ECOLOGICAL MODELLING Estimation of fire frequency

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Ecological Modelling 154 (2002) 103-120 /locate /ecolmode1
Estimation of fire frequency and fire cycle: a computational
Chao Li *
Northern Forestry Centre, Canadian Forest Service, 5320-122 Street, Edmonton, Alta, Canada T6H 3S5
Rece ived 31 J uly 2001 ; rece ived in revised form 28 Jan uary 2002; accepted 4 March 2002
This paper examines different concepts and methods of estimating fire frequency and fire cycle used in models of
forest dynamics, from a computational perspective. Fire frequency and fire cycle can be defined in point-based and
area-based ways, and the analytical results indicate that the different definitions are interrelated with each other,
except the fire number-based fire frequency definition. The point-based definitions can be seen as special cases of
area-based definitions when area is reduced to a single site. The generation of an ideal historical fire data set allowed
a comparison among different methods of estimating fire frequency and fire cycle in terms of associated bias. An
over-estimate of fire cycle could be generally expected by using stand-origin maps, and recent fire history could
influence the estimate significantly. The number of fire maps required to obtain a relatively stable fire cycle estimate
is also influenced by fire history. Point-based data obtained from random and systematic sampling designs displayed
almost equally good fire cycle estimates for the tested data set, when the sample sizes relative to the total number of
possible sample units are large. © 2002 Elsevier Science B.V. All rights reserved.
Keywords: F ire freq uency ; F ire cycle ; F ire reg ime ; Sampl ing design
1. Introduction
Forest dynamics, especially in the boreal region
of western Canada, have been historically shaped
by natural disturbance regimes such as fire (e.g.
Weber and Flannigan, 1997; Weber and Stocks,
1998). As a result, studies of local fire history and
fire regimes are important for understanding
forest dynamics as a whole, Thus, fire disturbance
has become one of the important components in
* Tel.: + 1-780-435-724 0 ; fax: + 1-780-435-7359.
E-mail address: [email protected] gc ca (C. L i).
forest dynamics models that account for the influ­
ence of a fire disturbance regime (e.g. Baker et aI.,
1991; Keane et aI., 1996; Li and Apps, 1996; Li et
aI., 1997; He and Mladenoff, 1999; Li, 2000a).
Fire frequency, cycle, size distribution, season,
intensity, and severity are among the descriptors
(Weber and Flannigan, 1997) of a fire regime.
Fire occurrence and burned area related descrip­
tors, such as fire frequency and fire cycle, are
commonly simulated in forest dynamics models.
In this section, the concepts and usefulness of fire
frequency and fire cycle will be described first, and
followed by a review of how the concepts are
implemented in existing fire models.
0304-3800/02/$ - see front matter © 2002 Else vier Science B.V. All r ights reserved.
PH: S 0 3 0 4-3 8 0 0 ( 0 2 ) 0 0 0 69- 8
C. Li / Ecological Modelling 154 (2002) 103-120
1.1. Concepts and usefulness of fire frequency and
fire cycle
Fire frequency is perhaps the most frequently
used terminology for describing a fire regime.
However, it has been used to address different
questions under various situations. Fire frequency
is originally defined in a point-based manner as
'the average number of fires that occur per unit
time at a given point' (Merrill and Alexander,
1987), FFp, but it is also applied to a specific area
(Agee, 1993) and referred to as 'fire incidence' or
'fire occurrence' (e.g. Heinselman, 1981; Johnson
and Van Wagner, 1985; Food and Agriculture
Organization of the United Nations, 1986). The
reciprocal of fire frequency is defined as fire inter­
val: 'the average number of years between the
occurrence of fires at a given point' (Merrill and
Alexander, 1987). The sizes of fires, nevertheless,
are not taken into account in the original fire
frequency concept.
Fire cycle, on the other hand, is originally
defined in an area-based sense as the number of
years required to burn over an area equal to the
entire area of interest, although some sites could
be burned more often than other sites within the
area, FCa, (e.g. Johnson and Van Wagner, 1985;
Merrill and Alexander, 1987; Turner and Romme,
1994). The fire cycle is also equivalent to 'fire
rotation' (Agee, 1993) and 'return period' (Hein­
selman, 1973; Food and Agriculture Organization
of the United Nations, 1986), as well as 'fire
return interval' (e.g. Apps et aI., 2000; Bhatti et
aI., 2001). The other definition of fire cycle by
Agee (1993) is that 'the average stand age of
forest whose age distribution fits a mathematical
distribution (negative exponential or Weibull)';
however, this definition will not be discussed in
this paper. The reciprocal of fire cycle is defined
as percentage of annual area burned, PAAB, and
the total area burned is the main focus in the fire
cycle concept:
In one special case, however, total area reduced
to only one site, the fire cycle is equal to the fire
interval, and the reciprocal of it is equal to fire
frequency (note that the calculation method also
applies to
Fire frequency can be used to indicate up to
three situations. One, how often a given site could
be burned, or how often a fire would recur on a
given site. Second, how many fire incidents could
occur per unit time on a given region under
investigation, or the density of fire incidents for
the region. Third, how often the cumulated
burned area could reach the total area of the
region under investigation, or the relative degree
of damage that fire could cause for the region.
In the first situation, fire frequency is concerned
with a particular site and hence it is considered as
a point-based representation. This fire frequency
can be described by one of the following descrip­
tors: (a) the average number of fires per year FNp,
usually less than 1; (b) the fire frequency FFp or
its reciprocal of FCp; and (c) the distribution of
intervals between two successive fire events at the
site under investigation. The data structure re­
quired for calculating FFp and FCp is the presence
or absence of a fire event in any given year over a
period of time. These events are usually consid­
ered as independent of each other, and not corre­
lated with the events that occur in their
The second situation is an expansion of the first
situation in which the spatial concern in size of a
study area is enlarged from a particular site to a
given region. Thus the average annual fire number
for the region could then describe the fire fre­
quency. Defined in this way, the number of fire
incidents in a single year might be more than one.
