Data-model integration:
Examples from belowground
ecosystem ecology
Kiona Ogle
University of Wyoming
Departments of Botany & Statistics
www.uwyo.edu/oglelab
Today’s Task
• What are some ecological questions to
which sensor network data could be
applied?
• How would those data be used in models?
• Overview modeling of ecological data and
processes.
Types of Questions
• What are some ecological questions to which
sensor network data could be applied?
– Spatial & temporal processes
• Improved ecological understanding
• More accurate prediction & forecasting
– Example problems
• “Biogeochemical exchanges between the
atmosphere & biosphere”
• How do environmental perturbations affect carbon
& water exchange?
• Partitioning ecosystem processes & components
• Linking processes & mechanisms operating at
multiple temporal & spatial scales
How to Address Such Questions?
• Couple data and models
– Sensor network data
• Very rich
– Real-time; large datasets; spatially extensive and/or temporally
intensive
• Heterogeneous
– Different locations, processes, and conditions
– Models & data analysis
• Less appropriate:
– “Classical” analyses that assume linearity and normality of data
– Design-based inference about patterns
• More appropriate:
– Coupling of process-based models with diverse and rich datasets
– Model-based inference about patterns and mechanisms
Why Couple Data & Process Models?
– Parameter estimation (or “model parameterization”)
• Quantification of uncertainty
• Improved predictions and forecasts
• Decision support, management, conservation
– Synthesize multiple types of data
• Relate different system components to each other
• Learn about important mechanisms
– Hypothesis generation
• Use data-informed models to generate testable hypotheses
• Inform sampling and network design
– Data analysis
• Go beyond simple “classical” analyses
• Explicit integration of multiple data types, diverse scales,
and nonlinear and non-Gaussian processes
How to Couple Data & Process Models?
– Multiple approaches, for example:
• Maximum likelihood-based models
• Least squares, minimization of objective functions
• Hierarchical Bayesian models
– Hierarchical Bayesian approach
• Recall, from Jennifer’s talk …
Unknown quantities
Observed data
P( D , P , Process | Data)
Posterior
Process parameters
Latent (or true) process
Data parameters
 P( Data | Process, D )  P( Process |  P )  P( D ,  P )
Likelihood
Probabilistic process model Prior(s)
Outline
• The process model:
– Types of ecological models
– Building process models
• Examples from belowground ecosystem ecology:
– Motivating issues
– Ex 1: Estimating components of soil organic matter
decomposition
– Ex 2: Deconvolution of soil respiration (i.e., CO2 efflux)
– In both examples, highlight:
• Data sources
• Process models
• Data-model integration
• Implications of data-model integration for sensor
network data & applications
Hierarchical Bayesian Model
Unknown quantities
Observed data
P( D , P , Process | Data)
Posterior
Process parameters
Latent (or true) process
Data parameters
 P( Data | Process, D )  P( Process |  P )  P( D ,  P )
Likelihood
Probabilistic process model Prior(s)
Data model (likelihood)
P( Data | Process, D ) :
Data = Latent process + observation error
Probabilistic process model
P( Process |  P ) :
Latent process = Expected process + process error
The “process model”
The Process Model
• Conceptual model:
– Systems diagrams
– Graphical models
• Model formulation:
– Explicit, mathematical eqn’s
• Systems equations
• State-space equations
Inputs
Unobserved
quantities
(parameters)
Observed
quantities
(driving variables)
Outputs
“Compare”
Conceptual
model
Mathematical
model
Simulation
model
The “process model”
Analytical
output
Numerical/
simulation
output
Observed
quantities
(data)
“Predict”
Unobserved
or latent
quantities
Types of Process Models
Deterministic
Stochastic
Compartment models
(differential or difference equn’s)
Matrix models
Reductionist models
(include lots of details & components)
Holistic models
(use general principles)
Static models
Dynamic models
Distributed models
(system depends on space & time)
Lumped models
Linear models
Nonlinear models
Causal/mechanistic models
Black box models
Analytical models
Numerical/simulation models
Jorgensen (1986) Fundamentals of Ecological Modelling. 389 pp. Elsevier, Amsterdam.
Upcoming Example:
Soil Carbon Cycle Model
Deterministic
Stochastic
Compartment models
(differential or difference equn’s)
Matrix models
Reductionist models
(include lots of details & components)
Holistic models
(use general principles)
Static models
Dynamic models
Distributed models
(system depends on space & time)
Lumped models
Linear models
Nonlinear models
Causal/mechanistic models
Black box models
Analytical models
Numerical/simulation models
Example Process Model
Pools or
state
variables
Simplified
systems diagram
of the soil
carbon cycle in
a temperate
forest
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.
Flows of
carbon
Model Formulation
d
X( t )  A X( t )  B u( t )
dt
 A: matrix of flux rates or “carbon
transfer coefficients” (parameters)
 u(t): flux of carbon into the system
(e.g., photosynthetic flux) (driving
variable or modeled quantity)
 B: vector of ‘allocation fractions’
(parameters)
 X: vector of state variables
(unobservable latent quantities,
outputs)
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.
Model Formulation
d
X( t )  A X( t )  B u( t )
dt
 X (t )
d
 X (t ) Observable
(data)
Expected  dt

