We used the process-based forest model Formind along with a “ForestFactory” MCMC approach to generate a sample of forest stands representing the variety of forest compositions potentially found in the field, building on the work by . Formind considers trees on the individual level and computes their gross and net productivity based on environmental conditions (e.g. temperature and precipitation), interactions between trees (e.g. light and water competition), as well as species-specific traits. As the model represents productivity and carbon fluxes explicitly and features both individual trees and their vertical leaf distribution, Formind is particularly suited to assess forest productivity and structure .

Before applying the model, we updated its parameterization for temperate forests in central Europe to better capture the carbon dynamics of heterogeneous forests. We considered forest inventory data collected at the “Hohes Holz” research site in central northern Germany (mean ann. temperature 10.8 °C; mean ann. precipitation 660 L m⁻²) and changed the temperate model parameterization in the following aspects: (1) we updated the allometric relationships according to data from ; (2) we adjusted parameters for light climate and respiratory losses to better model the carbon dynamics in structurally heterogeneous forests; (3) we chose parameters on soil properties that matched the soil at the research site (values taken from ); (4) we added the mechanistic defoliation and mortality mechanism introduced by ; and (5) we used a new approach to consider weather data on a daily time scale whereas forest dynamics such as growth and mortality remain modelled on the yearly time scale. Details on the parameterization can be found in Sect. S5 of the Supplement. We validated the parameterization by comparing the simulated GPP, NPP, and respiration with independent estimates based on Eddy-Flux measurements (; see Sect. S5.3). A more in-depth validation study of an older parameterization of Formind was conducted by at the Hohes Holz site and by at the Wetzstein site (Thuringia, Germany).

Typically, forest models such as Formind are initialized with existing forest inventory data or simulate forest successions starting from bare ground. However, since these simulations may not cover the entirety of forest states found in managed forests, we used a different approach here. Trees cannot survive if their GPP is insufficient to cover their respiratory needs. Hence, assuming that the GPP of any live tree found in a real forest must exceed its respiration , we randomly assembled a sample of states satisfying this property, starting from a uniform prior of forest states.

We considered forest patches of 20m×20m size, for each randomly selecting a one-year sequence based on the climate at the Hohes Holz site (years 2000–2017; ). Starting from bare ground, we now randomly added or removed trees of random sizes and species. Then we computed the yearly GPP and autotrophic respiration of the forest stand given the climate sequence and evaluated whether the stand was “feasible”, i.e., (1) each tree's GPP exceeded its respiration, and (2) the tree crowns did not exceed the available space. If the stand was feasible, we repeated the tree addition/removal procedure; otherwise we returned to the last feasible state and repeated the attempt to add and remove trees until a feasible state was found.

To increase the range of qualitatively different forest stands, we used a hierarchical sampling approach when adding new trees, randomly constraining the potential tree sizes and species in some stands . Furthermore, we facilitated computational efficiency by dynamically adjusting the “step size”, i.e., the number of individual tree additions or removals between two forest feasibility checks: we started at 5 additions/removals and increased or decreased this number by factor 2 if the last five feasibility checks were positive or negative, respectively. After 200 tree additions or removals, we terminated the process. Further details regarding this approach are provided in Sect. S2.

We generated 100000 forest stands for environmental conditions similar to those found at the Hohes Holz site. The forest stands differ, among others, in biomass, tree number, and species composition. For each stand, we computed the biomass, LAI, yearly GPP, and yearly NPP (mean values: biomass: 321 t ODM ha⁻¹, i.e., tons organic dry matter per hactar, LAI: 3, GPP: 28.8 t ODM ha⁻¹ yr⁻¹, NPP: 3.4 t ODM ha⁻¹ yr⁻¹). Furthermore, we determined the basal-area-weighted DBH entropy for the tree size distribution , which is a measure for the structural heterogeneity of forests (mean: -4.54). Here, a lower value indicates lower heterogeneity and/or stronger dominance of a single individual. We provide the formula of basal-area-weighted DBH entropy in Sect. S4. The distribution of the considered forest characteristics is visualized in Sect. S3. Lastly, we computed the forests' solar reflectance spectra using the radiative transfer model mSCOPE, as described below.

