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# The structure of tropical forests and sphere packings

Edited by Robert D. Holt, University of Florida, Gainesville, FL, and accepted by the Editorial Board October 20, 2015 (received for review July 8, 2015)

## Significance

Explaining the tree size structure of tropical forests is of crucial importance because other important forest attributes can be derived from this. The tree diameter distribution, for example, determines the amount of carbon stored in a forest. Here we present a simple and powerful approach based on stochastic geometry and tree allometries that can be used to predict tree diameter distributions.

## Abstract

The search for simple principles underlying the complex architecture of ecological communities such as forests still challenges ecological theorists. We use tree diameter distributions—fundamental for deriving other forest attributes—to describe the structure of tropical forests. Here we argue that tree diameter distributions of natural tropical forests can be explained by stochastic packing of tree crowns representing a forest crown packing system: a method usually used in physics or chemistry. We demonstrate that tree diameter distributions emerge accurately from a surprisingly simple set of principles that include site-specific tree allometries, random placement of trees, competition for space, and mortality. The simple static model also successfully predicted the canopy structure, revealing that most trees in our two studied forests grow up to 30–50 m in height and that the highest packing density of about 60% is reached between the 25- and 40-m height layer. Our approach is an important step toward identifying a minimal set of processes responsible for generating the spatial structure of tropical forests.

Forests are one of the world’s best investigated ecosystems (1⇓⇓–4). However, despite all of the studies devoted to forests, mechanistic connections among important features of forest physiognomy are not fully understood. Of crucial importance for this are tree diameter distributions, which have been used for decades in ecology and forestry to characterize the state of forests. Tree diameter distributions are available for many forests of the world and allow together with tree allometries prediction of other important forest attributes like leaf area (5), basal area (6), above-ground biomass (7), tree density, and the presence or absence of disturbances (8). Tropical forests usually include many small trees and far fewer large ones (1, 9). Diameter distributions can also be predicted from dynamic forest models (10⇓⇓–13) in which the forest structure emerges from the interplay between the dynamic processes of mortality, regeneration, competition, and growth.

In this study, we argue that tree diameter distributions of natural tropical forests can be predicted by stochastic packing theory—a method usually used in physics or chemistry—together with site-specific tree allometries. Packing systems have a long history of use across a wide range of disciplines, going back as far as Kepler’s analysis of regular packing of spheres (i.e., with a maximum of 74% filled volume). Irregular stochastic packing of spheres has been analyzed in the last decades and shows lower maximal packing densities (e.g., jammed sphere packing systems in physics, with 55–64% filled volume) (14, 15). We show here that simple principles of stochastic packing theory together with tree allometries and other simple structural rules allow for an accurate prediction of observed tree diameter distributions and related structural attributes for two tropical forests in Barro Colorado Island (BCI, Panama) and Sinharaja (Sri Lanka). Both forests show major differences in climate, soil, and topography that affect forest structure and dynamics (16, 17).

Our approach consists of three parts. First, we use tree allometries to transform size data of a tree (including the position **x** and the stem diameter *d*, which are available from forest inventories or from the packing model) into a simplified tree with a crown modeled as sphere with radius *c*(*d*) and height *h*(*d*) (Fig. 1). We use the simple allometric relationships *c*(*d*) *= c*_{0} *d** ^{pc}* and

*h*(

*d*)

*= h*

_{0}

*d*

^{ph}with parameters

*c*

_{0},

*h*

_{0},

*p*

_{h}, and

*p*

_{c}determined from independent datasets (18, 19) (for details, see

*Methods*). Other height-diameter relationships saturating in maximum tree height (7) could be used alternatively. However, because only few trees become large enough to experience this saturation, this has little influence on our results. The approximation of crowns as spheres is a simplification, given that crowns might shape differently during growth (20) (Fig. S1). In

*SI Results*, we examined the influence of the assumed crown shape on our results (Fig. S2

*C*).

