vegetation indices

Normalized Burned Ratio (NBR) and dNBR

Fire Intensity versus Burn Severity

Fire intensity represents the energy that is released from organic matter during the combustion process (Keeley, 2009). It also refers to the intensity of the fire while it is active. Burn severity, on the other hand, describes how the fire intensity affects the functioning of the ecosystem in the area that has been burnt. The observed effects often vary within the area and between different ecosystems (Keeley, 2009). Burn severity can also be described as the degree to which an area has been altered or disrupted by the fire. An illustration of the difference between fire intensity and burn severity is shown in Figure 1.

Normalized Burn Ratio (NBR)

The Normalized Burn Ratio (NBR) is an index designed to highlight burnt areas in large fire zones. The formula is similar to NDVI, except that the formula combines the use of both near infrared (NIR) and shortwave infrared (SWIR) wavelengths.

Healthy vegetation shows a very high reflectance in the NIR, and low reflectance in the SWIR portion of the spectrum (Figure 2) - the opposite of what is seen in areas devastated by fire. Recently burnt areas demonstrate low reflectance in the NIR and high reflectance in the SWIR, i.e. the difference between the spectral responses of healthy vegetation and burnt areas reach their peak in the NIR and the SWIR regions of the spectrum.

Comparison of the spectral response of healthy vegetation and burned areas. Source: U.S. Forest service.

To benefit from the magnitude of spectral difference, NBR uses the ratio between NIR and SWIR bands, according to the formula shown below. A high NBR value indicates healthy vegetation while a low value indicates bare ground and recently burnt areas. Non-burnt areas are normally attributed to values close to zero.
$NBR = \frac{NIR-SWIR}{NIR+SWIR}$

Burn Severity

The difference between the pre-fire and post-fire NBR obtained from the images is used to calculate the delta NBR (dNBR or ∆NBR), which then can be used to estimate the burn severity. A higher value of dNBR indicates more severe damage, while areas with negative dNBR values may indicate regrowth following a fire. The formula used to calculate dNBR is illustrated below:
$\text{dNBR} \,\, \triangle{NBR} = \text{prefireNBR} - \text{postfireNBR}$
dNBR values can vary from case to case, and so, if possible, interpretation in specific instances should also be carried out through field assessment; in order to obtain the best results. However, the United States Geological Survey (USGS) proposed a classification table to interpret the burn severity, which can be seen below (Table 1).
Table 1. Burn severity levels obtained calculating dNBR, proposed by USGS.

Burn severity data and maps can aid in developing emergency rehabilitation and restoration plans - post-fire. They can be used to estimate not only the soil burn severity, but the likelihood of future downstream impacts due to flooding, landslides, and soil erosion.

References
(1) Keeley, J. E. (2009). Fire intensity, fire severity and burn severity: A brief review and suggested usage. International Journal of Wildland Fire, 18(1), 116–126.
(2) Normalized Burn Ratio by Humbold State University (Link here).

relative differenced Normalized Burn Ratio (RdNBR)

A recent variation of the dNBR approach is the relative differenced Normalized Burn Ratio (RdNBR). While the dNBR algorithm measures absolute change between the pre- and post-fire images, the RdNBR algorithm determines burn severity based on pre-fire reflectance and calculates the relative change caused by fire.
$\text{RdNBR} = \frac{\text{NBR}_{\text{prefire}}-\text{NBR}_{\text{postfire}}}{\sqrt{|\text{NBR}_{\text{prefire}}/1000|}}$

Normalized Difference Vegetation Index

reference The Landscape Toolbox: Normalized Difference Vegetation Index
Description
The Normalized Difference Vegetation Index (NDVI) is an index of plant “greenness” or photosynthetic activity, and is one of the most commonly used vegetation indices. Vegetation indices are based on the observation that different surfaces reflect different types of light differently. Photosynthetically active vegetation, in particular, absorbs most of the red light that hits it while reflecting much of the near infrared light. Vegetation that is dead or stressed reflects more red light and less near infrared light. Likewise, non-vegetated surfaces have a much more even reflectance across the light spectrum.

Data for this graph courtesy of the Idaho Chapter of The Nature Conservancy
Reflectance of sunlight from four different land cover types in Hells Canyon, Idaho as measured by a field spectrometer.

By taking the ratio of red and near infrared bands from a remotely-sensed image, an index of vegetation “greenness” can be defined. The Normalized Difference Vegetation Index (NDVI) is probably the most common of these ratio indices for vegetation. NDVI is calculated on a per-pixel basis as the normalized difference between the red and near infrared bands from an image:
$NDVI = \frac{NIR - RED}{NIR + RED}$
where NIR is the near infrared band value for a cell and RED is the red band value for the cell. NDVI can be calculated for any image that has a red and a near infrared band. The biophysical interpretation of NDVI is the fraction of absorbed photosynthetically active radiation.

