1. Introduction
Leaf area index (LAI) is defined as half of the total leaf area per ground area [
1]. It is an important input parameter in land biogeochemical and biophysical ecosystems [
2,
3]. A variety of global LAI products have been produced from satellite data acquired by the advanced very high resolution radiometer (AVHRR) [
4,
5], the moderate-resolution imaging spectroradiometer (MODIS) [
6,
7], VEGETATION [
8,
9], and the multiangle imaging spectroradiometer (MISR) [
10], and so forth. However, the existing global LAI products’ spatial resolutions are medium or low and need to be improved. For example, the spatial resolution of the CYCLOPES LAI derived from VEGETATION and the global land surface satellite LAI (GLASS LAI) derived from MODIS are 1/112° and 1000-m, respectively. The spatial resolution of the MODIS LAI is 500-m (MCD15A2H, Version 6). Although these LAI products are globally available, the spatial resolutions of the global LAI products are coarse. This makes them mostly appropriate to global and regional studies [
11,
12,
13]. High-resolution LAI products are needed to monitor crop growth and study vegetation parameters and canopy structure at small scales.
Quantitative inversion methods for high-resolution LAI can be classified as the empirical approach, physical approach, and the mixed-model methods [
14,
15]. For these quantitative methods, the training samples and the inversion algorithms are the most important parts. Chen et al. [
16] established the relationships between the vegetation indices from high-resolution satellite imagery and ground-based LAI to improve the LAI inversion algorithms and validate Canada-wide LAI map products. Collecting training samples from field measurements is difficult and the measurements are often not sufficient. Duan et al. [
17] retrieved the LAI of three typical row crops (maize, potato, and sunflower) using a look-up table (LUT) based on the inversion of the PROSAIL model; the results indicated that the LUT-based inversion of the PROSAIL model was suitable for LAI estimation of these three crops. Some studies obtained training samples through radiative transfer model simulation, which needs many input parameters and the inverse problem is ill-posed [
18]. Currently, some global LAI products can provide more information regarding land surface vegetation. Many researchers have carried out studies based on these global LAI products [
19,
20]. At the same time, some research works show that various global LAI products can be used to choose representative training samples. For example, Xiao et al. [
7] developed a method to estimate GLASS LAI based on the MODIS and CYCLOPES LAI products using general regression neural networks (GRNNs). In this method, the MODIS and CYCLOPES LAI products were selected as training samples to train the GRNNs. However, the GLASS LAI products still have low spatial resolution, at 1000-m. In order to retrieve high-resolution LAI, Gao et al. [
21] used the MODIS LAI products as a reference to retrieve LAI from Landsat imagery. Results showed that the approach could produce accurate estimates of Landsat LAI for major crops. Nowadays, similar studies on generating training samples from existing global LAI products number too few, but it is a meaningful and important research for high-resolution LAI retrieval.
On the other hand, the inversion algorithm is also an important focus. There are many inversion algorithms for the inversion of LAI, such as the traditional regression model algorithms, which describe the linear or nonlinear relationships of LAI with surface reflectance or the derived vegetation indices (VIs). Vina et al. [
22] and Wang et al. [
23] both used mathematical statistical approaches to build the LAI–VI relationships to estimate LAI. However, mathematical statistical approaches have poor fitting ability and encounter difficulty in solving multidimensional problems. Chai et al. [
24] constructed recurrent neural networks by fusing the MODIS and VEGETATION products to estimate time series LAI. Although the results showed that the method can be helpful to improve the quality of LAI products of the typical vegetation types, the neural networks are too sensitive to the parameters in models [
25] and cannot overcome the phenomena of “over-learning” and “local minimum” [
26], which would affect the precision of the estimation. Support vector regression (SVR) is a machine-learning method which has a strong nonlinear fitting capability, and the kernel function can solve high-dimensional problems [
27]. Durbha et al. [
18] adopted a one-dimensional canopy reflectance model (PROSAIL) to retrieve LAI from MISR data using an SVR algorithm, and proposed a kernel-based regularization method to improve the SVR algorithm to solve the ill-posed problem.
The objective of this paper is to improve the high-resolution LAI retrieval approach using the SVR algorithm, especially focusing on the methods whereby to obtain high-quality training samples from the MODIS products. Three different sampling datasets for two methods were compared over two experimental sites. One site is located in the Heibei Province, China. The other site is located in central Iowa, U.S.
The second section introduces the SVR inversion algorithm and different high-quality training sample-acquiring methods. The third section introduces the research area and test data. The fourth section describes the experimental process and the results, followed by analysis and discussion.
