Sustainability & Environment (Agri-Food Systems) · Postgraduate & Professional Advanced · also Environmental Modelling, Crop and Ecosystem Simulation, Systems Analysis
Building and using models of crops, soils, catchments and ecosystems: model types, calibration and validation, and the main open models used in sustainability research.
In the exams: UGC NET Unit 9 (environmental modelling); ICAR AICE subjects 60 and 71; PhD research
A model is a simplified, formal representation of a system, built to understand it, to predict how it will respond to change, or to support a decision. Environmental and agro-ecosystem models range from a single mass-balance equation for a polluted lake to coupled simulations of crops, soils, catchments and climate across a continent. They are used to estimate how much nitrate a catchment will lose if fertiliser use changes, how wheat yields respond to warming, how fast soil carbon builds under a new rotation, which watersheds deliver most sediment to a reservoir, and how much carbon a landscape would lose under a development scenario. They also underpin national greenhouse-gas inventories at higher tiers, environmental impact assessments and many sustainability claims.
This manual starts with systems thinking and the classification of models, then builds the core mathematics through mass-balance and box models and population and ecosystem models. It then surveys the families of process-based models most used in sustainability research: crop simulation models, soil carbon and nitrogen models, catchment water-quality models and ecosystem-service models, naming the main open models in each family and explaining what they compute. The final chapters cover what separates trustworthy modelling from curve-fitting: calibration, validation and sensitivity analysis; uncertainty propagation and ensemble modelling; and coupling models with remote sensing and climate projections.
Statistics underlying calibration and the mathematics of differential equations are taught in the Quantitative Methods subject; climate model physics is taught in the Climate Models and Scenarios subject; this manual uses both. The emphasis is on understanding model structure well enough to choose a model, judge its output and explain its limits.
Examiners test this subject through definitions (empirical versus mechanistic, lumped versus distributed, verification versus validation, equifinality), short calculations (residence time, steady-state concentration in a mixed lake, first-order decay and half-life, logistic growth, thermal time, light interception, SCS curve-number runoff, Nash–Sutcliffe efficiency, Q10, uncertainty of a product), and judgement about which model fits which question. Environmental-science lecturer-eligibility and fellowship papers, agricultural meteorology and environmental science research-service examinations, environmental engineering licensing examinations (for mass balance and reactor models) and doctoral qualifying examinations all draw on this material.
A system is a set of interacting components within a boundary, exchanging matter, energy or information with its environment. Systems analysis begins by defining the purpose of the study, which determines the boundary, the components that must be included and the level of detail. The main elements of a dynamic model are state variables, which describe the condition of the system at a moment (biomass, soil carbon, water storage, population size); rate or flow variables, which change the states (photosynthesis, decomposition, discharge, births); parameters, constants that characterise processes (a decay rate, a radiation-use efficiency); and driving or forcing variables, external inputs not affected by the system (weather, management, emissions). The stock-and-flow notation of system dynamics, developed by Jay Forrester, draws states as boxes (stocks) connected by flows controlled by valves. Feedback loops give systems their characteristic behaviour. Negative (balancing) feedback opposes change and stabilises a system, as when a growing population depletes resources and slows its own growth. Positive (reinforcing) feedback amplifies change, as in the ice–albedo feedback, in which melting reduces reflectivity and causes more warming. Delays in feedback loops produce oscillations and overshoot. Systems exhibit emergent properties that cannot be predicted from components alone, and thresholds or tipping points where small changes in a driver cause large, sometimes irreversible shifts between alternative stable states, as in lake eutrophication or savanna–forest transitions. A conceptual model, a diagram of components, flows and feedbacks, is the first and often most useful product of systems analysis, because it exposes assumptions and data needs before any equations are written. George Box's remark that all models are wrong but some are useful summarises the attitude: a model's value depends on its purpose and on honest statements of its limits.
Models are classified along several axes. Empirical (statistical, correlative) models describe relations observed in data, such as a regression of yield on rainfall; they are simple and often accurate within the range of the data but unreliable when extrapolated to new conditions. Mechanistic (process-based) models represent causal processes, such as photosynthesis, water flow and decomposition, and are needed when predicting responses to conditions not yet observed, such as higher carbon dioxide; most practical models are mixtures, with empirical equations for sub-processes. Deterministic models give one output for a given input; stochastic models include random variation and give distributions of outputs. Static models describe a system at one time or at equilibrium; dynamic models track change through time. Lumped models treat an area such as a whole catchment as one unit with averaged properties; distributed models divide it into grid cells or units, and semi-distributed models use intermediate units such as sub-basins or hydrological response units. Individual-based (agent-based) models simulate many individuals, such as trees or farmers, each following rules, and system behaviour emerges from their interactions. Few dynamic models can be solved analytically, so most are solved numerically: the simplest scheme, Euler's method, updates each state as X(t + Δt) = X(t) + Δt × dX/dt, which is accurate only when the time step is small relative to the fastest process; higher-order schemes such as Runge–Kutta are more accurate for the same step. The modelling cycle runs from defining the purpose to conceptual model, mathematical formulation, coding, verification (checking that the code correctly implements the equations), calibration, validation (testing agreement with independent observations), sensitivity and uncertainty analysis, application and documentation; it is iterative, and models evolve with new data. Choosing the simplest model adequate for the purpose, the principle of parsimony, reduces the data needed and the risk of overfitting.
