## 1. Introduction

An important vegetation biophysical parameter, the leaf area index (LAI), is a dimensionless variable and a ratio of leaf area to per unit ground surface area. This ratio can be related to gas-vegetation exchange processes such as photosynthesis [

1], evaporation and transpiration [

2–

4], rainfall interception [

5], and carbon flux [

6–

8]. Long-term monitoring of LAI can provide an understanding of dynamic changes in productivity and climate impacts on forest ecosystems. Furthermore, LAI can serve as an indicator of stress in forests, thus, it can be used to examine relationships between environmental stress factors and forest insect damage [

9]. Emerging remote sensing platforms and techniques can complement existing ground-based measurement of LAI. Spatially explicit measurements of LAI extracted from remotely sensed data are an indispensible component necessary for modeling and simulation of ecological variables and processes [

10,

11]. Since LAI remains consistent while the spatial resolution changes, estimating LAI from remote sensing allows for a meaningful biophysical parameter, and a convenient and ecologically-relevant variable for multi-scale multi-temporal research that ranges from leaf, to landscape, to regional scales [

12].

The physiological and structural characteristics of leaves determine their typically low visible light reflectance except in green light. Past the visible, high near-infrared reflectance of vegetation allows optical remote sensing to capture detailed information about the live, photosynthetically active forest canopy structure, and thus begin to understand the mass exchange between the atmosphere and the forest ecosystem. Algorithms and models used as an input parameter to predict or estimate ecological variables have been developed using remotely sensed datasets based LAI [

13–

16]. For example, LAI obtained from optical remotely sensed data serves as a key parameter to estimate aboveground biomass of forest stands [

17]. Due to recent availability, fine resolution spatial and spectral (hyperspectral) remotely sensed data are being used to retrieve LAI and other biochemical contents such as chlorophyll in leaves of forests [

18–

20]. Also in recent years, due to the emergence of light detection and ranging (LiDAR) techniques and equipment, numerous methodologies are being developed for point cloud datasets obtained from LiDAR to assess vegetation and forest three-dimensional structures [

21–

26]. The explicit three-dimensional information contained in LiDAR point clouds offers the ability to investigate forest health [

27,

28], forest stand structure and biophysical parameters [

29–

33]. Particularly, terrestrial LiDAR, with very high density point clouds, allows for improved retrieval of forest stand structure information including LAI [

34,

35]. Meanwhile, factors influencing the accuracy of leaf area density estimation have been investigated [

31,

36] including attention to leaf-on and leaf-off conditions [

37,

38]. LiDAR has been used to monitor forest stands and environmental changes through the use of LAI as a key indicator parameter [

39]. Currentely, due to single spectral band information deficiency, LiDAR has been combined with other hyperspectral remotely sensed datasets to obtain more comprehensive information about biophysical characteristics of forest ecosystems [

40]. In recent years, a theory based on the spectral invariant property of leaves[

41] has been applied to retrieve LAI and physical canopy height from optical sensors including single- [

42,

43] and multiple-angles [

44]. The radiation budget theory characterizes the structural and spectral contribution in simulating the bidirectional reflectance factor in an efficient way and introduces new principles of photon-vegetation reflectance interaction, whereby one can characterize gap probability and gap fraction in terms of photon recollection probability and escape probability.

During past decades, efforts focused on LAI measurement strategy and theory, not only with ground-based field measurements, but also the retrieval of LAI based on array of remote sensors. In summary, there are two broad types of methods for estimation of LAI, either employing the “direct” measures involving destructive sampling, litter fall collection, or point quadrat sampling “indirect” methods involving optical instruments and radiative transfer models. The dynamic, rapid and large spatial coverage advantages of remote sensing techniques, which overcome the labor-intensive and time-consuming defect of direct ground-based filed measurement, allow remotely sensed imagery to successfully estimate biophysical and structural information of forest ecosystems.

A range of LAI definitions exist in the research literature, which complicates the comparison between works, and thus, the first focus of this paper is a compilation of LAI definitions. The second focus of the paper is the explanation of the gap fraction method theory. Thirdly, LAI estimation methods and sensors are discussed. Finally, remotely sensed LAI estimation and scaling issues associated with it are discussed.

