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Data-model integration:Data-model integration:Examples from Examples from
belowground ecosystem belowground ecosystem ecologyecology
Kiona OgleKiona OgleUniversity of WyomingUniversity of Wyoming
Departments of Botany & StatisticsDepartments of Botany & Statisticswww.uwyo.edu/oglelabwww.uwyo.edu/oglelab
Today’s TaskToday’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 QuestionsTypes 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?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 Why Couple Data & Process Models?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 How to Couple Data & Process Models?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 …
( , , | )
( | , ) ( | ) ( , )
D P
D P D P
P Process Data
P Data Process P Process P
Observed data
Latent (or true) process Data parameters
Process parametersUnknown quantities
Posterior
Likelihood Probabilistic process modelPrior(s)
OutlineOutline• 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 ModelHierarchical Bayesian Model
( , , | )
( | , ) ( | ) ( , )D P
D P D P
P Process Data
P Data Process P Process P
Observed data
Latent (or true) process Data parameters
Process parametersUnknown quantities
Posterior
Likelihood Probabilistic process modelPrior(s)
( | , ) :DP Data ProcessData = Latent process + observation error
( | ) :PP ProcessLatent process = Expected process + process error
Data model (likelihood)
Probabilistic process model
The “process model”
The Process ModelThe Process Model
• Conceptual model:– Systems diagrams– Graphical models
• Model formulation:– Explicit, mathematical eqn’s
• Systems equations• State-space equations
Conceptualmodel
Mathematicalmodel
Simulation model
Analyticaloutput
Numerical/simulation
output
The “process model”
Observedquantities
(data)
“Compare”
Unobservedor latent
quantities
“Predict”
Unobservedquantities
(parameters)
Outputs
Observedquantities
(driving variables)
Inputs
Types of Process ModelsTypes 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:Upcoming Example:Soil Carbon Cycle ModelSoil 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
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.
Example Process ModelExample Process Model
Simplifiedsystems diagramof the soilcarbon cycle ina temperateforest
Pools or state variables
Flows of carbon
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.
Model FormulationModel Formulation
( ) ( ) ( )d
t t u tdt
X AX B 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 FormulationModel Formulation
( ) ( ) ( )d
t t u tdt
X AX B
( )
( )
( )
( )
( )
X
X
AX
B
t
dt
Expected dttprocess
u t
u t
Observable(data)
How to Couple Data & Process How to Couple Data & Process Models?Models?
– Hierarchical Bayesian approach
( , , | )
( | , ) ( | ) ( , )
D P
D P D P
P Process Data
P Data Process P Process P
Observed data
Latent (or true) process Data parameters
Process parametersUnknown quantities
Posterior
Likelihood Probabilistic process modelPrior(s)
( | , ) :DP Data ProcessData = Latent process + observation error
( | ) :PP ProcessLatent process = Expected process + process error
Data model (likelihood)
Probabilistic process model
OutlineOutline• 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 ProcessesEcosystem Processes
Emphasis on aboveground
What about belowground?
NN
HH2200
HH2200
HH2200
CCCC
NN
PP
Biogeochemical CyclesBiogeochemical Cycles
NN
HH2200
HH2200
HH2200
CCCC
NN
PP
Biogeochemical CyclesBiogeochemical Cycles
Belowground system is critical
Tightly linked to aboveground system
BelowgroundBelowground• Little info• Difficult to measure• Aboveground measurements (helpful but limited)
Aboveground Aboveground • Lots of info• Easy to measure
Outstanding issuesOutstanding 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
Belowground “Issues”Belowground “Issues”
Motivating Questions: Soil Carbon Cycle 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 pulseprecipitationaffect sourcesof respiredCO2?
• Implications ofclimate changefor desert soilcarbon cycling?
Integrative ApproachIntegrative 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 TodayExamples 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
Objectives:Objectives: 1.1. Identify soil & microbial processes affecting Identify soil & microbial processes affecting
decompositiondecomposition
2.2. Learn how vegetation (i.e., microsite) controls these Learn how vegetation (i.e., microsite) controls these processesprocesses
Ex 1: Soil organic matter Ex 1: Soil organic matter decompositiondecomposition
Experimental DesignExperimental Design
Mesquite shrublandin southern Arizona
Microsite types:1. bare ground2. grass3. small mesquite4. big mesquite
Bare ground Grass Small mesquite Big mesquite
3 cores (reps)
...
8 depths (layers)
...
