## Abstract

### • Context

Forest resource projections are required as part of an appropriate framework for sustainable forest management. Suitable large-scale projection models are usually based on national forest inventory (NFI) data. However, sound projections are difficult to make for heterogeneous resources as they vary greatly with respect to the factors that are assumed to drive forest dynamics on a large spatial scale, e.g. geographically varying growth conditions (here represented by NFI regions), tree species composition (here broadleaf-dominated, conifer-dominated and broadleaf-conifer mixed stands) and stand structure (here high forest, coppice forest and high-coppice forest mixture).

### • Question and objective

Our question was how does the variance of forest dynamics parameters (i.e. growth, felling and mortality, and recruitment processes) and that of 20-year forest resource projections partition between these factors (NFI region, tree species composition and stand structure), including their interactions. Our objective was to capitalise on the suitability of an existing multi-strata, diameter class matrix model for the purposes of making projections for the highly heterogeneous French forest resource.

### • Methods

The model was newly calibrated for the entire territory of metropolitan France based on most recent NFI data, i.e. for years 2006–2008. The forest resource was divided into strata by crossing the factors NFI region, tree species composition and stand structure. The variance partitioning of the parameters and projections was assessed based on a model sensitivity analysis.

### • Results

Growth, felling and mortality varied mainly with NFI region and species composition. Recruitment varied mainly with NFI region and stand structure. All three factors caused variations in resource projections, but with unequal intensities. Factor impacts included first order and interaction effects.

### • Conclusions

We found, by considering both first order and interaction effects, that NFI region, species composition and stand structure are ecologically relevant factors that jointly drive the dynamics of a heterogeneous forest resource. Their impacts, in our study, varied depending on the forest dynamics process under consideration. Recruitment would appear to have a particularly great impact on resource changes over time.

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## Acknowledgements

We are particularly grateful to Nicolas Picard, Sylvie Gourlet-Fleury, Frédéric Mortier and Dakis-Yaoba Ouédraogo at the French CIRAD (Centre de Coopération Internationale en Recherche Agronomique pour le Développement) for valuable discussions on forest dynamics modelling. Moreover, we thank two reviewers for constructive and helpful comments on an earlier manuscript.

### Funding

Funding was provided by the French General Directorate for Education and Research DGER (Direction Générale de l'Enseignement et de la Recherche).

## Author information

### Affiliations

### Corresponding author

## Additional information

### Contribution of the co-authors

H.W. coordinated the presented work on the model, performed the model sensitivity analysis, and wrote the manuscript; A.C. and H.C. calibrated the model and ran the simulations, provided expertise on the model, and contributed to manuscript writing; J.-D.B. contributed to the set-up and performance of the scientific work on the model (scope and positioning, conceptual design), brought up structuring ideas in view of further model development, and contributed to manuscript writing; G.P. provided expertise on the model, wrote an NFI intern technical model description including approaches to further model development; S.C. brought up a thematic context that lead to the initiation of the work on the model; J.-M.L. contributed to the scientific positioning of the work on the model; J.-C. H. contributed to orientation and structure of work on the model, and provided expertise on the model; M.F. initiated the work on the model and contributed to its scientific positioning, and promoted the research project including funding.

**Handling Editor**: Matthias Dobbertin

## Appendices

### Appendix 1: Methods used to calculate model parameter values

### Growth parameter (*g*
_{s,i})

We calculated *g*
_{s,i}-values (1 / year) specifically for each stratum (*s*) and diameter class (*i*, where *i* < *k*) as

with

and

where *∆d*
_{s,i,y} = mean annual diameter increment (cm / year, over bark) of the trees inventoried in year *y*; *w*
_{s,i} = diameter class width (cm, over bark); *n*
_{s,i,y} = number of trees inventoried in year *y*; \( \frac{{\Delta r{c_{{s,i,j,y}}}}}{5} \)= mean annual radial increment (cm / year, over bark) over the 5-year period prior to the inventory year of tree *j* inventoried in year *y*; *∆rc*
_{s,i,j,y}-values were measured on increment cores (under bark) as the cumulated width of five annual rings, and an estimate of bark thickness increment was added, based on NFI bark thickness equations (C. Duprez and F. Morneau 2010, French NFI, unpublished data); inventory years (*y*) consisted of 2006, 2007 and 2008.