Second, we simulate the static forest packing system (FPM) based on the following rules:

*i*) A tree with a stem diameter*d*is randomly determined from a uniform distribution within the interval [*d*_{min},*d*_{max}] (with*d*_{min}= 1 cm and*d*_{max}corresponding to the maximal tree height). Fig. S3 shows that our results are insensitive to the selection of*d*_{min}and*d*_{max}.*ii*) We accept a tree of stem diameter*d*only with probability*p*(*d*) (i.e., the survival probability of a tree growing from stem diameter*d*_{min}to diameter*d*). This rule thus considers average tree mortality and growth. For simplicity, we assumed constant annual stem diameter growth rates*g*and constant annual mortality rates*m*that yield a probability*p*(*d*) =*p*_{0}exp[−(*m*/*g*)*d*] (*Methods*). For comparison, we investigated in this study two versions of our model: one without mortality (i.e.,*m*= 0) and one with mortality.*iii*) Based on the allometric relationships between tree height*h*(*d*) and crown radius*c*(*d*), the stem diameter*d*is translated into a crown with center*h*(*d*) −*c*(*d*) above ground (Fig. 1*A*and*B*). Thus, in contrast to irregular stochastic packing, each crown is located in a given height layer (depending on*d*). This height constraint will reduce the packing density.*iv*) We tentatively locate the stem and crown (and thus the tree) at a random position within the study area.*v*) The tree is retained if it does not overlap with an already placed tree (i.e., their crowns and stems).*vi*) The algorithm stops if the observed number of individual trees is placed.

Thus, the spatially explicit crown packing system emerges from model rules and simple site-specific tree allometries that are parameterized from the data or the literature (for details, see *Methods*). Results of a sensitivity analysis are provided in *SI Results*.

Finally, in the third step, the crown packing systems are analyzed with methods of classical packing theory in terms of the space occupied by crowns (Fig. 2*A*), local forest height (Fig. 2*B*), and the distribution of leaf area and packing density at different heights and spatial scales (Fig. 2 *C–E*). To calculate the leaf area of crowns, we assumed for simplicity that they are homogeneously filled with leaves (see *SI Results* for results if we modify this assumption; Fig. S2*D*).

## Results

On average only 15% of the available space is occupied by tree crowns at BCI (Fig. 2*A*) and 20% at Sinharaja (Fig. S4*E*; analysis of packing density φ at the local scale of 0.04 ha for 1,250 subplots at BCI and 625 subplots in Sinharaja). Nevertheless, in some plots, we observe higher densities, and in other plots, lower densities (e.g., due to former gaps of large fallen trees). For example, in the BCI forest, the crown packing density at the local scale ranges from 3% to 53% (Fig. 2*A*). The local packing densities are much smaller than the optimized regular packing of monosized spheres and below its irregular counterparts, which are known from stochastic geometry theory (see below) (14). Clearly, this is because tree crowns can only be located in their associated height layer and because we assumed a maximum tree height of 60 m, which is rarely attained (Figs. 1 and 2*B*). A reduced maximum tree height of 45 m would increase the overall packing density for BCI only from 15% to 18%. Ecologically more meaningful measures are therefore the crown packing density across different height layers (Fig. 2*C*) and the associated distribution of leaf area density (Fig. 2*D*). Essentially, we find a unimodal local crown packing distribution (Fig. 2*C*).

The densest layer in the BCI forest is reached at a height of about 25 m (40% of forest height) and yields a local mean leaf area density of *l*_{max} = 0.6 m^{2}/m^{3} (Fig. 2*D*, Fig. S5 *G* and *H*, and Table S1). Interestingly, the densest height layer is close to maximal stochastic packing densities (i.e., 60%; jammed packing), for which the filled space ranges from random loose (55%) to random close (64%) sphere packings (14). Note that *l*_{max} and the corresponding height also vary locally (Fig. S5 *G* and *H*). We obtained similar results for the Sinharaja forest (Figs. S4 *G* and *H* and S6*B* and Table S1). The packing density and leaf area distributions were unimodal at the Sinharaja forest (Fig. 3*E*), but that of BCI showed a small additional peak at about 17% (∼11 m) of forest height (Fig. 2*D* and Fig. S5*H*). We suspect that this is caused either by differences in soil quality or by former treefall gaps at the BCI forest. On a larger scale, the leaf area distribution is increasingly influenced by topography. We suspect that the small additional peaks in Fig. 2*E* are caused by the specific tree allometries for the BCI forest producing a relatively high number of mid- to large-size trees, which then allow only much smaller trees to be placed below (Fig. 1).