Many factors affect NDVI values like plant photosynthetic activity, total plant cover, biomass, plant and soil moisture, and plant stress. Because of this, NDVI is correlated with many ecosystem attributes that are of interest to researchers and managers (e.g., net primary productivity, canopy cover, bare ground cover). Also, because it is a ratio of two bands, NDVI helps compensate for differences both in illumination within an image due to slope and aspect, and differences between images due things like time of day or season when the images were acquired. Thus, vegetation indices like NDVI make it possible to compare images over time to look for ecologically significant changes. Vegetation indices like NDVI, however, are not a panacea for rangeland assessment and monitoring. The limitations of NDVI are discussed below.

Values of NDVI can range from -1.0 to +1.0, but values less than zero typically do not have any ecological meaning, so the range of the index is truncated to 0.0 to +1.0. Higher values signify a larger difference between the red and near infrared radiation recorded by the sensor - a condition associated with highly photosynthetically-active vegetation. Low NDVI values mean there is little difference between the red and NIR signals. This happens when there is little photosynthetic activity, or when there is just very little NIR light reflectance (i.e., water reflects very little NIR light).

Application
（1）Vegetation Dynamics / Phenology change over time
（2）Biomass production
（3）Grazing Impacts / Grazing Management
（4）Change Detection
（5）Vegetation / Land Cover Classification
（6）Soil Moisture Estimation
（7）Carbon Sequestration / CO2 flux

Limitations
The NDVI is correlated with a number of attributes that are of interest to rangeland ecologist and managers (e.g., percent cover of bare ground and vegetation, biomass). It is not, however, a direct measure of any of these things - it is a measure of “greenness” produced by the ratio of infrared and red light that is reflected from the surface. While the biophysical interpretation of NDVI is the fraction of absorbed photosynthetically active radiation (see fPAR wiki page) absorbed by the surface, there are a lot of factors that influence the strength of the relationship between NDVI and rangeland ecosystem attributes. These can include: atmospheric conditions, scale of the imagery, vegetation moisture, soil moisture, overall vegetative cover, differences in soil type, management, etc… It is important when using NDVI data in analyses that steps be taken to understand and, to the extent possible, control for factors that might be affecting NDVI values before interpretations of differences in NDVI between areas of within the same area over time can be made.

Light from the soil surface can influence the NDVI values by a large degree. This is of concern in rangeland applications because many semi-arid and arid environments tend to have higher cover of bare ground and exposed rock than other temperate or tropical habitats. Heute and Jackson (1988) found that the soil surface impact on NDVI values was greatest in areas with between 45% and 70% vegetative cover. This limitation was the reason for the development of the several different soil-adjusted vegetation indices (e.g., Soil-adjusted Vegetation Index, Modified Soil-adjusted Vegetation Index), and these indices tend to be preferred for rangeland applications.

In addition to the influence of soil surface at the low-end of vegetation cover, NDVI also suffers from a loss of sensitivity to changes in amount of vegetation at the high-cover/biomass end. This means that as the amount of green vegetation increases, the change in NDVI gets smaller and smaller. So at very high NDVI values, a small change in NDVI may actually represent a very large change in vegetation. This type of sensitivity change is problematic for analysis of areas with a high amount of photosynthetically active vegetation. This could be an issue in rangeland ecosystems if you were interested in assessing changes in riparian areas. In these situations, it may be advisable to use another vegetation index with better sensitivity to high-vegetation cover situations like the Enhanced Vegetation Index or the Wide Dynamic Range Vegetation Index.

Soil-adjusted Vegetation Index

reference The Landscape Toolbox: Soil-adjusted Vegetation Index
Description
In areas where vegetative cover is low (i.e., < 40%) and the soil surface is exposed, the reflectance of light in the red and near-infrared spectra can influence vegetation index values. This is especially problematic when comparisons are being made across different soil types that may reflect different amounts of light in the red and near infrared wavelengths (i.e., soils with different brightness values). The soil-adjusted vegetation index was developed as a modification of the Normalized Difference Vegetation Index (NDVI) to correct for the influence of soil brightness when vegetative cover is low.

The SAVI is structured similar to the NDVI but with the addition of a “soil brightness correction factor,”
$SAVI = \frac{NIR - RED}{NIR + RED + L} * (1 + L)$
where $NIR$ is the reflectance value of the near infrared band, $RED$ is reflectance of the red band, and L is the soil brightness correction factor. The value of $L$ varies by the amount or cover of green vegetation: in very high vegetation regions, $L=0$; and in areas with no green vegetation, $L=1$. Generally, an $L=0.5$ works well in most situations and is the default value used. When $L=0$, then $SAVI = NDVI$.

Calculating the SAVI requires a red and a near infrared band, and specification of the soil brightness correction factor, L. As such, SAVI can be performed on almost any type of imagery that has a red and near infrared band (e.g., Landsat, Ikonos, Quickbird, MODIS).
Limitations
Adjusting for the influence of soils comes at a cost to the sensitivity of the vegetation index. Compared to NDVI, SAVI is generally less sensitive to changes in vegetation (amount and cover of green vegetation), and more sensitive to atmospheric differences.

