ForEdgeClim models three temperature components. The first is the forest surface temperature (Tf, K), which represents an effective, density-weighted temperature of forest element surfaces within each voxel, including leaves, branches, and stems.

The model also simulates air temperature (Tair, K) and soil surface temperature (Ts, K). Air temperature relates to the individual voxels. As each voxel has a structural density value between 0 and 1, the remaining fraction (1 – density) represents the proportion of air. The soil surface temperature is treated as a single-layer value that characterises the ground surface temperature.

Air temperature is estimated through a linear interpolation approach similar to that used in the microclimate models microclimc and its successor microclimf , which simulate only vertical energy and radiation fluxes. In contrast, ForEdgeClim also resolves lateral fluxes: 2Tair=(wmX+wmZ)gmTm+wsgsTs+wfgfTf(wmX+wmZ)gm+wsgs+wfgf, where w represents a dimensionless weighting factor, g a convection coefficient (Wm-2K-1), and T a temperature (K). Subscripts m, s, and f refer to the macroenvironment, soil surface, and forest surface, respectively, while subscripts mX and mZ denote the lateral (x-axis) and vertical (z-axis) macroenvironmental boundaries.

In this formulation, air temperature is treated as a state variable defined on an Eulerian grid, with each voxel representing a fixed position in space. Heat exchange between neighbouring voxels and between air and surrounding surfaces is parameterised through effective exchange processes, which represent the net effect of turbulent mixing and small-scale air movement within the canopy. This approach is consistent with commonly used formulations in microclimate and canopy energy balance models , where turbulent transport is not explicitly resolved but represented through bulk exchange coefficients.

In this context, linearisation refers to approximating a non-linear relationship – typically between net energy balance and air temperature – by a local linear function. This allows the model to update air temperature using a simplified linear equation rather than repeatedly solving a full non-linear energy balance formulation, as is done for the forest surface temperature, making it computationally more efficient while still retaining high accuracy. Such linearised closures are commonly assumed to be appropriate when air–surface temperature differences remain small relative to the absolute temperature, such that higher-order non-linear terms can be neglected. These conditions are generally associated with sufficient air mixing, moderate radiation and humidity levels, and relatively homogeneous forest structures, under which turbulent transport can be reasonably approximated through bulk exchange processes.

In addition, vegetation density (ρ, dimensionless) directly scales the magnitude of surface–air energy exchange. In the model, ρ represents the effective fraction of vegetated surface within a voxel and enters explicitly in the formulations of sensible and latent heat fluxes (Eqs.  and ). Voxels with higher density therefore exhibit stronger coupling between surface and air temperatures, whereas low-density voxels represent more open air space with reduced exchange.

The weights w are defined as exponentially decaying functions of the distance to a given boundary: 3ws=eαsds. Here, Eq. () shows the weighting for the soil surface (ws), where ds is the distance to the soil (m) and αs is defined as: 4αs=ln⁡(0.5)is, where the parameter is denotes the distance of influence (m), defined as the characteristic distance over which the effect of the soil surface temperature on air temperature decreases by 50 %. Exponential decay is appropriate for microclimate modelling, as it captures the gradual attenuation of influence with distance, maintains numerical stability, and reflects the physics of diffusive and convective heat transfer.

For voxels without structural elements (and therefore without a defined Tf), a virtual forest surface temperature is assigned by averaging the temperatures of the corresponding x-, y-, and z-plane surfaces.

The convection coefficients g and distances of influence i are prescribed model parameters treated as effective bulk exchange coefficients controlling the magnitude and spatial reach of heat exchange within the canopy. In the current model implementation, the coefficients g are prescribed as spatially uniform semi-empirical parameters and are not dynamically calculated from local wind speed, turbulence, or voxel-scale canopy structure. Instead, they represent characteristic canopy-scale exchange efficiencies under typical forest conditions. This effective parameterisation is commonly used in microclimate and canopy models where metre-scale turbulent transport and airflow dynamics cannot be explicitly resolved computationally . In this context, the coefficients g implicitly represent the combined effects of unresolved turbulent mixing, boundary-layer exchange, and small-scale convective heat transport within the canopy.

The soil surface temperature is modelled using the one-dimensional heat conduction equation (i.e., Fourier's law): 5Ts=Tsoil+Gzks. Here, Tsoil (K) is the observed soil temperature at a reference depth. In our setup (see Sect. ), this is measured at a depth of 8 cm at 20 locations within the forest transect. The variable z (m) represents the measurement depth (8 cm), G is the ground heat flux (Wm-2), and ks is the thermal conductivity of the soil (Wm-1K-1). The formulation represents conductive heat transfer between the soil surface and a shallow subsurface reference layer. This type of formulation is commonly used as a first-order approximation of ground heat exchange in models that do not explicitly resolve vertical soil heat transport . The reference temperature Tsoil varies over time following the measured soil temperature dynamics and is prescribed as a boundary condition. In the current implementation, ks is treated as a constant parameter. In reality, soil thermal conductivity depends strongly on soil moisture content and soil composition, which are not explicitly represented in the model. As a result, spatial and temporal variability in soil thermal properties is not captured. The computation of G is described in Sect. .

