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I would to do pixel based 10 year NDVI time series analysis using MODIS data. I’ve 3 seasons in a year. Should I average all values in the study area per season?

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From the question, it is not totally clear if the MODIS data is daily or monthly, ... But from experience, I would suggest to keep it at the current level and then look for patterns and trends in the results. Then, if you see fit, try to go for seasonal patterns.

Averaging makes unwanted changes to the data and masks abnormalities (things that you might be interested). But it also reduces the noise and makes a good base for regression and correlation models.

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