Efficient means for assessment of the dynamics and the state of the stocks of renewable assets such as wood biomass are important for sustainable supplies satisfying current needs. So far attention has been paid mainly to the economic aspects of forest management while ecological problems are rising with the expected transfer from fossil to renewable resources supplies of which from forest being essential for traditional consumers of wood and for emerging biorefineries. Production of biomass is more reliant on assets other than money the land (territory) available and suitable for the purpose being the first in the number. Studies of the ecological impacts (the “footprint”) of sustainable use of biomass as the source of renewable energy encounter problems associated with the productivity of forest lands assigned to provide a certain annual yield of wood required by current demand for primary energy along with other needs.
Apart from a number of factors determining the productivity of forest stands, efficiency of land-use concomitant with growing forest depends on the time and way of harvesting (Thornley & Cannell, 2000). In the case of clear-cut felling the maximum yield of biomass per unit area is reached at the time of maximum of the mean annual increment (Brack & Wood, 1998; Mason, 2008). The current annual increment (rate of biomass accumulation by a forest stand or rate of growth) culminates before the mean annual increment reaches its peak value and there is a strong correlation between the maximums of the two measures. Knowing the time of growth-rate maximum (inflection point on a logistic growth curve) allows predicting the time of maximum yield (Brack & Wood, 1998). However, the growth-rate maximum is not available from field measurements directly. Despite the progress in development of sophisticated models simulating (Cournède, P. et al., 2009; Thürig, E. et al., 2005; Welham et al., 2001) and predicting (Waring et al., 2010; Landsberg & Sands, 2010) forest growth, there still remains, as mentioned by J. K. Vanclay, a strong demand for models to explore harvesting and management options based on a few available parameters without involving large amounts of data (Vanclay, 2010).
The self-consistent analytical model described here is an attempt to determine the best age for harvesting wood biomass by providing a simple analytical growth function on the basis of a few general assumptions linking the biomass accumulation with the canopy absorbing the radiation energy necessary to drive photosynthesis. A number of reports on employing remote sensing facilities (Baynes, 2004; Coops, et al., 1998; Lefsky et al., 2002; Richards & Brack, 2004; Tomppo E. et al., 2002 ; Waring et al.,2010 ) strongly support the optimism with regard to successful use of the techniques to detect the time of maximum yield of a stand well in advance by monitoring the expanding canopy.
According to the grouping of models suggested by K. Johnsen et al. in an overview of modeling approaches (Johnsen et al., 2001), the model described in this chapter belongs to simplistic traditional growth and yield models. It differs from other models of this kind by not incorporating mathematical representations of actual growth measurements over a period of time. Derived from a few essential basic assumptions the analytical representation rather provides the result that should be expected from measurements of growth under “traditional” (idealized) conditions. The chosen general approach of modeling the biomass production at the stand level allows obtaining compatible growth and yield equations (Vanclay, 1994) of a single variable – the age. Like with many other theoretical constructions the applicability of the model to reality is fairly accidental and restricted. However, since the derived equations are in good agreement with the universal growth curves obtained from measurements repeatedly confirmed and generally accepted as classic illustrations of biomass dynamics (Brack & Wood, 1998; Mason et al., 2008), it seems to offer a good approximation of the actual biomass accumulation by natural forest stands.
Equations representing the model are believed to reflect the simple assumptions made on the basis of common knowledge about photosynthesis and observations in nature: biomass is produced by biomass; the amount of produced biomass is proportional to the amount of absorbed active radiation; the absorbed radiation is proportional to effective light-absorbing area of the foliage (number and surface area of leaves) and limited by the ground area of the forest stand (the area determining the available energy flow). Projection of the canopy filling the ground area detectable by remote sensing is assumed to reflect dynamics and status (the stage) of forest growth. The height of the stand is another growth parameter accessible by remote sensing. Relationships of the latter with other measurable quantities determining the yield of accumulated biomass are well studied (Vanclay, 2009) and can be employed for remote assessment of the current annual increment and the state of forest stands (Lefsky et al., 2002; Ranson et al., 1997; Tomppo et al., 2002). The model presented hereafter has been developed to be aware of the current annual increment reaching the maximum merely from the data of remote observation of the dynamics of forest stand canopy while complemented by data of the average height would predict the yield.
