Improving local forest volume estimates by fusion of multi-temporal forest type maps

Environmental Modelling & Software 15 (2000) 373–385 www.elsevier.com/locate/envsoft

Improving local forest volume estimates by fusion of multi-temporal forest type maps Tom De Groeve *, Kim Lowell Center for Research in Geomatics, Pavillon Casault 0722, Laval University, Ste-Foy, QC, G1K 7P4, Canada Received 1 June 1999; accepted 27 November 1999

Abstract An alternative method for estimating standing wood volume based on the fusion of multi-temporal forest type maps and single reduced associated ground-based inventories is proposed. With the integration of photo-interpreted forest map realizations from different years into a single “Fused Map” resulting in an improved local estimate of the forest type, the proposed method offers more accurate estimates than the approach traditionally used by the Quebec Ministry of Natural Resources, even with reduced ground-based inventory effort. A Fused Map is basically an adapted mean of local forest type that accounts for differences among the classification systems used for each map, temporal differences between maps, and subjectivity associated with photo-interpreted data.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Spatial uncertainty; Wood volume estimation; Map overlay; Data integration; Geomatics

1. Introduction The traditional method of estimating wood volume that is employed by the Ministry of Natural Resources of Quebec (MNR) is designed to manage the productivity of the Quebec public forest (an area of some 757900 km2) on a large scale, i.e., a regional scale. Based on information about standing wood volume and forest growth rates, the MNR assigns annual harvest quotas to forest companies while ensuring the survival of the forest resource and its ecological communities. The traditional method of volume estimation consists of three steps (Be´langer et al., 1986). First, experienced photo-interpreters subjectively analyze large-scale aerial photographs (1:7500 to 1:15000) in order to segment the forest into homogeneous forest stands and identify the ecological parameters (i.e., the forest type) of those forest stands. The size of a typical forest stand is about 3 ha, but can be as low as 0.1 ha. For the identification of the forest type for each stand, the MNR considers four attributes: species composition, mean height, mean den-

* Corresponding author. Tel.: +1-418-656-5491; fax: +1-418-6563607. E-mail address: [email protected] (T. De Groeve).

sity (i.e., the ratio of the area occupied by tree crowns projected on the ground to the total forest stand area — note that other parts of the world label this entity “percent crown cover” or “proportional crown cover”) and mean developmental stage of the trees in the forest stand. All attributes are grouped into discrete classes using a standardized classification system. In the second step, a volume for each forest type appearing on a given map is obtained from a groundbased inventory. However, the number of possible forest types is high. For example, for the MNR classification system, the number of possible forest types is 960 (i.e., the product of the number of classes of all attributes). Since the volume for a given type is estimated by measuring all trees in ground-based sample plots distributed throughout stands of a given type, and since the number of sample plots per forest type must be sufficiently high to obtain a volume estimate of a specified precision, a ground survey of all 960 types would require an unreasonable amount of time and money. Therefore, similar forest types are combined into grouped types. This combining (or grouping) is the initial part of the forest-sampling procedure. The grouping is done so as to minimize the variance within a grouped type and to maximize the variance between different grouped types. In the MNR method, the 960 basic forest types are

1364-8152/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 4 - 8 1 5 2 ( 0 0 ) 0 0 0 1 7 - 7

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grouped into 14 grouped types. Following this grouping procedure within the forest-sampling scheme, a specific number of sample plots for each grouped type (proportional to the total area of the grouped type) is determined using an optimal spatial distribution (Bilodeau, 1991). In the subsequent ground survey, every tree larger than 9.1 cm dbh (diameter at breast height) on each sample plot (0.04 ha) is identified and measured. Finally, using this information, stock tables are compiled for a given grouped type showing the average volume by tree species and diameter. Note that, although it is necessary to conduct an economically viable forest inventory, a disadvantage of the forest type grouping is an averaging of the volume for each basic forest type over an entire grouped type thereby causing an increased associated variance within each grouped type. In the third step of the traditional forest volume estimate, the calculated volumes are related to the (mapped) forest types. For a given location, the associated volume estimate is the average volume of the grouped type into which the local forest type was grouped. Note that the reported variance of the estimated volume is dependent only on the statistical variability of the ground survey data. No allowance is made for (thematic or spatial) uncertainty or distortions in the spatial data — i.e., the forest type map. This method of estimating forest volume is unreliable for local volume estimates for four reasons. First, the method produces inaccurate estimates, since for a given location it is not the mean volume of the local forest type that is calculated, but the mean volume of the grouped type. Hence, the traditional method estimates the grouped type volume with a specified precision, but this grouped type volume is an inaccurate estimate of the local volume. Second, the reported variance is of limited value because it is biased. That is, by employing the grouped types, the variance of each individual forest type volume estimate is consistently overestimated. Third, the probability of an erroneous identification of a forest stand is not considered, yet the uncertainty on the polygon’s forest type is another source of variance of the local volume estimate. And finally, the possible positional uncertainty of polygon boundaries is not included. The latter two influences are not considered in large measure because they cannot be estimated from a single forest type map. Consequently, the volume estimator of the MNR is inaccurate for local estimates and the reported variance is inflated and incomplete (i.e., it does not describe the total variance). An alternative, potentially more accurate method for estimating local wood volume would rely less on ground surveys alone. Instead, the information on the forest type map might be used more optimally. The reason that this information is not used in the traditional method is the high associated spatial and thematic uncertainty and a

