# Introduction

The origins of correlation. Music by Bensound

Correlation is a statistical measure of how two, or more, variables vary together. In other words, correlation answers the question, “If I know the value of one variable can I predict the value of the other?” As we saw in the video above, in FIDUCEO we consider error correlation, that is the correlation between the error in one measured Earth radiance (one band/pixel) with that of another measured Earth radiance (another band/pixel). It is important to distinguish value correlation from error correlation.

In measurement, error correlation is introduced whenever there is a common error between measured values due to a common measurement.

In EO applications, the measured value in each pixel of an image comes from a separate measurement. In transforming Level 1 to Level 2 products, radiances from different spectral bands may combined for a given pixel. In Level 2 to Level 3+ processing, data from different pixels are combined. Thus, correlation between errors needs to be considered. In this regard, we distinguish three types of error:

• Random errors, (strictly, independent random errors) for which it is not possible to predict the error in a measured value based on knowledge of the error in another measured value. We could not correct for random errors by applying a correction, even in principle, and they are entirely uncorrelated from one measurement to the next.
• Systematic errors, which are errors that have a predictable relationship from one measured value to another. This predictable relationship means that in principle, but not in practice, the error could be corrected were additional information known (since the underlying effect is deterministic). Bias is a particular form of systematic error in which the error is in common between measured values, but not all systematic errors are simple biases. All systematic errors in EO are ‘structured’, in the sense that there is a pattern of influence on multiple data. They include, but are not limited to, effects that are constant for a significant proportion of a satellite mission.
• Structured random errors arise from random effects where the unknown error affects multiple pixels and/or bands (channels). An example is a scanning sensor for which calibration against a target is performed once per scan. Any random error in the measured target would affect all Earth radiance pixels for which that calibration were used. Such errors are structured because there are predictable relationships between the errors in different measured values, and are random because there is no information, even in principle, that would enable the errors to be corrected (since the underlying effect is random). In other words, the term ‘structured random’ implies that the error is both unpredictable and correlated across measurements.

that we have distinguished between random, systematic and structured random errors, let’s move on to look at correlation in different dimensions.