Modelling ingrowth in mediterranean pine forests : A case study from scots pine ( Pinus sylvestris L . ) and mediterranean maritime pine ( Pinus pinaster Ait . ) stands in Spain

National Forest Inventories (NFI) are a basic tool for forestry planning at the National level. A new two-step system for predicting ingrowth compatible with NFI data is presented in order to improve long-term estimation of stand condition. In growth and yield models, an ingrowth submodel is a key feature for long-term estimation. An accurate projection of ingrowth is needed to avoid model projection bias and inaccuracy. A two-step approach was used, which consisted of (I) estimating the probability of ingrowth occurrence on a sample plot and (II) quantifying the ingrowth in terms of basal area. Logistic regression was used for step I, while linear regression was used for step II. A good performance of the joint ingrowth model for Scots pine (Pinus sylvestris L.) and Mediterranean Maritime pine (Pinus pinaster Ait. Ssp mesogeensis) stands was observed. Logistic model include quadratic mean diameter as independent variables for both species while basal area is only included for Mediterranean Maritime. Quadratic mean diameter is the only independent variable in linear model for both species. The presented two-step modeling methodology for ingrowth is applicable to data from National Forest Inventories with concentric plots.

In uneven-aged stands or under continuous cover forestry systems the impact of ingrowth is higher than for even-aged stands where ingrowth ca be lacking during most of rotation.
Forest managers face uncertainty when they try to provide information to stakeholders.As Fox et al., (2001) state, accurate statements of prediction uncertainty are valuable for accounting for risk and uncertainty in forest management.Forest models can be classified into several different categories (empirical vs process models, spatial vs non-spatial models, deterministic vs stochastic models).Most biological processes have a stochastic component.Deterministic models are very useful to represent past growth conditions and give 'average' results but they do not provide much information on the expected variability of outcomes (Buongiorno and Gilles, 2003).Stochastic models attempt to cap-ture the variability and the uncertainty of the outcomes (Messier et al., 2003).Although there is an extensive literature incorporating stochasticity in stand and landscape level models, there is less information on how to incorporate stochasticity into distance-independent individual-tree growth models.Notable exceptions are Wykoff et al., (1982), Hamilton, (1991), Miina, (1993), Stage and Wykoff, (1993) and Fox et al., (2001).
Stochasticity impact on diameter and height growth and mortality is low because mortality rates are very small and trees grow in a fairly constant rate, given their size and stand conditions (density, productivity…).However, as we stated before ingrowth has a major stochastic component that contribute to the low performance of this component in distance-independent individual-tree models that do not consider it or include it in a deterministic manner.If stochasticity is not considered, a biased model outcome should be expected (Hamilton, 1991).This may not be important when foresters are interested in ranking different management alternatives.However, when they are focused on high precision growth and stand structure prediction, a stochastic approach must be considered.In addition, global change processes dramatically increase the stochasticity of forest responses to alternative silvicultural treatments (Bravo et al, 2008), so stochastic model approaches will become more important in the near future as environmental conditions become increasingly uncertain.
Ingrowth refers to those plot trees that during the time between two inventories of the plot have grown into a size that requires them to be measured by the inventory (Shiver andBorders, 1996) Ledermann (2002) and Muhairwe (2003) used this ingrowth definition in their models.However, because in most inventories at the National level, such as in the Finish (METLA, 2000), French (IFN, 1985), Spanish (Bravo et al., 2002), Swiss (Gertner and Kohl, 1992) and American (USDA, 2004), plots are nested concentric with different DBH thresholds, when an stochastic model approach is used, it is necessary to consider ingrowth in the different concentric plots.In a concentric plot, where there is a variable minimum plot radii and tree diameters to be inventoried, there may be a given time when adult trees are not measured which, due to their position with respect to the center of the plot and their size, cannot be inventoried in a specific subplot among those which make up a plot.Between two inventories even some adult trees may reach a size that makes it necessary to include them and to measure them at the next (upcoming) inventory.They therefore become part of the ingrowth, even though they are not young trees as an artifact of concentric plots with a variable minimum radii and diameters method.For our purposes, ingrowth concept includes young trees that become tally trees during the projection timeline (we will call this recruitment) and mature-trees that become tally trees during the projection timeline (we will call this upgrowth).Williams (1998) distinguished between change and growth in forest surveys.While growth is the biological increase in size between inventory times (t and t+1), change is the difference in aggregate quantity of all trees larger than a user-defined size threshold between times t and t+1.Both concepts growth and change, include an ingrowth component.According to Williams (1998) it would be better to have an intensive and specialized inventory of ingrowth, but in most cases it is too expensive for a large scale forest inventory.Ingrowth models can help us to solve, at least partially, the prediction bias when ingrowht is ignored by estimating it in a mid-point between inventory time t and t+1.
Traditional ingrowth models are inappropriate for a concentric plot design because they usually only predict the basal area for the in the innermost plot.Also, the DGB of ingrowth trees is frequently simply assigned to the lower diameter classes.When a deterministic model is used to predict future forest scenarios only a recruitment model is needed because the expansion factor is used as a modifier for mortality and growth impact (Vanclay, 1991).However, when a stochastic model is used to generate forest scenarios for concentric plot-based inventories, then a prediction of ingrowth in the different concentric plots is needed to make accurate predictions.Different strategies have been used to model ingrowth process, Markov chains (Valentine and Furnival, 1989), non-linear models (Moser, 1972;Moser, 1974;Hann, 1980), linear models (Trasobares et al., 2004a;Trasobares et al., 2004b) or a combination of logistic and linear or loglinear models.Two variants of this last approach have been used: (a) first predict through a linear model the amount of ingrowth and then use logistic regression to distribute the predicted amount between the stand species (Vanclay, 1989) or (b) first predict a probability of ingrowth then with probability proportional to the prediction assign the event 'ingrowth' to a stand (by a random draw from Bernoulli distribution) and then use the linear or log-linear model to predict the amount of ingrowth (Ledermann, 2002;Sánchez Orois and Rodríguez Soalleiro, 2002;Muhairwe, 2003).Ledermann (2002) who usesd this approach included functions to estimate ingrowth's specific composition and ingrowth's tree DBH and height.
Ingrowth models for Mediterranean forests are scarce.Models for Pinus sylvestris and Pinus nigra stands (Trasobares et al., 2004a) and for Pinus halepensis (Trasobares et al., 2004b) are exceptions.In these models, ingrowth was estimated from linear model with number of trees in the lowest diameter class as the explanatory variable.However, the coefficient of determination for these models was low (R-squared from 0.04 to 0.11).The ingrowth model developed here is part of a distance independent individual-tree model, developed for Scots pine and Mediterranean maritime pine in Spain (Bravo et al., 2005 and2007).In distance independent individual tree models the development of individual trees is modeled without explicit knowledge about the spatial distribution of the trees.Development of simple and accurate models that allow forest managers to predict, under alternative silviculture paths, different future scenarios is a prerequisite in sustainable forest management.
The objective of this study was to suggest and develop a new ingrowth model for Mediterranean stands of Pinus sylvestris L. and Pinus pinaster Ait. in central Spain that is compatible with a stochastic individual distance-independent tree growth model and with inventories using concentric plots as in the Spanish National Forest Inventory.The ingrowth model was developed according to a two-step approach.In addition, our results also serve to improve our understanding of the ingrowth process in this area of Spain.