Add water
Add sugar + water
Incubate at 25 oC
CO2
CO2
CO2 Measure CO2 efflux(soil respiration rate)at 24 & 48 hours
Experimental DesignExperimental Design
...
8 depths (layers)
...
Add water
Add sugar + water
Incubate at 25 oC
CO2
CO2
CO2 Measure CO2 efflux(soil respiration rate)at 24 & 48 hours
Measure:Microbial biomassSoil organic carbonSoil nitrogen
Experimental DesignExperimental Design
• Full-factorial design:
• Microsite• 4 levels: bare, grass, small mesq, big mesq
• Soil layer• 8 levels: 0-2, 2-5, ..., 40-50 cm
• Substrate addition type• 2 levels: water only, sugar + water
• Incubation time• 2 levels: 24, 48 hrs
• Soil core or rep• 3 cores per microsite
• Stochastic data:
• Soil respiration rate• N = 359 (25 missing)
• Microbial biomass• N = 18 (14 missing)
• Soil organic carbon• N = 89 (7 missing)
Design & Data OverviewDesign & Data Overview
Some DataSome Data
Soil d
ep
th
microbesmicrobes soil Csoil C COCO22 flux flux
?? ?? datadata
Estimate microbial respiration (decomposition) parameters (i.e.,
process parameters)
Carbon substrate Micro
bial
biom
assR
esp
irati
on
Analysis ObjectivesAnalysis Objectives
biomass&
activity
Estimate microbial respiration (decomposition) parameters (i.e.,
process parameters)
Carbon substrate Micro
bial
biom
assR
esp
irati
on
Microbial biomass (B)
Resp
irati
on (
R)
Saturating carbon (C)
Low C
Michaelis-Menton type model:
Assume Ac related to “substrate quality”:
Ab
Ac
0 1max 0,
Ab B Ac CR
Ab B Ac CAc c c N
0 1max 0,Ac c c N
microbial “base-line” metabolic rate
microbial carbon-use efficiency
Process Model: Soil RespirationProcess Model: Soil Respiration
• Full-factorial design:
• Microsite• Soil layer• Substrate addition type• Incubation time• Soil core or rep
• Stochastic data:
• Soil respiration rate• Microbial biomass• Soil organic carbon
Soi
l dep
th
microbesmicrobes soil Csoil C COCO22
?? ?? datadata
NN
fixedfixed
B C R N
• Things to consider:
• Multiple data types• Nonlinear model• Missing data• Experimental design
0 1max 0,
Ab B Ac CR
Ab B Ac CAc c c N
Data-Model IntegrationData-Model Integration
somedata
somedata
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
1. Let LR = log(R)
2. For microsite m, soil depth d, soil core r, substrate-addition type s, and time period t:
Observed rate Mean (“truth”)(latent process)
Observationprecision
(= 1/variance)
Data Model (Likelihood)Data Model (Likelihood)
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
1. Now, for the covariates...
2. For microsite m, soil depth d, and soil core r:
3. Note: the likelihoods are for both the observed and missing data
Observed Mean (“truth”)(latent process)
Observation precision(= 1/variance)
{ , , } { , }
{ , , } { , }
~ ,
~ ,
md r C md C
md r B md B
C Normal
B Normal
Data Model (Likelihood)Data Model (Likelihood)
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
Likelihood components
Data parameters
Latent processes
{ , , , , } { , , , }
{ , , } { , }
{ , , } { , }
~ ,
~ ,
~ ,
md r s t LR md r s LR
md r C md C
md r B md B
LR Normal
C Normal
B Normal
, ,D LR C B
{ , , , } { , } { , }, ,LR md r s C md B mdLatent processes
Data Model (Likelihood)Data Model (Likelihood)
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
Latent processes
Deterministic model for soil microbes & carbon contents
{ , , , } { , } { , }, ,LR md r s C md B mdLatent processes
*{ , } { , } { }
*{ , } { , } { }
C md md m
B md md m
c C
b B
Stochastic model for latent respiration
{ , , , } { , , }~ . ,LRLR md r s LR md sNormal
Probabilistic Process ModelProbabilistic Process Model
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
Specify expected process: Michaelis-Menten (process) model
{ , , , } { , , }~ . ,LRLR md r s LR md sNormal
{ , } { , } { , }
{ , } { , } { , }{ , , }
{ , }
water only.
