# EVI from First Principles

## Introduction: ARVI

In the 1980s and 1990s, atmospheric correction mechanisms were less developed than today, specifically in the case of aerosols. Aerosols are a problem because some red light, that might otherwise be absorbed by vegetation, gets reflected by aerosol particles and detected by sensors on satellites. This can lead to an overstated measure of red reflectance and hence an underestimate of vegetation levels.

Blue light however, is far less susceptible to aerosol effects. It tends to pass through aerosols with minimal interference. Therefore, to minimize aerosol effects, one might consider building a vegetation index using blue reflectance instead of red. The problem with that is the reflectance differential between soil and vegetation for blue light is not as significant as the difference for red light; i.e. its harder to tell the difference between vegetation and soil using blue light compared to red light:

However, you can still use blue light to help deal with aerosol effects. The idea is this: when there is zero aerosol interference, there will be some baseline difference between how much red light is reflected and how much blue light is reflected from the Earth’s surface. That difference will of course not be completely insensitive to the relative amounts of soil and vegetation present, but that difference will actually be more sensitive to aerosol interference.

When there is aerosol interference, red reflectance will increase and blue will not. Therefore the difference between red reflectance and blue reflectance will correlate with the amount of aerosol interference.

This leads to the key idea behind the Atmospherically Resistant Vegetation Index (ARVI): The increase in red reflectance due to aerosols will be proportional to the difference between red and blue reflectance. We can use this relationship to correct the value for RED:

$Red_{corrected} = RED - \gamma (BLUE - RED)$

Because RED is overstated due to aerosol effects, we apply a correction by reducing it by an amount proportional to the amount of aerosol interference. γ is determined by a calibration.

If we take the NDVI formula:

$NDVI = \frac{NIR - Red}{NIR + Red}$

And replace RED, with REDcorrected, we get:

$NDVI = \frac{NIR - Red_{corrected}}{NIR + Red_{corrected}}$

The above formula is the atmospherically resistant vegetation index, (ARVI).

## The Fundamental Innovation of ARVI

The key innovation of ARVI is the inclusion of the blue band in the index. Let’s rearrange the formula slightly to understand what is actually happening:

$ARVI = \frac{NIR - Red_{corrected}}{NIR + Red_{corrected}}$

Where:

$Red_{corrected} = RED - \gamma (BLUE - RED)$

Expanding:

$Red_{corrected} = (1 + \gamma)RED - \gamma BLUE$

Defining C1=1+γ, and C2=γ, we get:

$Red_{corrected} = C_{1}RED - C_{2}BLUE$

Therefore:

$ARVI = \frac{NIR - (C_{1}RED - C_{2}BLUE)}{NIR + (C_{1}RED - C_{2}BLUE)}$

Notice ARVI is inversely proportional to the C1RED-C2BLUE, i.e.:

$ARVI \propto \frac{1}{C_{1}RED - C_{2}BLUE}$

The math tells us, if you want to use the blue band in an index to achieve atmospheric aerosol correction, that index needs to be monotonically decreasing as C1RED-C2BLUE increases. We’ll see shortly that EVI harnesses this idea.

## The Fundamental Innovation of SAVI

Recall NDVI and SAVI:

$NDVI = \frac{NIR - Red}{NIR + Red}$

$SAVI = \frac{(NIR - RED)}{(NIR + RED + L)} \times (1 + L)$

The fundamental innovation of SAVI is the addition of a constant term in the denominator which mitigates soil effects. (See SAVI from First Principles.)

## Derivation of EVI

The idea of the Enhanced Vegetation Index (EVI) is to construct an index that is insensitive to soil effects (like SAVI) and insensitive to aerosol effects (like ARVI). To achieve this one would modify the basic NDVI formula such that it has a constant term in the denominator and is inversely proportional to C1RED-C2BLUE. This brings as directly to the formula for EVI:

$EVI = G \times \frac{NIR - RED}{NIR + C_{1}RED - C_{2}BLUE + L}$

The values for C1, C2 and L follow from calibrating against ground truth. G is a convenience scalar to get the index to fall between -1 and 1. From extensive calibrations since the index’s advent in the early part of this century, the best general purpose calibration for EVI is C1=6.5,C2=7.5, L=1.0 and G=2.5.

