# 3: Determining uncertainties

Link to the CEOS Cal/Val portal where these  http://www.fiduceo.eu/tutorial/introduction-0 ; http://www.fiduceo.eu/tutorial/introduction-1 will eventually live

## How do you define uncertainties?

For every source of uncertainty (every twig in the uncertainty diagram) it is important to evaluate the uncertainty. If you know something to be truly negligible (on all scales), then it is possible to state this, though ideally with some evidence for how negligible it is. Otherwise, some attempt should be made to evaluate or estimate the uncertainty. Note that there are two aspects of this: evaluating the uncertainty associated with a term in the measurement function, and evaluating the uncertainty associated with the measurand (Earth radiance) due to that uncertainty. The translation between these is done using the sensitivity coefficient.

## How do you correlate errors?

We need to understand the pattern in the errors (remember uncertainties describes the distribution around the measured value where the true value may lie, and the error is the unknown actual difference between the measured value and the true value). Although we can never know the actual error, we can say whether it is common or different from spectral band to spectral band or from pixel to pixel.

For each source of uncertainty we have identified, we should identify the error correlation form, i.e. how does the error vary from pixel to pixel in the image, over time and from spectral band to spectral band? For example:

• A noise in the Earth count will only affect one pixel. The noise may have the same associated uncertainty in the next pixel, but will be a different error (imagine a different draw from the probability distribution described by the uncertainty).
• Noise in the determination of a calibration coefficient measured post launch, will have a single error that will apply to all pixels that are based on that calibration.
• An uncertainty associated with a calibration parameter determined pre-flight or through harmonisation will create an unknown error in the calibration parameter that applies to all measurements at all times.

## What forms of error correlation are there?

To codify error correlation, FIDUCEO has defined the following error correlation forms :

### Random

In this there is no error correlation with any other measured value.

### Rectangular absolute (contains systematic)

In this the error correlation is constant for a particular range of values defined absolutely, rather than relative to the measured value. This includes the following cases:

• Where a single measured value is used over an explicit range, e.g. where a single calibration value is used for all measurements over several scanlines, or in a particular year, and a different calibration value is used for all measurements outside that range.
• For an effect that is fully systematic in that dimension (common to all measured values in that dimension). This is described with the dimensions  [-∞,+∞]

### Triangular relative

In this the error correlation drops linearly (in the dimension of interest) relative to a particular measured value. This comes from a running average with constant weights. (See Appendix B.4 for an explanation of this correlation form).

### Bell-shaped relative

In this the error correlation drops faster than linearly (in the dimension of interest) relative to a particular measured value. This can come from:

• A weighted running average (e.g. over neighbouring scanlines), which weights the central reading more than the others involved in the average.
• Any other form of weighted averaging (e.g. through a spline fit in geolocation)
• Other cases where our expectation is that the correlation drops off over distance in some way.

In none of these cases is the error correlation form exactly Gaussian, but a truncated Gaussian form is a practical approximation for the Bell-shaped form, and is used. See Appendix B.4.4 for a discussion of why a truncated Gaussian is appropriate for a weighted average. What this correlation form represents is the situation where “nearby” errors are relatively highly correlated, but this correlation drops off over a distance. By defining the Gaussian width and the truncation range (beyond which there is no error correlation), it is possible to define a reasonable range of realistic correlation forms

### Repeating rectangles

This comes from something for which the error correlation coefficient is constant within a small range (1 pixel or a range of pixels), then repeats on a regular cycle. It could come from a push-broom sensor where every nth scanlines are from a common detector element, or from a seasonal affect that occurs annually.

### Repeating bell-shapes

This is another repeating effect, but one where locally there is a drop off of correlation (partially correlated with neighbouring pixels/scanlines) which then has a repeating effect.

### Stepped triangle absolute

This accounts for the situation in HIRS where there is a calibration cycle, so that the instrument measures the calibration target once every 48 scanlines, and then there is a rolling average between scanlines. The correlation to neighbouring scanlines takes the form of a stepped triangle. See Appendix B.4.5 for a discussion of this correlation form.

### Other

Although true correlation structures may be more complicated than the ones given above, the above are sufficiently representative for the correlation structures encountered in FIDUCEO thus far. However, there may be situations where an FCDR producer needs to define a new error correlation form.

 Correlation form Parameters Description random none required For fully random effects there is no correlation with any other pixel rectangle_absolute [-a,+b] (rectangle limits). Provide these per pixel/scanline/orbit as required. Allow for a way of representing [-∞,+∞]   [rmax] States correlation coefficient for all pixel / scanline / orbit pixels. Default is rmax = 1 (fully correlated) An effect is systematic within a range and different outside that range. For each pixel / scanline / orbit in range say number of pixels / etc either side that it shares a correlation with. For fully systematic effects notation to say “systematic with all”. If rmax is defined, then the correlation coefficient is one for the pixel with itself, and is rmax with all other pixels. triangle_relative [n] – number of pixels/scanlines being averaged in simple rolling average (should be an odd number) Suitable for rolling averages over a window from (–n-1)/2 to (+n-1)/2 (i.e. for n pixels/scanlines being averaged) Assumes a simple mean, not a weighted mean. No rmax is needed, since it is always 1. bell_shaped_relative [n] – number of pixels being averaged in a weighted rolling average, from which truncation range and standard deviation for Gaussian representation follow (truncation beyond ±n pixels, ) (n should be odd)   OR   [n,sigma] n: truncation from –n to +n, sigma: width of Gaussian representation (n should be odd)   Typically provided once per orbit file (some further consideration needed about first/last scanlines in an orbit) Suitable for rolling averages over a window from (–n-1)/2 to (+n-1)/2 (i.e. for n pixels/scanlines being averaged). Assumes a weighted mean, for any weights (and thus also includes things like spline fitting). Also suitable for anything else where the assumption is that “closer pixels/scanlines are more correlated than further pixels”. This can use two terms – n gives the truncation range outside which the assumption is there is no (or negligible) correlation, and sigma gives how fast the correlation drops off. The derivation of the width sigma to use for a weighted rolling average is given in Appendix B.4.4. repeating_rectangles [-a,+b,rmax,L,h,imax] per pixel/scanline/orbit etc (rmax,L,h will be same for different pixels) Correlation coefficient assumed to be rmax for pixels/scanlines from –a to +b, and h for pixels/scanlines from L-a to L+b and from 2L-a to 2L+b and so on (iL-a to iL+b) for all integers i up to imax. repeating_bell-shapes [n,sigma,L,h, imax] Correlation coefficient assumed to drop off as a truncated Gaussian for local pixels/scanlines etc in the range defined by n and a similar Gaussian with a peak of h and the same width for pixels/scanlines iL pixels apart on either side, for all integers I up to imax. Stepped_triangle_absolute [-a,+b,n] per pixel/scanline/orbit etc (n will be same for different pixels) The step is a rectangular absolute from –a to +b with a correlation coefficient of one, after which the correlation coefficients drops for another a+b+1 lines, and then again. n is the number of calibration windows averaged. See Appendix B.4.5 Other A function describing the correlation Not yet implemented.

## Useful Documents

Woolliams, E., Mittaz, J., Merchant C. & Harris, P., 2017, D2_2a-Principles-of-FCDR-Effects-Tables