Propagating Uncertainty in Plot-based Estimates of Forest Carbon Stock and Carbon Stock Change

Ecosystem science increasingly relies on highly derived metrics to synthesize across large datasets. However, full uncertainty associated with these metrics is seldom quantified. Our objective was to evaluate measurement error and model uncertainty in plot-based estimates of carbon stock and carbon change. We quantified the measurement error associated with live stems, deadwood and plot-level variables in temperate rainforest in New Zealand. We also quantified model uncertainty for height–diameter allometry, stem volume equations and wood-density estimates. We used Monte Carlo simulation to assess the net effects on carbon stock and carbon change estimated using data from 227 plots from throughout New Zealand. Plot-to-plot variation was the greatest source of uncertainty, amounting to 9.1% of mean aboveground carbon stock estimates (201.11 MgC ha^?1). Propagation of the measurement error and model uncertainty resulted in a 1% increase in uncertainty (0.1% of mean stock estimate). Carbon change estimates (mean ?0.86 MgC ha^?1 y^?1) were more uncertain, with sampling error equating to 56% of the mean, and when measurement error and model uncertainty were included this uncertainty increased by 35% (22.1% of the mean change estimate). For carbon change, the largest sources of measurement error were missed/double counted stems and fallen coarse woody debris. Overall, our findings show that national-scale plot-based estimates of carbon stock and carbon change in New Zealand are robust to measurement error and model uncertainty. We recommend that calculations of carbon stock and carbon change incorporate both these sources of uncertainty so that management implications and policy decisions can be assessed with the appropriate level of confidence.     

Incorporating uncertainty in spatial structure for viability predictions: a case study of California sea lions (Zalophus californianus californianus)

In recent years, population viability analysis has become a popular tool to assess the relative risk of extinction among populations. Viability estimates for spatially structured populations require movement data that are often unavailable. In this paper, a diffusion approximation model was used to explore the effects of different spatial scenarios resulting from assumptions about movement rates. Census data for 13 breeding islands occupied by California sea lions Zalophus californianus californianus in the Gulf of California were used to explore three potential scenarios: unlimited movement between sites (panmictic population), limited movement (several clusters of populations) and no movement between islands (isolated islands). Predicted viability estimates were different for each scenario, but contrary to expectations, the mean extinction risk estimates were generally lowest when movement was unlimited (panmictic scenario). However, despite an extensive dataset, the confidence of the viability predictions for each scenario was low. In some cases, uncertainty in predictions within a scenario was greater than differences between scenarios. Therefore, it is recommended that in situations where movement rates and spatial structure are unknown, extinction risk estimates should reflect both the confidence intervals for each risk estimate and the uncertainty resulting from different spatial scenarios. This study also provides the first estimate of population viability (considering spatial structure) for California sea lions in the Gulf of California and an evaluation of population status based on the IUCN criteria for species listing.     

Current Practices in Reporting Uncertainty in Ecosystem Ecology

Ecosystem budgets of water and elements can be difficult to estimate and are often unreplicated, making it challenging to provide confidence in estimates of ecosystem pools and fluxes. We conducted a survey to learn about current practices in reporting uncertainties in precipitation, streamflow, soils, and vegetation. Uncertainty derives from natural variation, which is commonly characterized by replicate samples, and from imperfect knowledge, which includes measurement error and model error (model fit and model selection). We asked questions about whether researchers report uncertainties in these sources, whether they know how to do so, and how important they believe the sources to be. We also asked questions about identifying missing or unusable values, filling gaps in data, and dealing with analytical concentrations below detection limits. We obtained responses from 140 researchers representing 90 research sites around the world. Natural variation was the most important source of uncertainty in calculations of biomass and soil pools, according to respondents in these fields, and sampling error was the source they most often reported. In contrast, uncertainty in the chemical analysis of precipitation and stream water was the source most commonly reported by hydrologists, although they rated this one of the least important sources of uncertainty to calculations of hydrologic flux. Awareness of types of uncertainty can help identify sources of uncertainty that may have been overlooked, and quantifying them will help determine which sources are most important to report.     

When does parameter drift decrease the uncertainty in extinction risk estimates?

