# Accessibility

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Accessibility is a key factor for Land Use Change and the attractiveness of locations. A highly accessible location is well connected to other locations that are relevant for a specified purpose or set of purposes.

Accessibility of an origin location can be defined as an aggregation of Attraction Potentials of destinations.
Aggregation can be done by summation (1-norm if all attractions add up), or taking the maximum (+inf norm if only the best accessibility counts) or something in-betweeen (like 2-norm). See also Minkowski distance and p-norms.

# Attraction Potential

The attraction potential for an origin-destination relation is defined by the combination of the the #Travel Potential with the attraction of that destination.

\begin{align} p_{ij} &:= f(d_{ij}) \\ pot_{ij} &:= Attraction_{j} \cdot p_{ij} \\ Accessibility_{i} &:= \sum\limits_{j} { pot_{ij} } \end{align}

where:

• $$i =$$ index of origin zones
• $$j =$$ index of destination zones
• $$f( {d_{ij} } ) =$$ decreasing function (so that nearer or less expensive places are weighted no less than farther or more expensive places) of generalized travel costs, aka Impedance.

# Travel Potential

A distance decay function defines the relation between #Travel Impedance $$d_{ij}$$ and travel potential $$p_{ij}$$. The distance decay function must be decreasing. Travel Potential can be calculated with the GeoDMS from Travel Impedance Tables.

Examples of used distance decay functions are:

• inverse: $$d \to 1/d = d^{-1}$$
• inverse squared: $$d \to 1/d^2 = d^{-2}$$
• Power-b: $$d \to d^{-b}$$
• cut-off: $$d \to if ~ d < d_{max} ~ then ~1~ else ~0$$
• Log-logistic [d in minutes travel time]: $$d \to 1/(1+e^{a+b \ln d}) = 1/(1 + e^a d^b) = {e^{-a} d^{-b}}/(e^{-a} d^{-b} + e^0 d^0)$$

# Travel Impedance

The Travel Impedance of an origin-destination relation is the cost, time or distance (or a combination thereof) of travelling. It can be specific for one travel-mode but can also represent a combination of travel-modes.

Travel Impedance can be calculated with the GeoDMS:

• as the crow flies (Eucledian distance)
• over a road network, using graph theory, for different road users like
• cars
• bicycles
• pedestrians
• public transport
• through the air
• over water

Depending on the required accessibility measure (e.g. based on distance, time, travel costs), different GeoDMS Accessibility and Travel Impedance functions in the GeoDms can be applied.
The time of day/year can also be relevant for travel impedance and accessibility, examples: