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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 editor-in-chief 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 scale-dependence 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 long-term 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~. Cause-and-effect 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 long-term data sets (Arbaugh and Bednar 1996~. These approaches are characterized by: · comparison of "quasi-treatment" to "quasi-control" 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 long-term 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 ten-year 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 30-year 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 30-year 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 30-year cycle; even annual or biannual sampling would be too frequent. These considerations pose three problems: identifying reasonable long-term 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 long-term 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, fine-increment 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~. Tree-ring records are another source of data on the historical behav- ior of multiple, often interacting, ecosystem properties. For example, tree rings provide multi-century to multi-millennial histories of seasonally resolved climate as well as independent records of ecological responses to climate such as changes in fire regime and forest-stand 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 A-1. 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 individual-based models of forest ecosystems, known collectively as lABOWA-FORET 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 A-1. 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 A-1), 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 A-1) whose highest frequency occurs at B cycles per unit time (i.e., from Figure A-2) 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 A-2 Spectral analysis of the time series of productivity in Fig. A-1. Note the peak frequency at approximately 0.043 cycles per year (1 cycle per year = 1 Hz; x-axis in Hz/100), which corresponds to a period of 23 years. One half this period, or 11-12 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 11-12 years, is the Nyquist sampling frequency. Therefore, to characterize trends in forest productivity in a monitoring plot, one need sample no more often than 11-12 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.
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