lack of knowledge about the spatial distribution of the uncertainty across the forest type map. We now examine the forest type maps more closely. The accuracy of the MNR forest type maps is only guaranteed for the original purpose and scale for which the map was made, i.e., the managing of the forest resource on a regional scale. This means that summarizing data on a forest map over a large area — as is done by the MNR in its method of estimating forest type volume — offers information with an accuracy and precision acceptable for the purposes of the MNR. Conversely, on a local scale, volume estimates can be completely inaccurate, even though the forest inventory method used by the MNR is widely accepted, as are its map construction procedures. This inaccuracy is due to a combination of three things: (1) the complex spatial variation of the forest attributes at all scales, (2) the difficulty inherent in accurately identifying these attributes and (3) the limits of a thematic map in representing the forest accurately. We will briefly describe the three factors here. A detailed description of all of the causes of the inaccuracy in the map-making process in a general context can be found in Couclelis (1996), while De Groeve and Lowell, in review, described causes of uncertainty specifically for forest type maps. First, the spatial variation of the forest attributes is very complex. Any cartographic model of any forest is necessarily a simplification of reality and, consequently, causes uncertainty to be inherent in the model. The most frequently used model — also the one used by the MNR — is the forest stand model. This model implicitly supposes that the forest is divided into more or less homogeneous stands with well-defined boundaries — a condition, however, that is frequently violated. Second, the photo-interpretation process actually adds uncertainty to the forest stand delineation process and the identification of its forest type because of its subjectivity. Particularly in regions where the forest is not divided into clearly distinct and homogeneous stands, the subjectivity of the interpretation results in highly variable estimates of the forest attributes. Third, uncertainty is caused by constraints imposed by the mapping method. These constraints are spatial (the forest stand boundaries must be crisp) and thematic (each forest attribute belongs to one and only one class of the classification system used). Consequently, boundary transition zones cannot be represented correctly on the forest type map and each attempt to delimit a forest stand boundary crisply is subject to spatial uncertainty. Similarly, a forest stand with a heterogeneous height or a height on the boundary of two classes within the cartographic classification system cannot be represented correctly. Each forest type appellation (or label) will therefore be subjected to thematic uncertainty. The three factors described induce an elevated level of inaccuracy for polygon boundaries and the general

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content of a given map. In earlier work, we showed that only 5% to 40% of the total area of a typical forest type map contains consistently identified information dependent on how “severely” consistency is defined (De Groeve et al., 1998). In order to decrease the uncertainty inherent in a map, the solution we proposed was the integration of multiple interpretations of the same area. We believed that the integration of multiple map realizations provides a means to modify the forest inventory procedure of the MNR to provide improved local estimates of forest volume. Indeed, if we consider a map as a cartographic “measurement” of the forest attributes (as seen on aerial photographs), then we can make an analogy to other measurements. We can thereby conclude that the mean of different measurements is a better estimator of the true value than each measurement individually (see literature on Information Theory, such as Shannon and Weaver (1949)). The analogy is valid when objective measurements are replaced by cognitive measurements or “beliefs”, such as a subjective photointerpretation (BonJour, 1985; Lehrer, 1986).

2. Methodology The volume estimation methodology presented herein consists principally of two steps. First, we present the alternative volume calculation method based on the local forest type (i.e., the forest type of each polygon) that is estimated solely from a single photo-interpretation. Second, we present a method to update and integrate multi-temporal forest type maps into a single Fused Map having an improved estimate of the local forest type compared to any single map and demonstrate its use with the photo-interpretation based volume estimation method. 2.1. Study site Three forest type maps at the same scale (1:10000) for the Montmorency Forest — the research and teaching forest of Laval University, Quebec City — covering an area of 6625 ha were available for this study. The forest is located 80 km north of Quebec City, Canada and is dominated by boreal forest. The predominant tree species are balsam fir (Abies balsamea (L.) Mill.), white spruce (Picea glauca (Moench) Voss), and white birch (Betula papyrifera Marsh.). The forest has an intensive wood-harvesting regime, and suffered from an epidemic of the spruce budworm (Choristoneura fumiferana) in the late 1980s. Although most of the forest has been protected from this insect through spraying programs, some areas have been severely affected. The forest type maps available for the area are based on human interpretation of black-and-white infrared aerial photographs obtained for forest inventories conducted

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in 1973, 1984 and 1992. For each forest stand polygon, the dominant tree species, mean dominant height, mean age and stand density are identified in a context of three different classification systems (see further in text). From ground-based surveys of the forest, summary information is available including wood volume and the number of trees per hectare. 2.2. Volume calculation Because the traditional method of estimating wood volume based on forest type maps and an associated forest inventory sampling is not appropriate for obtaining locally reliable estimates, we developed an alternative method that focuses more on the local standing wood volume. The method is based on a model that estimates the volume per ha of a forest stand as a function of the photo-interpreted forest type (density, height and species composition) for each polygon. (The age of the forest stand is not considered in the volume calculation method since forest volume is more dependent on structural parameters.) Though this model is relatively simple compared to existing models in commercial packages (Dick and Jordan, 1990; Davis, 1995), it is sufficient to explain the concept and to support the methodology. The model is composed of two parts: first it estimates the number of trees per hectare t [ha⫺1] in a stand given the density, height, and species composition. Second, it estimates the volume u [m3] of a typical tree (see Fig. 1); the multiplication of both quantities yields the volume per hectare v=t·u [m3 ha⫺1]. The volume over a certain stand polygon is then calculated by multiplying the volume per hectare v by the area A [ha] of the polygon: V=v·A [m3]. The first part of the model addresses the number of trees per hectare having commercial value. More specifically, it estimates the number of trees per hectare having a diameter at breast height larger than 9 cm, which is the Quebec criterion for commercially valuable trees. The number of trees per hectare t is strongly related to the density of a forest stand. The photo-inter-

Fig. 1. Forest model to calculate volume in function of photo-interpreted attributes density d, height h and species composition H. Number of trees per hectare (t) multiplied by the volume of a representative tree (u) results in a volume per hectare.