2. Methodology
The empirical approach built by the SVR algorithm was developed in this research. Two training sample-selecting methods were developed to extract three different sampling datasets for the study. Namely, the MODIS reflectance products with MODIS LAI, the upscaled Landsat reflectance at the 500-m resolution with MODIS LAI, and unmixed MODIS reflectance and LAI at 30-m resolution were used as SVR training samples to retrieve Landsat LAI at the 30-m resolution from different periods.
Figure 1 shows the data processing framework. The high-quality training samples of SVR were selected by two methods: the homogeneous and pure pixel filter method (method A) and the pixel unmixing method (method B). For the homogeneous and pure pixel filter method, after the homogeneous and pure MODIS LAI pixels were selected, the upscaled Landsat reflectance (aggregate from 30-m to the 500-m resolution in the ENVI software, using a simple average method) of the corresponding pixels (the same location with the above homogeneous and pure MODIS LAI pixels) together with the LAI were selected as the training samples (dataset A1). Next, the corresponding MODIS reflectance of the pixels together with the LAI were also selected as the training samples (dataset A2). The pixel unmixing method took the unmixed LAI after quality control and reflectance of agricultural land pixels at the 30-m resolution as training samples (dataset B), which were obtained through the unmixing of MODIS LAI and reflectance products by linear models. The SVR models trained by the above three sets of training samples for the two methods were then applied to the Landsat surface reflectance to generate the 30-m resolution LAI. The retrieved LAI maps at the 30-m resolution and the temporal trends curves were generated and analyzed. The retrieval LAI results were then compared to the field measurements.
2.1. Methods of Training Sample Selection
2.1.1. The Homogeneous and Pure Pixel Filter Method
The homogeneous and pure pixel filter method chose the high-quality MODIS LAI pixels by three steps: (1) masking the LAI product images of the research area using MODIS land-cover products (MCD12Q1) to obtain the agricultural land area. According to the International Geosphere-–Biosphere Programme (IGBP) global vegetation type classification dataset (Land Cover Type 1), the pixels with the successive agricultural land type in every year during the research period are considered as the agricultural land area; (2) selecting the high-quality LAI pixels using quality control files (MODIS LAI QC layer) to ensure that the pixels’ LAI are retrieved from the main algorithm, namely, the look-up table (LUT) algorithm. Through the first two steps, we obtain high-quality LAI pixels of agricultural land, and these pixels are used as the input pixels of the third step: (3) filtering the remaining LAI pixels by the coefficients of variation (CV) to select homogeneous and pure pixels.
It is known that the relationship between the LAI and the spectral reflectance or vegetation index is nonlinear. The relationships vary for pure and mixed pixels at low spatial resolution. To model the relationship between the LAI and the spectral reflectance for different vegetation types, pure pixels should be selected as training samples. The spectral reflectance of different objects in a mixed pixel is different. We use the CV (ratio of standard deviation to the mean value) in a statistical model to represent this difference, similarly to in [
21].
The CV formula is as follows:
where CV is the coefficient of variation and
σ and
μ are the standard deviation and mean of the surface reflectance of the Landsat pixels within a MODIS pixel, respectively.
The CV of the reflectance of each band was calculated for all Landsat pixels at the MODIS pixel scale, and a threshold was determined based on the pixels’ quality and quantity. It is assumed that the MODIS pixels are pure pixels when the CV is small.
When the homogeneous and pure pixels of the MODIS LAI products were selected, the corresponding MODIS surface reflectances were also chosen together to act as the training samples (dataset A2). The MODIS reflectance products have a good correspondence with the MODIS LAI products because of having the same temporal and spatial resolution, with no geometric deviation. In addition, the upscaled Landsat reflectances at the 500-m resolution were aggregated to create another training sample (dataset A1), which allows comparison of these retrieved high-resolution LAI of the training samples for the different selection methods.
2.1.2. The Pixel Unmixing Method
At 500-m resolution, different land cover types may be mixed within a MODIS pixel. The surface reflectance is determined by the spectral characteristics of the different types. Pixel unmixing methods can decompose the pixels of coarse-resolution satellite imagery into different endmembers using high-resolution satellite imagery classification data. These methods can thus obtain the ratio of each type to the coarse-resolution pixel (as a proportion). Ichoku et al. summarized five mixture models. These models are the linear model, the probabilistic model, the geometric-optical model, the stochastic geometric model, and the fuzzy model [
28]. In this study, the linear model was used to unmix the MODIS surface reflectance and MODIS LAI. The linear model is a special case of a nonlinear model that ignores multiple scattering [
29]. In the linear model, it is assumed that the reflectance of a pixel is a linear combination of the reflectance of each endmember [
30]. The weight of the reflectance of the feature type is determined by the ratio of each type to the area of the pixel:
where
is the reflectance of the mixed pixel;
is the reflectance of the j
th endmember;
is the proportion of the j
th endmember in the mixed pixel, which can be calculated based on the 30-m land cover map from the Landsat image; and
ε is the error. In Equation (2),
R and
are known, and the unknown variable is the
. In the study, we assume that the neighboring pixels with the same land cover type have similar surface reflectance or LAI [
31]. When the number of endmembers is determined, the number of equations must be greater than or equal to the number of endmembers (
) in order to solve the equation. By using other pixels adjacent to the pixel being processed, one can avoid the ill-conditioning problem caused by too many unknown values, and this can be achieved by sliding the 3 × 3 window [
32]. For each mixed pixel, a maximum of nine equations can be derived from the 3 × 3 window, which traverses the entire study area. Each equation group was solved by the constrained least squares method.