Every environmental model rests on conservation of mass: for any control volume, accumulation = inputs − outputs + production − consumption. Drawing a boundary around a lake, a soil layer, an aquifer or the atmosphere and accounting for every flow across it is the basis of box (compartment) models. The simplest is the completely mixed box, also called a continuously stirred tank reactor (CSTR), in which the concentration is the same everywhere and equals the outflow concentration. For a lake of volume V receiving inflow Q at concentration Cin, losing water at the same rate Q, and removing a substance by first-order decay or settling with rate constant k, the balance is V dC/dt = Q Cin − Q C − k V C. At steady state (dC/dt = 0), C = Q Cin / (Q + kV) = Cin / (1 + kτ), where τ = V/Q is the hydraulic residence time. For a lake of 10⁷ m³ with an inflow of 10⁵ m³ per day, τ = 100 days; with k = 0.01 per day, the steady-state concentration is Cin / (1 + 1), half the inflow value. The time the system takes to respond to a change in input is governed by the combined rate (Q/V + k): after a step change the concentration approaches its new steady state exponentially, reaching about 63 percent of the change in one response time 1/(Q/V + k) and about 95 percent in three. A plug-flow system, such as an idealised river reach, has no mixing along its length, and concentration decays with travel time: C = Cin e^(−kτ), so plug flow removes more of a decaying substance than a mixed box of the same residence time. First-order decay with rate k has half-life t½ = ln 2 / k = 0.693 / k. Mixing two streams gives C = (Q₁C₁ + Q₂C₂)/(Q₁ + Q₂).
Box models have shaped environmental management. Vollenweider's phosphorus loading models for lakes, developed from data across many temperate lakes in the 1960s and 1970s, relate in-lake phosphorus to inflow concentration and residence time; one widely used form is P = Pin / (1 + √τ), with τ in years, and the predicted phosphorus is linked to chlorophyll and trophic state, giving lake managers a target load. In rivers, the Streeter–Phelps model combines first-order deoxygenation from decomposing organic matter (biochemical oxygen demand) with reaeration from the atmosphere to predict the dissolved-oxygen sag curve downstream of a discharge and the critical point where oxygen is lowest; it is the classical basis of water-quality models taught in environmental engineering. Multi-box models of pollutants in the environment, called fugacity or multimedia models, partition a chemical among air, water, soil, sediment and biota according to its properties, estimating where persistent organic pollutants accumulate. At planetary scale, the carbon cycle is described by reservoirs (atmosphere, ocean surface and deep layers, vegetation, soils, fossil reserves) and fluxes between them. Residence time, or turnover time, is stock divided by the flux in or out at steady state: a reservoir of 600 units with an outflow of 120 units per year has a residence time of 5 years. The concept explains why a pulse of carbon dioxide added to the atmosphere is partly removed within decades by ocean and land uptake but a fraction remains for many centuries, because the deep ocean and sediments exchange slowly. Box models also show that halting the growth of emissions does not immediately halt the growth of the stock: a stock rises as long as inflow exceeds outflow, a point that system-dynamics researchers found many educated adults misjudge. Mass balances are also the core of nutrient budgets for farms and catchments, material flow analysis and life-cycle inventories.