The forest packing model without mortality (i.e., *m* = 0) substantially overestimated the leaf area density at larger heights and the number of large trees (Fig. 3 *A* and *B* and Table S2) and can therefore be ruled out. In Sinharaja, this overestimation also holds but to a much lesser extent (Fig. S7 and Table S2). However, the model using the observed mean mortality and growth rates successfully reproduced the observed size structures of the tropical forest in BCI (Fig. 3*D*), the leaf area distribution (Fig. 3*C*), and the local scale variation of the packing density (0.04 ha; Fig. 4*A* and Table S1). We found at BCI slight differences in maximum leaf area density (Fig. S5*G*) and in the densest height layer at which this maximum is reached (Fig. S5*H*). This difference could be caused by canopy gaps in the data due to fallen larger trees, which are not considered in the model. The results for the Sinharaja forest are similar (Figs. 3 *E* and *F* and 4*B*, Fig. S4 *G* and *H*, and Table S1), but here our model slightly underestimates leaf area density at heights below 20% of forest height (Fig. 3*E*), and the observed local packing density at 20 × 20-m subplots is somewhat wider than the model prediction (Fig. 4*B*). We suspect that the mismatch for the Sinharaja data (especially for the local packing density; Fig. 4*B*) was caused by our assumed tree allometries, which are only approximations. Finally, the posterior tree diameter distributions emerging from our simulation (Fig. 3 *D* and *F*) are largely different from the prior *p*(*d*) used to propose tentative trees in the sphere packing algorithm (rule 2). This difference outlines the importance of the additional rule 5 of stochastic geometry to selectively accept proposed trees depending on their size.

## Discussion

Our tree crown packing simulations showed that simple principles of stochastic geometry combined with site-specific tree allometries and tree mortality are sufficient for accurately predicting important attributes of tropical forests such as tree diameter distribution and leaf area distribution (Fig. 3). Our static model is built around four major structural constraints of spatial architecture and dynamics of forests. First, the allometric relationships (rule 3) incorporate the essence of tree geometry found at each site that reflects local environmental conditions (16, 17) and physical constraints on canopies (21, 22). Second, rule 2 considers average tree growth and mortality of trees in the most basic way and therefore incorporates the essence of tree dynamics like recruitment, growth, and mortality. Third, rule 5 that prohibits overlap of crowns captures the essence of spatial tree competition in a structural way. Finally, random placement of trees (rule 4) introduces the important element of stochasticity that prevents the two tropical forests from being optimally packed. Notably, the interplay of these four structural constraints of spatial architecture generates the observed tree diameter distributions. Changes in tree geometry and mortality reflect the uniqueness of this interplay (Figs. S3, S8, and S9). We did here a first step in presenting a novel and simple hypothesis for explaining tree diameter distributions. Application of our approach to more forest sites will show if the simple mechanisms proposed here need to be modified and if additional rules are required.

Application of stochastic packing theory opens an inspiring new perspective on the architecture of space filling in forests, but this approach is embedded within a long tradition of modeling studies on forest dynamics and self-thinning. For example, the classical −3/2 power rule of self-thinning for even-aged forest stands has also been derived from simple geometric models of space filling together with tree allometries relating diameter to average height, average mass, or average crown radius (23). Forest simulation models also make assumptions about the geometry and space filling in forests. For example, several forest models include thinning of the forest in crowded stands (11). Other studies noted crown plasticity with different crown shapes of canopy trees, which are not anymore positioned above their stem location as trees grew toward light gaps in the forest canopy (20, 24) (Fig. S1*C*). Surprisingly, our study suggests that competition for light does not play a direct role in shaping tree diameter distributions, although this is also a central component of models designed to describe the dynamics of vegetation (25, 26). However, rule 5, which does not allow for overlap among tree crowns, and rule 2, which includes mortality effects, can be reinterpreted as considering indirect light effects on tree survival (Fig. 1 *B* and *D*).

Although irregular sphere packing models in physics maximize the packing density by rejecting objects (in case of overlapping with existing ones), the forest crown packing observed is a static view of forests where trees continuously die and grow. This approach leads to nonoptimal packing where often somewhat taller trees would fit below a large tree (Fig. 1*B*). Our sphere packing model is static. However, stochastic placement together with competition for space (rule 5) captures the outcome of this well because a newly placed tree will in general not have the optimal size that could be squeezed in but will be smaller (Fig. 1*B*). The effects of mortality and retarded regrowth are reflected in the packing density across height (Fig. 2*C*) and in the leaf area distribution (Fig. 2 *D* and *E*), which contrasts to approaches that assume equal space filling across tree size classes (27) (Fig. S1). However, our model cannot describe effects of the mortality of large trees that create canopy gaps that are then colonized by small trees. Fig. 4*A* shows that this leads only to a slight underestimation of 20 × 20-m subplots at BCI with low packing density.