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preted forest density is the ratio of the area covered by tree crowns to the total area of the stand. Hence, the higher the density (at a given height and species composition), the more trees in the stand. However, the height of the trees/stand also is related to the density/trees per hectare relationship since height is also related to the developmental stage of the stand. Given the same density in a “small-height” and a “large-height” stand, because small-height trees have smaller crowns than large-height trees, a small-height forest stand will contain more trees than a large-height forest stand (Scott et al., 1998). Finally, the species of the trees also plays a role. For example, a typical white birch (and hardwood trees in general) has a larger crown than a typical fir (and softwood trees in general). Thus, all other factors being equal, the number of trees per hectare will be less for a hardwood stand than for a softwood stand (Magnussen, 1997). Considering these relations, we express the trees per hectare t as: t⫽t(H, d, h)⫽g(H)·k(d, h)

(1)

with H the ratio of hardwood of the photo-interpreted species composition of the forest stand, d the photointerpreted density of the forest stand (expressed as a percentage) and h the photo-interpreted height of the forest stand (expressed in meters). For the sake of simplicity, we do not consider all tree species individually, but rather we consider only the percentage of hardwood and softwood of a forest stand. Since about 80% of the trees in the Montmorency Forest are either balsam fir (70%) or white birch (10%) (no other species represents more than 3% of the trees), this simplification does not affect the accuracy of the forest volume significantly. The difference in the number of trees per hectare due to the percentage of hardwood in the species composition H is modeled as a linear function g(H). Based on actual forest inventory data from Montmorency Forest, we estimate the number of trees per hectare for a given stand relative to the number of trees per hectare in a pure softwood stand (see Fig. 2). Note that at this point a distinction is made between regeneration and nonregeneration (young and mature) stands, since (as we will see later) different species composition classes are

Fig. 2. Difference in number of trees per hectare: number of trees per hectare for different species compositions (expressed as the hardwood concentration H) normalized by the number of trees for pure softwood in Montmorency Forest.

used for regeneration and non-regeneration stands on the forest type maps of Montmorency Forest. Using linear regression, we fit the function g(H) to those data, obtaining the number of trees per hectare for a forest stand as a function of its species composition. For nonregeneration stands (n=125), we have g(H)=1–0.07 H with R2=0.20. For regeneration stands (n=21), we have g(H)=0.23–0.05 H. Thus, all other factors being equal, the number of trees per hectare is less when more hardwood trees are in the stand. Also, the number of commercially valuable trees per hectare is much lower for regeneration stands than for mature stands (a finding that is evident). Further, the number of trees per hectare estimated to be present in a given stand as a function of density and height k(d, h) must be known. For these factors, k(d, h) is an estimate of all trees (with or without commercial value) since the filtering for commercially valuable trees is already included in the function g(H). The figures in Table 1 for each combination of a density class and a height class represent estimates for pure balsam fir stands (g(0%)=1). Since this type of data is not used in the traditional method of volume estimation, such data are not readily available in compiled forest inventory data. Therefore, the number of trees as a function of height and density of Table 1 are only rough estimates based on the expert opinions of Quebec forest engineers (Bilodeau, personnel communication; Boulianne, personnel communication) and a personal compilation of forest inventory data from Montmorency Forest. The estimated tabulated data of Table 1 were approximated by least square fitting a surface k(d, h) to them, having the following function (RMSE=205 compared to an average k of 996): k(d, h)⫽0.066(d⫺7.61)(h2⫺50.43h⫹775.64).

(2)

Upon examination of the form of the function k(d, h), we can see a linear variation of k with density and a quadratic variation with height. This behavior is to be expected. In the case of density, the area of the forest stand covered by tree crowns increases proportionally with an increase in density. Since the number of trees per hectare is directly related to the tree crown area, an increase in density is also proportional to an increase in trees per hectare. In the case of height, geometric and allometric considerations predict that the surface covered by the projection of a tree crown onto the ground increases in a quadratic manner when the height of the tree increases linearly. Therefore, the number of trees per hectare decreases at the same rate, i.e., a quadratic decrease compared to a linear increasing height. It must be stressed that the number of trees per hectare in Table 1 and from the function developed (Eq. (2)) is approximate. Since the majority of the volume of a forest is included in dense (classes A to B) forest stands with

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Table 1 The estimated number of trees per hectare for pure fir (SS) as a function of the height (h) and density (d) of a forest stand: k(d, h) Pure fir (SS) Height class (average height h) Density class 1 (24.5 m) 2 (19.5 m) (average density d) A B C D

(83%) (66%) (33%) (16%)

800 550 300 60

1000 700 350 70

3 (14.5 m)

1500 1000 500 100

tall (classes 1 to 3) trees, the estimates in the upper left corner of Table 1 are particularly critical to the accuracy of the method. Since it is difficult — and beyond the scope of this paper — to evaluate (and verify on the terrain) the accuracy of the estimates, we accept the model as it is. When discussing results, however, we will focus again on this critical point. For the sake of illustration of the use of the alternative volume method, we will employ an arbitrarily defined accuracy of the model for the number of trees per hectare. Therefore, we suppose — as an informed estimate — that the mean relative error of the number of trees per hectare from the function t=g(H)·k(d, h) is dt=20%. Upon implementation of the method in an actual decision context, this accuracy would need to be verified with ground-based samples. At this point, the first part of the model is complete — i.e., an estimate is available of the number of stems per hectare given the species composition, density, and height of a given stand. As mentioned earlier, the second part of the model addresses the average volume of a tree of given dimensions (see Fig. 1). Information from the forest inventories of Montmorency Forest contains individual tree data (such as height, volume and diameter at breast height of each measured tree) from which we derive a relationship between a tree’s volume (u), its height (h) and its diameter at breast height (dbh) for a given species at Montmorency Forest. For this purpose, we use commonly accepted methods as described in general forestry literature (e.g., Be´rard and Coˆte´, 1996; Holland et al., 1990). The relationships are derived by first fitting a polynomial of the third degree to measured (u, dbh) data. The resulting equations for white birch (BOP) and balsam fir (SAB) are: u(dbh)BOP⫽⫺0.0022dbh3⫹0.65dbh2⫹0.62dbh