Population models describe change in numbers through births, deaths, immigration and emigration. With unlimited resources, a population grows exponentially: dN/dt = rN, so N(t) = N₀ e^(rt), where r is the intrinsic rate of increase; the doubling time is ln 2 / r, so a population with r = 0.05 per year doubles in about 14 years. Where generations are discrete, Nₜ₊₁ = λNₜ, with λ = e^r the finite rate of increase; λ > 1 means growth. Resource limitation is captured by the logistic model, dN/dt = rN(1 − N/K), where K is the carrying capacity. Growth is fastest at N = K/2, where the rate equals rK/4; this is the maximum sustainable yield in the simplest harvest models of fisheries and forestry, a concept now regarded as risky because K and r fluctuate and are uncertain. Discrete-time versions of the logistic can show damped oscillations, cycles and chaos at high growth rates. Interactions between species are modelled by the Lotka–Volterra equations: in the predator–prey form, prey grow exponentially and are eaten in proportion to the product of prey and predator numbers, while predators convert prey into births and die at a constant rate, producing coupled oscillations; in the competition form, each species' growth is reduced by its own and its competitor's density, and coexistence requires that intraspecific competition be stronger than interspecific. Age- or stage-structured populations are modelled with matrices: a Leslie matrix contains age-specific fecundities in its first row and survival probabilities on its sub-diagonal; the dominant eigenvalue gives the long-term growth rate λ, and sensitivity and elasticity analysis show which vital rates most affect λ, guiding conservation of threatened species. Metapopulation models, introduced by Levins, track the fraction p of habitat patches occupied: dp/dt = cp(1 − p) − ep, with colonisation rate c and extinction rate e, giving an equilibrium p* = 1 − e/c, so a metapopulation persists only when c exceeds e.
Ecosystem models follow matter and energy rather than individuals. Compartment models represent pools such as phytoplankton, zooplankton, detritus and dissolved nutrients, connected by flows, with nutrient–phytoplankton–zooplankton (NPZ) models a classic example in limnology and oceanography. Food-web mass-balance models quantify biomass and consumption of each functional group so that production equals predation, fishing, other mortality and export; they are used to explore ecosystem effects of fishing. Primary production is often modelled with the light-use efficiency approach: gross primary production = ε × fAPAR × PAR, where PAR is incident photosynthetically active radiation, fAPAR the fraction absorbed by the canopy and ε the efficiency of conversion, reduced by temperature and water stress; it links directly to satellite data. Process-based biogeochemical models simulate photosynthesis (often with the Farquhar biochemical model), respiration, allocation, litterfall and decomposition, and are used to estimate carbon and water fluxes of forests and grasslands. Forest gap models simulate the establishment, growth and death of individual trees on small plots, capturing succession and species composition; they have been used to project forest responses to climate change. At the largest scale, dynamic global vegetation models (DGVMs) represent vegetation as plant functional types competing for light and water across a global grid, coupled to soil carbon and hydrology; they form the land component of Earth-system models and are compared annually in global carbon budget assessments, where their spread shows how uncertain the land carbon sink remains. Ecosystem models must balance detail against data: each added process needs parameters that may be poorly known, and more complex models are not necessarily more accurate. Testing against flux-tower measurements, inventory plots and long-term experiments is essential, and model intercomparison projects reveal structural differences between models that single-model studies hide.
Crop simulation models compute daily crop growth and development from weather, soil, management and cultivar characteristics. Most follow the hierarchy of production levels developed by the de Wit school at Wageningen. Potential yield is set by radiation, temperature, carbon dioxide and the genotype when water and nutrients are not limiting and pests are controlled; water-limited yield adds water supply; nutrient-limited yield adds nitrogen and other nutrients; actual yield is further reduced by weeds, pests, diseases and pollutants. The yield gap is the difference between potential (or water-limited, for rainfed crops) and actual farm yields, and mapping it shows where better management could raise production. Development (phenology), the progress through stages such as emergence, flowering and maturity, is usually driven by thermal time, the accumulation of growing degree-days: daily GDD = (Tmax + Tmin)/2 − Tbase when positive, with a crop-specific base temperature, often modified by daylength and vernalisation. A day with maximum 30 °C and minimum 16 °C contributes 13 degree-days above a base of 10 °C. Growth is commonly driven by intercepted radiation: the fraction of light intercepted by a canopy follows Beer's law, 1 − e^(−k × LAI), where k is the extinction coefficient and LAI the leaf area index; with k = 0.5 and LAI = 3 the canopy intercepts about 78 percent. Biomass production is intercepted radiation multiplied by radiation-use efficiency, in grams of dry matter per megajoule, with values for many C3 crops lower than for C4 crops such as maize. Some models instead calculate photosynthesis from leaf biochemistry. Assimilates are partitioned to leaves, stems, roots and storage organs according to stage, and yield is often derived from biomass and a harvest index. Water-driven models such as the FAO's AquaCrop compute biomass from transpiration and a normalised water productivity. The soil water balance uses reference evapotranspiration, commonly by the FAO-56 Penman–Monteith method, and crop coefficients or a transpiration routine, together with runoff, drainage and root water uptake.
