# Remote Sensing of Fractional Green Vegetation Cover Using Spatially-Interpolated Endmembers

^{*}

## Abstract

**:**

## 1. Introduction

_{i}is the VI value of pixel i, VI

_{s}is the VI of bare soil or dead vegetation pixels in the image (i.e., no green vegetation cover), and VI

_{v}is the VI of fully-vegetated pixels. Appropriate VI

_{s}and VI

_{v}endmembers are typically obtained from previous studies or soil databases [5,7], or directly from the imagery using various techniques [4,5,7]. Many studies estimate FVC under the assumption that VI

_{s}and/or VI

_{v}are invariant throughout an image (e.g., [3,4,7,8]). However, this assumption is often invalid [5] because differences in soil composition, grain size, and moisture content can cause the spectral characteristics of soil to vary [20], and differences in vegetation species, leaf water content, etc. can cause the spectral characteristics of green vegetation to vary [21]. In areas with rugged terrain, the effects of topography (i.e., differences in solar illumination on slopes oriented towards and away from the sun) also have an impact on spectral reflectance values of soil and green vegetation [22].

_{s}, a popular approach has been to use vegetation indices less sensitive to soil brightness than NDVI, such as the Soil Adjusted Vegetation Index (SAVI) [23]. In an alternative approach, [5] used a global soil reflectance database (2,906 soil reflectance samples), computed many possible FVC values for each pixel using different NDVI

_{s}values from the soil database (i.e., using all soil samples with NDVI values lower than the NDVI of the pixel), and assigned the mean FVC of these calculations to the pixel since it was the statistically most-likely. This method was able to increase the accuracy of FVC estimates for the continental United States, but for studies conducted at finer scales the majority of the soils in a global database may not be present, so the approach lacks a physical basis for these fine-scale studies.

_{v}, some studies have used land cover maps to derive separate VI

_{v}values for each type of vegetation [4,24]. This approach provides a way to account for some of the variability in VI

_{v}, but: (i) land cover maps of the desired classification accuracy or spatial resolution may not always be available, and (ii) it does not take into consideration the local climate and soil conditions which can cause spectral characteristics even at the species level to vary across space [25]. Thus, VI

_{v}values may still vary across space even if they are defined for each type of vegetation. In addition, it is possible that VI

_{s}may vary as well due to local differences in soil composition and soil moisture. For dealing with the positive spatial autocorrelation likely to exist in VI

_{s}and VI

_{v}, [10] suggest that subdividing the landscape into several units and developing/applying separate unmixing models for each unit may produce better results, but determining the number and size of the units for dividing the landscape would likely be difficult and subjective. On the other hand, a less subjective approach that also takes into account the likely spatial autocorrelation in VI

_{s}and VI

_{v}is to use spatial interpolation to predict the variation in VI

_{s}and VI

_{v}across space. Such an approach was developed in this paper.

_{s}and VI

_{v}in an image. Two interpolation techniques, Inverse Distance Weighting (IDW) and Ordinary Kriging (OK), were tested. The basic approach is as follows: (i) VI

_{s}and VI

_{v}values at sample locations were extracted from the imagery (i.e., from manually-selected “bare soil” and “full green vegetation” endmember pixels), (ii) VI

_{s}and VI

_{v}values at all other pixel locations were predicted by spatially interpolating the VI

_{s}and VI

_{v}values from the endmember pixels, and (iii) interpolated VI

_{s}and VI

_{v}replaced the VI

_{s}and VI

_{v}values used in the traditional linear FVC calculation shown in equation 1. The theoretical basis for the proposed approach is that, at a given location, the spectral characteristics of soil are likely to be more similar to nearer “bare soil” sample endmember pixels and the spectral characteristics of vegetation are likely to be more similar to nearer “full green vegetation” sample endmember pixels due to local similarities in soil and vegetation composition, moisture, etc. Therefore, local calculations of VI

_{s}and VI

_{v}should provide better estimates of the expected endmember values at each pixel location than globally-invariant VI

