Multi-type Random Patch Seeds (CARS)

• The paper demonstrates that integrating CARS in PLUS models leads to a 5.1% FOM improvement and better alignment with observed landscape metrics compared to traditional CA methods.
• CARS is a multi-type patch seeding mechanism that enables spontaneous, non-contiguous patch generation based on intrinsic suitability and probabilistic thresholds.
• Its algorithmic implementation, featuring threshold decay and Random Forest-estimated probabilities, effectively captures the spatial heterogeneity observed in real-world land use simulations.

The multi-type random patch seed mechanism (CARS) is a core component of contemporary patch-generating land use simulation (PLUS) models, designed to advance the simulation of land use and land cover (LULC) change within raster-based cellular automata (CA) frameworks. Unlike conventional CA approaches, which restrict new patch genesis strictly to contiguous, neighborhood-driven growth, this mechanism empowers each land-use class to initiate new spatially isolated patches (“seeds”) across the landscape, triggered by their intrinsic spatial suitability and a probabilistic threshold. This systematic expansion supports the dynamic simulation of highly fragmented, heterogeneous, and multi-type landscapes observed in both natural and anthropogenic systems (Liang et al., 2020).

1. Definition and Conceptual Framework

Multi-type random patch seeds are defined as the stochastic mechanism, embedded within a raster-based CA, whereby each land-use type $k$ (e.g., urban, cropland, forest) possesses its own set of seed-generation rules. At each simulation iteration, new “seeds” can initiate for any $k$ not only by typical neighbor propagation (i.e., where cells of type $k$ are adjacent), but also at cells $i$ where there are no $k$-type neighbors—provided that a uniform random variable $r \sim U(0,1)$ falls below the cell’s intrinsic conversion probability $P_{i,k}$. This approach bypasses the limitations of classic CA, wherein patch formation was strictly a function of spatial adjacency, and enables spontaneous, realistic emergence of fragmented patches representative of real-world land use dynamics.

Newly seeded cells subsequently propagate by conventional neighborhood-competition dynamics, producing irregular, mixed-type patch mosaics without manual seed specification or external deterministic rules.

2. Mathematical Description

The CA in PLUS integrates both neighbor-driven and seed-driven land use change. The overall probability that cell $i$ converts to land-use type $k$ at iteration $t$, $O_{i,k}^t$, is calculated as follows:

• Neighbor-Driven Growth (if $i$ has one or more type-$k$ neighbors):

$O_{i,k}^t = P_{i,k} \cdot n_{i,k}^t \cdot D_k^t$

where: - $P_{i,k}$ is the intrinsic probability (Random Forest-estimated) for cell $i$ to become $k$; - $n_{i,k}^t$ is the fraction of the $n \times n$ neighborhood in state $k$; - $D_k^t$ is the self-adaptive demand coefficient, enforcing global area constraints.

• Random-Seed Initiation (if $n_{i,k}^t = 0$ and $r < P_{i,k}$):

$O_{i,k}^t = (r \cdot \mu_k) \cdot D_k^t$

where $\mu_k \in (0,1)$ sets the rarity of new seeds for $k$.

Complete rule: $$O_{i,k}^t = \begin{cases} P_{i,k} n_{i,k}^t D_k^t & \text{if } n_{i,k}^t > 0, \ (r \mu_k) D_k^t & \text{if } n_{i,k}^t = 0 \wedge r < P_{i,k}, \ 0 & \text{otherwise,} \end{cases}$$ with further selection subject to a threshold $T$ and a transition-permitted matrix $\mathit{TM}_{k,c}$. Only $O_{i,k}^t > T$ and allowed $k \to c$ transitions may proceed, and $T$ decays after each conversion: $T \leftarrow \delta \cdot U(0,1)$ with $0 < \delta < 1$.

Roulette-wheel selection resolves competition among possible types when multiple $k$ satisfy these criteria.