## 2. Theory

In the early period of LAI research, due to the complicated distribution of foliage elements within the canopy, a modified Beer’s law light extinction model was developed. The model estimates LAI by mathematically analyzing light intercepting effect of leaves with different angular distribution based on a very common simplified assumption that all of foliage element and live parts within canopy are randomly distributed. The point quadrat method [

45,

46] was an early method used to mathematically analyze the relationship between projection area and foliage elements with all possible angular and azimuthal distributions. In this model, the extinction coefficient served as an important parameter to characterize the effect of leaves’ angular and spatial distributions on radiation interception. An algorithm was developed [

47] to calculate extinction coefficients based on the assumption that the angular distribution of leaf area in a canopy is similar to the distribution of area on the surface of prolate and oblate spheroids. Because of the assumption of randomly located foliage elements within canopy, the LAI obtained from gap fraction [

48] theory was not the true LAI, thus, a term called effective LAI was created to more accurately describe the result. However, gap fraction theory only applies to the percentage or proportion of gaps accounting for the whole hemispherical bottom-up view of a canopy. Gap size (dimensional information) is another very useful information to characterize clumping and overlapping effect, therefore, the gap size theory is a another stage for LAI ground-based filed indirect measurement development.

Recently LAI research focus has shifted from an empirical and statistical stage to process-based modeling stage due to the involvement of remotely sensed datasets and numerical ecological model implementation. The canopy structure in this paper is defined as the amount and spatial organization of aboveground plant materials including leaves, stems, branches, flowers and fruit, which affects the environmental factors such as air temperature, leaf temperature, atmospheric moisture, soil evaporation below the canopy, soil heat storage, leaf wetness duration and others [

48]. The physical dimension (size, shape), relative position, spatial arrangements between different canopy elements determine the amount and spatial distribution of fraction of photosynthetic radiation (fPAR) within and below the canopy, which control the absorption, reflectance, transmission, and scattering of solar radiation. A single live leaf reflects green light and near-infrared light due to its internal structure. When scalling to the individual tree or forest stand level, non-random distribution and multi-layer structure of canopy elements result in multiple scattering of radiation between the different layers of foliage elements and other parts of canopy. This results in the obvious difference in reflectance for the individual leaf, tree canopy and a stand at landscape level. The denser a canopy, the more absorption and reflectance of solar radiation occurs and less energy is transmitted to the ground surface below. In addition, the difference of reflectance properties at various scales is dependent on the field of view (FOV) and spatial resolution for various sensors. The shadows between tree canopy and hot spot result from the relative position of sensor and sun (the hotspot is a phenomenon that occurs when the sensor sees only sunlit elements). Geometrical optical (GO) models such as bidirectional radiative directional function (BRDF) [

49] and 5-SCALE radiative transfer model (RTM) [

50] were developed to simulate the reflectance properties at different scales. Clearly, vertical and horizontal canopy structures are becoming an indispensible input parameter for modeling ecological process such as photosynthesis, evaporation, transpiration and carbon sequestration of forest ecosystems. In terms of the economic value of a tree, the tree bole is linked to tree stem volume, timber production, and the characterization of forest inventory. From the ecological perspective, foliage is applied in modeling biological processes at leaf-level and the foliage distribution is a key factor affecting competition for resources, such as light, nutrients and moisture of intra- and interspecies of forest community stand. The most important canopy attributes affecting solar radiation penetration through canopy and indirect LAI measurements are leaf angular distribution and leaf spatial distribution. On one hand, leaf angular distribution affects radiation transmission through canopy at different angles; on the other hand, leaf spatial distribution affects the amount of radiation transmitted through the canopy.

#### 2.1. Definitions

During past decades, definitions of LAI have been provided by scientists from many disciplines for a range of purposes, such as determination of forest community succession, simulation of potential biological activities, and solar radiation regimes within plant canopies. The definition of LAI are summarized and compared in

Table 1. A common and acceptable definition of LAI needs to be addressed to make research results comparable.

As shown in

Table 1, total leaf area index (ToLAI) was first defined as the total one-sided area of photosynthetic tissue per unit ground surface area [

51,

52]. This definition is especially applicable to flat broad leaf condition with same area on both sides of leaf. In reality, the shape of leaves is not always of this type [

53], some leaves such as white spruce (

Picea glauca) are needle-shaped and arrangement is spiraled. Each needle has an approximate cylindrical shape which this definition cannot describe accurately, thus, projected area of leaves has been provided [

54]. Projected LAI (PLAI) is defined as the horizontal area that is cast beneath a horizontal leaf from a light at infinite distance above it. The cumulative LAI (unitless) of a canopy by calculating the sum of vertical projection of foliage area on a horizontal plane from ground (z = 0) to top of canopy (z = h) [

55]. LAI depends on the average surface density coefficient of the foliage (u) expressed in m

^{2}/m

^{2}:

