Geostatistical estimation of signal-to-noise ratios for spectral vegetation indices

ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27

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Geostatistical estimation of signal-to-noise ratios for spectral vegetation indices Lei Ji a,⇑, Li Zhang b,c, Jennifer Rover d, Bruce K. Wylie d, Xuexia Chen e a

ASRC InuTeq, Contractor to the U.S. Geological Survey (USGS) Earth Resources Observation and Science (EROS) Center, Sioux Falls, SD 57198-0001, USA Key Laboratory of Digital Earth Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China c State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China d U.S. Geological Survey (USGS) Earth Resources Observation and Science (EROS) Center, Sioux Falls, SD 57198-0001, USA e Sigma Space Corporation, VIIRS Characterization Support Team (VCST), Lanham, MD 20706, USA b

a r t i c l e

i n f o

Article history: Received 1 March 2014 Received in revised form 23 May 2014 Accepted 25 June 2014

Keywords: Geostatistics Nugget variance Semivariogram Signal-to-noise ratio Spectral vegetation index Standardized noise

a b s t r a c t In the past 40 years, many spectral vegetation indices have been developed to quantify vegetation biophysical parameters. An ideal vegetation index should contain the maximum level of signal related to specific biophysical characteristics and the minimum level of noise such as background soil influences and atmospheric effects. However, accurate quantification of signal and noise in a vegetation index remains a challenge, because it requires a large number of field measurements or laboratory experiments. In this study, we applied a geostatistical method to estimate signal-to-noise ratio (S/N) for spectral vegetation indices. Based on the sample semivariogram of vegetation index images, we used the standardized noise to quantify the noise component of vegetation indices. In a case study in the grasslands and shrublands of the western United States, we demonstrated the geostatistical method for evaluating S/ N for a series of soil-adjusted vegetation indices derived from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor. The soil-adjusted vegetation indices were found to have higher S/N values than the traditional normalized difference vegetation index (NDVI) and simple ratio (SR) in the sparsely vegetated areas. This study shows that the proposed geostatistical analysis can constitute an efficient technique for estimating signal and noise components in vegetation indices. Ó 2014 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

1. Introduction A spectral vegetation index is a non-dimensional measure of spectral reflectances using an algebraic operation such as ratio, difference, weighted difference, or normalized difference in two or more bands to quantify vegetation biophysical characteristics. Visible and near-infrared (NIR) bands are the most commonly used wavelengths in development of spectral vegetation indices. A vegetation index based on the visible and NIR (VNIR) bands is generally related to photosynthetically active radiation (PAR) absorbed by vegetation canopies and is therefore considered a proxy for photosynthetic activity or vegetation greenness (Gamon et al., 1995; Myneni et al., 1995; Sellers, 1985). Since the 1970s, various VNIR-based vegetation indices have been developed, which can be classified into five general groups: (1) simple vegetation indices using a ratio, a difference, or a

⇑ Corresponding author. Tel.: +1 6055946584; fax: +1 6055946529. E-mail address: [email protected] (L. Ji).

normalized difference of red and NIR reflectances, for example, the simple ratio (SR) (Jordan, 1969), the difference vegetation index (DVI) (Tucker, 1979), and the normalized difference vegetation index (NDVI) (Rouse et al., 1974); (2) vegetation indices developed to adjust for the influence of background soil, for example, the perpendicular vegetation index (PVI) (Richardson and Wiegand, 1977), the weighted difference vegetation index (WDVI) (Richardson and Wiegand, 1977), the soil-adjusted vegetation index (SAVI) (Huete, 1988), the transformed soil-adjusted vegetation index (TSAVI) (Baret and Guyot, 1991), the modified soiladjusted vegetation index (MSAVI) (Qi et al., 1994), the optimized soil-adjusted vegetation index (OSAVI) (Rondeaux et al., 1996), and the generalized soil-adjusted vegetation index (GESAVI) (Gilabert et al., 2002); (3) vegetation indices devised to compensate for atmospheric effects, such as the atmospherically resistant vegetation index (ARVI) (Kaufman and Tanré, 1992) and the global environmental monitoring index (GEMI) (Pinty and Verstraete, 1992); (4) vegetation indices combining corrections for both background soil influence and atmospheric effect, such as the soil atmospherically resistant vegetation index (SARVI) (Kaufman and Tanré,

http://dx.doi.org/10.1016/j.isprsjprs.2014.06.013 0924-2716/Ó 2014 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27