Halley (2003) proposed that parameter drift decreases the uncertainty in long‐range extinction risk estimates, because drift mitigates the extreme sensitivity of estimated risk to estimated mean growth rate. However, parameter drift has a second, opposing effect: it increases the uncertainty in parameter estimates from a given data set. When both effects are taken into account, parameter drift can increase, sometimes substantially, the uncertainty in risk estimates. The net effect depends sensitively on the type of drift and on which model parameters must be estimated from observational data on the population at risk. In general, unless many parameters are estimated from independent data, parameter drift increases the uncertainty in extinction risk. These findings suggest that more mechanistic PVA models, using long‐term data on key environmental variables and experiments to quantify their demographic impacts, offer the best prospects for escaping the high data requirements when extinction risk is estimated from observational data.     

Accounting for uncertainty in dormant life stages in stochastic demographic models

Dormant life stages are often critical for population viability in stochastic environments, but accurate field data characterizing them are difficult to collect. Such limitations may translate into uncertainties in demographic parameters describing these stages, which then may propagate errors in the examination of population‐level responses to environmental variation. Expanding on current methods, we 1) apply data‐driven approaches to estimate parameter uncertainty in vital rates of dormant life stages and 2) test whether such estimates provide more robust inferences about population dynamics. We built integral projection models (IPMs) for a fire‐adapted, carnivorous plant species using a Bayesian framework to estimate uncertainty in parameters of three vital rates of dormant seeds – seed‐bank ingression, stasis and egression. We used stochastic population projections and elasticity analyses to quantify the relative sensitivity of the stochastic population growth rate (log λs) to changes in these vital rates at different fire return intervals. We then ran stochastic projections of log λs for 1000 posterior samples of the three seed‐bank vital rates and assessed how strongly their parameter uncertainty propagated into uncertainty in estimates of log λs and the probability of quasi‐extinction, P_q(t). Elasticity analyses indicated that changes in seed‐bank stasis and egression had large effects on log λs across fire return intervals. In turn, uncertainty in the estimates of these two vital rates explained > 50% of the variation in log λs estimates at several fire‐return intervals. Inferences about population viability became less certain as the time between fires widened, with estimates of P_q(t) potentially > 20% higher when considering parameter uncertainty. Our results suggest that, for species with dormant stages, where data is often limited, failing to account for parameter uncertainty in population models may result in incorrect interpretations of population viability.     

Likelihood based population viability analysis in the presence of observation error

Population viability analysis (PVA) entails calculation of extinction risk, as defined by various extinction metrics, for a study population. These calculations strongly depend on the form of the population growth model and inclusion of demographic and/or environmental stochasticity. Form of the model and its parameters are determined based on observed population time series data. A typical population time series, consisting of estimated population sizes, inevitably has some observation error and likely has missing observations. In this paper, we present a likelihood based PVA in the presence of observation error and missing data. We illustrate the importance of incorporation of observation error in PVA by reanalyzing the population time series of song sparrow Melospiza melodia on Mandarte Island, British Columbia, Canada from 1975–1998. Using Akaike information criterion we show that model with observation error fits the data better than the one without observation error. The extinction risks predicted by with and without observation error models are quite different. Further analysis of possible causes for observation error revealed that some component of the observation error might be due to unreported dispersal. A complete analysis of such data, thus, would require explicit spatial models and data on dispersal along with observation error. Our conclusions are, therefore, two‐fold: 1) observation errors in PVA matter and 2) integrating these errors in PVA is not always enough and can still lead to important biases in parameter estimates if other processes such as dispersal are ignored.     

Protocols for listing threatened species can forecast extinction

Risk‐ranking protocols are used widely to classify the conservation status of the world's species. Here we report on the first empirical assessment of their reliability by using a retrospective study of 18 pairs of bird and mammal species (one species extinct and the other extant) with eight different assessors. The performance of individual assessors varied substantially, but performance was improved by incorporating uncertainty in parameter estimates and consensus among the assessors. When this was done, the ranks from the protocols were consistent with the extinction outcome in 70–80% of pairs and there were mismatches in only 10–20% of cases. This performance was similar to the subjective judgements of the assessors after they had estimated the range and population parameters required by the protocols, and better than any single parameter. When used to inform subjective judgement, the protocols therefore offer a means of reducing unpredictable biases that may be associated with expert input and have the advantage of making the logic behind assessments explicit. We conclude that the protocols are useful for forecasting extinctions, although they are prone to some errors that have implications for conservation. Some level of error is to be expected, however, given the influence of chance on extinction. The performance of risk assessment protocols may be improved by providing training in the application of the protocols, incorporating uncertainty in parameter estimates and using consensus among multiple assessors, including some who are experts in the application of the protocols. Continued testing and refinement of the protocols may help to provide better absolute estimates of risk, particularly by re‐evaluating how the protocols accommodate missing data.     

