Degradation risk is modelled using a truncated exponential decay function [Figure 1], where the independent variable d represents the Euclidean distance from the project boundary to the nearest deforested pixel. This exponential formulation follows well-established distance-decay relationships in landscape ecology, where the intensity of ecological processes decreases as a function of distance from disturbance sources^[27,30,39]. Empirical studies have consistently shown that proximity to forest edges influences microclimatic conditions, biomass dynamics, and biodiversity patterns, with effects attenuating nonlinearly with the distance^[20,40,41].

In this analysis, the dependent variable (R(d)) is the buffer radius, calculated for each point along the project edge, from which a risk zone of variable size is defined.

(1) $$ \begin{equation} \begin{aligned} R(d)=\left\{\begin{array}{cc} \alpha e^{-\beta_{d}} & d \leq \alpha \\ 0 & d > \alpha \end{array}\right. \end{aligned} \end{equation} $$

Where:

• R(d) is the buffer radius, defining the internal limit of degradation risk, expressed in meters;

• α is the maximum distance of anthropogenic matrix influence within the project area, expressed in meters;

• β is the exponential decay rate, governing how rapidly the influence attenuates with distance, expressed in m^-1.

• d is the distance between each point on the edge of the project area and the nearest deforested area within the landscape, expressed in meters;

• $$e^{-\beta_{d}}$$ is the exponential term, unitless.

The truncation ensures that external influence is bounded by a biologically plausible maximum distance, beyond which effects are assumed negligible. When d = 0 (minimum possible distance), the influence reaches its maximum (R(d) = α); when d > α, influence decays to zero. The model’s behavior is illustrated in Figure 1, which is based on hypothetical data.

This nearest-distance model is particularly applicable in landscapes where deforestation constitutes the primary and spatially continuous disturbance driver, exhibiting spatial autocorrelation and predominantly isotropic edge effects, such that distance to the nearest deforestation serves as a conservative proxy for degradation exposure.

The exponential decay model arises as the analytical solution of a first-order differential equation describing the proportional decay with distance, a property characteristic of diffusion-like ecological processes. As the exponential decay model is grounded in well-established mathematical properties of diffusion, attenuation, and boundary-driven processes, the parameters α and β are defined theoretically^[35,42].

Within this formulation, parameters emerge from the mathematical structure of the differential equation and from ecological first principles that govern the structure and dynamics of the landscape, not from data calibration. This is consistent with mechanistic modeling in landscape ecology, where parameters are derived from theoretical descriptions of spatial propagation, edge influence, fragment geometry, and landscape resistance, rather than solely from empirical regression.

The parameter α represents the maximum spatial extent of degradation influence generated by surrounding deforestation. It is interpreted as the upper bound of edge-driven degradation pressure acting on the project area. Alpha is obtained from the following steps: (i) RR delineation; (ii) Building of a binary matrix with deforestation data; (iii) Computing a Euclidean distance surface; (iv) Estimation of the Weighted Landscape Class Proportion (PL_w); and (v) Applying the equation to estimate the alpha.

The RR is defined as a buffer surrounding the project area, with radius scaled to project size. It provides the spatial data needed to set the parameters for the variable internal RACZ. The radius of the RR is calculated using a geometric expression that scales proportionally to the project area, ensuring consistent spatial representation of potential threats in different project contexts.

(2) $$ \begin{equation} \begin{aligned} R R_{{radius }}=\sqrt{\frac{P A / 2}{\pi}} \end{aligned} \end{equation} $$

Where:

• RR_radius is the buffer radius of the RR, expressed in meters;

• PA is the project area, expressed in square meters.

The term (PA/2) establishes a sufficiently large reference radius to capture the surrounding landscape context while avoiding an excessive spatial extent that would dilute the representation of local pressure.

Based on the calculated radius, we define an outer buffer zone to represent a RR, from which the data used to quantify external deforestation pressure is obtained.

We generated a binary land-cover matrix representing the presence or absence of deforestation within the RR, to quantify the spatial influence of deforestation over the area. Under this framework, deforestation refers to the current state of land cover, based on stable anthropogenic land-use classes, defined as the long-term or permanent conversion of forested land into non-forested land as a result of human activities. For this analysis, we used the land use and land cover (LULC) mapping from the MapBiomas platform, collection 10 (launched in August 2025) for 2024^[43]. MapBiomas provides annual, wall-to-wall LULC classifications for Brazil, based on the Random Forest algorithm for processing Landsat images, with a resolution of 30 m^[44].