Database
A database from the Sustainable Forest Management Group plot network (University of Valladolid) was used.The database contains detailed individual tree data from 103 plots located in Pinus sylvestris stands and 88 plots located in Pinus pinaster stands in central Spain.Plots were measured during the period from year 2002 to 2004.The spatial distribution of plots covers the entire range of site productivity, in accordance with the site index curves of Rojo and Montero (1996), Bravo and Montero (2001) and Bravo-Oviedo et al. (2004), density and age.As is usual, the oldest ages, especially at the best qualities, are not fully represented, because more intensive silviculture has been performed on the best sites of both species, and in the case of Maritime pine, due to the tapping of the high-quality stands 50 years ago.The plots are circular and composed of three subplots with radii of 5, 10 and 15 meters.The minimum DBH that qualifies a tree for measurement in the three plots is 75, 125 and 225 mm, respectively (Bravo et al., 2002).Diameter at breast height, total height and radial diameter increment were measured in all tally trees.On the basis of the increment data, auxiliary equations (site index curves and change in dominant height growth, etc.) and static equation (to estimate bark thickness), the dimensions of the trees were calculated backward to their measurements five years ago as a backdating procedure (Hann andHanus, 2001, Lizarralde, 2008).Such data provide unbiased estimates of permanent plot growth components (Wykoff, 1990), and they have been applied widely both in the evaluation of alternative measures of competition (Biging andDobbertin, 1992 and1995;Bravo et al., 2001), and in the development of different stand development models such as CACTOS (Wensel et al., 1987), FVS/Prognosis (Wykoff, 1990), and ORGANON (Hann and Ritchie, 1988;Hann and Larsen, 1991).None of the sampled stands had been treated silviculturally in the past five years.
After the basic tree variables had been backdated, a number of tree and stand variables were calculated for the start of the growth period, including total basal area (BA), trees per ha (N) and Site Index (SI).Site index was calculated from equations of Rojo and Montero (1996) and Bravo and Montero (2001) for Scots pine stands and from Bravo-Oviedo et al. (2004) for Maritime pine stands.The resulting data set contained 33 (32.04 %) plots showing ingrowth for Scots pine stands and 47 (53.41 %) plots showing ingrowth in Maritime pine stands.In Scots pine stands only 3 plots contained recruitment while in 33 plots upgrowth was found.In comparison, 7 Maritime pine plots included recruitment while 47 plots showed upgrowth.So we can see that the ingrowth generated by upgrowth is more important than recruitment which was not detected alone in both species.
The plots (table 1) cover a wide range of basal area (Scots pine stands: from19.19 to 78.89 m 2 /ha.Maritime pine: from 13.25 to 55.59 m 2 /ha), trees per ha (Scots pine: from 212 to 3958 trees/ha.Maritime pine: from 159 to 2465 trees/ha) and site index, dominant height at 100 years for Scots pine and at 80 years for Maritime pine (Scots pine 12.7 to 38 meters.Maritime pine 7.5 to 23.6 meters).