This fire frequency concept should also be consid­
ered as point-based because the sizes of burned
areas are not taken into account. From both the
forest dynamics and fire management perspec­
tives, however, this definition of fire frequency
merely counts the number of fire incidents and
does not provide useful information, either in fire
locations or fire sizes. Therefore, it will not be the
focus of this paper.
In the third situation, the meaning of fire fre­
quency refers to the total area burned. Thus the
definition of fire frequency can be considered as
C. Li I Ecological Modelling 154 (2002) 103-120
area-based. This fire frequency can be described
by one of following descriptors: (a) the fire cycle
FCa; (b) the FFa or PAAB, which is the reciprocal
of FCa; and (c) the fire size distribution. The data
structure required for calculating FCa is the an­
nual fire maps over a period of time for the region
under investigation.
With these diverse meanings of fire frequency,
one question could be raised: what are the distin­
guishable features among different fire frequency
concepts? The question can be addressed in terms
of bias estimation from a computational perspec­
tive. This paper examines this question by com­
paring different fire frequencies with regard to an
ideal historical fire data set.
1.2. Implementation of fire frequency concept in
fire models
There are different methods of simulating fire
frequency and fire cycle in forest dynamics mod­
els. For example, in small area or point-based
forest dynamics models, fire is represented as a
random event, which follows one of the following
three probability distributions. The first is a com­
plete random probability distribution in which the
probability of fire is the reciprocal of a predeter­
mined fire cycle. It applies the same fire probabil­
ity to each unit of a landscape, while discounting
the interactions among units (Van Wagner, 1978;
Keane et aI., 1989). The second uses a uniform
fire probability distribution in which the fire
would occur at either a fixed interval, or a ran­
dom interval with a mean value equal to a prede­
termined interval (e.g. Apps et aI., 2000; Bhatti et
aI., 2001). The third uses a forest age-dependent
fire probability distribution in which the probabil­
ity of fire is a function of forest age or time since
last burn (e.g. Clark, 1989; Prentice et aI., 1993;
McCarthy et aI., 2001). In these models, a com­
mon hypothesis that forest dynamics of large
regions can be represented by forest dynamics of
point-based sites was assumed, so that forest fire
processes that usually operating at a large spatial
scale were simplified as random events at a small
spatial scale. Such treatment might be reasonable
for investigating forest dynamics under intensive
management such as in Scandinavian boreal
forests; however, it might not \:>e suitable for the
situations in the boreal forests of Canada because
of the observed irregular sizes and shapes of
burned areas (Li and Apps, 1996).
In large area or area-based forest dynamics
models, however, fire frequency and fire cycle are
needed to run simulation models (e.g. in
DISPITCH of Baker et aI., 1991, LANDIS of He
and Mladenoff, 1999, and FLEP-X of Boychuk et
aI., 1997). These models are usually linked with
available geographical information system (GIS)
data sets, and represent a forested area (i.e. a
forest landscape) in either a vector or a raster
format. The models simulate the temporal and
spatial dynamics of forest stand mosaic patterns
through the creation of new stands by fires and
dissolution of old stands through forest growth
and subsequent decline and decay. The most com­
mon model design is to treat fire frequency and
fire size distribution separately, i.e. fire frequency
represents the annual fire number without consid­
ering their associated fire sizes, and thus the fire
size distribution is independent of fire frequency.
Therefore, two independent random numbers
were required to run these models. Once the size
of a fire was determined, the fire would spread to
this size regardless of the landscape and weather
conditions. However, different algorithms have
been employed in different models to realize the
predetermined fire sizes. The identification of the
two probability distributions from which the two
random numbers are drawn is then consequently
vital in determining the simulation results. While
the spatial correlation among burned forest areas
were taken into account in these large area or
area-based forest dynamics models, a pre-require­
ment for running these models is that users must
have a good understanding of the fire regimes in
order to determine the annual fire number and
sizes for each of the fires to be simulated. These
models were mainly designed and used for provid­
ing quick answers to what-if type of questions.
Other area-based forest dynamics models deter­
mine fire sizes according to landscape structure
and weather conditions. The Manitoba model
(Peterson, 1994; Holling et aI., 1996) was devel­
oped based on cellular automata algorithm where
a fixed rule of fire spread (i.e. forest age-depen-
C. Li / Ecological Modelling 154 (2002) 103-120
dent) and a fixed number of fires per season were
applied. The ON-FIRE model (e.g. Li et aI., 1997;
Li, 2000b) simulates fire size as a result of interac­
tions among fuel type, landscape topography, and
forest age. The annual fire number is a function of
the number of fire ignition sources and the site
conditions where the fire ignition sources were
presented. In the SEM-LAND model (e.g. Li,
2000a; Li et aI., 2000; Li and Barclay, 2001), both
fire initiation and spread probabilities are also a
function of weather conditions, and the relation­
ships summarized in the Canadian Fire Weather
Index System (FWI) (Van Wagner, 1987) and Fire
Behavior Prediction System (FBP) (Forestry
Canada Fire Danger Group, 1992; Hirsch, 1996)
were employed to calculate these probabilities.
Consequently, the simulated fire numbers and fire
sizes may or may not be the same as model input,
thus the fire frequency and fire cycle are the
output of the models. These models followed a
systems analysis approach to investigate forest
dynamics at a particular forest landscape as a
whole. The models attempt to incorporate con­
temporary knowledge and information about the
interactions among fire events and forest ecosys­
tem components from published results of fire
sciences. Due to the detailed simulation of fire
processes, applications of these models usually
require more computing power. However,
emerged properties from the forest dynamics at
the landscape scale such as self-organization of
forest dynamics (Holling et aI., 1996) could be
demonstrated, and hence natural fire regimes
could be simulated (Li, 2000a).