4 (9.5 m)

2000 1500 750 150

5 (5.5 m)

6 (2 m)

3000 2200 1100 200

4000 3000 1500 300

tories. Although a diversity of models have been proposed in the scientific literature (Huang and Titus, 1994; Be´gin and Raulier, 1995), we chose to employ the Chapman–Richards exponential model (Be´rard and Coˆte´, 1996) to derive the equations for the Montmorency Forest: h⬍15⇒dbh(h)BOP⫽⫺11.12 log(1⫺h/16) with R2

(5)

⫽0.86, n⫽113 h⬍20⇒dbh(h)SAB⫽⫺16.35 log(1⫺h/21) with R2

(6)

⫽0.33, n⫽796 For heights taller than the upper limits of Eqs. (5) and (6), we employ the maximum values dbh(15)BOP and dbh(20)SAB, respectively. This effectively provides an upper asymptote for height — something that is biologically realistic. Moreover, heights above the upper limit are very rare in the boreal forest type that covers Montmorency Forest. By substituting dbh from Eqs. (5) and (6) into Eqs. (3) and (4), we obtain tree volume as a function of height: u=u(h). Table 2 shows u (in dm3) for a tree of average height for each height class of the Montmorency Forest classification system for the two most abundant tree species (white birch and balsam fir) present in Montmorency Forest. We use this information subsequently for all hardwood or softwood species — a simplification that is justified for two reasons. First, as mentioned earlier, 80% of the trees in the Montmorency Forest are either balsam fir or white birch. Second, the forms of the remaining hardwood and softwood trees in the Montmorency Forest are very similar to that of white birch and balsam fir trees, respectively. We further note

(3)

⫺59.40 with R2⫽0.52 u(dbh)SAB⫽⫺0.0020dbh3⫹0.44dbh2⫹8.16dbh

(4)

⫺122.56 with R2⫽0.41 Besides these relationships, we also fit a relationship for each species to estimate a tree’s diameter at breast height (dbh) as a function of its height (h) — something that is also a general practice in the MNR forest inven-

Table 2 Volume per tree (dm3) for balsam fir (SAB) and white birch (BOP) as a function of its height u (dm3) Height class mid-point h (m) Tree species 24.5 19.5 14.5

9.5

SAB BOP

18 30

1129 514

946 514

210 500

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from Table 2 that the bulk of the forest volume is located in taller trees. Again, as for the number of trees per hectare, not only the estimate from the model is important, but also the accuracy of the model. Although the original, individual tree data are not available (only the compiled tree data such as stock tables are available), we suppose again for the sake of illustration that the mean relative error of the volume per tree function fitted is (by informed choice) du=20%. Finally, we can express the volume per hectare v as the sum of the volume of hardwood trees and the volume of softwood trees for a given stand, or v(H, d, h)⫽H·t(H, d, h)·uBOP(h)⫹

(7)

(1⫺H)·t(H, d, h)·uSAB(h). The total volume for a stand is then calculated as V=v·A. The error of the volume estimate of the alternative method is the mean absolute error of the calculated volume per hectare when all the independent variables (i.e., H, d, h and A) are without error, i.e., ⌬v. (Note that — in correspondence with the traditional scientific notation — the ⌬-symbol indicates an absolute error, while the d-symbol indicates a relative error, and that for a given variable x the following relation holds: dx=⌬x/x.) This error can be theoretically derived and expressed as the mean absolute error on the number of trees per hectare (⌬t) and the mean absolute error on the volume of a typical tree (⌬u) by means of the partial derivatives of v (see general literature about error theory for proof, e.g. Rabinovich, 1995): ⌬v⫽∂v/∂t·⌬t⫹∂v/∂uBOP·⌬uBOP⫹∂v/∂uSAB·⌬uSAB ⫽(H·uBOP⫹(1⫺H)·uSAB)·⌬t⫹(H·t)·⌬uBOP⫹

(8)

((1⫺H)·t)·⌬uSAB and if duBOP=duSAB=du, then the mean absolute error is ⌬v⫽v·(dt⫹du),

(9)

or, with the estimated relative errors of dt=20% and du=20%, the mean relative error on the volume estimation is dv⫽dt⫹du⫽20%⫹20%⫽40%

(10)

It must be noted that the mean relative error dv=40% is based on hypothetical accuracies for the number of trees per hectare and the volume of a typical tree and an assumption that the errors are cumulative and not compensatory. However, when summing the volume of multiple polygons in order to calculate the standing wood volume in a given region, it is assumed that the errors are compensatory and not cumulative. Therefore, the standard deviation of the mean of the volume per hectare over n polygons, each with a mean relative error

dv, is dv⬘=dv/n. Furthermore, it must be clear that, next to the errors related directly to the method of volume calculation, additional errors on the volume estimation of a given polygon can occur when the photo-interpreted variables height, density and species composition are inaccurate and uncertain. Before concluding this section, it must be clear that the ground-sample data necessary to calibrate the alternative method consists of only a fraction of the groundsample data necessary for the traditional method. Indeed, for the traditional method, ground-sample data are necessary for all forest types (or grouped types). Conversely, for the alternative method, only the volume of a typical tree must be known as a function of its height (information that can be extracted from a single traditional sample plot) in addition to the number of trees for a hardwood and a softwood stand. Moreover, once this information is known, it can be used for any future forest inventory, since the information does not change over time. Consequently, since the number of sample plots can be significantly reduced, the alternative method offers standing wood volume information for a significantly reduced cost. 2.3. Fused map The alternative method for volume calculation described in the previous section leans heavily on the photo-interpreted values of density, height, species composition and area of each forest stand. Since the reliability of these photo-interpreted variables is generally low in a single forest type map (for various reasons explained in the introduction), the local volume estimation with the alternative method would tend to be unreliable too. However, if the photo-interpreted variables can be estimated with an improved accuracy and precision, the alternative method for volume calculation may yield more reliable local volume estimates than the traditional method. This section describes a method to obtain a better local estimate of density, height and species composition through the integration of thematic information from different map realizations into a single Fused Map. The integration of thematic information of multiple map realizations would seem to be a straightforward operation. Indeed, the basic principle is the simple comparison of equivalent (forest) attributes (species composition, height, density and age) on different maps for every polygon resulting from the overlay of the different map realizations. If the attributes were continuous values, for each overlay polygon this comparison could take the form of the calculation of an expected value such as the arithmetic or adapted mean. However, the attributes are categorical (rather than point values) implying that the class is known only within a certain range of values and not the actual point value of the