3. Algorithmic Implementation

The patch seeding and growth process may be summarized as a single CARS module iteration:

Inputs: L^(t−1)[i] ← current land-use per cell P[i,k] ← RF-estimated type-conversion probabilities Q_target[k] ← global target areas μ[k], δ, Step ← user parameters (seed threshold, decay, demand step) TM[k,c] ← conversion permissions neighborhood size n 1. Compute initial area gap G_k = current_area(k) - Q_target[k] 2. For each iteration t: a. Update D_k^t from area gap ΔG_k b. For all cells i and types k ≠ current type: i. Compute neighbor fraction n_i,k ii. Draw r ~ U(0,1) iii. If n_i,k > 0: O_i,k ← P[i,k] × n_i,k × D_k^t Else if (n_i,k=0 and r < P[i,k]): O_i,k ← (r × μ[k]) × D_k^t Else: O_i,k ← 0 c. For cell i: - Candidate types C_i = {k | O_i,k > T and TM[current,k]=1} - If C_i ≠ ∅: Update threshold T ← δ × U(0,1) Choose c ∈ C_i via roulette selection weighted by O_i,k L^(t)[i] = c Else: L^(t)[i] = L^(t−1)[i] d. Recompute area errors G_k; terminate if sum below tolerance

4. Parameterization and Calibration

In the PLUS Wuhan case study (2003–2013), the following settings were adopted:

• Neighborhood: $3 \times 3$ Moore window (all 8 neighbors equally weighted)
• RF model: 50 trees, 14 predictor variables, 5% random sampling
• Seed thresholds: $\mu_k = 0.1$ for all $k$, established via trial and error to render spontaneous seeding rare but consequential
• Threshold decay: $\delta = 0.9$, yielding a gradual relaxation of $T$
• Demand regulation: area-correction step of 500 ha/iteration
• Transition matrix: restricted to plausible transitions (e.g., cropland $\to$ urban allowed; water $\to$ urban disallowed)
• All parameters underwent sensitivity analysis to maximize the Figure-of-Merit (FOM) over the calibration period

5. Comparison with Pixel-Based Cellular Automata

Classical pixel-based CA can only facilitate outward growth of pre-existing land-use patches. The genesis of new patches, particularly for land types absent in the current neighborhood, requires either manual seeding or external prescriptive logic. In contrast, the CARS mechanism:

• Enables automatic, probabilistic initiation of new patches for any land-use type in locally suitable, non-contiguous map areas
• Effectively captures the emergence of small, fragmented, and isolated land-use features (e.g., forest fragments, wetlands)
• Addresses edge-bias by allowing distant, high-suitability cells to convert absent spatial adjacency
• Preserves temporal coherence by applying a descending probability threshold, mitigating stochastic flip noise

Empirical evaluation in Wuhan revealed a FOM improvement from 0.2514 (CA without seeding) to 0.2642 (with CARS), a 5.1% relative gain. Seven of fifteen landscape metrics (encompassing number of patches, patch size distribution, spatial contiguity, and adjacency statistics) achieved their closest matches to observed reality under the CARS approach.

6. Empirical Performance and Insights from the Wuhan Case

The application of CARS within PLUS produced a FOM of 0.2642 for 2003–2013, outperforming the benchmark FLUS model (FOM = 0.1895). Landscape error patches were notably smaller and more evenly distributed, with fragmentation patterns in forest and cropland classes more faithfully reproduced.

Across 15 landscape-level metrics, PLUS with CARS ranked first or second in all, and first in 7 metrics. Analysis of transition rules using the random forest component evidenced that grassland expansion was favored in remote, low-anthropogenic-impact zones, deciduous forest patch origination clustered near arterial roads, and urban growth was strongly associated with proximity to tertiary roads—patterns discernible only when spontaneous patch seeding is operative.

Scenario simulations to 2035 under sustainable development objectives demonstrated that CARS-generated landscapes featured more contiguous forest corridors and compact urban morphologies than pixel-CA models, all while honoring multi-objective demand constraints.

7. Implications and Broader Significance

The multi-type random patch seed mechanism in the PLUS framework permits a mathematically parsimonious yet algorithmically robust representation of both contiguous and non-contiguous land expansion. This facilitates quantitative reproduction of real-world spatial heterogeneity, enables nuanced attribution of LULC transitions to underlying drivers, and advances the fidelity of policy-supporting scenario analyses. The approach, substantiated by multi-metric empirical gains in the Wuhan case, highlights the importance of incorporating stochastic seed initiation within raster CA for realistic land use simulations (Liang et al., 2020).