional pedagogy: instead of “learn math, then apply it,” we pursue “observe phenomena, develop mathematics to understand it.”

Computational implementation follows derivation. Every mathematical concept is implemented in code, demonstrating that equations are executable instructions. Students see that mathematical models are programs, bridging abstract notation and concrete computation.

Learning Objectives

By completing this series, learners will be able to:

Translate spatial phenomena into mathematical models using variables, units, and functional relationships
Derive and solve differential equations for growth, decay, and equilibrium processes
Implement numerical solutions for systems without analytical solutions
Interpret model behavior through phase plots, stability analysis, and parameter sensitivity
Apply vector calculus to terrain analysis and spatial gradients
Understand Earth’s geometry through spherical coordinates and rotational dynamics
Model human spatial behavior using gravity models and diffusion processes
Critique model assumptions and identify failure modes

Mathematical Progression

The series builds mathematical sophistication systematically:

Models 1-4 (Cluster A): Single-variable calculus

Linear functions and rates of change
Exponential and logarithmic functions
First-order differential equations
Stability and equilibrium concepts

Models 5-6 (Cluster B): Applications of exponential processes

Beer-Lambert law and vertical profiles
Trigonometric functions for solar geometry
Vector dot products

Models 7-9 (Cluster C): Multivariable calculus and optimization

Functions of two variables (terrain elevation)
Partial derivatives and gradient vectors
Graph theory and flow routing
Local optimization

Models 10-11 (Cluster D): Spatial interaction models

Inverse power laws and distance decay
Discrete-time dynamical systems
Network diffusion processes

Model 12 (Cluster E): Coordinate transformations

Spherical coordinate systems
Latitude/longitude to Cartesian
Rotational kinematics

Computational Skills Developed

Function implementation and visualization
Numerical integration (Euler, Runge-Kutta)
Gradient calculation from raster data
Graph algorithms (flow routing, shortest path)
Interactive parameter exploration
Vector field visualization

Prerequisites

Assumed: Algebra I (linear equations, exponents, basic graphing)

Not assumed: Calculus, differential equations, programming (taught from scratch)

Entry Points by Background

High school students (advanced): Start at Model 1, work sequentially. Expect 3-4 hours per model.

Undergraduates (STEM): Skim Models 1-3 if calculus background. Focus on modelling perspective.

Geography students (quantitative gap): Start Model 1. Series designed to build mathematical confidence.

Engineering students (spatial gap): Models 7-9 (terrain analysis) and 10-11 (human systems) provide geographic context for familiar mathematics.

Model Sequence

Cluster A: Modelling Change (Models 1-4)

Model 1: What Is a Spatial Model? Introduction to variables, units, dimensional analysis. Scalar fields. First computational implementations.

Model 2: Linear Change and Rate Slope as rate of change. Linear functions in geographic contexts (elevation profiles, population growth). Introduction to derivatives.

Model 3: Exponential Growth and Logarithms Population growth, compound interest. First differential equation: dP/dt = rP. Analytical vs numerical solutions.

Model 4: Logistic Growth and Equilibrium Carrying capacity and saturation. Phase plots. Stability analysis. Numerical integration.

Cluster B: Energy and Attenuation (Models 5-6)

Model 5: Light Attenuation in a Canopy Beer-Lambert law. Exponential decay with depth. Vertical profiles in forests and water columns.

Model 6: Solar Geometry and Projection Trigonometry for solar angles. Dot product as geometric projection. Aspect and hillshade.

Cluster C: Terrain and Spatial Derivatives (Models 7-9)

Model 7: Digital Elevation Models as Functions Two-variable functions. Finite differences. Raster data structures as discretized continuous functions.

Model 8: Gradient, Aspect, and Direction of Steepest Descent Partial derivatives. Gradient vector magnitude and direction. Slope and aspect calculation.

Model 9: Hydrological Flow as Optimization Steepest descent as local optimization. Graph construction from DEMs. Flow accumulation and drainage networks.

Cluster D: Human Systems as Mathematical Systems (Models 10-11)

Model 10: Gravity Models of Trade and Migration Inverse square and power laws. Distance decay. Calibration and parameter estimation. Human mobility as physics analogy.

Model 11: Spatial Diffusion of Innovation Discrete-time models. Logistic curves in space and time. Network adjacency. Contagion processes.

Cluster E: Observing the Earth (Model 12)

Model 12: Earth as a Rotating Sphere Spherical coordinates. Angular velocity and rotational period. Coordinate transformations. Earth observation geometry foundation.

Assessment Strategies

Formative:

Interactive parameter manipulation (immediate feedback)
Worked examples with detailed solutions
“What Could Go Wrong?” sections for common errors

Summative suggestions:

Derive model for novel geographic phenomenon
Implement and validate numerical solution
Sensitivity analysis on real dataset
Critical evaluation of model assumptions

Extensions and Pathways

For mathematical depth: Continue to Series 2 (energy balance, hydrology) for PDEs and conservation laws.

For spatial analysis: Jump to Series 3 (map algebra, interpolation, spatial statistics).

For remote sensing: Series 5 after completing Series 1 and basics of Series 2.

For hazards: Series 6 (atmospheric) requires Series 1 + selected Series 2 models.

Key Insights Students Gain

Geography requires quantitative reasoning. Spatial patterns demand mathematical description.
Models are simplifications with purpose. Perfect accuracy isn’t the goal; insight is.
Differential equations describe rates of change. Most geographic processes are rate-driven.
Numerical methods enable solutions. Most real equations lack analytical solutions.
Parameters control behavior. Small parameter changes can produce qualitatively different outcomes.
Human systems follow mathematical regularities. Social phenomena exhibit quantitative patterns.
Computation makes mathematics tangible. Code reveals what equations mean operationally.

Further Resources

All models are self-contained with complete derivations. Additional resources:

Interactive visualizations (ECharts/Canvas)
Raspberry Pi compatible implementations
Python/Julia code examples
Dataset links for real-world applications

Estimated Time Investment

Per model: 2-4 hours (reading + working example + implementation)

Full series: 30-50 hours for thorough engagement

Accelerated path: 20-30 hours (skim familiar material)

Next: Proceed to Series 2 (Environmental Systems) for energy balance, hydrological processes, and ecosystem modelling. Or skip to Series 3 (Spatial Analysis) for GIS-oriented techniques.