Radiative processes are simulated in two dimensions (vertical and lateral) using a two-stream radiative transfer model (RTM) based on the formulation implemented in the ED2.2 model . A detailed description of the ForEdgeClim RTM can be found in Appendix .

Briefly, the shortwave RTM simulates multi-scatter radiative transfer along a single column or row of voxels, where direct and diffuse sunlight interact with layered structures defined by voxel density (see Fig.  for a visualisation). Direct-beam radiation (Ib↓, Wm-2) follows an exponential decay: 6dIb↓dρ=-KbIb↓, while diffuse upward and downward components (I↑, Wm-2; I↓, Wm-2) are governed by the following coupled system of linear ordinary differential equations, which is subsequently discretised and solved as a linear system using a direct matrix solver based on LU decomposition: 7dI↓dρ=-[1-(1-β)ω]KdI↓+βωKdI↑+(1-β0)ωKbIb↓,8dI↑dρ=[1-(1-β)ω]KdI↑-βωKdI↓-β0ωKbIb↓. In these equations (derived as Eqs.  and in Appendix ), dρ denotes the change in forest density with depth in canopy (dimensionless), ω the shortwave scattering coefficient (dimensionless), and β and β0 the fractions of scattered diffuse and scattered direct-beam radiation in the backward direction (dimensionless). Kd and Kb are the diffuse and direct-beam extinction coefficients (dimensionless), respectively. All parameter definitions are given in Table . The two-stream formulations (Eqs. –) are presented in their continuous form for clarity. In ForEdgeClim, however, they are solved in a discretised form across the voxel grid, using finite-difference updates of Ib↓, I↓, and I↑ layer by layer along the radiative path.

To build physical intuition for the extinction coefficients: in dense canopies, vertical direct-beam extinction coefficients (Kb) typically approach 0.9, reflecting strong attenuation by leaves and branches. Diffuse radiation is attenuated less strongly (Kd<Kb), as it arrives from multiple directions and can pass through canopy gaps. Lateral attenuation coefficients are generally smaller than vertical ones because radiation can travel longer path lengths through the canopy, a consequence of leaves typically being more horizontally inclined . Seasonal variability in attenuation can be represented by adjusting Kb and Kd to account for changes in leaf phenology.

The shortwave RTM is implemented as a one-dimensional column model. To obtain a two-dimensional radiative field, it is applied sequentially to each vertical column (fixed x and y) and each horizontal row (fixed y and z) in the 3D voxel grid. Direct-beam solar radiation is partitioned between the vertical and lateral directions according to the solar elevation angle, whereas diffuse radiation is assumed to be isotropic, such that vertically and laterally incident diffuse fluxes are equal.

This formulation represents the three-dimensional radiation field as a set of coupled one-dimensional vertical and lateral radiative transfer problems. While this introduces a simplified representation of the angular distribution of radiation, it enables the model to capture the dominant radiative gradients associated with vertical attenuation and lateral radiation penetration at forest edges in a computationally efficient manner. The radiative transfer scheme is therefore directionally resolved within the vertical and lateral domains, but does not explicitly discretise the full three-dimensional angular radiation field. Anisotropy is thus only partially represented through the separation of vertical and lateral fluxes and the dependence of direct radiation on solar elevation angle.

Net radiation (Rn) represents the net radiative energy available at a surface, resulting from the balance between incoming and outgoing shortwave and longwave radiation. It comprises three shortwave RTM components – direct-beam (Ib↓), diffuse downward (I↓), and diffuse upward (I↑) – and two longwave RTM components – longwave downward (L↓) and longwave upward (L↑): 9Rn=Ib↓+I↓-I↑+L↓-L↑.

Longwave radiation is computed using an analogous two-stream formulation to that of the shortwave RTM, but without direct-beam terms and including thermal emission from forest surfaces. The longwave upward and downward components (L↑, Wm-2; L↓, Wm-2) are governed by the following coupled system of linear ordinary differential equations, which is subsequently discretised and solved as a linear system using a direct matrix solver: 10dL↓dρ=-[1-(1-β)ω]KlL↓+βωKlL↑+ϵfσTf4Kl,11dL↑dρ=[1-(1-β)ω]KlL↑-βωKlL↓-ϵfσTf4Kl. Here, Kl is the longwave extinction coefficient (dimensionless). The forest emissivity (ϵf, dimensionless) and Stefan–Boltzmann constant (σ, Wm-2K-4) define the emitted longwave flux. All model constants are given in the appendix, Table . As with the shortwave RTM, these longwave equations are numerically implemented in discretised form, with fluxes updated sequentially across voxels.