1992), the modified normalized difference vegetation index (MNDVI) (Liu and Huete, 1995), the enhanced vegetation index (EVI) (Huete et al., 2002), and the two-band enhanced vegetation index (EVI2) (Jiang et al., 2008); (5) vegetation indices designed to increase the index’s linearity with biophysical parameters, for example, the nonlinear vegetation index (NLI) (Goel and Qi, 1994), the renormalized difference vegetation index (RDVI) (Roujean and Breon, 1995), the modified simple ratio (MSR) (Chen, 1996), the green normalized difference vegetation index (GNDVI) (Gitelson et al., 1996), the green atmospherically resistant vegetation index (GARI) (Gitelson et al., 1996), the wide dynamic range vegetation index (WDRVI) (Gitelson, 2004), the linearized vegetation index (LVI) (Ünsalan and Boyer, 2004), and the linearized normalized difference vegetation index (LNDVI) (Jiang and Huete, 2010). In addition to the VNIR-based indices, indices using longer wavelengths (e.g., shortwave infrared) and narrow bands derived from hyperspectral sensors are also widely used. An ideal vegetation index should meet two criteria. First, the index should be sensitive to the ‘‘signal’’ of a given biophysical parameter such as leaf area index, green vegetation fraction, PAR, or leaf chlorophyll concentration. Moreover, the index’s sensitivity to signal should be consistent over the entire range of the biophysical parameter, requiring that the index not saturate for dense vegetation cover. Second, the index should be insensitive to ‘‘noise’’ such as the effects of background soil, atmosphere, canopy structure, sun-target-sensor geometry, and ground topography. Evaluation and comparison of various vegetation indices have received great attention in the remote sensing community (e.g., Bannari et al., 1995; Gong et al., 2003; Xu et al., 2003; Silleos et al., 2006; Ji and Peters, 2007; Jiang and Huete, 2010). Some basic statistical techniques, such as correlation, regression, and analysis of variance, are useful in evaluating vegetation indices (e.g., Lawrence and Ripple, 1998; Purevdorj et al., 1998; Gong et al., 2003; Haboudane et al., 2004). To specifically estimate the signal and noise of a vegetation index, investigators devised several statistical metrics including the relative equivalent noise (Baret and Guyot, 1991), the vegetation equivalent noise (Huete et al., 1994), and the sensitivity function (Ji and Peters, 2007). These metrics, despite their different forms, can quantify the signal and noise components for a vegetation index based on the statistical relationship of the index to a biophysical variable. However, these methods for estimating signal and noise components require field-based measures of biophysical parameters, laboratory experiments, or model simulations. Because a vegetation index consists of both desired signals and unwanted background noise, evaluation of a vegetation index can be simplified into an estimation of signal-to-noise ratio (S/N). In the remote sensing area, Smith and Curran (1999) summarized several image-based S/N evaluation methods: the homogeneous area method, the nearly homogeneous area method, the geostatistical method, the homogeneous block method, and the multiple waveband method. The geostatistical method was proposed by Curran and Dungan (1989) and later adapted or modified by Eklundh (1995), Atkinson et al. (1996), Atkinson (1997), Chappell et al. (2001), Foody et al. (2004), Atkinson et al. (2005), Atkinson et al. (2007), Guo and Dou (2008), Asmat et al. (2010), and others. The geostatistical method has been applied to estimate S/N of airborne and satellite images. For example, Curran and Dungan (1989) used this method to estimate S/N for the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data; Eklundh (1995), Chappell et al. (2001) performed S/N analysis for Advanced Very High Resolution Radiometer (AVHRR) NDVI data; Atkinson et al. (2005), Asmat et al. (2010) evaluated image noise for the hyperspectral images derived from Compact Airborne Spectrographic Imager; and Guo and Dou (2008) applied a modified geostatistical method to estimate S/N for the visible and infrared data acquired from China-launched FY-2 geostationary meteorological satellite series.

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Although the geostatistical method has been used to estimate S/ N of remotely sensed images, vegetation indices are different from regular images where each pixel records a brightness value that can be further converted to radiance or reflectance. The magnitude and sign of a vegetation index are normally irrelevant to the signal strength but are functions of land surface characteristics such as land cover types, vegetation density and condition, and soil properties. Therefore, not all the geostatistical metrics previously developed are suitable for vegetation indices. In this study, we applied the existing geostatistical techniques and proposed a procedure for estimating and comparing S/N values of different vegetation indices. In a case study in the grasslands and shrublands in the western United States we used several vegetation indices derived from the Moderate Resolution Imaging Spectroradiometer (MODIS) data to demonstrate the effectiveness of our proposed S/N estimation method. 2. A brief review of geostatistical methods for S/N estimation S/N in electronics is defined as a measure of signal strength relative to background noise. S/N can be expressed using different formulas because signal and noise can be defined in different ways (Schowengerdt, 1997). The most common definition of S/N in image processing is given by