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attribute. Moreover, for the multi-temporal maps available for Montmorency Forest the categories/classes for a given attribute are not necessarily defined in the same way in different classification systems, and thereby not fit for direct comparison. Furthermore, the calculation of a mean attribute value is not always a solution. In order to account for the evolution of attributes between the epochs of multi-temporal maps, an explicit comparison of classes is necessary. Indeed, the problem of attribute evolution can be formulated as the necessity of finding the most likely class of the “old” classification system at the time of the “new” classification system given the class of the “old” classification system at the time of the “old” system and the class of the “new” classification system at the time of the “new” system. When class boundaries are defined differently in different classification systems, this is no easy task. Simply put, it is necessary to be able to compare classes from different classification systems in a meaningful way. The solution herein for the comparison of classes of different classification systems is the standardization of all classification systems to a single common normalized classification system. This standardization will be applied to the height attribute and — to a lesser extent — the species composition attribute since we include these two attributes in a growth model to account for evolution among the epochs of the Montmorency Forest maps (from 1973, 1984 and 1992). 2.3.1. Adapted means for attributes So as to avoid comparing entities that are defined in a different way, we must examine the different classification systems used from 1973 to 1992 for Montmorency Forest in relation to the underlying data. For Quebec forest type maps, forest type is defined using four characteristics or attributes: height, density, age and tree species composition. Height, density and species composition are defined in the same manner (although some individual class limits are defined differently) in all three systems — i.e., 1973, 1984, and 1992. Conversely, in the 1973 classification system, the age attribute is expressed qualitatively as a developmental stage (i.e., regeneration, young or mature stand), while in more recent systems tree age is described quantitatively (by the age of the stand since the last disturbance or clear-cut). Since the developmental stage is not related only to the age of trees, but also to the species composition, height, density and other environmental factors, “age” from 1973 and “age” from 1984 and 1992 are not defined in the same manner (Table 3). Consequently, we ignore the “age” attribute for the creation of the Fused Map. The omission of the age attribute for the fusion of maps does not cause a problem for the methodology, since the age of trees is not used explicitly in our model for volume calculation (described earlier). For the other three attributes (species composition,

379

height and density), for each polygon we calculate an adaptive mean (a context-dependent average) of all the interpretations for each overlay polygon. For density (an interval-ratio quantity), for each polygon we calculate the best estimate as an average of the class midpoints of all interpreted classes (Table 3). For height (an intervalratio quantity), we do the same thing. However, for the height attribute, the midpoints are not well defined for all classes. In particular, height class 1 represents height from 22 m to infinity. Since 22 m is already close to the maximum tree height in boreal forests, for this class we define an upper limit of 27 m to obtain a mid-point of 24.5 m. Using this adaptation, the adapted mean for height and density attributes can then be calculated. Averaging classifications for the species composition attribute is more complex. The classification systems for all three maps are based on the respective concentrations of each tree species in a forest stand. The concentration of a given tree species is calculated as the number of trees of that species over the total number of trees in the forest stand. Each tree species is represented by a letter code (e.g., “Bb”=white birch). A tree species is termed “dominant” when it consists of more than 50% of the trees in a forest stand and co-dominant when it consists of more than 25%, but less than 50% of the trees in a forest stand. The classification of species composition is different for regeneration and non-regeneration forest stands. For regeneration forest stands, the tree species mix cannot be determined adequately by photo-interpretation. Therefore, species composition classes for regeneration stands are based on the concentration of hardwood and softwood species classes only instead of the concentration of individual tree species; the classes are “F” (pure hardwood), “M” (mixed) and “R” (pure softwood) and are identical for the three classification systems (1973, 1984 and 1992). For non-regeneration forest stands, a stand is classified into a species composition class based on the dominant and co-dominant tree species, and their respective letter-codes form the species composition class name. Although the classification systems of 1973, 1984 and 1992 for species composition for non-regeneration (or mature) stands are based on the same principle (dominant and co-dominant tree species), they differ in the classification of individual tree species. The Montmorency Forest is largely dominated by three tree species each with their respective well-defined class (balsam fir “S”, white spruce “E” and white birch “Bb”). Depending on the year, secondary tree species with a lesser presence (such as yellow birch “Bj”, red pine “Pr”, poplar “Pe”, etc.) are sometimes placed in individual classes, sometimes grouped under a unified label (shade intolerant hardwood “Fi”, shade tolerant hardwood “Ft”, etc.), or else joined to a more abundant tree species. Nevertheless, when these secondary species are joined to a larger class, small changes in objectives of the classification

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Table 3 Matching of classification systems of the Montmorency Forest 1973 Height classes 1 2 3 4 – – – Density A B C D Age r, y, m.

