...many environmental organizations projections about global trends are exaggerations or myths Aug many environmental organizations projections about global trends are exaggerations or myths Aug

Challenges of (terrestrial) ecosystem modeling Processes on many scales Interactions complex, non-linear Dynamics too slow to observe Cannot do large enough experiments to quantify relationships

Reconstruction of past temperature Projections based on scenarios IPCC 2000 Global change brings novelty

Northern spotted owl and old growth logging Confidence intervals enter policy Human impacts on populations?

forest dynamics with aridity and elevated CO 2 Consequences for Biodiversity

Episodic events have long-term consequences

Forecasting future consequences Conditions change? Confidence envelope? Tilman et al., Science (2001) Projecting trends

Small and large experiments Satellites Observational data Heterogeneous information, obscure variables

- instantaneous - eight years Thomas and DeLucia Mohan, Clark, & Schlesinger, Ecol Appl (2007) Response depends on scale

Processes are subgrid/subpixel, context is regional Complexity is O(n 2 ): the N-body problem Finite memory Solutions are approximate: how much error is acceptable? Simulating large processes

How much detail?

The N-body problem Complexity is O(trees 2 ) or O(length 2 ) Map of elm trees (dots) and seed density (blue shading) at Blackwood, NC

Mean seed density CV (over years) Estimated process variability Acceptable error?

Goals of (terrestrial) ecosystem modeling Understanding –Implications of complex interactions –Long-term behavior Prediction –Exploit observed relationships, in the context of a model, to learn about unobserved phenomena

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)

Types of ecosystem models population ecology –demography to population dynamics –e.g., how do changes in birth or death rates affect future demand for resources? community ecology –population dynamics in the context of other species and environment determines diversity –e.g., do biotic interactions, climate fluctuations threaten spp with extinction? ecosystem ecology –water, nutrient, energy cycles –e.g., how does climate change interact with rising atmospheric CO2 to influence C uptake by plants?

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)

Formulating a model linear vs nonlinear (statistics vs mathematics) multiple state variables discrete vs continuous states discrete vs continuous time statistical vs theoretical vs simulation

Linear vs non-linear Statistics: linear in parameters –If linear and Gaussian, easy to solve for estimators –If not, traditionally only asymptotic approximations Ecology (mathematics) –Linear in state variables –Non-linear models can have complex behavior

Continuous vs discrete time Continuous time: rate equations –Assumes process is continuous –Often easier to formulate, easier to solve –Simpler behavior –If stochastic, then hard Discrete time –Assumes process steps forward in discrete time steps –Solve only the simplest linear eqns –Readily accommodate stochasticity

Continuous vs discrete time Ecological models traditionally continuous –But can only solve/approximate very simple dynamics anyway With simulation modeling, divergence –Computer models discrete time –Theoretical models continuous (but exceptions to this) Why Im discrete –Combine theoretical process with inference and do it with computers

Combinations of time/state (no structure) Discrete timeContinuous time (usually deterministic) Discrete state (usually stochastic) Cellular automata neighborhood {x}: Birth-death process (Pr of abundance y): Continuous state Difference eqn:Differential eqns:

Combinations of time/state (with discrete structure) Discrete timeContinuous time Discrete state Stochastic state transitions: Continuous time Markov process Continuous state Difference eqns: Matrix models: y t+1 = Ay t Differential eqns:

Combinations of time/state (with continuous structure) Discrete timeContinuous time Discrete state Not important Continuous state Integrodifference eqns: Partial differential eqns:

Discretize a continuous time model Ex: Logistic population growth

Northern spotted owl and old growth logging Linear example from population ecology Continuous state, continuous time Continuous state, discrete time: With stage structure: – is the dominant eigenvalue of A. What is the value of ?

juvenilessubadultsadults ss1s1 s1s1 b Stage structured model of NSO Approach: estimate parameters in A Calculate dominant eigenvalue of A Propogate error in parameters to uncertainty in Is < 1?

Inference on Northern spotted owl growth rate Clark, Ecology (2003) Spread due to estimation error Growth rate 1990s estimates Overconfidence intervals

Eleven literature estimates of population growth rate Clark, Ecology (2003) Overfitting? Fit here doesnt predict there

Clark, Ecology (2003) error propagation Confidence intervals controlled by sample size No individual variation behavior, health? In space? In time? Estimation error Propagated estimation error ss1s1 b Traditional inference

Clark, Ecology (2003) Parameters have distributions Estimation error classical Hierarchical Bayes Admit process level variation

Discretize a space-time model Ex: advection

Diffusion

A diffusion example from population ecology House finch – eastern populations from 1940 release on LI, NY Records of spread from breeding bird survey

Maps of spread Wikle, Ecology (2003)

