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located within the selected case study. The indicator is then calculated according
to the formula
X
X
A
i
¼
M
i
exp
ð
b
t
ii
Þþ
M
j
exp
ð
b
t
ij
Þþ
M
k
exp
ð
b
t
ik
Þ
ð
4
Þ
j
k
where i and j are municipalities located within the Mazovia region, k is any other
Polish municipality (outside the Mazovia region), and Mi,
i
, M
j
and M
k
are the
populations of municipalities i,
j and k, respectively. In consequence, Mi
i
exp(
-β
t
ii
)
is the value of the self-potential of municipality i, and
∑
j
M
j
exp(
-β
t
ij
)+
∑
k
M
k
exp
(
-β
t
ik
) represents the sum of the potential resulting from the opportunity to access
all other Polish municipalities.
2. Grid model (M2). The second model is calculated similarly to the previous one
however it uses grid cells in the calculation process instead of the municipalities
of the Mazovia region. The indicator values received for particular grid cells are
further aggregated to the municipal level using the population weighted average:
P
a
ðð
M
a
exp
ð
b
t
aa
Þþ
P
b
ð
M
b
exp
ð
b
t
ab
Þþ
P
c
ð
M
c
exp
ð
b
t
ac
Þþ
P
k
ð
M
k
exp
ð
b
t
ak
ÞÞ
M
a
Þ
P
a
M
a
A
i
¼
ð
5
Þ
where a and b are grid cells located within a municipality i, c is a grid cell located in
municipality j, but outside of the municipality where a and b are located, M
a
, M
b
and M
c
are the population of grid cells a, b and c, respectively, while k and M
k
are
described as above. The difference between the results received from the
rst and
the second models is a factor of the scale dimension of the MAUP, i.e. it is a
consequence of the application of different spatial resolutions.
3. Population-weighted average travel time model (M3). The last accessibility
model differs from the second one by the method of data aggregation from grid
into municipal resolution. While the previous one aggregates the results of
potential accessibility indicator values, in the third model the distance decay
function includes the population-weighted average travel times between all pairs
of grid-cell-nodes located in the municipalities analysed. As a result,
the
potential accessibility for administrative unit
i
is obtained using the following
formula:
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