_{s}and VI

_{v}values. Two well-established vegetation indices, the Normalized Differential Vegetation Index (NDVI) [26] and the Modified Soil Adjusted Vegetation Index (MSAVI) [27], were used for the FVC calculations. Very high spatial resolution imagery was used for assessing the accuracy of FVC estimates, and FVC estimates calculated using the spatially interpolated VI

_{s}and VI

_{v}approach were compared with estimates calculated using the traditional approach (i.e., with invariant VI

_{s}and VI

_{v}values) to assess the impact that the interpolation approach had on the accuracy of estimates. The accuracy of FVC estimates was assessed for an entire validation dataset (100 validation samples), and also for edge- and non-edge validation samples separately in order to provide a broad view of the spatial distribution of FVC estimation errors in a patchy landscape (i.e., a landscape in which large changes in vegetation cover can occur over short distances).

## 2. Study Area and Data

## 3. Methods

#### 3.1. Atmospheric and Geometric Correction

#### 3.2. Calculation of Vegetation Indices

#### 3.3. Calculation of VI_{s} and VI_{v} at Each Sample Location

_{s}and MSAVI

_{s}or NDVI

_{v}and MSAVI

_{v}values (depending on whether the sample was of gv_0 or gv_1) for that location. To determine if the VI

_{s}and VI

_{v}values exhibited positive spatial autocorrelation for either vegetation index, we computed Global Moran’s I (MI) [34], a metric that calculates, on average, how similar a sample is to other nearby samples. MI was calculated as

_{ij}is a measure of spatial proximity (measured using IDW), y

_{i}is the value of sample i, and ȳ is the mean value of all of the samples. $MI\hspace{0.17em}\hspace{0.17em}\ge \hspace{0.17em}\hspace{0.17em}\frac{-1}{\text{n}-1}$ indicates positive spatial autocorrelation, and $MI\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\frac{-1}{\text{n}-1}$ indicates negative spatial autocorrelation. In addition to MI, p values, which give the confidence level for the MI calculations, were also calculated (e.g., $MI\hspace{0.17em}\hspace{0.17em}\ge \hspace{0.17em}\hspace{0.17em}\frac{-1}{\text{n}-1}$ and p < 0.05 indicates positive spatial autocorrelation at a 95% confidence level). As shown in Table 1, NDVI

_{s}and MSAVI

_{s}both showed strong positive spatial autocorrelation at a 95% confidence level, while NDVI

_{v}and MSAVI

_{v}showed moderate positive spatial autocorrelation for both indices at a 90% (for NDVI

_{v}) and 80% (for MSAVI

_{v}) confidence level. It is likely that the lower spatial autocorrelation in NDVI

_{v}and MSAVI

_{v}were due to the fact that, in some cases, nearby gv_1 sample locations contained different types of vegetation with quite different spectral characteristics, resulting in low positive (or even negative) local spatial autocorrelation for these samples. Spectral changes in soil, on the other hand, were more gradual, resulting in a higher degree of spatial autocorrelation for gv_0 samples. The fact that all MI calculations for VI

_{s}and VI

_{v}showed positive spatial autocorrelation provides supporting evidence that it may be preferable to calculate VI

_{s}and VI

_{v}locally rather than globally in an image, and that spatial interpolation may be useful for this purpose.

#### 3.4. Spatial Interpolation of VI_{s} and VI_{v}

_{s}and MSAVI

_{s}values of the gv_0 sample locations, and the NDVI

_{v}, and MSAVI

_{v}values of the fv_1 sample locations were used to predict the values at all other locations by spatial interpolation. The first spatial interpolation method tested was IDW, a deterministic interpolation method that assigns nearer samples greater weight for predictions, and the IDW exponent value controls this weight [35]. A full explanation of IDW is not given in this paper due to its wide availability from other sources, so readers interested in more details are encouraged to refer to [36] or other works in the literature that discuss IDW. IDW is a relatively simple interpolation method, but one advantage is that it requires little parameter calibration, making it fast to implement and quite objective. Many IDW exponent values between 1.0 and 3.0 were tested, and the optimal NDVI

_{s}, MSAVI

_{s}, NDVI

_{v}, and MSAVI

_{v}values were optimized through cross-validation of the sample data. As an example, maps of predicted NDVI

_{s}and NDVI

_{v}(i.e., IDW-NDVI

_{s}and IDW-NDVI

_{v}) are shown in Figure 3.