Modified Soil-adjusted Vegetation Index

Description
The modified soil-adjusted vegetation index (MSAVI) and its later revision, MSAVI2, are soil adjusted vegetation indices that seek to address some of the limitation of NDVI when applied to areas with a high degree of exposed soil surface. The problem with the original soil-adjusted vegetation index (SAVI) is that it required specifying the soil-brightness correction factor (L) through trial-and-error based on the amount of vegetation in the study area. Not only did this lead to the majority of people just using the default L value of 0.5, but it also created a circular logic problem of needing to know what the vegetation amount/cover was before you could apply SAVI which was supposed to give you information on how much vegetation there was. Qi et al. (1994a) developed the MSAVI, and later the MSAVI2 (Qi et al. 1994b) to more reliably and simply calculate a soil brightness correction factor.

The formula for calculating MSAVI itself is the same as the formula for calculating SAVI:
$SAVI = \frac{NIR - RED}{NIR + RED + L} * (1 + L)$
where RED is the red band reflectance from a sensor, NIR is the near infrared band reflectance, and L is the soil brightness correction factor. The difference between SAVI and MSAVI, however, comes in how L is calculated. In SAVI, L is estimated based on how much vegetation there is (but it’s generally left alone at a compromise of 0.5). MSAVI uses the following formula to calculate L:
$L = 1 - \frac{2 * s * (NIR - RED) * (NIR - s * RED)} {(NIR + RED)}$
where s is the slope of the soil line from a plot of red versus near infrared brightness values.

“Feature Space” images like this one are created by graphing the red band reflectance value against the near infrared band values for every pixel in an image. The colors in the image represent how many pixels have that RED:NIR value combination - warmer colors mean more, cooler colors mean fewer pixels. When a red vs. near infrared feature space plot is created, a soil-line can be identified by the combinations of red and near-infrared pixel values where vegetation no longer occurs. The slope of this soil line is used in calculating L in the MSAVI equation.

Qi et al. (1994b), starting with the MSAVI equation, substituted 1-MSAVI(n) for a range of n and then solved the equation recursively until MSAVI(n)=MSAVI(n-1). This yields the following formula, commonly called MSAVI2, which eliminates the need to find the soil line from a feature-space plot or even explicitly specify the soil brightness correction factor:

$MSAVI2=\frac{\left(2 * NIR+1-\sqrt{(2 * NIR+1)^{2}-8 *(NIR - RED)}\right)}{2}$
Limitations
One significant limitation of the MSAVI is that it sacrifices some overall sensitivity to changes in vegetation amount/cover to correct for the soil surface brightness. Hence, MSAVI may not be as sensitive to vegetation change as another index like NDVI. MSAVI would also be more sensitive to differences in atmospheric conditions between areas or times.

Soil-adjusted Total Vegetation Index

Description
The soil-adjusted total vegetation index (SATVI) is a modification of several earlier vegetation indices that correlates with the amount of green and senescent vegetation present on the ground. Commonly used vegetation indices like NDVI and SAVI are sensitive to the amount of green (i.e., photosynthesizing) vegetation, but generally do not correlate well with the amount of senescent or dead vegetation. This prompted Qi et al. (see Marsett et al. 2006) to develop the normalized difference senescent vegetation index (NDSVI) to map dried (non-photosynthesizing) vegetation. They then combined the NDSVI with SAVI to create the SATVI to index both green and senescent vegetation.

The SATVI can be calculated from only image information without any empirical field data. With field data, though, the SATVI can be converted into percent cover estimates of green and senescent vegetation (see Fractional Cover). The SATVI requires bands from three different wavelength regions: red (~630 to 690nm), short-wave infrared #1 (~1,550 to 1,750nm) and short-wave infrared #2 (~2,090 to 2,350nm). These correspond to Landsat TM bands 3, 5, and 7, respectively. The formula for calculating SATVI from Landsat data is:
$SATVI = \frac{\rho_{band5} - \rho_{band3}}{\rho_{band5}+\rho_{band3} + L}*(1+L)-\frac{\rho_{rand7}}{2}$
where $\rho$ is the reflectance value for the TM bands and $L$ is constant (related to the slope of the soil-line in a feature-space plot) that is usually set to 0.5.

Output
The output of SATVI calculation is an index of the amount of green and senescent (i.e., brown) vegetation for each pixel. SATVI values range from -1 (no green or senescent vegetation) to +1 (complete coverage by green vegetation).

Enhanced Vegetation Index

Description
The enhanced vegetation index (EVI) was developed as an alternative vegetation index to address some of the limitations of the NDVI. The EVI was specifically developed to:

• be more sensitive to changes in areas having high biomass (a serious shortcoming of NDVI),
• reduce the influence of atmospheric conditions on vegetation index values, and
• correct for canopy background signals.

EVI tends to be more sensitive to plant canopy differences like leaf area index (LAI), canopy structure, and plant phenology and stress than does NDVI which generally responds just to the amount of chlorophyll present. With the launch of the MODIS sensors, NASA adopted EVI as a standard MODIS product that is distributed by the USGS (see below).

EVI is calcualted as
$EVI=2.5 * \frac{(NIR - RED)}{\left(NIR+C_{1}^{*} RED-C_{2} * BLUE+L\right)}$
where NIR, RED, and BLUE are atmospherically-corrected (or partially atmospherically-corrected) surface reflectances, and C1, C2, and L are coefficients to correct for atmospheric condition (i.e., aerosol resistance). For the standard MODIS EVI product, L=1, C1=6, and C2=7.5.