We simulated the solar reflectance spectra of the generated forest patches for wavelengths between 400 and 2400nm. To that end, we used an extended version of the radiative transfer model mSCOPE, designed for application in forests with vertically heterogeneous leaf traits . Our extended model version adds on to the original in also considering vertically heterogeneous leaf densities. Furthermore, we adjusted the probability of observing sunflecks on the ground, as the structure and covering of the ground are difficult to model exactly.

We parameterized the radiative transfer model similar to but adjusted individual parameters based on data from the TRY plant trait database . A list of the parameters can be found in Sect. S1. We set the view zenith angle to 0° and the sun zenith angle to 31.5°, corresponding to maximal zenith angle observed at the Hohes Holz site. Furthermore, we set the relative azimuth angle between sun and view direction to a value of 140° , but note that this value is only relevant if the view zenith is changed from its default 0°. We set the probability to observe direct sunlight (sunflecks) at the ground to 0, which reduced the simulated reflectance values in the visible range. This improved the agreement between model results and field observations at the study site.

We simulated forests and corresponding hyperspectral data for 9 scenarios with different dimensions of variation each. First, we considered the scenario without any parameter variations. Second, we varied the LAI input to the radiative transfer model in each simulated forest patch by a random factor ε∈23,32 from a log-uniform distribution: ln⁡ε∼Uln⁡2/3,ln⁡3/2. Here, Ua,b is the uniform distribution in a,b. Using a log-uniform multiplicative perturbation ensures that the parameters remain in the positive domain and in their respective order of magnitude.

Third, we perturbed all species' leaf trait parameters (Chlorophyll a+b content, leaf mass per unit area, equivalent water thickness, senescence material, carotenoid content) independently by random factors ε∈23,32 (log-uniform distribution). Fourth, we altered the leaf structure parameter in each patch by a random factor ε∈23,32 (log-uniform distribution), but constrained the resulting values to the admissible range 1,3 where necessary. Fifth, we changed the leaf inclination parameters by random values ϵ∼U-0.2,0.2 and constrained them to the admissible range -1,1 if required. Furthermore, we normalized these parameters by their joint Euclidean norm if the norm exceeded 1.

Sixth, we drew the soil wetness parameter in each simulated patch randomly from U0,1. Seventh, we changed the sun and view zenith angles as well as the relative azimuth by independent random values from U-15,15°. Eighth, we imposed independent random perturbations to all simulated reflectance values. The perturbations followed normal distributions with mean 0 and standard deviation σi=0.05Ri+0.005, where i indicates the wavelength and Ri the corresponding reflectance value. Finally, ninth, we combined all the perturbations listed above.

For each considered variation scenario, we generated 100000 forest states with corresponding reflectance spectra and filtered out all forest states with a biomass below 50 t ODM ha⁻¹, as the hyperspectral model was primarily designed for areas with full forest cover. This concerned less than 2% of the forest patches. We split the datasets into 75% training and 25% validation data. Then we selected all possible pairs of two wavelengths and fitted a model estimating either biomass, LAI, GPP, or NPP based on the corresponding reflectance values of these wavelengths in the training data.

Here, we considered two types of models: (1) an affine-linear model of the form f=a0+a1Rw1+a2Rw2, where f is the forest attribute, w1 and w2 are the wavelengths, and Rw1 and Rw2 are the corresponding reflectance values, and (2) a parameter-free model based on regular binning of the data. The shape of parameter-free models is not given by a simple formula but directly derived from the data, making these models particularly useful for data-driven vegetation indices. To obtain the parameter-free model, we binned the data into 25 regular intervals along each wavelength axis, resulting in 625 potential bins for the tuples Rw1,Rw2. Then we determined the mean value of the considered forest property in each bin and assigned this value to the middle point of this bin. That is, for a bin aw1,bw1×aw2,bw2, we set 1faw1+bw12,aw2+bw22:=1B∑i∈Bfi, where B is the set of all data points in the bin and B its cardinality. We then applied a bi-linear interpolation between these points to obtain model predictions for all other reflectance values. For reflectances outside the convex hull of the training data, we applied a nearest neighbour extrapolation based on the closest bin centre.