_{s}and OK-NDVI

_{v}are shown in Figure 4. Comparison of the interpolated surfaces in Figures 3 and 4 show that OK interpolation produced a smoother result in general.

#### 3.5. Calculation of FVC Using Invariant and Interpolated VI_{s} and VI_{v}

_{s}and VI

_{v}) and the proposed methods. For the traditional FVC calculation using NDVI, NDVI

_{s}in equation 1 was set as the mean (−0.16) of the gv_0 samples and NDVI

_{v}was set as the mean (0.77) of the gv_1 samples. For the traditional FVC calculation using MSAVI, MSAVI

_{s}was set as the mean (−0.14) of the gv_0 samples and MSAVI

_{v}as the mean (0.49) of the gv_1 samples. For the proposed approach, the interpolated NDVI and MSAVI values replaced the invariant values. For all calculations, FVC was constrained to the physically possible 0–1 range by assigning saturated non-vegetated or fully vegetated pixels (i.e., pixels with an estimated FVC < 0 or > 1) a value of 0 or 1, respectively. The number of pixels with FVC < 0 or > 1 was typically between 20 and 30%. For example, when the traditional NDVI approach was used, 11.16% of pixels had FVC < 0 and 18.06% had FVC > 1, while for the OK-NDVI approach, a similar percentage of pixels had FVC < 0 (11.24%) and fewer pixels had FVC > 1 (15.53%).

_{s}values calculated in other FVC studies, which were typically between 0.05–0.20 [11], the mean NDVI

_{s}in our study area was lower. This was most likely due to fact that some of the sample non-vegetation pixels in the image contained some frost or snow (which typically has negative NDVI values). We tried to avoid using endmember pixels that contained snow, but this could not be totally avoided because the GeoEye-1 image was taken before the ASTER image (it may have snowed in some areas after the GeoEye-1 image was taken), and it was not always clear in the ASTER image whether or not a pixel was at least partially covered by snow. On the other hand, our mean NDVI

_{v}was similar to the NDVI

_{v}values of the past studies, also provided in [11]. We did not compare our MSAVI

_{s}and MSAVI

_{v}values with those of past studies because we could not find reported values.

#### 3.6. Validation of ASTER FVC Estimates

_{i}is the predicted FVC of sample k, and o

_{i}is the reference FVC. MAE provides the expected FVC error per pixel in the image, while RMSE provides an estimate in which greater weight is given to large errors. In addition to performing accuracy assessment for the validation dataset as a whole, we also subdivided the validation data into “edge pixels” and “non-edge pixels” and assessed the accuracy of each separately to obtain a broader picture of the spatial distribution of errors. A validation pixel was considered to be an edge pixel if the 3 × 3 window it was found within contained a clear boundary between sparsely- and heavily-vegetated land cover. Non-edge pixels were all other pixels in the image, and were typically surrounded by other pixels with relatively-similar spectral characteristics (and thus relatively-similar FVC).

## 4. Results and Discussion

_{s}and VI

_{v}typically led to a modest reduction in MAE and RMSE, (ii) NDVI estimates of FVC were more accurate than MSAVI estimates, (iii) the improvement in FVC estimates due to spatial interpolation of NDVI

_{s}and NDVI

_{v}was greater than the improvement due to interpolation of MSAVI

_{s}and MSAVI

_{v}, and (iv) OK interpolation lead to a greater reduction in MAE and RMSE than IDW interpolation. The overall better performance of OK over IDW in this study is consistent with results from past studies [37], while the better performance of NDVI over MSAVI for FVC estimation, even for the traditional invariant FVC calculation method, was somewhat unexpected due to MSAVI’s intended design to be less sensitive to differences in soil brightness than NDVI. The lower accuracy of MSAVI estimates may have been due to a more non-linear relationship between MSAVI and FVC in our study area, but this could not be confirmed since we did not test non-linear estimation methods. To allow for a visual inspection of results, scatterplots of reference and estimated FVC using the traditional and proposed methods are shown in Figure 5, and the map of FVC produced by the most accurate estimation method, OK-NDVI, is shown in Figure 6.

