Chapter 6 Understanding Uncertainty in Environmental Modeling

A model is never right—it can only be useful, trustworthy enough, and transparent about its uncertainty. This chapter introduces the vocabulary and habits of mind needed to evaluate model quality, acknowledge uncertainty, and justify the assumptions behind a model.

None of your projects are complete they never will be because a model is never complete.

Uncertainty is not a flaw in a model; it is a fundamental part of describing complex natural systems. Every measurement, parameter, and equation carries uncertainty. The goal of this chapter is to understand where that uncertainty comes from, how to represent it, and how to quantify its effects on predictions.

Environmental systems are inherently variable and only partially observable. Because of this, uncertainty cannot be eliminated. Instead, it must be characterized, communicated, and incorporated into model evaluation and decision-making.

6.1 Why Uncertainty Matters in Modeling

Environmental models are simplifications of real systems such as forests, rivers, nutrient cycles, and climate dynamics. These systems are shaped by interacting processes, irregular disturbances, and feedback loops that cannot be fully captured by any single mathematical description. As a result, every model leaves something out—and those omissions create uncertainty.

Uncertainty matters because a model’s purpose is not to reproduce reality perfectly, but to provide useful guidance about how a system behaves under different assumptions or scenarios. A model that hides or ignores uncertainty can give a false sense of precision, leading to overconfident decisions and misinterpretation of results. A model that is transparent about uncertainty, in contrast, equips students and scientists to reason carefully, compare alternatives, and make informed judgments.

Uncertainty arises from multiple sources: imperfect measurements, unknown parameters, natural variability in the environment, and structural assumptions in the model itself. Understanding these sources allows us to distinguish between uncertainty we can reduce (through better data or calibration) and uncertainty we must simply acknowledge (due to randomness or incomplete knowledge).

Incorporating uncertainty enables modelers to:

  • evaluate whether predictions are credible and understand their limits
  • compare competing models on a fair and transparent basis
  • identify which assumptions or parameters have the greatest influence on outcomes
  • build models that behave realistically under extreme or unexpected conditions
  • communicate uncertainty clearly to stakeholders, policymakers, and scientists

Ultimately, incorporating uncertainty is what allows a model to be trustworthy enough for its intended purpose—even though it will never be “right” in an absolute sense.

6.1.1 The Art of Modeling

The art of modeling is not defined by coding or by the ability to implement equations in software. The real skill lies in critically evaluating what is included in a model and, equally importantly, what is left out. Every exclusion, simplification, and assumption creates uncertainty. Understanding these choices—why they were made, how they shape model behavior, and where they limit model applicability—is the foundation of building useful, credible, and transparent models. This is where uncertainty originates, and where thoughtful modelers must focus their attention.

6.2 Major Sources of Uncertainty

Uncertainty enters a model from multiple places. Even when the coding is perfect, the representation of the system is never complete. Each layer of uncertainty reflects a different kind of judgment about what matters, what can be measured, and what can be ignored. The most common categories are measurement uncertainty, process uncertainty, parameter uncertainty, and structural uncertainty. Each affects the credibility and usefulness of a model in different ways.

6.2.1 Measurement Uncertainty

All environmental observations contain error, and these errors propagate through any model that uses the data. Even the most carefully designed monitoring programs face unavoidable limitations in how precisely they can measure the real world.

Examples include:

  • Rain gauges under-catch snowfall because wind turbulence deflects snow away from the opening. A hydrologic model calibrated with these data will underestimate snow-water equivalent and runoff.
  • CO₂ analyzers drift over time, producing slow biases in atmospheric carbon datasets. Without frequent calibration, long-term trends may be misestimated.
  • Satellite NDVI varies with viewing angle and atmospheric conditions, introducing noise that can resemble ecological change even when vegetation is stable.