A X (t )
process

 u (t )

 B u (t )
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.
How to Couple Data & Process Models?
– Hierarchical Bayesian approach
Unknown quantities
Observed data
P( D , P , Process | Data)
Posterior
Process parameters
Latent (or true) process
Data parameters
 P( Data | Process, D )  P( Process |  P )  P( D ,  P )
Likelihood
Probabilistic process model Prior(s)
Data model (likelihood)
P( Data | Process, D ) :
Data = Latent process + observation error
Probabilistic process model
P( Process |  P ) :
Latent process = Expected process + process error
Outline
• The process model:
– Types of ecological models
– Building process models
• Examples from belowground ecosystem ecology:
– Motivating issues
– Ex 1: Estimating components of soil organic matter
decomposition
– Ex 2: Deconvolution of soil respiration (i.e., CO2 efflux)
– In both examples, highlight:
• Data sources
• Process models
• Data-model integration
• Implications of data-model integration for sensor
network data & applications
Ecosystem Processes
Emphasis on aboveground
What about belowground?
Biogeochemical Cycles
N
H 20
N
H 20
C
C
P
H 20
Biogeochemical Cycles
N
H 20
Belowground
system is critical
N
Tightly
linked to aboveground system
H 20
C
C
P
H 20
Belowground “Issues”
Aboveground
• Lots of info
• Easy to measure
Belowground
• Little info
• Difficult to measure
• Aboveground measurements (helpful but limited)
Outstanding issues
•
•
•
•
Partitioning above- & belowground
Quantifying & partitioning belowground
Implications for ecosystem function
Examples: arid & semiarid systems
Figure from Kieft et al. (1998) Ecology 79:671-683
Motivating Questions: Soil Carbon Cycle
•
•
•
•
From where in the soil is CO2 coming from?
What are the relative contributions of autotrophs vs. heterotrophs?
What factors control decomposition rates & heterotrophic activity?
How does pulse
precipitation
affect sources
of respired
CO2?
• Implications of
climate change
for desert soil
carbon cycling?
Integrative Approach
• Diverse data sources
–
–
–
–
Experimental & observational
Lab & field studies
Multiple scales
Varying “amounts” & “completeness”
• Process-based models
– Key mechanisms, processes, components
– Balance detail & simplicity
– Multiple scales & interactions
• Statistical models: data-model integration
– Hierarchical Bayesian framework
– Mark chain Monte Carlo
Examples Presented Today
Deterministic
Stochastic
Compartment models
(differential or difference equn’s)
Matrix models
Reductionist models
(include lots of details & components)
Holistic models
(use general principles)
Static models
Dynamic models
(implicit dependence on time)
Distributed models
(implicit dependence on space & time)
Lumped models
Linear models
Nonlinear models
Causal/mechanistic models
Black box models
Analytical models
Numerical/simulation models
Ex 1: Soil organic matter decomposition
Objectives:
1. Identify soil & microbial processes affecting decomposition
2. Learn how vegetation (i.e., microsite) controls these processes
Experimental Design
Mesquite shrubland
in southern Arizona
Microsite types:
1. bare ground
2. grass
3. small mesquite
4. big mesquite
Bare ground
3 cores (reps)
Grass
Small mesquite
Big mesquite
Experimental Design
Add water
...
...
Incubate at 25 oC
Add sugar + water
CO2
Measure CO2 efflux
(soil respiration rate)
at 24 & 48 hours
CO2
8 depths (layers)
CO2
Experimental Design
Add water
...
...
Measure:
Microbial biomass
Soil organic carbon
Soil nitrogen Incubate at 25 oC
Add sugar + water
CO2
Measure CO2 efflux
(soil respiration rate)
at 24 & 48 hours
CO2
8 depths (layers)
CO2
Design & Data Overview
• Full-factorial design:
• Microsite
• 4 levels: bare, grass, small mesq, big mesq
• Stochastic data:
• Soil respiration rate
• N = 359 (25 missing)
• Soil layer
• Microbial biomass
• Substrate addition type
• Soil organic carbon
• 8 levels: 0-2, 2-5, ..., 40-50 cm
• 2 levels: water only, sugar + water
• Incubation time
• 2 levels: 24, 48 hrs
• Soil core or rep
• 3 cores per microsite
• N = 18 (14 missing)
• N = 89 (7 missing)
Some Data
Analysis Objectives
microbes
soil C
CO2 flux
Soil depth
Estimate microbial respiration (decomposition)
parameters (i.e., process parameters)
?
biomass
&
activity
?
data
Process Model: Soil Respiration
Estimate microbial respiration (decomposition)
parameters (i.e., process parameters)
Respiration (R)
Saturating carbon (C)
Low C
Microbial biomass (B)
Michaelis-Menton type model:
Ab  B  Ac  C
Ab  B  Ac  C
Ac  max  0, c 0  c 1  N 
R
Ab  microbial “base-line” metabolic rate
Ac  microbial carbon-use efficiency
Assume Ac related to “substrate quality”:
Ac  max  0, c 0  c 1  N 
Data-Model Integration
Ab  B  Ac  C
R
Ab  B  Ac  C
Ac  max  0, c 0  c 1  N 
• Full-factorial design:
• Microsite
• Soil layer
• Substrate addition type
• Incubation time
• Soil core or rep
• Things to consider:
• Multiple data types
• Nonlinear model
• Missing data
• Experimental design
microbes
Soil depth
• Stochastic data:
• Soil respiration rate
• Microbial biomass
• Soil organic carbon
B
?
some
data
C
soil C
?
some
data
R
N
CO2
N
data
fixed
Data Model (Likelihood)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
1. Let LR = log(R)
2. For microsite m, soil depth d, soil core r, substrateaddition type s, and time period t:
Observed rate
Mean (“truth”)
(latent process)
Observation
precision
(= 1/variance)
Data Model (Likelihood)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
1. Now, for the covariates...
2. For microsite m, soil depth d, and soil core r:
C{m ,d ,r } ~ Normal  C {m ,d } , C 
B{m ,d ,r } ~ Normal  B{m ,d } , B 
Observed
Mean (“truth”)
(latent process)
Observation precision
(= 1/variance)
3. Note: the likelihoods are for both the observed
and missing data
Data Model (Likelihood)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Likelihood components
LR{m ,d ,r ,s ,t } ~ Normal  LR{m ,d ,r , s } , LR 
C{m ,d ,r } ~ Normal  C {m ,d } , C 
B{m ,d ,r } ~ Normal  B{m ,d } , B 
Data parameters
D   LR , C , B 
Latent processes
Latent processes   LR{m ,d ,r ,s } , C {m ,d } , B{m ,d }
Probabilistic Process Model
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Latent processes
Latent processes   LR{m ,d ,r ,s } , C {m ,d } , B{m ,d }
Deterministic model for soil microbes & carbon contents
C {m ,d }  c{m ,d }  C{*m }
 B{m ,d }  b{m ,d }  B{*m }
Stochastic model for latent respiration
 LR{m ,d ,r ,s } ~ Normal   . LR{m ,d ,s } , 
LR