sugar + water
B md md C md
B md md C mdLR md s
B md
Ab Acif s
Ab Ac
Ab if s
{ , } 0 1 { , }max 0,md mdAc c c N
Stochastic model for latent respiration
Microbial biomass (B)
Res
pir
atio
n (R
)
Saturating carbon (C)
Low C
Probabilistic Process ModelProbabilistic Process Model
Process components
Process parameters
* *{ , } { , } { } { } 0 1, , , , , , ,
LRP md md m mc b C B Abc c
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
{ , , , } { , , }
{ , } { , } { , }
{ , } { , } { , }{ , , }
{ , }
{ , } 0 1 { , }
*{ , } { , } { }
*{ , } { , } { }
~ . ,
.
max 0,
LRLR md r s LR md s
B md md C md
B md md C mdLR md s
M md
md md
C md md m
B md md m
Normal
Ab Acwater
Ab Ac
Ab sugar
Ac c c N
c C
b B
Probabilistic Process ModelProbabilistic Process Model
Data parameters
Process parameters
* *{ , } { , } { } { } 0 1, , , , , , ,
LRP md md m mc b C B Abc c
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
, ,D LR C B
Conjugate, relatively non-informative priors for precision terms
, , , ~ 0.01,0.001LRLR C B Gamma
Parameter Model (Priors)Parameter Model (Priors)
Data parameters
Process parameters
* *{ , } { , } { } { } 0 1, , , , , , ,
LRP md md m mc b C B Abc c
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
, ,D LR C B
Non-informative Dirichlet priors for relative distributions of microbes and carbon
{ ,.} { ,.}, ~ 1,1,1,...,1m mc b Dirichlet
Multivariate version of the beta distribution(with all parameters set to 1: multidimensional uniform)
Parameter Model (Priors)Parameter Model (Priors)
Data parameters
Process parameters
* *{ , } { , } { } { } 0 1, , , , , , ,
LRP md md m mc b C B Abc c
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
, ,D LR C B
Relatively non-informative (diffuse) normal priors for the rest:
* *0 1 { } { }, ,ln ,ln ,ln ~ 0,0.0001m mc c C B Ab Normal
Parameter Model (Priors)Parameter Model (Priors)
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
4 8 3 2 2 2
{ , , , , } { , , , }1 1 1 1 1
4 8 3 2 2
{ , , } { , } { , , } { , }1 1 1
exp2 2
exp exp2 2 2 2
ex2
( | , )
LR
LR LRmd r s t LR md r s
m d r s t
C C B Bmd r C md md r B md
m d r
D
LR
C B
P Data Process
0.0010.00
4 8 3 2 2
{ , , , } { , , }1 1 1
10.99 0.001 0.99 0.99 0.001 0.99
20 1
1
e e e e
0.0001 0.0001 0.
p
0001 0.0001exp 0 exp
.
02 2 2 2
2
CLR B LR
LR
LR
LR md r s LR md sm
C B
d r s
LR
c c
2
2
4 2 2* *{ } { }
1
0.0001 0.0001exp 0
2 2
0.0001 0.0001 0.0001 0.0001exp 0 exp 0
2 2 2 2m mm
Ab
B C
The PosteriorThe Posterior
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
4 8 3 2 2 2
{ , , , , } { , , , }1 1 1 1 1
4 8 3 2 2
{ , , } { , } { , , } { , }1 1 1
exp2 2
exp exp2 2 2 2
ex2
( | , )
LR
LR LRmd r s t LR md r s
m d r s t
C C B Bmd r C md md r B md
m d r
D
LR
C B
P Data Process
0.0010.00
4 8 3 2 2
{ , , , } { , , }1 1 1
10.99 0.001 0.99 0.99 0.001 0.99
20 1
1
e e e e
0.0001 0.0001 0.
p
0001 0.0001exp 0 exp
.
02 2 2 2
2
CLR B LR
LR
LR
LR md r s LR md sm
C B
d r s
LR
c c
2
2
4 2 2* *{ } { }
1
0.0001 0.0001exp 0
2 2
0.0001 0.0001 0.0001 0.0001exp 0 exp 0
2 2 2 2m mm
Ab
B C
No analytical solution for the joint posterior distribution
No analytical solution for most of the marginal distributions
Approximate the posterior: Markov chain Monte Carlo methods,implemented in WinBUGS
The PosteriorThe Posterior
Model Implementation: Model Implementation: WinBUGSWinBUGS
Model Goodness-of-fitModel Goodness-of-fit
[1,1]
[1,2]
[1,3]
[1,4]
box plot: parms.m[1,]
1.00E+3
2.00E+3
3.00E+3
4.00E+3
5.00E+3
[2,1]
[2,2][2,3]
[2,4]
box plot: parms.m[2,]
0.0
5.0
10.0
C* (total soil carbon, g C/m2) B* (microbial biomass, g dw/m2)
Bare Bigmesq.