The concept of silhouette leaf area index (SLAI) was introduced and defined as the area of leaves inclined to the horizontal surface, and was compared with TLAI and PLAI to investigate the effect of leaf orientation on radiation interception, it was shown that the leaf orientation effects, or shading, or both, caused more variation in the interception of solar radiation than did variation in leaf geometry [

56]. Effective leaf area index (ELAI) was defined as one half of the total area of light intercepted by leaves per unit horizontal ground surface area based on the assumption that foliage elements randomly distributed in space, and was introduced to precisely describe the shortwave and long wave irradiance condition under a Douglas fir forest stand [

57]. Trees usually have differently-shaped canopies and foliage elements which means that a general definition of LAI needs to be obtained. The most popular and widely accepted definition of the true leaf area index (TLAI) is defined as one half of total leaf area per unit surface ground area [

58,

59] by mathematically analyzing mean projection coefficient for various perfect geometrical objects representing the different shapes of real leaves.

#### 2.2. Canopy Distribution and Leaf Inclination

Based on the assumption that the forest canopy is randomly-distributed, the solar radiation regime was simulated to obtain the amount of penetrated beam radiation through the canopy structure and develop the algorithms to estimate the LAI. According to Beer’s Law [

60], when a beam of monochromatic radiation passes through a compound, absorbance and transmittance takes place and the radiation will be attenuated. Likewise, when a beam of solar radiation transmits through the canopy, the leaves will absorb some of the visible light and reflect some infrared light which results in the changes between the solar radiation before and after passing through leaves. The extinction coefficient [

47] was developed to describe the canopy function when shifting the beam radiation.

The extinction coefficient represents the area of shadow cast on a horizontal surface by the canopy divided by the area of leaves in the canopy or the average projection of leaves onto a horizontal surface [

47]. Among the many geometrical objects, the sphere, cylinder, and cone models provided simple methods to calculate the extinction coefficient, the

Figure 1 shows an example of extinction coefficient calculation of ellipsoid, and the following equations calculates the shadow area of ellipsoid under parallel light source:

where A

_{s} is the shadow area of ellipsoid under the illumination of parallel light source, a and b are the long and short axis of ellipsoid respectively, ϕ is the inclination angle of direct solar beam. The density functions for foliage inclination angle are generally crude approximations to actual foliage inclination angle densities. Thus, the probate and oblate spheroids were proposed to approximate the actual foliage spatial distribution and a more flexible and general equation was developed to calculate a more accurate extinction coefficient of a forest canopy [

47]. Based on the ellipsoid model, a leaf angle density function for canopies has been provided and the leaf inclination angle density function is a fundamental property of plant canopy structure and is needed for computing distribution of leaf irradiance [

61].

#### 2.3. Gap Fraction

Among the indirect methods to estimate LAI, one popular way is to measure light penetration and the amount and distribution of openings in the canopy which is often referred to as gap fraction [

48,

62]. Gap fraction describes the possibility of sun light rays not penetrating into the understory through the canopy. The spatial position of each single leaf in reality is determined by its spatial and angular distribution, which is shown in the

Figure 2.

The measurement of gap fraction is generally an acceptable way to analyze the structure of a tree canopy and often parameterized with the LAI and leaf angle distribution. P (θ), which denotes the gap fraction at the zenith angle θ of incoming direct sun light, which can be expressed mathematically as the Poisson model:

where G(θ) is the projection coefficient of the foliage onto a plane (normal) perpendicular to incoming radiation [

63,

48], and L is the LAI of the forest canopy including all aboveground structural components (branches, boles, cones, and epiphytes).

K represents the average projected area of the canopy components on a horizontal plane. It is assumed to link to the extinction coefficient discussed above.

Due to the multilayered structure of a forest canopy, many gaps form within the canopy and allow solar beams to penetrate through and provide enough light for understory growth (

Figure 3). This method is based on an assumption that the spatial distribution of foliage is random; the overlapping and clumping of leaves within the canopy has not been considered, thus, the LAI obtained in this way is not the true LAI, but underestimated LAI. Based on the Miller theory [

64] and Chen definition [

57], the definition of effective LAI is:

where

P(θ) is the measured canopy gap fraction at zenith angle θ and L

_{e} is the effective LAI.

Chen also pointed out that an important consideration implicitly expressed in (4) is that LAI can be calculated without knowledge of foliage angle distribution if the gap fraction is measured at several zenith angles covering the full range from 0 to π/2 [

65].