Diversity Dynamics in Nymphalidae Butterflies: Effect of Phylogenetic Uncertainty on Diversification Rate Shift Estimates

The species rich butterfly family Nymphalidae has been used to study evolutionary interactions between plants and insects. Theories of insect-hostplant dynamics predict accelerated diversification due to key innovations. In evolutionary biology, analysis of maximum credibility trees in the software MEDUSA (modelling evolutionary diversity using stepwise AIC) is a popular method for estimation of shifts in diversification rates. We investigated whether phylogenetic uncertainty can produce different results by extending the method across a random sample of trees from the posterior distribution of a Bayesian run. Using the MultiMEDUSA approach, we found that phylogenetic uncertainty greatly affects diversification rate estimates. Different trees produced diversification rates ranging from high values to almost zero for the same clade, and both significant rate increase and decrease in some clades. Only four out of 18 significant shifts found on the maximum clade credibility tree were consistent across most of the sampled trees. Among these, we found accelerated diversification for Ithomiini butterflies. We used the binary speciation and extinction model (BiSSE) and found that a hostplant shift to Solanaceae is correlated with increased net diversification rates in Ithomiini, congruent with the diffuse cospeciation hypothesis. Our results show that taking phylogenetic uncertainty into account when estimating net diversification rate shifts is of great importance, as very different results can be obtained when using the maximum clade credibility tree and other trees from the posterior distribution.     

Primate diversification inferred from phylogenies and fossils

Biodiversity arises from the balance between speciation and extinction. Fossils record the origins and disappearance of organisms, and the branching patterns of molecular phylogenies allow estimation of speciation and extinction rates, but the patterns of diversification are frequently incongruent between these two data sources. I tested two hypotheses about the diversification of primates based on ～600 fossil species and 90% complete phylogenies of living species: (1) diversification rates increased through time; (2) a significant extinction event occurred in the Oligocene. Consistent with the first hypothesis, analyses of phylogenies supported increasing speciation rates and negligible extinction rates. In contrast, fossils showed that while speciation rates increased, speciation and extinction rates tended to be nearly equal, resulting in zero net diversification. Partially supporting the second hypothesis, the fossil data recorded a clear pattern of diversity decline in the Oligocene, although diversification rates were near zero. The phylogeny supported increased extinction ～34 Ma, but also elevated extinction ～10 Ma, coinciding with diversity declines in some fossil clades. The results demonstrated that estimates of speciation and extinction ignoring fossils are insufficient to infer diversification and information on extinct lineages should be incorporated into phylogenetic analyses.     

Estimating Uncertainty in Ecosystem Budget Calculations

Ecosystem nutrient budgets often report values for pools and fluxes without any indication of uncertainty, which makes it difficult to evaluate the significance of findings or make comparisons across systems. We present an example, implemented in Excel, of a Monte Carlo approach to estimating error in calculating the N content of vegetation at the Hubbard Brook Experimental Forest in New Hampshire. The total N content of trees was estimated at 847 kg ha⁻¹ with an uncertainty of 8%, expressed as the standard deviation divided by the mean (the coefficient of variation). The individual sources of uncertainty were as follows: uncertainty in allometric equations (5%), uncertainty in tissue N concentrations (3%), uncertainty due to plot variability (6%, based on a sample of 15 plots of 0.05 ha), and uncertainty due to tree diameter measurement error (0.02%). In addition to allowing estimation of uncertainty in budget estimates, this approach can be used to assess which measurements should be improved to reduce uncertainty in the calculated values. This exercise was possible because the uncertainty in the parameters and equations that we used was made available by previous researchers. It is important to provide the error statistics with regression results if they are to be used in later calculations; archiving the data makes resampling analyses possible for future researchers. When conducted using a Monte Carlo framework, the analysis of uncertainty in complex calculations does not have to be difficult and should be standard practice when constructing ecosystem budgets.     