Probabilistic Process Model
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Stochastic model for latent respiration
 LR{m ,d ,r ,s } ~ Normal   . LR{m ,d ,s } , 
LR

Specify expected process: Michaelis-Menten (process) model
 Ab  B{m ,d }  Ac{m ,d }  C {m ,d }
 Ab  
B{ m ,d }  Ac{ m ,d }  C { m ,d }
 . LR{m ,d ,s }  

Ab  B{m ,d }

s  water only
if
s  sugar + water
Saturating carbon (C)
Respiration (R)
Ac{m ,d }  max  0, c 0  c 1  N{ m ,d } 
if
Low C
Microbial biomass (B)
Probabilistic Process Model
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Process components
 LR{m ,d ,r ,s } ~ Normal   .LR{m ,d ,s } , 
LR

 Ab  B{m ,d }  Ac{m ,d }  C {m ,d }

 .LR{m ,d ,s }   Ab  B{m ,d }  Ac{m ,d }  C {m ,d }

Ab   M {m ,d }

water
sugar
Ac{m ,d }  max  0, c 0  c 1  N{m ,d } 
C {m ,d }  c{m ,d }  C{*m }
B{m ,d }  b{m ,d }  B{*m}
Process parameters
 P    , c{m ,d } , b{m ,d } , C{*m } , B{*m } , Ab , c 0 , c 1
LR
Parameter Model (Priors)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Data parameters
D   LR , C , B 
Process parameters
 P    , c{m ,d } , b{m ,d } , C{*m } , B{*m } , Ab , c 0 , c 1
LR
Conjugate, relatively non-informative priors for precision terms
 LR , C , B ,  ~ Gamma  0.01, 0.001
LR
Parameter Model (Priors)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Data parameters
D   LR , C , B 
Process parameters
 P    , c{m ,d } , b{m ,d } , C{*m } , B{*m } , Ab , c 0 , c 1
LR
Non-informative Dirichlet priors for relative
distributions of microbes and carbon
c{m ,.} , b{m ,.} ~ Dirichlet 1,1,1,...,1
Multivariate version of the beta distribution
(with all parameters set to 1: multidimensional uniform)
Parameter Model (Priors)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Data parameters
D   LR , C , B 
Process parameters
 P    , c{m ,d } , b{m ,d } , C{*m } , B{*m } , Ab , c 0 , c 1
LR
Relatively non-informative (diffuse) normal priors for the rest:
c 0 , c 1 , ln C{*m }  , ln  B{*m}  , ln  Ab  ~ Normal  0, 0.0001
The Posterior
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
P ( Data |Process , D ) 