The main open crop modelling systems include the Decision Support System for Agrotechnology Transfer (DSSAT), which hosts the CERES models for cereals and CROPGRO for legumes; the Agricultural Production Systems sIMulator (APSIM), developed in Australia, which emphasises farming systems, rotations and soil processes; WOFOST from Wageningen, used in European crop monitoring; AquaCrop from FAO, designed for water-limited conditions with relatively few parameters; STICS from France; and rice models such as ORYZA. They differ in how they represent photosynthesis, water stress, nitrogen and phenology, which is why multi-model intercomparisons, coordinated by the Agricultural Model Intercomparison and Improvement Project (AgMIP), have become standard: for wheat, maize and rice, the spread among models in simulated yield response to temperature is large, and the median of an ensemble of models usually agrees better with observations than most individual models. Applications include yield forecasting within a season, testing management options (sowing dates, cultivars, irrigation and nitrogen schedules), estimating yield gaps, assessing climate-change impacts and adaptation, designing ideotypes for breeding and estimating nitrogen losses and greenhouse-gas emissions. Running a model for a new site needs daily weather (radiation, maximum and minimum temperature, rainfall, and for some models humidity and wind), soil profile properties (water limits, organic carbon, texture, depth), management records and cultivar coefficients. Cultivar or genetic coefficients, such as thermal time to flowering, are calibrated from field experiments; using default coefficients for unknown local cultivars is a major error source. Limits are important: most models assume well-managed fields free of pests and diseases, represent extreme heat at flowering, waterlogging and nutrient interactions crudely, and are tested on few environments in many tropical regions. Results should therefore be framed as responses relative to a baseline rather than as precise yields, unless validated locally.
Soil organic matter models treat decomposition as first-order: the rate of loss from a pool is proportional to its size. For a single pool with annual input I and decay constant k, dC/dt = I − kC. At steady state C* = I/k, and the turnover time is 1/k; with inputs of 2 t C ha⁻¹ per year and k = 0.02 per year, the equilibrium stock is 100 t C ha⁻¹ and the turnover time 50 years. After a change in management, the stock approaches its new equilibrium exponentially: C(t) = C* + (C₀ − C*)e^(−kt), so half the eventual change occurs in ln 2 / k years. Real soil organic matter is a continuum of materials with turnover from days to millennia, so models use several pools. The Rothamsted carbon model (RothC), developed from the long-term experiments at Rothamsted, has decomposable and resistant plant material, microbial biomass, humified organic matter and a small inert pool, runs on a monthly step and needs only clay content, monthly weather, plant cover and carbon inputs; it is widely used for national inventories and carbon projects. The CENTURY model, developed for grasslands of the North American Great Plains, has active, slow and passive pools linked to nitrogen, phosphorus and sulphur cycles and a plant-growth submodel; its daily version DayCent adds trace-gas emissions. The denitrification–decomposition model (DNDC) emphasises soil redox chemistry and simulates nitrous oxide, methane and ammonia emissions. Decomposition rates in all of them are modified by temperature, moisture, clay and tillage. Temperature sensitivity is often expressed as Q₁₀, the factor by which a rate increases for a 10 °C rise: rate(T) = rate(Tref) × Q₁₀^((T − Tref)/10); with Q₁₀ = 2, a 10 °C rise doubles decomposition and a 5 °C rise multiplies it by about 1.41. Because initial pool sizes are rarely measured, models are initialised by a spin-up run to equilibrium under historical land use, or by fractionation of soil samples into measurable pools.
Nitrogen models track mineralisation of organic nitrogen to ammonium, immobilisation by microbes, nitrification of ammonium to nitrate, denitrification of nitrate to nitrous oxide and dinitrogen under low oxygen, ammonia volatilisation, plant uptake and leaching of nitrate with drainage water. The carbon-to-nitrogen ratio of decomposing material governs whether nitrogen is released or immobilised: residues with C:N above roughly 25 to 30 cause net immobilisation in the short term, while those below release nitrogen, although the exact threshold depends on the microbial biomass C:N and the efficiency with which microbes use carbon. Nitrous oxide is produced in both nitrification and denitrification and is strongly controlled by soil water-filled pore space, nitrate availability, carbon supply and temperature; its emissions are episodic, with large pulses after rain on fertilised soils, which makes them hard to measure and model. The IPCC guidelines for national inventories define three tiers. Tier 1 uses global default emission factors and stock-change factors, such as the default direct N₂O emission factor for fertiliser nitrogen and default factors for land use, tillage and inputs applied to reference soil carbon stocks over a 20-year default transition period. Tier 2 replaces defaults with country-specific factors. Tier 3 uses process models such as those above, combined with measurements and activity data at fine resolution, and requires verification against measured data. Soil carbon models are also used to estimate sequestration from changed practices, such as cover crops, reduced tillage, biochar, organic amendments and agroforestry, in voluntary carbon markets. There, model error, baseline choice, saturation as stocks approach a new equilibrium, reversibility if practice stops and the equivalent-soil-mass correction for changes in bulk density are central questions. Model-predicted sequestration should be checked against repeated soil sampling, which itself needs many samples because soil carbon is highly variable.
