View Angle-Dependent Clumping Indices for Indirect LAI Estimation

Abstract

The leaf area index (LAI) of forest canopies can be rapidly estimated by the canopy gap fraction measured using in situ optical instruments. Gap fraction analysis, however, will routinely underestimate LAI when forest canopies exhibit a strong pattern of aggregation at one or more spatial scales. Recent advances in optical-based measurement techniques and gap fraction analyses show that inclusion of a view angle-dependent clumping index Ω(θ) may substantially improve optical-based estimates of LAI. Here we introduce four different estimates of Ω(θ) and demonstrate how these indices can be extracted directly from hemispherical canopy photographs. Our findings suggest that estimates of Ω(θ) can improve LAI estimation in the presence of spatial non-randomness, although their overall effectiveness was strongly dependent on the clumping index chosen, quantity of leaf area, severity and complexity of foliage clumping, and the view angles used for LAI integration.

Appendix: Lacunarity as a Scale-Dependent Measure of Spatial Non-randomness in Forest Canopies

6.1.1 Gliding-Box Algorithm

The gliding-box algorithm requires five main steps to compute the lacunarity of a binary set composed of 1s (occupied/foliage) and 0s (unoccupied/gap) (Allain and Cloitre 1991; Plotnick et al. 1996). First, a ‘gliding’ or moving window of box size r (in grid cell units) is shifted one position i at a time along the entire length n of the set. At each grid cell position i along the set, the sum or box mass s of grid cell values x _j contained within the box of size r is computed until all positions i along the set have been sampled. The result is a frequency distribution of box mass n(s, r) sampled at r. Second, each box mass s of the frequency distribution n(s, r) sampled at r is divided by the total number of sampled boxes of size r to generate a probability distribution Q(s, r). Third, the first and second statistical moments of the probability distribution Q(s, r) are computed (Cheng 1997; Dale 1999) as:

$$Z_{Q}^{(1)} (r) = \frac{1}{{\left( {n + 1 - r} \right)}}\sum\limits_{i = 1}^{n + 1 - r} {\sum\limits_{j = 1}^{i + 1 - r} {x_{j} } }$$

(6.11)

$$Z_{Q}^{(2)} (r) = \frac{1}{{\left( {n + 1 - r} \right)}}\sum\limits_{i = 1}^{n + 1 - r} {\left( {\sum\limits_{j = 1}^{i + r - 1} {x_{j} } } \right)^{2} }$$

(6.12)

Fourth, lacunarity at r is estimated by dividing the second moment of the probability distribution Q(s, r) by the square of the first moment (Plotnick et al. 1996):

$$\Lambda (r) = \frac{{Z_{Q}^{(2)} (r)}}{{\left[ {Z_{Q}^{(1)} } \right]^{2} }} = 1 + \frac{{{\sigma}_{Q}^{2} (r)}}{{\left[ {Z_{Q}^{(1)} } \right]^{2} }}$$

(6.13)

where $\sigma_{Q}^{2} (r)$ and $Z_{Q}^{(1)}$ are, respectively, the sample variance and mean of the probability function Q(s, r). Last, repeating the lacunarity calculations at box sizes 1 ≤ r ≤ n reveals a distinct pattern of variation in the logarithm of Λ(r) when it is plotted against the logarithm of r.

6.1.2 Scale-Dependency and Interpretation of Λ(r)

Three general patterns of spatial dispersion are recognized in binary sets composed of occupied (black = 1) and unoccupied (white = 0) sites, viz., random, uniform, and aggregated (Dale 1999). Random patterns arise when occupied sites occur randomly and independently of one another along the length of a set (e.g., set A, Fig. 6.7). Uniform and aggregated patterns, in contrast, are neither random nor independent and exhibit characteristics of spatial dependence or interaction. For example, uniform patterns form a ‘regular’ spacing of occupied sites, where the presence of one occupied site reduces the probability that another will be found in close proximity to it (e.g., set B, Fig. 6.7). Aggregated patterns appear ‘clumped,’ and the presence of an occupied site increases the probability that another will be found close by (e.g., sets C–F, Fig. 6.7). Spatial pattern, however, is strongly scale-dependent and those patterns that appear random, uniform, or aggregated at one measurement scale may be very different at finer or at coarser scales (Plotnick et al. 1996; Dale 2000).