Re-assessing current extinction rates

There is a widespread belief that we are experiencing a mass extinction event similar in severity to previous mass extinction events in the last 600 million years where up to 95% of species disappeared. This paper reviews evidence for current extinctions and different methods of assessing extinction rates including species–area relationships and loss of tropical forests, changing threat status of species, co-extinction rates and modelling the impact of climate change. For 30 years some have suggested that extinctions through tropical forest loss are occurring at a rate of up to 100 species a day and yet less than 1,200 extinctions have been recorded in the last 400 years. Reasons for low number of identified global extinctions are suggested here and include success in protecting many endangered species, poor monitoring of most of the rest of species and their level of threat, extinction debt where forests have been lost but species still survive, that regrowth forests may be important in retaining ‘old growth’ species, fewer co-extinctions of species than expected, and large differences in the vulnerability of different taxa to extinction threats. More recently, others have suggested similar rates of extinction to earlier estimates but with the key cause of extinction being climate change, and in particular rising temperatures, rather than deforestation alone. Here I suggest that climate change, rather than deforestation is likely to bring about such high levels of extinction since the impacts of climate change are local to global and that climate change is acting synergistically with a range of other threats to biodiversity including deforestation.     

Impact of taxon sampling on the estimation of rates of evolution at sites

The function of individual sites within a protein influences their rate of accepted point mutation. During the computation of phylogenetic likelihoods, rate heterogeneity can be modeled on a site-per-site basis with relative rates drawn from a discretized Gamma-distribution. Site-rate estimates (e.g., the rate of highest posterior probability given the data at a site) can then be used as a measure of evolutionary constraints imposed by function. However, if the sequence availability is limited, the estimation of rates is subject to sampling error. This article presents a simulation study that evaluates the robustness of evolutionary site-rate estimates for both small and phylogenetically unbalanced samples. The sampling error on rate estimates was first evaluated for alignments that included 5-45 sequences, sampled by jackknifing, from a master alignment containing 968 sequences. We observed that the potentially enhanced resolution among site rates due to the inclusion of a larger number of rate categories is negated by the difficulty in correctly estimating intermediate rates. This effect is marked for data sets with less than 30 sequences. Although the computation of likelihood theoretically accounts for phylogenetic distances through branch lengths, the introduction of a single long-branch outlier sequence had a significant negative effect on site-rate estimates. Finally, the presence of a shift in rates of evolution between related lineages can be diagnostic of a gain/loss of function within a protein family. Our analyses indicate that detecting these rate shifts is a harder problem than estimating rates. This is so, partially, because the difference in rates depends on two rate estimates, each with an intrinsic uncertainty. The performances of four methods to detect these site-rate shifts are evaluated and compared. Guidelines are suggested for preparing data sets minimally influenced by error introduced by sequence sampling.     

Verification of external exposure assessment for the upper Techa riverside by luminescence measurements and Monte Carlo photon transport modeling

An area located in the Southern Urals was contaminated in 1949–1956 as a result of radioactive waste releases into the Techa river by the Mayak Production Association. The external dose reconstruction of the Techa river dosimetry system (TRDS-2000) for the exposed population is based on an assessment of dose rates in air (DRA) obtained by modeling transport and deposition of radionuclides along the river for the time before 1952 and by gamma dose rate measurements since 1952. The aim of this paper is to contribute to a verification of the TRDS-2000 external dose assessment. Absorbed doses in bricks from a 130-year-old building in the heavily exposed Metlino settlement were measured by a luminescence technique. By the autumn of 1956 the population of Metlino had been evacuated, and then a water reservoir was created at the village location, which led to a change in the radioactive source geometry. Radiation transport calculations for assumed environmental sources before and since 1957 were performed with the MCNP Monte Carlo code. In combination with TRDS-2000 estimates for annual dose rates in air at the shore of the Techa river for the period 1949–1956 and contemporary dose rate in air measurements, absorbed doses in bricks were calculated. These calculations were performed deterministically with best estimates of the modeling parameters and stochastically by propagating uncertainty distributions through the calculation scheme. Assessed doses in bricks were found to be consistent with measured values within the uncertainty bounds, while their best estimates were approximately 15% lower than the luminescence measurements.An erratum to this article can be found at     