Our tree allometries include the geometric constraint that trees of a given size can only be filled into a given height layer and that large tree crowns cannot occur near the forest floor. This geometric constraint explains partly the low crown packing density of the two analyzed forests compared with physical packing systems. Additionally, more than half of the nonoccupied space (∼57%) is found above 30-m height on BCI. This large proportion is because the majority of trees in the forests, whether simulated by our packing model or derived from inventory data, are much smaller than our assumed maximum tree height (of 60 m; Fig. 2*B*) and only a small number of emergent trees exceed heights of 30–50 m (1, 28). This finding also contributes to the old debate on stratifications and layers in tropical forests (29, 30).

Our findings provide important insights into the heterogeneous structure of forests that might be useful also for remote sensing of forests (28, 31). Lidar scanning technologies send beams through the canopy and measure the forest structure (e.g., leaf and branch density as the emitted beams interact with them) (32). Hereby, the density of signals returned at a specific height may be comparable to the observed pattern of leaf area density across height (31). Knowledge on the heterogeneous crown structure of forests derived from this study can potentially enhance the interpretation of such remote sensing measurements.

## Methods

### Study Areas.

For this study, we use forest inventory data on two tropical forests: one in Panama (BCI; 50 ha) (33⇓–35) and one in Sri Lanka (Sinharaja; 25 ha) (36). The plot on BCI is located within the Panama Canal and represents a lowland tropical forest with an annual rainfall of 2,570 mm (1). The plot in Sinharaja is located in Sri Lanka and constitutes an everwet lowland forest with an annual rainfall of 5,016 mm on average (1, 36).

The 50-ha plot on BCI hosts ∼230,000 trees at least 1 cm in diameter at breast height, belonging to more than 300 species (1). The size of the forest plot in Sri Lanka is half that of the forest plot in Panama (i.e., 25 ha), with ∼180,000 trees and 209 species (1). Each tree has been identified to species and tagged, and its location on the forest plot and stem diameter measured. For both forest plots, the topography is recorded at discrete 5-m steps.

### Creating Crown Packings from Forest Inventories.

Forest inventories traditionally include information on stem diameter and species, but are increasingly supplemented by the location of trees (33⇓–36). In this study, we reconstruct the packing of tree crowns by using data on stem location and stem diameter at breast height. Additionally, relationships between stem diameter *d* (cm) and other tree characteristics [like tree height *h* (m) or crown radius *c* (m)] are used (18, 19).

We analyzed forest inventories at BCI (census 2010) in Panama and Sinharaja (census 2001) in Sri Lanka. For both forests, we exclude tree measurements if (*i*) the position of a tree is not within the forest area, (*ii*) the measured stem diameter is missing or zero, or (*iii*) a tree has been declared dead. Stem diameter measurements, which were not taken at breast height (1.30 m above the forest floor), are not corrected (i.e., we assume cylindrical stems for simplification). We assume a maximum tree height of 60 m (1). The few trees whose height exceeded the assumed maximum height have been excluded (tree height is derived from the stem diameter measurements).

As allometric relationships between tree height *h*(*d*) (m), as well as crown radius *c*(*d*) (m) and stem diameter *d* (cm), we used for Panama, relationships from the literature [i.e., *c*(*d*) = 0.37*d*^{0.67} and *h*(*d*) = 2.74*d*^{0.6}] (18). Site-specific allometric relationships for tree geometry are not available for the Sinharaja forest. We therefore use allometric relationships for the *Dipterocarpaceae* family [i.e., *c*(*d*) = 0.4*d*^{0.66} and *h*(*d*) = 2.78*d*^{0.69}] (19). This family shows the highest abundance of individuals, number of tree species, and basal area at Sinharaja (1, 36).

### Parameterization of the Forest Packing Model.

For BCI, tree diameters between *d*_{min} = 1 cm (the minimal size in the census) and a maximal *d*_{max} = 1.7 m has been considered in the forest packing model (for Sinharaja, we used *d*_{min} = 1 cm and *d*_{max} = 0.86 m). Average tree mortality rates of *m*_{BCI} = 2.2%/y (for BCI) and *m*_{S} = 1.3%/y (for Sinharaja) were both determined on the basis of the most recent census data (BCI, 2005 and 2010; Sinharaja, 1996 and 2001). To keep the model simple, we assumed constant growth rates typical for tropical forests (37) of 0.5 (BCI) (1) and 0.3 cm/y (Sinharaja) (38).