The diverse model implementation of fire im­
pact on forest dynamics results from, among
other factors, the different concepts of fire fre­
quency and fire cycle employed by model design­
ers and developers. Thus, a theoretical question is
raised: are fire frequency and fire cycle possibly
related to each other? The answer to this question
may be important in improving the simulations of
how fire influences forest dynamics. Van Wagner
(1978) discussed the relationship through a theo­
retical negative exponential probability distribu­
tion that links fire cycle and forest age
distribution. This paper examines a hypothesis
that point-based fire frequency and fire interval
concepts are the special case of area-based PAAB
and fire cycle concepts from a computational
This paper starts with the derivation of fire
frequency and fire cycle concepts, followed by the
presentation of empirical fire data structure and
the structure of the ideal data set of fire history
from a computational point of view. One such
ideal fire data set is therefore, generated by using
a generic spatially explicit model for landscape
dynamics (SEM-LAND) on a study area in west­
central Alberta. This data set is then used to
examine the hypothesis stated above and the ac­
curacy of different methods of estimating fire
frequency and fire cycle. A general discussion on
methodology of fire frequency and fire cycle esti­
mation, and a number of suggestions on simulat­
ing fire regimes will close the paper.
2. Derivation and estimation of fire frequency and
fire cycle
As mentioned in the previous section, fire oc­
currence and burned area related fire regime de­
scriptors such as fire frequency and fire cycle are
commonly used to characterize a fire regime. The
concepts of fire frequency and fire cycle are funda­
mentally derived from the structure of available
empirical fire data. In this section, the structures
of historical fire data shall be reviewed, and the
derivation of the concepts of fire frequency and
fire cycle shall be described.
2.1. Structure of empirical fire data
Historical fire data can be categorized into the
following five categories according to their
Category 1 comprises the point-based fire scar
data obtained from the dendrographical analysis
of fire scar records. In fire history studies, this is
the most common method of collecting field sam­
ples to determine when fires occurred at the sam­
pling sites. The fire scar data usually represent the
occurrence history of non-stand replacement fires
(Johnson and Gutsell, 1994; Johnson et aI., 1999;
Weir et aI., 2000), but the sampled tree could also
C. Li / Ecological Modelling 154 (2002) 103-120
be right on the edge of a historical stand-replace­
ment fire. Examples of such fire data are illus­
trated in Fig. l c.
Category 2 comprises the sediment pollen and
charcoal data, which are mainly point-based with
influences from surrounding conditions. Different
from fire scar records, the pollen and charcoal
records collected from lake-sediment represent the
occurrence of historical fires around the sampling
locations, i.e. indirect evidence. It can be the
Sample I
Sample ..
Fig. I. Fire history data struct ure. (a) A yearly fire map shows the b urned area within a st udy area under investigation. (b) A series
of yearly fire maps collected o ver time. The samples I to 5 indicate the random samples located within the st udy area. (c) Fire
incidences identified from the random samples.
C. Li / Ecological Modelling 154 (2002) 103-120
indication of either stand-replacement fires or
non-stand-replacement fires. However, the exact
locations of historical fires can not be precisely
determined because strong winds can blow pollen
and charcoal from locations far away (e.g. Swain,
1978; Green, 1981; MacDonald et al., 1991; Clark
and Royall, 1996).
Category 3 comprises the individual fire records
from fire report systems under fire management,
which include the date, location, size, cause, age
at burn, vegetation, etc. for each recorded fire.
The statistical summary of these records usually
appear in various government documents. The
individual fire records from the fire report systems
have detailed descriptions of historical fires. How­
ever, the data are non-spatial, i.e. the spatial
locations of the burned areas are not described.
The data have been widely used for demonstrat­
ing the number of fires and the area burned for
given geographical regions.
Category 4 comprises the annual fire maps
compiled by fire management agencies, represent­
ing the areas affected by fires. The maps show the
exact spatial locations of fires, especially larger
fires, within the concerned regions; consequently,
the impact of fires can be accurately estimated.
The fire map compilation over a long period of
time for a given region will provide a data set
closer to the ideal fire history data set described in
Fig. l a and b and Section 4.
Category 5 comprises the area-based stand­
origin data maps obtained from whole inventory
sampling, thus indicating time-since-last-burn for
each forest stand. This type of data relies on the
underlying assumption that all stands result from
the presence of fire. This assumption may or may
not be true, because stand-replacement could also
be caused by other factors such as severe insect
attacks and blowdown by strong winds. The con­
struction of a stand-origin map requires consider­
able time and resource investment, but results in a
fuller understanding of stand-replacement history
(e.g. Heinselman, 1973; Johnson et al., 1999; Weir
et al., 2000). However, stand-origin maps cannot
provide any information about surface fire
regimes. A forest age layer of the forest resource
inventory data can be used to approximate the
map when required time and resources are un-
available (e.g. Li, 2000a). The stand-origin maps
in category 5 also contain spatial data represent­
ing the results of overlay footprints from histori­
cal fires. The maps illustrate the fire history for
given geographical regions (e.g. Johnson and Gut­
sell, 1994; Johnson et al., 1999; Weir et al., 2000).
The basic structure of the historical fire data
from the five categories of data sources can be
classified into two types: point-based fire data and
area-based fire data. The point-based fire data
include those from categories I and 2. The area­
based fire data include those from categories 4
and 5. Categories 3 fire data have characteristics
of both point-based (for fire ignition location) and
area-based (for final size of each fire in a non-spa­
tial sense) fire data. Fig. I shows the two types of
historical fire data. Fig. l a represents an annual
fire map within a study area. The two-dimensional
fire map indicates all the areas burned within the
fire season without distinguishing how many fires
actually caused those burns. When the annual fire
maps accumulated over a period of time, a three­
dimension historical fire data map can be con­
structed as showed in Fig. lb. The longer the
period, the better and more complete the data set
will be. The five samples in Fig. I b represent
point-based fire data from the dendrographical
study of fire scars. Samples were taken from
different locations within the study area. Fig. I c
illustrates possible results obtained from the five
samples, with 4, 4, 5, 5, and 3 fire scars,
2.2. Fire frequency and fire cycle estimated from
point-based fire scar data
In each of the samples illustrated in Fig. I c, a
time period between any two adjacent fire scars
represents a fire interval. From the long-term
perspective, these fire intervals would usually have
different lengths even when considering a single
sample. Thus, the mean value of the fire intervals
should be an appropriate estimate of true fire
interval for the sample site. This concept was
characterized in the original definition of fire in­
terval (Merrill and Alexander, 1987). As described
in Section 1.1, the fire interval has been consid­
ered as a special case of the fire cycle when the
C. Li / Ecological Modelling 154 (2002) 103-120
size of the study area is reduced to a single site.