Measurement uncertainty shapes the quality of:

  • initial conditions (e.g., the true forest biomass at model start)
  • calibration datasets (e.g., streamflow used to estimate parameters)
  • validation (e.g., comparing predictions to noisy observations)

A common representation is:

\[ y_{\text{observed}} = y_{\text{true}} + \epsilon, \qquad \epsilon \sim \text{Normal}(0,\sigma^2) \]

A simple example is a temperature sensor with ±0.5°C accuracy. Even if the model is perfect, poor measurements will introduce variability in predictions.

6.2.2 Process Uncertainty

Process uncertainty reflects the inherent randomness in environmental systems—variability that exists even if we knew the true equations and parameters. Many environmental processes cannot be perfectly predicted because they depend on countless hidden drivers.

Examples:

  • Daily rainfall fluctuates due to atmospheric instability, making storm timing and intensity uncertain even with good forecasts.
  • Wildfire ignition depends on chance events: a spark, a lightning strike, an ignition source. Models can estimate risk, but not exact ignition times.
  • Biological growth varies with microclimate, shading, genetics, and competition. Even genetically identical plants grown a few meters apart may follow different growth paths.

A simple mathematical representation is:

\[ x_{t+1} = f(x_t) + \eta_t, \qquad \eta_t \sim \text{Normal}(0, \sigma_\eta^2) \]

For example, logistic population growth with environmental noise might produce year-to-year fluctuations even when the average growth rate is stable.

Process uncertainty reminds us that models predict distributions, not single futures.

6.2.3 Parameter Uncertainty

Parameters are rarely known with certainty. They must be estimated from data, fit from past observations, borrowed from literature, or inferred indirectly. Limited data, unmeasured drivers, and changing environmental conditions all contribute to parameter uncertainty.

Examples:

  • Growth rate \(r\) in the logistic model varies with climate, food availability, density dependence, and species interactions. Using a single value ignores this variability.
  • Carrying capacity \(K\) changes seasonally, after disturbance, or with land-use change.
  • Reaction rates in biogeochemical models depend on temperature and microbial communities, both of which fluctuate unpredictably.
  • Streamflow recession constants depend on soil properties, geology, and moisture—variables that cannot be measured everywhere.

Because of this, parameters are often represented as distributions:

\[ r \sim \text{Normal}(0.35, 0.05^2) \]

This allows simulations to explore a range of plausible system behaviors. For instance, a population forecast may show wider uncertainty bands if \(r\) is poorly known, but narrower bands when it is well constrained by data.

Parameter uncertainty highlights how small uncertainties in assumptions can cascade into large uncertainties in predictions.

6.2.4 Structural Uncertainty

Structural uncertainty arises from the model’s architecture: the equations, compartments, simplifications, and assumptions used to represent the system. It is the uncertainty we introduce when we decide how to model the world.

Examples:

  • Logistic vs. Gompertz growth: both represent density dependence, but they differ in curvature and long-term behavior. Choosing one shape over the other embeds a structural assumption about biological processes.
  • Linear vs. nonlinear runoff relationships: a linear rainfall–runoff model cannot capture threshold behavior such as infiltration excess, leading to systematic bias in flood prediction.
  • Compartment models of carbon cycling often reduce complex soil and vegetation processes into two or three pools. This simplification creates uncertainty because real systems have many interacting fluxes.

Structural uncertainty is difficult to quantify because it sits in the form of the model rather than its parameters. It is the uncertainty that comes from what the model leaves out:

  • missing feedback loops
  • unrepresented processes
  • assumed relationships
  • simplified spatial structure

Structural choices define what the model is capable of capturing. A wildfire model without wind cannot reproduce realistic fire spread. A population model without age structure cannot capture demographic effects.

Among all uncertainty types, structural uncertainty has the greatest potential to mislead if ignored, because it can cause systematic errors rather than random variation.

6.3 Additional Contributors to Uncertainty

The major categories of uncertainty—measurement, process, parameter, and structural—form the core of how uncertainty enters environmental models. But real-world modeling also involves several additional contributors that can meaningfully influence predictions. These forms of uncertainty arise not from the system itself, but from the way we run simulations, prepare inputs, and initialize the model. Recognizing these additional sources allows modelers to diagnose unexpected behavior, reconcile discrepancies across models, and build a more complete picture of uncertainty.