Limitations
One of the biggest current limitations to implementing EVI is that it needs a blue band in order to be calculated. Not only does this limit the sensors that EVI can be applied to (e.g., ASTER has no blue band), but the blue band typically has a low signal-to-noise ratio. Research is ongoing to develop a two-band EVI that can be calculated from just red and near infrared bands (see Jiang et al. 2008).

Fractional vegetation cover

The fractional vegetation cover (FVC) is defined as the vertical projection areal proportion of the landscape occupied by green vegetation [Gitelson et al., 2002]. Many studies have demonstrated that the fractional vegetation cover (FVC) has a linear relationship with the normalized difference vegetation index (NDVI) and shows a continuous scale from 0% (bare soil) to 100% (pure green vegetation) [ Enßle et al., 2014; Gitelson et al., 2002].
$FVC = \frac{NDVI − NDVI_s}{NDVI_v − NDVI_s}$
where $NDVI$ is given by: $NDVI$ = ($\rho_{nir}$ − $\rho_{red}$)/($\rho_{nir}$ − $\rho_{red}$); $\rho_{nir}$ and $\rho_{red}$ are corrected reflectance obtained from the sensor bands located in the near infrared (NIR) and the red spectral regions for each pixel within an image. The $NDVI_s$ and $NDVI_v$ are values of the NDVI for bare soil (FVC = 0) and pure green vegetation (FVC = 1) within an image, respectively.

The accuracy of FVC estimates mainly depends on selection of the end members; endmember variables include bare soil and dense vegetation pixels, the saturation effect of vegetation indices such as NDVI at high leaf area index (LAI) levels as well as effects on satellite imagery due to changes in season and variations in topography and atmosphere [Enßle et al., 2014].

Radiative Transfer Modeling

All radiative transfer modeling is ultimately based on the fundamental equation of radiative transfer, which relates the change in radiation intensity $I_\nu$ along a ray path to local absorption $k_\nu$ and volume emission $j_\nu$,
${1\over k_\nu}{dI_\nu\over ds}=-I_\nu+{j_\nu\over k_\nu}$

For a nonscattering atmosphere in local thermodynamical equilibrium, $j_\nu/k_\nu$ equals the Planck function $B_\nu$ at the local temperature T. The solution to the above equation is then given by
$I_\nu(s_0)=I_\nu(0)e^{-\int_0^{s_0} k_\nu(s)\,ds}+\int_0^{s_0} K_\nu(s)B_\nu(T(s))e^{-\int_s^{s_0} k_\nu(s')\,ds'}\,ds$

where $s_0$ is the optical path above the surface and $I_\nu(0)$ is the surface brightness. For observations of emission from planetary atmospheres, the integration is instead begun at the top of the atmosphere and continued downward until either the surface or some very large optical depth is reached. In the latter case, the ``surface emission’’ term can be neglected.
For numerical evaluation of the absorption due to multiple absorbers in a real atmosphere divided into N layers, it is useful to rewrite the solution in a discretized form,
$I_\nu=\sum_{i=1}^N B_\nu(T_i)(1-e^{-\Delta\tau_i/\mu_i})e^{-\sum_{j=1}^i (\Delta \tau_j/\mu_j)}$
where $T_i$ is the temperature of layer $i$,
$\Delta\tau_i\equiv \int k_\nu(p){dz\over dp}\,dp$

is the total optical depth in layer $i$, $k_\nu(p)$ is the absorption coefficient at pressure level $p$ due to all contributing species (with units of cm-1), and
$\mu\equiv \cos\theta = {dz\over ds}$

is the cosine of the emission angle $\theta$ (the angle between the local vertical and the line-of-sight).
In our atmospheric model, the opacity is computed starting at some top pressure level. The equations of radiative transfer are then applied to find the contribution to the brightness temperature and optical depth from this level. Calculation then proceeds to the next pressure level, and so on, until a specified (large) optical depth is reached. The atmospheric levels are treated as locally spherical for the purposes of the radiative transfer model (since spheroidal layers become geometrically unmanageable for oblique ray paths), but a full Darwin-de Sitter spheroid formalism is used for calculating the net gravitational acceleration at each layer. In order to compute $\mu$, the planetocentric latitude is converted to planetographic latitude as in de Pater and Massie (1985).

It is often useful to view () as a sum of contributions of various atmospheric layers to the total radiation emitted at the top. This can be done by defining the so-called weighting function by
$W_i\equiv (1-e^{-\Delta\tau_i/\mu_i})e^{-\sum_{j=1}^i (\Delta \tau_j/\mu_j)}.$

Then solution can be written in the particularly simple form
$I_\nu=\sum_{i=1}^N W_iB_\nu(T_i)$

and a plot of $W_i$ versus altitude immediately shows the atmospheric levels from which the majority of the observed flux originates.
In order to calculate the radiation emitted by an atmosphere, it is necessary to specify the temperature and compositional structure as a function of altitude.

Linear Spectral Unmixing

The linear spectral unmixing is used to determine the relative abundance of materials that are depicted in multispectral or hyperspectral imagery based on the materials’ spectral characteristics.