Therefore, the fire interval is referred to as a
point-based estimate of fire cycle, FCp, and its
reciprocal as a point-based estimate of fire fre­
quency, FFp.
The FCp is usually estimated by using the for­
mula (3) with two exceptions. The usual situa­
tion for fire scar data sets is that no fire scar, is
located at either end of the data set, such as the
samples I, 3, and 5 illustrated in the Fig. I c. In
this case, three full FCp and two incomplete FCp
would appear, and the best estimate of FCp can
be represented as:
FCp = TINo. of fire scar
where T is the total time period.
The first exception is that of two fire scars
located at both ends of the data set (e.g. sample
4 in Fig. Ic), hence only three full FCp are pre­
sented. Thus, the estimate of FCp would be:
FCp= TI(No. of fire scar
The second exception is that of one fire scar
located at one end of the data set (e.g. sample 3
in Fig. l c), hence there are three full FCp and
one incomplete FCp in the data set, so the FCp
estimate would be between the Eqs. (3) and (4):
FCp = TI(No. of fire scar - 0.5)
It is worth noting that a reliable estimate of
FCp requires long-term observations covering a
period of several FCp• In the Fig. Ic, assume
that the total time period is 200 years, then the
FCp calculated from the samples 1 , 3, and 5 are
50, 40, and 66.7 years (with 4, 5, and 3 fire
scars), respectively. If the three samples are the
only data available for the study area, then the
FCp estimated for the study area would be 55.2
years, which is the average of the three fire cy­
SN i= I No. of fire scar
where SN is the total number of samples, and
the subscript i indicates the ith sample. How­
ever, if all five samples are taken into account,
the estimated FCp for the study area would be
47.6 years. This also indicated that the FCp esti­
mate depending upon the criteria of determining
the sample sizes. In reality, point-based fire scar
data are biased because only trees that survived
a past fire will be available today to give data.
At a given location, one can only look back as
far as the time when the previous tree at the site
was killed by fire.
2.3. Fire frequency and fire cycle estimated from
area-based fire maps
From a fire map illustrated in Fig. Ia, annual
total area burned can be calculated, so that an
area-based fire cycle, FCa, can be estimated from
any start year until the year when a cumulative
burned area equals the total area under investi­
gation (see Fig. I b). Assume the data set ex­
pressed in the Fig. 1 b last T years that cover the
length of several FCa, then the FCa can be cal­
culated as:
where B indicates total area burned during the T
years, and A is the size of the area under inves­
tigation. The reciprocal of FCa is the PAAB,
which is an area-based estimate of fire frequency
FFa indicating how fast the total area could be
burned once.
2.4. Fire frequency and fire cycle estimated from
records contained in fire report systems
Government fire agencies in Canada have kept
historical individual fire event records in fire re­
port systems in an effort to improve future fire
management. The records are a summation of
non-spatial annual fire maps and numerous
small fire records that are usually not shown on
annual fire maps.
The summation of sizes of these fires is the B,
thus the FCa and FFa can be calculated from
Eqs. (7) and (2). The data provided in these
records is not as detailed as those used in fire
scar analysis, and consequently the FCp cannot
be adequately estimated.
C. Li / Ecological Modelling 154 (2002) 103-120
2.5. Fire ji-equency and fire cycle estimated from
government statistics
Forest fire statistics from government reports
often contain information on annual total number
of fires FN and the total area burned B. The
information is usually organized by province or
region (Canadian Council of Forest Ministers,
1 997). Thus, FCa and FFa can be estimated from
the Eqs. (7) and (2) for different provinces or
Fire size can be ideally represented by its distri­
bution. The sizes associated with individual fires
are generally available from these government
statistics. There is a mean fire size (MFS) that can
be calculated as
Caution should be used when calculating MFS.
The accuracy of the FN could change over time
due to the increasing human capacity to detect fire
in remote areas, and B could be underestimated
because of undetected fires. The uncertainty asso­
ciated with estimating FN would have a relatively
large influence in calculating MFS, but less effect
on FCa and FFa calculation because undetected
fires are usually small. For example, if 1 00 fires
(FN) burned a total area of 1 0 000 ha (B) over
1 00 years (T) within a landscape of 1 0 000 ha (A),
then the MFS will be 1 00 ha and the FCa is 1 00
years. However, if five fires were unreported, then
the FN is actually 1 05, and the MFS should be
95.24 ha. Consequently, the relative error is about
5%. Assuming the five unreported fires were small
(such as 20 ha each), then estimated FCa would be
99.01 years, and this leads to about 1 % relative
error. This example also suggests that area-based
fire descriptors FCa and FFa might be less sensi­
tive to possible instrumental errors in detecting
fire incidence.
In addition to the mathematical concern indi­
cated above, some theoretical issues need to be
resolved for the MFS. Statistical MFS is usually
meaningful if the fire size distribution follows a
Normal probability distribution. In reality, fire
size distributions are not even close to such a
distribution, and a negative exponential probabil­
ity distribution is usually assumed for fire size
distributions (e.g. Van Wagner, 1 978; Baker,
1 995). This assumption might be expected if
enough natural (i.e. without human intervention)
fire records (i.e. over a long period) are included
in the fire size distribution. Therefore, the MFS
might be meaningful when it serves as a parame­
ter of the theoretical negative exponential fire size
distribution (Li et aI., 1 999).
A mathematical description for different fire
frequency and fire cycle concepts from an ideal
data set shall be presented in the next section.