 4 8 3 2 2   LR
2 
 LR
exp

LR


 {m ,d ,r ,s ,t } LR{m ,d ,r ,s }   
 
2

2
 m 1 d 1 r 1 s 1 t 1 
 




 4 8 3  C
2  
2 
C
B
B
exp

C



exp

B


 {m ,d ,r } C {m ,d }    2
 {m ,d ,r } B{m ,d }   
 
2

2
2
 m 1 d 1 r 1 
 
 
4
8
3
2  
2 
  LR
 LR
exp 

 LR{m ,d ,r ,s }   .LR{m ,d ,s }    

 2
m 1 d 1 r 1 s 1 
 2


LR
0.99 0.001 LR
e

  
C
0.99 0.001 C
e
  


B
0.99 0.001 B
e
  


 LR
0.99 0.001 LR
e


 0.0001
0.0001
0.0001
0.0001
2  
2 
exp 
exp 
c 0  0   
c1  0   

2
2
2
2

 


 0.0001
0.0001
exp 
 Ab  0

2

2


2





 0.0001
2  
2 
0.0001 *
0.0001
0.0001 *
exp

B

0

exp

C

0










{m}
{m}
2

2
2

2
m 1 

 
 
4
The Posterior
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
P ( Data |Process , D ) 


 4 8 3 2 2   LR
2 
 LR
exp

LR


 {m ,d ,r ,s ,t } LR{m ,d ,r ,s }   
 
2

2
 m 1 d 1 r 1 s 1 t 1 
 




 4 8 3  C
2  
2 
C
B
B
exp

C



exp

B


 {m ,d ,r } C {m ,d }    2
 {m ,d ,r } B{m ,d }   
 
2

2
2
d 1 r 1 
 m 1 analytical
 
 
No
solution for the joint
posterior distribution
4
8
3
2  
2 
  LR
 LR
exp 

 LR{m ,d ,r ,s }   .LR{m ,d ,s }    

No analytical
solution
 2for most
m 1 d 1 r 1 s 1 
 2 of the marginal distributions

e
e
 the
   e Markov
   chain
 Monte
 e Carlo
  methods,
Approximate
posterior:
LR
0.99 0.001 LR

C
0.99 0.001 C
B


0.99 0.001 B


 LR
0.99 0.001 LR

 0.0001
implemented
WinBUGS
0.0001
0.0001
0.0001
2   in
2 
exp 
exp 
c 0  0   
c1  0   

2
2
2
2

 


 0.0001
0.0001
exp 
 Ab  0

2

2


2





 0.0001
2  
2 
0.0001 *
0.0001
0.0001 *
exp

B

0

exp

C

0










{m}
{m}
2

2
2

2
m 1 

 
 
