Accessibility

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

Accessibility of a 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-between. 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_i**j and travel potential p_i**j. 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 → 1/d = d**⁻¹
• inverse squared: d → 1/d² = d⁻²
• Power-b: d → d^−b
• cut-off: d → if d < d_max then 1 else 0
• Log-logistic [d in minutes travel time]: d → 1/(1+e^{a + b ln d}) = 1/(1+e^ad^b) = e^−ad^−b/(e^−ad^−b+e⁰d⁰)
  • with a =  − 6.899 and b = 2.326 for potential to jobs and employees, see Geurs & van Eck, 2001

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 can be applied. The time of day/year can also be relevant for travel impedance and accessibility, examples:

• How congestion in the Netherlands influences travel times during the day
• The effect of historical railroad construction on traveltimes in the Netherlands
• Surinam's New Infrastructure.

Centrality by random walker distributions (markov modelling)

When an outgoing link of node \(i\) to node \(j\) has a probability \(P_{ij}\) of being chosen by a random walker at node \(i\) with \(\sum\limits_{j} {P_{ij}} = 1\) for each \(i\), a random walker starting at node \(i\) with probability state vector \(\pi_{0}\) with \(\pi_{i0}\) indicating the probability of the walker being in node \(i\) at stage 0, will in stage \(t\) (i.e. after \(t\) steps) be in node \(j\) with probability \(\left(P^t \cdot \pi_{0}\right)_j\), which converges to a stationary markov distribution which is independent of the starting state for a fully connected network. This stationary distribution can be considered as a measure of centrality for a connected set of nodes.

See also: Markov chains on wikipedia or here

If a network has \(n\) nodes with an average of \(k\) outgoing links, and having distribution \(\pi_{it}\), calculating \(\pi_{j,t+1} := \sum\limits_{i} {P_{ij} \cdot \pi_{it}}\) can be done easily with a simple array lookup(index), multiplication and partitioned summation in \(O\left(nk\right)\) time.

 attribute link_probability(link) := 1.0 / float64(pcount(link_F1); // P_ij  attribute node_probability_0(node)):= 1.0 / pcount(node);  attribute node_probability_1(node)e:= sum(  node_probabilty_0[ link_F1 ] * link_probability, link_F2);

By applying the last line iteratively, we can calculate p_i**t iteratively with for_each t.

 ...  unit t: nrofrows = 100 { attribute name := 't'+string(ID(t)); }  container iteration := for_each_nedv(t/name,      'sum(  '+MakeDefined(t/name[ID(iter)-1], 'node_probability_0')+'[ link_F1 ] * link_probability, link_F2)'  ,   node, float64);

external links

• Accessibility measures: review and applications. Evaluation of accessibility impacts of land-use transportation scenarios, and related social and economic impact, Geurs KT ; Ritsema van Eck JR, pbl.