S=Nvar ¼

r2signal r2noise

ð1Þ

where r2signal and r2noise are the variances of signal and noise, respectively (Schowengerdt, 1997). The geostatistical estimation of S/N was developed based on the concept of the semivariogram, a key function in geostatistics. The sample semivariogram is defined as mðhÞ

cðhÞ ¼

1 X 2 ½zðxi Þ  zðxi þ hÞ 2mðhÞ i¼1

ð2Þ

where z(xi) is the value of a pixel location (xi), h is the lag distance between pairs of pixels, m is the numbers of pairs of pixels at lag h, and c(h) is the estimate of the semivariogram at lag h (Isaaks and Srivastava, 1989). For semivariogram, there is an intrinsic stationarity assumption that the mean is a constant and the variance of the difference is the same everywhere in the region of interest. The sample semivariogram usually displays a characteristic shape, increasing from smaller to larger lags. The shape of a sample semivariogram is characterized by three parameters: sill variance (c), range (a), and nugget variance (c0). In general, a semivariogram c(h) increases with large lags and levels off asymptotically. The semivariogram c(h) value and the lag h value at the asymptote are referred to as sill variance (c) and range (a), respectively. The nugget variance is the c(h) value when the lag h is zero. Ideally, c0 = 0 for h = 0, but in reality, data noise or measurement error can cause a discontinuity at the origin of the semivariogram resulting in a positive nugget variance (Isaaks and Srivastava, 1989; Schowengerdt, 1997). A sample semivariogram can be fitted by a mathematical model, such as nugget effect model, spherical model, exponential model, and Gaussian model, to determine the semivariogram parameters c, a, and c0. A complicated semivariogram model may contain two or more nested structures that combine multiple mathematical models (Isaaks and Srivastava, 1989). The use of the nugget variance to estimate random noise for images was proposed by Curran and Dungan (1989). They justified this method by these two arguments: (1) the variance of an image is the sum of signal variance (or underlying variance) and the noise variance, and (2) when the lag approaches zero, the signal variance will be nearly zero, so the semivariogram of the image will consist of nearly pure noise variance. Curran and Dungan (1989) defined S/ Nmean as

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L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27

z S=Nmean ¼ pffiffiffiffiffi c0

ð3Þ

where z is the mean of the pixel values, and c0 is the nugget variance. Eklundh (1995) developed the standardized noise to indicate the percent of noise observed in the total variance of an image.

SN ¼

pffiffiffiffiffi c0  100% s

ð4Þ

where SN is the standardized noise, s is the standard deviation of the image. Atkinson (1997) reviewed Curran and Dungan’s method and indicated that the nugget variance consists of not only measurement error (or image noise) but also other effects, especially the underlying variance near the origin of the semivariogram (i.e., the underlying variance that exists within a sub-pixel distance, or 0 < h < 1 pixel). However, Atkinson et al. (2005) considered that the amount of sub-pixel underlying variance is small because remotely sensed observations are usually abutting or overlapping, so the nugget variance can be assumed to be due primarily to measurement errors. Atkinson et al. (2007) modified the geostatistical S/N estimation method and proposed a newer expression for S/N, given by

S=Nvar ¼

c c0

ð5Þ

where c0 is the nugget variance, and c is the sill to indicate the signal component variance.

3. Geostatistical estimation of S/N for vegetation indices As mentioned above, using the nugget variance to estimate noise component has been commonly accepted (e.g., Curran and Dungan, 1989; Eklundh, 1995; Chappell et al., 2001; Atkinson et al., 2007). The estimation of signal component, however, seems more complicated than the estimation of noise component. Curran and Dungan (1989), Chappell et al. (2001), Atkinson et al. (2005), Asmat et al. (2010) used the mean of pixel values (i.e., digital numbers, digital counts, or brightness values) to indicate signal (Eq. (3)). In this study, we argue that the mean pixel values do not indicate the signal level of vegetation indices. In general, a vegetation index is a function of vegetation characteristics, including vegetation type, density, and greenness. Variation in vegetation index values is determined mainly by surface characteristics, although noise induced by sensors, atmosphere, and targets affects the index values. Thus, the mean pixel value within an area of interest does not necessarily indicate the signal level of a vegetation index image. Moreover, most vegetation indices contain zero and negative values. Any value of a vegetation index, regardless of the magnitude or the sign, measures vegetation biophysical status. Zero or negative values are not uncommon in vegetation index images, but these values do not infer no-signal or negative signal for the index. In this study, we used the standardized noise developed by Eklundh (1995) (Eq. (4)) to quantify the level of image noise. We also suggest a procedure for estimating vegetation index S/N using this geostatistical method, including the following five steps: (1) Creating sample windows. A sample window is an image subset with all pixels within the subset used for the semivariogram analysis. Water, snow, clouds, and shadows need to be masked out from the sample windows. (2) Trend removal. Semivariogram modeling requires mean stationarity (i.e., mean is constant) for the observations. If the data are mean non-stationary, the spatial trend must be removed before the semivariogram is created (Schabenberger and Pierce, 2002). The trend can be detected and removed using a linear or a polynomial trend model. If a