_{s}and VI

_{v}for these edge pixels, since spatial interpolation of endmember values will not reduce the effects of coregistration errors nor the effects of diffuse reflection. To our knowledge, no past studies have assessed the accuracy of FVC estimates for edge and non-edge pixels separately. However, since we found a large variation in both (a) the accuracy of FVC estimates for edge and non-edge pixels, and (b) the degree to which errors were reduced by the proposed methods for each of these types of pixels, we recommend that future studies in patchy environments report accuracy for both types of pixels to provide better estimates of (a) the spatial distribution of errors and (b) the degree to which new techniques improve FVC estimates for each type of pixel.

## 5. Conclusions

_{s}) and green vegetation (VI

_{v}) caused by local differences in soil and vegetation conditions. This approach was used to estimate FVC in an Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) image of a forested area. Inverse Distance Weighting (IDW) and Ordinary Kriving (OK) were tested for spatial interpolation, and both the Normalized Differential Vegetation Index (NDVI) and the Modified Soil Adjusted Vegetation Index (MSAVI) were tested for calculation of FVC using the linear Vegetation Index (VI) model described in [3]. We found that spatial interpolation of VI

_{s}and VI

_{v}reduced the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) of FVC estimates by up to 5.1% and 2.7%, respectively, for the validation data set as a whole, and by up to 8.7% and 6.2% for non-edge pixels in the validation data set. Use of NDVI in the VI model and OK for interpolation produced these most accurate FVC estimates. The results of this study indicate that spatial interpolation of endmembers’ spectral properties can improve sub-pixel estimates of land cover.

_{s}and VI

_{v}values, incorporating a larger number of endmember pixels for interpolation (ideally using automated endmember extraction algorithms like the pixel purity index [40]) or use of more sophisticated spatial interpolation methods is likely to result in further improvement in FVC estimates. The main limitation of our proposed method is that endmember bare soil and green vegetation pixels must be widely distributed across an image to allow for interpolation (rather than extrapolation) of their spectral characteristics, so it may not be applicable when endmembers can be found only within a small subregion of an image. It should be noted that in this study we only used a linear approach for FVC estimation, which is the most commonly-used approach. In addition to non-linear unmixing methods, regression and machine learning algorithms have also been used for FVC estimation (e.g., [10,13]), so a comparison of the accuracy of the proposed methods with the accuracy achieved using spatially-interpolated values for these other FVC estimation methods should be tested in the future. Especially in study areas where endmembers do not exist or are not widely distributed, regression-based methods may provide a better alternative for estimating FVC since mixed pixels can be used to derive the unmixing formula [10]. Finally, in future studies, spatial interpolation of the spectral properties of endmembers should be tested for images containing a higher number of spectral bands (e.g., hyperspectral images) and using a larger number of endmembers.

## Acknowledgments

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**Figure 1.**False color Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) image, with the study area outlined in black (centered at 36°57′N, 140°38′E). NIR, R, and Green ASTER bands are shown in red, green, and blue color, respectively.

**Figure 2.**Normalized Differential Vegetation Index (NDVI) (

**a**) and Modified Soil Adjusted Vegetation Index (MSAVI) (

**b**) images of the study area. The topographic effects evident in Figure 1 have been largely removed using the vegetation indices.

**Figure 3.**Predicted NDVI

_{s}(

**a**) and NDVI

_{v}(

**b**) values using Inverse Distance Weighting (IDW) interpolation. Yellow points show the locations of gv_0 (a) and gv_1 (b) samples. Optimal IDW exponent value was 1.12 for (a) and 2.45 for (b).