4
Model Implementation: WinBUGS
Model Goodness-of-fit
Example Results
C* (total soil carbon, g C/m2)
B* (microbial biomass, g dw/m2)
box plot: parms.m[1,]
5.00E+3
box plot: parms.m[2,]
10.0
[1,2]
[2,2]
[2,3]
4.00E+3
[1,3]
[1,4]
3.00E+3
[2,4]
5.0
[1,1]
[2,1]
2.00E+3
1.00E+3
0.0
Bare
Big
Med. Grass
mesq. Mesq.
Bare
Big
Med. Grass
mesq. Mesq.
Example Results
Bare ground
Big mesquite
Relative amount of
microbial biomass
box plot: parms.md[5,1,]
box plot: parms.md[5,2,]
0.2
0.2
[5,2,1]
0.15
0.15
0.1
0.1
[5,1,1]
[5,2,2]
[5,1,2]
[5,1,3]
0.05
[5,2,4]
0.05
[5,1,4]
[5,1,7]
[5,1,5]
[5,2,3]
[5,2,5]
[5,1,8]
[5,1,6]
[5,2,8]
[5,2,6]
[5,2,7]
0.0
0.0
Surface
Deep
Surface
Soil depth (or layer)
Deep
Sensitivity to Data Sources
Ex 2: Deconvolution of Soil Respiration
•
•
•
•
From where in the soil is CO2 coming from?
What are the relative contributions of autotrophs vs. heterotrophs?
What factors control decomposition rates & heterotrophic activity?
How does pulse
precipitation
affect sources
data
of respired
data
data
CO2?
data
data
data
data
• Multiple data
sources
• lots
• limited
data
The Field Sites
San Pedro River Basin
Sonoran Desert
Santa Rita Experimental Range
Stable Isotope Tracers
CO2
Respired CO2
signature
CO2
12C
13C
Source
isotope
signatures
12C
12C
Important data source:
facilitates “partitioning”
Data Source Examples
stochastic data
Pool Isotopes (δ13Ci)
(roots, soil, litter;
Keeling plots)
Soil CO2 flux
(automated chambers)
Soil samples
(carbon content,
C:N, root mass)
Literature data
Soil Isotopes (δ13CTot)
(automated chambers
& Keeling plots)
Root mass
(manual chambers)
(arid systems;
total mass)
Datasets:
field/lab pubs
Soil carbon
(arid systems;
total C)
(in situ gas exchange)
(root-free,
carbon substrate,
microbial mass,
heterotrophic activity)
(arid systems; total mass,
carbon, microbes)
Soil CO2 flux
Root respiration
Soil incubations
Litter
covariate data
Soil temp & water
(automated,
multiple locations,
many depths)
Root distributions
(arid systems,
different functional
types)
Microbial mass
(arid systems;
total mass)
Root respiration
(arid systems,
different functional
types)
Potential sensor network data
Example Data
Santa Rita pulse experiment
San Pedro automated flux measurements
5
Respiration (mol / m2 / s)
Pre-monsoon
Dry Monsoon
Wet Monsoon
2
Respiration (mol / m / s)
6
4
3
2
1
0
-1 0
2
6
14
Day
San Pedro incubation experiment
-1
Open
Grass
Medium Mesquite
Big Mesquite
0.010
-1
-17
24 hours post-sugar addition
-19
-21
-23
-25
0.008
0.006
0.004
0.002
-2
0
2
4
6
8
Day
10
12
14
16
0.010
0.008
0.006
0.004
0.002
0.000
0.000
-27
0.012
-1
-15
Respiration (mol g s )
13
o
d C of respired CO2 ( /oo)
0.012
-1
-13
Respiration (mol g s )
Santa Rita pulse experiment – d13C
0-2 2-5 5-10 0-15 5-20 0-30 0-40 0-50
1
1
2
3
4
Depth (cm)
0-
Hierarchical Bayesian Model:
Deconvolution Approach
•
•
•
Integrate multiple sources of information
• Diverse data sources
• Different temporal & spatial scales
• Literature information
• Lab & field studies
Detailed flux models
• Respiration rates by source type & soil depth
• Dynamic models
Mechanistic isotope mixing models
• Multiple sources
Data Source Examples
stochastic data
Pool Isotopes (δ13Ci)
(roots, soil, litter;
Keeling plots)
Soil CO2 flux
(automated chambers)
Soil samples
(carbon content,
C:N, root mass)
Soil Isotopes (δ13CTot)
(automated chambers
& Keeling plots)
(arid systems; total mass,
carbon, microbes)
Root mass
(manual chambers)
(arid systems;
total mass)
P ( j | Data )  L (Data | j )  P ( j )
Microbial mass
Soil carbon
(arid systems;
total C)
(in situ gas exchange)
(root-free,
carbon substrate,
microbial mass,
heterotrophic activity)
Litter
Soil CO2 flux
Root respiration
Soil incubations
Literature data
covariate data
Soil temp & water
(automated,
multiple locations,
many depths)
Root distributions
(arid systems,
different functional
types)
(arid systems;
total mass)
Root respiration
(arid systems,
different functional
types)
Bayesian Deconvolution
The Hierarchical Bayesian Model
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Obs
Obs
Data  d 13CTot
(t ), RTot
(t ),SWC (z , t ),T (z , t ), M i (z , t )
Some Likelihood Components
Obs
d 13CTot
(t ) ~ No d 13CTot (t ),  C2 
Obs
2
RTo
t (t ) ~ No  RTot (t ),  R 
Observations
(data)
Likelihood of data
(isotopes & soil flux)
Latent processes: from
isotope mixing model &
flux models
Define process models…
Functions of
parameters 
The Deconvolution Problem
Theory & Process Models
d 13CTot (t ) 
N source

i 1
 B 13

d
C
(
z
,
t
)

p
(
z
,
t
)
dz


i
i
0

Isotope mixing model
(multiple sources & depths)
??
Contributions by source (i ) and depth (z )? Temporal variability?
pi (z , t ) 
ri (z , t )
RTot (t )
Relative contributions
(by source & depth)
??
Source-specific respiration? Spatial & temporal variability?
RTot (t ) 
N source

i 1
??
B

r
(
z
,
t
)
dz
 i

0

ri (z, t )  f i ,SWC(z, t ),T (z, t ), M i (z, t )
(Q10 Function, Energy of Activation)
Total flux
(at soil surface)
Flux model
(source- & depth- specific)
From previous “incubation/decomposition” study (Ex 1)
M i (z , t )  known /measured /estimated
Mass profiles
(substrate, microbes, roots)
The Deconvolution Problem
Objectives
ri (z, t )  f i ,SWC(z, t ),T (z, t ), M i (z, t )
Covariate data
Flux model
(source- & depth- specific)
What is i?
(source-specific parameters)
i  ri (z , t )
 Component fluxes
ri (z , t )  RTot (t )
 Total soil flux
RTot (t )  pi (z , t )
 Contributions
How to estimate i?
Bayesian Deconvolution
The Parameter Model (Priors)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Example: Lloyd & Taylor (1994) model
ri (z , t )  f i ,SWC (z , t ),T (z , t ), M i (z , t )