A study of statistical error in isothermal titration calorimetry

In isothermal titration calorimetry (ITC), the two main sources of random (statistical) error are associated with the extraction of the heat q from the measured temperature changes and with the delivery of metered volumes of titrant. The former leads to uncertainty that is approximately constant and the latter to uncertainty that is proportional to q. The role of these errors in the analysis of ITC data by nonlinear least squares is examined for the case of 1:1 binding, M+X right arrow over left arrow MX. The standard errors in the key parameters-the equilibrium constant Ko and the enthalpy DeltaHo-are assessed from the variance-covariance matrix computed for exactly fitting data. Monte Carlo calculations confirm that these "exact" estimates will normally suffice and show further that neglect of weights in the nonlinear fitting can result in significant loss of efficiency. The effects of the titrant volume error are strongly dependent on assumptions about the nature of this error: If it is random in the integral volume instead of the differential volume, correlated least-squares is required for proper analysis, and the parameter standard errors decrease with increasing number of titration steps rather than increase.     

Species coexistence in a variable world

The contribution of deterministic and stochastic processes to species coexistence is widely debated. With the introduction of powerful statistical techniques, we can now better characterise different sources of uncertainty when quantifying niche differentiation. The theoretical literature on the effect of stochasticity on coexistence, however, is often ignored by field ecologists because of its technical nature and difficulties in its application. In this review, we examine how different sources of variability in population dynamics contribute to coexistence. Unfortunately, few general rules emerge among the different models that have been studied to date. Nonetheless, we believe that a greater understanding is possible, based on the integration of coexistence and population extinction risk theories. There are two conditions for coexistence in the presence of environmental and demographic variability: (1) the average per capita growth rates of all coexisting species must be positive when at low densities, and (2) these growth rates must be strong enough to overcome negative random events potentially pushing densities to extinction. We propose that critical tests for species coexistence must account for niche differentiation arising from this variability and should be based explicitly on notions of stability and ecological drift.     

ESTIMATING DIVERSIFICATION RATES: HOW USEFUL ARE DIVERGENCE TIMES?

The dynamics of species diversification rates are a key component of macroevolutionary patterns. Although not absolutely necessary, the use of divergence times inferred from sequence data has led to development of more powerful methods for inferring diversification rates. However, it is unclear what impact uncertainty in age estimates have on diversification rate inferences. Here, we quantify these effects using both Bayesian and frequentist methodology. Through simulation, we demonstrate that adding sequence data results in more precise estimates of internal node ages, but a reasonable approximation of these node ages is often sufficient to approach the theoretical minimum variance in speciation rate estimates. We also find that even crude estimates of divergence times increase the power of tests of diversification rate differences between sister clades. Finally, because Bayesian and frequentist methods provided similar assessments of error, novel Bayesian approaches may provide a useful framework for tests of diversification rates in more complex contexts than are addressed here.     

Effects of uncertain cost-surface specification on landscape connectivity measures

Estimates of landscape connectivity are routinely used to inform decision-making by conservation biologists. Most estimates of connectivity rely on cost-surfaces: raster representations of landscapes in which cost values represent the difficulty involved with traversing an area. However, there is considerable uncertainty in the generation of cost-surfaces that have not been widely explored. We investigated the effects of four potential sources of uncertainty in the creation of cost-surfaces: 1) number of landscape classes represented; 2) spatial resolution (grain size); 3) misclassification of edges between landscape classes; and 4) cost values selected for each landscape class. Following a factorial design we simulated multiple cost-surface pairs, each comprising one true surface with no errors and one surface with uncertainty comprised of some combination of the four error sources. We evaluated the relative importance of each source of uncertainty in determining the difference between the least-cost paths (LCPs) costs and resistance distances generated for the true and erroneous cost-surfaces, using four model evaluation metrics. Errors in the underlying geospatial layers produced larger inaccuracies in connectivity estimates than those produced by cost-value errors. Incorrect grain size had the largest overall effect on the accuracy of connectivity estimates. Though the removal of an element class was found to have a large effect on the configuration of connectivity estimates, and the addition of an element class had a large effect on estimates configuration. Our results highlight the importance of minimising and quantifying the uncertainty inherent in the geospatial data used to develop cost-surfaces.     

