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Appendixes
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150
Ecological Indicators for the Nation
Michael L. Rosenzweig is professor and former head of ecology and evolutionary
biology at the University of Arizona. He is editorinchief of Evol2~tior~ary Ecology. He received
an A.B. (1962) and a Ph.D. (1966) in zoology from the University of Pennsylvania. Dr.
Rosenzweig uses mathematical modeling to study species diversity, habitat selection, and
population interactions. With the late Robert MacArthur, he help to found modern dynamical
predation theory. His work on desert rodent ecology in the United States and Israel established
these systems as valuable models for the investigation of general ecological questions. His isoleg
theories of habitat selection and their tests in birds and mammals have been among the first to link
the study of behavior to population dynamics and community ecology. Dr. Rosenzweig's recent
text, Species Diversity in Space and Time (Cambridge University Press, 1 995) synthesizes the
patterns and processes operating on diversity at scales of up to the entire planet and all of
Phanerozoic time.
Milton Russell is a senior fellow at the Joint Institute for Energy and Environment,
professor emeritus of economics at the University of Tennessee, and collaborating scientist, Oak
Ridge National Laboratory. He was a member of the National Research Council Board on
Environmental Studies and Toxicology, chairs the Committee to Assess the North American
Research Strategy for Tropospheric Ozone (NARSTO) Program and is a member of the National
Academy of Sciences/National Academy of Engineering Joint Committee on Cooperation in the
Energy Futures of China and the United States. Dr. Russell served as Assistant Administrator for
Policy at the U.S. Environmental Protection Agency from 1983 to 1987. His current research
activities are concentrated on environmental policy in China and on waste management, especially
cleanup of U.S. Department of Energy sites. He received his Ph.D. in economics from the
University of Oklahoma in 1963.
Susan Stafford is Professor and Department Head of Applied Statistics and Research
Information Management, Colorado State University. Her current interests are research
information management, applied statistics, multivar~ate analysis and experimental design,
scientific databases, GIS applications, and other data management topics. Dr. Stafford received
her Ph.D. in applied statistics in 1979 Dom the State University of New York, College of
Environmental Science and Forestry.
Project Director
David Policansly has a B.A. in biology from Stanford University and an M.S. and Ph.D.,
biology, from the University of Oregon. He is associate director of the Board on Environmental
Studies and Toxicology at the National Research Council. His interests include genetics,
evolution, and ecology, particularly the effects of fishing on fish populations, ecological risk
assessment, and natural resource management. He has directed approximately 25 projects at the
National Research Council on natural resources and ecological risk assessment.
PREPUBLICATION COPY
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Appendix A
Variability, Complexity, and
the Design of Sampling Procedures
No general laws exist that allow us to predict the relative magni
tude of temporal and spatial variability of different types of
parameters across the diversity of ecological systems. Variability
is highly dependent on the temporal and spatial scales of the data sets and
on the level of aggregation of the parameter of interest (e.g., species level
versus community level [Allen and Starr 1982, Frost et al. 1988, Wood et
al. 1990~. This scaledependence raises a complication in comparative
studies because it is possible to confound differences in patterns of vari
ability with differences in scales of measurement made at two or more
systems.
To compute a nationwide indicator, a variety of different data must
be aggregated, and the choice of aggregation level may be one of the most
important decisions in the design of an indicator and of the monitoring
program that Generates the necessary input data.
For biological data,
levels of aggregation (e.g., species, guilds, major groups) has greater effect
on observed variability than does spatial or temporal extent of the data
(Kratz 1995~.
Parameters useful in monitoring programs designed to detect trends
and patterns often have two, potentially conflicting characteristics. On
the one hand, they must be sufficiently sensitive to environmental condi
tions to indicate changes that occur. On the other hand they must not
exhibit so much natural variability as to mask detection of significant
changes in environmental conditions. Thus, understanding the relative
151
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52
APPENDIX A
variability and sensitivity of parameters is important in choosing optimal
parameters for a monitoring program.
In addition, use of data from longterm monitoring programs is often
made difficult because such monitoring programs are typically not designed
as experiments. Monitoring is usually conducted to assess the effects of
certain agents, procedures, or programs that cannot be subjected to con
trolled experimentation (Cochran 1983~. Causeandeffect relationships
usually cannot be inferred from such data because confounding variables
(identified or not) that were not controlled during the study may be
responsible for the observed differences (Ramsey and Schafer 1997~.