ri (z , t )  r i (z , t )  exp Eo

1
 1


   To T (z , t )  To



Informative priors for Eo and To:
Eo ~ No  308.56,2
304
To ~ No  227.13,10 
308
312
316
215 220 225 230 235 240
Implementation
•
Markov chain Monte Carlo (MCMC)
• Sample parameters (θi ) from posterior
• Posteriors for: θi’s, ri(z,t)’s, pi(z,t)’s, etc.
• Means, medians, uncertainty
•
WinBUGS
Results: Dynamic Source Contributions
San Pedro Site – Monsoon Season
Proportional Contribution
0.025
Proportional Contribution of Respiration Sources
Heterotrophs (0-5 cm)
Grass Roots (5-50 cm)
Mesquite Roots (5-50 cm)
0.020
0.015
0.010
0.005
VWC (v/v)
0.000
0.12
Soil
Moisture
0.08
Zoom-in
0.04
o
Soil T ( C)
30
Soil
Temperature
25
20
15
190
200
210
220
230
240
Day of Year
250
260
270
Results: Root Respiration Responses
Rain (mm)
Zoom-in: July 27 – August 4
30
25
20
15
10
5
0
5.0
4.0
0.30
Mesquite (C3 shrub)
0.25
0.20
3.0
0.15
2.0
1.0
Soil water
Sacaton (C4 grass)
0.05
0.0
209
Jul 27
0.10
0.00
210
211
212
213
Date
214
215
216
217
Aug 4
Soil water (v/v)
Total root respiration
(umol m-2 s-1)
205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
5.0
0.30
Mesquite (C3 shrub)
4.0
0.20
3.0
0.15
2.0
Soil water
1.0
Depth (cm)
0.10
0.10
Sacaton (C4 grass)
0.05
0.0
0.00
209
0.00
0.25
Soil water (v/v)
Total root respiration
(umol m-2 s-1)
Results: Contributions Vary by Depth
210
211
212
213
214
215
216
Date
Relative contributions by depth
0.20
0.00
0.10
0.20
0.00
0-5
5-10
10-15
15-20
20-25
25-30
0-5
5-10
10-15
15-20
20-25
25-30
0-5
5-10
10-15
15-20
20-25
25-30
30-40
30-40
30-40
40-50
40-50
40-50
Day 210
Day 213
217
0.10
0.20
Day 216
Summary
• Sources of soil CO2 efflux
• Mesquite (shrub): major contributor, stable source
• Sacton (grass): minor contributor, threshold response
• Microbes (bare): minor contributor, coupled to pulses
• Deconvolution & data-model integration
•
•
•
•
Soil depth (including litter)
By species or functional groups
Quantify spatial & temporal variability
Incorporate environmental drivers
• Implications & applications
• Identify mechanisms
• Predictions & forward modeling
Outline
• The process model:
– Types of ecological models
– Building process models
• Examples from belowground ecosystem ecology:
– Motivating issues
– Ex 1: Estimating components of soil organic matter
decomposition
– Ex 2: Deconvolution of soil respiration (i.e., CO2 efflux)
– In both examples, highlight:
• Data sources
• Process models
• Data-model integration
• Implications of data-model integration for sensor
network data & applications
Implications for Sensor Networks
– Parameter estimation (or “model parameterization”)
• Process models related to “biogeochemical exchanges
between the atmosphere & biosphere”
• Quantification of uncertainty
• Improved predictions and forecasts
– Synthesize data
•
•
•
•
Go beyond simple “classical” analyses
Explicit integration of multiple data types & scales
Relate different system components to each other
Learn about important mechanisms
– Hypothesis generation & sampling design
• Use data-informed models to generate testable hypotheses
• Inform sampling and network design
– Where (spatial), when (temporal), what (components)?
Questions?
Photo by Travis Huxman
Monsoon flood, San Pedro River Basin; Sonoran desert
Results: Dynamic Source Contributions
Proportional Contribution
0.025
Proportional Contribution of Respiration Sources
Heterotrophs (0-5 cm)
Grass Roots (5-50 cm)
Mesquite Roots (5-50 cm)
0.020
0.015
0.010
0.005
VWC (v/v)
0.000
0.12
Soil
Moisture
0.08
0.04
o
Soil T ( C)
30
Soil
Temperature
25
20
15
190
200
210
220
230
240
Day of Year
250
260
270
Example WinBUGS Output
parms[1] chains 1:2
parms[1] chains 1:2 sample: 100
315.0
EO
0.4
0.3
0.2
0.1
0.0
310.0
305.0
302.5
300.0
1
2000
4000
305.0
307.5
310.0
6000
iteration
parms[2] chains 1:2
parms[2] chains 1:2 sample: 100
230.0
TO
0.2
0.15
0.1
0.05
0.0
220.0
210.0
200.0
190.0
190.0
1
2000
4000
195.0
6000
iteration
Posterior statistics
parameter node
Eo
parms[1]
To
parms[2]
....
contribution pSc[1,1,1]
by
pSc[1,1,2]
[date,
pSc[1,1,3]
plot,
pSc[1,2,1]
source]
pSc[1,2,2]
pSc[1,2,3]
....
mean
307.7
201.1
sd
1.205
2.335
0.00145
0.8854
0.1132
0.00158
0.9721
0.02636
0.002904
0.009694
0.009998
0.003157
0.008952
0.008649
2.5%
305.2
195.8
-0.03455
0.8433
0.06992
-0.03353
0.9297
-0.01523
median 97.5%
307.7
309.9
201.0
205.1
sample
1000
1000
6.34E-4
0.8853
0.1131
6.685E-4
0.9725
0.02622
1000
1000
1000
1000
1000
1000
0.03565
0.9288
0.1558
0.03607
1.013
0.0654
200.0
205.0
The Inverse Problem
Plant water uptake
Soil respiration
B
d Dstem (t )   d D (z )  q (z , t )dz
0
??
B
d 18Ostem (t )   d 18O (z )  q (z , t )dz
Isotope
mixing
model
d 13CTot (t ) 
N source