Ground heat flux (G) represents the transfer of energy between the ground surface and the underlying soil. It is modelled as a fixed proportion of the net radiation at the ground surface: 12G=p(1-ρ)Rn, following the approach implemented in the SCOPE 2.0 model . Here, p is the fraction of Rn that is absorbed by the soil surface and ρ is the forest structural density of the voxel layer directly above the soil (dimensionless). Ground heat flux is therefore reduced under dense vegetation cover, where less radiation reaches the soil surface. Throughout this manuscript, ρ refers to voxel vegetation density (dimensionless), whereas ρair denotes air density (kgm-3).

Equation () provides a simplified local closure of the surface energy balance and implicitly assumes that ground heat flux responds instantaneously to net radiation. As such, the formulation does not explicitly resolve temporal phase shifts associated with heat storage and delayed conductive heat transport within deeper soil layers.

The resulting flux is used to estimate soil surface temperature (Sect. ), which in turn influences near-surface air temperature (Sect. ) and the overall energy balance.

Sensible heat flux (H) represents the transfer of thermal energy between forest surfaces and the surrounding air. In ForEdgeClim, this process is simulated in three dimensions and includes two components: (i) heat exchange between adjacent air voxels: 13D=hAΔTairΔx,14Tair,new=Tair,old-DcpρairV, and (ii) heat exchange between forest elements and the air: 15H=ρgf(Tf-Tair), where h (Wm-1K-1) is an effective heat transfer coefficient governing air–air exchange between adjacent voxels, and gf (Wm-2K-1) is a bulk forest–air sensible heat transfer coefficient. Here, ρ represents the voxel-scale vegetation density (dimensionless), such that sensible heat exchange increases proportionally with the amount of vegetated surface present within a voxel.

Boundary conditions allow heat exchange with the macroenvironment at the canopy and forest edge, with the soil at the lower boundary, and impose a no-flux condition at the forest core.

In Eqs. ()–(), A is the surface area of one voxel face (m2), ΔTair is the air temperature difference between adjacent voxels (K), Δx is the voxel size (m), Tair,new and Tair,old are the updated and previous air temperatures (K), cp (Jkg-1K-1) is the specific heat capacity of air, ρair (kgm-3) is air density, V is the voxel volume (m3), and ρ is the voxel's forest structural normalised density (dimensionless).

The air–air heat exchange term (Eqs.  and ) represents effective thermal transport driven by local temperature gradients. This parameterisation does not explicitly resolve the underlying transport processes, but instead captures their combined effect at the voxel scale through the coefficient h. In practice, heat transport is expected to be dominated by turbulent mixing under most conditions, while molecular diffusion contributes only marginally.

The forest–air heat exchange term (Eq. ) represents the net sensible heat transfer between vegetation surfaces (e.g., leaves, branches, and stems) and the surrounding air. The coefficient gf is treated as an effective bulk sensible heat exchange parameter representing the characteristic efficiency of heat transfer between vegetation surfaces and surrounding air under typical canopy conditions. It implicitly accounts for unresolved turbulent mixing, boundary-layer convection, and small-scale air movement within the canopy.

This formulation allows sensible heat fluxes to respond to local temperature gradients and forest structural density, while maintaining computational efficiency within the voxel-based framework.

Latent heat flux (LE) represents the transfer of energy associated with phase changes of water, including both evaporation and transpiration (i.e., evapotranspiration) from forest surfaces to the atmosphere. In ForEdgeClim, LE is estimated using the empirical Priestley–Taylor method: 16LE=ραRns(Tf)s(Tf)+γ, which provides a simplified form of the Penman–Monteith equation by representing evapotranspiration as a function of available radiative energy . Here, ρ refers to voxel vegetation density (dimensionless), α (dimensionless) is the Priestley–Taylor coefficient, γ is the psychrometric constant (kPaK-1), and s(Tf) is the slope of the saturation vapour pressure curve.

In this formulation, evapotranspiration is represented as a bulk canopy flux. Stomatal regulation and canopy resistance are not explicitly resolved; instead, their combined effects, together with atmospheric controls, are implicitly captured in the empirical coefficient α. As a result, this approach provides a first-order approximation of latent heat flux under conditions where evapotranspiration is primarily energy-limited . Consequently, the current formulation does not explicitly represent dynamic stomatal responses to environmental drivers such as vapour pressure deficit, soil moisture limitation, or drought stress. The latent heat flux formulation should therefore be interpreted as a simplified first-order approximation of canopy evapotranspiration.

The slope of the saturation vapour pressure curve is defined as: 17s(Tf)=4098es(Tf)(Tf-35.85)2.

The saturation vapour pressure es(Tf) is estimated using the empirical formulation of Tetens: 18es(Tf)=0.6108⋅exp⁡17.27(Tf-273.15)Tf-35.85, which provides a reliable approximation of the Clausius–Clapeyron relationship for temperatures up to 50 °C .

To calibrate the model, we used the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) , a stochastic, derivative-free evolutionary optimisation algorithm well-suited for non-linear and non-convex objective functions. It can be used when the search space is complex, discontinuous, or contains multiple local optima . These properties make it highly appropriate for calibrating process-based environmental models, where objective functions are often non-smooth and derivative information is unavailable or unreliable . Moreover, CMA-ES has demonstrated strong and robust performance in relatively low-dimensional parameter spaces , which aligns with the scope of our calibration problem.

