Upkeep - Expected area error amplitude
Contents
Basic concepts
Background
A technical tolerance is an expression of uncertainty or precision. Its value depends on the measurement methodology and conditions such as source material and the nature of the object measured.
A tolerance should be known for any particular method and can only be applied to that specific measurement. It then serves a pass/fail function: if there is a significant difference between that specific measurement and an older one of the same object, the new measurement value should be used, else, one must accept the old value as remaining true and valid.
This article approaches such tolerance from its two main component:
- the nature of the object measured, which is stable and independent from the actual measurement. It can be calculated (semi-)automatically in the LPIS for each parcel
- the effects of the actual measurement, in case a delineation on digital orthoimagery. This component is measurement dependent and has to be entered on a case by case basis.
Incertainty (or error components)
The main contributors to incertainty are:
- Normal variation: Even in ideal conditions, no two measurements of an object will provide exactly the same result: A series of repetitions will follow the normal distribution so the probability of a result can be estimated by gaussian theory. Significance is then based on a threshold probability of the value being measured. Traditionally, this probability is set to 0.95 (two sided) meaning that, if 20 repetitions of the measurement would be made, 19 of these would be in the range. Clear instructions and trained operators are the instruments to keep the normal variation to a minimum.
- Parcel conditions: In the real world, ideal parcels are rare.
- not all measurement vertices (boundary markers) can be measured with the same precision: fence pole, crop boundary corner, hedge each has a different "identification accuracy".
- the area of simple square polygons are measured more precisely than irregular or elongated polygons
- Image resolution: Image resolution influences the precision similar to marker spacings of a ruler: Lower resolution (=larger pixel size) brings in more incertainty. Other image qualities are not considered in the tolerance as the imagery is assumed "fit for purpose".
Setting the tolerance for CAPI operations
Main formula
It's scientifically possible, but rather impractical to establish the probability for each individual image, land cover feature and landscape. Rather a simplified formula is proposed.
The expected area error amplitude of the measurements is then calculated as
- EAEA = C (RMSp) * B [m2]
Where:
- RMSp: the map scale represented by the pixel size of the map (note that the guidelines impose a relation between GSD and pixel size, dependent on the sensor type
- C: the shape of the parcel (regular, elongated, irregular…):
- B: the nature of the parcel boundary perimeter and its length
Supporting formulas
C shape, size and resolution
C is calculated from the Gaussian formula and is a function of the pixel size RMSp:
RMSp = pixel size in meter
B: border properties
B is estimated by the following table, based on the abundance of border quality:
Perimeter composition | >60% precise borders | >25% vague borders | all other conditions |
B value | 1.96 | 2.94 | 2.45 |
where:
- precise borders: those composed of ground level features that have an identification precision better than 50 cm. Examples are: crop boundaries, base of fences, base of walls, road sides,....
- vague borders: are composed of features NOT at ground level (raised or sunk for 2 meters or more), that block visibility to the ground level and that are subject to cyclical dynamics. Examples are: Lines of threes, high hedges, deep ditches, terraces....
- intermediate borders can be identified for border qualities which do not qualify as either precise or vague: e.g. low hedges, degraded walls.
- Note: Any third party boundary (e.g. cadastral boundary, topographic map line) is considered a precise border IF it represents a physical (fence, boundary marker) rather than a virtual (legal) feature.
Figure 4f.1 illustrates the concepts of precise border and vague border.
figure 4f.1: left: parcel with predominantly precise border (red); right: parcel with predominantly vague borders (green).
The determination of the B-value for a given parcel solely depends on the abundance of these precise and vague borders along its perimeter, intermediate borders play no role at all.
In general, there should be no need to measure the abundance of precise and vague borders in detail. An estimation procedure adapted to the landscape should do in the vast majority of cases. If doubt remains, the 2.45 as B-value can be taken. Figure 4f.2 provides examples and details of cases where the abundance of border classes prevents the use of this 2.45 B value.
figure 4f.2: The green shaded shape indicates vague borders that can not be delineated accurately. Left example: 301m precise border (red) on a 375m perimeter or 80.2% > 60% and the B value is set to 1.96). Right example: 240m vague border (green) on a 295m perimeter or 81.3% > 25% and the B value is set to 2.94.
Illustration of polygon error by shape and size
The results from the above formula are calculated for various sizes of four simple and symmetric shapes: a square (w=l), a rectangle (l=4w), an elongate rectangle (1=16w) a 5-pointed star .
The resulting area errors are calculated for shapes with precise borders (B = 1.96). Vague borders cause an increase of these results with 50% (= 2.94 / 1.96).