For both model types, we computed the Pearson R2 of each fitted model based on the validation data. Furthermore, we created heatmaps for the R2 values as function of the wavelengths. Afterwards, we determined for each forest characteristic the maximal obtained R2 and the percentage P0.9 of models achieving an R2 of at least 90% of the maximum. We computed P0.9 by determining the surface area in the heatmap where models achieved a sufficiently large R2 and dividing it by the total surface area of all considered bands (Table ). Specifically, we defined a function f0.9w1,w2 with f0.9w1,w2=1 for wavelength pairs with sufficient R2 and f0.9w1,w2=0 otherwise, Then, P0.9=∫∫f0.9w1,w2dw1dw2, which we approximated via a midpoint Riemann sum (numerical integration) of f0.9 with those wavelength pairs as sampling points for which we had computed reflectance values.

To measure how important each individual band (Table ) is for obtaining a large R2, we determined the fraction of all models that use a wavelength from the respective band and achieve a high R2 (90 % of max). We plotted this quantity (colour coded by band) along with the respective maximal R2 for each considered forest characteristic and variation scenario.

To understand the impact of forest structure on these results, we repeated the analysis in the scenario without variations after grouping the data by forest structure. Specifically we split the forest set into sparse and dense forests (biomass below and above 200 t ODM ha⁻¹, respectively) and structurally heterogeneous forests vs. forests dominated by few individuals (DBH entropy SDBH less or greater than -2.5, respectively; cf. ). That way, we obtained the maximal R2 values and the most significant wavelength bands for forests with qualitatively different structural properties. To streamline our study, we focused our analysis on the NPP.

We visualized the functional form of the best models with heatmaps (scenario without variations). We set the obtained R2 values into context with existing vegetation indices by determining the correlations of classical vegetation indices with biomass, LAI, GPP, and NPP based on our simulated data without noise. Here, we considered all vegetation indices listed by that use reflectances from clearly defined wavelengths or bands and do not depend on a site-specific parameterization. By squaring the correlation coefficients, we obtained the R2 values that could potentially be achieved with linear models predicting the forest properties based on the vegetation indices.

Neither a large number of different wavelengths nor a complicated functional form were necessary to estimate forest attributes with relatively high accuracy. In fact, linear models superseded parameter-free models without pre-imposed functional form in some wavelength regions. However, focusing on a specific functional form limited the range of wavelengths suitable for analyzing forest attributes and increased the challenge of identifying the optimal wavelengths. This agrees with earlier findings by , who evaluated different wavelength combinations and functional forms to estimate LAI based on reflectance data. The better suitability of linear models in some wavelength regions is due to the limited resolution of the binning method we applied in the parameter-free case, where we computed average values of forest characteristics for individual reflectance intervals. Increasing the resolution (i.e., using smaller intervals) or using more general parametric models will therefore lead to even better R2 values than we presented, but may require larger datasets.

Despite the relatively high R2 values we obtained, ranging from 0.6 for the NPP to 0.97 for the LAI, our results show that identifying the right wavelengths for estimating specific forest properties is key. In the absence of environmental and physiological variability, the classical vegetation indices rarely utilized wavelength pairs optimal for estimating any of the considered forest characteristics. Exceptions were the vegetation indices using red and NIR/red edge light (e.g. NDVI, EVI, NDVIre, NIRv), which were, however, only in the optimal range for estimating GPP and neither of the other forest characteristics. This is surprising, since LAI and GPP are typically assumed to be strongly correlated , and several vegetation indices were developed by considering the reflectance profile of leaves .

In the presence of environmental or physiological variability, the range of suitable wavelengths decreased significantly, and some bands lost their suitability. For example, visible light became sub-optimal when leaf traits were uncertain, and NIR light became unsuited for estimating NPP in this case. This suggests that the choice of wavelengths for estimating forest characteristics should take into account the type and magnitude of variations expected in the field data. Identifying and quantifying these sources of variation will hence be an important preparational step to deriving data-driven vegetation indices for forest characteristics.

