Methodology & Mathematical Formulations

This manual details the mathematical equations and spatial algorithms underlying Landuse Analyst’s computational solvers.

I developed these formulations to bring agronomic precision to landscape archaeology. Below are the explicit definitions for Annual Caloric Demand, Arable Land Requirements, Von Thünen Land Zoning, Octagonal Cost-Grid Propagation, Tobler Hiking Velocity, and Minetti Metabolic Costing.

Three-panel schematic. Left: concentric Von Thünen rings around a settlement — intensive horticulture, arable grain, extensive grazing. Centre: a 3x3 grid of square cells with eight arrows from the centre cell, four orthogonal at cost d and four diagonal at cost root-two d, inside the regular octagon those step costs produce, with the four-neighbour Manhattan diamond dotted inside it. Right: the Minetti curve of metabolic energy cost against slope. — Figure 1 — Von Thünen ring zoning (`A1`), 8-neighbour octagonal cost propagation (`A2`), and Minetti slope-energy costing (`A3`).

1. Annual Caloric Requirement Equation

The annual caloric requirement $E_{\text{annual}}$ for a settlement of population $P$ with mean per-capita daily energy demand $E_{\text{daily}}$ (default $2,100 \text{ kcal/person/day}$) equals:

$$E_{\text{annual}} = P \cdot E_{\text{daily}} \cdot 365$$

For a benchmark settlement of $P = 1,200$ inhabitants:

$$E_{\text{annual}} = 1,200 \times 2,100 \times 365 = 9.198 \times 10^8 \text{ kcal/year}$$

2. Crop Mix & Land Requirement Equation

The question, then, is how much cultivated land is required to yield this energy. The required arable area $A_{\text{req}}$ (in hectares) — accounting for crop proportions $f_c$, crop yields $Y_c$ ($\text{kg/ha}$), caloric densities $K_c$ ($\text{kcal/kg}$), and a biennial fallow multiplier $\beta$ ($\beta = 2.0$ for 1:1 fallow) — is:

$$A_{\text{req}} = \beta \cdot \frac{E_{\text{annual}}}{\sum_{c} \left( f_c \cdot Y_c \cdot K_c \right)}$$

Default Crop Baseline Values:

Cereals (Wheat/Barley): $f_{\text{cereal}} = 0.65$, $Y_{\text{cereal}} = 1,200 \text{ kg/ha}$, $K_{\text{cereal}} = 3,300 \text{ kcal/kg}$.
Legumes (Pulses): $f_{\text{pulse}} = 0.20$, $Y_{\text{pulse}} = 800 \text{ kg/ha}$, $K_{\text{pulse}} = 3,400 \text{ kcal/kg}$.
Horticulture (Olives/Vines): $f_{\text{hort}} = 0.15$, $Y_{\text{hort}} = 600 \text{ kg/ha}$, $K_{\text{hort}} = 2,100 \text{ kcal/kg}$.

3. Octagonal Cost-Grid Propagation Algorithm

Doing it this way allowed me to eliminate Manhattan grid distortion without incurring the heavy computational cost of full continuous vector propagation. Landuse Analyst uses an 8-neighbor octagonal cost-propagation algorithm (A2):

Orthogonal Step Cost ($\Delta x = 0$ or $\Delta y = 0$): $$d_{\text{ortho}} = \text{pitch} \times f_{\text{friction}}$$
Diagonal Step Cost ($\Delta x \ne 0$ and $\Delta y \ne 0$): $$d_{\text{diag}} = \sqrt{2} \cdot \text{pitch} \times f_{\text{friction}}$$

Accumulated cost $T(u, v)$ to reach cell $(u, v)$ from origin $(u_0, v_0)$ is the minimum path cost:

$$T(u, v) = \min_{\text{paths } P} \sum_{e \in P} d(e)$$

4. Terrain Slope Cost Functions

Tobler Hiking Function (1993)

Predicts walking speed $W$ ($\text{km/h}$) as a function of slope gradient $s = \tan(\theta)$:

$$W(s) = 6 \cdot \exp\left(-3.5 \cdot \left| s + 0.05 \right|\right)$$

Travel-time cost per metre $t_{\text{step}}$ (seconds/metre):

$$t_{\text{step}}(s) = \frac{3.6}{W(s)}$$

Minetti Metabolic Cost Equation (1987)

Calculates metabolic energy expenditure $C_w$ ($\text{J / kg}\cdot\text{m}$) as a 5th-order polynomial of slope gradient $i$ (A3):

$$C_w(i) = 280.5 i^5 - 58.7 i^4 + 33.2 i^3 + 1.8 i^2 + 7.8 i + 2.5$$

On level ground ($i = 0$), $C_w = 2.5 \text{ J/kg}\cdot\text{m}$. On steep inclines ($i = +0.20$), $C_w$ increases to $>7.5 \text{ J/kg}\cdot\text{m}$.

Agricultural Catchment Model — Application of caloric equations.
Barriers & Travel-Time Sandbox — Application of Tobler and Minetti functions.
Glossary — Mathematical definition index.