6.3.1 Initial Condition Uncertainty

Initial conditions determine how the model begins its simulation, but these values are almost always uncertain. Even small errors in the starting state can propagate, especially in nonlinear or chaotic systems.

A common statistical representation is:

\[ x_0 \sim \text{Normal}(x^{\text{obs}}, \sigma_0^2) \]

Examples:

  • Forest biomass models often begin with an estimated starting carbon stock that is derived from plot sampling. Two stands of similar age can have very different biomass, and a 10–20% error in starting biomass may produce significantly different long-term projections.
  • Hydrologic models require initial soil moisture, snowpack, or groundwater storage. These values are notoriously hard to measure and vary dramatically across a watershed. A small error in initial soil saturation can lead to large differences in simulated storm runoff.
  • Ocean circulation models depend on initial temperature and salinity profiles. Slight mismatches between observed and modeled initial ocean states can grow over time and lead to different long-term circulation patterns.
  • Population models starting with imperfect census data (e.g., fish surveys, wildlife counts) may produce divergent trajectories even when parameters are identical.

Initial condition uncertainty is especially important in systems with sensitive dependence on starting conditions, where tiny differences can lead to diverging outcomes.

6.3.2 Forcing Uncertainty

Forcing uncertainty comes from uncertainty in the inputs that drive the model—future rainfall, temperature, nutrient loading, emissions scenarios, management actions, or other external pressures. These uncertainties reflect the fact that we often do not know what the environment will do next.

Common examples include:

  • Rainfall forecasts that vary across weather models, producing different flood or drought predictions when used as inputs to hydrologic models.
  • Future temperature projections that differ depending on greenhouse gas emissions trajectories (e.g., SSP1 vs. SSP5 scenarios). A heatwave model may respond very differently under different climate futures.
  • Nutrient loading into lakes and estuaries, where agricultural practices, fertilizer application rates, and land-use changes are uncertain. Water-quality models depend heavily on these inputs.
  • River flow regulation, where future water releases depend on policy decisions, demand, and hydropower needs. Ecological flow models must therefore consider a range of possible operating regimes.
  • Sea-level rise projections, which combine uncertainty in ice-sheet dynamics, global climate response, and regional land subsidence.

Forcing uncertainty is especially prominent in long-term projections, where the choice of input scenario often matters more than the structure of the model itself.

6.3.3 Numerical / Computational Uncertainty

Even if a model is conceptually perfect and all inputs are known, numerical computation introduces additional uncertainty. These forms of uncertainty emerge because we use computers to approximate continuous processes.

Examples include:

  • Floating-point precision errors:
    Computers store numbers with finite precision. Very small rounding errors can accumulate over thousands of time steps, particularly in iterative population or climate models.
  • Solver tolerances in differential equation models:
    Many models use ODE or PDE solvers that approximate solutions using step sizes. Changing the step size can alter the predicted temperature of a lake or the trajectory of a predator–prey model.
  • Discretization error in spatial models:
    Landscape fire models or pollutant dispersion models break space into grids. Predictions can change if the grid is finer or coarser.
  • Monte Carlo sampling variability:
    When running stochastic simulations, each Monte Carlo run samples randomness differently. Small sample sizes can produce misleadingly narrow or wide uncertainty estimates.
  • Convergence issues in optimization or calibration:
    If a calibration routine does not converge fully, parameter estimates may differ between runs, even with the same data.

These numerical uncertainties are usually small compared to other uncertainty sources, but they can matter in complex models, especially those with strong nonlinearity or chaotic behavior.

Understanding initial condition, forcing, and computational uncertainties broadens the modeler’s perspective beyond equations and parameters. It helps identify why two models that appear identical may behave differently, and why precise-looking outputs often mask deeper instability or variability.

6.4 Representing Uncertainty in Models

Once uncertainty has been identified, the next step is to represent it explicitly in the model. This requires both conceptual and mathematical tools that allow uncertainty to be expressed, manipulated, and propagated through a simulation. The goal is not to eliminate uncertainty, but to encode it honestly so that predictions reflect a range of plausible outcomes rather than a single deterministic trajectory.