The unmixed pixels of MODIS LAI and surface reflectance products at the 30-m resolution using the above method was obtained. Then, choose the pixels of agricultural land cover type and the LAI retrieved from the main algorithm using quality control. Finally, the unmixed MODIS surface reflectance and LAI of the above chosen corresponding pixels were taken as the training samples for SVR.
2.2. Support Vector Regression
Support vector regression (SVR) is a supervised machine-learning method based on the principles of the Vapnik–Chervonenkis dimension theory and structural risk minimization [
33]. SVR is more suitable for small samples and nonlinear, high-dimensional problems. SVR aims to construct an optimal super-pipe so that the pipeline can provide as much data as possible under a given accuracy ε, and so that the distance from the sample point to the pipe edge is not larger than ε [
34]. This can be expressed by Equation (3):
where
is the normal vector (or the “support vector”),
stands for the dataset, and
is the bias.
The loss function in ε-SVR can be expressed as Equation (4):
Operating on the basis of the structural risk minimization theory, SVR finally evolves into a convex optimization problem. Here, the slack variables
and
are introduced to represent the fitting error:
where
is the margin parameter.
SVR has the advantage of solving the nonlinear problem by introducing a kernel function, which is a good solution to the problem in high-dimensional space and is the core of SVR. The radial basis function (RBF) has been found to be superior to other kernel functions for reflecting the nonlinear relationship between LAI and reflectance. Therefore, the RBF was used in this research. After that, the choice of hyperparameters determines the quality of the model, which affects the quality of the SVR algorithm. The range of hyperparameters and kernel parameters was [–10, 10], and the six-fold cross-validation method was used to find the optimal parameters. As is well known, the higher the proportion of training samples, the closer the inversion results will be to the measured values. Next, the proportion of training samples was fixed at 80% for optimal parameter determination.
Three inversion models were built through training the SVR algorithm using the three training samples datasets obtained by the homogeneous and pure pixel filter method (datasets A1 and A2, method A) and the pixel unmixing method (dataset B, method B), both of which were described in detail in
Section 2.1. In the inversion process, Landsat surface reflectance was used as the input to the three SVR inversion models to retrieve high-resolution LAI at 30-m resolution.
5. Discussion
A comparative analysis of accuracy between the LAI retrievals and the field measurements (or MODIS LAI) was performed for the homogeneous and pure pixel filter method (method A) and the pixel unmixing method (method B). The results demonstrated that inversion of high-resolution LAI combining MODIS products based on the SVR algorithm was feasible. Compared to physical modeling methods, the empirical model using the SVR algorithm did not need to consider the physiological characteristics of vegetation and was easy to implement. Compared to other methods of obtaining training samples, our methods can obtain a greater number of high-quality samples based on global LAI products.
Gao et al. [
21] used the decision tree with the upscaled Landsat reflectance at the 500-m resolution and MODIS LAI to retrieve LAI at the 30-m resolution. The comparison (R
2 = 0.79, mean bias error = −0.18, mean absolute difference = 0.58) between Landsat retrievals (30-m) of Gao’s and field measurements at the field scale (10-m) showed that there was a good agreement with low to moderate LAI (0–3), but retrievals were underestimated for high LAI (3–5). In this paper, the R
2 was 0.79 and the RMSE was 0.73 for the homogeneous and pure pixel filter method when the MODIS LAI and upscaled Landsat reflectance at the 500-m resolution were used as the training samples (dataset A1). The R
2 was 0.81 and RMSE was 0.69 for the homogeneous and pure pixel filter method when the MODIS LAI and reflectance were used as the training samples (dataset A2). In addition, the retrievals of the unmixing method with unmixed MODIS reflectance and LAI at the 30-m resolution as the training samples (dataset B) were slightly higher, with an R
2 value of 0.82 and an RMSE value of 0.65. Compared with Gao et al., we selected the multiyear MODIS products for obtaining training samples to ensure the richness and representativeness of the samples. In addition, the MODIS LAI and reflectance products were also used as training samples to build the SVR inversion model, and yielded a good result. Moreover, the pixel unmixing method was used to obtain the SVR inversion model, and resulted in the highest accuracy.