Catchment models predict how much water, sediment, nutrients and other pollutants leave a landscape and reach streams, lakes and coasts. They begin with the water balance: precipitation is partitioned into interception, infiltration, surface runoff, evapotranspiration, soil storage, groundwater recharge and baseflow. A widely used method for daily surface runoff is the curve number method of the US Natural Resources Conservation Service (formerly the Soil Conservation Service). The potential maximum retention is S = 25,400/CN − 254 in millimetres, where the curve number CN (0 to 100) reflects soil hydrologic group, land cover and antecedent moisture; initial abstraction is commonly taken as 0.2S; and runoff Q = (P − 0.2S)² / (P + 0.8S) when rainfall P exceeds 0.2S, and zero otherwise. For CN = 80, S = 63.5 mm; a 75 mm storm then produces about 31 mm of runoff. A pollutant load is the mass transported per unit time, concentration × discharge: 2 mg L⁻¹ of nitrate-nitrogen in a flow of 5 m³ s⁻¹ is 10 g s⁻¹, or 864 kg per day. Because concentration and flow both vary, annual loads are estimated from frequent sampling or regression of concentration on flow, and most of the annual load of sediment and particulate phosphorus often moves in a few high-flow events. The simplest catchment models are export-coefficient models, which multiply the area of each land use by an annual export rate, for example kilograms of nitrogen per hectare per year, and add point sources; they need little data and are useful for screening and scenario comparison, although export coefficients vary widely with climate, soils and management and must be chosen from comparable catchments. Soil erosion is commonly estimated with the Universal Soil Loss Equation, A = R × K × LS × C × P, and its revised versions, combined with a sediment delivery ratio for the fraction that reaches the stream.
Process-based catchment models simulate hydrology and water quality continuously. The Soil and Water Assessment Tool (SWAT), developed by the US Department of Agriculture's research service and Texas A&M University, is a semi-distributed, daily model that divides a catchment into sub-basins and further into hydrological response units, unique combinations of land use, soil and slope; it simulates runoff by the curve number or infiltration methods, crop growth, nutrient cycling, sediment, pesticides and channel routing, and is the most widely applied catchment water-quality model worldwide, used on every continent. Other families include the Hydrological Simulation Program – FORTRAN (HSPF), the integrated catchment (INCA) models developed in Europe, and hybrid statistical-mechanistic models such as SPARROW, which relate measured stream loads to sources and landscape characteristics by nonlinear regression. In-stream models such as QUAL2K simulate dissolved oxygen, biochemical oxygen demand, nutrients, algae and temperature along a river, building on the Streeter–Phelps oxygen-sag principle. Model performance for streamflow and loads is judged with several statistics. The Nash–Sutcliffe efficiency, NSE = 1 − Σ(Oᵢ − Sᵢ)² / Σ(Oᵢ − Ō)², equals 1 for a perfect fit, 0 when the model is no better than the observed mean and is negative when it is worse; percent bias (PBIAS) measures systematic over- or underestimation; the Kling–Gupta efficiency combines correlation, bias and variability. Moriasi and colleagues (2007) proposed widely cited guidelines in which monthly streamflow simulations are satisfactory when NSE exceeds 0.5 and PBIAS is within ±25 percent, with wider limits for sediment and nutrients; such thresholds are conventions that depend on data quality and purpose. Catchment models support total maximum daily load programmes, river basin management plans, targeting of conservation practices to critical source areas and assessment of land-use and climate scenarios.
Ecosystem service models map and quantify the benefits people obtain from ecosystems, such as carbon storage, water yield, sediment and nutrient retention, pollination, habitat for biodiversity, flood regulation, coastal protection and recreation, usually for current conditions and alternative land-use or management scenarios. They connect three steps: the ecological supply of a service (what the landscape produces), its flow to beneficiaries (who receives it and where) and its value (in biophysical, monetary or social terms). The most widely used open suite is InVEST (Integrated Valuation of Ecosystem Services and Tradeoffs), developed by the Natural Capital Project, a partnership of universities and conservation organisations. Its models are spatially explicit and deliberately simple, using land-use and land-cover maps and lookup tables of biophysical parameters. The carbon model multiplies the area of each land-cover class by its carbon stocks in four pools (above-ground, below-ground, soil and dead matter), so converting 1000 ha of forest at 150 t C ha⁻¹ to cropland at 50 t C ha⁻¹ releases 100,000 t C, about 367,000 t CO₂. The annual water yield model uses a Budyko-type curve, which relates the ratio of actual evapotranspiration to precipitation to the aridity index, the ratio of potential evapotranspiration to precipitation; water yield is precipitation minus actual evapotranspiration. The sediment delivery ratio model applies the Universal Soil Loss Equation to every pixel and estimates the fraction of eroded soil reaching streams, and retention by vegetation is the difference between erosion with and without vegetation cover. Habitat quality models combine habitat suitability with the intensity, distance and decay of threats. Pollination models estimate the abundance of wild pollinators reaching farms from nesting and floral resources in the surrounding landscape. Other tools include ARIES, which uses artificial intelligence to assemble models, and LUCI, which emphasises fine-resolution hydrology and trade-offs.