### Analysis Tools for Crown Packed Forests.

Based on the crown packing system, either created from forest inventory data or simulated using the forest packing model, we calculated how much space is filled (crown packing density φ). For this purpose, a cuboid representing the border of the analyzed forest is used. This cuboid is defined by the ground area and the maximum possible tree height of 60 m (for both forests; *SI Methods*).

We further divided the forest into distinct height layers to calculate the crown packing density and the leaf area distribution. We presumed linear binning (using 1-m layer widths). Logarithmic binned height layers are also possible, but our results are independent of the chosen discretization for the height layers (Fig. S2 *A* and *B*). For each height layer, the leaf area of each tree crown part reaching that height layer is summed up and divided by the volume of the specific layer (whereby we assume 1 m^{2} of leaf area per 1 m^{3} of tree crown volume).

### Sensitivity Analyses.

We investigated whether the congruence of simulated forests (using the forest packing model) and the observed forest inventories (exemplary for BCI) would change if the assumed input parameters vary: (*i*) in minimum and maximum tree sizes (Fig. S3), (*ii*) in tree geometry relationships (i.e., allometries; Fig. S8), (*iii*) in the mortality of trees (Fig. S9*A*), and (*iv*) in the stopping rule of the forest packing model (Fig. S9*B*).

Assumptions concerning (*i*) the crown shape (e.g., spheres or cylinder; Fig. S2*C*) and (*ii*) the distribution of leaf area within the tree crown (e.g., full sphere vs. upper semisphere; Fig. S2*D*) were additionally examined.

## SI Methods

### Details on the Analysis Tools for Crown Packed Forests.

#### Preparation: Including the topography of the forest floor.

For both forest plots (BCI and Sinharaja), elevation data with a resolution of 5 × 5 m are available. We add to each tree’s height the elevation value that is nearest to the tree’s observed location and exclude the space below the forest floor.

#### Crown packing density.

Based on the derived tree crowns, we calculated the density φ of packed crowns within the forest. For this purpose, we first define a cuboid that represents the maximal available volume *V*_{max} that can be filled by tree crowns. This cuboid is characterized by (*i*) a ground area given by the forest area, (*ii*) a height given by the assumed maximum tree height *h*_{max} (here 60 m, adjusted by the topography as mentioned above), and (*iii*) the excluded space below the forest floor (according to topographic information). Next we sum up the crown volumes of all trees and relate them to the total available space^{3}) of a tree crown *i*, and *V*_{max} denotes the maximal available space (m^{3}).

Note that Eq. **S1** does not include tree crown overlapping. Although the forest packing model (FPM) avoids overlapping of trees, the crown packed forest derived from the forest inventories can include such overlapping. To keep our approach simple, we include the entire crown volume of trees whose crown passes over the side borders of the cuboid (but not those above *h*_{max}).

## SI Results

### Sensitivity Analysis.

#### Height discretization (Fig. S2 A and B).

We compared calculated leaf area distributions using linearly binned height layers (using a bin width of 1 m) and exponentially increasing height layers (i.e., logarithmic binning by e^{+0.1} m). We observed no notable effect in the shape or quantity of leaf area density across height.

#### Crown shape (Fig. S2C).

In the standard model, crown shapes were approximated by spheres. Here we analyzed the outcome of leaf area distribution for the field data on BCI if crowns are assumed as cylinders (Fig. S1*D*; with modified allometric relationships) (18). The radius of the cylinder was determined as in the standard case (i.e., crown sphere radius). The length of the crown cylinder was determined by an additional allometric relationship [i.e., *cl*(*d*) *=* 0.7*d*^{0.7}, with *d* (cm) as stem diameter and *cl* (m) as crown length] (18). For crowns approximated by cylinders, we observe a more pronounced peak in the vertical leaf area distribution of the forest inventory, but the shape of distribution remains the same.

#### Individual tree leaf area distribution within crown (Fig. S2D).

The variation of leaf area distribution within an individual tree crown (either homogenous in the full crown or only in the upper semisphere) showed mainly a reduction in leaf area density.