trend exists, the residuals (the difference between the original value and trend model predictions) are used in semivariogram modeling. (3) Creating sample semivariogram. The original values (if mean stationary) or the residuals (if mean non-stationary) are used to create a sample semivariogram, following Eq. (2). However, if data are anisotropic (i.e., semivariograms are different at different directions), several angle regions are divided and the anisotropic semivariograms are built separately by angle region. (4) Modeling sample semivariogram. A mathematical model is used to fit the sample semivariogram. The most common models are the exponential model (Eq. (6)) and the spherical model (Eq. (7)):





cðhÞ ¼ c0 þ c 1  exp  (

cðhÞ ¼

 h a

 3 1:5 ha  0:5 ha

if h 6 a

c0 þ c

otherwise

ð6Þ

ð7Þ

Nugget variance (c0), sill (c), and range (a) are obtained from the model. If anisotropy exists, more models are needed to fit several sample semivariograms corresponding to different angle regions. (5) Estimation of standardized noise. Standardized noise can be calculated using nugget variance (c0) and standard deviation (s) using Eq. (4). In case of anisotropy, several nugget variance values are obtained from different sample semivariograms corresponding to different angle regions. As a result, the mean of the nugget variance values is used to calculate the standardized noise that indicates the signal and noise levels of the sample window. 4. A case study of S/N analysis for vegetation indices 4.1. Selection of vegetation indices In areas with sparse vegetation, surface reflectance and vegetation indices are influenced by exposed soil influences. Several vegetation indices incorporated with soil adjustment factors, such as SAVI, were designed to reduce the impact of soil brightness on canopy spectral signatures and increase the index sensitivity to canopy biophysical characteristics (Huete, 1988; Qi et al., 1994; Huete et al., 2002). To verify the improvement of soil-adjusted indices over traditional indices such as NDVI and SR in low vegetated areas, we applied the geostatistical method to estimate the S/N values for seven VNIR-based vegetation indices as listed in Table 1. SAVI was developed to minimize the influence of background soil color and brightness by incorporating a soil-adjustment factor L into the NDVI form (Huete, 1988). The dynamic range of the index is reduced with constant L between 0 (very high vegetative cover) and 1 (very low vegetative cover). Huete (1988) suggested using L = 0.5 for intermediate vegetation cover. Unlike SAVI, which requires users to pre-specify the soil-adjustment factor L, the MSAVI equation contains a calculation of L using available red and NIR bands (Qi et al., 1994). OSAVI is a generalized form of all indices in the SAVI family (SAVI, TSAVI, and MSAVI) (Rondeaux et al., 1996). Rondeaux et al. (1996) suggested a soil-adjustment coefficient X of 0.16 as the optimal value to reduce the background soil effect. Huete et al. (1994) developed SARVI combining SAVI and ARVI to minimize both soil noise and atmospheric effect. Huete et al. (1997) further combined SARVI and the modified normalized difference vegetation index (MNDVI) to create the second version of SARVI (i.e., SARVI2), which was later renamed EVI (Huete et al., 2002). In the EVI formula, G is the gain factor, L is

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L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27 Table 1 Definitions of the VNIR-based vegetation indices analyzed in the study. Index

Equationa

Single ratio (SR) or ratio vegetation index (RVI)

SR ¼ qqNIR red qred NDVI ¼ qqNIR  NIR þqred ð1þLÞðqNIR qred Þ SAVI ¼ q þq þL NIR red

Normalized difference vegetation index (NDVI) Soil-adjusted vegetation index (SAVI) Modified soil-adjusted vegetation index (MSAVI) Optimized soil-adjusted vegetation index (OSAVI) Enhanced vegetation index (EVI) Two-band enhanced vegetation index (EVI2) a

MSAVI =

Constant

Jordan (1969) Rouse et al. (1974) L = 0–1; L = 0.5 in this study

Huete (1988)

X = 0.16

Rondeaux et al. (1996)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

2qNIR þ1

Qi et al. (1994)

ð2qNIR þ1Þ 8ðqNIR qred Þ 2

q q OSAVI ¼ q NIRþq redþX NIR red Gðq q Þ EVI ¼ ðq þC1 qNIR Cred 2 qblue ÞþL NIR red qred Þ EVI2 ¼ ðqGðqNIR þC q ÞþL NIR red

Author(s)

G = 2.5, L = 1, C1 = 6, C2 = 7.5

Huete et al. (2002)

G = 2.5, L = 1, C = 2.4

Jiang et al. (2008)

In the equations, qblue, qgreen, qred, and qNIR are the reflectance of blue, green, red, and NIR bands, respectively.