Med.Mesq.
Grass Bare Bigmesq.
Med.Mesq.
Grass
Example ResultsExample Results
Example ResultsExample Results
[5,1,1][5,1,2]
[5,1,3]
[5,1,4]
[5,1,5]
[5,1,6]
[5,1,7] [5,1,8]
box plot: parms.md[5,1,]
0.0
0.05
0.1
0.15
0.2
[5,2,1]
[5,2,2]
[5,2,3]
[5,2,4]
[5,2,5]
[5,2,6][5,2,7]
[5,2,8]
box plot: parms.md[5,2,]
0.0
0.05
0.1
0.15
0.2
Bare ground Big mesquite
Soil depth (or layer)
Surface Deep Surface Deep
Rela
tive a
mou
nt
of
mic
rob
ial b
iom
ass
Sensitivity to Data SourcesSensitivity to Data Sources
• 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 pulseprecipitationaffect sourcesof respiredCO2?
• Multiple datasources
• lots• limited
Ex 2: Deconvolution of Soil Ex 2: Deconvolution of Soil RespirationRespiration
data
datadata
data
datadata
data
data
The Field SitesThe Field Sites
Sonoran Desert
San Pedro River Basin
Santa Rita Experimental Range
Stable Isotope TracersStable Isotope Tracers
CO2
CO2
1212CC
1212CC
1212CC1313CC
Source isotope
signatures
Respired CO2
signature
Important data source:facilitates “partitioning”
stochastic data Literature data
Data Source ExamplesData Source Examples
Datasets: field/lab pubs
Soil Isotopes (δ13CTot)(automated chambers
& Keeling plots)
Soil CO2 flux(manual chambers)
Pool Isotopes (δ13Ci)(roots, soil, litter;
Keeling plots)
Soil CO2 flux(automated chambers)
Root respiration(in situ gas exchange)
Root distributions(arid systems,
different functionaltypes)
Soil carbon(arid systems;
total C)Root respiration
(arid systems,different functional
types)
Microbial mass(arid systems;
total mass)
Root mass(arid systems;
total mass)
Litter(arid systems; total mass,
carbon, microbes)
Soil temp & water(automated,
multiple locations,many depths)
covariate data
Soil samples(carbon content,C:N, root mass)
Soil incubations(root-free,
carbon substrate,microbial mass,
heterotrophic activity)
Potential sensor network data
Example DataExample Data
Day-1 0 2 6 14
Resp
iratio
n (
mol /
m2 /
s)
0
1
2
3
4
5
6Pre-monsoon Dry MonsoonWet Monsoon
Santa Rita pulse experiment
Res
pira
tion
(m
ol /
m2
/ s)
San Pedro automated flux measurements
San Pedro incubation experiment
-27
-25
-23
-21
-19
-17
-15
-13
-2 0 2 4 6 8 10 12 14 16
Day
d13C
of
resp
ired
CO
2 (
o/ o
o)
Santa Rita pulse experiment – d13C
Hierarchical Bayesian Model:Hierarchical Bayesian Model:Deconvolution ApproachDeconvolution Approach
• Integrate multiple sources of Integrate multiple sources of informationinformation
• Diverse data sources
• Different temporal & spatial scales
• Literature information
• Lab & field studies
• Detailed flux modelsDetailed flux models• Respiration rates by source type & soil depth
• Dynamic models
• Mechanistic isotope mixing modelsMechanistic isotope mixing models• Multiple sources
stochastic data Literature data
Data Source ExamplesData Source Examples
( | ) ( | ) ( )j j jP Data L Data P
Soil Isotopes (δ13CTot)(automated chambers
& Keeling plots)
Soil CO2 flux(manual chambers)
Pool Isotopes (δ13Ci)(roots, soil, litter;
Keeling plots)
Soil CO2 flux(automated chambers)
Root respiration(in situ gas exchange)
Root distributions(arid systems,
different functionaltypes)
Soil carbon(arid systems;
total C)Root respiration
(arid systems,different functional
types)
Microbial mass(arid systems;
total mass)
Root mass(arid systems;
total mass)
Litter(arid systems; total mass,
carbon, microbes)
Soil temp & water(automated,
multiple locations,many depths)
covariate data
Soil samples(carbon content,C:N, root mass)
Soil incubations(root-free,
carbon substrate,microbial mass,
heterotrophic activity)
Bayesian DeconvolutionBayesian Deconvolution
d 13 ( ), ( ), ( , ), ( , ), ( , )Obs ObsTot Tot iData C t R t SWC z t T z t M z t
The Hierarchical Bayesian ModelThe Hierarchical Bayesian Model
Likelihood of data
(isotopes & soil flux)
d d
113 2
2
3 ( )
( )
( )~ ,
( )~ ,
ObsTot CTot
ObsTo Tot t R
C t No
R t N
C
R to
t
Latent processes: from isotope mixing model &
flux models
Functions of parameters
Some Likelihood ComponentsSome Likelihood Components
Define process models…
Observations(data)
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
The Deconvolution ProblemThe Deconvolution Problem
Isotope mixing model(multiple sources &
depths)
Relative contributions
(by source & depth)
Total flux(at soil
surface)
Flux model(source- & depth-
specific)
Mass profiles(substrate, microbes,
roots)
(Q10 Function, Energy of Activation)
( , )
( , )( )
ii
Tot
r z tp z t
R t
1 0
( ) ( , )source BN
Tot ii
R t r z t dz
/ /( , )i known measured estimatedM z t
????