Fig. 6.7

Simulated binary sets displaying six unique scale-dependent patterns of foliage (black) and gap (white) dispersion: (A) random, (B) uniform, (C) randomly distributed gaps of random width, (D) randomly distributed gaps of equal width, (E) uniformly distributed gaps of equal width, and (F) single gap. Each set has the same number of occupied sites (a = 630), total length (n = 1000), and density p (a/n = 0.63)

Scatterplots of the logarithm of Λ(r) against the logarithm of r reveal the scale-dependent spatial structure, heterogeneity, and non-randomness that is associated with each of the binary sets described below (Fig. 6.8). The magnitude of Λ(r) is a measure of the spatial heterogeneity that is found in the set at scale r (Plotnick et al. 1996). For example, values of Λ(r) that are equal or close to 1 indicate translational invariance or spatial homogeneity at r, while increasingly larger values of Λ(r) denote increasing spatial heterogeneity. Lacunarity curves A to F show an overall decline in Λ(r) with increasing r; however, the rate and pattern of this decline differs according to the set’s scale-dependent spatial structure or ‘gappiness’. All six curves share the same X and Y intercepts, because each set has the same number of occupied sites (a = 630), total length (n = 1000), and pattern density (p = a/n = 0.63). The magnitude of Λ(r) at r = 1 will vary inversely with p, since Λ(1) is equal to 1/p (Plotnick et al. 1996).

Fig. 6.8

Natural logarithm of Λ(r) plotted against the natural logarithm of box size r. Lacunarity curves A–F correspond directly to binary sets A–F in Fig. 6.7. Lacunarity curves that lie above or fall below the Λ(r) that is expected under CSR (curve A) indicate the kind, magnitude, and scale of non-randomness. For example, curve B falls substantially below CSR and thus indicates spatial uniformity across all measurement scales. Curves E, D, C, and F, on the other hand, all lie above CSR and display, in ascending order, increasing quantities and scales of spatial aggregation or clumping

Lacunarity curves that were generated for sets A and B approach translational invariance (Λ(r) = 1) more rapidly than curves representing the coarser-grained sets C–F (Fig. 6.8). The steepest decline in Λ(r) with increasing r occurs when discrete occupied sites are regularly spaced (set B). When occupied sites are dispersed randomly and independently of one another, as is the case in set A, the decline in Λ(r) occurs at a rate predicted by the Binomial Theorem (Dale 2000):

$$\Lambda_{CSR} (r) = 1 + 1/rp - 1/r$$

(6.14)

where Λ _CSR (r) is the expected lacunarity statistic at box size r under the condition of complete spatial randomness (CSR), and p is the pattern density of the binary set. Lacunarity curves that lie above or which fall below Λ(r) expected under CSR (set A) indicate the type, magnitude, and scale of non-randomness. For example, curves E, D, C, and F all lie above CSR and display, in ascending order, increasing quantities and scales of spatial aggregation (clumping). Curve E exhibits the least amount of coarse-scale aggregation, because gaps are relatively small, homogeneous in size, and uniformly dispersed across the set. Set F is composed of one large gap, and displays the greatest amount of aggregation at all spatial scales. Curve B falls substantially below CSR and, therefore, indicates the presence of spatial uniformity at all measurement scales.