3. Mathematical description for fire frequency
concepts from an ideal data set
Let a forest landscape be represented by a grid
of cells (the size of a cell can be determined by the
representation of a single sample) with m rows
and n columns. The fire occurrence for each cell
can then be expressed as x(k)
in a three-dimen­
sional array, which indicates whether a fire occurs
at the ith row, jth column, and kth year. The
is defined as follows:
(if a fire occurs)
(if no fire occurs)
The combination of ith row and jth column
determines a location within the area under inves­
tigation. For a fixed combination of i and j, x(k)
is a time series of fire incident (see Fig. I c)
suitable for estimating point-based FFp and FCp
(similar to historical fire data sources in categories
1 and 2 in the Section 2. 1 ). They can be calculated
(1 0)
For a fixed k, x(k)
is a fire map indicating the
locations burned by fires during the year k (see
Fig. l a). Therefore, the summation of these loca­
tions is the annual area burned in year k. A
collection of such fire maps constitutes an ideal
fire history data set (Fig. I b). The area-based fire
frequency, FFa, which can be expressed as PAAB,
can be estimated by the mean value of annual
summations over the period of investigation. The
C. Li / Ecological Modelling 154 (2002) 103-120
FCa is the cumulated area burned from year 1
until it reaches the total area under investigation.
Due to its dynamic properties, the FCa for a given
area should be represented by its long-term aver­
age, i.e. over a period that may contain a number
of FCa. Thus, the FCa and FFa can also be
calculated as
FCa=I/FFa=m xnxT/I I I xij(k)
i� l.i� I k� I
The hypothesis that point-based fire frequency
and fire interval concepts are the special case of
area-based PAAB and fire cycle concepts, is
equivalent to the hypothesis that point-based FFp
and FCa are the special case of area-based FFa
and FCa when m and n are equal to 1, from a
computational perspective. To test this hypothe­
sis, an ideal fire data set needs to be prepared. The
method of generating such an ideal fire data set
shall be described in the next section, and the
statistical results of the data set shall be used as
the true value of FCa for comparison from a
computational perspective.
4. Generation of an ideal data set
A data set with a structure illustrated in Fig. 1b
was constructed by running the SEM-LAND
model. The model was designed to perform de­
tailed temporal and spatial simulation of forest
landscape dynamics subjected to natural distur­
bance regimes (primarily fire). A forest landscape
is simulated as a grid of rectangular cells, and
each cell represents a forest stand with homoge­
neous site condition, vegetation cover type, and
tree age. The raster-based model uses current
landscape conditions such as GIS data layers of
land and vegetation cover type, forest age, site
index, terrain, as well as fire and weather condi­
tions as input. The model outputs simulated indi­
vidual fire records and fire regime statistics, as
well as forest conditions over time and space such
as forest age, volume, basal area, and height.
Forest growth without disturbance is described
by standard forest growth formulas for Alberta,
Canada (Alberta Forest Service, 1984). The for­
mulas describe the basal area and volume of
major tree species as a function of their ages.
A forest fire process was simulated in two
stages: initiation and spread. The annual fire igni­
tion sources were randomly allocated within the
study area. The presence of a fire ignition source
may cause a fire initiation (most of the trees in a
I ha forest are killed) depending on a fire initia­
tion probability, which is a function of forest age,
forest type hence fuel type, and weather condi­
tions. Once a fire is initiated, it will have the
potential to spread to adjacent forest stands de­
pending on a fire spread probability, which is a
function of forest age, forest type, terrain, and
weather conditions. The relationships summarized
in the FWI (Van Wagner, 1987) and FBP
(Forestry Canada Fire Danger Group, 1992) sys­
tems describe how weather conditions will influ­
ence the moisture code in different types of fuel,
and how the fuel moisture code will affect the rate
of fire spread. Based on an assumption that fire
initiation and spread probabilities are propor­
tional to the rate of fire spread, the relationships
are used in the SEM-LAND model to simulate
detailed fire spread process at the landscape scale.
The fire is allowed to spread until it is stopped
in all its directions or reaches the boundary of the
study area. In yearly time steps, forest stands age
1 year if not burned, or are reset to zero if they
are burned. A self-replacement of forest cover was
assumed in the model. The detailed model de­
scription can be found in Li (2000a,b), Li et al.
(2000), and Li and Barclay (2001).
The Athabasca Working Circle Compartments
28, 29, 31, and 32 of Weldwood of Canada (Hin­
ton Division) in west-central Alberta, Canada was
chosen as the study area for current research
purposes. The study area has 31 444 ha in total
and forest covered 30 974 ha. Within this area,
more than half of the forest (50.4°A» is dominated
by lodgepole pine (Pinus contorta Dougl. Ex
Loud. var. lalifolia Engelm.), and the rest of the
area is covered by black spruce (Picea mariana
(Mill.) B. S. P.) ( 13.73%), white spruce (Picea
glauca (Moench) Voss) (10.42%), aspen (Populus
tremuloides Michx.) (12.99%), and non-productive
land (12.47%). A total of 10 000 ha within the
area were selected as the base for the data set
generation. The GIS data layers of land and
vegetation cover type, forest age, site index, as
C. Li I Ecological Modelling 154 (2002) 103-120
2 r-----�����--��---�
F ig . 2. S im ulated ann ual fire n umber (a), area b urned (b), and mean forest age (c) of the constructed ideal fire data set.
well as weather conditions for the study area were
used as SEM-LAND model input. The fire and
forest dynamics were simulated for a period of
500 years, and the simulated fire maps for the last
300 years were used as the ideal fire data set.
Fig. 2a and b show the simulated annual fire
number and area burned over the 300 years.
While the mean annual fire number was almost
constant, the annual area burned fluctuated sig­
nificantly. There were 12 years (4% of the 300
years) that had an annual area burned larger than
500 ha, which is 5% of the total area. The largest
annual area burned was 7141 ha in year 36, the
second largest was 5156 ha in year l OO. The
annual area burned in all the rest of the years was
small. The irregular fire sizes and annual area
burned over time demonstrated in this data set are
typical in Canadian boreal forests.
The constructed data set consists of 300 years'
fire maps. Each fire map is a grid of 100 by 100
cells, and each cell represents an area 1 ha in size.