d d
13 13
1 0
( ) ( , ) ( , )source BN
Tot i ii
C t C z t p z t dz
( , ) , ( , ), ( , ), ( , )i i ir z t f SWC z t T z t M z t
Contributions by source (i ) and depth (z )? Temporal variability?
Source-specific respiration? Spatial & temporal variability?????
????
Theory & Process ModelsTheory & Process Models
From previous “incubation/decomposition” study (Ex 1)
What is i?(source-specific
parameters)
The Deconvolution ProblemThe Deconvolution ProblemObjectivesObjectives
Flux model(source- & depth-
specific)
( , ) , ( , ), ( , ), ( , )i i ir z t f SWC z t T z t M z tCovariate data
( , )
( , ) ( )
( ) ( , )
i i
i Tot
Tot i
r z t
r z t R t
R t p z t
Total soil flux
Contributions
How to estimate How to estimate ii??
Component fluxes
Bayesian DeconvolutionBayesian DeconvolutionThe Parameter Model (Priors)The Parameter Model (Priors)
Example: Example: Lloyd & Taylor (1994) model
( , ) , ( , ), ( , ), ( , )
1 1( , ) ( , ) exp
( , )
i i i
i i oo o
r z t f SWC z t T z t M z t
r z t r z t ET T z t T
Informative priors for EEoo and TToo:
304 308 312 316 215 220 225 230 235 240
~ 308.56,2Eo No ~ 227.13,10To No
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
ImplementationImplementation
• Markov chain Monte Carlo (MCMC)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
• WinBUGSWinBUGS
Soil Temperature
Day of Year
190 200 210 220 230 240 250 260 270
Soi
l T (
oC
)
15
20
25
30
Soil Moisture
VW
C (
v/v)
0.04
0.08
0.12
Proportional Contribution of Respiration SourcesP
ropo
rtio
nal C
ontr
ibut
ion
0.000
0.005
0.010
0.015
0.020
0.025Heterotrophs (0-5 cm)Grass Roots (5-50 cm)Mesquite Roots (5-50 cm)
Results: Dynamic Source ContributionsResults: Dynamic Source ContributionsSan Pedro Site – Monsoon SeasonSan Pedro Site – Monsoon Season
Zoom-inZoom-in
Results: Root Respiration ResponsesResults: Root Respiration ResponsesZoom-in: July 27 – August 4Zoom-in: July 27 – August 4
05
1015202530
205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220
0.0
1.0
2.0
3.0
4.0
5.0
209 210 211 212 213 214 215 216 217
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Date
Tota
l ro
ot
resp
irati
on
(um
ol m
-2 s
-1)
Soil w
ate
r (v/v
)
Rain
(m
m)
Mesquite (C3 shrub)
Sacaton (C4 grass)
Soil water
Jul 27 Aug 4
Results: Contributions Vary by DepthResults: Contributions Vary by Depth
0.0
1.0
2.0
3.0
4.0
5.0
209 210 211 212 213 214 215 216 217
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Date
Tota
l ro
ot
resp
irati
on
(um
ol m
-2 s
-1)
Soil w
ate
r (v/v
)
0.00 0.10 0.20 0.00 0.10 0.20 0.00 0.10 0.20
Day 210 Day 213 Day 216
0-5
5-10
10-15
15-20
20-25
25-30
30-40
40-50
Dep
th (
cm)
0-5
5-10
10-15
15-20
20-25
25-30
30-40
40-50
0-5
5-10
10-15
15-20
20-25
25-30
30-40
40-50
Relative contributions by depth
Mesquite (C3 shrub)
Sacaton (C4 grass)
Soil water
SummarySummary
• Sources of soil COSources of soil CO22 efflux efflux• Mesquite (shrub): major contributor, stable source
• Sacton (grass): minor contributor, threshold response
• Microbes (bare): minor contributor, coupled to pulses
• Deconvolution & data-model Deconvolution & data-model integrationintegration
• Soil depth (including litter)
• By species or functional groups
• Quantify spatial & temporal variability
• Incorporate environmental drivers
• Implications & applicationsImplications & applications• Identify mechanisms
• Predictions & forward modeling
OutlineOutline• 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 Implications for Sensor NetworksNetworks– 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)?