Fig. 1. The map of the study area showing the 50-km by 50-km sample windows and the associated land cover types and ecoregions. The ecoregion names for the four labeled ecoregions are Mojave Basin and Range (14), Chihuahuan Deserts (24), Southwestern Tablelands (26), and Northwestern Great Plains (43). The land cover map was simplified based on the NLCD 2001. The ecoregion polygons were derived from the EPA’s Level III ecoregions.

the soil-adjustment coefficient, and C1 and C2 are the coefficients to correct aerosol resistance (Huete et al., 1997; Huete et al., 2002). The coefficients G = 2.5, L = 1, C1 = 6, and C2 = 7.5 are adapted in the standard MODIS EVI product. As a variety of three-band EVI, EVI2 was designed to use only red and NIR bands, but it retains all merits of EVI (Jiang et al., 2008). EVI2 can be used for sensors without a blue band, such as AVHRR, to produce an EVI-like vegetation index.

4.2. Study area, data, and methods In the case study, we used the Terra MODIS product ‘‘MOD09A1’’ (version 5), an 8-day composite 7-band surface reflectance, 500-m resolution dataset (https://lpdaac.usgs.gov/). The original gridded tiles were reprojected from the sinusoidal system to the Albers Equal Area system. The ‘‘500-meter surface reflectance data state quality assurance’’ layers were used to flag clouds, cloud

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L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27

Fig. 2. A sample-window NDVI image (a), quadratic surface trend (b), and NDVI residuals (c) in the grasslands of the Southwestern Tablelands ecoregion. The geostatistical analysis is conducted on the NDVI residuals. The black pixels are non-grasslands that are excluded in the geostatistical analysis.

Fig. 3. Sample semivariograms and the exponential model fits for an NDVI sample window (13–20 September, 2004) located in the Southwestern Tablelands ecoregion. Because of anisotropy, four semivariograms were created corresponding to four angle regions. Nugget (c0), sill (c), and range (a) for each semivariogram were calculated based on the exponential model.

shadows, and snow/ice pixels, that were excluded from the data analysis. We selected four sample windows defined as a 50 km by 50 km square containing 10,000 pixels from the MODIS images. The sample windows are located in four grassland or shrubland ecoregions in the western United States, centered at 35°190 N/ 117°290 W, 31°330 N/103°550 W, 36°160 N/104°30 W, and 44°580 N/ 102°430 W (Fig. 1). The land cover within the sample windows was identified with the 2001 National Land Cover Database (NLCD 2001) (http://www.mrlc.gov/nlcd.php). We degraded the 30-m NLCD data to 500-m resolution using a majority algorithm to match the pixel size of the MODIS data. The ecoregions were

determined by the U.S. Environmental Protection Agency (EPA) Level III ecoregions of the Continental United States (2007 version) (http://www.epa.gov/bioiweb1/html/usecoregions.html). Temporally, we selected six 8-day intervals of the MODIS data from 2001, 2004, and 2007 for each sample window. The six intervals are evenly distributed throughout a year: days of year 9–16, 73–80, 129–136, 193–200, 257–264, and 313–320. For the S/N analysis, we obtained 71 sample windows that included four sites (ecoregions), and each site has 18 dates (one sample window was excluded due to a large snow cover). Following the procedure described in Section 3, we performed the geostatistical analysis

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L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27

of 13–20 September, 2004, located in the Southwestern Tablelands ecoregion with grasslands as the dominant cover type (Fig. 2a). First, we visually detected a surface trend pattern from the NDVI image, where NDVI values are generally higher in the north-central portion and lower in the south. We used a 2nd-order polynomial model to fit the surface trend (Fig. 2b):

NDVI ¼ b0 þ b1 x þ b2 y þ b3 x2 þ b4 y2 þ b5 xy

Fig. 4. Box plot of the standardized noise values for the vegetation indices. The dotted and solid lines within the box indicate the mean and median values, respectively; the upper and lower boundaries represent the 75th and 25th percentiles; the whiskers above and below the box represent the 90th and 10th percentiles; the solid circles above and below the whiskers are the 95th and 5th percentiles.

for each sample window containing only grassland or shrubland pixels, with snow, cloud, and cloud shadows removed using the quality assurance layer. The S/N analysis was conducted for all seven vegetation indices listed in Table 1. To understand the dependency of the S/N of each vegetation index on the MODIS channels, we performed S/N analysis on each individual VNIR band. The Statistical Analysis Software (SAS) program was used in the semivariogram analysis.

4.3. Results Following the procedure mentioned in Section 3, here we demonstrate an example of geostatistical S/N estimation for a sample window. This sample window was the MODIS composite