The MapBiomas dataset was subsequently reclassified into a binary deforestation matrix, in which anthropogenic land uses are assigned a value of 1 (deforested), while all remaining land-cover classes are assigned a value of 0 (non-deforested). The resulting matrix serves as the fundamental layer from which we calculate the Euclidean distance (d) to the nearest deforested pixel, generating a continuous "distance to deforestation" surface. This distance raster is then used in the remaining steps.

After generating the binary deforestation matrix, the next preparatory step consists of computing Euclidean distance surface to quantify how far each location within the study area lies from the nearest deforested pixel. The Euclidean distance algorithm measures the shortest straight-line distance between every non-deforested cell (value 0) and the nearest cell classified as deforested (value 1), producing a continuous raster in which each pixel contains a distance value expressed in map units. This approach follows the standard convention in landscape ecology, where proximity to edges and disturbance sources is represented as straight-line distance due to its geometric clarity and established empirical performance^[27,33,37]. Because edge influence and degradation processes tend to diffuse outward from deforestation boundaries in approximately radial patterns, distance, from Euclidean distance algorithm, has long been recognized as the most appropriate metric for capturing spatial exposure to anthropogenic pressure^[19,30,38]. The resulting distance raster captures the spatial gradient of deforestation intensity and, in addition to being used to determine alpha, serves as the main input variable (d) for the exponential degradation model.

The Weighted Landscape Class Proportion (PL_w) is an original metric proposed in this study, designed to quantify deforestation pressure, in terms of magnitude and proximity, by integrating class proportion with a distance-based weighting function. It is constructed to translate the quantity and spatial configuration of deforestation within the RR into an effective measure of external anthropogenic pressure.

The calculation of the PL_w is performed based on the following steps:

(i) Creation of a weight raster w(d), in which each pixel receives an individual weight based on its distance from the edge (Euclidean distance). To achieve this, the RR is divided into concentric circles reflecting the spatial distribution of deforestation in terms of the distance of each deforested pixel from the project edge. Each deforested pixel then receives a weight from 1 at the edge of the project to 0 at the outer limit of the ring, decaying with distance.

(3) $$ \begin{equation} \begin{aligned} w\left(d_{i}\right)=1-\frac{d_{i}}{R R_{{radius }}} ; ~0 \leq d_{i} \leq R R_{{radius }} \end{aligned} \end{equation} $$

Where:

• w(d_i) is the distance-based weight assigned to pixel i, (unitless);

• d_i is the Euclidean distance between the deforested pixel i and the project boundary, expressed in meters;

• RR_radius is the buffer radius of the RR, expressed in meters.

The result provides a spatial representation of the landscape structure, based on deforestation distribution.

(ii) Estimation of a weighted average that favors what is closest to the edge, regardless of the geometry or size of the RR.

(4) $$ \begin{equation} \begin{aligned} P L_{w}=\frac{\sum_{i=1}^{n} w\left(d_{i}\right) b_{i}}{\sum_{i=1}^{n} w\left(d_{i}\right)} \end{aligned} \end{equation} $$

Where:

• PL_w is the Weighted Landscape Class Proportion, ranging from 0 to 1 (unitless);

• w(d_i) is the distance-based weight assigned to pixel i, unitless;

• n is the total number of pixels within the RR;

• b_i is a binary variable equal to 1 if pixel i is deforested and 0 otherwise.

Lower PL_w values reflect negligible or spatially distant deforestation relative to the project boundary, whereas higher values denote greater deforestation intensity and increasing edge-proximal concentration.

Unlike a simple proportional metric, PL_w captures not only how much deforestation exists in the surrounding landscape, but also how spatially proximate it is to the project area, thereby providing a more meaningful representation of pressure gradients.

The parameter α defines the maximum extent to which degradation can spread from the edge of a project area. Its value is derived empirically by applying a pressure adjustment factor (Fp) to the radius of the RR (RR_radius) (see Equation 5), ensuring that α responds proportionally to the level of external disturbances to which the area is exposed. Ultimately, α represents the structural attributes of the landscape in the surroundings of the area, in relation to the deforestation class.