Below are four common ways uncertainty is represented in environmental modeling: stochastic formulations, parameter distributions, scenarios, and Bayesian approaches.

6.4.1 Stochastic Models

Stochastic models incorporate randomness directly into their structure. They assume that part of the system’s behavior is inherently unpredictable due to environmental variability, demographic noise, or unmeasured processes. Instead of producing one trajectory, these models produce a distribution of possible trajectories.

A common form is:

\[ x_{t+1} = f(x_t) + \eta_t, \qquad \eta_t \sim \text{Normal}(0, \sigma^2) \]

Examples:

  • Stochastic logistic growth:
    Population growth varies with environmental conditions. Adding noise to the logistic model produces fluctuating population sizes even under stable average conditions.
  • Rainfall-runoff models that include noise in infiltration or evapotranspiration rates to account for small-scale heterogeneity not captured by the model.
  • Wildfire spread models where random ignition probabilities or stochastic wind gusts influence daily burn area.
  • Fishery models that incorporate random recruitment (the number of new fish entering the population each year), reflecting uncertainty in spawning success.

Stochastic models are essential whenever variability is an inherent feature of the system rather than simply a measurement artifact. They help modelers visualize uncertainty as a natural outcome of system dynamics.

6.4.2 Parameter Distributions

Instead of assigning a single value to a parameter, we treat it as a random variable drawn from a distribution. This reflects the idea that parameters are not truly known—they are estimated with error, vary in space or time, or differ across individuals or populations.

An example representation:

\[ K \sim \text{Uniform}(450, 550) \]

Examples:

  • Hydrologic models may treat soil hydraulic conductivity as a distribution because soils vary spatially and are never measured everywhere.
  • Carbon cycling models may treat decomposition rates as uncertain because they depend on microbial communities, temperature, and moisture—variables that fluctuate unpredictably.
  • Predator–prey models may draw attack rates or mortality rates from distributions to reflect uncertainty in ecological interactions.
  • Climate energy-balance models often represent climate sensitivity (°C warming per doubling of CO₂) as a distribution, not a single number, because it is inferred from complex and uncertain historical data.

Parameter distributions are typically used in Monte Carlo simulations, where thousands of model runs are generated by sampling parameters repeatedly. The spread of outcomes reveals how parameter uncertainty affects forecasts.

6.4.3 Scenario Approaches

Scenario approaches are used when uncertainty is driven by external forces for which probability distributions are unknown or unknowable. These are not “random variables” in the usual sense; instead, they reflect different plausible futures, often shaped by policy choices, socio-economic shifts, or climate pathways.

Examples:

  • Climate emissions scenarios (e.g., SSP1, SSP3, SSP5) for projecting future temperature, precipitation, and sea-level rise.
  • Land-use change scenarios, such as increased urbanization vs. expanded conservation areas, which influence water quality or habitat availability.
  • Forest management scenarios, such as frequent thinning vs. fire suppression, used to assess long-term carbon storage or fire risk.
  • Agricultural nutrient-loading scenarios exploring high fertilizer use, low fertilizer use, or best-management practices to evaluate potential impacts on lake eutrophication.

Scenario modeling is especially valuable when uncertainty is tied to human decisions or large-scale processes that cannot be represented probabilistically. It emphasizes transparency and clearly communicates the assumptions behind each model run.

6.4.4 Bayesian Concepts

Bayesian approaches represent uncertainty in a mathematically rigorous way by combining prior knowledge with observational data. Parameters are treated as distributions (priors), and data update those distributions (posteriors).

The central concept is:

\[ \text{Posterior} \propto \text{Likelihood} \times \text{Prior} \]

This framework allows uncertainty to decrease as new information becomes available.