Learn about parameters of spread Construct full probability model –Focus on conditionals Hierarchical structures

Rxn-diffusion model of spread Process: Parameter variability Parameters: Estimation error local diffusion, growth Distributions for diffusion, growth parameters Hyperparameters: Process error Spatiotemporal variability Actual spread Observation error Bird counts Latent variables: Data: Wikle, Ecology (2002) A hierarchical model

Wikle, Ecology (2002) Rate of diffusion

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)

Stochasticity for the unknown Statistical models traditionally not dynamic: time and space too hard yet most data collected to understand spatio-temporal processes deterministic part plus stochastic part (known, unknown)

Traditional uses of stochasticity Statistical models: observation error Process models: –transitions among discrete states (e.g., birth, death) –stochastic regulation: viewed as a process Alternative view: the unknown

Stochasticity for unknowns at all levels Counts in sample j Births to individual i Fecundity depends on resources r Variation in response to r

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)

Inference vs forward simulation traditional approach: –write down a process model –scavenge literature for parameter values –compare model output with data Alternative: –write additional models for data –estimate parameters –formal prediction (mix over all sources of uncertainty) model Predicted data Parameter values model Data Parameter estimates

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)

GCMs Temp, pressure, velocity 1° to 4° lat/long grid About 20 vertical levels >10 6 variables Biosphere for surface feedback, C/H2O/energy exchange

Predict the biosphere response to changing climate and CO 2 The importance of vegetation for accurate climate forecasts

Heat flux in climate models S- insolation - albedo G- ground heat flux H- sensible heat flux E- evapotranspiration - latent heat of vaporization

Atmosphere-Biosphere exchange 1 st generation: latent heat flux depends on surface soil moisture, through aerodynamic resistance

Atmosphere-Biosphere exchange 2 nd generation: separate canopy and soil Heat fluxes thru Evap and Trans VPD Leaf H 2 O potential

Atmosphere-Biosphere exchange 3 rd generation: C, heat, and H 2 O fluxes Net PSN for both C and H 2 O

Global patterns

Canopy physiology in BIOME-BGC Water balance –Daily time step Conductance to H 2 O –big leaf –Maximum transpiration from LAI –Reduced by VPD, light, temp, soil moisture Soil evaporation

Parameter values for jack pine (BOREAS Experiment)

Data for BOREAS implementation Data inputs: –Temp and Prec: 15 min –Missing data interpolated or substituted –Calculated daily min/max Strategy: compare simulations with data –Canopy water & latent heat flux (eddy flux data, integrated over a full day) –Soil moisture TDR, neutron probe 3 to 6 locations 25 to 120 m from flux towers 0 to 120 cm depth Averaged over day and depth –Snow depth gauge

Evapotranspiration (calculated from eddy flux)

Soil moisture

Some results ET largely driven by VPD and solar radiation –No water stress

Land-surface biophysics –Terrestrial C: hourly C/energy/H 2 O fluxes integrated over year Predict leaf area, biomass of nine main functional types –Compete for water, light –C 3 shrubs, C 4 (warm-season) grasses Water balance NPP 2° lat x 2° long

Two time scales

Carbon balance calculated first (full year)

Land surface: H 2 O/C exchange

Canopy physiology Leaf-level physiology: Light, CO2, kinetic limitations on A n Respiration –Leaf-level physiology –Proportional to stem, root biomass Light limitation CO 2, Kinetic limitation

Canopy physiology Stomatal conductance –Proportional at A n –Declines with [CO 2 ], Rel Humidity –Declines with soil moisture (empirical) Note: A n does not appear to depend on conductance, but the converse Phenology –Winter deciduous –Drought deciduous

PFT prediction Differential capacity to accumulate C, depending on climate Determined by summing hourly C balance over entire year, for each PFT at each location! Annual Gross 1° prod: Annual Net 1° prod:

Inputs Monthly averaged climate variables –Interpolated to daily –Transformed to diurnal variation –Stochastic precip Topography, Soil texture Parameter values –Leaf-level physiology –Crude guesswork for respiration

Dynamics in time

PFTs Competition for light, moisture explains global distribution of vegetation types Temporal dynamics reasonable Missing fire and climate extremes implicated in errors

Big leaf models The way to get biosphere feedback to atmosphere for climate prediction One dimensional Struggle with time scales for different processes Deterministic (zero uncertainty) forward mode: do outputs look like data?

Outline Types of ecosystem models Formulating a model –Linear/non-linear –Continuous/discrete time –Continuous/discrete states Stochasticity (Inverse) inference vs (forward) prediction Three examples –The terrestrial big leaf (today) –Patches to continents (next time) –Stand dynamics (time after that)