(5) $$ \begin{equation} \begin{aligned} \alpha=F p * R R_{{radius }}, \quad F p=\sqrt{P L_{w}} \end{aligned} \end{equation} $$

Where:

• α is the maximum spatial extent over which external anthropogenic pressure exerts influence on the Project Area, expressed in meters;

• RR is the radius is the radius of the RR, expressed in meters;

• Fp is the pressure adjustment factor, unitless.

The parameter β controls the rate at which the degradation influence attenuates with distance in the exponential model. It represents the structural susceptibility of the forested landscape to edge-driven degradation and is determined by fragment size and fragment shape^[40], two well-established controls on edge exposure and core-area buffering in landscape ecology. Fragments with higher edge-to-area ratios, typically associated with smaller size or irregular geometry, are more exposed to external influences and therefore, exhibit a smoother decay curve (lower β). Conversely, larger or more compact habitats show steeper attenuation of pressure (higher β). In these formulations, β acts as a modulator of the response, reflecting the degree of ecological resilience or susceptibility of the landscape to external stress.

Operationally, beta is obtained from two independent components:

(i) a shape-based circularity index (CI), representing geometric exposure to edge effects; and

(ii) an area-based resistance factor (k), representing the buffering effect of fragment size.

Fragment shape is quantified using the CI, a dimensionless metric that relates area to perimeter and provides a proxy for edge susceptibility^[45]. Shape metrics are strongly linked to fragmentation and edge exposure^[33]. CI measures how closely a fragment approximates a perfect circle, which minimizes edge length for a given area.

(6) $$ \begin{equation} \begin{aligned} \mathrm{CI}=\frac{4 \pi \mathrm{A}}{\mathrm{p}^{2}} \end{aligned} \end{equation} $$

Where:

• CI is the circularity index, unitless;

• A is the project area, expressed in square meters;

• p is the project perimeter, expressed in meters.

CI ranges from 0 to 1. Values approaching 1 indicate compact geometries with low edge exposure, while values near 0 indicate elongated or irregular shapes with high edge density and increased susceptibility to external disturbance.

Fragment size influences degradation dynamics primarily through the proportion of interior (core) habitat because no standardized size index exists at the single-fragment level^[41]. We defined an area factor (k) that captures the buffering capacity associated with fragment size. The relationship between area (A) and factor (k) is monotonically increasing and non-linear in nature—that is, larger areas have higher k values, but with decreasing growth (smaller marginal gains as A increases). This behavior is represented using a smooth, continuous function derived from monotonic cubic Hermite interpolation by segments (splines)^[46], following established approaches for preserving monotonicity in ecological scaling^[47]. k(A) is defined as the continuous function interpolated in logarithmic space, from previously defined anchor points.

(7) $$ \begin{equation} \begin{aligned} \mathrm{k}(\mathrm{A})=10^{\mathrm{f}\left(\log _{10} \mathrm{A}\right)} \end{aligned} \end{equation} $$

Where:

• k(A) is the area factor, unitless;

• A is the project area, expressed in the same units used to define the anchor points;

• f is a monotonic cubic Hermite spline function fitted to predefined anchor points in log-area space.

This formulation yields a continuous, analytically tractable estimate of k for each project, avoiding discontinuities at class boundaries, and allows recalibration if additional reference points are introduced [Figure 2].

The parameter β integrates the effects of fragment size and shape into a single structural control on degradation propagation. This combination allows β to reflect both a fragment's intrinsic capacity to dampen disturbances (area factor) and its geometric propensity to amplify edge effects (CI), resulting in a biologically significant and landscape-responsive parameter in the exponential degradation model.

Beta is calculated as:

(8) $$ \begin{equation} \begin{aligned} \beta=\beta_{0}\left(1-\mathrm{e}^{(-\mathrm{k} * \mathrm{Cl})}\right), \beta_{0}=1~\mathrm{m}^{-1} \end{aligned} \end{equation} $$

Where:

• β is the decay rate of the exponential model, expressed in m^-1;

• β₀ is a scaling coefficient to preserve dimensional consistency, expressed in m^-1;

• k is the Area Factor, unitless;

• CI is the circularity index, unitless.