Nonetheless, there are approaches that can reduce the likelihood of draw
ing incorrect conclusions from longterm data sets (Arbaugh and Bednar
1996~. These approaches are characterized by:
· comparison of "quasitreatment" to "quasicontrol" groups to
resemble a designed experiment;
· comparison of "treatment" groups with more than one "control"
group to develop different contrasts with the "treatment" group;
· comparison of "treatment" and "control" groups with important
exogenous variables; if this is unfeasible, groups should be adjusted for
differences using covariates in the analysis; and
· use of a variety of measures/components to reduce the depen
dence of study results on a single aspect of the data and on assumptions
inherent for single methods of analysis (Strauss 1990~.
It is impossible to optimize both experimental and monitoring goals
simultaneously, but longterm monitoring data can be used as key inputs
in the development of estimates of the states of the systems being moni
tored, together with confidence levels associated with those estimates.
However, careful consideration must be given to statistical treatment of
correlated observations when variance estimates and confidence intervals
are calculated (Conquest 1993~.
TEMPORAL BEHAVIOR OF ECOSYSTEMS AND
SAMPLING DESIGN
The natural world oscillates. Such oscillations are well known to
ecologists and include many classic population cycles such as the hare
lynx and lemming cycles (Elton 1942, Keith 1963~. The frequency and
amplitude of such oscillations, which are characteristic properties of any
complex system, pose problems in sampling to detect trends in a particular
property. For example, suppose that an annual sampling strategy over
ten years has revealed a steady decline (or increase) in productivity. Such
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APPENDIX A
153
a tenyear record is relatively long for most monitoring programs. Can
we say with confidence that the system is deteriorating (or improving) as
a result of worse (or better) growing conditions? We cannot unless we
can separate such a trend from normal oscillations: we may simply have
caught the system in the descending (or ascending) portion of a 30year
oscillation whose mean is stationary.
To detect trends, we must sample at a frequency from which we can
characterize the oscillatory character of systems. However, to determine
the natural frequencies from real data, we must sample for a long time.
For example, to determine whether a system has a 30year oscillation, we
must sample for 60 years to obtain data on at least two repeated oscilla
tions. In many cases, by the time we have a sufficiently long time series,
the monitoring strategy will simply confirm what may already be obvi
ous. Conversely, we may have spent much time and effort (and money)
sampling the system too frequently to detect a relatively long cycle. For
example, it would make no sense whatsoever to sample at daily frequen
cies to detect trends embedded within a 30year cycle; even annual or
biannual sampling would be too frequent.
These considerations pose three problems: identifying reasonable
longterm surrogates for current ecosystems to serve as first approxima
tions to characterize oscillatory behavior, determining the optimal sam
pling frequency of an ecosystem property from these surrogates, and
detecting changes in the behavior of an oscillating system.
Simulation models and longterm paleorecords can be used as first
approximation surrogates of the real system to obtain a long time series
about attributes of interest (species composition, biomass, productivity,
soil nitrogen, etc.) under changing conditions for which we desire to make
policy (climate warming, etc.~.
The advantage of paleorecords is that they provide real data on the
historical behavior of particular ecosystem properties at a given location.
Traditionally, pollen analyses have been the main tool of paleoecologists,
yielding data on changes in species composition surrounding a catch
ment basin. However, recent analyses include correlating changes in
vegetation with charcoal abundance (Clark 1988a and b, Clark et al.1989)
and biogeochemical analyses of sediment (MacDonald et al. 1993~. A
particularly useful paleorecord can be obtained from cores taken from
varved lakes. A varved lake sediment is one in which annual cycles of
deposition are clearly visible in the sediment column, enabling paleo
ecologists to reconstruct an annual time series of changes in species com
position and associated biogeochemical properties and fire regimes (e.g.,
Clark 1988a and b). In the absence of varved sediments, fineincrement
sampling and description of layers also enables a paleoecologist to recon
struct a detailed time series, particularly when layers are radiocarbon
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54
APPENDIX A
dated to anchor the chronology (MacDonald et al.1993, Cooper and Brush
1991~. Treering records are another source of data on the historical behav
ior of multiple, often interacting, ecosystem properties. For example, tree
rings provide multicentury to multimillennial histories of seasonally
resolved climate as well as independent records of ecological responses to
climate such as changes in fire regime and foreststand structure and
composition (Swetman and Betancourt 1998, Lloyd and Graumlich 1997~.
