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I am using images from two different sensors, Landsat 8 OLI (January 2017) and ETM+ (January 2007) to compute various vegetation indices.

These indices involve RED and NIR band to scale vegetation density and intensity. I have noticed the difference in wavelengths of both sensors’ required bands as follows;

Wavelength (micrometers)

Landsat 7 ETM+

Band 3 – Red : 0.63-0.69

Band 4 – Near Infrared (NIR) : 0.77-0.90

Landsat 8 OLI

Band 4 – Red : 0.64-0.67

Band 5 – Near Infrared (NIR) : 0.85-0.88

See Reference https://landsat.usgs.gov/what-are-band-designations-landsat-satellites

Keeping in view this wavelength difference of both sensors' RED & NIR Bands, I am interested to know whether this difference will affect my intended results?

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This question really is a bit too large for a post here, but thankfully, the research has already been done (multiple times) and the conclusion is usually something along the lines of:

The difference of vegetation indices comparison showed that there were subtle differences between both sensors, which demonstrated high similarity. LSWI, NBR and NDVI had average difference values less than ±0.05 and a smaller standard deviation.

And goes on to state:

Additionally, correlation analysis of the vegetation indices indicated that both sensors had a very high linear correlation coefficient, with R2 greater than 0.96. The subtle differences and high correlation of vegetation indices demonstrated that ETM+ and OLI imagery can be used as complementary data.

Source: http://www.mdpi.com/2072-4292/6/1/310/pdf

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