1984

1992

Normalized

Class Mid-point

Explanation of symbols

1 2 3 4 5 6 –

1 2 3 4 5 6 Origin

1 2 3 4 5 – –

24.5 19.5 14.5 9.5 5.5 2 0

⬎22 m ⬍22 m ⬍17 m ⬍12 m ⬍7 m ⬍4 m ⬍1.5 m

A B C D

A B C D

A B C D

87.5 62.5 37.5 12.5

⬎75% ⬎50% ⬎25% ⬍25%

10, 30, 50, 70, 90

15, 30, 45, 60, 75, 90

–

–

Non-comparable (maturity classes: r=regeneration, y=young, m=mature)

system cause them to be joined with different classes for different classification systems. For example, the code “Bj” is used for yellow birch and “Bb” for white birch in the 1992 classification system. Conversely, in the 1973 classification system “Bb” is employed as “all birch species, including white and yellow birch”. Furthermore, in the 1984 classification system, yellow birch belongs to the general shade intolerant hardwood “Fi” class. Thus because there is overlap between the species composition classes among different classification systems, the classification systems cannot be compared directly. In order to compare the classification systems, we must step back from the classes themselves and consider what is of interest to us, i.e., the actual concentrations of tree species. Since the less-frequently appearing tree species are of limited interest to us, we do not consider those classes explicitly. Therefore, we define herein a Hardwood species class and a Softwood species class. The Hardwood species class (“Hw”) contains white birch, yellow birch, poplar and all other hardwood trees that can be present in a boreal forest. Similarly, the boreal Softwood species class (“Sw”) contains balsam fir, white spruce, black spruce and all other softwood trees. Since the two dominant tree species in Montmorency Forest are balsam fir and white birch, we attribute their characteristics to the softwood and hardwood species class, respectively. Consequently, we consider only two tree species (balsam fir and white birch) and suppose that all the forest stands are a mixture of these two tree species. Dominance and co-dominance of the two species classes determine the possible species composition classes for non-regeneration forest stands. Added to the four possible non-regeneration species composition classes are the three species composition classes for regeneration forest stands (F, M and R). Since the species composition attribute can now be expressed as the

tree tree tree tree

crown crown crown crown

coverage coverage coverage coverage

proportion of hardwood of the total trees, it is also a ratio-interval attribute. The hardwood proportion midpoints of the classes are listed in Table 4. Finally, we obtain a best estimate of the species attribute as the average of the respective species concentrations of all interpreted species composition classes. 2.3.2. Temporal changes: height and species composition To this point, the creation of the Fused Map has consisted of multiple map realizations of the same epoch. However, the method can also be adapted to multi-temporal map realizations (i.e., maps of the same territory but created on different dates). This is the case, for example, for the three Montmorency Forest maps utilized in this study. Since attributes change over time, temporal changes must be considered for the integration of maps. Clear-cuts, pathological epidemics, forest growth and other natural or man-made changes in the forest affect the integration of multi-temporal maps. In general, for undisturbed areas, the natural evolution must be compensated for from one epoch to another, while for disturbed areas the pre-disturbance information must be ignored. Using information on disturbances from the database, consistency-retaining rules from growth models and an associated adaptation of the best attribute estimate, a Fused Map can be created using multi-temporal map realizations rather than multiple map realizations. We discuss this now in detail. First, disturbed areas are generally documented in the map’s associated database. Man-made disturbances such as clear-cuts are recorded in the database of the Montmorency Forest type maps using the year of the clear-cut or the year of the “origin” of the forest stand or polygon. For these polygons that have no match in the older layers, the information from the older maps is unrelated

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381

Table 4 Species composition classification systems and the class midpoints (expressed as the percentage of hardwood in the class) Normal

1973

1984

1992

H = Hw/(Hw + Sw)

F M R HwHw HwSw SwHw SwSw

F M R Bb, Fi FiR(F), BbR(F) FiR(R), BbR(R) S, E

F M R Fi BbS(F), FiS(F) BbS(R), FiS(R) S(S), S(E), E(S), E(E)

F M R

83.3% 50.0% 16.6% 87.5% 62.5% 37.5% 12.5%

to the present map and must not be considered in the comparison. Rather, such areas are labeled as “new” or “clear-cut” on the subsequent map. The forest type estimate from the Fused Maps must be based on the information of the maps dated after the clear-cut. Second, for undisturbed areas, the explicit comparison of classes from maps of a different epoch is necessary. The comparison must be done using expert knowledge (such as growth models) on the growth of different tree species in the studied region. Map attribute growth models can be formulated as consistency-retaining transition rules between classes. In this way, we can evaluate if an interpretation from one epoch is consistent with an interpretation from another epoch given an existing growth model. For the calculation of the adapted mean of an attribute, we replace an “old” attribute with the value of the “new” attribute if and only if both attributes are consistent with the growth model. For the Montmorency Forest, we developed and applied consistencyretaining rules (described subsequently) for height and species composition. The natural evolution of the density of a forest stand is too complex to be included in our growth model and can be ignored for our purposes. While our growth model is very simplistic in comparison to those implemented in commercially available forest management software (such as Davis, 1995), it is sufficient nonetheless to demonstrate the volume estimation methodology. In order to compare classes from different classification systems explicitly, the classification systems must be normalized. The classification systems for the height attribute are similar for classes 1 through 4 (see Table 3). However, for heights less than 7 m (trees without commercial value), the three systems differ due to changes in objectives among epochs. Therefore, all classes under 7 m must be merged into a single class in the normalized classification system. Using the normalized classification system for the height attribute, we can apply growth models from expert knowledge in order to create a Fused Map at the epoch of the most recent map realization. For the height attribute, there is an average annual tree-height growth of 0.5 m in Montmorency Forest. This means that tree height can increase up to 5 m over a 10-year period,

FiS, BjS, PeS, BbS SFi, SBj, SPe, SBb SS, SE, ES, EE

which is equivalent to a change of 0 or 1 height class. Since the maps for Montmorency Forest were made in 1973, 1984 and 1992, there are about 20 years between the first and last maps. Thus, when H is a height class number for classes ordered in increasing height, the following transitions retain consistency: 앫 앫 앫 앫

H→H+1, between two maps, and H→H→H+1, H→H+1→H+1 and H→H+1→H+2, between the three maps.

