📍Location Scale

General ideas

The location scale is one of the multipliers to calculate a station's reward. Its goal is to distribute rewards in areas with a high number of base stations.

It is a number between 1 and 0, obtained after multiplying all the reduction factors ($RF$) of its neighbor stations within a 70km radius. The neighboring stations $i=1\ldots n$ are sorted by distance to your own station.

$$LocationScale = {RF}_1 \cdot RF_2 \cdot \ldots \cdot RF_n$$

A location scale of 1 will mean you get full rewards regarding the location, a location scale of 0 means you will get no rewards regarding our location.

Each Reduction Factor $RF$ is a function of the distance and the quality of the stream so that:

• The two nearest stations will not affect your location scale ($RF_1 = 1$and $RF_2 = 1$), which is to promote redundancy

• The further away a neighbor is, the lower the distance penalty ($DP$)

• The higher your signal quality and availability compared to that of your neighbors, the higher your share of the rewards. A Share Factor ($SF$) is calculated for each of the neighbors. This means that a low quality stream does not heavily affect the rewards of high quality streams. It is still profitable to install a high quality base station in an area already covered by low quality base stations

$$RF_i=1-(DP_i\cdot SF_i)$$

Distance Penalty ($DP$)

For each neighboring station, a Distance Penalty will be calculated. The distance penalty of each neighboring base station is defined using the following rules:

1. The two closest base stations are ignored, because we want to incentivize network redundancy

2. Stations within 15 kilometers from your own base station will have a distance penalty of 1

3. The distance penalty for stations located between 15 and 50 kilometers declines quadratically from 1 to 0, where at 15 km it is 1 and at 50 km is 0

4. Stations further away than 50 kilometers are ignored

The equation for the Distance Penalty between 15 and 50 km is

$$DP = (1-\frac{DistanceToStation - 15 \text{km}}{50\text{km} - 15 \text{km}})^2$$

This plot shows the Distance Penalty in a graph. The first two stations are ignored, hence the graph will only apply to those stations after the first 2. Between 15 and 50 km, the quadratic decline applies.

Graphical display of the Distance Penalty (DP). The farther the neighbor, the lower the distance penalty.

Share Factor ($SF$)

Similarly, for each neighboring station, a Share Factor is calculated. It is taking into account the signal quality and the availability (Qual factor) of each station:

$$SF=\frac{Qual_{\text{Neighbor}}}{Qual_{\text{Neighbor}} + Qual_{\text{Your Station}}}$$

An example

Let’s look at a practical example of how the location scale is calculated for each station.

This is the widget displayed in your Reference Stations tab. On the right side of the map, we can see your exemplary base station, in green color, surrounded by 3 third party base stations scattered through the surrounding land, in purple color.

Find this map of neighbors in your Reference Stations tab.

Next to your exemplary base station, we can click on each of the four neighboring stations to get the details.

The first two stations are ignored. The third neighbor's reduction factor, Qual factor and distance to the example station is shown

Calculating Reduction Factors ($RF$)

If we sort the neighboring stations simply by the distance to your own station, we can create a plot showing the relation between the distance and the distance penalty $DP$:

The dashed gray line represents the quadratic function so that you can see the behavior of the penalty as a function of the distance.

In this example, the distance penalties have been scaled up from 0% to 100% but for the calculation purposes the punctuations range from 0 to 1.

Let's take station ESPBARSAB4 as an example and calculate $DP$, $SF$ and $RF$.

ESPBARSAB4 has $Qual_{\text ESPBARSAB4} = 0.934$ and a distance to your exemplary station of 25.522 km. In comparison, $Qual_{\text Your Station} = 0.99$.

As picked from the graph,

$$DP_{\text{ESPBARSAB4}} = (1-\frac{DistanceToStation - 15 \text{km}}{50\text{km} - 15 \text{km}})^2=0.489$$

Next, we calculate $SF$ and $RF$.

$$SF_\text{ESPBARSAB4}=\frac{0.934}{0.934+0.99}=0.485$$

$$RF_\text{ESPBARSAB4}=1-(0.489\cdot 0.485)=0.763$$

This process is repeated for all stations between 15 and 50 km radius:

Now we can calculate your station's final Location scale by multiplying all the Reduction Factors. As ESPBARSAB4 is the only station in 50 km range that is not ignored, the Location Scale is simply

$$LocationScale=0.763$$

This Location Scale factor will then be used for the multiplication with the other reward factors, resulting in your total Reward Scale.