1. Data-model integration: Examples from belowground ecosystem ecology
Kiona Ogle
University of Wyoming
Departments of Botany & Statistics

2. 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.

4. 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

5. 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

6. 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

7. 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
Data parameters
Process parameters
Latent (or true) process
Posterior
Likelihood
Probabilistic process model
Prior(s)

8. 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

9. Hierarchical Bayesian Model
Observed data
Latent (or true) process
Data parameters
Process parameters
Unknown quantities
Posterior
Likelihood
Probabilistic process model
Prior(s)
Data model (likelihood)
Probabilistic process model
The “process model”

10. The Process Model Conceptual model: Model formulation:
Systems diagrams
Graphical models
Model formulation:
Explicit, mathematical eqn’s
Systems equations
State-space equations
Inputs
Outputs
“Compare”
Unobserved
quantities
(parameters)
Conceptual
model
Mathematical
model
Observed
quantities
(data)
Analytical
output
Observed
quantities
(driving variables)
Numerical/
simulation
output
“Predict”
Simulation model
Unobserved
or latent
quantities
The “process model”

11. 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.

12. Soil Carbon Cycle Model
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

13. Example Process Model Simplified systems diagram of the soil
Pools or state variables
Simplified
systems diagram
of the soil
carbon cycle in
a temperate
forest
Flows of carbon
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.

14. Model Formulation
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.

15. Model Formulation Observable (data)
Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007.

16. How to Couple Data & Process Models?
Hierarchical Bayesian approach
Unknown quantities
Observed data
Data parameters
Process parameters
Latent (or true) process
Posterior
Likelihood
Probabilistic process model
Prior(s)
Data model (likelihood)
Probabilistic process model

17. 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

18. Ecosystem Processes
Emphasis on aboveground
What about belowground?

19. Biogeochemical CyclesN
H20 N
H20 C C P

20. Biogeochemical CyclesN
H20 C P
Belowground system is critical
Tightly linked to aboveground system

21. Belowground “Issues” Aboveground Belowground Outstanding issues
Lots of info
Easy to measure
Belowground
Little info
Difficult to measure
Aboveground measurements (helpful but limited)
As plant ecologists, we know quite a bit about aboveground responses of plants and ecosystems to environmental effects, which can be attributed to our ability to easily measure aboveground responses. For example, we can measure leaf-level photosynthesis and transpiration, we can monitor the phenology of leaf production and fruiting, we can quantify shifts in species composition, we can install sapflow gauges on plant stems and continuously measure whole-plant transpiration.
But, much less is known about belowground processes, yet they are critical to understanding ecosystem-level behavior. For example, soils, soil microfauna, and plant roots are critical players in ecosystem carbon and water fluxes. Now, we can use aboveground measurements to help infer belowground processes to some extent. For example, we can use soil respiration chambers to measure soil CO2 efflux, we can install eddy flux towers to measure net ecosystem carbon and water extreme at relatively large scales. Currenlty these data only provide limited insight into belowground dynamics, but they have the potential to provide much greater insight if complimented by other data sources and models.
In summary, two challenges facing plant and ecosystem ecologists are:
Partitioning the different components contributing to ecosystem carbon and water fluxes
And, indentify key belowground processes and their effects on ecosystem fluxes.
Outstanding issues
Partitioning above- & belowground
Quantifying & partitioning belowground
Implications for ecosystem function
Examples: arid & semiarid systems
Figure from Kieft et al. (1998) Ecology 79:

22. 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?
Some questions related to ecosystem carbon dynamics, and specifically, soil respiration, include:
I will address these questions in this talk, but I want to note that this project emerged out of conversations with Jessica Cable and Travis Huxman and use data from Jessie’s disseration work.

23. Integrative Approach Diverse data sources Process-based models
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

24. 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

25. Ex 1: Soil organic matter decomposition
Objectives:
Identify soil & microbial processes affecting decomposition
Learn how vegetation (i.e., microsite) controls these processes

26. Experimental Design Mesquite shrubland in southern Arizona Bare ground
Microsite types:
bare ground
grass
small mesquite
big mesquite
Bare ground
Grass
Small mesquite
Big mesquite
3 cores (reps)

27. Experimental Design ... ... CO2 Add water Add sugar + water
Incubate at 25 oC
CO2
Measure CO2 efflux
(soil respiration rate)
at 24 & 48 hours
CO2
8 depths (layers)
CO2

28. Experimental Design ... ... CO2 Add water Add sugar + water Measure:
Microbial biomass
Soil organic carbon
Soil nitrogen
Incubate at 25 oC
CO2
Measure CO2 efflux
(soil respiration rate)
at 24 & 48 hours
CO2
8 depths (layers)
CO2

29. Design & Data Overview Full-factorial design: Microsite Soil layer
4 levels: bare, grass, small mesq, big mesq
Soil layer
8 levels: 0-2, 2-5, ..., 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)

30. Some Data

31. ? ? Analysis Objectives Soil depth microbes soil C CO2 flux
Estimate microbial respiration (decomposition) parameters (i.e., process parameters) ?
Soil depth ?
data
Respiration
biomass
&
activity
Microbial
biomass
Carbon substrate

32. Process Model: Soil Respiration
Estimate microbial respiration (decomposition) parameters (i.e., process parameters)
Microbial biomass (B)
Respiration (R)
Saturating carbon (C)
Low C
Respiration
Microbial
biomass
Carbon substrate
Michaelis-Menton type model:
microbial “base-line” metabolic rate
microbial carbon-use efficiency
Assume Ac related to “substrate quality”:

33. Data-Model Integration
Full-factorial design:
Microsite
Soil layer
Substrate addition type
Incubation time
Soil core or rep
B C R N
Stochastic data:
Soil respiration rate
Microbial biomass
Soil organic carbon
Things to consider:
Multiple data types
Nonlinear model
Missing data
Experimental design
some
data
some
data

34. Data Model (Likelihood) Observation precision
Let LR = log(R)
For microsite m, soil depth d, soil core r, substrate-addition type s, and time period t:
Observed rate
Mean (“truth”)
(latent process)
Observation precision
(= 1/variance)

35. Data Model (Likelihood) Observation precision
Now, for the covariates...
For microsite m, soil depth d, and soil core r:
Note: the likelihoods are for both the observed and missing data
Observed
Mean (“truth”)
(latent process)
Observation precision
(= 1/variance)

36. Data Model (Likelihood) Likelihood components
Data parameters
Latent processes

37. Probabilistic Process Model
Latent processes
Deterministic model for soil microbes & carbon contents
Stochastic model for latent respiration

38. Probabilistic Process Model
Stochastic model for latent respiration
Specify expected process: Michaelis-Menten (process) model

39. Probabilistic Process Model
Process components
Process parameters

40. Parameter Model (Priors)
Data parameters
Process parameters
Conjugate, relatively non-informative priors for precision terms

41. Parameter Model (Priors)
Data parameters
Process parameters
Non-informative Dirichlet priors for relative distributions of microbes and carbon
Multivariate version of the beta distribution
(with all parameters set to 1: multidimensional uniform)

42. Parameter Model (Priors)
Data parameters
Process parameters
Relatively non-informative (diffuse) normal priors for the rest:

43. The Posterior

44. The Posterior
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

45. Model Implementation: WinBUGS

46. Model Goodness-of-fit

47. Example Results C* (total soil carbon, g C/m2)
B* (microbial biomass, g dw/m2)
Bare
Big mesq.
Med. Mesq.
Grass
Bare
Big mesq.
Med. Mesq.
Grass

48. Relative amount of microbial biomass
Example Results
Bare ground
Big mesquite
Relative amount of microbial biomass
Surface Deep
Surface Deep
Soil depth (or layer)

49. Sensitivity to Data Sources

50. 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 of respired CO2?
Multiple data sources
lots
limited
data
data
data
data
data
data
data
Some questions related to ecosystem carbon dynamics, and specifically, soil respiration, include:
I will address these questions in this talk, but I want to note that this project emerged out of conversations with Jessica Cable and Travis Huxman and use data from Jessie’s disseration work.
data