Regionalization of methane emissions in the Amazon Basin with microwave remote sensing

Wetlands of the Amazon River basin are globally significant sources of atmospheric methane. Satellite remote sensing (passive and active microwave) of the temporally varying extent of inundation and vegetation was combined with field measurements to calculate regional rates of methane emission for Amazonian wetlands. Monthly inundation areas for the fringing floodplains of the mainstem Solimões/Amazon River were derived from analysis of the 37 GHz polarization difference observed by the Scanning Multichannel Microwave Radiometer from 1979 to 1987. L‐band synthetic aperture radar data (Japanese Earth Resources Satellite‐1) were used to determine inundation and wetland vegetation for the Amazon basin (<500 m elevation) at high (May–June 1996) and low water (October 1995). An extensive set of measurements of methane emission is available from the literature for the fringing floodplains of the central Amazon, segregated into open water, flooded forest and floating macrophyte habitats. Uncertainties in the regional emission rates were determined by Monte Carlo error analyses that combined error estimates for the measurements of emission and for calculations of inundation and habitat areas. The mainstem Solimões/Amazon floodplain (54–70°W) emitted methane at a mean annual rate of 1.3 Tg C yr^?1, with a standard deviation (SD) of the mean of 0.3 Tg C yr^?1; 67% of this range in uncertainty is owed to the range in rates of methane emission and 33% is owed to uncertainty in the areal estimates of inundation and vegetative cover. Methane emission from a 1.77 million square kilometers area in the central basin had a mean of 6.8 Tg C yr^?1 with a SD of 1.3 Tg C yr^?1. If extrapolated to the whole basin below the 500 m contour, approximately 22 Tg C yr^?1 is emitted; this mean flux has a greenhouse warming potential of about 0.5 Pg C as CO₂. Improvement of these regional estimates will require many more field measurements of methane emission, further examination of remotely sensed data for types of wetlands not represented in the central basin, and process‐based models of methane production and emission.     

Phylogenetic extinction rates and comparative methodology

Species are not independent points for comparative analyses because closely related species share more evolutionary history and are therefore more similar to each other than distantly related species. The extent to which independent-contrast analysis reduces type I and type II statistical error in comparison with cross-species analysis depends on the relative branch lengths in the phylogenetic tree: as deeper branches get relatively long, cross-species analyses have more statistical type I and type II error. Phylogenetic trees reconstructed from extant species, under the assumptions of a branching process with speciation (branching) and extinction rates remaining constant through time, will have relatively longer deep branches as the extinction rate increases relative to the speciation rate. We compare the statistical performance of cross-species and independent-contrast analyses with varying relative extinction rates, and conclude that cross-species comparisons have unacceptable statistical performance, particularly when extinction rates are relatively high.     

Observer variation in field assessments of vegetation condition: Implications for biodiversity conservation

Summary   Uncertainty in assessments of vegetation condition that are used to inform land management and planning decisions for biodiversity conservation in Australia may lead to unexpected outcomes, including loss of biodiversity. This study investigates observer error in field estimates of vegetation attributes, one component of uncertainty in assessments of vegetation condition. Ten observers conducted vegetation condition assessments using two assessment protocols (BioMetric and Habitat Hectares) on 20 sites in a grassy woodland community. Observers' estimates varied substantially across multiple scoring categories for all vegetation attributes on almost all sites. Across all sites, the average coefficient of variation in total vegetation condition scores was 15–18% for both protocols, with a maximum of 60%. The primary cause of variation in total vegetation condition scores was random error in raw estimates of vegetation attributes, although sensitivity of some highly weighted attributes to error exacerbated variation in some cases. Observers generally agreed on the total scores and ranks of highly degraded (pasture) sites, but were less consistent on other sites. Rank correlations between pairs of observers were stronger for Habitat Hectares, suggesting BioMetric may be slightly more sensitive to observer error. It is recommended that: (i) research is undertaken into methods for reducing observer error; (ii) review is made of the sensitivity of index scoring structures to observer error; (iii) field observers estimate uncertainty around point estimates of vegetation condition; and, (iv) decision-makers explicitly incorporate uncertainty into the decision-making processes and aim for outcomes that are robust to this uncertainty.