ð8Þ

where x and y are the geographic coordinates in kilometers with origin (0, 0) located at the southwest corner of the NDVI image, and b0, b1, . . ., b5 are the regression coefficients. Because the model and all five coefficients were all significant (p-value < 0.05), the model depicted the trend surface of the NDVI window (Fig. 2b). After removing the trend surface from the NDVI image, we performed the geostatistical analysis for the NDVI residuals (Fig. 2c), i.e., the difference between the NDVI image and the polynomial trend surface. Second, we generated a sample semivariogram for the NDVI residual image. However, because of the anisotropic issue, we needed four sample semivariograms, each corresponding to one of the four angle regions, 0° ± 22.5°, 45° ± 22.5°, 90° ± 22.5°, 135° ± 22.5°, representing four geographic directions: N–S, NE– SW, E–W, and NW–SE, respectively (Fig. 3). Third, we fitted the sample semivariograms using an exponential model (Eq. (6)), from which we obtained three basic semivariogram parameters: sill variance (c), range (a), and nugget variance (c0) (Fig. 3). Last, because the four directions have different nugget variances, we calculated the mathematic mean of the four nugget variances, which resulted in a mean c0 of 0.000167. As the standard deviation (s) of the NDVI image was 0.0514, we obtained standardized noise value of 25.2% based on Eq. (4). For all 71 sample windows across four ecoregions from 18 dates, we calculated the standardized noise values for seven vegetation indices. The box plot in Fig. 4(a) demonstrates the distribution of the standardized noise values of the seven vegetation indices for the 71 sample windows. Table 2(a) indicates the mean and the standard deviation of the standardized noise values. Based on the standardized noise values we ranked vegetation indices from high to low S/N: EVI, EVI2, SAVI, MSAVI, OSAVI, NDVI, and SR. Comparing to NDVI and SR, all the soil-adjusted indices were deemed to reduce the soil-induced noise to some extent in the sparsely vegetated areas. We calculated the standardized noise values for the MODIS surface reflectances of bands 1 (red), 2 (NIR), 3 (blue), and 4 (green) for all 71 sample windows. The mean, standard deviation, and distribution of the standardized noise values of the four bands are shown in Table 2(b) and Fig. 4(b). Thus, the rank of the four bands from higher to lower S/N is NIR, red, green, and blue. For the seven

Table 2 Standardized noise of the vegetation indices and spectral bands for the 71 grassland and shrubland images: summary statistics and correlation coefficients. (a) Vegetation index Vegetation index Mean standardized noise Standard deviation

NDVI 30.5 10.6

SR 30.4 11.1

SAVI 27.7 11.4

MSAVI 28.5 12.4

OSAVI 29.0 11.1

EVI 26.6 11.7

EVI2 27.6 11.5

(b) Spectral band Spectral band Mean standardized noise Standard deviation

Blue 36.6 12.5

Green 34.4 10.8

Red 33.8 11.5

NIR 32.0 11.3

(c) Correlation Blue Green Red NIR

NDVI 0.573 0.577 0.521 0.586

SR 0.471 0.459 0.424 0.459

SAVI 0.544 0.552 0.575 0.549

MSAVI 0.483 0.482 0.514 0.494

OSAVI 0.541 0.547 0.533 0.548

EVI 0.546 0.563 0.594 0.573

EVI2 0.541 0.550 0.578 0.546

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L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27

vegetation indices mentioned above, only EVI uses red, NIR, and blue bands and all other indices use only red and NIR bands. Because EVI has the lowest standardized noise and the blue band has the highest standardized noise, we believe that the S/N level of a vegetation index is irrelevant of the band input. Table 2(c) shows a correlation matrix between the standardized noise of vegetation indices and the standardized noise of spectral bands. It is noted that all vegetation indices are correlated to the spectral bands with regard to the standardized noise values. However, all four spectral bands have very close correlations to vegetation indices, implying that the S/N of a vegetation index does not depend on the specific spectral bands used in the vegetation index.

5. Conclusions In this study, we used the geostatistical method to estimate and compare the S/N levels for different vegetation indices. Based on the concept that the nugget variance derived from the semivariogram model is related to image noise, we used the standardized noise to evaluate the performance of vegetation indices. We proposed a geostatistical analysis procedure including creating sample windows, surface trend removal, developing sample semivariograms (both isotropic and anisotropic cases), regression modeling of semivariograms, and estimation of standardized noise. We demonstrated a case study in the grasslands and shrublands in the western United States using the 500-m, 8-day composite MODIS data. From 71 sample windows across four ecoregions and 18 dates, we analyzed five soil-adjusted indices (SAVI, MSAVI, OSAVI, EVI, and EVI2) and two traditional indices (NDVI and SR). The geostatistical analyses of the indices show that all the soiladjusted indices have lower standardized noise values than NDVI and SR. The case study suggested that vegetation indices perform better when they are adjusted for the soil factor in low vegetation cover areas.

Acknowledgements This study was funded by the National Natural Science Foundation of China (Grant No. 41271372), the National High Technology Research and Development Program of China (863 Program) (Grant No. 2012AA12A301), and the U.S. Geological Survey Climate Effects Network and Global Change Research & Development Programs. The work by L. Ji was performed under USGS contract G13PC00028. We thank Eugene Fosnight and Thomas Adamson for reviewing the manuscript and providing valuable comments. Any use of trade names is for descriptive purposes only and does not imply endorsement by the U. S. Government.