EVI thus promises to be a more robust index than NDVI because it leverages the soil effect mitigation technique of SAVI and the aerosol mitigation technique of ARVI.

## EVI and Saturation

EVI has the desirable qualities of SAVI. However, it actually has better saturation characteristics than SAVI. This is a mathematical artifact of the EVI formula.

Recall from SAVI from First Principles that saturation is a consequence of RED reflectance falling to near zero once there is a certain amount of chlorophyll present; often well before vegetation is at its maximum quality or quantity.

Now also recall from NDVI from First Principles that near-infrared reflection (NIR) is correlated with the quantity of plant matter present and red reflection (RED) is negatively correlated with the amount chlorophyll present. This suggests an alternative way to mitigate saturation: make the index more sensitive to NIR and less sensitive to RED.

Mathematically, EVI is less sensitive to RED than SAVI is which is why it has better saturation characteristics. To understand why, let’s compute the sensitivity of EVI to RED, ie ∂EVI/∂RED:

$\frac{\delta EVI}{\delta RED} = G \times \frac{-1(NIR + C_{1}RED - C_{2}BLUE + L) - C_{1}(NIR - RED))}{(NIR + C_{1}RED - C_{2}BLUE + L)^2}$

= $\frac{\delta EVI}{\delta RED} = G \times \frac{(-1(NIR - C_{2}BLUE + L) - C_{1}(NIR))}{(NIR + C_{1}RED - C_{2}BLUE + L)^2}$

The thing to notice here is that RED appears only in the denominator of the derivative. This means that, as C1 increases, ∂EVI/∂RED decreases; i.e. As you scale RED up in the denominator of the EVI formula, EVI’s sensitivity to it decreases and thus its saturation threshold increases.

## EVI2

Since the advent of EVI, atmospheric correction technologies have improved, especially with respect to aerosols. Today, the best atmospheric correction mechanisms, like those used by NASA and ESA, mitigate aerosol effects eliminating the need to even introduce the blue band into an index. (The quality of atmospheric correction on commercial satellites varies; practitioners have to be alert to this and use blue band information when needed.)

So today, EVI’s only advantage over SAVI is its better saturation characteristics. But that is entirely attributable to scaling RED in the denominator. The SAVI formula can easily be modified and generalized to include such a scalar:

$SAVI_{improved} = G \times \frac{NIR - RED}{NIR + C \times RED + L}$

Recalibration of this modified formula to maximize correlation with ground truth is the definition of EVI2. (The calibration will give C=2.4, L=1 and G=2.5.) The EVI2 formula is thus:

The EVI2 formula is $EVI2 = 2.5 \times \frac{ NIR - RED}{NIR + 2.4 \times RED + 1}$

When red and near-infrared measurements derive from modern remote sensors, EVI2 is equivalent to EVI.

## Conclusion

EVI leverages the blue band to mitigate aerosol effects. But modern atmospheric correction mechanisms do that anyway making the use of the blue band redundant today. The one valuable innovation from EVI worth keeping, however, is the idea of designing an index with slightly less sensitivity to the red band than NDVI or SAVI. EVI2 is a simple evolution of SAVI, that does just that.

In real world situations where soil effects are a factor and saturation a risk, EVI2 is unequivocally superior to NDVI and SAVI.

## Epilogue

This article takes a straight path from NDVI to SAVI to ARVI to EVI to EVI2 because a first principles understanding of the formulas is the objective. Were this an article on the actual path from NDVI to EVI2, there would be many more steps along the way and lots of deadends. SARVI (soil and atmosphere adjusted vegetation index) and MNDVI (Modified NDVI) are two (now redundant) indices that were steps on the path to EVI2. There are dozens of other indices that were contemplated but were simply less effective than the ones already mentioned.