The reflectance at each pixel of the image is assumed to be a linear combination of the reflectance of each material (or endmember) present within the pixel. For example, if 25% of a pixel contains material A, 25% of the pixel contains material B, and 50% of the pixel contains material C, the spectrum for that pixel is a weighted average of 0.25 times the spectrum of material A, plus 0.25 times the spectrum of material B, plus 0.5 times the spectrum of material C.

The number of endmembers must be less than the number of spectral bands, and all of the endmembers in the image must be used.

Accuracy Metrics

There are many different ways to look at the thematic accuracy of a classification. The error matrix allows you calculate the following accuracy metrics:

• Overall Accuracy and Error
• Errors of omission
• Errors of commission
• User’s accuracy
• Producer’s accuracy
• Accuracy statistics (e.g., Kappa)
  
  We will use the same error matrix show above to calculate the various accuracy metrics.

Overall Accuracy

Overall Accuracy is essentially tells us out of all of the reference sites what proportion were mapped correctly. The overall accuracy is usually expressed as a percent, with 100% accuracy being a perfect classification where all reference site were classified correctly. Overall accuracy is the easiest to calculate and understand but ultimately only provides the map user and producer with basic accuracy information.

To calculate the overall accuracy, you sum up the number of the diagonal elements represent the correctly classified sites and divide it by the total number of reference site.

Example based on the above error matrix:

Number of correctly classified site: 21 + 31+ 22 = 74
Total number of reference sites = 95
Overall Accuracy = 74/95 = 77.9%

We could also express this as an error percentage, which would be the complement of accuracy (error + accuracy = 100%. In the above example the error would be the number of sites incorrectly classified divided by 95 or 21/95 = error.

Error Types

Errors of Omission

Errors of omission refer to reference sites that were left out (or omitted) from the correct class in the classified map. The real land cover type was left out or omitted from the classified map. Error of omission is sometime also referred to as a Type I error. An error of omission in one category will be counted as an error in commission in another category. Omission errors are calculated by reviewing the reference sites for incorrect classifications. This is done by going down the columns for each class and adding together the incorrect classifications and dividing them by the total number of reference sites for each class. A separate omission error is generally calculated for each class. This will allow us to evaluate the classification accuracy for each class.

Omission Error Example based on the above error matrix:

Water: Incorrectly classified reference sites: 5 + 7 = 12 Total # of reference sites = 33 Omission Error = 12/33 = 36%
Forest: Incorrectly classified reference sites: 6 + 2 = 8 Total # of reference sites = 39 Omission Error = 8/39 = 20%
Water: Incorrectly classified reference sites: 0 + 1 = 1 Total # of reference sites = 23 Omission Error = 1/23 = 4%

Errors of Commission

Errors of omission are in relation to the classified results. These refer sites that are classified as to reference sites that were left out (or omitted) from the correct class in the classified map. Commission errors are calculated by reviewing the classified sites for incorrect classifications. This is done by going across the rows for each class and adding together the incorrect classifications and dividing them by the total number of classified sites for each class.

Commission Error Example based on the above error matrix:

Water: Incorrectly classified sites: 6 + 0 = 6 Total # of classified sites = 27 Omission Error = 6/27 = 22%
Forest: Incorrectly classified sites: 5 + 1 = 6 Total # of classified sites = 37 Omission Error = 6/37 = 16%
Water: Incorrectly classified sites: 7 + 2 = 9 Total # of classified sites = 31 Omission Error = 9/31 = 29%

Other Accuracy Metrics

Producer’s Accuracy

Producer’s Accuracy is the map accuracy from the point of view of the map maker (the producer). This is how often are real features on the ground correctly shown on the classified map or the probability that a certain land cover of an area on the ground is classified as such. The Producer’s Accuracy is complement of the Omission Error, Producer’s Accuracy = 100%-Omission Error. It is the number of reference sites classified accurately divided by the total number of reference sites for that class.

Producer’s Accuracy Example based on the above error matrix:

Water: Correctly classified reference sites = 21 Total # of reference sites = 33 Producer’s Accuracy = 21/33 = 64%
Forest: Correctly classified reference sites = 31 Total # of reference sites = 39 Producer’s Accuracy = 31/39 = 80%
Water: Correctly classified reference sites = 22 Total # of reference sites = 23 Producer’s Accuracy = 22/23 =96%

User’s Accuracy

The User’s Accuracy is the accuracy from the point of view of a map user, not the map maker. the User’s accuracy essentially tells use how often the class on the map will actually be present on the ground. This is referred to as reliability. The User’s Accuracy is complement of the Commission Error, User’s Accuracy = 100%-Commission Error. The User’s Accuracy is calculating by taking the total number of correct classifications for a particular class and dividing it by the row total.

