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The issue for accumulated fields is that the packing parameters - and hence the discretisation and the packing error - change as the range of values increases.  Because the bits per value remains constant, the effect is to increase the packing error by a factor of two periodically as the range increases.  This is illustrated in the figure below which shows how the packing error (the red line) increases as the values of the sunshine duration field increase throughout a 10-day forecast

Packed values depend on the packing error

The increase in packing error also has an effect on the packed values themselves.  This is illustrated in the diagrams below.

How small spurious negative - and positive - values appear when de-accumulating an accumulated field

The example below for a small grid of 9 points shows how the packing error and hence the packed values change as the range of the values increases.

Step=0

• All accumulations start with a zero value at step=0
• The values at all points are packed exactly

Step=1

• The model values increase at all points except the north west
• Not all values can be represented exactly using 8 bits per value
• In particular, the value of 0.031250 at the southern point is exactly the packing error and its packed value becomes 0.062500
• In this case, the packed values are all multiples of 2*packing error=0.0625 so values of 4.55, 5.9 and 2.2 are not stored precisely

Step=2

• Only the model value at the north east point increases

• The increase in the range of values increases the packing error by a factor of 2

• Now all packed values are multiples of 2*packing error=0.125

• The value at the central point of 4.55 is now packed as 4.5 compared to a packed value of 4.5625 at the previous step - a negative accumulation of -0.0625
• The value at the south west point of 2.2 is now packed as 2.25 compared to a packed value of 2.1857 at the previous step - a positive accumulation of +0.0625

Step=3

• The model value at the north east point increases further to 40.0
• Model values at all other points remain the same as at step=3
• The increase in the range of values increases the packing error.
• Now all packed values are multiples of 2*packing error=0.25
• The model value at the eastern point is unchanged at 5.9 but now its packed value is 6.0 giving an apparent positive accumulation  of +0.125

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For accumulated fields, such as precipitation, the GRIB packing can have a strange affect on the totals, even when the data are used on the native model reduced Gaussian grid.  In some cases, the values concerned are sufficiently small that the issue can be ignored, but for others - for example when looking at the frequency of zero rain, or light drizzle on forecast day 10 - action is needed. The figures below highlight the issue is and its impacts:

On the map plot example (b) one can see that a large part of the world (blue & pink areas) is affected by the issue.  The graph (a) shows how the capacity of the model output to represent small values in 12h (or indeed any) periods diminishes at longer lead times. The discretisation will naturally be worse at the end of, say, the monthly forecast. This also means that the total areas of zero rain in a given period will become larger as the forecast progresses, purely because of the GRIB packing.

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