Statistical Methods
The ingrowth model has two parts: i) a logistic model to predict the probability of ingrowth occurrence in a specific plot and ii) a linear model to predict the basal area (m 2 /ha) of ingrowth trees conditional on P(ingrowth>determined threshold).
The logistic model was developed for both species.The logistic model has the following general structure: In which P is the probability that there will be a positive ingrowth in the plot during the next five years, α is the constant term, is a linear combination of parameters ∑b i X i and independent variables X i .The independent variables tested for both species are the site index (SI), the number of trees per ha (N), the basal area (BA) and the quadratic mean diameter (QMD).The "backward" variable selection procedure was used with a probability of 0.05.Hosmer and Lemeshow test (Hosmer and Lemeshow, 1989) and the Akaike information criterion (Zhang et al., 1997) were used to assess goodness of fit.If the predicted probability of ingrowth occurrence was greater than or equal to a threshold, the ingrowth in the plot was considered realized; otherwise it was considered absent.The threshold value, to determine whether the stands have ingrowth or not, was chosen in such a way that the percentage of correctly classified plots was maximized.The logistic equation can be formulated to accept a binary dependent variable, such as a survival indicator or presence/absence of regeneration or ingrowth, and the parameters can be estimated by either weighted nonlinear regression or by maximum likelihood methods (Monserud and Sterba, 1999).The resulting predictions are bounded by 0 and 1.The logistic equation form has been widely applied in the past for modeling different binary events in forest research (Hamilton and Edwards, 1976;Hamilton, 1986;Hann and Wang, 1990;Monserud and Sterba, 1999;Cao, 2000;Bravo et al., 2001;Hann and Hanus, 2001;Álvarez González et al., 2004;Bravo-Oviedo et al., 2005).
The linear component of this model allows us to quantify ingrowth (in basal area terms) in those stands in which, on the basis of the logistic component, it is determined that there is ingrowth.The linear component tested is as follows (eq.2): In which BA ing is the ingrowth in m 2 /ha, QMD is the quadratic mean diameter (cm), BA is the basal area (m 2 /ha), N is the number of trees per ha, and SI is the site index (m) according with Rojo and Montero (1996), Bravo and Montero (2001) and Bravo-Oviedo et al. (2004).Selection of the variables was performed using the backward procedure (probability equal to 0.05).The adjusted coefficient of determination was used to determine the liner model fit's accuracy.
The joint model's adequacy was assessed by estimating a coefficient of determination (eq.3), through the fit of a straight line between real value and predicted values of the ingrowth (eq.4), as well as the basal area at the end (BA 2 ) of the projection period (eq.5) and by calculating bias of the model (eqs.6 and 7) to determine the accuracy of the joint two-step model (Huang et al., 2003).