Conversely, when the goal is a Fused Map for the 1973 epoch, newer maps can be “down-dated” or evolved back to that epoch by inverting the consistency-retaining transformations. The species composition attribute presents us with a different problem. The normalization of the three systems is solved by considering only the hardwood and softwood proportions of forest stands, such as explained before. However, evolution among species composition classes occurs as transitions from regeneration forest stands to non-regeneration (or mature) forest stands. These transitions can retain or violate consistency, depending on the ratio of hardwood versus softwood in each stand. However, since the regeneration species composition classes are overlapping with the non-regeneration species composition classes, multiple consistency-retaining transitions are possible. The following transitions retain consistency: 앫 F (hardwood)→HwHw, HwSw; 앫 M (mixed)→HwSw, SwHw; 앫 R (softwood)→SwHw, SwSw. Again, these results can be reversed in order to “downdate” a map realization to an epoch of interest.

3. Results and discussion The proposed method for obtaining local and regional wood volume estimates has been implemented for the three forest type maps of the Montmorency Forest. To

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Fig. 3. Total standing wood volume of Montmorency Forest for the three epochs with the traditional and the alternative method.

evaluate regional accuracy (for volume summarized over a large area), we compared the volume of the total area of the Montmorency Forest as estimated by the proposed method to similar summarized volumes from the traditional method. Conversely, in order to evaluate local accuracy of the method, we compared local volume estimates to the measured volume of ground-based sample plots. First, we examine the volume estimates for the entire Montmorency Forest for the three individual map realizations. Fig. 3 and Table 5 show the total standing wood volume of the Montmorency Forest for the three individual maps for both the traditional and the alternative methods. We can see that the alternative method provides volume estimates that are very similar to those obtained from the traditional method. The root mean square error (RMSE) among volume estimates from the traditional method and the alternative method for the three epochs is 33420 m3, or about 5.9% of the total volume. Since this mean error falls within the accuracy of volume estimates of the traditional method of about 10% as reported in the MNR forest inventory (Be´langer

et al., 1986), the alternative method is consistent with the traditional method on a regional scale. However, the alternative method was not developed to be used with individual forest type maps but with the integration of multiple map realizations, i.e., the Fused Map. Because of disturbances and temporal evolution of the forest among the three different epochs of the Montmorency Forest maps, the same areas are not covered by forest on the three maps. Clear-cuts and disturbances needlessly complicate the Fused Map creation since, in those areas, it is not based on all three interpretations. Therefore, for simplicity’s sake, we exclude them from the volume calculation. Rather, for each of the three maps we consider only the area of Montmorency Forest that is covered by forest and undisturbed since the first map realization of 1973. In order to compare the three maps in those areas, growth models developed earlier have been applied to the older maps (1973 and 1984) in order to create a Fused Map for the 1992 epoch. For this 4145 ha forested and undisturbed part (67%) of the Montmorency Forest, for each overlay polygon, three attribute estimates are available. These attribute estimates are averaged in the Fused Map as explained previously in the methodology. The volumes calculated for this part of the forest are shown in Fig. 4 and in Table 5. First we observe that the volumes based on the individual maps are less than volumes for equivalent quantities in Fig. 3. This is expected since we do not consider the entire forest in the present procedure. The 1992 volume is 18% less than for the complete map. Because no traditional volume estimates were available for the undisturbed forest only, we approximated it by taking 82% of the total traditional 1992 volume (i.e., a similar reduction as for the alternative method). This volume will be used as a reference subsequently. We can further observe that the volume based on the Fused Map is relatively centered in the range of volumes from individual maps. This is expected, since the map is based on

Table 5 Wood volumes from the traditional and the alternative method for the total Montmorency Forest

1973 (m3) 1984 (m3) 1992 (m3) Fused Map (m3) a

Traditional method

Alternative method

Alternative method for maps evolved to the 1992-epocha

543701 561788 594765 –

505217 603917 585025 –

438354 503765 490702 454061

This information is strictly based on undisturbed forest.

Fig. 4. Alternative method applied on the individual maps evolved to the 1992 epoch and on the Fused Map at the 1992 epoch. Note that these are based on a subset of stands used in Fig. 3.

T. De Groeve, K. Lowell / Environmental Modelling & Software 15 (2000) 373–385

adapted means of all the attributes. Nevertheless, the volume based on the Fused Map is not the average volume of the three maps (Table 5) since the volume is a nonlinear function of the photo-interpreted parameters height, density and species composition. Therefore, the volume calculated based on the Fused Map must be considered as a better estimate of the volume than just the mean volume of the three maps and the creation of the Fused Map is hereby not only justified but necessary. Considering the accuracy of the volume estimate based on the Fused Map, the RMSE between individual map realizations and the Fused Map as truth or reference is 8%. We consider this RMSE as the mean error (and hence an estimate of the accuracy) of the Fused Map estimate. This is slightly smaller than the mean error for the traditional method of 10%. Since both the traditional volume estimate (i.e., 82% of the 1992 estimate) and the alternative volume estimate based on the Fused Map fall in each other’s accuracy intervals, both measures are consistent. Therefore, we can conclude that the alternative method performs at least as well as the traditional method for volume calculation on a regional scale. One must note, however, that thematic and spatial map uncertainty both are considered for the confidence interval for the alternative method, while they are not for the traditional method. Moreover, as seen before, the groundsample data set necessary to calibrate the alternative method is smaller than the one necessary for the traditional method. Consequently, since the number of sample plots can be significantly reduced, the alternative method offers standing wood volume information with comparable accuracy as the traditional method for a highly reduced cost. The second part of the evaluation of the alternative method of volume estimation is the comparison of the local volume estimation using ground-based sample plots. The wood volume compiled from 55 0.04 ha sample plots measured in 1984 is compared to estimates from the traditional method and from our alternative method. The 55 sample plots have different photo-interpreted forest types. The wood volume compiled from the 1984 sample plots is comparable to the estimates from the alternative method using maps of different epochs since the 55 sample plots are chosen in undisturbed forest stands and since the individual map attributes have been updated (for the 1973 map) or “down-dated” (for the 1992 map) to the 1984 epoch. We applied the alternative method to both the Fused Map and the individual maps separately. In Fig. 5 and Table 6, we can observe that the alternative method performs better than the traditional method for the estimation of local wood volume. Note that all points in Fig. 5 should fall on the diagonal line in the figures, since the true volume and the estimated volume should be the same for each polygon. When this is the case, the correlation between the true volume and the estimated volume

