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Instruments used to estimate Pgap commonly do not distinguish between wood and leaf elements, therefore the metric estimated is PAI not LAI. α enables conversion from PAI to LAI as LAI = PAI.α. α values have been found to account for a significant proportion of total plant area for different species and forest types. Therefore, its accurate retrieval is a critical step in the estimation of LAI from indirect LAI estimation methods applying the physical model (Table 1). Several authors have employed direct or indirect methods to estimate α, which are summarised below.

5.3.1.1 DIRECT RETRIEVAL METHODS

Similar to direct sampling of LAI, direct estimates of woody area have traditionally been regarded as the most accurate due to the potential to completely sample all plant material. Importantly, both total leaf area and total woody area metrics are required for the derivation of α, because it is the proportion of wood-to-total plant area. Complete woody area sampling was conducted by Hagihara & Yamaji (1993), who estimated the woody silhouette area, or longitudinal section area, of the branches and stems by multiplying the diameters of woody material by the length. Such manually intensive methods are usually only applicable to a sub-sample of trees in a plot (Hagihara & Yamaji, 1993). Approximations of total woody area are made to avoid destructively harvesting trees. For example, Deblonde et al. (1994) and Gower et al. (1997) calculated stem area for each tree by assuming that the stem from DBH to the base of live crowns was a paraboloid, and from DBH to the soil surface was a cylinder. The area of the main stem was used as the metric of total woody area, assuming the contribution from branches was negligible. However, Hagihara & Yamaji (1993) found 80% of woody area was from branches for

Chamaecyparis obtuse (α = 0.11), thus indicating that depending on tree species and tree form, the

implication of this assumption on total woody area accuracy will vary.

5.3.1.2 INDIRECT RETRIEVAL METHODS

Indirect estimates of α have the advantage of being comparatively efficient and non-destructive. A number of indirect methods have been employed which mainly vary in instrumentation used and applicability to deciduous or evergreen species to name a few. Sea et al. (2011) estimated α as a simple proportion of woody cover to total tree cover from classified HP images. α was determined as the slope of the linear fit between wood cover fraction and total tree cover fraction for all classified images. This method assumes that woody material and leaf material are distributed approximately evenly throughout the canopy. The classified material proportions from the images were stable in a savannah

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environment spanning a 900 km transect in Northern Australia, thus indicating the applicability of a single α correction factor across an ecological region. TLS offers another means to separate wood and leaf interceptions, similar to classified HPs, via intensity thresholding of a single wavelength (Béland et al., 2014; Malenovský et al., 2008) or specially designed dual wavelength scanners (Danson et al., 2014; Douglas et al., 2012). Analogous to dual-wavelength TLS, the multiband vegetation imager (MVI) instrument was developed to separate non-photosynthetic from photosynthetic material and sunlit from shaded material (Kucharik et al., 1998; Kucharik et al., 1997).

A commonly employed method in deciduous forests is to estimate PAI in leaf-on conditions, and then repeat the measurements in leaf-off conditions to estimate the Woody Area Index ‘WAI’; where α = WAI:PAI (Chen et al., 1997a; Leblanc & Chen, 2001). Leblanc & Fournier (2014) found that this technique provided reasonable estimates of α in a 3D modelling framework, due to errors cancelling out from estimating PAI in leaf-on conditions and WAI in leaf-off conditions. An alternative to this method is to estimate PAI, and then mask out woody material for applicable methods to estimate an approximation of LAI, and then to apply Eqn. 26 (Liu et al., 2012). However, the spatial distribution of wood with respect to leaf material, such as mutual shading or occlusion, needs to be carefully considered. The importance of mutual occlusion is potentially enhanced if instruments or methods are only operating at a narrow view zenith angle or low angles, such as at nadir when stems are not visible, yet are known to contribute significantly to PAI. However, theoretically the WAI:PAI technique (Eqn. 26), albeit from direct or indirect methods, is the soundest methodology for estimating α, assuming both metrics are accurately estimated. Indirect techniques would benefit from benchmarking against precisely known or ‘true’ values to evaluate method accuracy and better elucidate potential limitations.

5.3.1.3 REFERENCE WOOD-TO-TOTAL PLANT AREA BASED ON 3D MODELING

Another potential α estimation method is through 3D reconstruction of trees (Section 3.5), and subsequent querying of total woody facet area. A challenge of this method is to have accurately determined the total leaf area applicable to the branching structure of the reconstructed tree in order to determine α (Eqn. 26).