In particular, the SVR algorithm had an advantage in solving the nonlinear problem because it overcomes the phenomena of “over-learning” and “under-learning” [
41]. After the kernel function and the hyperparameters were chosen, the relationship between the MODIS LAI and surface reflectance was fitted. It was shown that the SVR algorithm can represent the relationship between LAI and reflectance and had good generalization ability.
Moreover, the quality of the training samples, including sample distribution, can seriously affect the quality of all empirical approaches, including the SVR. In this study, the training samples spanned multiple years and included the whole crop-growing season. Therefore, the data quality was good and representative. The homogeneous and pure pixel filter method took the quality control file (QC layer) of the MODIS LAI products into consideration and ensured the quality of the MODIS LAI. This was the precondition for taking the MODIS LAI as training samples. The ratio of the standard deviation to the mean value (CV) in a statistical model can represent the difference of spectral reflectance of different objects in a mixed pixel. The MODIS pixels can be considered homogeneous and pure pixels when the thresholds of the QC and CV settle within a certain range. The pixel unmixing method decomposed the pixels of coarse-resolution satellite imagery into different endmembers and obtained the high-resolution LAI (or reflectance). It can obtain higher-quality training samples than other methods.
However, there were also a few limitations to be improved upon. First, when the LAI was greater than 3, the inversion results were underestimated during July. A main reason for this was that the relationship between the surface reflectance and LAI tended to become saturated when the LAI was greater than 3. In this paper, when the MODIS LAI is high, the probability of using the main algorithm is low. In other words, if the training samples were selected by the main algorithm, many samples with LAI > 3 were eliminated, resulting in fewer training samples with high LAI values.
Figure 10 shows the histogram of LAI from the LAI-SR samples of SMEX02. As we can see from the histogram, there are fewer large LAI values (>3.0). In the SMEX02, there are 21,500 LAI-SR samples for dataset A1; 2028 for dataset A2, using the homogeneous and pure pixel filter method; and 17,889 for the pixel unmixing method. In Baoding, the LAI-SR samples are 6909, 8797, and 8838, respectively. In addition, the scaling effect may cause the under-representation of high LAI values. The high values can be smoothed out in the coarse-resolution image. For both training sample-acquiring methods, we used the MODIS quality control flags to select the highest-quality retrievals derived from the MODIS LAI main algorithm. We relied on the MODIS LAI data quality flags and have not considered the effect of noise associated with the main algorithm. The noise from the main algorithm retrieval may need to be considered for other regions, such as the tropical area, where clouds are always present.
Furthermore, when the upscaled Landsat reflectance and MODIS LAI were used to build the relational model, geometrical registration between Landsat and MODIS was not performed in this study. This led to a bias when calculating the CV of the Landsat surface reflectance in a MODIS pixel. The CV threshold of the homogeneous MODIS pixel was determined based on LAI sample quality and quantity from subjective experience at present. The threshold may vary with different study sites or landscapes. In order to compare the inversion results from these two sites in the paper, the same threshold of 0.15 was used. Additional study and analysis are needed to quantify the threshold. In this paper, we used the eight-day MODIS LAI products. This means that the MODIS LAI product for the period was made up from eight independent days. However, the Landsat imagery reflected the instantaneous optical characteristics of vegetation for the Landsat acquisition date. Therefore, differences between the Landsat and MODIS data products were observed in time and space.
Note that the spectral response function of the sensor is different, and the wavelengths of the corresponding bands are also different. The spectral response function of the sensor is a function of the wavelength. It is the ratio of the radiance received by the sensor at each wavelength to the radiance of the incident. In this study, we compared the bands of Landsat and MODIS and chose the three bands with small difference. Although the relational model was built using MODIS reflectance and MODIS LAI, the Landsat reflectance was used as the input parameter to retrieve the LAI. There exists a large gap in the wavelength range, particularly in the near-infrared band4 of Landsat5. Thus, the use of a spectral response function to convert the Landsat reflectance and MODIS reflectance will be the next key work.
Finally, an empirical model has advantages, but also limitations. In this study, the verification experiments were conducted over two study areas: the Hebei Province, China and Des Moines, Iowa, United States. Crops included corn, soybean, and winter wheat. Although the inversion model proposed here achieved good results in the study area, applications in other areas need to be further verified. To apply the model to large regions, not only must the method of screening high-quality data be improved, but the biophysical mechanisms of vegetation must also be studied. In the present study, the relationship between LAI and reflectance had a certain scope of application. Moreover, inversion accuracy was high in the main growth period, but was reduced in other periods, especially in the late stages of crop growth. Climate change may impact vegetation growth. The accuracy of LAI retrieval may be decreased if LAI-SR samples cannot cover this variation. In our study sites, climate change from the study period is not significant to enable testing of this hypothesis. Future research may consider different inversion models at different times.