383

would be unity if the correlation is calculated as r=Cov(x, y)/(sxsy), with covariance Cov(x, y)=⌺(xi ⫺mx)(yi⫺my)/n. The volume obtained from the traditional method shows not much correlation (r=0.22) with the measured volume on the sample plots (Fig. 5(a)). Conversely, the alternative method shows a very significant relation to the measured volume and has high determination coefficients in general (r⬎0.4) with a clear superiority of the Fused Map (Fig. 5(b)) (r=0.59) compared to the individual maps (Fig. 5(c), (d) and (e)). This means that the alternative method estimates the local volume better than the traditional method. This finding is related to the use of grouped types in the traditional method. Indeed, the use of grouped types averages the volume information to such a degree so as to eliminate any significant relation between forest type and volume estimation. The alternative method developed in this article, however, does not group forest types and thereby retains a relationship among local forest type parameters and the local volume estimates. Considering absolute errors of volume estimation (compared to the truth from the sample plots), the Fused Map provides an RMSE of 57.6 m3/ha for no significant reduction over the traditional method with an RMSE of 58.2 m3/ha. Compared to the mean volume per hectare of the sample plots (113 m3/ha), the mean errors are relatively large (51%). The mean error of 51% is composed of the methodological error (earlier hypothesized as dv=40%) and the error due to erroneous photo-interpretation of the local forest type. While the methodological error component can be decreased by a better analysis of field data (abandoning the simplicity adopted herein), the photo-interpretation error can be decreased by adding more interpretations to the Fused Map. Indeed, by using the Fused Map based on three map realizations instead of individual map realizations, the mean error has been reduced by 20%. This is strong evidence for the thesis that additional map realizations improve the local standing wood estimation. This not only corresponds to the main purpose for the creation of the Fused Map but also implies that the mean error can be reduced even more by integrating a fourth or a fifth map realization into the Fused Map (without having a larger ground sample set). As a final point in our discussion, we return to the disadvantages of the traditional method of volume estimation, and evaluate them relative to the alternative method. First, the alternative method estimates the mean of the forest type for an individual forest stand polygon rather than using the mean of a grouped forest type (see Fig. 5). Second, the estimated variance of the wood volume for the alternative method consists of the true sampling variance of individual polygons and not the sample variance within the grouped type. Third, thematic uncertainty is included in the variance estimation of the volume estimate from the alternative method by considering the mean and the variance of map attributes for three

384

Fig. 5.

T. De Groeve, K. Lowell / Environmental Modelling & Software 15 (2000) 373–385

Comparison of measured volume of 55 sample plots and the volume calculated using the traditional method and the alternative method.

interpretations. Finally, the positional uncertainty of polygon boundaries is implicitly accounted for in the alternative method. Indeed, on different interpretations, uncertain boundaries will be slightly displaced and will create sliver polygons (long, thin and small polygons) upon being overlaid. The forest type of these sliver polygons will be an average of the forest types from the neighboring “real” polygons and, hence, will represent a transition zone in an intuitive manner and therefore

account for spatial boundary uncertainty. Consequently, it is apparent that the alternative volume estimation method that we propose offers a solution for all of these weaknesses of the traditional method. 4. Conclusions The alternative method for estimating standing wood volume based on the integration of multiple maps into

T. De Groeve, K. Lowell / Environmental Modelling & Software 15 (2000) 373–385

Table 6 Comparison of local volume estimates with measured volumes in 55 sample plots Correlation ra

RMSEb

RMSE/mx

0.22

58.15

51%

0.44 0.44 0.41 0.59

79.56 74.74 78.55 57.58

70% 66% 70% 51%

385

thank also Dr Geoffrey Edwards for fruitful discussions improving the quality of this work.

References Traditional method 1973 1984 1992 Fused Map

a When the true (measured) volume is x and the estimated volume is y, then Correlation r=Cov(x, y)/(sxsy), with covariance Cov(x, y)=⌺(xi⫺mx)(yi⫺my)/n. b When the true (measured) volume is x and the estimated volume is y, the Root Mean Square Error=√⌺(xi⫺yi)2/n.

a single Fused Map performs better in many ways than the traditional method used by the MNR. It offers not only comparable results for volume data summarized over large areas with a well estimated unbiased variance on the volume, but it offers also relatively accurate unbiased estimates of the local standing wood volume. Moreover, the technique permits the integration of information from older forest inventories in the Fused Map, thereby improving the accuracy of the information. Since the Fused Map is constructed completely automatically using only the most basic database management features, the alternative volume estimation method can be a valuable alternative for traditional forest type maps without increasing costs — provided that more than one map is available. Moreover, since only a reduced set of the ground-based sample plots are necessary in order to obtain a similar accuracy, the cost of the alternative method can be lower in the end. Furthermore, since an increased knowledge of the reliability and accuracy of the local information is available, it can broaden the possible applications of photo-interpreted forest type maps substantially.

Acknowledgements This work is part of a doctoral thesis prepared at Laval University, which is funded partially by the Association des Industries Forestiers de Que´bec (AIFQ), the Ministry of Natural Resources (MRN), Rexfor, the Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche (FCAR) and the Natural Sciences and Engineering Research Council (NSERC). The authors would like to

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