Following the Sobol sensitivity analysis (Sect. ), only parameters that together explain more than 65 % of the total variance of any condition were optimised, whereas all others were fixed at their mean literature values (Table ). Calibration was conducted separately for each season using three 24 h periods that captured distinct radiation regimes: the sunniest day, the cloudiest day, and the day with the strongest solar fluctuations over the period July 2023–April 2025 (Table  and, e.g., Fig. ). Cloudy hours were defined as those with a difference of less than 5 Wm-2 between total and diffuse radiation. These three days were selected to represent the dominant microclimatic contexts each season experiences and to ensure strong constraints on model processes.

In addition, a year-round calibration was performed using all 12 selected days (three per season). This allowed evaluation of whether a single parameter set can describe microclimatic processes consistently across the full annual cycle, and how its performance compares to the seasonal calibrations.

The objective function was the root mean square error (RMSE) between the simulated and observed air temperature. Observations were obtained from 15 TOMST TMS-4 sensors at 15 cm height (ten positions along the central horizontal transect line and five vertical positions along the tower, purple dots in Fig. ). Each seasonal calibration thus used 1080 observations (3d×24h×15sensors). To ensure a balanced influence of horizontal and vertical gradients, the five tower sensors were upweighted by a factor of two in the RMSE calculation.

To allow a consistent comparison of model performance across temporal scales, for all seasonal calibrations and the year-round calibration, the following performance metrics were computed against observations from TOMST TMS-4 temperature sensors: RMSE, mean error (ME), standard deviation of the residuals (SD), coefficient of determination (R2), and Nash–Sutcliffe Efficiency (NSE).

The TOMST TMS-4 sensors used for air temperature observations can occasionally be affected by sunflecks, where direct sunlight locally heats the sensor and biases the measurements due to overheating of the radiation shield. For all 24 h periods listed in Table , data was manually checked for sunflecks, by inspecting for abnormal temperature increases relative to neighbouring sensors. No sunflecks were detected on the most cloudy or the most solar fluctuating days; however, two affected measurements (10:00 UTC and 11:00 UTC) were identified on the sunniest summer day (7 July 2023) for one ground sensor located in an open canopy area. These data points were excluded from the calibration.

CMA-ES was run until convergence, with a maximum of 50 generations (7 offspring per generation). The number of generations required for convergence varied among seasons, reflecting seasonal differences in parameter sensitivity and model behaviour.

Although model calibration is commonly performed on a larger dataset than model validation, we adopted the opposite strategy. The three seasonally selected calibration days represent distinct and highly informative radiation regimes – sunny, cloudy, and strongly solar fluctuating – which provide stronger constraints on the underlying processes than a larger number of less diagnostic days. Using a small but diverse calibration set also reduces the risk of overfitting to the conditions of a particular month or synoptic situation. Validation was instead carried out on full representative months for each season (Sect. ), allowing a much more stringent assessment of parameter robustness under the full range of seasonal meteorological variability. This design ensures that parameters are calibrated on physically meaningful situations while being evaluated on all conditions relevant for model application.

Model validation was performed separately for each season using all days of a representative month: July 2023 (summer), October 2023 (autumn), January 2025 (winter), and April 2025 (spring). These months capture a wide range of microclimatic conditions and include the sunniest calibration day for each season. The seasonal validation sets contained 10 800 (or 11 160) observations (30/31d×24h×15sensors).

A year-round validation was also conducted using the combined dataset of all four representative months. This enabled assessment of how well the parameter set from the year-round calibration generalises across the full annual cycle.

Validations were run using both calibrated and uncalibrated parameter sets. In calibrated runs, only the most influential parameters identified through the Sobol analysis were replaced by their optimised values; all other parameters were kept at their mean literature values. Uncalibrated runs used mean literature values for all parameters.

For all validations – seasonal and year-round – the same performance metrics were computed by comparing modelled air temperatures with measurements from TOMST TMS-4 sensors (RMSE, ME, SD, R2, and NSE), enabling direct comparison of model accuracy and robustness under different calibration strategies.

Comparison of calibrated and uncalibrated runs allowed assessment of the representativeness of literature-derived parameter ranges. Parameter estimates falling near the boundaries of their prior distributions indicate that published ranges may be insufficiently constrained, whereas calibrated values lying well within these bounds suggest that literature-based parameters provide a reliable basis for model application.

To assess the potential influence of wind-driven processes not explicitly represented in the model, model residuals were analysed as a function of observed wind speed and distance to the forest edge.

In addition, to evaluate the uncertainty associated with forest surface temperature predictions, the Sobol parameter ensemble was propagated through the model, and the resulting distributions of simulated surface temperature were analysed across all validation conditions.

The calibrated parameter values and associated performance metrics for each season and for the full year are summarised in the upper part of Table , including the root mean square error (RMSE, the objective function), mean error (ME), standard deviation of the residuals (SD), coefficient of determination (R2), and Nash–Sutcliffe efficiency (NSE).