[eq. 3]
Where and are, respectively, the sample's variance of errors committed and the sample's variance in the dependent variable (basal area).
BA ing = c 10 + c 11 BA ing_pred [eq. 4] BA 2 = c 20 + c 21 BA 2_pred [eq.4] [eq. 6] [eq. 7] In which, n is the number of observations, BA ing is the actual ingrowth, BA ing_pred is the predicted ingrowth, BA 2 is the basal area at the end of the project period, 5 years, BA 2_pred is the predicted basal area at the end of the projection period, all in m 2 /ha, while c i0 and c i1 are the parameters to adjust which, if equal to 0 and 1, respectively, demonstrate that the model is unbiased.

Results
The proposed two-step model allows us to reach a joint model adequacy equal to 63.5 % for Scots pine and to 70.0 % for Maritime pine.

Step I. Logistic Component
On the basis of the complete model (eq.1), the statistically non significant variables were gradually eliminated by using a backwards elimination procedure, leaving us with the final model for Scots pine only including QMD as independent variable, while in the case of the Maritime pine both the QMD and BA were significant.The sign of the QMD with expectations while BA sign not but with significance close to the limit (p-value equal to 0.0416).
In the case of the Scots pine (table 2), the final model showed a value for the Akaike information criterion equal to 75.709 (if the independent variables are not considered, it is equal to 131.195), and the Hosmer and Lemeshow test (Pr >0.8279) confirm a good fit.On the other hand, for Maritime pine (table 2), the Akaike information criterion value is equal to 49.648 (123.584 in the event of not considering the independent variables), whereas the Hosmer and Lemeshow test also shows that there is no lack of fit (Pr >0.9199).Threshold values were 0.38 for Maritime pine and to 0.43 for Scots pine.These threshold values allow us to classify correctly 88.6 % of Maritime pine plots (sensitivity equal to 95.7 % and specificity equal to 80.5%) and 86.4 % of Scots pine plots (sensitivity equal to 78.8 % and specificity equal to 90.0 %).

Step II. Linear Component
For the two species studied, the linear component that allows us to estimate total basal area quantity incorporated into the stand was reduced to a function of the quadratic mean diameter.As expected, the higher the QMD, the lower is the basal area contribution of ingrowth trees (table 2).For Scots pine plots, the adjusted coefficient of determination was 52.57%.For Maritime pine it was 51.38 %

Validation of the Joint Model
The model for Scots pine shows a joint determination coefficient equal to 63.5%, whereas the case of Maritime pine stands is equal to 70.0%.The result of the fitted validation straight lines between real and predicted values is shown on Table 3.In all cases, the parameter representing the independent term contains the zero in its confidence interval, and the independent variable parameters contain the one in their confidence interval.So, the joint model does not show bias or lack of accuracy (fig 1).The absolute bias for the Pinus sylvestris and Pinus pinaster ingrowth was very low (-0.095 and -0.079 m 2 /ha respectively).In relative terms the Pinus pinaster ingrowth bias is low (7.37 %) while the Pinus sylvestris is low enough (17.51%) to allow us to accept it.However, additional data and validation is needed to help us reach a better ingrowth model for Scots pine.