References Asmat, A., Atkinson, P.M., Foody, G.M., 2010. Geostatistically estimated image noise is a function of variance in the underlying signal. Int. J. Remote Sens. 31 (4), 1009–1025. Atkinson, P.M., 1997. On estimating measurement error in remotely sensed images with the variogram. Int. J. Remote Sens. 18 (14), 3075–3084. Atkinson, P.M., Dunn, R., Harrison, A.R., 1996. Measurement error in reflectance data and its implications for regularizing the variogram. Int. J. Remote Sens. 17 (18), 3735–3750. Atkinson, P.M., Sargent, I.M., Foody, G.M., Williams, J., 2005. Interpreting imagebased methods for estimating the signal-to-noise ratio. Int. J. Remote Sens. 26 (22), 5099–5115. Atkinson, P.M., Sargent, I.M., Foody, G.M., Williams, J., 2007. Exploring the geostatistical method for estimating the signal-to-noise ratio of images. Photogrammetric Eng. Remote Sens. 73 (7), 841–850. Bannari, A., Morin, D., Bonn, F., 1995. A review of vegetation indices. Remote Sens. Rev. 13 (1–2), 95–120. Baret, F., Guyot, G., 1991. Potentials and limits of vegetation indices for LAI and APAR assessment. Remote Sens. Environ. 35 (2–3), 161–173.

Chappell, A., Seaquist, J.W., Eklundh, L., 2001. Improving the estimation of noise from NOAA AVHRR NDVI for Africa using geostatistics. Int. J. Remote Sens. 22 (6), 1067–1080. Chen, J.M., 1996. Evaluation of vegetation indices and a modified simple ratio for boreal applications. Canadian J. Remote Sens. 22 (3), 229–242. Curran, P.J., Dungan, J.L., 1989. Estimation of signal-to-noise: a new procedure applied to AVIRIS data. IEEE Trans. Geosci. Remote Sens. 27 (5), 620–628. Eklundh, L.R., 1995. Noise estimation in NOAA AVHRR maximum-value composite NDVI images. Int. J. Remote Sens. 16 (15), 2955–2962. Foody, G.M., Sargent, I.M.J., Atkinson, P.M., Williams, J.W., 2004. Thematic labelling from hyperspectral remotely sensed imagery: trade-offs in image properties. Int. J. Remote Sens. 25 (12), 2337–2363. Gamon, J.A., Field, C.B., Goulden, M.L., Griffin, K.L., Hartley, A.E., Joel, G., Penuelas, J., Valentini, R., 1995. Relationships between NDVI, canopy structure, and photosynthesis in three Californian vegetation types. Ecol. Appl. 5 (1), 28–41. Gilabert, M.A., González-Piqueras, J., García-Haro, F.J., Meliá, J., 2002. A generalized soil-adjusted vegetation index. Remote Sens. Environ. 82 (2–3), 303–310. Gitelson, A.A., 2004. Wide dynamic range vegetation index for remote quantification of biophysical characteristics of vegetation. J. Plant Physiol. 161 (2), 165–173. Gitelson, A., Kaufman, Y.J., Merzlyak, M.N., 1996. Use of a green channel in remote sensing of global vegetation from EOS–MODIS. Remote Sens. Environ. 58 (3), 289–298. Goel, N.S., Qi, W., 1994. Influences of canopy architecture on relationships between various vegetation indices and LAI and Fpar: a computer simulation. Remote Sens. Rev. 10 (4), 309–347. Gong, P., Pu, R., Biging, G.S., Larrieu, M.R., 2003. Estimation of forest leaf area index using vegetation indices derived from Hyperon hyperspectral data. IEEE Trans. Geosci. Remote Sens. 40 (6), 1355–1362. Guo, Q., Dou, X., 2008. A modified approach for noise estimation in optical remotely sensed images with a semivariogram: principle, simulation, and application. IEEE Trans. Geosci. Remote Sens. 46 (7), 2050–2060. Haboudane, D., Miller, J.R., Pattey, E., Zarco-Tejada, P.J., Strachan, I.B., 2004. Hyperspectral vegetation indices and novel algorithms for predicting green LAI of crop canopies: Modeling and validation in the context of precision agriculture. Remote Sens. Environ. 90 (3), 337–352. Huete, A.R., 1988. A soil-adjusted vegetation index (SAVI). Remote Sens. Environ. 25 (3), 295–309. Huete, A., Justice, C., Liu, H., 1994. Development of vegetation and soil indices for MODIS–EOS. Remote Sens. Environ. 49 (3), 224–234. Huete, A.R., Liu, H.Q., Batchily, K., van Leeuwen, W.J.D., 1997. A comparison of vegetation indices over a global set of TM images for EOS–MODIS. Remote Sens. Environ. 59 (3), 440–451. Huete, A., Didan, K., Miura, T., Rodriguez, E.P., Gao, X., Ferreira, L.G., 2002. Overview of the radiometric and biophysical performance of the MODIS vegetation indices. Remote Sens. Environ. 83 (1–2), 195–213. Isaaks, E.H., Srivastava, R.M., 1989. An Introduction to Applied Geostatistics. Oxford University Press, New York. Ji, L., Peters, A., 2007. Performance evaluation of spectral vegetation indices using a statistical sensitivity function. Remote Sens. Environ. 106 (1), 59–65. Jiang, Z., Huete, A.R., Didan, K., Miura, T., 2008. Development of a two-band enhanced vegetation index without a blue band. Remote Sens. Environ. 112 (10), 3833–3845. Jiang, Z., Huete, A.R., 2010. Linearization of NDVI based on its relationship with vegetation fraction. Photogrammetric Eng. Remote Sens. 76 (8), 965–975. Jordan, C.F., 1969. Derivation of leaf-area index from quality of light on the forest floor. Ecology 50 (4), 663–666. Kaufman, Y.J., Tanré, D., 1992. Atmospherically resistant vegetation index (ARVI) for EOS–MODIS. IEEE Trans. Geosci. Remote Sens. 30 (2), 261–270. Lawrence, R.L., Ripple, W.J., 1998. Comparisons among vegetation indices and bandwise regression in a highly disturbed, heterogeneous landscape: Mount St. Helens, Washington. Remote Sens. Environ. 64 (1), 91–102. Liu, H.Q., Huete, A., 1995. A feedback based modification of the NDVI to minimize canopy background and atmospheric noise. IEEE Trans. Geosci. Remote Sens. 33 (2), 457–465. Myneni, R.B., Hall, F.G., Sellers, P.J., Marshak, A.L., 1995. The interpretation of spectral vegetation indexes. IEEE Trans. Geosci. Remote Sens. 33 (2), 481–486. Pinty, B., Verstraete, M.M., 1992. GEMI: A non-linear index to monitor global vegetation from satellites. Vegetatio 101 (1), 15–20. Purevdorj, Ts., Tateishi, R., Ishiyama, T., Honda, Y., 1998. Relationships between percent vegetation cover and vegetation indices. Int. J. Remote Sens. 9 (18), 3519–3535. Qi, J., Chehbouni, A., Huete, A.R., Kerr, Y.H., Sorooshian, S., 1994. A modified soil adjusted vegetation index. Remote Sens. Environ. 48 (2), 119–126. Richardson, A.J., Wiegand, C.L., 1977. Distinguishing vegetation from soil background information. Photogrammetric Eng. Remote Sens. 43 (12), 1541– 1552. Rondeaux, G., Steven, M., Baret, F., 1996. Optimization of soil-adjusted vegetation indices. Remote Sens. Environ. 55 (2), 95–107. Roujean, J.-L., Breon, F.-M., 1995. Estimating PAR absorbed by vegetation from bidirectional reflectance measurements. Remote Sens. Environ. 51 (3), 375–384. Rouse Jr., J.W., Haas, R.H., Shell, J.A., Deering, D.W., 1974. Monitoring vegetation systems in the Great Plains with ERTS-1. Third Earth Resources Technology