2. General approach and basic equations
The analytical model offered to describe dynamics of the standing stock of wood biomass in natural forests is based on the obvious relationship between the rate of growth (rate of accumulation of biomass)
By turning to common knowledge that biomass is produced by biomass the rate of accumulation of new biomass in the first approximation can be assumed being proportional to the amount of biomass already accumulated:
and integrating it provides and exponential growth of the stock of biomass:
which is unrealistic in the long run because of finite resources of nutrients and other limiting factors not taken into account in Eq. (2). The problem can be solved by setting an asymptotic limit to growth:
The rate of biomass accumulation
It seems to be reasonable to assume that accumulation of biomass in a forest stand occupying a large enough land area follows the same law as the rate at which the light-absorbing area (the canopy) of the growing stand expands with time. As noticed, the number and total surface area of leaves absorbing radiation is proportional to the accumulated biomass approaching some asymptotic limit
the rate of expansion of the light-absorbing area expressed as:
can be written in the form:
Dimension of the constant in Eq. (8) is the reciprocal of the product of area and time. Since area
Assuming that the rate of biomass accumulation follows the rate of expansion of the light-absorbing area it can be described by equation similar to Eq. (8):
where the value of the constant factor (dimension of which here is the dimension of current increment) can be chosen to satisfy some selected normalizing condition, as will be done further.
The time-dependent part of Eq. (9) has a maximum at time
The normalized rate of biomass accumulation expressed by current annual increment in time scale
Returning to Eq. (1) the biomass stored by time x = xc is expressed by definite integral:
By normalizing the stock choosing its asymptotic limit as the normalized unit
3. Mean annual increment and productivity
The mean annual increment of a forest stand is an essential factor illustrating the overall productivity of the stand at a given age and is expressed by the ratio of stock to age of the stand (Brack & Wood, 1998). The stock being presented by Eq. (16) the mean annual increment
In Fig. 4 the current annual increment (rate of biomass accumulation) and the mean annual increment are presented together wherefrom the mean annual increment is seen to reach the maximum value (equal to ≈ 0.8 of the peak value of current annual increment) at cross-point of the two curves.
The reciprocal of the mean annual increment is a parameter characterizing the size of plantation for sustainable supply of biomass. The total area of a plantation for sustainable annual supply comprised of equal lots of stands of ages in sequence from one year to the cutting age is directly proportional to cutting age
The constant is equal to the required annual yield of biomass; function
is the reciprocal of the mean annual increment at cutting age.
As follows from Eq. (22), felling the forest at age corresponding to 1.8 units of the normalized time scale provides the maximum yield per unit area of a particular stand and hence of the whole plantation. In other words, the maximum productivity of land area under a forest is achieved when felling at the time of the mean annual increment peak.
4. Validation of the model
Neither the value of the current annual increment at maximum, nor the real time when a forest stand reaches the maximum is known
The growth-rate function given by Eq. (14) cannot be used directly to compare the model equation with experimental growth-rate data. For that purpose a different exponential equation can be employed containing variable parameters related to the quantities not measurable directly. The values of the variable parameters providing the best fit of the measured annual increments with the equation are chosen to evaluate the unknown quantities. A rather abundant database available for natural grey alder (
The 5-year mean annual increments available from field measurements (Daugavietis, 2006) are a good approximation for the current annual increment value at mid-time of the respective 5-year period (Fig. 5, a). By choosing a function of the type
to describe the current annual increment it is possible to assign physical sense to variable parameters
It should be noticed here that dimension of constant
By varying parameters
The values of increments calculated from Eq. (23) coincide with the set of experimental data (Daugavietis, 2006) (Fig. 5) within standard deviation of 2.5 % of the maximum value, the correlation between the sets of calculated and experimental data being better than 0.99.
The normalized time scale is introduced by choosing variablex to satisfy condition
By substituting the normalized time variable
By defining new constant parameters and Eq. (25) is rewritten as:
Normalizing function (26) with respect to ym = (
The mean annual increment
reaches maximum under condition
After finding the age of the maximum of current annual increment, the set of experimental points (Fig. 5, a) can be put on the normalized time scale
5. Rate of growth as function of light-absorbing area
Equation (9) describing the rate of biomass accumulation derived from Eq. (7) in section 1 is based on the assumption that dynamics of current annual increment follows dynamics of the expansion of light-absorbing area of the canopy. Returning to Eq. (7) it can be assumed to describe the relationship between the normalized rate of growth (
shown in Fig. 6.
It has to be noticed that the pace at which the biomass is stored is not necessarily equal to the pace at which the light-absorbing area increases. The uptake of biomass (photosynthesis) depending on the effective light-absorbing area obviously should follow with some delay, which means that the normalized (intrinsic or specific) time scale of the equation derived from Eq. (8) to describe the rate of expansion of the light-absorbing area:
is different from that of Eq. (14) describing the rate of biomass accumulation.