Ecosystem service models are used in strategic environmental assessment, spatial planning, the design of payments for ecosystem services, natural capital accounting under the UN System of Environmental-Economic Accounting – Ecosystem Accounting (SEEA EA), corporate nature-related disclosures, and prioritisation of restoration and protected areas. Their strength is in comparing scenarios and showing where services and trade-offs are concentrated; their weakness is that many are rarely validated, rely on lookup values transferred from other regions and ignore dynamics. Good practice includes defining the decision context and the beneficiaries first, choosing the simplest model that answers the question, using local data for the most sensitive parameters, running sensitivity analysis on lookup values, validating against measured streamflow, sediment loads or field carbon stocks where possible, and reporting results as relative differences between scenarios with uncertainty rather than as precise absolute values. Benefit transfer, applying monetary values per hectare estimated in one place to another, is common but error-prone, especially between countries with different incomes and preferences; it is acceptable for screening but not for decisions requiring precision. Supply maps should not be presented as benefits where no beneficiaries exist: a watershed's sediment retention matters to a downstream reservoir, not to an empty basin. Spatial mismatches between where services are produced and where they are used, and between who bears the costs of conservation and who benefits, are central to equity in payments for ecosystem services. Scenario development, in which stakeholders define plausible futures, such as business as usual, agricultural expansion or restoration, links models to participatory planning. Finally, ecosystem service models should be combined with the biophysical and social evidence they simplify; a single map of total monetary value hides trade-offs among services and among groups of people.
Calibration is the adjustment of uncertain model parameters, within physically plausible ranges, so that simulated outputs match observations. It requires an objective function, such as the sum of squared errors, root-mean-square error or Nash–Sutcliffe efficiency, and a search method: manual trial and error, automatic optimisation (for example shuffled complex evolution or genetic algorithms), or Bayesian methods such as Markov chain Monte Carlo that return probability distributions of parameters rather than single values. Not all parameters should be calibrated: those that can be measured should be measured, and the number calibrated should be small relative to the information in the data, or the model will be overfitted, matching noise in the calibration data and predicting poorly elsewhere. A parameter is identifiable if the data constrain it; parameters that compensate for each other cannot be separately identified. Keith Beven described equifinality: many different parameter sets, and even different model structures, can reproduce observations equally well, so a single best set is an illusion; his generalised likelihood uncertainty estimation (GLUE) retains all behavioural parameter sets and uses them to produce prediction bounds. Validation tests the calibrated model against data not used in calibration. Klemeš (1986) proposed a hierarchy of tests for hydrological models: the split-sample test (calibrate on one period, validate on another); the differential split-sample test (calibrate in, for example, wet years and validate in dry years, essential when the model will be used for climate change); the proxy-basin test (validate on a different catchment); and combinations. Common statistics are RMSE, mean absolute error, mean bias, the coefficient of determination R², NSE and Willmott's index of agreement. R² alone is insufficient because a model that systematically doubles every observation can have R² = 1. A model is never validated in general, only shown to be adequate for a purpose within tested conditions.
Sensitivity analysis asks how much model outputs change when inputs or parameters change. It identifies which parameters most need careful measurement or calibration, which can be fixed at default values, and whether the model behaves plausibly. Local, one-at-a-time (OAT) analysis changes each parameter by a small amount around a base value while holding others constant; relative sensitivity is S = (ΔY / Y) / (ΔP / P), so if a 10 percent increase in a parameter raises output by 5 percent, S = 0.5, and |S| > 1 means the output amplifies the parameter's change. OAT is cheap but misses interactions and depends on the base point, which matters in nonlinear models. Global sensitivity analysis varies all parameters across their full ranges simultaneously. The Morris method (elementary effects) screens many parameters with few runs, ranking them by the mean of the absolute effects (overall influence) and their standard deviation (nonlinearity or interactions). Variance-based methods, particularly Sobol indices, decompose output variance into contributions: the first-order index measures the share of variance due to a parameter alone, and the total-order index includes all its interactions; a large gap between them signals interactions. Variance-based methods need many model runs, often thousands, so emulators (statistical surrogates trained on a limited set of runs) are used for expensive models. Regression-based measures from Monte Carlo samples, such as standardised regression coefficients, work well for nearly linear models. Sensitivity results depend on the parameter ranges chosen, which should reflect real uncertainty, and on the output examined: a crop model's yield may be most sensitive to radiation-use efficiency, while its nitrate leaching is most sensitive to soil drainage parameters. Reporting sensitivity analysis is part of good modelling practice, recommended in guidance for models used in environmental regulation.