L. Ji et al. / ISPRS Journal of Photogrammetry and Remote Sensing 96 (2014) 20–27 Satellite-1 Symposium, Vol. 1. NASA Scientific and Technical Information Office, Washington, D.C., pp. 309–317, 10-14 December 1973. Schabenberger, O., Pierce, F.J., 2002. Contemporary statistical models for the plant and soil sciences. CRC Press, Boca Raton, FL. Schowengerdt, R.A., 1997. Remote Sensing: Models and Methods for Image Processing, 2nd ed. Academic Press, San Diego, CA. Sellers, P.J., 1985. Canopy reflectance, photosynthesis and transpiration. Int. J. Remote Sens. 6 (8), 1335–1372. Silleos, N.G., Alexandridis, T.K., Gitas, I.Z., Perakis, K., 2006. Vegetation indices: advances made in biomass estimation and vegetation monitoring in the last 30 years. Geocarto Int. 21 (4), 21–28.

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Smith, G.M., Curran, P.J., 1999. Methods for estimating image signal-to-noise ratio (SNR). In: Atkinson, P.M., Tate, N.J. (Eds.), Advances in Remote Sensing and GIS Analysis. John Wiley and Sons, Chichester, pp. 61–74. Tucker, C.J., 1979. Red and photographic infrared linear combinations for monitoring vegetation. Remote Sens. Environ. 8 (2), 127–150. Ünsalan, C., Boyer, K.L., 2004. Linearized vegetation indices based on a formal statistical framework. IEEE Trans. Geosci. Remote Sens. 42 (7), 1575– 1585. Xu, B., Gong, P., Pu, R., 2003. Crown closure estimation of oak savannah in a dry season with Landsat TM imagery: comparison of various indices through correlation analysis. Int. J. Remote Sens. 9 (24), 1811–1822.