Photo by Travis HuxmanPhoto by Travis HuxmanMonsoon flood, San Pedro River Basin; Sonoran desertMonsoon flood, San Pedro River Basin; Sonoran desert
Questions?Questions?
Soil Temperature
Day of Year
190 200 210 220 230 240 250 260 270
Soi
l T (
oC
)
15
20
25
30
Soil Moisture
VW
C (
v/v)
0.04
0.08
0.12
Proportional Contribution of Respiration Sources
Pro
port
iona
l Con
trib
utio
n
0.000
0.005
0.010
0.015
0.020
0.025Heterotrophs (0-5 cm)Grass Roots (5-50 cm)Mesquite Roots (5-50 cm)
Results: Dynamic Source ContributionsResults: Dynamic Source Contributions
Example WinBUGS OutputExample WinBUGS Outputparms[1] chains 1:2
iteration
1 2000 4000 6000
300.0
305.0
310.0
315.0
EO
parms[2] chains 1:2
iteration
1 2000 4000 6000
190.0
200.0
210.0
220.0
230.0
TO
Posteriorstatisticsparameter node mean sd 2.5% median 97.5% sampleEo parms[1] 307.7 1.205 305.2 307.7 309.9 1000To parms[2] 201.1 2.335 195.8 201.0 205.1 1000
....contribution pSc[1,1,1] 0.00145 0.002904 -0.03455 6.34E-4 0.03565 1000by pSc[1,1,2] 0.8854 0.009694 0.8433 0.8853 0.9288 1000[date, pSc[1,1,3] 0.1132 0.009998 0.06992 0.1131 0.1558 1000plot, pSc[1,2,1] 0.00158 0.003157 -0.03353 6.685E-4 0.03607 1000source] pSc[1,2,2] 0.9721 0.008952 0.9297 0.9725 1.013 1000
pSc[1,2,3] 0.02636 0.008649 -0.01523 0.02622 0.0654 1000....
The Inverse ProblemThe Inverse ProblemPlant water uptake Soil respiration
Isotope mixing model
Fractional contributions
Total flux
Fluxmodel
Substrate orroot profiles
( , )
( , )( )Tot
U z tq z t
U t
1 1 2 2( ) ( , ) (1 ) ( , )RA z Ga Ga
0
( ) ( , )B
TotU t U z t dz
(Q10 Function, Energy of Activation)
( , ) ( , )( , )
( )i i
iTot
r z t M z tp z t
R t
1 0
( ) ( , ) ( , )source BN
Tot i ii
R t r z t M z t dz
/ ?( , )i known measuredM z t
????
????
d d
d d
0
18 18
0
( ) ( ) ( , )
( ) ( ) ( , )
B
stem
B
stem
D t D z q z t dz
O t O z q z t dz
d d
13 13
1 0
( ) ( , )source BN
Tot i ii
C t C p z t dz
( , ) ( ) ln ( )
( , ) ( , ) ( ) ( )root root
U z t RA z a RA z
z t k z t t k t
( , ) , ( , ), ( , )i ir z t f SWC z t T z t
The Inverse ProblemThe Inverse Problem
Isotope mixing model(multiple sources &
depths)
Relative contributions
(by source & depth)
Total flux(at soil
surface)
Flux model(source- & depth-
specific)
Mass profiles(substrate, microbes,
roots)
(Q10 Function, Energy of Activation)
( , ) ( , )( , )
( )i i
iTot
r z t M z tp z t
R t
1 0
( ) ( , ) ( , )source BN
Tot i ii
R t r z t M z t dz
/ ?( , )i known measuredM z t
????
d d
13 13
1 0
( ) ( , ) ( , )source BN
Tot i ii
C t C z t p z t dz
( , ) , ( , ), ( , )i ir z t f SWC z t T z t
Contributions by source (i ) and depth (z )? Temporal variability?