In 3D tree model methodologies, the reference α value for each scene can be calculated precisely from querying every tree model’s leaf and wood facet area:

_{∑( ) ∑( )}∑( ) [26]

As Eqn. 26 provides an exact quantification of α, it can be considered a reference value. A key distinguishing feature of the ‘reference’ α is that it is the precise value of the 3D model or scene, as opposed to any in-situ retrieval method which are only estimates or approximations of the ‘true’ value that can never be known in the field environment.

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5.3.2 C

Canopy element clumping retrieval methodologies are used to determine the degree of non- randomness for input into the physical model formulation (Eqn. 15). A clumping factor of 1 is associated with a theoretically random spatial distribution of canopy elements, and therefore provides an estimate of effective PAI or LAI (Chen & Cihlar, 1995a). However, in forests a clumping factor of 1 is typically invalid due to the aggregation of canopy elements into clumps at all scales. Clumping factors less than 1 mean that there is some degree of spatial aggregation of canopy elements occurring, where more gaps are found than in a randomly distributed pattern. Conversely, clumping factors greater than one are associated with regularly distributed canopy elements, meaning less gaps are found than for a random distribution of canopy elements. Canopy element clumping retrieval methodologies are an area of ongoing research (Gonsamo et al., 2011; Leblanc, 2014; Leblanc & Fournier, 2014; Zhao et al., 2012). The following sections present a categorisation and description of clumping retrieval methods, typically applied from measurements of gap size and gap size distribution. As such, the clumping factor is a function of view zenith angle and can be retrieved over narrow angle ranges from instrument measurements such as Hemispherical Photography images (Leblanc & Chen, 2001; Leblanc & Fournier, 2014; Pisek et al., 2011).

5.3.2.1 LOGARITHMIC AVERAGING (ΩLX)

Lang and Xiang (1986) proposed a method to retrieve clumping based on logarithmic averaging, using finite segments ‘k’ of gap size distribution measurements, hereafter referred to as LX:

( ) ( ̅) ⁄ ̅̅̅̅̅̅̅( ) ( ̅) ∑⁄ ( ) [27] where ( ̅) is the logarithm of averaged gap fraction over a predefined area (i.e. a proportion of a single measurement of gap sizes or multiple measurements), ̅̅̅̅̅̅̅( ) is the average logarithm of gap fraction over the same area. ( ) is the gap fraction of segment k relating to a sub-domain of the gap size measurement. The size of k should preferably be at least 10 times the mean canopy element width, typically corresponding to leaf size. Two assumptions are made: (i) the canopy elements at the k scale are distributed randomly, and (ii) segments contain gaps, due to the undefined logarithm of Pgapk = 0. Gonsamo et al. (2010) proposed a clumping retrieval method based on the LX principle,

that utilises the minimum segment size for which a gap is present, as opposed to a fixed segment size, which could lead to null gap segments. The implications of null gap segments and segment size are discussed later in Section 5.5.2.4.

LX with a segment size ‘k’ ≈ 270-360° (azimuth view cap specific) is the clumping method typically applied to the LAI-2000/2200 instrument Pgap estimates (LI-COR, 2011; Section 2.2.3). However, multiple measurements at the same location while rotating the view cap can be used to restrict ‘k’ to mimic HP azimuthal segment capabilities from a single measurement (Chianucci et al., 2014).

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5.3.2.2 GAP SIZE DISTRIBUTION (ΩCC)

The corrected ‘CC’ Chen and Cihlar (1995b) method by Leblanc (2002) enables clumping retrieval through gap size distribution information, given as:

( ) [ ( ) ( ) ] [

( )

( )] [28]

where Pgap() is the overall gap probability, Pgapr() is the gap probability after removing large ‘non- random’ gaps. Pgapr() is obtained by the sequential removal of large non-random gaps that are not statistically possible for a given Pgap, until the pattern of gap size distribution resembles that of an equivalent canopy with a random spatial distribution of canopy elements. This is a key assumption of the method. CC is the clumping method typically employed by the TRAC instrument (Leblanc, 2002; Leblanc et al., 2005).

Variations of the CC method include: an approximation of the CC method by Walter et al. (2003), termed ‘CCW’; and a modified CC method termed ‘CMN’ by Pisek et al. (2011), based on the original equation by Miller & Norman (1971), which does not consider a normalization factor after the removal of large gaps. However, Leblanc & Fournier (2014) demonstrated the CMN method produced similar results to the CC method, and also found it to be less reliable at low PAI values (<≈2-3 PAI) in the virtual scenes tested.

5.3.2.3 COMBINED LOGARITHMIC AVERAGING AND GAP SIZE DISTRIBUTION (ΩCLX)

Leblanc et al. (2005) combined the logarithmic averaging and gap size distribution methods, termed ‘CLX’, to address the potentially limiting assumption of a random distribution of canopy elements at the segment scale associated with the LX method. The overall clumping index is then calculated over n segments:

( ) [ ( ̅)]

∑ ( ) ⁄ ( ) [29]

where ΩCCk () is the CC method (Eqn. 28) applied to the segment scale ‘k’.