The parameters CI and k are descriptors of the area’s structure and do not carry spatial units. Their role in the model is to modulate the rate at which the influence of external pressure decays across space. The dimensional component of the decay parameter is provided by β₀. This formulation allows the model to maintain dimensional consistency while preserving its ecological interpretation as a measure of its susceptibility to degradation.

Mathematically, the equation ensures that β decreases with both a decrease in project area and an increase in shape irregularity (i.e., lower CI). Conversely, beta increases as the project area grows and CI increases. Consequently, lower decay rates are produced in structurally susceptible areas, while higher rates are produced in more stable areas.

The RACZ framework was applied to two contrasting case-study landscapes to demonstrate model behavior under differing disturbances. In both cases, land-cover data were reclassified into a binary deforestation matrix, from which Euclidean distance-to-deforestation surfaces were generated and used as inputs to the exponential degradation model.

The first case represents a low-pressure landscape, characterized by limited surrounding deforestation and weak fragmentation dynamics. The second represents a high-pressure frontier landscape, where extensive deforestation generates strong spatial gradients of degradation risk. Together, these cases demonstrate the flexibility, scalability, and structural sensitivity of the RACZ framework.

Case study 1 [Figure 3] is a tropical forest called Kokolopori Bonobo Natural Reserve, located between the provinces of Tshuapa and Tshopo in the Democratic Republic of Congo, within the Congo Basin tropical rainforest, part of the Afrotropical moist broadleaf forest biome. The region lies in the Af classification of the Köppen-Geiger system and is characterized by high precipitation, consistently high temperatures, and the absence of a dry season. These conditions sustain dense evergreen forests with high biomass and complex vertical structure, forming part of one of the largest contiguous tropical forests in the world.

Case study 2 [Figure 4] is located within the Brazilian Amazon and falls under the Am classification of the Köppen-Geiger climatic system, characterized by uniformly high temperatures, a pronounced wet season, and a short dry period. The area corresponds to the indigenous land Parakanã in Pará, an officially recognized Indigenous Territory containing extensive tracts of mature evergreen forest. Although these conditions support the functional integrity of dense tropical forests, with their high levels of biomass accumulation and year-round ecological productivity, the project area is located within an expanding agricultural frontier and is subject to persistent deforestation due to logging, the expansion of ranching and regional infrastructure. These conditions generate high edge exposure and strong spatial gradients of degradation risk along the project boundary.

In Kokolopori Bonobo Natural Reserve, Democratic Republic of the Congo, deforestation within the RR was minimal [Figure 5A-1]. The weighted landscape class proportion (PL_w) was 0.0008, indicating that less than 0.1% of the surrounding areas contribute to deforestation on the project boundary. Such an extremely low value reflects the scarcity of consolidated deforestation around the site and confirms that the project area is embedded in a largely intact forest matrix. Consequently, the correction factor applied to the initial α parameter, derived from PL_w, was also extremely small, resulting in a maximum ecological influence distance (α) of only 719 m. This short maximum reach already signals that anthropogenic effects do not propagate far into the landscape, consistent with an environment under minimal external disturbance.

The structural characteristics of the project area reinforce this pattern of low vulnerability. The site encompasses more than 400,000 hectares and exhibits a relatively compact geometry (CI = 0.61), resulting in a relatively high decay rate (β = 0.0043m^-1). Therefore, whatever disturbance pressure exists diminishes rapidly with increasing distance from deforested pixels. When this β value is combined with the α parameter, the resulting exponential decay function produces a RACZ of only 1,150 hectares, around 0.27% of the entire area. Spatially, this RACZ is concentrated in narrow strips adjacent to isolated deforestation points and does not extend far into the forest interior [Figure 5A-2 and A-3].

In contrast, the indigenous land Parakanã exhibited extensive surrounding deforestation [Figure 5B-1]. The resulting PL_w value of 0.83 indicates that more than 80% of the surrounding landscape contributes directly to deforestation-driven pressure on the project boundary. This level of exposure is characteristic of frontier contexts where deforestation is widespread, spatially aggregated, and strongly connected. Consequently, the correction factor applied to the initial α parameter was extremely high, producing a maximum external influence distance (α) of 19,407 m.