Area and Perimeter of the shapes
The relation between area and perimeter (the basis for buffer tolerance) is provided below for 4 different shapes. Each line holds the area and perimeter for each of the 4 shapes. Area is expressed in square meter (and ha), perimeter entries are in meter.
Area | w=l | 4w=l | 16w=l | Star |
100 (0.01ha) | 40 | 50 | 85 | 55 |
1000 (0.1ha) | 126 | 158 | 269 | 173 |
5000 (0.5ha) | 283 | 353 | 601 | 387 |
10000 (1ha) | 400 | 500 | 850 | 548 |
50000 (5ha) | 894 | 1118 | 1901 | 1225 |
200000 (20ha) | 1789 | 2236 | 3801 | 2450 |
Area errors for the shapes using measurement based on 50cm RMSp
The table below lists the expected area error amplitude as calculated for each shape for a given area and an RMSp of 0.5 meter. All entries are expressed in square meter.
Area | w=l | 4w=l | 16w=l | Star |
100 | 9.80 | 14.29 | 27.77 | 8.40 |
1000 | 30.99 | 45.18 | 87.83 | 26.57 |
5000 | 69.30 | 101.02 | 196.38 | 59.40 |
10000 | 98.00 | 142.86 | 277.73 | 84.00 |
50000 | 219.13 | 319.44 | 621.02 | 187.83 |
200000 | 438.27 | 638.88 | 1242.03 | 375.67 |
Area errors for the shapes using measurement based on 20cm RMSp
The table below lists the expected area error amplitude as calculated for each shape for a given area and an RMSp of 0.2 meter. All entries are expressed in square meter.
Area | w=l | 4w=l | 16w=l | Star |
100 | 3.92 | 5.71 | 11.11 | 3.36 |
1000 | 12.40 | 18.07 | 35.13 | 10.63 |
5000 | 27.72 | 4.41 | 78.55 | 23.76 |
10000 | 39.20 | 57.14 | 111.09 | 33.60 |
50000 | 87.65 | 127.78 | 248.41 | 75.13 |
200000 | 175.31 | 255.55 | 496.81 | 150.27 |
Effect of additional vertices
The EAEA formula relies on meaningful polygon vertices. Adding collinear points inbetween primary vertices improves the measurement and reliability and so reduces the amplitude.
For a 1 hectare square polygon
- with only 4 corner vertices, EAEA = 98m²
- with 20 vertices (1 every 20m on the perimeter), EAEA = 58.80m²
both calculated using RMSEp = 50cm and B = 1.96
Further discussion
Some trivial observations
- By using such technical tolerance, there still remains a 5% chance that a the new measurement will indicate a difference where there is none.
- As the old value is taken as reference, there is no tolerance applied to this value. The appropriate tolerance is only applied to the new measurement .
- The advances in technology generally cause the new measurements to be more precise (i.e. with a smaller tolerance) than the methodologies used to obtain the old value. However, this is not guaranteed in these update processes, as recent imagery may be coarser than the reference image used for the reference values.
- Technical tolerances should not be confused with thresholds, which are conventional baselines controlling a decisions.
- There is no uncertainty involved with thresholds, but it does imply that any method used to collect information is precise enough to make such threshold meaningful. Setting a threshold lower than technical tolerance used for the measurement is in fact nonsense. On the other hand, using a method that comes with a tolerance larger than the threshold is equally unacceptable.
- A well known threshold is the limit for the area difference of 3%/5%/7%, pending on parcel size. This threshold determines area non-conformity during the LPIS QA and is dependent of parcel-size.
Implications regarding the image acquisition
Dividing the AE derived from the formula by the parcel area results in the relative area error. If follows that, in all but the very small reference parcels, the 3% threshold is met by applying orthophoto of:
- 0.5 m pixel size for regular parcels
- 0.2 m pixel size for elongated parcels
- 1.0 m pixel size for regular parcel of area greater than 2000 m2 and for elongated parcels of area greater than 10000 m2.
This knowledge is crucial when planning your data acquisition. If measurements consistently (consider more than 5 % of the cases) overshoot the thresholds, the imagery is not appropriate for that particular zone.
Implications regarding the LPIS design
The ability to correctly quantify by delineation the MEA for direct aids is a key functional requirement of the LPIS reference parcel. If, despite using optimal imagery, the AE derived from the formula keeps overshooting the 3% threshold, one should check if the parcel design is truly appropriate in the given landscape.
Some landscapes and parceling pattern pose real challenges to the LPIS design, but in some cases the sub-parceling and parcel aggregation can cause small or irregular parcel borders/perimeters. As a general rule, reference parcels that closely reflect the actual land management are less likely to cause problems (see slides 24-34 of this presentation
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