Directional RDF-Functions

The last extension to RDFs discussed in greater detail are the direction dependent RDFs. I shall explain them by directly using an example with real data.

**Figure 7.13:** Directional RDF configurations. Left: angular configuration (slice). Right: transect configuration (transect).
$\includegraphics[scale=0.56]{images/xy.eps}$ $\includegraphics[scale=0.4,angle=-30]{images/rdf_directional2.eps}$

In the previous RDFs a circular configuration was used for calculations. This basically uses the assumption of a circular perception and use of the environment without any preferences for specific directions. Sometimes animals do not use their environment equally in all directions. Güttinger (1997) showed for example that Greater Mouse-eard bats Myotis myotis of one colony used the habitat around their roost preferentially in the two directions NNW and SSE.

Such findings made it clear that in some cases RDFs need to be extended by a directional component. This should allow for a (spatially) finer analysis of the environment, but it also introduces greater complexity.

There are several ways how this can be achieved. Two of them are illustrated in figure 7.13. In the case of a static RDF the calculations can be extended by dividing the surrounding area into different slices or transects for which the calculations are performed separately (figure 7.13). To be able to interpret the results, again the data have to be edited and compiled into an appropriate graphical representation due to the complexity of the data. Such a procedure was performed for one of the badger setts mentioned earlier (Do Linh San, 1997) showing the relative amount of meadows in its surrounding areas.

**Figure 7.14:** Spatial configuration of the meadows around one of the badger setts in the Knonaueramt. This data was used to calculate the following directional RDFs. The red dot indicates the sett.
$\includegraphics[scale=0.56]{images/xy.eps}$ $\framebox{ \includegraphics[scale=0.2,angle=0]{images/d_rdf_wiesen2.ps}}$

In figure 7.14 the meadows around one of the before mentioned badger setts are shown. The red dot in the middle indicates the sett.

The main focus here is to overcome the vague qualitative interpretations of the surrounding area. Normally we would only be able to make statements like there seems to be more in the upper left part or the upper left part has a relatively small amount of meadows. The other approach that was possible up to now is to draw a circle around the sett and then state that there are e.g. 30 percent meadows around it within 800 meters. To overcome these restrictions a much finer and much more quantitative approach needs to be applied.

From the configuration in figure 7.14 four directional RDFs have been created. In figure 7.15 the results are shown for the RDF-area (a) and RDF-percent-area (b) plots. The RDF-area plot gives a very detailed picture of the total amount of meadows around the badger sett. As it is normally the case in RDF-area plots, higher amounts are found at larger distances^7.3. But here there are two exceptions to the general behavior. At medium distances in the sectors 1 and 4 (counted clockwise from the north direction) there is a zone with high amounts of meadows. The RDF-percent-area plot (shown in figure 7.15b) provides a completely different picture. The highest concentrations of meadows occur in the closer vicinity in the sectors 4 and 8.

**Figure 7.15:** Directional RDFs of the meadows around one of the badger setts in the Knonaueramt. Left: the absolute amount of meadows within a zone. Right: the percentage of meadows within a zone. Yellow = small amount, green = medium amount, blue = large amount of meadows. The length of a side of the square is 2000m. The number of sectors used to divide the area (8 in this example) can be chosen as needed.
$\includegraphics[scale=0.56]{images/xy.eps}$ a $\includegraphics[scale=0.2]{images/d_rdf_zonepc2.ps}$ b $\includegraphics[scale=0.2]{images/d_rdf_zonesum2.ps}$

Figure 7.16 shows the cumulative versions of the RDF-area plots. The cumulative RDF-area plot (figure 7.16: left) shows a characteristic which might result in difficulties for an animal. The directions with larger amounts of meadows in closer vicinities (4 and 8) do not or only partially correspond to the directions where large amounts of meadows are available at larger distances (sectors 1, 2 and 6). As a hypothesis this may result in larger distances which have to be traveled by the animal when compared to a configuration where closer and further areas are in the same directions. This may result in pathways describing loops or hook shapes with start and end points at the sett. This plot might be very helpful when evaluating minimum resource requirements of animals.

In figure 7.16 (right) the cumulative percentage of meadows around the badger sett is shown. The areas with high values are clearly in the directions of sectors 4 and 8 and partly sector 6. This was partly recognizable before in the RDF-percent-area plot (figure 7.15). In the example used here (meadows as feeding places for badgers) the direction of an animal is probably much influenced by the chance to find another suitable feeding patch. A second factor influencing this direction is the amount of patches already found in this direction. If the interval of finding patches declines the animal will change its direction, whereas when the interval is becoming shorter it will keep going in that direction^7.4. For this aspect the cumulative RDF-percentage-area plot may be very useful. It shows the cumulative relative amount of meadows around the badger sett.

**Figure 7.16:** Directional RDFs of the meadows around one of the badger setts in the Knonaueramt. Left: the cumulative absolute amount of meadows within zones and sectors. Right: the cumulative percentage of meadows within a zone. Yellow = small amount, green = medium amount, blue = large amount of meadows.
$\includegraphics[scale=0.56]{images/xy.eps}$ $\includegraphics[scale=0.2]{images/d_rdf_cumzonesum2.ps}$ $\includegraphics[scale=0.2]{images/d_rdf_cumzonepc2.ps}$

This would suggest that if travel path length is limited to this scale level directions in sectors 4, 8 and to some lesser extent in sectors 5-7 should preferably be used. This contrasts completely the intuitive rating when looking at figure 7.15 that the preferred directions would be towards the upper right and lower left part of the area.

Maybe the reader is still a little bit sceptic about the value of RDF plots. I have now introduced RDF-plots in different applications and with some examples. So I assume the reader got accustomed to reading and interpreting them in a relatively easy way. To illustrate their power I would like to ask the reader to find out which badger sett shown in figure 7.8 is corresponding to the meadows configuration presented in figure 7.14.

With these examples I would like to finish the introduction of the basic forms of radial distance functions. They were all illustrated using the areal extent as the base measurement. As it was shown in figure 7.7, the application of RDFs can be used in a variety of other measurements. Since the basic principle is well explained above and the applications with other measurements is straight forward, I would like to go on to discuss some further extensions of RDFs in the next section.