Examples:

  • Estimating wildlife population size, where prior information from past surveys is updated with new trap or camera data.
  • Hydrologic model calibration, where uncertain parameters such as baseflow recession constants are updated as additional flow data become available.
  • Estimating CO₂ fluxes from eddy covariance towers, where noisy measurements are combined with process-based constraints to infer true carbon exchange rates.
  • Species distribution models, where prior ecological knowledge informs where a species is expected to occur, and presence/absence data refine those predictions.

Bayesian methods are powerful for quantifying uncertainty formally, but they can be computationally intensive and require careful interpretation. They are particularly useful when new data arrive continually, allowing the model to learn and adapt over time.

Together, these methods—stochastic formulations, parameter distributions, scenario approaches, and Bayesian tools—provide a flexible toolkit for representing uncertainty realistically. Each serves a different purpose, and the choice depends on the questions being asked, the data available, and the nature of the environmental system being modeled.

6.5 Quantifying Uncertainty

Quantifying uncertainty means translating sources of variability or ignorance into numerical estimates that can be explored, compared, and communicated. This is where uncertainty becomes visible: instead of a single forecast, we produce a distribution of possible outcomes. Quantification does not eliminate uncertainty; it simply makes it explicit, measurable, and actionable.

The three most common approaches are Monte Carlo simulation, sensitivity analysis, and uncertainty propagation. Each addresses a different question:

  • Monte Carlo: What outcomes arise when parameters and processes vary randomly?
  • Sensitivity analysis: Which inputs matter most?
  • Uncertainty propagation: How does uncertainty in inputs translate into uncertainty in outputs?

Together, these tools form the core of modern environmental forecasting.

6.5.1 Monte Carlo Simulation

Monte Carlo simulation repeatedly runs a model using different combinations of parameters, noise realizations, or input scenarios. Each run represents one plausible version of the world. The ensemble of runs reveals how uncertainty affects predictions.

Common use cases:

  • exploring uncertain growth rates in logistic or predator–prey models
  • evaluating flood risk under uncertain rainfall inputs
  • estimating pollutant concentration ranges under uncertain emission rates
  • calculating probability of population extinction under environmental stochasticity

Outputs typically include:

  • mean trajectory (central estimate)
  • prediction intervals (e.g., 5th–95th percentile)
  • confidence bands around curves
  • probability of hitting a threshold (e.g., exceeding a flood stage)

Example: Monte Carlo logistic growth (escaped R chunk):

Monte Carlo methods are simple to implement but powerful in practice. They reveal not just the expected outcome, but the full range of possibility.

6.5.2 Sensitivity Analysis

Sensitivity analysis asks: Which inputs matter the most?
This is crucial when:

  • data collection is expensive
  • parameters are poorly known
  • resources for calibration are limited

It helps decide where to spend effort and prioritize future measurements.

6.5.2.1 Local Sensitivity

Local sensitivity examines how small changes in parameters near a baseline value affect outputs:

\[ S_r = \frac{\partial f}{\partial r} \]

Examples:

  • increasing growth rate \(r\) by 5% in logistic growth to see how quickly population equilibria shift
  • increasing infiltration capacity in a runoff model to assess changes in peak flow
  • changing decomposition rate by 1% in a carbon model to observe soil carbon response

Local sensitivity is intuitive but may miss nonlinear or threshold effects.

6.5.2.2 Global Sensitivity

Global sensitivity explores wide parameter ranges simultaneously, capturing interactions and nonlinear responses. Methods include:

  • Latin hypercube sampling (LHS): stratified random sampling of parameter space.
  • Variance-based methods: decompose output variance into contributions from each parameter.
  • Sobol indices: quantify how much each parameter (and each interaction) contributes to output variability.

Examples:

  • identifying which climate parameters drive uncertainty in carbon-climate feedback models
  • determining which soil or vegetation variables control uncertainty in fire behavior predictions
  • assessing which life-history traits (mortality, fecundity, growth) most influence fish population stability

Global sensitivity provides a holistic view of model behavior—not just how outputs change, but why.

6.5.3 Uncertainty Propagation

Uncertainty propagation links input uncertainty to output uncertainty. If rainfall, temperature, growth rate, or initial conditions are uncertain, how uncertain is the prediction?

