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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

An extended Fourier approach was presented to improve the retrieved leaf area index (LAI_{r}) of herbaceous vegetation in a time series from an alpine wetland. The retrieval was performed from the Aqua MODIS 8-day composite surface reflectance product (MYD09Q1) from day of year (DOY) 97 to 297 using a look-up table (LUT) based inversion of a two-layer canopy reflectance model (ACRM). To reduce the uncertainty (the ACRM inversion is ill-posed), we used NDVI and NIR images to reduce the influence of the soil background and the priori information to constrain the range of sensitive ACRM parameters determined using the Sobol’s method. Even so the uncertainty caused the LAI_{r} _{r} results, obtaining LAI_{F}. We note that the level of precision of the LAI_{F} potentially may increase through removing singular points or decrease if the LAI_{r} data were too noisy. To further improve the precision level of the LAI_{r}, the Fourier model was extended by considering the LAI_{r} uncertainty. The LAI_{r}, the LAI simulated using the Fourier model, and the LAI simulated using the extended Fourier approach (LAI_{eF}) were validated through comparisons with the field measured LAI. The R^{2} values were 0.68, 0.67 and 0.72, the residual sums of squares (RSS) were 3.47, 3.42 and 3.15, and the root-mean-square errors (RMSE) were 0.31, 0.30 and 0.29, respectively, on DOY 177(early July 2011). In late August (DOY 233), the R^{2} values were 0.73, 0.77 and 0.79, the RSS values were 38.96, 29.25 and 27.48, and the RMSE values were 0.94, 0.81 and 0.78, respectively. The results demonstrate that the extended Fourier approach has the potential to increase the level of precision of estimates of the time varying LAI.

The use of remotely sensed optical data over large areas to quantitatively infer key biophysical and biochemical parameters of vegetation, such as the leaf area index (LAI), has been a popular application of remote sensing [

Vegetation parameters, including LAI, that are retrieved simultaneously may not meet the requirements of specific applications, such as the classification of vegetation types [

One way to smooth these curves is the data assimilation technique [

If the dataset is periodic, another way to smooth multi-temporal curves or solve missing data problems is the Fourier model [_{r}) of herbaceous vegetation in a one-year time series from an alpine wetland. The study includes three major steps. The LAIs from different time periods are retrieved using a two-layer canopy reflectance model (ACRM), the Aqua MODIS 8-day composite surface reflectance product (MYD09Q1) from DOY 97 to 297, and a look-up table (LUT) algorithm. The Fourier model is then fitted to simulate the variation of LAI values in the time series using the LAI_{r} and the least squares method. Finally, the Fourier model is extended to quantify the uncertainty in the LAI_{r}. Therefore, the precision level of the fitted LAI curve is improved. The LAI_{r}, fitted LAI, and the improved LAI are validated using the LAI values measured on DOY 177 (early July 2011) and 233 (late August). Details about the method are given next.

The inversion is by nature ill-posed [

The ACRM radiative transfer model is used [

With the Ǻith the turbidity equation [_{z}_{soil}_{i}_{1}_{2}

In addition, weight _{m}_{l}_{m}_{m}_{m}_{l}_{l}_{l}_{l}_{l}_{m}_{l}_{l}_{l}

The leaf optics model PROSPECT [_{ab}_{w}_{m}_{bp}_{w}_{bp}

The NDVI is generally sensitive to LAIs between 0 and 4.0 (_{r}, one can use NDVI images to retrieve the LAI when LAI < 4.0, and NIR images to retrieve the LAI of high values. Based on field measurement, the NDVI = 0.8 is selected as the threshold. When 0 < NDVI < 0.8, the LAI is retrieved by the NDVI, while 0.8≤NDVI < 1.0 the LAI is retrieved by NIR.

The sensitivities of different model parameters are different at RED and NIR wavelengths. To mitigate the ill-posed inversion problem, we set the sensitive parameters as free variables and the insensitive parameters as empirical values. Sobol’s method [

The main idea behind Sobol’s method for the computation of sensitivity indices is to decompose the function or model of _{1}_{n}_{1}_{n}_{0}^{n}_{i}_{i}_{i}_{i}_{i}_{1}_{n}_{Ti}_{i}

The LUT algorithmis generally computationally efficient [_{r} increases because some cases may not be true. For example, the LAI is always positive for vegetated surfaces. A zero or negative value of LAI in the LUT can cause large uncertainty in the LAI_{r}. The priori information has been shown to be useful in constraining the range of sensitive parameters.