Across seasons and throughout the year, several parameters converged towards the upper bounds of their uniform prior distributions during calibration. In the winter, spring, and annual calibration, the parameter ks approached its maximum allowed value of 2.2 Wm-1K-1. In the winter, summer, and annual calibration, the parameter im similarly converged towards its upper bound of 60 m. These patterns indicate that the optimum lies at, or beyond, the upper bounds of the prescribed prior ranges for ks and im and may not adequately capture the effective values required to reproduce the observed microclimate dynamics at the study site. By contrast, the parameter is remained close to the centre of its literature-derived parameter range, across the annual and all seasonal calibrations.

The annual residual-based performance metrics (RMSE, ME, and SD) converged to values comparable to the seasonal metrics' mean (resp. 1.24, 0.06, and 1.24 °C), reflecting the integration of contrasting seasonal dynamics. In contrast, both efficiency-based metrics (R2 and NSE) attained their highest values for the annually calibrated parameter set (resp. 0.97 and 0.97). This indicates that, while annual calibration does not minimise absolute errors for individual seasons, it exposes the model to a wider range of thermal conditions and spatial gradients. This leads to parameter values that better capture the structure and variability of the temperature field, as reflected by higher R2 and NSE values.

Over all seasons and throughout the year, the convergence behaviour of the CMA-ES optimisation process reveals a clear reduction in the objective function (RMSE) during the early generations, followed by stabilisation as the algorithm approaches its optimum (e.g., Fig. a). This pattern indicates that no further improvement in model fit is achieved once the RMSE curve has levelled off.

An additional view of convergence is obtained by examining the distribution of sampled parameter sets in a principal component space. A principal component analysis (PCA) performed across all generations and offspring shows that the spread of parameter sets narrows progressively, with points clustering more tightly as optimisation proceeds (e.g., Fig. b). This contraction of the parameter cloud reflects a strong concentration of sampling around the converged region of the parameter space.

To provide context for the aggregated validation statistics, we first examine model behaviour for a representative warm summer day (8 July 2023) in the Aelmoeseneiebos forest by comparing simulated horizontal and vertical air temperature gradients from annually and seasonally calibrated ForEdgeClim runs with TOMST TMS-4 observations (Fig. ). Model outputs (solid and dashed lines) are evaluated against sensor measurements (dots). Both calibration strategies capture the main observed temperature patterns and spatial gradients. For this summer day, the annually calibrated simulation shows closer agreement with the observed gradients, whereas the seasonally calibrated model likely overfits season-specific noise. This suggests that annual calibration captures more robust, process-level behaviour.

At noon (12:00 UTC), a decrease in air temperature from the forest edge towards the core was observed, driven by canopy shading that limits radiative heating of the lower forest strata (Fig. a). The forest structure further buffers heat transfer by restricting the penetration of warm macroclimatic air entering near the forest edge into the interior. While most of the absolute temperatures are overestimated, the model reproduces the magnitude and direction of the observed edge-to-core gradient.

During the night (01:00 UTC), spatial temperature differences diminished considerably, with a relatively uniform and cool temperature field extending from edge to core (Fig. a). This reflects the reduced thermal contrast under low-radiation conditions. The small positive bias during this period suggests that nocturnal cooling processes may be underestimated in the current model configuration.

In the morning (08:00 UTC), a horizontal gradient reappeared as lateral radiation entered the forest from the east, resulting in increased warming near the edge (Fig. a). This highlights the role of lateral radiation in shaping edge microclimates. The temperature contrast between the forest interior and the external environment was strongest during this period, after which heating became more spatially uniform later in the day. The relatively large discrepancy between modelled air temperatures and TOMST observations during the morning period, particularly in the vicinity of forest gaps (distance from the forest edge of approximately 110 m), is likely amplified by differences in effective measurement height. Model outputs represent mean air temperature within 1 m3 voxels, whereas TOMST observations are recorded at 15 cm above the ground. During the morning transition, radiative forcing and heat transfer may already have warmed the upper part of the lowest 1 m forest layer, while temperatures closer to the ground remain cooler, resulting in larger apparent differences between modelled and observed values.

Vertical air temperature profiles reveal distinct stratification patterns across the three selected time points (Fig. b). During the night (01:00 UTC), temperatures remain nearly uniform throughout the vertical column, reflecting stable conditions in the absence of radiative forcing. As the sun rises (08:00 UTC), a vertical gradient develops, with warming concentrated in the upper canopy. This vertical stratification persists through midday (12:00 UTC), indicating sustained thermal differentiation within the canopy during daytime conditions.