Limits of the paleorecord are that only certain types of data are avail
able, and the causes of temporal patterns must be inferred from the record
itself or from ancillary data. Simulation models as surrogates of real
systems may allow the researcher to overcome these problems, but the
simulation models themselves must be extensively tested first. Even then,
the model behavior is often extended into domains beyond that for which
the model was originally intended or parameterized. Nonetheless, simu
lation models are useful tools in many cases the best available tool
that the researcher can use to generate a long time series of a particular
ecosystem property and examine its expected behavior under different
stressors, such as climate change (Shugart 1984; Solomon 1986; Pastor and
Post 1988, 1993; Agren et al. 1991; Cohen and Pastor 1991; Post and Pastor
1996; Prentice et al. 1993~. Such simulation models can help test hypoth
eses regarding causes of changes seen in the paleorecord (Solomon and
Shugart 1984~.
To determine the optimal sampling frequency and to examine how
the system responds to rapid changes, the surrogate time series of data
can then be analyzed using techniques of signal processing (Shannon and
Weaver 1964, Simpson and Houts 1971, Wickerhauser 1994~. Fourier
analysis and wavelet analysis are the two main techniques of interest.
A system may oscillate with many different frequencies, each with
different amplitudes. Fourier analysis sometimes called spectral analy
sis decomposes this complex behavior into a sum of sine and cosine
waves of specific frequencies and amplitudes (Shannon and Weaver 1964,
Simpson and Houts 1971, Wickerhauser 1994~. A particularly good review
of the application of this techniques to ecological problems, including the
paleorecord and output from simulation models, is given by Platt and
Denman (1975~. One objective of a Fourier decomposition is to identify
the frequencies with the greatest amplitude because these are the fre
quencies that most strongly govern the system behavior. To accomplish
this, frequencies of the sine and cosine waves are plotted against their
amplitude. These plots are often called spectral plots.
Once a spectral plot is constructed, a powerful sampling theorem
the Nyquist theorem can be used to specify the optimal sampling
frequency. This theorem states that the time between samples of an oscil
latory signal need be no greater than half the frequency of the shortest
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APPENDIX A
155
cycle. This frequency is known as the Nyquist frequency. Sampling more
frequently than the Nyquist frequency does not yield any additional infor
mation and simply adds to the cost of the sampling program. By first
determining the Nyquist sampling frequencies for different properties in
the surrogate data sets, the researcher can determine the optimal time that
current sites need to be resampled to characterize their behavior.
Although Fourier analysis determines the relative contributions of
different oscillatory frequencies to system behavior, it is less useful for
determining when sudden changes in frequencies occur. These sudden
changes may be early warning signs of changes in the status of a system.
For example, both simulation models (Solomon 1986, Pastor and Post
1988) and the paleorecord (MacDonald et al. 1993) have often shown that
productivity and species composition of forests change rapidly in response
to gradual changes in climate once threshold temperatures for particular
species are exceeded.
To detect these sudden changes, the related and newer tool of wavelet
analysis is useful (Wickerhauser 1994, Hubbard 1996~. Instead of deter
mining the contribution of each frequency to the entire time series signal
as Fourier analysis does, wavelet analysis determines when there are sud
den changes in frequency, times of rapid change in system behavior. This
would then alert the researcher to investigate further whether this change
is a normal aspect of system functioning or whether it is imposed by
changes in a stressor. Applying wavelet analysis to surrogate data sets
with known behavior to changes in a stressor would help us recognize the
symptoms of suc~cren change In real word systems. Because of the
recent development of this mathematical technique, the application of
wavelet analysis to ecological problems (e.g., Bradshaw and McIntosh
1993, Bradshaw and Spies 1992) would benefit from a focused research
program. If the wavelet behavior of paleorecords or simulation models
can be characterized with respect to stressors such as global warming,
then we would have a set of likely "symptoms" to alert policy makers to
impending problems.
v
,, .. ~ . . . .
AN EXAMPLE OF FOURIER ANALYSIS OF SIMULATION MODEL
OUTPUT TO DETERMINE OPTIMAL SAMPLING FREQUENCIES
We present here an example of using a simulation model to generate
a long time series of output as a first approximation of the behavior of a
forested ecosystem. The output is then analyzed using Fourier analysis to
determine the optimal frequency to sample an oscillating system under
current climates.
