...many environmental organizations projections about global trends are exaggerations or myths Aug 7...

‘‘...many environmental organizations ...many environmental organizations projections about global trends are projections about global trends are

exaggerations or myths’exaggerations or myths’Aug 7 2001Aug 7 2001

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 basedon scenarios

IPCC 2000

Global change brings novelty

http://earthobservatory.nasa.gov/Study/SpottedOwls/

Northern spotted owl and old growth logging

• Confidence intervals enter policy

Human impacts on populations?

forest dynamics with aridity and elevated CO2

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(n2): the N-body problemFinite memory

Solutions are approximate: how much error is acceptable?

Simulating large processes

QuickTime™ and aCinepak decompressor

are needed to see this picture.

How much detail?

The N-body problem

• Complexity is O(trees2) or O(length2)

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)

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





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

€

dndt

= f n;θ( )

€

nt+dt = f nt ,dt;θ( )

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 I’m discrete– Combine theoretical process with inference and do

it with computers

Combinations of time/state (no structure)

Discrete time Continuous time

(usually deterministic)

Discrete state

(usually stochastic)

Cellular automata neighborhood {x}:

Birth-death process (Pr of abundance y):

Continuous state

Difference eqn: Differential eqns:

€

yx,t+1 ~ p y{x},t( )

€

dpy dt = f py−1, py+1( )

€

yt+1 = f yt( )

€

dy dt = f y(t)( )

Combinations of time/state (with discrete structure)


Discrete state

Stochastic state transitions:

Continuous time Markov process

Continuous state

Difference eqns:

Matrix models: yt+1 = Ayt

Differential eqns:

€

y1,t+1 = f1 y1,t , y2,t( )

y2,t+1 = f2 y1,t , y2,t( )

€

dy1 dt = f1 y1, y2( )

dy2 dt = f2 y1, y2( )

€

yt+1 ~ Multinom f yt( )( )

Combinations of time/state (with continuous structure)


Discrete state

Not important Not important

Continuous state

Integrodifference eqns:

Partial differential eqns:

€

yt+1(x) = c f (x − x' )yt (x' )dx'∫

€

∂y ∂t = −c∂y ∂x

Discretize a continuous time model

€

dndt

= rn 1−nK

⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

nt+dt −ntdt

€

nt+dt ≈ nt + dn

€

=nt +dn

dt

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt +ε t

€

=nt + rn 1−n

K

⎛ ⎝ ⎜

⎞ ⎠ ⎟dt +ε t

• Ex: Logistic population growth

http://earthobservatory.nasa.gov/Study/SpottedOwls/

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.

€

dndt

= rn

€

n t( ) = n(t)ert

€

nt+1 = ntλ

€

nt = n0λt

€

nt+1 = ntA

€

nt = n0At

What is the value of ?

juveniles

€

A =

0 0 b

s 0 0

0 s1 s1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

subadults adultss s1

s1

b

€

nt+1 = ntA

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

1990’s estimates

• ‘Overconfidence intervals’

€

A =

0 0 b

s 0 0

0 s1 s1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Eleven literature estimates of population growth rate


Overfitting?

• Fit here doesn’t predict there


error propagation

Confidence intervals controlled by sample size

No individual variation behavior, health?

In space?

In time?

Estimation error

Propagated estimation error

€

A =

0 0 b

s 0 0

0 s1 s1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

s s1b

Traditional inference


Parameters have distributions

Estimation error

classical

Hierarchical Bayes

€

A =

0 0 b

s 0 0

0 s1 s1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Admit process level variation

Discretize a space-time model

• Ex: advection

€

∂c∂t

= −u∂c∂x

€

cx,t+dt ≈ cx,t +ucx+dx,t − cx−dx,t

2dx

⎛

⎝ ⎜

⎞

⎠ ⎟dt + εx,t

Diffusion

€

∂u∂t

= ru+∂

∂xD x, y( )

∂u∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟+

∂∂y

D x, y( )∂u∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=ru+∂D∂x∂u∂x

+D∂2u

∂x2+

∂D∂y∂u∂y

+D∂2u

∂y2

€

∂u∂t

≈ux,y,t+Δt −ux,y,t

Δt

€

∂u∂x

≈ux+Δx,y,t − ux−Δx,y,t

2Δx€

∂2u

∂x2≈

1

Δx

ux+Δx,y,t − ux,y,tΔx

−ux,y,t − ux−Δx,y,t

Δx

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=ux+Δx,y,t −2ux,y,t + ux−Δx,y,t

Δx2

€

∂D∂x

≈Dx+Δx,y −Dx−Δx,y

2Δx

€

ux,y,t+Δt −ux,y,tΔt

= rux,y,t

€

+Dx+Δx,y −Dx−Δx,y

2Δx

⎡

⎣ ⎢

⎤

⎦ ⎥

€

ux+Δx,y,t − ux−Δx,y,t

2Δx

⎡

⎣ ⎢

⎤

⎦ ⎥

€

+Dx,yux+Δx,y,t −2ux,y,t + ux−Δx,y,t

Δx2

⎡

⎣ ⎢

⎤

⎦ ⎥

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)

1988 1999

1966 1977

Learn about parameters of spread

• Construct full probability model– Focus on conditionals

• Hierarchical structures

( )dataparametersp

( ) ( ) ( )hypphypparameterspparametersp ∝

( )parametersprocessp×

( )parametersprocessdatap ,∝

( )parametersp×

Rxn-diffusion model of spread

Process:

Parameter variabilityParameters:

Estimation error

local diffusion, growth

Distributions for diffusion, growth parameters

Hyperparameters:

Process error

Spatiotemporal variabilityActual spread

Observation errorBird counts

Latent variables:

Data:


A hierarchical model


Rate of diffusion

Outline





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

€

bi,t ~ Pois mi,t( )

€

mi,t = f (ri,t ;θi )+ εi,t

€

θi ~ N θ ,v( )

€

y j,t ~ Pois mi,ti∈A j∑ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Outline





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

Parametervalues

model

Data

Parameterestimates

Outline





GCMs• Temp, pressure, velocity• 1° to 4° lat/long grid• About 20 vertical levels• >106 variables• Biosphere for surface

feedback, C/H2O/energy exchange

• Predict the biosphere response to changing climate and CO2

• 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

• 1st generation: latent heat flux depends on surface soil moisture, through aerodynamic resistance


• 2nd generation: separate canopy and soil• Heat fluxes thru Evap and Trans

VPD

Leaf H2O potential


• 3rd generation: C, heat, and H2O fluxes

• Net PSN for both C and H2O

Global patterns

Canopy physiology in BIOME-BGC

• Water balance– Daily time step

• Conductance to H2O

– “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/H2O fluxes integrated over year

• Predict leaf area, biomass of nine main functional types– Compete for water, light– C3 shrubs, C4 (warm-season) grasses

• Water balance• NPP• 2° lat x 2° long

Two time scales

Carbon balance calculated first (full year)

Land surface: H2O/C exchange

Canopy physiology

• Leaf-level physiology: Light, CO2, kinetic limitations on An

• Respiration– Leaf-level physiology– Proportional to stem,

root biomass

Ligh

t lim

itatio

n CO2, Kinetic limitation

Canopy physiology

• Stomatal conductance– Proportional at An

– Declines with [CO2], Rel Humidity– Declines with soil moisture (empirical)

• Note: An 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:

€

GPPi,y = Ag,i,h,yh=1

8760

∑

€

NPPi,y = 1−η( ) Ag,i,h,y − Rg,i,h,y( )h=1

8760

∑

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





...many environmental organizations projections about global trends are exaggerations or myths Aug 7...

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