For the winter season, the differences in statistical metrics are most pronounced when comparing the calibrated validation set with the uncalibrated validation set. A modest improvement is observed when the calibrated parameter set is applied, as evidenced by higher values of both the coefficient of determination (R2) and the Nash–Sutcliffe efficiency (NSE). Consistent with this, the residuals exhibit smaller deviations for the calibrated simulations, with notably reduced root mean square error (RMSE), mean error (ME), and standard deviation of the residuals (SD) indicating an overall increase in model accuracy (Fig. ). An analysis of the residuals grouped by hour of the day shows that the smallest discrepancies between observed and modelled temperatures occur during daytime hours (Fig. b). In addition, calibration leads to a marked improvement in model performance at locations closer to the forest core, where canopy gaps are more common (Fig. c), as well as at positions near the forest floor and just above the canopy (Fig. d). Overall, these results indicate that the calibration improves model performance in a process-specific manner, particularly for radiation–structure interactions, rather than through a uniform error reduction across space and time.

In spring, summer, autumn, and for the annual period, the distinction between calibrated and uncalibrated performance is less pronounced, indicating that the application of the calibrated parameter set leads to only marginal changes in model performance. A full overview of validation statistics for all seasons and throughout the year is provided in Table .

The validation statistics show that residual-based performance measures for the annual validation sets (both calibrated and uncalibrated) converge towards the mean of the seasonal residuals. In contrast, both R2 and NSE consistently attain their highest values for the annual period, indicating a more consistent representation of observed temperature variability across independent conditions. While seasonally calibrated parameter sets can improve agreement with observations under specific conditions, their performance across independent validation periods is more variable. This suggests that seasonal calibration optimises model behaviour for targeted conditions, whereas annual calibration yields a more generalisable parameterisation across contrasting meteorological regimes.

Overall, both the calibrated and uncalibrated configurations reproduce observed air temperature patterns with modest errors across seasons and throughout the year. RMSE, ME, and SD values remain low, while R2 and NSE indicate that a substantial fraction of the observed variability is captured. For comparison, RMSE values reported during the validation of microclimf range between 0.69 and 2.9 °C , whereas in this study RMSE values across both configurations range between 0.98 and 2.01 °C.

Across the full simulation period, modelled and observed edge–core temperature gradients show a moderate level of agreement, with modelled gradients broadly following the observed direction (Fig. a). However, substantial scatter and clear seasonal differences indicate that the model has limited ability to accurately reproduce the magnitude of these gradients across all conditions, even though the presence of spatial gradients is generally captured.

This variability in model performance can be explained by differences in the magnitude of the observed gradients. The proportion of cases in which model error is smaller than the observed gradient increases markedly with gradient magnitude (Fig. b). For larger gradients, the majority of cases fall below the threshold where model error exceeds the observed gradient, indicating that the model is generally able to resolve pronounced spatial temperature differences. In contrast, for small gradients (e.g., below approximately 1 °C), the proportion of cases where model error exceeds the signal is substantially higher, indicating limited ability to resolve weak spatial contrasts.

This pattern reflects the fact that edge–core temperature differences approach zero under near-isothermal conditions, where even small absolute model errors can exceed the signal. As a result, the model's ability to resolve weak spatial gradients is inherently limited under these conditions due to a low signal-to-noise ratio, rather than a systematic misrepresentation of spatial patterns.

Overall, these results indicate that the model is capable of resolving spatial temperature gradients when they are sufficiently pronounced, but that caution is required when interpreting simulated gradients under conditions where observed differences are small. This behaviour is consistent with the calibration strategy, which targets point-wise air temperature (RMSE) rather than explicitly constraining the edge–core temperature gradient.

In the current implementation of ForEdgeClim, wind-driven processes are not explicitly represented. To assess the potential influence of this omission, model residuals were analysed as a function of observed wind speed and distance to the forest edge (Fig. ).

Overall, the influence of wind speed on model residuals, both during night- and daytime, appears limited, as indicated by the relatively small slopes across all spatial positions (1×10-3-1×10-1°Cm-1s in absolute order of magnitude). This suggests that wind-driven processes are not a dominant control on model performance under the conditions considered here. However, across all distances, residuals consistently increase with wind speed during both daytime and nighttime, indicating that wind-related processes are not yet fully captured by the current model formulation. The analysis spans four seasons, each represented by one month, suggesting that these patterns are robust across a range of environmental conditions.

Model performance nevertheless shows a clear but non-monotonic spatial pattern. Residuals are relatively small near the forest edge, where exchange with the external macroenvironment (represented by gm and im) directly constrains local conditions. This indicates that the parameterised macroenvironmental forcing is sufficient to capture first-order edge effects.

Further into the forest, discrepancies increase, with the largest slope occurring during nighttime at approximately 90 m from the edge, corresponding to a canopy gap. At this location, residuals clearly increase with wind speed. In the forest core, higher residuals are also observed at higher wind speeds, particularly during the day. This suggests that, while wind penetration is reduced, the simplified parameterisation of heat exchange between air, macroenvironment (gm, im), soil (gs, is), and forest structure (gf, if) does not fully capture the complex and heterogeneous flow regimes that characterise dense forest interiors.