User’s Accuracy Example based on the above error matrix:

Water: Correctly classified sites = 21 Total # of classified sites = 27 Omission Error = 21/27 = 78%
Forest: Incorrectly classified sites = 31 Total # of classified sites = 37 Omission Error = 31/37 = 84%
Water: Incorrectly classified sites = 22 Total # of classified sites = 31 Omission Error = 22/31 = 70%

Comparing User’s and Producer’s Accuracy
The user and producer accuracy for any given class typically are not the same. In the above examples the producer’s accuracy for the Urban class was 96% while the user’s accuracy was 71%. this means that even though 96% of the reference urban areas have been correctly identified as “urban”, only 71% percent of the areas identified as “urban” in the classification were actually urban. Water (7) and forest (2) areas were mistakenly classified as urban. By analyzing the various accuracy and error metrics we can better evaluate the analysis and classification results. Often you might have very high accuracy for certain classes, while others may have poor accuracy. The information is important so you and other users can evaluate how appropriate it is to use the classified map.

Kappa Coefficient

The Kappa Coefficient is generated from a statistical test to evaluate the accuracy of a classification. Kappa essentially evaluate how well the classification performed as compared to just randomly assigning values, i.e. did the classification do better than random. The Kappa Coefficient can range from -1 t0 1. A value of 0 indicated that the classification is no better than a random classification. A negative number indicates the classification is significantly worse than random. A value close to 1 indicates that the classification is significantly better than random.

McNemar’s test

it was first published in a Psychometrika article in 1947. It was created by Quinn McNemar, who was a professor in the Psychology and Statistics department at Stanford University. This non-parametric (distribution-free) test assesses if a statistically significant change in proportions have occurred on a dichotomous trait at two time points on the same population. It is applied using a 2×2 contingency table with the dichotomous variable at time 1 and time 2. In medical research, if a researcher wants to determine whether or not a particular drug has an effect on a disease (e.g., yes vs. no), then a count of the individuals is recorded (as + and – sign, or 0 and 1) in a table before and after being given the drug. Then, McNemar’s test is applied to make statistical decisions (using the Chi Square test statistic) as to whether or not a drug has an effect on the disease.
Procedure:
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We assume that the row total is equal to the column total. In other words:
$\begin{array}{l} A + B = A + C\\ C + D = B + D \end{array}$
In this case, we will cancel the A and D equation and this implies that B=C. By using this equation, we will calculate the test as:
$\chi^2 = \frac{(B-C)^2}{B+C}$
Here, $\chi^2$ statistic has with one degree of freedom.

Basic definitions

GIS definition

There are several definitions of GIS (Geographic Information Systems), which is not simply a program. In general, GIS are systems that allow for the use of geographic information (data have spatial coordinates). In particular, GIS allow for the view, query, calculation and analysis of spatial data, which are mainly distinguished in raster or vector data structures. Vector is made of objects that can be points, lines or polygons, and each object can have one ore more attribute values; a raster is a grid (or image) where each cell has an attribute value (Fisher and Unwin, 2005). Several GIS applications use raster images that are derived from remote sensing.

Remote Sensing definition

A general definition of Remote Sensing is “the science and technology by which the characteristics of objects of interest can be identified, measured or analyzed the characteristics without direct contact” (JARS, 1993).

Usually, remote sensing is the measurement of the energy that is emanated from the Earth’s surface. If the source of the measured energy is the sun, then it is called passive remote sensing, and the result of this measurement can be a digital image (Richards and Jia, 2006). If the measured energy is not emitted by the Sun but from the sensor platform then it is defined as active remote sensing, such as radar sensors which work in the microwave range (Richards and Jia, 2006).

The electromagnetic spectrum is “the system that classifies, according to wavelength, all energy (from short cosmic to long radio) that moves, harmonically, at the constant velocity of light” (NASA, 2013). Passive sensors measure energy from the optical regions of the electromagnetic spectrum: visible, near infrared (i.e. IR), short-wave IR, and thermal IR (see Figure Electromagnetic-Spectrum).

The interaction between solar energy and materials depends on the wavelength; solar energy goes from the Sun to the Earth and then to the sensor. Along this path, solar energy is (NASA, 2013):

Transmitted - The energy passes through with a change in velocity as determined by the index of refraction for the two media in question.

• Absorbed - The energy is given up to the object through electron or molecular reactions.
• Reflected - The energy is returned unchanged with the angle of incidence equal to the angle of reflection. Reflectance is the ratio of reflected energy to that incident on a body. The wavelength reflected (not absorbed) determines the color of an object.
• Scattered - The direction of energy propagation is randomly changed. Rayleigh and Mie scatter are the two most important types of scatter in the atmosphere.
• Emitted - Actually, the energy is first absorbed, then re-emitted, usually at longer wavelengths. The object heats up.

Sensors

Sensors can be on board of airplanes or on board of satellites, measuring the electromagnetic radiation at specific ranges (usually called bands). As a result, the measures are quantized and converted into a digital image, where each picture elements (i.e. pixel) has a discrete value in units of Digital Number (DN) (NASA, 2013). The resulting images have different characteristics (resolutions) depending on the sensor. There are several kinds of resolutions:

• Spatial resolution, usually measured in pixel size, “is the resolving power of an instrument needed for the discrimination of features and is based on detector size, focal length, and sensor altitude” (NASA, 2013); spatial resolution is also referred to as geometric resolution or IFOV;
• Spectral resolution, is the number and location in the electromagnetic spectrum (defined by two wavelengths) of the spectral bands (NASA, 2013) in multispectral sensors, for each band corresponds an image;
• Radiometric resolution, usually measured in bits (binary digits), is the range of available brightness values, which in the image correspond to the maximum range of DNs; for example an image with 8 bit resolution has 256 levels of brightness (Richards and Jia, 2006);
• For satellites sensors, there is also the temporal resolution, which is the time required for revisiting the same area of the Earth (NASA, 2013).