Aside the above-mentioned cases, the best band combinations for estimating forest characteristics remained remarkably stable throughout the considered scenarios. In particular, combinations of longer wavelengths (NIR, SWIR1, and SWIR2) were generally well suited. This is in line with previous studies , yet contrasts with the most common choices of wavelengths for vegetation indices (only 4 out of 23 vegetation indices listed in the review of combined two wavelengths from these bands). However, these vegetation indices were typically not designed to quantitatively estimate the forest characteristics considered in this study. Instead, many are applied to assess the presence of vegetation, e.g. classify areas into forest landscapes and other landscape types, whereas in our study, we considered forests only, i.e., presumed that the landscape had been filtered to only include forests before. Nonetheless, a classification into forested and unforested land could be easily conducted in an independent preparational step, making forest-specific vegetation indices applicable in combined multi-step procedures in practice.

We found that combinations of NIR and SWIR wavelengths were particularly useful for estimating forest characteristics. Reflectance in the SWIR range is strongly related to leaf water content, lignin, proteins, nitrogen and cellulose , and the combination of LAI and nitrogen content has been found well suited for estimating forest productivity . The significance of NIR/SWIR combinations may be further understood by considering the sensitivity of forest reflectance to LAI, which is low in the NIR range but high in the SWIR range (; sensitivity analysis of radiative transfer models). Furthermore, NIR and SWIR reflectances are much less sensitive to leaf constituents such as chlorophyll or carotenoid than reflectances from the visible spectrum . This may also explain why visible light became unsuitable for estimating GPP when the leaf properties were uncertain. We note, however, that a discussion of potential technical limitations in measuring NIR/SWIR light or required atmospheric corrections is beyond the scope of this study.

Since variations in leaf constituents may be induced by high species richness, reflectances from the visible spectrum may be unsuitable for assessing particularly diverse forests. The significance of visible light decreased similarly in forests with high biomass, potentially due to the low penetration depth of visible light leading to a quick saturation of reflectance-LAI relationships . These findings suggest that visible light is sub-optimal to assess the considered forest characteristics in structurally complex forests.

Noteworthily, an increase in the heterogeneity of tree sizes had a contrasting effect: it increased the significance of visible light for estimating NPP. The tree size diversity, measured by the DBH entropy in this study, is low if forest patches are dominated by individual large trees, overshadowing understorey trees. If multiple differently sized trees contribute equally to a forest's basal area, it is more likely that their canopies are directly visible from above, making it easier and potentially optimal to use visible light to assess the forest state. This relationship may not hold in strongly stratified (e.g. tropical) forests, though.

A major aim of this study was to establish a simulation-based methodological framework for systematically evaluating two-band vegetation indices. Though building on previous work on generating ecologically feasible forests with simulation models and combining forest models with radiative transfer models , we introduced several methodological advancements. Our Markov Chain Monte Carlo approach for sampling forests allows a rigorous statistical interpretation of the distribution of the resulting forests. Furthermore, the algorithm's simplicity reduces the risk of model artifacts potentially favouring specific forest structures, makes it easier to incorporate climatic effects on forest states and to exchange the underlying forest model with other process-based forest models. The updates of the forest model Formind and its parameterization for temperate forests improved the model's efficiency and accuracy in heterogeneous forests.

The adjusted version of mSCOPE disentangles the vertical space and LAI dimensions, thereby permitting the simulation of forests with heterogeneous leaf densities in different layers of the canopy. The new soil sunfleck parameter reduces the impact of uncertain soil properties on reflectance and may lead to more accurate results in scenarios with large solar zenith angles.

The parameter-free vegetation indices proposed in this study circumvent the need to limit the predictive power of vegetation indices by pre-imposing specific functional forms. This makes this class of vegetation indices particularly useful for data-driven vegetation indices, as the optimal functional form can be directly derived from the data. At the same time, their methodological simplicity makes data-driven indices well-suited for considering large data sets and analyzing many vegetation indices systematically. However, if less data are available or more than two or three wavelengths are regarded jointly, other methods, such as machine learning approaches, should be considered.