Across all locations, the largest residuals predominantly occur during daytime conditions and at intermediate wind speeds. However, these high residuals are mainly associated with a limited number of extreme values. In contrast, the majority of daytime observations form a dense cluster with relatively small residuals, generally lower than those observed during nighttime periods, consistent with Fig. b.

Overall, these results indicate that the use of fixed convection parameters and distances of influence is sufficient to represent first-order macroenvironmental exchange and edge-to-core temperature gradients under the temperate forest conditions considered in this study, but insufficient to capture the full spatial variability in wind-driven turbulence and mixing associated with canopy gaps and structurally complex forest interiors. At the same time, the relatively weak sensitivity of residuals to wind speed suggests that explicitly resolving wind processes would likely lead to only incremental improvements under the tested conditions. However, the transferability of the current parameterisation to more open, windy, drought-stressed, or structurally contrasting forest systems remains uncertain and would require additional validation.

To assess the implications of parameter uncertainty for forest surface temperature, the Sobol parameter ensemble was propagated through the model, and the resulting distributions of simulated surface temperature were analysed across all conditions along both the horizontal and vertical transect lines (resp. Figs.  and ).

The resulting distributions are approximately bell-shaped, indicating a smooth and stable model response to parameter variation, without evidence of abrupt thresholds or regime shifts. However, the width of these distributions varies substantially between conditions, demonstrating that forest surface temperature remains weakly constrained in several cases, with the central 95 % range spanning approximately 0.02–11.29 °C for the horizontal transect line and 0.02–4.05 °C for vertical transect line.

This behaviour is consistent with the Sobol sensitivity analysis, which identified the forest heat transfer parameter gf as a dominant contributor to surface temperature variance (8.7 %–32 %). As gf does not influence air temperature and is therefore not constrained during calibration, its variability directly translates into uncertainty in surface temperature predictions.

The resulting uncertainty envelopes thus provide a quantitative measure of this lack of constraint and highlight the need for additional observational constraints to better resolve surface energy exchange processes.

ForEdgeClim v1.0 represents forest microclimates using a spatially explicit three-dimensional voxel framework that resolves radiative and thermal energy exchange processes within complex canopy structures derived from TLS data. The model explicitly simulates shortwave and longwave radiative transfer, evapotranspiration, and sensible heat exchange, and is designed to reproduce spatial temperature gradients along forest edge-to-core transitions.

This modelling approach reflects a balance between physical realism and computational tractability, allowing key microclimate processes to be represented mechanistically while remaining applicable to high-resolution forest structure data. External macroclimatic forcing is prescribed based on observed meteorological data, and canopy structure is represented using voxel-level density derived from TLS measurements and scaled using plant area index (PAI).

Several limitations arise from the simplified representation of physical processes in the current model version.

First, canopy–atmosphere exchange processes are only partially represented. Wind-driven turbulence, advection, and momentum transfer are not explicitly simulated, although these processes play a key role in shaping forest microclimates, particularly at forest edges . Sensible heat exchange is parameterised using a bulk heat transfer coefficient, and evapotranspiration is represented using a Priestley–Taylor formulation, which assumes radiation-driven latent heat fluxes and neglects explicit aerodynamic control. While this assumption is often reasonable for forest interior conditions, it may be less appropriate near forest edges where horizontal advection of heat and moisture can modify evaporative demand. This limitation is also reflected in the residual analysis, which indicates that model–observation discrepancies increase under higher wind speeds, particularly in structurally open areas where atmospheric coupling is enhanced. Consequently, the current parameterisation is expected to be most applicable to temperate forest conditions characterised by moderate wind exposure and relatively closed canopy structure, similar to those represented in this study.

Second, the representation of canopy structure is simplified. Vegetation elements within each voxel are treated as a bulk medium using a structural density (ρ), without explicit separation of foliage and woody components. While the voxel-based representation derived from TLS data captures large-scale structural heterogeneity and gap fraction variability, sub-voxel variability (e.g., leaf area density profiles or canopy clumping) is not explicitly resolved. In addition, although seasonal variation in canopy density is partially represented through PAI scaling and separate leaf-on and leaf-off optical parameter sets, this currently represents only a first-order seasonal approximation. Continuous phenological transitions and explicit separation of foliage and woody canopy components are not yet represented within the voxel-based framework, which may limit the realism of seasonal radiative and evapotranspiration dynamics during transitional periods. In addition, the representation of canopy structure is inherently dependent on the chosen voxel resolution. While a 1 m resolution was adopted in this study as a compromise between structural detail and computational efficiency, changes in spatial resolution may alter the representation of canopy heterogeneity and thereby influence radiative transfer and heat exchange processes. The sensitivity of model results to voxel resolution was not evaluated in the current study and remains an important source of structural uncertainty.

Third, the representation of subsurface and ecohydrological processes is limited. Soil heat exchange is represented using a simplified conductive formulation with a shallow reference layer, and the model assumes quasi-steady-state conditions for each simulated time step. As a result, temporal heat storage and the characteristic phase lag between surface forcing and subsurface heat flux are not explicitly resolved . In addition, water transport processes and plant hydraulics are not currently represented. Feedbacks between soil moisture, plant water status, and energy exchange – such as stomatal regulation of transpiration – are therefore not captured, which may influence microclimate dynamics under water-limited conditions.