6.1.3 Derivation of Ω_LAC(θ)

From the brief description and examples of lacunarity analysis that were outlined in the previous sections, it is evident that the integrated cross-scale differences between observed estimates of Λ(r) and those expected under CSR provide both the conceptual and analytical bases for the derivation of Ω(θ). Three important analytical steps were necessary to develop this index. First, binary sets extracted from fisheye photographs have no logical start or end due to the circular nature of the sample transect. Nevertheless, there are implications to and rules for selecting a starting point: (i) sets must always start at a gap edge to avoid the fragmentation of discrete gaps or clumps; (ii) Λ(r) can be sensitive to the set’s starting point, because at coarse measurement scales the gliding-box algorithm samples pixels (grid cells) in the middle of the set more often than ones near the start or end (Cheng 1997). It is, therefore, imperative to resample the set using randomly or regularly placed starting points and to compute the mean observed lacunarity statistic for all resampled sets:

$$\overline{{\Lambda_{Obs} (r)}} = \frac{1}{m}\sum\limits_{t = 1}^{m} {\Lambda_{t} (r)}$$

(6.15)

where Λ _t (r) is the lacunarity statistic estimated at r for each resampled transect t, and m is the total number of resampled transects. Short, spatially heterogeneous sets require more resampling than long, homogeneous sets; however, we found that resampling the binary set four to eight times using regularly spaced starting points was usually sufficient to stabilize $\overline{{\Lambda _{Obs} (r)}}$ .

Second, the magnitude of Λ(r) is a function of p and its spatial variability or dispersion across the binary set. Lacunarity curves generated for sets with identical lengths n and different densities p are therefore not directly comparable unless the effect of density p on Λ(r) is first eliminated. Normalization to a common Y intercept can be accomplished by dividing each value of Λ(r) by Λ(r = 1). In doing so, the effect of the first statistical moment $Z_{Q}^{(1)}$ on all values of Λ(r) is removed, which allows direct comparison of the scale-dependent variation in lacunarity for sets with identical lengths n and different densities p (Plotnick et al. 1996; Feagin, 2003).

Last, the total integrated area found between the normalized $\overline{{\Lambda _{Obs} (r)}}$ and Λ _CSR (r) curves across all measurement scales r gives a quantitative estimate of a set’s departure from CSR, where areas below and above Λ _CSR (r) indicate the amount of spatial uniformity and clumping, respectively. Also knowing the area beneath a hypothetical curve of ‘maximal clumping’ allows us to derive an estimate of Ω(θ):

$$\Omega_{LAC} (\theta ) = \left[{\frac{{A_{Max} (\theta) - A_{Obs} (\theta)}}{{A_{Max} (\theta) - A_{CSR} (\theta)}}} \right]$$

(6.16)

where A _Obs (θ), A _CSR (θ), and A _Max (θ) are respectively the integrated areas under the normalized $\overline{{\Lambda_{Obs} (r)}}$ Λ _CSR (r), and ‘maximally clumped’ Λ _Max (r) lacunarity curves computed for a binary set extracted at θ (Fig. 6.9). Values of Ω _LAC (θ) range from greater than unity for uniform sets, unity at CSR, and less than unity for clumped sets. Previous work by Plotnick et al. (1996) indicates that a binary set defining Λ _Max (r) will have two important properties: (i) all occupied sites (black pixels) will be completely separate from all unoccupied sites (white pixels), and (ii) p will approach zero. To understand this conceptually, imagine that all shoots or leaves visible in a fisheye photograph at θ were rearranged so that they appear at only one point in the hemispherical object region, with a single foliage unit obstructing the view of all other units that are stacked behind it. Although this kind and degree of clumping would never occur in natural forests, Λ _Max (r) does represent a reasonable endpoint for spatial clumping from a purely geometrical perspective. We found that Λ_Max(r) reached its maximum value when p was equal to 0.01.

Fig. 6.9

Normalized lacunarity curves identifying the position of the mean ‘observed’ curve (set C in Fig. 6.7) relative to CSR and maximal clumping. We divide the integrated area between maximal clumping and the mean ‘observed’ curve by the integrated area between maximal clumping and CSR to estimate Ω_LAC(θ). The range of variation in the 14 ‘observed’ curves is due to the resampling of transect starting and end points. The estimated mean and standard deviation of Ω_LAC(θ) for set C are 0.36 ± 0.01 (0.34 ≤ Ω_LAC(θ) ≤ 0.40) using all 14 possible transect start and end points