We used a model-based approach for a systematic analysis of the vegetation indices under controlled conditions. This is a typical method in theoretical studies see e.g. and facilitates a general understanding of the studied relationships, as a variety of different scenarios can be considered easily. Nonetheless, the results depend on the assumptions underlying the applied models and are hence subject to associated limitations.

The applied forest model Formind considered climatic drivers such as droughts and their effects, but the relationships between these stressors with the optical leaf properties were not modelled explicitly. Similarly, the effects of shadows occurring on the surface of heterogeneous tree canopies , which could affect the estimability of forest characteristics by inducing a relationship between forests' height heterogeneity and their reflectance, are not accounted for in mSCOPE. We addressed these simplifications by modelling and investigating the effect of different sources of variation on the results, and our findings are in good agreement with earlier analyses of wavelength combinations for estimating LAI .

We simulated the analyzed forest stands considering environmental conditions at the Hohes Holz site, where the utilized forest model had been extensively validated in multiple respects . The “ForestFactory” approach to generate forests yields stands with a broad variety of structures . This makes the simulated dataset representative for a large set of forests varying in site conditions and management. We accounted for weather-induced variability by additionally randomizing the utilized weather data. Nonetheless, the model results may not be representative for forests with significantly different site conditions or climate (e.g. montane forests). Additional studies are required to further assess the impacts of a wider range of environmental site characteristics on vegetation index performance (e.g. soil type, rooting depth, slope and aspect, mean temperature and precipitation).

Our systematic in-silicio analysis of forests and their reflectance provides the basis for deriving data-driven vegetation indices from field data. Field data for structural forest properties such as biomass and LAI could be derived from LIDAR measurements and orthophotos. These data could then be combined with reflectance data from multi- and hyperspectral satellite missions. Data-driven vegetation indices for biomass, LAI, and yearly GPP could be derived based on data from the Landsat 8/Landsat 9 satellites (most promising band combination: band 5: NIR and band 7: SWIR 2), the Sentinel 2 satellite (band 8: NIR/band 12: SWIR 2), or the MODIS instrument (band 5: NIR/band 7: SWIR). Here, the MODIS data may be particularly well suited due to the sensor's relatively small bandwidth in the SWIR range.

Deriving a data-driven vegetation index for NPP may be more challenging, as corresponding field data are difficult to acquire. Furthermore, the wavelengths we found to be best suited for estimating NPP are often not considered by multispectral satellite missions, because these wavelengths are close to the atmospheric water absorption bands. Here, data from hyperspectral satellites such as EnMAP and CHIME could prove particularly useful, as they record a significantly larger number of wavelength channels.

Since we found the sensitivity towards wavelength choice to be minimized by parameter-free vegetation indices, we suggest taking a corresponding approach when developing data-driven vegetation indices. Once data-driven indices are established for some areas where field data are available, the results could be used to recalibrate and fine-tune our model-driven analysis pipeline, e.g., by providing insights into the variability of environmental conditions and plant traits. The refined model could then be used to generalize the results from the field studies or to develop indices specifically tailored towards specific forest types.

These indices could be derived by simulating forests with structural properties mimicking the those of the respective region of interest. Tailor-made vegetation indices could yield more precise results than presented in this study, where we considered a broad mixture ranging from even-aged monocultures to uneven-aged heterogeneous forests. Moreover, since data-driven vegetation indices require that the training data are representative for the area of interest, using carefully constrained data for fitting may even be a prerequisite for accurate estimates.

Besides generalizing results from spatially constrained field studies, the ability to jointly simulate forests and their reflectance can support the development of advanced vegetation indices that incorporate more wavelengths simultaneously. To that end, machine learning methods may be applied, which require sufficiently large training datasets. This increases the importance of simulation approaches. Our process-based modelling approach can furthermore be extended to consider variables beyond those analyzed in this study. For example, it might be possible to develop specialized vegetation indices for forest properties such as species diversity, structural diversity, net ecosystem exchange, forest health or disturbance.