^{1}

^{2}

^{3}

^{4}

^{2}

^{3}

^{5}

^{*}

^{6}

^{4}

This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (

Directional gap probability or gap fraction is a basic parameter in the optical remote sensing modeling. Although some approaches have been proposed to estimate this gap probability from remotely sensed measurements, few efforts have been made to investigate the scaling effects of this parameter. This paper analyzes the scaling effect through aggregating the high-resolution directional gap probability (pixel size of 20 meters) estimated from leaf area index (LAI) images of VALERI database by means of Beer's law and introduces an extension of clumping index, Ĉ, to compensate the scaling bias. The results show that the scaling effect depends on both the surface heterogeneity and the nonlinearity degree of the retrieved function. Analytical expressions for the scaling bias of gap probability and Ĉ are established in function of the variance of LAI and the mean value of LAI in a coarse pixel. With the VALERI dataset, the study in this paper shows that relative scaling bias of gap probability increases with decreasing spatial resolution for most of land cover types. Large relative biases are found for most of crops sites and a mixed forest site due to their relative large variance of LAI, while very small biases occur over grassland and shrubs sites. As for Ĉ, it varies slowly in the pure forest, grassland and shrubs sites, while more significantly in crops and mixed forest.

Directional gap probability or gap fraction is defined originally as the probability of a beam transferring at a given incident zenith angle through the vegetative canopy without any interception. As a key variable describing canopy structure and biomass spatial distribution, it is used to simplify the 3- D light interception problem to a 1-D problem (

where

Through observation and studies in different scales including foliage (

This study focuses on the analysis of the scaling effect on the directional gap probability by means of a simple scaling-up scheme and LAI derived from high resolution spatial data. The second section provides the theoretical framework to estimate the scaling effect of directional gap probability raised by two different aggregation schemes from local scale to larger scale. In the third section, we present the different types of remotely sensed LAI images obtained from VALERI database (Validation of Land European Remote sensing Instruments). In section 4, the scaling effect associated with the non- linear relationship between LAI and gap probability is quantified over several types of landscape. Conclusion is given in section 5.

There are two different schemes generally used to aggregate the parameters/variables from the local scale to regional or global scale (

The aggregation of the results which are derived from a distributed model f using distributed input variables. Spatially distributed variables _{pixel}

The aggregation of input variables before use in an aggregative model F (here _{pixel}_{pixel}

As it concerned to gap probability, supposing that the pixel whose area is S is composed by N homogeneous sub-pixels, each sub-pixel i has an area of s_{i}

where

The directional gap probability can also be aggregated following the second aggregation scheme (see right flowchart of

Then computing the directional gap probability with help of the same formula as

Since the distributed model related LAI to P is nonlinear (see _{pixel}_{pixel}_{LAI}

The relative scaling bias (RE) is therefore obtained

From

In order to take into account the scaling effects of spatial heterogeneity of LAI on estimate of the directional gap fraction and to make the estimation of the directional gap fraction independent on the observation scale and the aggregation schemes used, a parameter Ĉ is introduced in

Following the same development made by

As shown by this equation, the parameter Ĉ is directly proportional to the mean LAI and inversely proportional to the spatial heterogeneity of LAI (

It should be noted that the parameter Ĉ introduced in

Foliage clumping affects the gap probability for the same LAI by delaying the occurrence of the saturation in reflectance as LAI increases. There have been some studies mostly concentrated on the estimation of clumping index with multi-angular data.

The data used here are part of the VALERI database which provides high spatial resolution (20 m) SPOT-HRV scenes for several landscapes sampled (including crops, forest, grassland and shrubs) around world (Baret et al., 2005). This wide coverage of landscape makes the conclusion of this study more general. Each site has an enough sampling size (about 3km by 3km). Detailed information about each site (including land cover type, location and the date of measurement) is given in

In this study, we adopt a simple formula proposed by _{L}

Inserting

_{L}

As shown in _{L}

In order to see the magnitude of the spatial scaling bias of directional gap probability with real scenarios, the VALERI dataset is used in this study. Three assumptions are made in the following calculations:

Beer's law used to retrieve gap probability from LAI (

Incident beam is assumed to be vertical, i.e. cos(

A spherical leaf angle distribution is assumed, i.e. G=0.5, which is a reasonable assumption for many conifer shoots and closed, broad-leaved canopies (

Following the schemes proposed and showed in

From this figure, we notice that the relative scaling bias of gap probability increases with decreasing spatial resolution for most of land cover types. Larger relative bias occurs at crops (104%, 50%, 26%, 14%, at pixel size of 1280m, respectively) than pure forest sites (≤ 20% at pixel size of 1280m except for the mixed forest (Larose-August03) which has relative bias of 120% at pixel size of 1280m), grassland and shrubs (≤ 0.5% at pixel size of 1280m), demonstrating that our crops sites are relatively more heterogeneous than forest, grassland and shrubs sites. Previous research conducted by

As a result, a large uncertainty (bias) is introduced in estimate of the gap probability from low spatial resolution data such as NOAA-AVHRR or MODIS over large heterogeneous sites if the scaling effects are not considered.