Model predictions are uncertain for several reasons. Input uncertainty comes from errors and gaps in weather, soil, land-use and management data. Parameter uncertainty reflects imperfect knowledge of rate constants and coefficients. Structural uncertainty arises because different models represent the same processes differently, or omit some. Scenario uncertainty concerns future drivers such as emissions, land use, prices and policies, which cannot be assigned probabilities in the usual sense. Natural variability, such as weather from year to year, is irreducible. A useful distinction is between aleatory uncertainty (inherent randomness) and epistemic uncertainty (lack of knowledge, reducible with research). Uncertainty is propagated through a model in two main ways. Analytical error propagation uses rules for simple equations: for a sum or difference of independent quantities, absolute standard uncertainties add in quadrature; for a product or quotient, relative (percentage) uncertainties add in quadrature. The IPCC inventory guidelines use these rules for Approach 1: if activity data have 30 percent uncertainty and the emission factor 40 percent, the emission estimate has √(30² + 40²) = 50 percent uncertainty. For a total of independent sources, the combined uncertainty is √Σ(Uᵢxᵢ)² / Σxᵢ, so totals are proportionally less uncertain than their parts. These rules assume independence, small uncertainties and near-normal distributions. Monte Carlo simulation (IPCC Approach 2) handles larger, skewed or correlated uncertainties: values of uncertain inputs are drawn repeatedly from probability distributions, the model is run for each draw, and the distribution of outputs gives a mean and confidence interval, typically the 2.5th and 97.5th percentiles for a 95 percent interval. Correlations between inputs must be represented, or uncertainty will be misestimated. Bayesian calibration provides posterior parameter distributions that can be propagated directly.
An ensemble is a set of model runs designed to represent uncertainty. Initial-condition ensembles run one model from slightly different starting states to sample internal variability, as in weather forecasting. Perturbed-parameter (perturbed-physics) ensembles vary uncertain parameters within one model. Multi-model ensembles combine different models; climate projections from the Coupled Model Intercomparison Project (CMIP) and crop, hydrological and ecosystem impact projections from intercomparisons such as AgMIP and ISIMIP are examples. Scenario ensembles run the same models under several future pathways. Multi-model ensemble means or medians often outperform most individual members because independent errors partly cancel, but ensembles of opportunity, assembled from whatever models happen to be available, are not designed samples of uncertainty; models share components and biases, so their spread may understate true uncertainty. Approaches beyond 'one model, one vote' include weighting models by performance and independence, and Bayesian model averaging. Separating sources of uncertainty, for example with variance decomposition, shows whether climate models, impact models or scenarios dominate, which tells researchers where effort will reduce uncertainty most: in many crop-yield projections the impact-model spread is comparable to or larger than the climate-model spread. A projection, conditional on a scenario, differs from a prediction or forecast, which aims to be the most likely outcome. Communicating uncertainty clearly is part of the modeller's job. The IPCC uses calibrated language: confidence (very low to very high) reflects evidence and agreement, and likelihood terms carry defined probabilities, such as likely (66 to 100 percent), very likely (90 to 100 percent) and virtually certain (99 to 100 percent). Decision-makers often need to know which outcomes are plausible and which decisions are robust across them, rather than a single number; robust decision-making and scenario discovery methods use large ensembles for exactly this purpose.