The cost function that links the simulated and observed values is then established. The function is expressed as
_{j}^{*} is the _{i}_{i}_{j}^{*}. The retrieved result is generally in a range of LAI values with different frequencies. The mean LAI or the LAI with the maximum frequency can be used to represent the LAI_{r} if the retrieved LAI range is normally distributed. However, for the case of a non-normal distribution, it is more reliable to use the LAI with the maximum frequency (LAI_{max-fre}) to represent the LAI_{r}. It should be noted that more combinations of model parameters meet _{r} than for the LAI_{r} value closest to the given boundary. This indicates that the scheme of LAI retrieval used in this study should have higher precision for low and high LAI values than values in the middle of the range.

Due to the periodicity in the LAI chronological series, the LAI _{0}_{i}_{i}

The Fourier model coupled with polynomials (non-classical harmonic methods) can reduce the “roughness” in the fitting procedure [_{j}_{p,n}_{s,t}_{F}), and _{r,t} is the dataset of LAI_{r} on the _{r} dataset normally contains singular points. The fitting procedure can remove these points and improve the level of precision of the LAI_{r}. However, if the dataset contains large uncertainty, the fitting procedure may reduce the level of precision of the LAI_{r}.

Using the data assimilation technique, the uncertainty in the observed data can be taken into account to improve the simulation results. Thus, the uncertainty can be included in the fitting procedure to possibly improve the precision level of the LAI_{r} in the time series. Based on this consideration, _{r} as _{e}_{p,n,u}_{s,t}_{r} updated through the extension (LAI_{eF}), _{uncertainty,t}_{s,t}_{r} values in one year. Because the _{max-fre}_{r} (Section 2.1.4), the _{uncertainty}_{r,i}_{r,t}

The study area is located in the Wutumeiren prairie (latitude 36°46′ to 37°30′N, longitude 92°18′ to 93°24′E), Qinghai Province, China (^{2} and has an average elevation of 2900 m. The prairie is surrounded by the Gobi Desert. The Wutumeiren River is the main water source in the area. The climate includes annual rainy and dry seasons; the rainy season is normally from July to August, and the rest of the year is dry. The soil is characterized by high salinity and alkalinity. Thus, only a few types of adaptable and strong vegetation can grow. The vegetation is primarily composed of reeds. Due to variations in the soil moisture content, the reeds are clumped in some areas and are homogenous in others. Mixtures of grasses, shrubs, and trees are located at the edges of the prairie. They were planted to prevent the spread of the desert into the prairie. The edges of the prairie are not included in the study.

Thirty MYD09Q1 images(

The parameters LAI, _{L}_{z}_{m}_{L}_{z}_{m}_{ab}_{L}_{m}_{l}_{L}_{z}_{L}_{z}_{ab}

Based on the qualitative and quantitative priori information acquired from the field measurements, empirical values were used to determine the parameters that were insensitive to NIR and RED wavelengths (_{z}_{ab}_{z}_{ab}_{ab}_{ab}_{s}_{v}_{raz}

The NDVI images from DOY 97 to 177 and from DOY 265 to 329 (the arid periods) were used to retrieve relatively lower LAI values, and both the NIR and NDVI images from DOY 185 to 257 were used to retrieve relatively higher LAI values to alleviate the influence of the soil background (Section 2.1.2). As stated in Section 2.1.4, LAI_{max-fre} was used as the LAI_{r}, but it may be not reliable at low values of _{mean} was 0.49, whereas LAI_{max-fre} was 0. This singular case was avoided in _{r} values would increase as well (_{mean} or LAI_{max-fre} can be regarded as the LAI_{r}. However, if the LAI was near the boundaries of the given range, LAI_{mean} and LAI_{max-fre} might differ greatly. For example, in _{mean} = 6.14, whereas LAI_{max-fre} = 7.0 because the LAI distribution was no longer normally distributed. In this case, it was more reliable to set LAI_{max-fre} as the LAI_{r}, as was assumed in Section 2.1.4. Therefore, in the study, _{max-fre} was set as the LAI_{r}.

_{r} and ground-measured LAI (LAI_{m}) values from July. The dashed line is the 1:1 line, and the solid line is the least-squares regression line. Of 38 points, the residual sum of squares (RSS), coefficient of determination(R^{2}), and root-mean-square error (RMSE) values were 3.47, 0.68, and 0.31, respectively. Because the points were clustered at low LAI values (≤1.0), the RMSE value of 0.31 could signal a low level of precision of the retrieval. In August (_{r} and LAI_{m} points at high and low LAI values were less scattered than those in the middle range of LAI values. In other words, the LAI retrieval procedure might be more sensitive at the low and high ends of the LAI values. Of 46 points, the RSS, R^{2}, and RMSE values were 38.96, 0.73, and 0.94, respectively. The higher RSS and RMSE values in August compared to those in July could be caused by the increased LAI values as the reeds grew during this part of the year. This interpretation was supported by the high measured LAI values (_{r} values had large uncertainty in the retrieval; this is illustrated in _{r} values in the study period and from three pixels (locations) decrease from (a) to (b) to (c). Therefore, the LAI_{r} values were smoothed to attempt to improve the precision level of LAI_{r}.

