## 2 Materials and Methods
### Uncertainty Expression  
As stated in [PERSON] [6], [PERSON] [7], [PERSON] et al. [8], and [PERSON] et al. [9], several sources of uncertainty can be identified for farm machinery field measurements:  
* harvesting dynamics:
* lag time;
* filling time and emptying time of the combine harvester;
* measurement errors:
* related directly to yield;
* related to moisture observations;
* accuracy of the positioning system:
* collocated observations;
* harvesting strategy (decisions and actions of the harvester operator):
* higher than the recommended harvesting speed;
* sudden changes of speed;
* harvest turns and headlands;
* overlapping (crossings) and partially overlapping trajectories;
* raising the cutting bar due to obstacles or unevenness of terrain.  
The computation of uncertainty for the purposes of this paper was defined according to the following equation:  
\[N0=\left|\left(1-\frac{v}{\widetilde{x_{1}}}\right)\right|+\left|\left(1-\frac {v}{\widetilde{x_{2}}}\right)\right| \tag{1}\]  
where  
* index of uncertainty variation in a given point;
* value of yield production in a given point;
* mean value of yield production of predeceasing \(n\) points (field measurements);
* mean value of yield production of following \(n\) points (field measurements);
* defined as 1/20 of relative density of measurements per hectare (see the more detailed explanation below).  
Equation (1) was defined based on the following process:  
* This analysis is preconditioned on the fact that the variation of yield production is a continuous process that follows environmental and soil characteristics; therefore, sudden and small coverage area differences in yield production compared with the surrounding areas are not probable/expected, that is, are considered an uncertainty.
* An analysis of yield variations for 15 crossings of farm machinery trajectory when taking into account variable rates of preceding and following points (see Figure 3). Crossings were used as a model owing to the known uncertainty in those areas.  
* On the basis of the analysis, it was discovered that the suitable area for calculating uncertainty is mainly dependent on the frequency of measurements and distance. Both of these variables are included in the density of measurements per hectare. The value of 1/10 of the density (i.e., half of that on each side of the point) was taken into account as representative surroundings. Other environmental aspects (relief, water distribution, soil conditions, and so on) of the plot might be taken into account as well.
* Values of preceding and following points are calculated separately owing to different possible behaviour on each side. For example, in Figure 3, points at the beginning and at the end of the \"U\" shape are affected by uncertain values only from one side, while the other is balancing the final uncertainty towards a lower value. On the other hand, a point in the middle of the \"U\" shape is different from the neighbours on both sides and, additionally, emphasizes the final value of uncertainty.
* Absolute values are taken into account owing to possibly overestimated uncertain values (above the mean value of surroundings).  
Figure 4 presents some of the situations that affected uncertainty owing to the harvesting strategy: crossings, overlapping and partially overlapping trajectories, raising the cutting bar causing gaps in data, harvester turns, and headlands (beginning of a new row).  
Figure 3: The mean and the range of yield values in crossings of a farm machinery trajectory: a demonstration of point measurements for a proof-of-concept uncertainty definition.