Radiance and Reflectance

Sensors measure the radiance, which corresponds to the brightness in a given direction toward the sensor; it useful to define also the reflectance as the ratio of reflected versus total power energy.

Spectral Signature

The spectral signature is the reflectance as a function of wavelength (see Figure Spectral Reflectance Curves of Four Different Targets); each material has a unique signature, therefore it can be used for material classification (NASA, 2013).

Spectral Reflectance Curves of Four Different Targets (from NASA, 2013)

Land Cover

Land cover is the material at the ground, such as soil, vegetation, water, asphalt, etc. (Fisher and Unwin, 2005). Depending on the sensor resolutions, the number and kind of land cover classes that can be identified in the image can vary significantly.

Land Cover Classification

This chapter provides basic definitions about land cover classifications.

Supervised Classification

A semi-automatic classification (also supervised classification) is an image processing technique that allows for the identification of materials in an image, according to their spectral signatures. There are several kinds of classification algorithms, but the general purpose is to produce a thematic map of the land cover.

Image processing and GIS spatial analyses require specific software such as the Semi-Automatic Classification Plugin for QGIS.

A multispectral image processed to produce a land cover classification (Landsat image provided by USGS)

Color Composite

Often, a combination is created of three individual monochrome images, in which each is assigned a given color; this is defined color composite and is useful for photo interpretation (NASA, 2013). Color composites are usually expressed as:

$“R\,\, G\,\, B = B_r\,\, B_g\,\, B_b”$

where:
R stands for Red;
G stands for Green;
B stands for Blue;
Br is the band number associated to the Red color;
Bg is the band number associated to the Green color;
Bb is the band number associated to the Blue color.
The following Figure Color composite of a Landsat 8 image shows a color composite “R G B = 4 3 2” of a Landsat 8 image (for Landsat 7 the same color composite is R G B = 3 2 1; for Sentinel-2 is R G B = 4 3 2) and a color composite “R G B = 5 4 3” (for Landsat 7 the same color composite is R G B = 4 3 2; for Sentinel-2 is R G B = 8 4 3). The composite “R G B = 5 4 3” is useful for the interpretation of the image because vegetation pixels appear red (healthy vegetation reflects a large part of the incident light in the near-infrared wavelength, resulting in higher reflectance values for band 5, thus higher values for the associated color red).

Spectral Angle Mapping

Therefore a pixel belongs to the class having the lowest angle, that is:
The Spectral Angle Mapping calculates the spectral angle between spectral signatures of image pixels and training spectral signatures. The spectral angle $\theta$ is defined as (Kruse et al., 1993):
$\theta(x, y)=\cos ^{-1}\left(\frac{\sum_{i=1}^{n} x_{i} y_{i}}{\left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} *\left(\sum_{i=1}^{n} y_{i}^{2}\right)^{\frac{1}{2}}}\right)$
Where:

• x = spectral signature vector of an image pixel;
• y = spectral signature vector of a training area;
• n = number of image bands.
  Therefore a pixel belongs to the class having the lowest angle, that is:
  $x \in C_{k} \Longleftrightarrow \theta\left(x, y_{k}\right)<\theta\left(x, y_{j}\right) \,\,\,\,\forall k \neq j$
  where:
• $C_k$ = land cover class $k$;
• $y_k$ = spectral signature of class $k$;
• $y_j$ = spectral signature of class $j$.
  
  Spectral Angle Mapping example
  In order to exclude pixels below this value from the classification it is possible to define a threshold $T_i$
  $\begin{array}{r}{x \in C_{k} \Longleftrightarrow \theta\left(x, y_{k}\right)<\theta\left(x, y_{j}\right) \forall k \neq j} \\ {\qquad \text { and }} \\ {\theta\left(x, y_{k}\right)<T_{i}}\end{array}$
  Spectral Angle Mapping is largely used, especially with hyperspectral data.

Multiple Endmember Spectral Mixture Analysis

Description
Multiple Endmember Spectral Mixture Analysis (MESMA) is a technique for estimating the proportion of each pixel that is covered by a series of known cover types - in other words, it seeks to determine the likely composition of each image pixel. Pixels that contain more than one cover type are called mixed pixels. “Pure” pixels contain only one feature or class. For example, a mixed pixel might contain vegetation, bare ground, and soil crust. A pure pixel would contain only one feature, such as vegetation. Mixed pixels can cause problems in traditional image classifications (e.g., supervised or unsupervised classification) because the pixel belongs to more than one class but can be assigned to only a single class. One way to address the problem of mixed pixels is to use subpixel analysis with hyperspectral imagery.
Spectral Endmembers
Subpixel analysis methods determines the component parts of mixed pixels by predicting the proportion of a pixel that belongs to a particular class or feature based on the spectral characteristics of its endmembers. It converts radiance to fractions of spectral endmembers that correspond to features on the ground.