Figure 3 shows the theoretical model used to calculated gap fraction for multi-layer canopy structure forest stand. The canopy is divided into two levels: branch level and leaf level. For each layer, each branch is composed by a sub-branch with attached leaves and perforations. The probability of penetrating the perforation between leaves P

_{Lj}(θ) and branched P

_{bj}(θ) in j

^{th} layer are calculated respectively, thus, the probability of solar beam penetrate the j

^{th} layer is the product of probability of penetrating leaves and branches respectively P

_{j}(θ) = P

_{bj}(θ) × P

_{Lj}(θ), P

_{L1}(θ, β) and P

_{L1}(θ) represent the probability of direct solar beam penetrate the leaves in first layer of canopy where the incident angle of solar beam is θ and azimuthal angle is β.

Besides the theoretical formula and analytical expression described in

Figure 3, an improved algorithm has been developed by Nilson to estimate the canopy indices and LAI from gap fraction data [

66], the method used the eigenvectors and eigenvalue of the covariance matrix to describe the random variation of gap fraction at the near-zenith view direction and showed good performance in relatively open boreal and sub-boreal forest environments.

Gap fraction is usually obtained automatically using optical radiation measurement instrument such as hemispherical photograph, or LAI-2000 (Li-Cor, Inc). A key component of this method is to set up the optimal threshold to separate the leaves from sky. Usually overexposure will result in an overestimated projected LAI and underexposure will make the projected LAI much higher. Different digital hemispherical photographs which were collected under a range of sky brightness conditions for an array of forest species and openness have been compared [

67]. Zhang [

67] found that the automatic exposure is apt to underestimate the effective LAI and provides a protocol for taking the digital hemispherical photography in different open-canopy conditions.

Many commercial optical instruments based on the gap fraction theory are available to estimate the effective LAI. All of the instruments can be divided into two broad types including linear sensors such as DEMON, line quantum sensors, and the other type are hemispherical sensors such as LAI-2000 (Li-Cor, Inc), the leaf laser, hemispherical photography and the CI-100. Unfortunately, these usually underestimate the LAI of forest trees due the assumption of random distribution of foliage.

#### 2.3.1. Clumping and Gap Size

There are two causes that affect the accuracy of LAI estimation. The first is the non randomly-distribution of tree foliage resulting in overlapping and clumping between the leaves within canopies. If we want to obtain true LAI, these effects should be carefully considered and incorporated into the LAI estimate. The other cause is light obstruction from canopy components such as branches, boles and stems, especially for conifers on which needles with a shoot will be significantly clumped.

The fraction characteristic of sunfleck for obtaining effective LAI has been well studied under the “gap fraction theory”. Because the sunflecks’ size and their spatial distribution under canopy result from the gaps in the non-randomly distributed overlaying canopy in the Sun’s direction, the structural characteristics of the sunfleck are an important information source. If the quantitative correlation between sunflecks’ distribution and frequency and foliage clumping and overlapping effect can be identified, such information is sufficient to translate effective LAI to true LAI based on this relationship denoted by gap size theory. This procedure is summarized below.

In order to quantitatively describe the sunfleck dimensional information, a theoretical model focused on the size and shape of sunflecks under forest canopy needs to be developed. The model uses the sunlit segments along a straight-line transect under the forest canopy to represent the sunfleck size distribution [

68], and the probability distribution of shadow-edge angles information (penumbral effects of the finite solar disc) to predict the shape of sunflecks [

69]. By combining the gap-size theory and penumbral effect, the light intensity under the plant canopy can be predicted quantitatively and used to accurately and spatially estimate moisture evaporation and photosynthesis of leaves. The sunfleck distribution, direct solar radiation and diffuse skylight are related to the geometrical structure of plant stand, thus Nilson [

70] proposed a theoretical model to analyze the gap frequency of forest plant stands based on Possion, positive and negative binomial distributions. The Markov processes theory was also presented by Nilson [

70] to predict the gap frequency for stand geometry. Nilson [

70] recommended that the binomial and Markov model be used for practical use side by side in order to avoid the unrealizable Poisson model. All three models are based on the assumption of randomly distribution of foliage elements.

Two different gap-size theories were developed by Chen and Black [

71] (hereinafter referred to as theory one) and Chen and Cihlar [

72] (hereinafter referred to as theory two) to evaluate the effect of foliage clumping at scales larger than the shoot, and the term “clumping index” was given for this effect. The clumping index can be measured by using the sunfleck-LAI instrument Tracing Radiation and Architecture of Canopies (TRAC, 3rd Wave Engineering) [

73]. The major difference between these two methods is the dependence on randomly spatially distributed foliage element. Theory one developed a Poisson model to describe sunfleck size distribution under clumped plant canopies based on the assumption that foliage clumps are randomly distributed in space and foliage elements are randomly distributed within each clump. Although it improves the result of LAI estimation without considering the canopy architectural information, it is still not reliable due to this assumption. Theory two developed a gap-size measurement model which can be used for any heterogeneous canopies and is the theoretical foundation of prototype sunfleck-LAI measurement instrument TRAC. It’s an improvement on the finite-transect method because it avoids the assumption of local randomness. Thus, theory two avoids making the assumption for a spatial distribution pattern of foliage clumps used in the theory one and is applicable to various of plant canopies.