Remote sensing and models are complementary: satellites observe large areas frequently but measure only some variables, often indirectly, while models simulate many variables continuously but drift from reality as errors accumulate. Remote sensing enters models in four ways. As forcing, it supplies inputs: land cover for catchment and ecosystem-service models, leaf area or fAPAR for light-use efficiency models, satellite rainfall and radiation for crop and hydrological models in regions with few weather stations. As parameterisation, it provides spatial fields such as soil properties derived from digital soil mapping, elevation and vegetation height. As evaluation data, it allows models to be tested across space, for example simulated evapotranspiration against energy-balance estimates, simulated leaf area against satellite products or simulated snow cover and soil moisture against microwave retrievals. And through data assimilation, observations update the model's state variables as the simulation runs. In sequential assimilation, the model forecasts the state, an observation arrives, and the state is corrected towards the observation in proportion to their relative uncertainties; the Kalman filter does this optimally for linear systems with Gaussian errors, the ensemble Kalman filter extends it to nonlinear models by estimating error statistics from an ensemble of runs, and particle filters handle non-Gaussian cases. Variational methods adjust initial conditions or parameters to fit observations over a time window. Assimilating satellite leaf area index or soil moisture into crop models improves regional yield estimates, and assimilating soil moisture into hydrological models improves flood and drought forecasts. Scale mismatch is a constant problem: a 10 km soil-moisture pixel represents the top few centimetres averaged over many fields, while a crop model simulates one field's root zone, so observation operators and bias correction are needed before assimilation.
Impact studies drive crop, hydrology, soil and ecosystem models with climate projections. Global climate models (GCMs) from the Coupled Model Intercomparison Project, currently its sixth phase (CMIP6), simulate the climate under scenarios combining Shared Socioeconomic Pathways with radiative forcing levels, such as SSP1-2.6 (strong mitigation), SSP2-4.5 (intermediate), SSP3-7.0 and SSP5-8.5 (high emissions); the earlier generation used Representative Concentration Pathways (RCPs). GCM grid cells, typically 100 km or more across, are too coarse and too biased for local impact models, so outputs are downscaled and bias-corrected. Dynamical downscaling nests regional climate models within GCMs over a region at finer resolution, coordinated internationally by the CORDEX programme; statistical downscaling relates large-scale predictors to local observations. Bias correction adjusts model output to match observed statistics in a historical period. The simplest approach, the delta-change method, adds the projected change in temperature to an observed baseline series (25 °C plus a projected 2.5 °C gives 27.5 °C) and multiplies observed precipitation by the projected ratio (800 mm × 0.9 = 720 mm); it preserves observed variability but ignores changes in variability. Quantile mapping corrects the whole distribution, including extremes, but assumes that the biases are stationary and can distort trends. Good practice is to use several GCMs, chosen to span the range of projected change, more than one scenario and, where possible, more than one impact model, and to report the resulting cascade of uncertainty. Coupling can be loose (offline: one model's output files become another's inputs) or tight (online: models exchange information every time step, allowing feedbacks, such as irrigation cooling the local climate). Integrated assessment models link energy, land use, economics and climate to explore mitigation pathways, while Earth-system models and emerging digital-twin initiatives combine observations and models continuously.
A student uses this subject to understand what a model in a paper actually computes and how far to trust it. Core skills are drawing a conceptual stock-and-flow diagram, writing and solving a mass balance for a mixed lake, computing logistic growth, thermal time and light interception, running an open crop or catchment model with supplied data, and calculating NSE and bias against observations. Typical errors are reporting only R², calibrating many parameters on one short record, and treating a single model run as a prediction.
A practitioner five years into a consultancy, agency or research project runs models for decisions: SWAT for a river basin nutrient plan, an ecosystem-service model for a land-use plan, RothC or DayCent for a soil-carbon project, or a crop model for an adaptation study. Daily choices include which parameters to calibrate, how to split data for validation, which climate models and scenarios to use, how to bias-correct them, and how to present uncertainty ranges to a client or regulator without hiding them.
A senior professional with twenty-five years of experience commissions and reviews modelling for policy and investment: national inventories at Tier 3, catchment pollution targets, carbon-credit methodologies and climate-risk assessments. Judgements include whether a model is fit for the purpose and data available, whether multiple models are needed, how to audit model-based claims, when measurements must replace or check models, and how to manage institutional modelling capacity and data infrastructure over decades.
A researcher with a PhD and long experience develops and tests models: new process representations for heat stress, nitrous oxide or soil carbon stabilisation, data-assimilation systems, emulators for global sensitivity analysis and multi-model intercomparisons. Such researchers lead model evaluation protocols, act as authors in international assessments, supervise doctoral modellers and examine model-based theses, and so must understand both the mathematics and the long history of over-confident model use in environmental science.
This manual explains model structures and simple calculations but does not teach programming, numerical analysis beyond Euler's method, or the internal equations of any named model in full; users must read each model's documentation. Groundwater, air-quality dispersion, ocean and atmospheric chemistry models, agent-based land-use models and economic integrated assessment models are mentioned only briefly. Thresholds quoted for performance statistics, C:N ratios and Q₁₀ are conventions or typical values that vary with system and purpose. Model names, versions and maintaining institutions change; check current documentation.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
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