????Source-specific respiration? Spatial & temporal variability?
What is i?(source-specific
parameters)
Likelihood of data
(isotopes & soil flux)
d d
113 2
2
3 ( )
( )
( )~ ,
( )~ ,
ObsTot CTot
ObsTo Tot t R
C t No
R t N
C
R to
t
From isotope mixing model & flux models
The Deconvolution ProblemThe Deconvolution ProblemData-Model IntegrationData-Model Integration
Flux model(source- & depth-
specific)
( , ) , ( , ), ( , ), ( , )i i ir z t f SWC z t T z t M z tCovariate data
( , ) ( )
( ) ( , )i Tot
Tot i
r z t R t
R t p z t
Total soil flux
Contributions
Depend on
i
stochastic data Literature data
Data Source ExamplesData Source Examples
( | ) ( | ) ( )j j jP Data P Data P
Soil Isotopes (δ13CTot)(automated chambers
& Keeling plots)
Soil CO2 flux(manual chambers)
Pool Isotopes (δ13Ci)(roots, soil, litter;
Keeling plots)
Soil CO2 flux(automated chambers)
Root respiration(in situ gas exchange)
Root distributions(arid systems,
different functionaltypes)
Soil carbon(arid systems;
total C)Root respiration
(arid systems,different functional
types)
Microbial mass(arid systems;
total mass)
Root mass(arid systems;
total mass)
Litter(arid systems; total mass,
carbon, microbes)
Soil temp & water(automated,
multiple locations,many depths)
covariate data
Soil samples(carbon content,C:N, root mass)
Soil incubations(root-free,
carbon substrate,microbial mass,
heterotrophic activity)
The Deconvolution ProblemThe Deconvolution ProblemPlant water uptake Soil respiration
Isotope mixing model
Fractional contributions
Total flux
Fluxmodel
Substrate orroot profiles
( , )
( , )( )Tot
U z tq z t
U t
1 1 2 2( ) ( , ) (1 ) ( , )RA z Ga Ga
0
( ) ( , )B
TotU t U z t dz
(Q10 Function, Energy of Activation)
( , ) ( , )( , )
( )i i
iTot
r z t M z tp z t
R t
1 0
( ) ( , ) ( , )source BN
Tot i ii
R t r z t M z t dz
/ ?( , )i known measuredM z t
????
????
d d
d d
0
18 18
0
( ) ( ) ( , )
( ) ( ) ( , )
B
stem
B
stem
D t D z q z t dz
O t O z q z t dz
d d
13 13
1 0
( ) ( , )source BN
Tot i ii
C t C p z t dz
( , ) ( ) ln ( )
( , ) ( , ) ( ) ( )root root
U z t RA z a RA z
z t k z t t k t
( , ) , ( , ), ( , )i ir z t f SWC z t T z t
Plant water uptake Soil respiration
1 1 2 2( ) ( , ) (1 ) ( , )
( , )
( , )
RA z Ga Ga
U z t
q z t
( , ) , ( , ), ( , )
( )
( , )
i i
Tot
i
r z t f SWC z t T z t
R t
p z t
What areω, 1, 1, 2, 2?
What isi?