#### Minimum and maximum tree sizes (Fig. S3).

Here, we analyzed the influence of the assumed minimum and maximum tree height on the simulation results (exemplary for BCI). Investigated minimum tree heights range from *h*_{min} = 1, 2, 5, to 10 m and maximum tree heights from *h*_{max} = 10, 20, 40, to 60 m. We observe more or less the same leaf area and tree diameter distribution for the field data and the forest packing simulation.

#### Allometric relationships (Fig. S8).

We vary (exemplary for BCI) the pre-exponential factor (*c*_{0} and *h*_{0}) and the scaling exponent (*p _{c}* and

*p*) of the functional relationships between crown radius

_{h}*c*(

*d*) and stem diameter

*d*[i.e.,

*c*(

*d*)

*= c*

_{0}

*d*

^{pc}] as well as tree height

*h*(

*d*) and stem diameter

*d*[i.e.,

*h*(

*d*)

*= h*

_{0}

*d*

^{ph}].

If we change the parameters values of *p*_{c} by ±30%, we observe a mismatch between field data and forest packing simulation in the leaf area and tree diameter distribution (Fig. S8*B*). Changing *c*_{0} by ±30% has similar effects, but to a lesser extent, than changing *p*_{c}. Varying *h*_{0} or *p*_{h} in the height-stem diameter relationship by ±30% has in comparison a smaller and opposite effect (Fig. S8*A*). For example, an increase of *p*_{h} by 30% allows more trees to be packed, whereas a decrease by 30% largely reduce successfully packed tree crowns compared with the observed field data. In addition, changes in *h*_{0} and *p*_{h} mainly influence the shape of leaf area distribution, whereas *c*_{0} and *p*_{c} mostly affect the shape of the tree diameter distribution.

#### Mortality of trees (Fig. S9A).

An increase in mortality (i.e., by up to 400%; exemplary for BCI) results in a large mismatch between field inventories and simulated crown packing. An increase of mortality reduces mainly large trees in the upper canopy, resulting in a steeper decaying tree diameter distribution.

#### Stopping rule (Fig. S9B).

Using a generic stopping rule (moving window of a number of steps, in which the acceptance rate of packed trees is tested) instead of a fixed total stem number yields only similar results for testing if less than 1% trees are successfully packed in a moving window of 1,000 steps. Reducing the number of steps within a moving window shows less small trees while reducing the acceptance rate has an opposite effect (i.e., increasing the number of smaller trees).

## Acknowledgments

We thank especially two anonymous reviewers, R. D. Holt, J. Brown, and D. Coomes for their helpful comments. We thank N. Gunatilleke and S. Gunatilleke for their support and cooperation concerning the Ecological Research Project at Sinharaja World Heritage Site. The 25-ha Long-Term Ecological Research Project at Sinharaja World Heritage Site is a collaborative project of the University of Peradeniya, the Center for Tropical Forest Science of the Smithsonian Tropical Research Institute, and the Arnold Arboretum of Harvard University, with supplementary funding received from the John D. and Catherine T. MacArthur Foundation, the National Institute for Environmental Science, Japan, and the Helmholtz Centre for Environmental Research-UFZ for past censuses. The principal investigators of the Research Project at Sinharaja World Heritage Site gratefully acknowledge the Forest Department and the Post-Graduate Institute of Science at the University of Peradeniya for supporting the project, and the local field and laboratory staff who tirelessly contributed in the repeated censuses of the plot. The Barro Colorado Island forest dynamics research project was founded by S. P. Hubbell and R. B. Foster and is now managed by R. Condit, S. Lao, and R. Perez under the Center for Tropical Forest Science and the Smithsonian Tropical Research in Panama. Numerous organizations have provided funding, principally the US National Science Foundation, and hundreds of field workers have contributed. A.H., T.W., and F.T. were supported by an Advanced Grant of the European Research Council (ERC) (233066) and by the Helmholtz Allianz Remote Sensing and Earth System Dynamics.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: franziska.taubert{at}ufz.de.

Author contributions: F.T. and A.H. designed research; F.T. and M.W.J. performed research; F.T., M.W.J., H.-J.D., T.W., and A.H. contributed new reagents/analytic tools; F.T. analyzed data; and F.T., M.W.J., H.-J.D., T.W., and A.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. R.D.H. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1513417112/-/DCSupplemental.

Freely available online through the PNAS open access option.

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