This is essential for:

  • risk assessment
  • decision-making under uncertainty
  • communicating credible intervals around forecasts

Common examples:

  • Rainfall uncertainty → flood height uncertainty in hydrologic models
  • Temperature variability → stream temperature uncertainty in thermal habitat models
  • Uncertain emissions trajectories → wide distribution of future CO₂ concentrations
  • Uncertain initial snowpack → large spread in spring runoff forecasts

In nonlinear models, uncertainty may amplify:

  • small input uncertainty → large output uncertainty (e.g., chaotic food webs)

Or it may dampen:

  • input uncertainty averages out across processes (e.g., large watersheds integrating small-scale rainfall errors)

Understanding propagation helps clarify what predictions are robust and what predictions require caution.

6.6 Communicating Uncertainty

Representing uncertainty is only useful if the results are communicated clearly. Communication must be transparent, honest, and appropriately calibrated to audience needs. This is a core scientific skill, especially in environmental contexts where policy decisions depend on model output.

Three modes of communication—visual, numerical, and verbal—work together to convey the full picture.

6.6.1 Visual

Visual communication makes uncertainty visible and intuitive. Common approaches include:

  • Prediction bands showing uncertainty around a line or trajectory
  • Fan plots displaying widening uncertainty over time (common in climate projections)
  • Spaghetti plots overlaying many Monte Carlo trajectories
  • Histograms or density plots summarizing final outcomes or key indicators

Examples:

  • showing a 5–95% band around a forecast hydrograph during storm modeling
  • presenting a fan plot of sea-level rise under multiple climate scenarios
  • layering 1,000 population trajectories to illustrate extinction risk
  • displaying a density plot of predicted carbon storage across different parameter draws

Visuals often communicate uncertainty more effectively than tables or equations.

6.6.2 Numerical Summaries

Numerical summaries quantify uncertainty concisely. Common metrics include:

  • mean, median (central tendency)
  • interquartile range (IQR) (spread of the middle 50%)
  • 5th–95th percentile interval (confidence or prediction interval)
  • exceedance probabilities (probability of crossing a threshold)

Examples:

  • “There is a 22% probability that peak flow exceeds 1,000 m³/s this year.”
  • “The median predicted carbon storage is 105 Mg/ha, with a 90% interval of 75–140 Mg/ha.”
  • “Extreme heat days increase by 6–14 days depending on the scenario.”

Numbers make uncertainty operational—they can guide actions, risk mitigation, and policy design.

6.6.3 Verbal Statements

Many agencies, including NOAA and the IPCC, use calibrated language to communicate uncertainty qualitatively. These verbal categories help stakeholders understand the degree of confidence behind model results.

Common verbal indicators:

  • “likely,” “very likely,” “unlikely”
  • “low confidence,” “medium confidence,” “high confidence”
  • “under plausible assumptions”
  • “supported by multiple lines of evidence”

Examples:

  • “It is very likely that average annual temperature will continue to rise.”
  • “There is low confidence in projections of local precipitation patterns due to model disagreement.”
  • “Forest biomass recovery under Scenario A is likely, but outcomes vary depending on disturbance frequency.”

Verbal statements complement visual and numerical approaches, especially when communicating with non-experts or decision-makers.

6.7 Modeler’s Checklist

A practical reference for evaluating uncertainty throughout a modeling project:

  1. Identify uncertainty
    • What are the sources? Measurement? Process? Parameters? Structure?
  2. Represent uncertainty
    • Should uncertainty be modeled stochastically? Through parameter distributions? Scenarios?
  3. Quantify uncertainty
    • What methods are appropriate? Monte Carlo? Sensitivity analysis? Bayesian updating?
  4. Interpret uncertainty
    • Which uncertainties dominate model behavior? Which are negligible?
  5. Communicate uncertainty
    • Are results shown with clear visual, numerical, and verbal expressions of uncertainty?

Together, these steps ensure that models are not only technically correct but scientifically trustworthy and transparent.