Letting

As shown in

As far as sites with the same land cover type are concerned, the magnitude of “clumping index” also varies at different aggregated sizes, and mostly is inversely proportional to the spatial heterogeneity of LAI (

Therefore “clumping index” redefined by

In this study, spatial scaling effect of the gap probability based on Beer's law for different types of land cover is analyzed and corrected for by introducing an extension of the “clumping index”, Ĉ which accounts for the spatial heterogeneity.

Analytical expressions developed in this paper show that:

relative scaling bias is only dependent on the G function and the spatial heterogeneity of LAI, but independent on the LAI value itself, and

extension of “clumping index” Ĉ is directly proportional to the mean value of LAI and inversely proportional to the spatial heterogeneity of LAI for given G function and direction.

With the VALERI dataset, this study shows that relative scaling bias of gap probability increases and “clumping index” value decreases with decreasing spatial resolution for most of land cover types. Large relative biases and large variation of “clumping index” Ĉ are found for most of crops sites and a mixed forest site due to their relative large variance of LAI, while very small biases and small variation of clumping index are found for grassland and shrubs sites.

The parameters introduced in this paper has endowed a new significance to traditional clumping index and provided evidence to the utility of clumping index as an improvement of the estimate of gap probability from LAI. The results exhibit the capability of clumping index for scaling Beer' law and representing spatial heterogeneity, as well as the feasibility of the inversion approach for gap probability from remote sensing data. Meanwhile a simple and feasible method to estimate “clumping index” from remote sensing data is also explored from the above experiment, which will provide a support to global mapping of the vegetation clumping index.

This research is partly supported by 973 program (Grant No 2007CB714402) and partly supported by the Knowledge Innovation Program of Chinese Academy of Sciences through contract No. KGCX3- SYW-408.

The data used in this study are acquired from VALERI project (

General schemes of two aggregation schemes.

Relative scaling bias of gap probability versus the variance of LAI for different mean of leaf inclination angles _{L}

Relative scaling bias of gap probability against pixel size for different landscapes: six forest sites, five crops sites, one grassland site and one shrubs site.

same as

Detailed information of remote sensing images used in this research. The last two columns represent the mean (m) and the standard deviation (σ) of LAI respectively.

_{LAI} |
_{LAI} | |||||
---|---|---|---|---|---|---|

Aekloba-May01 | Palm tree plantation | 1/Jun./2001 | 2.63 | 99.58 | 3.54 | 0.671 |

Alpilles-March01 | Crops | 15/Mar./2001 | 43.81 | 4.74 | 0.93 | 1.15 |

Barrax-July03 | Cropland | 3/Jul./2003 | 39.07 | -2.10 | 0.97 | 1.41 |

Fundulea-May02 | Crops | 9/Jun./2002 | 44.41 | 26.59 | 1.53 | 1.30 |

Gilching-July02 | Crops and forest | 8/Jul./2002 | 48.08 | 11.32 | 5.39 | 1.79 |

Hirsikangas-August03 | Forest | 2/Aug./2003 | 62.64 | 27.01 | 2.55 | 1.14 |

Jarvselja-June02 | Boreal forest | 13/Jul./2002 | 58.30 | 27.26 | 4.20 | 1.09 |

Laprida-November01 | Grassland | 3/Nov./2001 | -36.99 | -60.55 | 5.66 | 2.07 |

Larose-August03 | Mixed forest | 18/Sep./2003 | 45.38 | -75.21 | 5.87 | 2.00 |

Larzac-July02 | Grassland | 12/Jul./2002 | 43.94 | 3.12 | 0.81 | 0.20 |

Nezer-April02 | Pine forest | 21/Apr./2002 | 44.57 | -1.04 | 2.38 | 1.11 |

Rovaniemi-June04 | Forest | 23/Jul./2004 | 66.46 | 25.35 | 1.25 | 0.52 |

Turco-August02 | Shrubs | 29/Aug./2002 | -18.24 | -68.19 | 0.04 | 0.03 |