i 1

d 13Ci

B

   pi (z , t )dz  
0
 
??
0
q (z , t ) 
U (z , t )
U Tot (t )
Fractional
contributions
pi (z , t ) 
B
U Tot (t )   U (z , t )dz
Total flux
RTot (t ) 
0
U (z , t )  RA(z )  ln a  RA(z ) 
 (z , t )  k (z , t )  root (t )  kroot (t )
RA(z )    Ga(1, 1 )  (1   )  Ga(2 , 2 )
Flux
model
N source

i 1
ri (z , t )  M i (z , t )
RTot (t )
B

r
(
z
,
t
)

M
(
z
,
t
)
dz
 i

i
0

ri (z, t )  f i ,SWC(z, t ),T (z, t )
(Q10 Function, Energy of Activation)
M i (z , t )  known /measured ?
Substrate or
root profiles
The Inverse Problem
d CTot (t ) 
13
N source

i 1
 B 13

  d Ci (z , t )  pi (z , t )dz 
0

Isotope mixing model
(multiple sources & depths)
??
Contributions by source (i ) and depth (z )? Temporal variability?
pi (z , t ) 
ri (z , t )  M i (z , t )
RTot (t )
??
Relative contributions
(by source & depth)
Source-specific respiration? Spatial & temporal variability?
RTot (t ) 
N source

i 1
B

r
(
z
,
t
)

M
(
z
,
t
)
dz
 i

i
0

ri (z, t )  f i ,SWC(z, t ),T (z, t )
(Q10 Function, Energy of Activation)
M i (z , t )  known /measured ?
Total flux
(at soil surface)
Flux model
(source- & depth- specific)
Mass profiles
(substrate, microbes, roots)
The Deconvolution Problem
Data-Model Integration
ri (z, t )  f i ,SWC(z, t ),T (z, t ), M i (z, t )
Covariate data
Flux model
(source- & depth- specific)
What is i?
(source-specific parameters)
ri (z , t )  RTot (t )
 Total soil flux
RTot (t )  pi (z , t )
 Contributions
Obs
d 13CTot
(t ) ~ No d 13CTot (t ),  C2 
Obs
2
RTo
t (t ) ~ No  RTot (t ),  R 
From isotope mixing
model & flux models
Likelihood of data
(isotopes & soil flux)
Depend on
i
Data Source Examples
stochastic data
Pool Isotopes (δ13Ci)
(roots, soil, litter;
Keeling plots)
Soil CO2 flux
(automated chambers)
Soil samples
(carbon content,
C:N, root mass)
Soil Isotopes (δ13CTot)
(automated chambers
& Keeling plots)
(arid systems; total mass,
carbon, microbes)
Root mass
(manual chambers)
(arid systems;
total mass)
P ( j | Data )  P (Data | j )  P ( j )
Microbial mass
Soil carbon
(arid systems;
total C)
(in situ gas exchange)
(root-free,
carbon substrate,
microbial mass,
heterotrophic activity)
Litter
Soil CO2 flux
Root respiration
Soil incubations
Literature data
covariate data
Soil temp & water
(automated,
multiple locations,
many depths)
Root distributions
(arid systems,
different functional
types)
(arid systems;
total mass)
Root respiration
(arid systems,
different functional
types)
The Deconvolution Problem
Plant water uptake
Soil respiration
B
d Dstem (t )   d D (z )  q (z , t )dz
0
??
B
d 18Ostem (t )   d 18O (z )  q (z , t )dz
Isotope
mixing
model
d 13CTot (t ) 
N source

i 1

d 13Ci

B

   pi (z , t )dz  
0
 
??
0
q (z , t ) 
U (z , t )
U Tot (t )
Fractional
contributions
pi (z , t ) 
B
U Tot (t )   U (z , t )dz
Total flux
RTot (t ) 
0
U (z , t )  RA(z )  ln a  RA(z ) 
 (z , t )  k (z , t )  root (t )  kroot (t )
RA(z )    Ga(1, 1 )  (1   )  Ga(2 , 2 )
Flux
model
N source

i 1
ri (z , t )  M i (z , t )
RTot (t )
B

r
(
z
,
t
)