Burned cells are denoted as 1 and unburned cells
have a value of O. Therefore, the summation of a
fire map is the total area burned in the particular
year, and the summation of the whole data set is
the total area burned in 300 years. The fire cycle
FCa, which can be calculated by using Eq. (7), is
about 131.3313 years, i.e. the total area burned
during the 300 years is equal to 2.2843 times of
the total size of the area. This FCa shall be treated
as the true value for this data set.
The SEM-LAND model also outputs forest
stand age mosaic maps at the end of each year's
simulation, in order to be compatible with the
model output of fire maps. Fig. 2c shows the
dynamics of mean forest age under such a fire
regime. The mean forest age sharply decreased
after the large fires at the year 36 and the year
l OO, as well as after a period of relatively large
fires from year 169 to year 172.
In the next three sections, estimates of FCa
from fire maps, stand-origin maps, and point­
based sampling designs, are presented. The results
are compared with the true value of FCa, and thus
the hypothesis that point-based fire frequency and
fire interval concepts are the special case of area­
based PAAB and fire cycle concepts is tested from
a computational perspective.
C. Li / Ecological Modelling 154 (2002) 103-120
5. Fire map-based
Fea estimate
There are 300 continuous fire maps in the gen­
erated ideal fire data set, and each fire map con­
tains fire records in 100 by 100 cells, either
burned or unburned. If all 300 fire maps are
taken into account, then the mean value of the
300 annual area burned would provide sufficient
information to estimate FCa according to Eq.
(11), and the result would be equal to the true
value of 131.3313 years exactly (see description in
Section 4). In reality, no such long period of fire
maps is available. Therefore, the question is how
many fire maps would be sufficient for a satisfac­
tory estimation of FCa? Two issues need to be
considered in answering this question: ( l ) influ­
ence of different lengths of data sets on the FCa
estimation; and (2) influence of the starting year
of compiling fire maps on the FCa estimation.
For the first issue, one could expect that once
the number of fire maps (i.e. data length) reaches
a certain threshold, the estimated FCa would tend
to be stable and closer to the true value of FCa•
For the second issue, one could expect that when
fire maps are compiled from different starting
years, i.e. different times during a fire cycle, the
estimated FCa would still tend to be stabilized
and closer to the true value of FCa, as long as the
data set is long enough. Furthermore, the varia­
tion associated with FCa estimates would change
with different data lengths.
-; 2.5
- TnHvalue
Start alyear 1
Start atyear 40
- - Slart at year 105
.,,' . '
Length of data set (year)
F ig. 3. The Fea estimated from d ifferent n umbers of fire maps
and three d ifferent start ing years.
To examine these two issues, FCa values were
calculated using different numbers of fire maps
for three different starting years. The first starting
year is year I of the data set, so that 300 FCa
values could be obtained by using the first year's
fire map, the first 2 years' fire maps, the first 3
years' fire maps, etc. Fig. 3 shows the results
where an overestimate of FCa appeared before the
first large fire in year 36, an underestimate of FCa
appeared after that large fire, and the estimates of
FCa thereafter were gradually approaching the
true value of FCa• For this particular data set, less
than 34 years of data would generate a FCa
estimate that could be more than four times the
true value of FCa•
The second starting year is year 40 (see Fig. 3),
which is after the first large fire in year 36. Thus,
only 260 FCa estimates were produced. Although
a relatively stable FCa estimate could be reached,
after the overestimate of FCa for the first 60 years
(i.e. before the second large fire occurred at year
100), the required number of fire maps has
The third starting year is year 105 (see Fig. 3),
which is after the second large fire in year 100. In
this case, 195 FCa values can be estimated. There
was no large fire during this period, and the
estimated FCa gradually approached a relatively
stable value of FCa. The required number of fire
maps is reduced to about 25.
The results suggested that the number of fire
maps required to obtain a relatively stable FCa
might not be constant, and it could be influenced
by the fire history of the region under investiga­
tion. Furthermore, fewer fire maps might be re­
quired if less extreme fire behavior occurred
during the past two to three decades.
The results presented in this section clearly
indicate that the length of a set of fire maps or
the starting point of compiling fire maps can
significantly influence the estimate of FCa• This
explains that different FCa estimates for the same
region could be obtained if different continuous
numbers of fire maps are used for FCa estimation.
The very large number of fire maps may also
needed in some cases, which will not generally be
C. Li / Ecological Modelling 154 (2002) 103-120
probability that a stand survives to time t, Set),
can be described by a negative exponential prob­
ability distribution:
-.:- 600
Estimated fire cycle
True fire cycle
where A is the parameter of the exponential
probability distribution, defined as the reciprocal
of its mean value. Consequently, FCa can be
calculated as:
.. .... '
.. .... I
Time step (year)
Fig. 4. The Fe. estimated from stand-origin maps at IO-year
intervals by a non-linear curve fitting procedure.
6. Stand-origin map-based Fea estimate
A stand-origin map represents a snapshot of
the forest patch age mosaic, which results from
overlapping stand-replacement fires in the past.
The method of constructing a stand-origin map
usually requires air photo interpretation of stand
boundaries combined with sampling each stand
within the whole region to determine stand age.
This means that considerable time and resources
are required to perform such analysis. However,
the map of forest stand age in forest inventory
data can approximate the stand-origin map (Li,
2000a). The focus of investigation here is
whether a reliable estimate of FCa can be ob­
tained by constructing such a stand-origin map,
in other words, whether a stable FCa can be
obtained if the stand-origin map were con­
structed at different years. To address this issue,
an analysis of forest age mosaic maps, another
SEM-LAND model output from the same simu­
lation, was carried out.
The estimation of FCa based on such a stand­
origin map is to perform a survival analysis for
estimating the expected length of stand
longevity. A standard method for such analysis
has been described in detail (e.g. Johnson and
Van Wagner, 1985; Johnson, 1992; Johnson and
Gutsell, 1994; Reed et aI., 1997), based on an
assumption that forest age across a landscape
follows a negative exponential probability distri­
bution under a constant fire hazard. Thus, the
A non-linear curve fitting procedure was ap­
plied to every 10-year interval of stand-origin
maps (SAS Institute Inc., 1989). The curve
fitting was performed for determining the best fit
of A value for the cumulative percentage of area
burned against forest age, in order to calculate
the FCa• The estimated FCa values at different
years were showed in Fig. 4.