Spectral endmembers are the “pure” spectra corresponding to each of the land cover classes. Ideally, spectral endmembers account for most of the image’s spectral variability and serve as a reference to determine the spectral make up of mixed pixels. Thus the definition of land cover classes, and selection of appropriate endmembers for each of these classes, are both critical in MESMA. Endmembers obtained from the actual image are generally preferred because no calibration is needed between selected endmembers and the measured spectra. These endmembers are assumed to represent the purest pixels in the image.

Spatial Autocorrelation

Spatial autocorrelation in GIS helps understand the degree to which one object is similar to other nearby objects and can be measured by Moran’s I (Index). Spatial autocorrelation definition measures how much close objects are in comparison with other close objects. Moran’s I can be classified as positive, negative and no spatial auto-correlation. Positive spatial auto-correlation occurs when Moran’s I is close to +1. Negative spatial auto-correlation occurs when Moran’s I is near -1.

Maximum value composite

Purpose: utilize the Cell Statistics in the ArcGIS to realize the composition with maximum pixels of NDVI of many rasters.
Procedure: ArcToolbox>>Spatial Analyst Tools>>Local>>Cell Statistics
Problem: The type of pixel value in NDVI raster is float, if directly using the Cell Statistics, the calculation result is Int( 1 or 0), not float.
Resolve method: 運用Cell Statistics求取最大值運算前先將浮點值NDVI值乘以10000變成Int型，再用Cell Statistics求最大值，得到Int型最大值後再除以10000.0還原成float浮點型。flotat->Int型及Int->float型轉換可利用Raster Calculator實現。

multi-collinearity

Multicollinearity generally occurs when there are high correlations between two or more predictor variables. This creates redundant information, skewing the results in a regression model.

It’s more common for multicollineariy to rear its ugly head in observational studies; it’s less common with experimental data. When the condition is present, it can result in unstable and unreliable regression estimates.
What Causes Multicollinearity?
The two types are:

• Data-based multicollinearity: caused by poorly designed experiments, data that is 100% observational, or data collection methods that cannot be manipulated. In some cases, variables may be highly correlated (usually due to collecting data from purely observational studies) and there is no error on the researcher’s part.
• Structural multicollinearity: caused by the researcher, factitiously creating new predictor variables.

Causes for multicollinearity can also include: (1) Insufficient data; (2)Dummy variables may be incorrectly used; (3) Including a variable in the regression that is actually a combination of two other variables; (4) Including two identical (or almost identical) variables.

variance inflation factor

A variance inflation factor(VIF) detects multicollinearity in regression analysis. The VIF estimates how much the variance of a regression coefficient is inflated due to multicollinearity in the model.
VIFs are calculated by taking a predictor, and regressing it against every other predictor in the model. This gives you the R-squared values, which can then be plugged into the VIF formula. “$i$” is the predictor you’re looking at (e.g. $x_1$ or $x_2$):
$\mathrm{VIF}=\frac{1}{1-R_{i}^{2}}$
Variance inflation factors range from 1 upwards. The numerical value for VIF tells you (in decimal form) what percentage the variance (i.e. the standard error squared) is inflated for each coefficient. For example, a VIF of 1.9 tells you that the variance of a particular coefficient is 90% bigger than what you would expect if there was no multicollinearity — if there was no correlation with other predictors.
A rule of thumb for interpreting the variance inflation factor:

• 1 = not correlated.
• Between 1 and 5 = moderately correlated.
• Greater than 5 = highly correlated.

Exactly how large a VIF has to be before it causes issues is a subject of debate. What is known is that the more your VIF increases, the less reliable your regression results are going to be. In general, a VIF above 10 indicates high correlation and is cause for concern. Some authors suggest a more conservative level of 2.5 or above. Sometimes a high VIF is no cause for concern at all.

Reference

• Marsett, R.C., Qi, J., Heilman, P., Biedenbender, S.H., Watson, M.C., Amer, S., Weltz, M., Goodrich, D., Marsett, R. 2006. Remote sensing for grassland management in the arid southwest. Rangeland Ecology and Management 59:530-540.
• Jiang, Z., A.R. Huete, K. Didan, and R. Miura. 2008. Development of a two-band enhanced vegetation index without a blue band. Remote Sensing of the Environment 112(10):3833-3845.
• https://wiki.landscapetoolbox.org/doku.php/remote_sensing_methods:soil-adjusted_vegetation_index
• https://wiki.landscapetoolbox.org/doku.php/remote_sensing_methods:normalized_difference_vegetation_index
• Enßle, F., Heinzel, J., Koch, B., 2014. Accuracy of vegetation height and terrain elevation derived from ICESat/GLAS in forested areas. Int. J. Appl Earth Observ. Geoinf.
  31, 37–44.
• Gitelson, A.A., Kaufman, Y.J., Stark, R., Rundquist, D., 2002. Novel algorithms for remote estimation of vegetation fraction. Remote Sens. Environ. 80, 76–87.