INTRODUCTION

LAI is defined as a one-sided leaf area per unit area of soil background. LAI is a critical variable of vegetation at the canopy level, and an essential indicator for crop growth status monitoring [@fu2013]. However, the traditional field measurement of LAI could be time-consuming and uncertain, which cannot satisfy the requirements of quickly monitoring and analyzing on a large scale. Traditional remote sensing methods involve extracting LAI from spectral curves by analyzing changes or absorption in sensitive bands. This process often combines empirical models, creating relationships that rely on specific site, time, and sample characteristics [@liang2015].

The utilization of remote sensing data in estimating LAI has been explored in the past few years [@fang2019].Several methodologies, rooted in the PROSAIL radiative transfer model, are extensively employed for estimating LAI. Iterative optimization, lookup table, and hybrid inversion are noteworthy approaches among them[@liang2015]. However, hybrid inversion is the most efficient and easy method due to the computational complexity of iterative methods and the uncertainty of lookup tables. This method determines the relationship between spectral features and canopy features, the relationship between vegetation indices and LAI in this paper, by the regression method [@liang2015]. In addition, using the full spectrum estimation may lead to computational complexity and susceptibility to atmospheric interference factors. Therefore, this study used vegetation indices to simplify calculations and improve robustness to atmospheric impacts.

Furthermore, with a dedicated focus on the research question, this paper endeavors to meticulously investigate the correlation between vegetation indices and LAI. The study utilizes Bidirectional Reflectance Factor (BRF) and employs hyperspectral sensor datasets to enhance the assessment of vegetation variables through the strategic application of vegetation indices. This method, poised to offer a comprehensive understanding of the nuanced relationship between these variables, forms the crux of our research.

MATERIALS AND METHODS {#sec:MATERIALS&METHODS}

This paper seeks to leverage the simulated spectral dataset produced by the PROSAIL model for the generation of diverse simulated vegetation indices. The primary objective is to develop model-fitting equations that correlate various vegetation indices with LAI values. Subsequently, this study used RMSE and $R^{2}$ to evaluate the goodness of fit of regression models.

PROSAIL

The estimation of this paper is based on the PROSAIL radiative transfer model to analyze crops’ structural and biochemical properties.The PROSAIL model (leaf-canopy model) is the combination of the PROSPECT model (leaf model) and the SAIL model (canopy model). PROSPECT simulates leaf properties by building the connection between Chlorophyll II content, structural parameters, water content, and reflectance and transmittance spectra. The SAIL model sets several input parameters, encompassing leaf-related factors, such as LAI, Leaf Inclination Angle Distribution Function, Leaf Reflectance, and Transmittance. Additionally, it considers variables, such as solar Zenith Angle, Zenith View Angle, hot spot size parameter, sun-view azimuth difference, and soil reflection. These parameters collectively facilitate the simulation of canopy reflectance [@allen1968].

This paper exclusively delves into the precision of vegetation indices concerning their accuracy in LAI estimation. To maintain consistency in our analyses, other parameters for the PROSAIL model were uniformly set during our investigation. For leaf-related parameters, we adopted reasonable default settings as follows: Chlorophyll content = 40, structure parameter N = 1.5, Equivalent water thickness = 0.01, carotenoid content = 8, dry matter content = 0.009, brown pigment content = 0. Concerning atmospheric parameters, the values were chosen as follows: hotspot parameter = 0.01, solar zenith angle = 30, observer zenith angle = 10, and relative azimuth angle = 0. To mitigate the influence of soil background on spectral data, soil brightness was set to 0.5. Additionally, LIDF (the Leaf Inclination Distribution Function) was configured as ‘spherical’ to enhance the accuracy of our analysis.

Selected Vegetation Indices {#sec:Selected Vegetation Indices}

To explore which vegetation index is most suitable for estimating LAI, many different types of vegetation indices need to be used, including simple ratio of calculation, normalized difference ratio, triangular vegetation index, modified versions for these three ratios, and derivative spectral index [@liang2015]. The required wavelengths are all from the visible to near-infrared part of spectra. This paper selectively selected a representative from each type of vegetation index and added TCARI and OSAVI due to their excellent performance in previous papers [@liang2015].

The following six vegetation indices have been selected: SR (Simple Ratio), NDVI (Normalized Difference Vegetation Index), TVI (Triangular Vegetation Index), MTVI1 (Modified Triangular Vegetation Index), TCARI (Transformed Chlorophyll Absorption in Reflectance Index), and OSAVI (Optimized Soil-adjusted Vegetation Index). Table 1 provides the specific formulations and references for their sources. In these formulations, the subscript of R represents the wavelength (in nm) of hyperspectral data. However, this article will use BRF instead of reflectance to calculate.

This study employs the simulated dataset to depict the variation curves of six distinct vegetation indices across LAI values spanning from 1 to

1. The values of various vegetation indices are computed using the formulations outlined in Table 1, with the vegetation index designated as the dependent variable (y) and the LAI value as the independent variable (x). Employing optimal curve-fitting techniques, this article endeavors to establish linear regression, exponential regression, logarithmic regression, and power regression models [@liang2015], adjusting the regression orders to identify the most fitting formulas for diverse vegetation indices, as summarized in Figure 1.