The structural characteristics of the project area further modulate this spatial influence. Despite encompassing over 340,000 hectares, the site exhibits a shape factor of 0.49, reflecting a moderately irregular geometry with a relatively high perimeter-to-area ratio. Such configurations reduce the stability of interior habitats by increasing the proportion of forest areas exposed to edge influence. Under these conditions, the model estimated a decay rate of β = 0.0027 m^-1, which corresponds to a slower decline of disturbance intensity, i.e., higher external influence penetration, relative to the very large α. The result of applying the model to each point on the area's edge is shown in Figure 5B-2. The various overlapping buffers are merged and inserted into the area to generate the RACZ [Figure 5B-3]. When combined in the exponential decay formulation, the large α and relatively lower β produced a RACZ of 337,985 hectares [Figure 5B-2 and B-3], extending across most of the project area.

This expansive RACZ reflects the ecological reality of a landscape under intense anthropogenic pressure. Disturbance does not dissipate quickly and edge effects penetrate deeply into forest interiors due to both strong external forcing and structural fragmentation. The resulting pattern demonstrates how the RACZ framework responds sensitively to the joint effects of pressure and susceptibility, capturing a broad spatial footprint of vulnerability in contexts where degradation is both pervasive and spatially far-reaching.

The RACZ framework should not be understood as a methodological complement to existing ICVCM-approved REDD+ standards, but rather as a distinct and additional scope within nature-based solutions. REDD+ methodologies estimate avoided emissions by projecting future deforestation and quantifying the carbon losses that would occur in the absence of the project. By contrast, RACZ does not model deforestation trajectories; instead, it delineates the spatial extent of forest area that is vulnerable to degradation driven by proximity to anthropogenic land uses.

Whereas REDD+ methodologies model the risk of forest loss using distance to remaining forest as a predictor—consistent with its objective of forecasting where new clearing might occur—the RACZ model uses distance to deforestation to represent how already-established edges propagate structural and functional degradation into standing forests. This distinction aligns directly with empirical evidence, showing that edge-driven degradation reduces biomass, ecosystem function, biodiversity, and growth rates, even in forests that remain nominally intact.

In this sense, RACZ introduces a new conceptual scope for nature-based climate solutions, designed to protect the ecological integrity and growth potential of vulnerable forest margins. Rather than extending or substituting existing REDD+ frameworks, RACZ expands the methodological landscape by formalizing the spatial component of degradation risk and enabling crediting approaches that reflect the well-documented influence of fragmentation and anthropogenic proximity on long-term forest carbon dynamics.

The following are the potential implications of the RACZ approach for conservation-based carbon removal credits and the main innovations of the methodology.

The RACZ framework offers a pathway for substantially improving the environmental integrity of carbon crediting by embedding ecological risk directly into the spatial definition of creditable areas. Current nature-based methodologies, particularly those relying on avoided deforestation, typically delineate project boundaries using projected deforestation risk or administrative units, implicitly assuming that vulnerability is uniform across landscapes that are, in reality, highly heterogeneous^[5,52]. Such assumptions can lead to crediting forest areas that face little or no disturbance pressure, thereby weakening additionality and overstating climate benefits.

By contrast, RACZ introduces risk-adaptiveness as a core design principle. By integrating external anthropogenic pressure (operationalized as distance to deforestation) with internal ecological susceptibility (derived from fragment size, shape, and structural configuration), the model identifies the portions of a project area where degradation is most likely to occur. This ensures that crediting zones are geographically restricted to areas with demonstrable, ecologically justified vulnerability, rather than areas selected through uniform or administratively convenient rules.

In doing so, RACZ improves crediting accuracy by aligning carbon benefits with actual landscape processes, reducing the risk of over-crediting, and increasing transparency for both project developers and standard-setting bodies. The result is a crediting framework that is more scientifically defensible, more consistent with contemporary understanding of forest degradation dynamics, and better suited to support high-integrity nature-based climate solutions. Therefore, RACZ provides a more defensible and scientifically coherent basis for avoiding over-crediting while strengthening environmental integrity across carbon portfolios.

The proposed approach provides a scientifically grounded delineation of the areas that crediting areas reflect actual zones of vulnerability. By deriving spatial risk directly from the α-β parameterization, RACZ translates decades of empirical knowledge on distance-decay processes, landscape ecology, and structural susceptibility^{[20,23,26,33,50,53]} into an operational framework.