As previously discussed, the Fourier model can remove singular points in time series. For example, in _{r} values were then smoothed using the polynomial included in the Fourier model (_{F}) and LAI_{m} were then compared again. Of the 38 points from July (^{2}, and RMSE values were 3.42, 0.67, and 0.3, respectively. Of the 46 points from August (^{2}, and RMSE values were 29.25, 0.77, and 0.81, respectively. Compared to the July data, the precision of the LAI_{r} assessed by the RSS, R^{2}, and RMSE values were similar before and after the Fourier model smoothing. No meaningful improvement in the precision was achieved for the LAI_{r} values. The precision level of the August LAI_{r} data might have improved after the Fourier smoothing based on the decrease in the RSS and RSME values and the increase in R^{2} value compared with the values before the smoothing, but the improvement was slight. The extended Fourier approach was investigated next to determine whether the approach could increase the precision level in a meaningful way.

As shown in _{eF}), but the shapes of the LAI_{F} and LAI_{eF} curves were different. The LAI_{eF} curves were farther away from the LAI_{r} points with higher uncertainty and were closer to the LAI_{r} points with lower uncertainty. For example, in _{r} value had a much lower uncertainty on DOY 217 than on DOY 225. The LAI_{eF} curve was similar to the LAI_{r} value on DOY 217 but far from that on DOY 225. _{F} curves did not. The LAI_{eF} and LAI_{m} values were then compared again (^{2}, and RMSE values were 3.15, 0.72, and 0.29, respectively. For the 46 points in August (^{2}, and RMSE values were 27.48, 0.79, and 0.78, respectively. The precision level of the LAI_{eF} values was improved compared with the LAI_{r} and LAI_{F} (_{r} curves in the time series but also improved the precision level of the LAI_{r} values by taking into account the uncertainty in the LAI_{r}.

The Fourier model can be used to simulate the variation of LAI values in a time series. The Fourier model includes 11 free variables, which are fitted by the LAI_{r} at different periods and the least squares method. However, the set of free variables in the Fourier model is much smaller than that in physical dynamic models, such as crop growth models. Physical dynamic models can better simulate the growth of crops because they are structured based on the physical mechanisms of vegetation growth. However, more work needs to be done on the selection of the appropriate model to simulate or approximate the growth of wild plants in alpine wetlands and to determine the large sets of input parameters. Therefore, it is infeasible to use a physical dynamic model to improve LAI estimates.

The Fourier model has been widely used to smooth the curves of parameters that are periodic. However, the model might not improve the precision level of the dataset. In this study, the Fourier model used in the fitting process was extended by taking into account the uncertainty in the LAI_{r} values. As explained in Section 4.4, the improved fitted curves will be close to the points that have lower uncertainty and avoid points with larger uncertainty. The LAI_{r} will have the best results and the precision level should be improved. As in the Fourier model, the extended Fourier approach can be used if the dataset is periodic, but the model will improve the precision of the dataset only if the uncertainty in the dataset is expressed correctly.

The LAI_{uncertainty} imported in _{r} values because the uncertainty is affected by many factors, such as cloud cover, the LAI retrieval algorithm, and the ill-posed inversion problem.

Characterized by high altitude and cold and dry weather, the alpine wetland is extremely fragile and vulnerable. The vegetation is typically composed of a single type of annual herb. In this study, an extended Fourier approach was presented to improve the LAI_{r} values in a 2011 time series from an alpine wetland located in western China. The LAI was retrieved on DOYs based on the ACRM, the MYD09Q1 product from DOY 97 to 297, and the LUT algorithm. To alleviate the ill-posed inversion problem, three strategies were implemented. (I) To reduce the influence of the soil background, the NDVI and NIR reflectance of wavebands were used to retrieve the LAI for sparse vegetation and dense vegetation, respectively;(II) A sensitivity analysis of the ACRM in the RED and NIR wavelengths was performed to determine the free parameters using Sobol’s method;(III) The priori information was imported to constrain the range of free parameters in the ACRM. The LAI_{r} in the time series did not form smooth curves due to the effects of the ill-posed inversion problem. A normalized Fourier model with two additional polynomials was used to smooth the LAI_{r} curves in the time series because of the periodicity of the vegetation in the study area. The Fourier model is able toimprove the level of precision of the LAI_{r} if singular points are present, but it may also reduce the level of precision if the datasets that the model is based on contain large uncertainty. An extended Fourier approach was presented by taking into account the uncertainty in the LAI_{r} in the Fourier model to improve the level of precision of the LAI_{r}. The LAI_{r}, LAI_{F}, and LAI_{eF} values were validated through a comparison with field measured LAI (LAI_{m}) values, which resulted in R^{2} values of 0.68, 0.67 and 0.72, RSS values of 3.47, 3.42 and 3.15, and RMSE values of 0.31, 0.30 and 0.29, respectively, on DOY 177 (early July). On DOY 233 (late August), the R^{2} values were 0.73, 0.77 and 0.79, the RSS values were 38.96, 29.25 and 27.48, and the RMSE values were 0.94, 0.81 and 0.78, respectively. The results demonstrated that this approach has the potential to improve the level of precision of periodically varying parameters in a time series.