M
(
z
,
t
)
dz
 i

i
0

ri (z, t )  f i ,SWC(z, t ),T (z, t )
(Q10 Function, Energy of Activation)
M i (z , t )  known /measured ?
Substrate or
root profiles
The Deconvolution Problem
Plant water uptake
Soil respiration
What are
ω, 1, 1, 2, 2?
What is
i?
RA(z )    Ga(1, 1 )  (1   )  Ga( 2 , 2 )
ri (z , t )  f i ,SWC (z , t ),T (z , t )
 U (z , t )
 q (z , t )
 RTot (t )
 pi (z , t )
Obs
d Dstem
(t ) ~ No d Dstem (t ),  D2 
Obs
d 13CTot
(t ) ~ No d 13CTot (t ),  C2 
Obs
18
2
d 18Oste
m (t ) ~ No  d Ostem (t ),  O 
Obs
2
RTo
t (t ) ~ No  RTot (t ),  R 
Likelihood of data
From isotope mixing
model & flux model
Types of data provides by sensor
networks
• high-frequency tunable diode laser (TDL)
measurement of the stable isotope
• eddy covariance for measuring concentrations
and fluxes of gases (e.g., water vapor and CO2)
• soil environmental data: temperature, water
content, water potential, etc.
• micro-met data: air temp, RH, vpd, light, wind
speed, etc.
• plant ecophys/ecosystem data: sapflux, ET,
albedo & reflectance
Key components
ID
TABLENM
229 TREE
407 TREE
408 TREE
1059 TREE
6768 TREE
7019 TREE
7111 TREE
8105 TREE
8539 TREE
12808 TREE
12810 TREE
13315 TREE
19399 TREE
19445 TREE
22050 TREE
22053 TREE
22060 TREE
22137 TREE
23519 TREE
26415 TREE
26783 TREE
28623 TREE
29299 TREE
29320 TREE
30129 TREE
30139 TREE
32119 TREE
34017 TREE
34329 TREE
34716 TREE
35041 TREE
36375 TREE
36410 TREE
36411 TREE
"PLT_CN"
PHYSCLCD "STATECD" "CYCLE""SUBCYCLE""UNITCD""COUNTYCD" "PLOT"
"SUBP"
"TREE" "CONDID"
23854928010661.00
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8
1
Data
6
6
a c
dP
 A   Ri  Si  E   i i
dt
i 1
i 1 ci  d i
dSi ai  E

 d i  Si
dt ci  d i
Process models
P ( | X ) 
P ( X |  )  P( )
 P ( X |  )  P( )d
P( | X )
State
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA
VA

Statistical tools
data-model integration
The Process Model
• Conceptual models:
– Systems diagrams
– Graphical models
Observations
of real system
Conceptual
model
• Model formulation:
– Explicit, mathematical eqn’s
• Systems equations
• State-space equations
Mathematical
model
Analytical
output
“Compare”
Observational
data
Simulation
model
Numerical/
simulation
output
Examples Presented Today
Deterministic
Stochastic
Compartment models
(differential or difference equn’s)
Matrix models
Reductionist models
(include lots of details & components)
Holistic models
(use general principles)
Static models
Dynamic models
(implicit dependence on time)
Distributed models
(implicit dependence on space & time)
Lumped models
Linear models
Nonlinear models
Causal/mechanistic models
Black box models
Analytical models
Numerical/simulation models
Jorgensen (1986) Fundamentals of Ecological Modelling. 389 pp. Elsevier, Amsterdam.
Data Model (Likelihood)
P( D , P , Process | Data)  P( Data | Process, D )  P( Process |  P )  P( D , P )
Likelihood components
LR{m ,d ,r ,s ,t } ~ Normal  LR{m ,d ,r , s } , LR 
C{m ,d ,r } ~ Normal  C {m ,d } , C 
B{m ,d ,r } ~ Normal  B{m ,d } , B 
Assuming conditional independence,
likelihood of all data is:


 4 8 3 2 2   LR
2 

P ( Data | Process ,  D )   
exp  LR  LR{m ,d ,r ,s ,t }   LR{m ,d ,r ,s }    
2
 m 1 d 1 r 1 s 1 t 1  2
 




 4 8 3  C
2  
2 
C
B
B
exp  C{m ,d ,r }  C {m ,d }    
exp   B{m ,d ,r }  B{ m ,d }   
 
2
2
 m 1 d 1 r 1  2
 
  2