Likelihood of data
d d
113 2
2
3 ( )
( )
( )~ ,
( )~ ,
ObsTot CTot
ObsTo Tot t R
C t No
R t N
C
R to
t
d
d
d
d
2
18 8 21
( )( )~ ,
( ))~ ,(
Obsstem D
Obsste
stem
stemm O
D t
O t
D t No
O t No
From isotope mixing model & flux model
The Deconvolution ProblemThe Deconvolution Problem
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
6 6
1 1
i ii i
i i i i
i ii i
i i
a cdPA R S E
dt c
dS a Ed S
dt c
d
d
Process modelsProcess models
State ID TABLENM "PLT_CN" PHYSCLCD "STATECD" "CYCLE""SUBCYCLE""UNITCD""COUNTYCD" "PLOT" "SUBP" "TREE" "CONDID"VA 229 TREE 23854928010661.00 21 51 3 2 1 1 13 2 1 1VA 407 TREE 23958646010661.00 21 51 3 4 1 1 19 1 5 1VA 408 TREE 23958646010661.00 21 51 3 4 1 1 19 1 6 1VA 1059 TREE 23958861010661.00 21 51 3 4 1 1 54 2 6 2VA 6768 TREE 23997965010661.00 22 51 3 5 2 7 9 2 4 1VA 7019 TREE 23906005010661.00 22 51 3 3 2 7 15 2 9 1VA 7111 TREE 23857072010661.00 22 51 3 2 2 7 16 2 7 1VA 8105 TREE 23805545010661.00 22 51 3 1 2 7 39 2 1 1VA 8539 TREE 23906968010661.00 22 51 3 3 3 9 9 1 3 1VA 12808 TREE 23807296010661.00 23 51 3 1 4 15 29 3 8 1VA 12810 TREE 23807296010661.00 23 51 3 1 4 15 29 3 10 1VA 13315 TREE 23858135010661.00 22 51 3 2 4 15 54 1 1 1VA 19399 TREE 23809332010661.00 22 51 3 1 2 19 23 3 4 1VA 19445 TREE 23909859010661.00 22 51 3 3 2 19 26 1 8 1VA 22050 TREE 23861227010661.00 23 51 3 2 5 21 8 2 5 1VA 22053 TREE 23861227010661.00 23 51 3 2 5 21 8 2 2 1VA 22060 TREE 23861227010661.00 23 51 3 2 5 21 8 1 4 1VA 22137 TREE 23910676010661.00 23 51 3 3 5 21 12 2 6 1VA 23519 TREE 23910590010661.00 23 51 3 3 5 21 39 2 5 1VA 26415 TREE 23911957010661.00 22 51 3 3 1 25 1 2 8 2VA 26783 TREE 23863007010661.00 21 51 3 2 1 25 7 2 3 1VA 28623 TREE 23862849010661.00 22 51 3 2 1 25 41 2 17 1VA 29299 TREE 24002221010661.00 24 51 3 5 1 25 54 3 6 1VA 29320 TREE 24002221010661.00 24 51 3 5 1 25 54 4 14 1VA 30129 TREE 23862787010661.00 22 51 3 2 1 25 69 3 6 1VA 30139 TREE 23862787010661.00 22 51 3 2 1 25 69 3 11 1VA 32119 TREE 23913201010661.00 23 51 3 3 5 27 42 3 2 1VA 34017 TREE 23813210010661.00 22 51 3 1 2 29 23 4 9 1VA 34329 TREE 23913514010661.00 22 51 3 3 2 29 29 1 4 1VA 34716 TREE 23914198010661.00 22 51 3 3 2 29 35 1 20 1VA 35041 TREE 24003030010661.00 22 51 3 5 2 29 41 3 2 1VA 36375 TREE 23813999010661.00 22 51 3 1 2 29 68 2 5 1VA 36410 TREE 23813999010661.00 22 51 3 1 2 29 68 3 7 1VA 36411 TREE 23813999010661.00 22 51 3 1 2 29 68 3 8 1
DataData
P
(|
X )
Statistical toolsStatistical toolsdata-model integrationdata-model integration
( | )
( | ) ( )( | ) ( )
P X
P X PP X P d
Key componentsKey components
The Process ModelThe Process Model
• Conceptual models:– Systems diagrams– Graphical models
• Model formulation:– Explicit, mathematical eqn’s
• Systems equations• State-space equations
“Compare”Observations
of real systemConceptual
modelMathematical
model
Simulation model
Analyticaloutput
Numerical/simulation
output
Observationaldata
Examples Presented TodayExamples 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.
( , , | ) ( | , ) ( | ) ( , ) D P D P D PP Process Data P Data Process P Process P
Assuming conditional independence,likelihood of all data is:
4 8 3 2 2 2
{ , , , , } { , , , }1 1 1 1 1
4 8 3 2 2
{ , , } { , } { , , } { , }1 1 1
( | , ) exp2 2
exp exp2 2 2 2
LR LRD md r s t LR md r s
m d r s t
C C B Bmd r C md md r B md
m d r
P Data Process LR
C B
Likelihood components
{ , , , , } { , , , }
{ , , } { , }
{ , , } { , }
~ ,
~ ,
~ ,
md r s t LR md r s LR
md r C md C
md r B md B
LR Normal
C Normal
B Normal
Data Model (Likelihood)Data Model (Likelihood)
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