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I am using Google Earth Engine to compute NDWI time series, in particular by plotting "Landsat 7 Collection 1 Tier 1 8-Day NDWI Composite" data values

The product description says: These Landsat 7 Collection 1 Tier 1 composites are made from Tier 1 orthorectified scenes, using the computed top-of-atmosphere (TOA) reflectance.
It is derived from the Near-IR band and a second IR band, ≈1.24μm when available and the nearest available IR band otherwise.

This seems strange to me, since Landsat 7 has no 1.24μm band, but only a SWIR band at 1.55-1.75 μm https://landsat.usgs.gov/what-are-band-designations-landsat-satellites

How does the Google algorithm calculates it?

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It is computed as (NIr1 - NIr2) / (NIr1 + NIr2). For ETM+, NIr1 is B4 and NIr2 is B5. Note that this is the Gao (1996) version of NDWI (which may or may not be suitable for Landsat) and should not be confused with the McFeeters (1996) NDWI.

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    This confirms my guess. But it is not Gao Version. Gao used 0.86-μm and the 1.24-μm wavelenght, but considering ETM+ the computation occurs with 0.85 and 1.65. It should be labeled as NDII - Normalized Difference Infrared Index. See here for more info:hydrol-earth-syst-sci.net/20/3361/2016
    – giulio
    Commented Jul 13, 2018 at 12:03

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