The results in Fig. 4 suggest that the FCa
estimated from the stand-origin maps might also
not be constant over time. In other words, the
time for constructing the stand-origin map
would have a significant influence on estimated
FCa• The reason for such an unstable estimate
of FCa is probably due to the changes in forest
age structure, and the forest age structure is in
turn determined by the fire regime (Li and Bar­
clay, 2001). However, the estimate could be im­
proved by mapping stand age on a larger area.
The results also suggest that the standard
method might generally overestimate the fire cy­
cle. This phenomenon can be explained by the
historical overlapping of burned areas. With the
increased degree of overlapping, the observed
patch sizes (which are used as the estimated
sizes of burned areas in the standard method),
especially the sizes of old patches, will be de­
creased. Consequently, the estimated A value
would be decreased, hence the increased fire cy­
cle estimate.
The results presented in Fig. 4 also suggest
that the FCa estimated from a stand-origin map
might or might not be stable, depending on the
fire history. Therefore, estimation of FCa using
this standard method should be done with cau­
! 15
and their
(B). with different levels of
different lire numbers.
If all 10 000
\vere taken into account.
then the mcan value of the
10 (Jon estimated F("
13 J.:n 13 years
the true \alue of
. however.
a small perceDt­
could be taken (Johnson and
the issues arc: (I) how
arc needed to ubtain a
estimate of F(
'vvhat "am-
is the best
with the least
this purposL'
size needccL' In this sectilll1.
7. f.
Simple random
unit has an
chance of selection
C. Li / Ecological Modelling 154 (2002) 103-120
� 140
t +---+
- - -
- - - + - - - -!- 130
Sample size
Fig. 6. The Fe" estimated from a random sampling design
with different sample sizes. The horizontal dished line indi­
cates the true value of Fe..
underestimated with smaller variations. Neverthe­
less, the range of estimated FCa is not very large.
In reality, the randomness was not implemented
fully, with a higher probability to sample on older
forest stands than that on younger stands. Conse­
quently, an overestimated FCa might appear be­
cause older forest stands tend to have burned
fewer times during the 300 years.
7.2. Under a systematic sampling design
The systematic sampling design is to take sam­
ples at a fixed interval in space. It is easier to
locate and execute the sampling without mistakes,
and intuitively, it seems likely to be more precise
or representative than simple random sampling
(Cochran, 1 977). This method has been popular
for assessing timber and range conditions (Avery
and Burkhart, 1 994). Under this sampling design,
once the initial sample unit is selected, other units
can be mechanically spaced at uniform intervals
throughout the landscape.
Four different spatial intervals were used in the
computer simulation of this sampling design on
the landscape. The first interval was a distance of
1 0 units; therefore, 1 00 sample units were selected
started at the upper left corner. A total of 1 00
possible designs could be implemented, when the
coordination of the initial sample unit changed
across the landscape. The second interval was a
distance of 1 5 units, thus the sample size of 49
could be selected in each sampling design, and the
maximum possible designs totaled 1 00. The third
interval was a distance of 20 units, with a simple
size of 25 for each design, and a total of 400
possible designs could be determined. The fourth
interval was a distance of 30 units and 1 6 sample
units in total is selected in each sampling design.
A total of 1 00 sampling designs could be imple­
mented. Table I shows the estimated FCa and
their associated standard errors for different spa­
tial intervals.
Results in Table I also suggested that with the
increase of sample size, the estimated FCa would
be closer to the true value, and the associated
variations could become smaller. However, the
sampled spatial data cannot be analyzed statisti­
cally in general (Southwood and Henderson,
2000), because it might coincide with some unsus­
pected systematic distribution pattern. However,
some studies have shown that the resulting statis­
tics could be 'at least as good, if not rather better'
than those obtained from random sampling (e.g.
Miline, 1 959). From a cost-related perspective, it
could be an advantage for the systematic sam­
pling design because it may be carried out more
quickly than the random method. In this particu­
lar data set, the systematic sampling design might
Table 1
Fea and associated standard errors estimated from a systematic sampling design
Spatial interval
Sample size
Fire cycle (year)
Standard error
C. Li I Ecological Modelling 154 (2002) 103-120
be seen as equally good as the simple random
design, because 16-20 samples could provide a
similar estimate on FCa.
8. Discussion
The results presented in this paper confirmed
that: (1) the hypothesis that point-based fire fre­
quency and fire interval concepts are the special
case of area-based PAAB and fire cycle concepts;
(2) despite the theoretical confirmation of the
hypothesis, biased estimates of fire frequency and
fire cycle could arise from existing methods of
estimation; and (3) the biased estimates can be a
result of sampling design, sample size, and distur­
bance history of the study area under investiga­
tion. Therefore, to obtain an accurate estimate of
fire frequency and fire cycle for a particular area
needs to consider the stand-replacement distur­
bance history of the area. There are minimum
requirements of data quality that are needed to
derive unbiased estimates.
FCa and FFa estimations from area-based fire
and landscape data such as fire maps and stand­
origin maps might be significantly influenced by
recent fire history, especially by large and extreme
fire disturbances (see Figs. 3 and 4). At least
35-40 continuous annual fire maps would be
required to obtain relatively stable FCa and FFa
estimates for this particular data set. The stand­
origin map would provide overestimates of FCa
and FFa in general, but a significant estimation
bias could arise if the map is constructed shortly
after a large or extreme fire year (see Fig. 4). The
area-based data, especially stand-origin maps,
usually describe the effect of stand-replacement
fires. Therefore, the estimated FCa and FFa are for
crown fire regimes. For fire maps that include
severe surface fire effect, the estimated FCa and
FFa could represent a mixture of crown and sur­
face fire regimes.