Fourth, model evaluation is constrained by the available observations. Calibration and validation are primarily based on air temperature measurements, which represent the main target variable of the model. As a consequence, surface temperatures and individual energy balance components are less directly constrained by observations.

Finally, the representation of the external macroenvironment is simplified. Boundary conditions are derived from standard meteorological observations over short grass following World Meteorological Organization guidelines . This implicitly assumes homogeneous surrounding land cover, whereas adjacent land-use types (e.g., cropland or urban areas) can differ substantially in albedo, roughness, heat storage, and moisture availability, thereby influencing lateral radiative and thermal fluxes at forest edges . In addition, while solar geometry is represented, the sensitivity of simulated microclimates to alternative edge orientations was not systematically explored, despite its known influence on edge microclimate gradients .

These limitations define a clear roadmap for future model development (Fig. ).

A first priority is to improve the representation of canopy–atmosphere exchange by incorporating wind-dependent aerodynamic processes. This can be achieved within the existing voxel-based framework by linking the current bulk heat transfer coefficients (gm, gs, gf) and distances of influence (im, is, if) to wind speed and canopy structural properties . Such an extension would allow turbulent mixing and advective heat transport to be represented more explicitly, while retaining the current parameterisation structure. Future developments should additionally improve the representation of transpiration processes and canopy resistance. In the current formulation, evapotranspiration is represented as a bulk energy-limited flux using the Priestley–Taylor approach, without explicitly resolving stomatal regulation or dynamic canopy responses to environmental drivers such as vapour pressure deficit, soil moisture limitation, and drought stress. Incorporating more mechanistic transpiration formulations, including Penman–Monteith-type approaches, would enable more explicit representation of canopy resistance, atmospheric demand, and humidity-related state variables such as vapour pressure deficit and relative humidity.

A second development pathway concerns structural refinement of the canopy representation. This could be implemented by partitioning voxel-level density into leaf and woody components, allowing distinct optical and physiological properties to be assigned within each voxel while preserving the existing spatial discretisation. Incorporating TLS-based leaf–wood separation and voxel-level structural metrics (e.g., leaf area density or clumping indices) would improve the representation of radiative transfer and evapotranspiration processes, similar to approaches used in ecosystem models such as ED2.2 and DART . While seasonal variation in optical properties is already represented through separate leaf-on and leaf-off parameter sets, this currently represents a simplified first-order seasonal treatment. Future developments could enable explicit separation of foliage and woody canopy components together with a more continuous and mechanistic representation of phenological transitions within the voxel framework. In addition, the choice of voxel resolution is closely linked to the representation of canopy structure. While the current study adopts a 1 m resolution, changes in spatial resolution directly affect how structural heterogeneity is represented within the voxel grid, with potential implications for radiative transfer, heat exchange, and parameter sensitivity. Future work should therefore systematically evaluate the influence of voxel resolution on model behaviour and performance, in order to guide scale-aware application of the model.

A third priority is the integration of soil and plant water dynamics. Implementing a multi-layer soil heat and water balance scheme would explicitly represent vertical heat diffusion and soil moisture dynamics, building on the current vertical heat exchange formulation within the voxel grid. Coupling this with a plant hydraulic framework would enable feedback between soil moisture, plant water status, and stomatal conductance, thereby improving the representation of evapotranspiration and energy partitioning under varying environmental conditions.

A fourth development direction is to improve boundary condition realism and model evaluation. Incorporating spatially explicit representations of adjacent land-use types would allow edge effects to be simulated more realistically across heterogeneous landscapes. In addition, future studies could include multi-variable validation using surface temperature, humidity, and radiative flux measurements. In particular, the inclusion of thermal infrared observations, such as canopy radiometers or thermal imaging cameras, would enable direct validation of forest surface temperature and provide constraints on surface energy balance processes, thereby reducing uncertainty associated with parameters that primarily control surface temperature dynamics. Sensitivity analyses exploring edge orientation effects could further improve understanding of directional microclimate responses.

Finally, advances in radiative transfer modelling could be explored within the current framework by replacing or complementing the current coupled vertical–lateral two-stream approximation with more detailed angular radiation schemes, such as discrete ordinates methods , spherical harmonics expansions , or Monte Carlo ray tracing approaches , where computationally feasible. Such developments would allow a more explicit representation of the full three-dimensional angular radiation field and radiative anisotropy within heterogeneous forest canopies.

Together, these developments illustrate that ForEdgeClim v1.0 should be viewed as an initial modelling framework. The current radiative–thermal formulation provides a flexible basis upon which progressively more mechanistic representations of canopy microclimate processes can be integrated, without requiring fundamental restructuring of the voxel-based architecture. This design enables the model to evolve towards a more comprehensive three-dimensional microclimate modelling system for heterogeneous forest landscapes.