This calculation method meant that annual radial increments in some years were more represented than those in other years because the 5-year increment periods partly overlapped. However, different trees were inventoried in each of the years in 2006–2008 (see French National Forest Inventory method), and our aim was to obtain robust parameter values representing mean growth over several years.

If for a given stratum diameter class *n*
_{s,i,y} = 0 for *y* = 2006–2008, we calculated its *g*
_{s,i}-value through interpolation based on the two *g*
_{s,i}-values of the next lower and next higher diameter classes.

### Felling + mortality parameter (*fm*
_{s,i})

We calculated stratum (*s*) and diameter class (*i*)-specific *fm*
_{s,i}-values (1 / year) as

where *nfm*
_{s,i,y} = number of trees that disappeared (due to felling or mortality) from the stratum diameter class (*s*,*i*) over the 5-year period prior to the inventory year *y*; *nfm*
_{s,i,y}-values were assessed based on the decomposition status of tree stumps and dead trees; and *cf* = global correction factor to tune *fm*
_{s,i}-values in such a manner to approximately meet the total stem volume that disappeared in 2007 on the country scale (FAO 2010; MAAPRAT 2011). If *n*
_{s,i,y} = 0 for *y* = 2006–2008, then *fm*
_{s,i} = 0.

### Recruitment parameter (*r*
_{s,1})

We calculated stratum (*s*)-specific *r*
_{s,1}-values (number of trees / year) as

where

and *n*
_{s,y} = number of trees inventoried in year *y*; *d*
_{s,j,y} = diameter (cm, over bark) of tree *j* inventoried in year *y*; *∆rc*
_{s,j,y} = radial increment (cm, over bark) over the 5-year period prior to the inventory year of tree *j* inventoried in year *y*; *∆rc*
_{s,j,y}-values were measured on increment cores (under bark) as the cumulated width of 5 annual rings, and an estimate of bark thickness increment was added, based on NFI bark thickness equations (C. Duprez and F. Morneau 2010, French NFI, unpublished data).

### Appendix 2: Method used to calculate sensitivity measurements

We calculated sensitivity measurements separately per diameter class for a given model parameter or output variable *Y*. The sensitivity of *Y* to one model input factor (here: stratification factor) *X*
_{a} (first order effect) was measured as the ratio between the *Y*-variance *V*
_{a}, due to *X*
_{a}, and the total *Y*-variance *V*(*Y*) (Saltelli et al. 2004; Wernsdörfer et al. 2008):

Similarly, the sensitivity of *Y* to two input factors *X*
_{a}, *X*
_{b} (second order effect) and three input factors *X*
_{a}, *X*
_{b}, *X*
_{c} (third order effect) was measured as

where *V*
_{a,b} and *V*
_{a,b,c} are the *Y*-variances due to *X*
_{a}, *X*
_{b} and *X*
_{a}, *X*
_{b}, *X*
_{c}, respectively. The variances *V*
_{a}, *V*
_{a,b} and *V*
_{a,b,c} were calculated as

where the expectation *E* was approximated as a mean. For a deterministic model, the sum of the sensitivity measurements for all orders is equal to one.

In this study, *V*(*Y*) was the empirical variance across the 54 values of *Y*, i.e. across the 54 strata subjected to the global sensitivity analysis (see Global sensitivity analysis). Considering the 54 values of *Y* as the outcome of a three-factor pseudo-experiment without replicates, the decomposition of the variance of *Y* was basically the same as for a three-way analysis of variance (though here unbalanced). As there was no replicate, the number of degrees of freedom for the residuals was equal to zero. However, the residual variation may be identified using the third order interaction. This means that the first-order effects and the second-order interactions could be interpreted as a ratio over the third-order interaction.

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Wernsdörfer, H., Colin, A., Bontemps, JD. *et al.* Large-scale dynamics of a heterogeneous forest resource are driven jointly by geographically varying growth conditions, tree species composition and stand structure.
*Annals of Forest Science* **69, **829–844 (2012). https://doi.org/10.1007/s13595-012-0196-1

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### Keywords

- Forest resource
- Forest dynamics
- Stratification
- Matrix model
- Tree diameter class
- National forest inventory