$R^{2}$ and NRMSE

To evaluate the accuracy of the fitted curves in simulating trend changes, this paper computes $R^{2}$ and NRMSE to assess the performance of each fitted model. The $R^{2}$ value, ranging from 0 to 1, signifies the model’s fit quality, with closer proximity to 1 indicating better fitting. Conversely , NRMSE, a non-negative value, increases with a poorer fit.

$R^{2}$ and Root Mean Square Error (RMSE) are widely recognized metrics for evaluating the precision of predictive models, particularly in the field of regression analysis. Furthermore, this paper adopts NRMSE instead of RMSE. NRMSE, as the standardized counterpart, offers a normalized perspective by accounting for the inherent scale of the data. Consequently, NRMSE emerges as a pivotal tool for a more nuanced and comprehensive evaluation of model performance.

RESULTS

Figure 1 illustrates the variations in six different vegetation indices in response to the increase in LAI. The equations of the best-fitting curves can be observed in the graph. Meanwhile, with the increase of LAI, the value of vegetation index reaches a saturation state when LAI =

1. Meanwhile, while LAI=5, there is a decline in the vegetation index value for each image, contrary to the expected pattern.

$R^{2}$ and RMSE serve as indicators of the accuracy of fit for the fitted curves, providing a means to assess the effectiveness of different vegetation indices in estimating LAI. Based on the findings from Figure 2 and Figure 3, certain conclusions can be drawn. SR and NDVI perform average in both evaluation metrics. The vegetation indices with the highest $R^{2}$ values are TVI and MTVI1, while the lowest $R^{2}$ values is TCARI, which is extremely lower than other vegetation indices. The vegetation indices with the highest NRMSE values are TCARI , while TVI and MTVI1 exhibit the lowest RMSE values. MTVI1 and TVI consistently receive the highest evaluation in both metrics. Conversely, TCARI demonstrates the smallest $R^{2}$ value and simultaneously a large NRMSE.

DISCUSSION

The primary objective of this paper is to develop model-fitting equations that correlate various vegetation indices with LAI values. Although result of analysis of the simulated dataset in this study suggests that MTVI1 and TVI hold promise as vegetation indices for estimating, it cannot be pointed out that they are the best. Due to the multitude of intervening factors, such as variations in leaf content and structure, geometric parameters, and the influence of soil brightness, the optimal selection of a vegetation index necessitates a demonstration of robust resistance to the fluctuations induced by these interfering elements. After identifying vegetation indices in this study, the subsequent conduct of sensitivity analysis on these vegetation indices can yield higher accuracy in LAI estimation. Because of comparative analysis of selected vegetation indices, LAI estimation can be carried out by fully leveraging the advantages of vegetation indices in providing a sensitive, quantitative representation of changes in vegetation growth status.

Simultaneously, at higher LAI values—specifically, in this study, when LAI reaches 6—the vegetation structure is likely to be already sufficiently dense, and leaf cover may be ample. Further increments in LAI may not lead to a significant rise in the vegetation index. During the estimation of LAI values, it is crucial to exclude this non-linearly growing curve area from being incorporated as part of the estimation model to ensure the accuracy and interpretability of the model. Meanwhile, when LAI is equal to 5, there is a certain decrease in the vegetation index value for each image, which goes against normal expectations. Typically, with an increase in vegetation density, the vegetation index should exhibit a linear growth. Due to the aforementioned saturation phenomenon, in subsequent estimations, the value corresponding to LAI=5 can be deemed an outlier and thus excluded from the analysis.

Compared to existing studies, Liang’s work involved the computation of 43 different vegetation indices [@liang2015]. The optimal vegetation indices obtained were SR, OSAVI, and MTVI2. In their subsequent sensitivity verification, the SR vegetation index was excluded. In Fu’s research, OSAVI was also identified as one of the well-performing vegetation indices [@fu2013]. Interestingly, this differs from the present study’s findings, where MTVI and TVI emerged as the optimal vegetation indices. In this paper, OSAVI performed well in both vegetation types, its performance was consistently below that of MTVI1. The significance of this study lies firstly in establishing the credibility of our results and, secondly, in emphasizing the importance of conducting sensitivity analyses on various factors. Moreover, the different optimal vegetation indices identified in our research underscore the importance of considering diverse factors and datasets in vegetation index selection. These nuances contribute to developing and refining scientific understanding in this domain. Expanding on these foundations, the next logical step would be to compare the estimated LAI values with actual measurements. Accomplishing this comparison will validate the feasibility of utilizing vegetation indices for estimating LAI values.

CONCLUSION

MTVI1 and TVI consistently receive the highest evaluation in both metrics, establishing it as the most suitable vegetation index in this study. Conversely, TCARI demonstrates the smallest $R^{2}$ value and simultaneously a large RMSE, indicating it is the least suitable vegetation index for LAI estimation in the simulated dataset of this research. Subsequently,sensitivity analysis on these indices may enhance accuracy of LAI estimation. Leveraging these vegetation indices, LAI estimation can be achieved with precision, showcasing the potential of selected indices in capturing dynamic vegetation changes.