Many ecosystems are now well described by extensively tested simu
lation models that incorporate fundamental biological processes respon
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56
1 4 
u'
O 1 0
11111~1
6
a
._

v _
3
° 2—
APPENDIX A
o
1 0
Year
1 5 2 0 x 1 0
FIGURE A1. Time series of model output of productivity of a forest in north
eastern Minnesota on a silty clay loam soil under current climate. Model runs
were made with LINKAGES. Adapted from Pastor and Cohen 1997.
sible for system oscillations. Among the best known are individualbased
models of forest ecosystems, known collectively as lABOWAFORET
models (Shugart 1984~. Emanuel et al. (1978) provide an excellent ex
ample of how to apply Fourier analysis to the output from such models to
determine optimal sampling rates in the field. We use this approach here
to analyze cycles of productivity from a forest ecosystem model under
current climate for northeastern Minnesota (Pastor and Cohen 1997), from
which we calculate the optimal sampling rate to characterize these cycles.
We first generate a long time series of productivity under current
climate for northern Minnesota for a single plot of 0.01 ha on a silty clay
loam soil with high water holding capacity (Figure A1. Such a plot
represents a standard forest inventory plot that would be resampled in a
monitoring program such as the USES Continuous Forest Inventory and
Analysis (CFIA) System, which supplies the raw data for timber policy in
this country.
We then use Fourier analysis to decompose this output stream or
signal into a sum of sine and cosine functions each of characteristic fre
quency and amplitude according to:
2n ~
00
~ A(6~)e(i2~)dt
—00
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APPENDIX A
157
where At) is the data or signal at time t (Figure A1), A(~) is a coefficient
(often called a Fourier coefficient) that specifies the power or amplitude of
a cycle of frequency of, and i = I. The coefficients and frequencies are
determined such that each exponential function and its associated coeffi
cient is orthogonal to the others. The oscillatory nature of the above
equation can be clarified by recalling that
ei2~= = cost + i sink.
We essentially thereby obtain a series of cosine functions each of char
acteristic frequency and amplitude that describe the time series of pro
ductivity under northeastern Minnesota climate. The next step is to plot
the amplitude against the frequency of the data to obtain Graphs known
as spectral density plots (Figure Am.
to 1 '
We are now ready to apply the Nyquist sampling theorem to this
spectral density plot. This theorem (Nyquist 1928) states that a signal (i.e.,
Figure A1) whose highest frequency occurs at B cycles per unit time (i.e.,
from Figure A2) is uniquely determined by sampling at a uniform rate of
0.25—
0.20—
`,, 0.1 5—
o
Q 0.10
0.05—
o.oo

\
l
l
0 2 4 6 8 1 OmHz
Frequency (cycleslyear)
FIGURE A2 Spectral analysis of the time series of productivity in Fig. A1. Note
the peak frequency at approximately 0.043 cycles per year (1 cycle per year = 1
Hz; xaxis in Hz/100), which corresponds to a period of 23 years. One half this
period, or 1112 years, is the optimal sampling frequency according to the Nyquist
sampling theorem. Adapted from Pastor and Cohen 1997.
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158
APPENDIX A
at least 2B samples per unit time. In other words, the time between
samples need be no greater than half the frequency of the shortest cycle,
known as the Nyquist frequency.
For the above plot, the highest frequency occurs at approximately
0.043 cycles per year. The inverse of the frequency is the period, or
23 years. This period is a result of the dynamics of canopy death and
replacement. One half of this, or 1112 years, is the Nyquist sampling
frequency. Therefore, to characterize trends in forest productivity in a
monitoring plot, one need sample no more often than 1112 years. Inter
estingly, the CFIA plots are sampled on average once every 10 years, and
Emanuel et al. (1978) found a Nyquist sampling frequency of 10 years for
an Appalachian hardwood forest using an earlier version of this model.
Therefore, the current sampling frequencies of the CFIA and Forest Health
Monitoring Programs appear to sample forests sufficiently often to char
acterize productivity, at least in northern Minnesota on clay soils.
The above model run was for a silty clay loam soil. Similar runs could
be made for other soil types, such as sands. One might find that other
soils require different sampling frequencies to characterize productivity.
One might also apply the technique to other model output of interest (soil
nitrogen, biomass, foliage height diversity, species diversity, etc.) to deter
mine optimal sampling rates for other properties. Similar approaches
could be used to determine changes in sampling frequency under differ
ent climates, with increasing or decreasing acidity of precipitation, with
different fertilizer application levels or other parameters of interest.