‘ ‘ ...many environmental organizations ...many environmental organizations projections about global trends are projections about global trends are exaggerations or myths’ exaggerations or myths’ Aug 7 2001 Aug 7 2001
Mar 27, 2015
‘‘...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)
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
€
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 time Continuous time
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 time Continuous time
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
Clark, Ecology (2003)
Overfitting?
• Fit here doesn’t 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
€
A =
0 0 b
s 0 0
0 s1 s1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
s s1b
Traditional inference
Clark, Ecology (2003)
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:
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
€
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
• 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
Parametervalues
model
Data
Parameterestimates
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• >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
Atmosphere-Biosphere exchange
• 2nd generation: separate canopy and soil• Heat fluxes thru Evap and Trans
VPD
Leaf H2O potential
Atmosphere-Biosphere exchange
• 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
• 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)