As mentioned in Section 1.1, transition probabilities govern changes in land cover by reflecting the economic, sociological, and ecological influences on landscape structure and function. These probabilities are derived empirically through a time series analysis of changes in land cover, while considering road networks, population density, and physical attributes of the landscape. Using multinomial logit models [13], the transition probabilities reflecting changes in the land cover of the Little Tennessee River Basin of western North Carolina can be structured as a matrix with individual rows in the matrix representing probabilities of transition from one land-cover category to any possible category for a given landscape condition [16] . During a LUCAS simulation, the landscape condition labels in the composite map are matched with equivalent landscape condition values in the transition probability matrix (TPM). The appropriate set of transition probabilities are applied and the resulting landscape category is assigned to the appropriate pixels in order to generate a new output map of land cover.
Figure 2: Sample landscape condition label for a forested grid cell
in the Little Tennessee River Basin .
For example, the landscape condition label of
a given forested (vegetation category 1) grid cell
of the Little Tennessee River Basin which is privately-owned
(ownership category 2), 512 meters above sea level (elevation),
of 30 degree slope, 120 meters from the nearest road,
1,300 meters from the nearest town, and reflects a population density
of 15,000/acre can be represented by the 
attribute
column vector 
shown in Figure 2.
The value of each 
is used in a multinomial logit
equation [13] (suitable for regression analyses of
continuous and discrete independent variables) to generate the
probability of transition to another vegetation cover class
(unvegetated or grassy/brushy). Specifically, this probability
is given by
where 
is a 
column vector composed of the last
5 elements (i.e., 
) of 
from
Figure 2, 
is an estimated constant
(intercept), n is the number of vegetation types (see
Table 2), and 
is the probability
of land cover at the privately-owned forested grid cell at time t
having the same cover class 
at time t+1 (i.e.,
j=1) or changing to another cover class (i.e., j=2,3). Since each
(for j=1,2,3) is a 
column vector of estimated coefficients from [14], 3
probabilities of transition can be calculated using Equation
(1) for j=1,2,3. A random number is then chosen from a
uniform distribution between 0 and 1. If the random number
falls within an interval associated with a transition probability
to a different land cover, the grid cell is changed; otherwise,
the grid cell remains in its present land cover. For the grid cell
whose landscape condition label is given by Figure 2,
the probability of transition to grassy/brushy and unvegetated
land covers is easily computed using the logit coefficients
(
's) provided in Table 3. For this grid cell,
the probability of transition to unvegetated land cover
is 
(i.e., very likely), and
the probability of transition to grassy/brushy land cover
is 
(i.e., not very likely).
These probabilities constitute 2 of the 3 elements of a single
row of the 
TPM for transitions from privately-owned
forest in the Little Tennessee River Basin.
In general, the various sets of logit coefficients
for transitions from each vegetation
cover (j) with different ownership classes are estimated separately
by maximizing their respective likelihood functions using
a nonlinear optimization method [14] to match
observed (historical) land cover changes. The null transition or
probability of no land cover change (
from above example)
is simply determined by
Table 3: Logit coefficients for transitions from privately-owned
forest in the Little Tennessee River Basin
based on 1986--1991 historical land-cover transitions.
Within the LUCAS Socioeconomic Model module (see Figure 1), this process is then repeated for each grid cell (having public or private ownership) in order to produce a new map of land cover. The spatial pattern of land cover and any associated impacts (see Section 1.1) is analyzed at the end of each time step, and the simulation is continued for a specified duration of time.