Discussion
Usually, ingrowth is not taken in account in growth and yield models (Curtis et al., 1981;Curtis et al., 1982;Wensel et al., 1987;Rojo and Montero, 1996;Hann et al., 1997;Bravo and Montero, 2001;Montero et al., 2001) or, when considered, the resulting prediction is poor or not compatible with concentric plot design.Usually, the low performance of ingrowth models is due to the fact of not considering the stochastic nature of this event and fitting linear models based on database containing a large number of zero events (no ingrowth).Stiff (1979) obtained a low R 2 (0.14) for Appalachian hardwoods.Hyink and Moser (1983) dealing with Northern hardwoods found just 0.08 value of R 2 .Trasobares et al. (2004a and2004b) obtained low determination coefficient for ingrowth models for Pinus sylvestris (R-squared equal to 0.11), Pinus nigra (R-squared equal to 0.11) and Pinus halepensis (Rsquared equal to 0.04) in a Mediterranean area.On the other hand, some authors have been succesfull in modeling ingrowth.Moser (1972) and Ek (1974), working on Northerns hardwoods, obtained a R 2 over 0.70.Although the determination coefficient cannot be used to compare the performance of models dealing with different dependent variables, it is useful to study the percentage of variance explained by different models.Other authors (Ledermann, 2002;Sánchez Orois and Rodríguez Soalleiro, 2002) have used a compound model with two independent parts (logistic plus linear model) but these do not consider the impact of concentric plot design on ingrowth.The two-step methodology allows us to obtain adequate projections and surpass the results obtained with linear functions, and it is applicable under the widespread concentric plot method.
Density related variables (basal area, just in Maritime pine, and quadratic mean diameter, in both species) sents just the 3.03 %.The bias values (absolute and relative) should be used considering the total ingrowth account for both species that is equal to 3.3 % (Scots pine) and 5.2% (Mediterranean Maritime pine) of total basal area at the end of the considered five-year growth period.Low growth rate has influence upon the small basal area amount that ingrowth represents.Parameters signs match the expected in accordance with current knowledge.
Finally, an allocation rule to distribute the ingrowth between the different concentric plots is needed.The past observed ingrowth rate for each concentric plot can be used.This procedure is similar to proportional growth allocation in stand growth distribution between trees in disaggregation models (e.g., Clutter and Jones, 1980;Pienaar and Harrison, 1988;Ritchie and Hann, 1997).Observed ingrowth rates in the different concen-are significant in our model and show the importance of stand density driving the ingrowth process (figs. 2 and 3).
Joint validation shows that the two-stage model is useful for predicting the ingrowth in stands of Scots pine and Maritime pine (Fig 1 and table 3) inventoried by using concentric plots.Bias values (absolutes and relatives) allow us to recommend the use of the developed model instead of ignoring the fact that ingrowth occurs in targeted forest ecosystems.However a validation procedure, with an independent data set, should be done to confirm the validity of developed model.Most of the ingrowth detected follows from the concentric plot sample design but, in addition, there are small amounts of basal area that correspond with recruitment.In the Pinus pinaster stands recruitment represents the 7.69 % of total ingrowth while in Pinus sylvestris repre- Ingrowth rate is smaller in the inner plot and grows as plot size increase.Pinus pinaster ingrowth rates were 8.08, 32.63 and 59.28 percent in each concentric plot while in Pinus sylvestris database they were 3.84, 27.18 and 68.98 percent.It is clear that this allocation rule is only a first approximation.However, this simple rule can serve as the starting point.As far as we know there are no experiences on allocating ingrowth under concentric plot design so new research is needed to refine our approach.Different allocation approaches that have been suggested for growth equations (Ritchie and Hann, 1997) can be explored.Together, logistic and linear models accurately predict ingrowth under concentric plot design forest inventories.Results on Mediterranean species show the importance of density to model ingrowth process.
Developed two-step model will be included in a growth simulator to project Spanish National Inventory.In addition, the ingrowth model developed jointly with diameter and height growth, mortality and other relevant equations (taper, site index equations…) composed a model that can be used to optimize stand management practice and to accurately evaluate alternative management regimes.

Figure 1 .
Figure 1.Actual values (y-axis) vs predicted values (x-axis) ingrowths (upper graphs) and.Actual values (y-axis) vs predicted values (x-axis) of basal area at the end of five-year growth period (lower graphs) for Scots pine (left graphs) and Mediterranean Maritime pine (right graphs).

Figure 2 .Figure 3 .
Figure 2. Ingrowth probabilities predicted by the logistic model for different basal area and mean quadratic diameter for Maritime pine (Pinus pinaster Ait.) in Central Spain and for different mean quadratic diameter for Scots pine (Pinus sylvestris L.).

Table 1 .
Descriptive statistics for studied Scots pine and Mediterranean Maritime pine stands in Central Spain.SI: Site index, N: Trees per ha, BA: basal area, QMD: Quadratric mean diameter and BA ing : ingrowth expresed as basal area

Table 2 .
Logistic and linear components of the ingrowth models for Pinus sylvestris L. and Pinus pinaster Ait.stands in Central Spain.BA: basal area, m 2 /ha, QMD:quadratic mean diameter, cm

Table 3 .
Ingrowth model validation.Linear models show, for both ingrowth and final basal area values, a high determination coefficient and the independent term is not different from zero and the slope is not different from one.In brackets the parameter standard errors are shown tric plots follow the same pattern for both species.