The Poisson model was first modified by considering the non-random spatial distribution of canopy elements and expressed as [

63]:

where G (θ) characterizes the leaf angle distribution. Ω is a parameter determined by the spatial distribution pattern of leaves. When the foliage spatial distribution is random, Ω is 1. If leaves are regularly-distributed (extreme case: leaves are laid side by side), Ω is larger than 1. When leaves are clumped (extreme case: leaves are stacked on top of each other), Ω is less than 1. Foliage in plant canopies is generally clumped, and hence Ω is often referred to as the clumping index [

74]. L is the LAI of a forest canopy including all aboveground structural components (branches, boles, cones, and epiphytes). In terms of conifer trees, the clumping index Ω can be separated into two parts:

where Ω

_{E} is the stand-level clumping factor at scales larger than shoot and γ

_{E} is the clumping at shoot level and was named after “needle-to-shoot area ratio” [

74]. As for deciduous trees, γ

_{E} is equal to 1.

In order to quantify the clumping effects, Chen thought that there are two underlying assumptions of this correction method: shoots are the basic unit responsible for light interception, and shoots randomly distribute within a canopy [

72]. In addition, the research found that the non-randomness of shoot position reduces indirect measurement of LAI by approximately 35% for a Douglas-fir canopy. In this situation, there are three components that constitute the percentage of measurement of LAI. Non-randomness of shoot position accounts for 35%, the indirect measurement through destructive sampling captures 31%; the remaining 34% can be explained by the clumping of needles with shoots. The effect of needle clumping with shoots can be obtained by measuring the ratio of half the total needle area in a shoot to the shoot intercepting area.

By combining gap fraction and gap size theory, true LAI can be obtained for individual trees. The regional, landscape, or even global LAI spatial distribution or variation can be acquired using an airborne, or satellite platform based sensors, along with the specific algorithms applicable to the characteristics datasets collected by these platforms.

## 4. Conclusions

Direct and indirect terrestrial methods are the foundation and basis for obtaining accurate LAI estimates. The destructive sampling method for various forest tree species provides valuable data to validate results obtained from remote sensing platforms and algorithms. It is necessary to develop optical sensors that capture the light environment within the canopy which not only contains direct light from sunlight through the gaps of trees, but also diffused light and environmental light affections. Most of the optical instruments based on gap fraction theory infer effective LAI by recording the Photosynthetic Active Radiation (PAR), with these techniques it’s difficult to visualize the site environmental such as density and canopy height expect by utilizing hemispherical photography approach. Although a hemispherical photograph can permanently capture the light conditions at the moment that the picture is taken, it is difficult to control the exposure values and acquire every detail due to the resolution of which the digital camera is capable. Furthermore, the information obtained from digital hemispherical photographs is only two dimensional, thus, terrestrial LiDAR is an ideal instrument to complement photography by permanently recording the three-dimensional structural information. Some terrestrial LiDAR scanners are also capable of recording 360 degrees color images by setting up an appropriate exposure value (example: Leica ScanStation 2), however, the optical distortions in such imagery still needs to be reconciled. An instrument which not only records the Photosynthesis Photo Flux Density (PPFD), but also permanently captures the light conditions at the moment when the measurement is taken is ideal. Due to the non-random spatial distribution of leaves within a canopy, overlapping, and clumping effects, it is difficult to acquire true LAI despite removing partial effects based on the gap size analysis theory. Another problem of estimating LAI is the effect of branches and stems blocking the light. This blockage leads to the overestimation of LAI; thus far, the approach to remove this effect employs species-dependent empirical estimation of the fraction of non-photosynthetic portions of a tree. With the development of remote sensing technology, accurate, timely and dynamic acquisition of LAI at the landscape level requires us to develop new algorithms optimized for the characteristics of the remote sensors. Especially, with the emergence of LiDAR, we should synthesize the power to estimate the biophysical parameters of trees from forest stand level to landscape level, regional level, and even to global level by combing the spectral information of optical remote sensing and three-dimensional structure information from LiDAR. In such way we can provide more-accurate, timely and meaningful information for the development of ecological models and thus the dynamic monitoring of changes in an ecosystem.