A schematic diagram depicting the LX, CC, CLX method principles of operation are shown in Figure 33. It demonstrates how each method is applied to a number of fixed segments ‘k’ of a hemispherical photo gap size distribution transect (the first row). The exemplar transect consists of a number of sections of varying density, with three large gaps ‘LG’ representative of between-crown gaps. The logarithmic mean ‘LX’ of Pgap of a measured transect is performed on the fixed segment length ‘k’. Conversely, the CC method does not operate on segments. Instead, it attempts to remove large non-random gaps ‘LG’ from the entire transect to estimate Pgapr(). It then determines Pgap() from the combined segment length of Pgapr()and the segment length corresponding to the sum of large gaps ‘LG’. The CLX method applies the logarithm of Pgap to segments after the removal of large non-random gaps following the CC method.

100 FIGURE 33: CLUMPING RETRIEVAL METHOD SCHEMATIC DIAGRAM

Figure 33. Schematic diagram modified from (Leblanc, 2014) depicting the principles of the LX, CC and

CLX methods.

5.3.2.4 OTHER CLUMPING RETRIEVAL METHODS

A number of other clumping retrieval methods have been proposed, which vary predominantly in measurement or parameterisation method and frequency of implementation in the literature. The previously outlined clumping methods are more prevalent in the literature (Gonsamo & Pellikka, 2009; Gonsamo et al., 2011; Pisek et al., 2011).

The Pielou coefficient of segregation (1962) was applied to HP imagery to explore spatial dispersion of canopy elements based on Pgap and gap size distribution (Gonsamo et al., 2011; Pisek et al., 2011; Walter et al., 2003). Pisek et al. (2011) suggested others methods such as CLX and CMN in preference to this method due to abnormal clumping retrieval results and a poor degree of matching with five other clumping retrieval methods tested for their study sites. Jonckheere et al. (2006) utilised a fractal- dimension based modelling approach to improve clumping factors of theoretical foliage distribution models. Frazer et al. (2005) developed a technique to determine fine-scale forest heterogeneity, based on lacunarity analysis. However, the authors concluded further research is required to understand how statistical estimates and gradients of measured heterogeneity relate to other ecological metrics such as forest structure. Clumping factors have also been retrieved from empirical equations (Kucharik et al., 1999) and theoretical models (Nilson, 1999).

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Methods incorporating ranging information from instruments such as TLS have been more recently developed to aid in clumping retrieval or to minimise the impact of clumping on indirect PAI estimation. Jupp et al. (2009) developed a PAI inversion method from TLS ranging data that subsumes some clumping effects via logarithmic averaging. They suggested multiple areas for improvement including separating between-crown gaps from within-crown gaps. Zhao et al. (2012) developed a nominal spatial extent index (NSEI) to characterise within-crown and between-crown gaps and then applied the results to the CC method (Eqn. 28).

5.3.2.5 CLUMPING BASED ON 3D MODELING

Canopy element clumping can be calculated as the ratio of effective PAI ‘PAIe’ to total PAI. In real forested environments the total PAI is not known a priori. However, using virtual 3D forested environments, the total element area of both leaf and woody facets are known precisely. Therefore, the ratio of PAIe to PAI can be estimated in the virtual forest environment using the known virtual forest parameters and simulated measurements to derive Pgap. The PAIe/PAI ratio can therefore be treated as the ‘reference’ clumping factor (Leblanc & Fournier, 2014). Reference clumping values can be used to benchmark and directly assess the accuracy of the clumping retrieval methods tested in the same virtual scenes using the same measurements of Pgap, following Leblanc & Fournier with a substitution of GT for

GL (2014):

( ) ( ) ( ̅) ( ) ⁄ [30]

In order to calculate the angular reference clumping value ( ) , the parameters of mean Pgap ( ̅)’ derived from simulated scene measurements and the projection coefficient of all canopy elements ‘GT()‘ must first be derived. Using ‘GT()’ instead of the leaf-only projection coefficient ‘GL()’ avoids errors associated with the incorrect G() application due to the presence of woody elements as

outlined in Chapter 4.

All clumping retrieval methods, including the reference clumping, do not distinguish between foliage and woody components, thus the clumping metric derived is referred to as total element clumping ΩT() (Eqn. 15, Eqn. 17).

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5.4

M

M

5.4.1 V

The 24 virtual scenes comprising four PAI levels and six stem distributions described in Section 3.8 were