Relationship between the units of the two normalized time variables –
wherefrom, remembering that
It means that a unit of the normalized time scale of the rate of expansion of the light-absorbing area is about 0.56 of the unit of the normalized time scale describing the rate of biomass accumulation. The units of the two normalized time scales presented in Fig. 7 are approximately equated by
As seen from Fig. 7, expansion of the light-absorbing area of the canopy (curve 1) proceeds ahead of the rate of biomass accumulation (curve 2) complying with the assumption that higher rates of the increase of the surface area (and the number) of leaves require a greater proportion of the gross product of photosynthesis lost after seasonal vegetation.
The size of the effective light-absorbing area expressed by the ratio to its asymptotic limit is presented in Fig. 7 on the lower time axis. The maximum rate of expansion
The basic components of the model – equations presenting current and mean annual increments, stock, and the rate of expansion of the light-absorbing area as functions of age expressed in the intrinsic time units are summarized in Fig. 8.
6. Conceptual remarks
The analytical expressions comprising the model are derived from rather general principles of biomass production by photosynthesis in living stands without taking into account factors affecting forest growth other than the effective light-absorbing area of the canopy. However, since dynamics of the latter is strongly dependent on availability of nutrients, water, and some other crucial factors, the model reflects the cumulative effect of all of them through the relationship between the rate of growth and the capacity to capture the active radiation. Therefore, monitoring the canopy dynamics can provide reliable information for conclusions about that capacity and the expected end product of photosynthesis.
Determining the best time for harvesting by observing expansion of the canopy from satellites is one of attractive practical applications of the model for management of even-age stands in concert with remote sensing. Even though the canopy projection measureable by remote sensing instruments is not quite equal either to the light-absorbing area or the leaf area index, the correlation between the three is strong enough to make corrections necessary for detecting the time (age) of growth-rate maximum from remote observations of the dynamics of canopy expansion.
with parameter values
A generalized differential form of sigmoid growth (the growth-rate function) has been considered by C. P. D. Birch (Birch, 1999) and a detailed formalistic analysis of the family of sigmoid growth equations is given by O. Garcia (Garcia, 2005). The sigmoid shape of the yield (stock) curve Eq. (17) in the present case is predetermined by the shape (the maximum) of the obtained growth-rate function Eq. (14).
The normalized time unit introduced to provide a dimensionless common measure to match the model with experimental data is the same intrinsic time unit suggested by B. Zeide as a unit provided by organisms themselves and clarifying the meaning of parameters of growth functions (Zeide, 2004). A number of other growth factors, such as biological potential of a particular species, the site quality, changing climate, etc. are reflected in the real-time equivalent of the intrinsic time unit. For instance, comparison of best fits to available measured data of grey alder stands at sites of different quality (Daugavietis, 2006) show the stands at sites of higher quality reaching the growth-rate maximum earlier (Kosmach, 2010). Since climate change is a factor affecting forest growth (Nakawatase & Peterson, 2006), the real-time equivalents of the intrinsic time unit obtained from monitoring the growth of stands of a given species hold information for potential assessment of the changing environment accessible by remote observations and retrospective studies of forest growth.
The Richards equation (37) predicts diminishing of the current increment to zero with the age of the stand while the effective light-absorbing area given by Eq. (6) approaches a constant maximum and, therefore, should be expected to provide a constant maximum increment of biomass. However, the real growth curves (at least of natural forest stands) rather comply with Richards equation even if the underlying models do not take into account factors, such as respiration or partition, diminishing the annual above-ground biomass production. In the present case they are somehow implied in the factor (
The simple logistic analytical model of biomass accumulation by forest stands derived on the basis of general assumptions about photosynthesis comprises compatible equations of growth and yield as functions of time. The function describing dynamics of the rate of growth derived as function of the effective light-absorbing area of the canopy provides a growth function representing particular case of Richards equation and is in good agreement with data obtained from experimental measurements. The model contains two related parameters: the unit of the intrinsic (normalized with respect to peak current annual increment) time scale and the effective light-absorbing area of the canopy not equal but closely related to the leaf area index or to projection of the canopy. The latter accessible by remote sensing opens the use of remote sensing data for monitoring the growth of forest stands to predict the culmination of current annual increment the age of the stand at which being known allows predicting the optimum age for harvesting.
The model has been developed for determining the land area and the optimum harvesting age of even-aged natural stands for sustainable supply of firewood and wood biomass to satisfy the needs of paper mills and biorefineries. It can be extended to consider solutions of the same problems with regard to timber products such as boards and other construction elements of buildings.
Some further studies are necessary to find out the relationship between remote observations of canopy dynamics and dynamics of the effective light-absorbing area to realize the benefits of using the model with the opportunities provided by remote sensing to forest management.
The presented model has been derived on the basis of studies supported by the National Research Programs of Renewable energy resources and rational management of deciduous tree forests.