This evidence-based approach has direct implications for assessments of project performance. Because α captures the external pressure envelope and β encodes internal structural resistance, RACZ identifies zones where biomass, canopy structure, and ecological function are most vulnerable to degradation, areas that traditional deforestation-only baselines fail to recognize^[14,45]. In doing so, RACZ enhances not only project-level accuracy but also portfolio-level conservativeness, an increasingly important criterion in international carbon standards aiming to improve environmental integrity.

Finally, the RACZ framework also strengthens conservation practice by identifying where forests are most vulnerable to edge-driven degradation, enabling more targeted deployment of restoration, fire prevention, and connectivity interventions. By operationalizing these insights into a practical spatial product, RACZ strengthens the capacity of conservation programs to anticipate degradation before it becomes irreversible, ultimately improving the long-term resilience of forest landscapes.

Although the RACZ framework provides a mechanistic, ecologically grounded approach for delineating degradation-risk-adaptive conservation zones, several important limitations should be acknowledged.

First, the current approach infers spatial vulnerability from structural and contextual predictors—distance to deforestation, landscape configuration, and pressure indices—without yet incorporating temporal validation of degradation processes. While extensive literature supports the assumption that RACZ-identified areas are more susceptible to biomass loss, microclimatic alteration, and biodiversity decline, the model has not been explicitly tested using time-series observations of canopy height, above-ground biomass, spectral degradation indices, or species-level responses within RACZ zones. Longitudinal validation using LiDAR, radar backscatter, repeat optical imagery, or ecological plot networks will be essential to confirm whether RACZ-delineated areas indeed undergo accelerated structural or functional degradation, and to refine the decay parameters accordingly.

Second, some components of the model, particularly the area factor, shape-based adjustments, and landscape-specific decay rates, were calibrated using a combination of ecological theory and generalized empirical regularities. While these formulations capture broad ecological patterns, they may not fully represent biome-specific or region-specific dynamics. Forests differ widely in their sensitivity to fragmentation. For example, humid tropical forests often show deep structural degradation gradients, whereas dry forests, savannas, and temperate systems may exhibit different decay rates or disturbance propagation pathways. Future work should therefore explore biome-specific parameterizations, including alternative α and β distributions, distinct formulations for shape and area-derived susceptibility, and possibly additional modifiers such as climatic seasonality, fire regimes, hydrological connectivity, or species composition.

A related limitation lies in the structure of the decay function. The model assumes a simple exponential model, which is consistent with diffusion-based ecological theory but may not capture multimodal or threshold-driven behavior observed in some ecosystems, such as abrupt biomass collapse following fire incursion, nonlinear feedbacks from edge-driven mortality, or spatial contagion reinforced by human activities. Likewise, the assumption that the nearest deforested pixel is the dominant source of degradation pressure may underestimate risks in landscapes with multiple interacting disturbances dynamically over time, such as selective logging, chronic understory fire, or edge-induced drought stress.

Revalidation of the model with updates of the RACZ area’s delineation, incorporating multi-source disturbance layers and probabilistic representations of disturbance interactions, may help refine RACZ estimates. The framework assumes that deforestation is the primary and spatially continuous driver of disturbance, such that distance to the nearest deforestation provides a proxy for degradation risk. This assumption may be less appropriate in landscapes dominated by multi-source or strongly directional disturbances, where cumulative or anisotropic effects are not explicitly captured. Integrating multi-source disturbance into the model, as well as with predictive models of land-use change, fire spread, or road expansion could strengthen its utility for long-term conservation planning and dynamic carbon-crediting.

At a broader level, an important limitation is the dependence on the quality of deforestation and land-cover data. Spatial errors in deforestation detection, classification bias, or temporal inconsistencies may propagate uncertainty into the risk surface.

Future research should therefore focus on three key areas:

(1) Temporal validation, using independent multi-sensor datasets (GEDI, ICESat-2, Sentinel-1/2, Landsat harmonics) to test whether RACZ-designated zones systematically undergo greater biomass, canopy, or biodiversity loss.

(2) Biome and region-specific parameterization, ensuring that α, β, and the area-factor spline reflect structural and ecological realities of distinct forest types.

(3) Model expansion and integration, incorporating additional disturbance pathways, alternative decay functions, and links to predictive spatial models.

Despite these limitations, the RACZ framework represents an important step toward more ecologically realistic, empirically grounded, and risk-adaptive zoning for carbon crediting.