FCa and FFa estimations from point-based sam­
pling data require at least 0.16-0.20% of the total
possible samples, i.e. 16-20 samples for this par­
ticular data set. A simple random sampling design
and a systematic sampling design demonstrated a
similar result for this particular data set (see Fig.
6 and Table 1). The point-based sampling data
(e.g. from fire scar records) mainly describe the
effect of non-stand-replacement fires; thus, the
estimated FCa and FFa are for surface fire
regimes. The fires in this particular data set were
assumed to have a stand-replacement effect, and
thus the estimations are still for a crown fire
The results presented in this paper suggest that
the estimates of fire frequency and fire cycle are
not static, but change over time (see Figs. 2-4).
This is consistent with the contemporary under­
standing of fire regimes. Many factors may influ­
ence the dynamics of fire regimes. For example,
weather or climate is apparently a major factor
that determines the fire behavior in any specific
fire event, thus the final fire size and in turn the
fire cycle. Vegetation type is also important in
determining fuel type and the succession of vege­
tation may lead to a different fuel type, corre­
sponding to a different development stage of
vegetation. In the FBP systems, for example, im­
mature and mature jack pine forests have been
recognized with different fuel types. Different fuel
types may indicate different fire behavior and thus
different probabilities of fire. Landscape structure
may also play a notable role in determining final
fire sizes and shapes, through the spatial configu­
ration of physical fire barriers and different fuels.
Topography can influence fire-spread process,
hence the final fire sizes and shapes (Van Wagner,
1977, 1988). Finally, the patterns of fire ignition
sources, such as level of fire ignition sources and
spatial-temporal presence within the area under
investigation, also influence fire dynamics (Li,
2000c). Considering the interactions among these
factors over time and space, it may be unrealistic
to expect a static estimate for either fire frequency
or fire cycle.
The estimated fire frequency and fire cycle
could vary for the same geographical region, if
different data sets representing different time peri­
ods are used (Fig. 3). The time periods that
available fire data sets covered are usually much
shorter than a fire cycle, thus a reliable estimate of
fire cycle is unlikely to be obtained. Therefore, it
would be appropriate to reaffirm that long-term
observations are needed to obtain reliable esti­
mates of fire cycles.
C. Li / Ecological Modelling 154 (2002) 103-120
The fact that different estimates of fire fre­
quency and fire cycle could be obtained for the
same geographical region has raised a concern
about the usefulness of the fire cycle concept (S.
Cumming, Boreal Ecosystems Research Ltd., per­
sonal communication, April 2001). While a reli­
able estimate of fire cycle cannot be obtained
from available methods, it is suggested that the
concept of fire cycle might still be useful, e.g.
serving as statistics in making comparisons among
different geographical regions or different vegeta­
tion types, if the same methods are applied to
different data sets for a given time period.
The results presented in this paper also sug­
gested that considerations on designing fire regime
models are needed. In the models of forest dy­
namics, the modelling purpose determined model
structure for either a small area or a large area.
Although the concepts of fire frequency and fire
cycle may be apparent to model developers, it is
strongly recommended that definitions be in­
cluded in model documentation. This could pre­
vent any possible confusion or misinterpretation
of results.
While incorporating the fire regime effect in
forest dynamics models is important, the method­
ology of the incorporation is crucial to ensure the
fire regime effect be represented reasonably. A
forest fire event is a meso-scale spatial process,
and a fire regime is the result of long-term interac­
tions among fire events, weather or climate, vege­
tation, topography, landscape structure, and fuel
conditions. Every representation of fire regimes in
forest dynamics models is a simplification of real
fire regimes, however, to what extent the simplifi­
cation can be referred to as appropriate is of great
interest to forest dynamics modelers.
There could be a spectrum of levels of fire
regime simplification, from the simplest one in
which a fire regime were represented as a periodic
destruction of a forest stand in non-spatial simu­
lation models (e.g. Apps et aI., 2000; Bhatti et aI.,
200 I ), to the most complex one in which a fire
regime were simulated as a series of spatial pro­
cesses occurring within a forest landscape where
weather and fuel conditions interacted with land­
scape structure according to the relationships de­
veloped in fire sciences (e.g. Keane et aI., 1996; Li,
The results presented in this paper suggest that
it might be appropriate for both model developers
and users to evaluate the reasonableness of fire
regime effect in forest dynamics models. Despite
difficulties in model validation due to the
availability of suitable empirical data (Rykiel,
1996), some qualitative indications of a fire regime
based on current understanding can be used to
evaluate the reasonableness of the level of fire
regime effect simplification. The qualitative indi­
cations may include spatial pattern formation and
dissolution, irregular fire shape, remnants within a
burned area, fire size distribution, and so on. The
evaluation could help to avoid misleading conclu­
sions on forest dynamics and fire regimes.
For non-spatial models, especially site based
models, it is recommended that the evaluation of
representative individual sampling sites might also
be required, in order to obtain a reasonable esti­
mate of fire regime effect. This is because the
probability of fire for a specific unit could be
influenced by whether it is adjacent to a burning
site, i.e. a function of its distance to fire ignition
locations (Li and Apps, 1996; McCarthy et aI.,
2001). This simplification of fire disturbances in
forest dynamics models of isolated units does not
take into account the adjacency fire effect. It also
does not consider the effect of hierarchical struc­
ture of forest ecosystems (Allen and Starr, 1982).
A representative of individual sampling sites
could be evaluated from a sampling design
The Alberta Vegetation Inventory data were
provided by the Weldwood of Canada Limited
(Hinton Division); weather data were from Mike
Flannigan of the Canadian Forest Service. The
author thanks Yonghe Wang for help on curve
fitting with non-linear procedure for generating
Fig. 4 and Harinder Hans for GIS assistance. The
manuscript was improved from the critical read­
ing and helpful comments from Ian Corns, Marty
Alexander, Mike Flannigan, Brenda Laishley, and
two anonymous reviewers on an earlier version of
this manuscript. The encouragement from Ian
C. Li / Ecological Modelling 154 (2002) 103-120
Corns is much appreciated. Funding for this re­
search was partially contributed from the Climate
Change Impact on Energy Sector (CCIES) pro­
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