This work was funded by the Fundamental Research Funds for the Central Universities (No. ZYGX2012Z005) and the National High-Tech Research and Development Program of China (863 Program, No. 2013AA12A302). The authors are grateful to Yongshuai Yan, Bo Zhang, Hongzhang Dai, Xiaojing Bai and Ningning Wang for their assistance during the field campaigns.

XingwenQuan designed and performed experiments, analysed data, and wrote the manuscript; Binbin He gave comments and suggestions to the manuscript and performed the experiments; Yong Wang commented the manuscript and checked the writing; Zhi Tang and Xing Li performed part of the experiments.

The authors declare no conflict of interest.

Sensitivity of normalized difference vegetation index (NDVI) to leaf area index (LAI) (

The study area (Wutumeiren prairie). The color composite Landsat5 image is TM4 (red), TM3 (green), and TM2 (blue). Green points represent the sampling plots from 6 to 9 July 2011, and the yellow points represent sampling plots between 26 and 29 August 2011.

Sensitivity analysis of key input parameters of ACRM at near-infrared (NIR) and RED wavebands using Sobol’s method. TSI stands for total sensitivity index.

Distribution of LAI_{r}. Because of the ill-posed inversion problem, LAI_{r} is not a single value but a wide range of different frequencies.

LAI_{r} compared to the field-measured LAI on DOY 177 (July) (

LAI_{r} _{r}, and the error bars on each DOY represent _{uncertainty}_{r} of vegetation near Wutumeiren river; (_{r} of vegetation in drought area. The peak LAI_{r} values in the study period decrease from (

Comparison of the LAI_{r} and LAI_{F} for three pixels from dense vegetation to sparse vegetation in the time series. (_{r} of vegetation near Wutumeiren river; (_{r} of vegetation in drought area.

Results of LAI_{F} compared to LAI_{r} in early July (

Comparison between the LAI_{r}, LAI_{F}, and LAI_{eF} values of three pixels from dense vegetation to sparse vegetation in the time series. (_{r} of vegetation near Wutumeiren river; (_{r} of vegetation in drought area.

Results of the LAIe_{F} values compared to the LAI_{m} values in early July (

Inputs to the ACRM.

Sun zenith angle | (°) | _{s} |
- |

View zenith angle | (°) | _{v} |
- |

Relative azimuth angle | (°) | _{raz} |
- |

Ǻngström turbidity coefficient | 0.12 | ||

Leaf area index | m^{2}/m^{2} |
LAI | / |

LAI of ground level | m^{2}/m^{2} |
LAI_{g} |
0.05 |

Mean leaf angle of Elliptical LAD | (°) | _{l} |
60.0 |

Hot spot parameter | _{L} |
0.5/LAI | |

Markov clumping parameter | _{z} |
/ | |

Refractive index | 0.9 | ||

Weight of the first basis function | / | ||

Leaf mesophyll structure | / | ||

Chlorophyll |
μg·cm^{−2} |
_{ab} |
/ |

Leaf equivalent water thickness | cm | _{W} |
0.015 |

Dry matter content | g·m^{−2} |
_{m} |
90 |

Brown pigment | _{bp} |
0.4 |

Ranges and steps (dx) of sensitive parameters in the ACRM for the NDVI and NIR image-based retrieval schemes.

LAI | 0–4 | 0.2 | 2–7 | 0.2 |

_{z} |
0.4–1.0 | 0.2 | 0.4–1.0 | 0.2 |

0.25–0.5 | 0.03 | 0.05–0.25 | 0.03 | |

1.0–2.0 | 0.2 | 1.5–2.5 | 0.1 | |

_{ab} |
30–90 | 20 | 60 | 0 |

Levels of precision of LAI_{r}, LAI_{F}, and LAI_{eF} compared to LAI_{m} on DOY 177 and 233.

DOY 177 | LAI_{r} |
3.47 | 0.68 | 0.31 |

LAI_{F} |
3.42 | 0.67 | 0.30 | |

LAI_{eF} |
3.15 | 0.72 | 0.29 | |

| ||||

DOY 233 | LAI_{r} |
38.96 | 0.73 | 0.94 |

LAI_{F} |
29.25 | 0.77 | 0.81 | |

LAI_{eF} |
27.48 | 0.79 | 0.78 |