**Figure 4.**Predicted NDVI

_{s}(

**a**) and NDVI

_{v}(

**b**) values using Ordinary Kriging (OK) interpolation. Yellow points show the locations of gv_0 (a) and gv_1 (b) samples.

**Figure 5.**Scatterplots of Reference and Estimated FVC using the NDVI (

**a**), OK-NDVI (

**b**), MSAVI (

**c**), and OK-MSAVI (

**d**) approach. OK-NDVI and OK-MSAVI show a modestly better linear fit than NDVI and MSAVI (based on R

^{2}values).

**Table 1.**Moran’s I (MI) and p values for NDVI

_{s}, NDVI

_{v,}MSAVI

_{s}, and MSAVI

_{v}. All endmembers exhibit positive spatial autocorrelation.

NDVI | MSAVI | |||
---|---|---|---|---|

MI | p | MI | p | |

Vi_{s} | 0.23 | 0.03 | 0.30 | 0.004 |

VI_{v} | 0.15 | 0.07 | 0.10 | 0.20 |

**Table 2.**Mean absolute error (MAE) and root mean square error (RMSE) for each FVC estimation method. For the proposed spatially interpolated estimation methods, relative changes in MAE and RMSE (%) show the changes compared to the invariant FVC calculation method.

FVC Estimation Method | MAE | Relative Change in MAE (%) | RMSE | Relative Change in RMSE (%) |
---|---|---|---|---|

NDVI (invariant) | 0.136 | 0.182 | ||

OK-NDVI | 0.129 | −5.1% | 0.177 | −2.7% |

IDW-NDVI | 0.131 | −3.7% | 0.179 | −1.6% |

MSAVI (invariant) | 0.164 | 0.200 | ||

OK-MSAVI | 0.16 | −2.4% | 0.196 | −2.0% |

IDW-MSAVI | 0.164 | 0% | 0.202 | +1.0% |

**Table 3.**Mean absolute error (MAE), root mean square error (RMSE), and their relative changes in edge and non-edge pixels. Much more improvement in estimation accuracy is seen in non-edge pixels.

Pixel Type | FVC Estimation Method | MAE | Relative Change in MAE (%) | RMSE | Relative Change in RMSE (%) |
---|---|---|---|---|---|

Edge | NDVI | 0.178 | 0.221 | ||

OK-NDVI | 0.175 | −1.7% | 0.219 | −0.9% | |

IDW-NDVI | 0.174 | −2.2% | 0.22 | −0.5% | |

MSAVI | 0.183 | 0.218 | |||

OK-MSAVI | 0.182 | −0.5% | 0.219 | +0.5% | |

IDW-MSAVI | 0.186 | +1.6% | 0.226 | +3.6% | |

Non-edge | NDVI | 0.104 | 0.145 | ||

OK-NDVI | 0.095 | −8.7% | 0.136 | −6.2% | |

IDW-NDVI | 0.098 | −5.8% | 0.139 | −4.1% | |

MSAVI | 0.149 | 0.184 | |||

OK-MSAVI | 0.144 | −3.4% | 0.178 | −3.3% | |

IDW-MSAVI | 0.147 | −1.3% | 0.182 | −1.1% |

## Share and Cite

**MDPI and ACS Style**

Johnson, B.; Tateishi, R.; Kobayashi, T.
Remote Sensing of Fractional Green Vegetation Cover Using Spatially-Interpolated Endmembers. *Remote Sens.* **2012**, *4*, 2619-2634.
https://doi.org/10.3390/rs4092619

**AMA Style**

Johnson B, Tateishi R, Kobayashi T.
Remote Sensing of Fractional Green Vegetation Cover Using Spatially-Interpolated Endmembers. *Remote Sensing*. 2012; 4(9):2619-2634.
https://doi.org/10.3390/rs4092619

**Chicago/Turabian Style**

Johnson, Brian, Ryutaro Tateishi, and Toshiyuki Kobayashi.
2012. "Remote Sensing of Fractional Green Vegetation Cover Using Spatially-Interpolated Endmembers" *Remote Sensing* 4, no. 9: 2619-2634.
https://doi.org/10.3390/rs4092619