51. The Field Sites Sonoran Desert San Pedro River Basin
Santa Rita Experimental Range
Sonoran Desert

52. Source isotope signatures
Stable Isotope Tracers
Respired CO2 signature
CO2
12C
13C
Source isotope signatures
Aspects of both of these belowground processes can be infered from stable isotope measurements made on both aboveground and belowground components.
For example, water is composed of hydrogen and oxygen, both of which occur in different isotopic forms. E.g., some water molecules may contain the heavy stable isotope of hydrogen (deuterium) or the heavy stable isotope of oxygen (O-18). The relative abundance of the heavy isotopes in soil water often varies with soil depth.
Likewise, the distribution of plant roots vary with depth, and the plant takes-up water from various depths in the soil. The amount of water taken-up from different depths depends on the soil water availity at each depth and the distribution of roots that are actively acquiring water.
Thus, water in the plant stem is derived from different soil water sources, and its isotopic composition or signature reflects the soil layers from which the roots are extracting water.
Likewise, consider soil respiration and the CO2 escaping from the soil.
Respired CO2 is derived from substrates such as decomposing organic matter or actively respiring roots, and these substrates or sources vary in their composition of light and heavy carbon.
And, the distribution of these substrates varies with depth, and their respiration rates at different depths depend on environmental conditions such at temperature and soil water content.
As for the stem isotope signature, the 13C isotope signature of the respired CO2 reflects the weighted or average signature of the different sources.
Important data source:
facilitates “partitioning”

53. (arid systems; total mass, heterotrophic activity)
Data Source Examples
stochastic data
Literature data
Pool Isotopes (δ13Ci)
(roots, soil, litter;
Keeling plots)
Soil Isotopes (δ13CTot)
(automated chambers
& Keeling plots)
Litter
(arid systems; total mass,
carbon, microbes)
Soil CO2 flux
(automated chambers)
Soil CO2 flux
(manual chambers)
Root mass
(arid systems;
total mass)
Datasets: field/lab pubs
Soil samples
(carbon content,
C:N, root mass)
Microbial mass
(arid systems;
total mass)
Soil incubations
(root-free,
carbon substrate,
microbial mass,
heterotrophic activity)
Root respiration
(in situ gas exchange)
Soil temp & water
(automated,
multiple locations,
many depths)
covariate data
Root distributions
(arid systems,
different functional
types)
Soil carbon
(arid systems;
total C)
Root respiration
(arid systems,
different functional
types)
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.
Potential sensor network data

54. Example Data Santa Rita pulse experiment
San Pedro automated flux measurements
Respiration (mmol / m2 / s)
Santa Rita pulse experiment – d13C
San Pedro incubation experiment

55. 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

56. (arid systems; total mass, heterotrophic activity)
Data Source Examples
stochastic data
Literature data
Pool Isotopes (δ13Ci)
(roots, soil, litter;
Keeling plots)
Soil Isotopes (δ13CTot)
(automated chambers
& Keeling plots)
Litter
(arid systems; total mass,
carbon, microbes)
Soil CO2 flux
(automated chambers)
Soil CO2 flux
(manual chambers)
Root mass
(arid systems;
total mass)
Soil samples
(carbon content,
C:N, root mass)
Microbial mass
(arid systems;
total mass)
Soil incubations
(root-free,
carbon substrate,
microbial mass,
heterotrophic activity)
Root respiration
(in situ gas exchange)
Soil temp & water
(automated,
multiple locations,
many depths)
covariate data
Root distributions
(arid systems,
different functional
types)
Soil carbon
(arid systems;
total C)
Root respiration
(arid systems,
different functional
types)
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.

57. Bayesian Deconvolution
The Hierarchical Bayesian Model
Some Likelihood Components
Likelihood of data
(isotopes & soil flux)
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.
Observations
(data)
Latent processes: from isotope mixing model & flux models
Functions of parameters 
Define process models…

58. The Deconvolution Problem
Theory & Process Models
Isotope mixing model
(multiple sources & depths) ??
Contributions by source (i ) and depth (z )? Temporal variability?
Relative contributions
(by source & depth) ??
Source-specific respiration? Spatial & temporal variability?
Total flux
(at soil surface)
Flux model
(source- & depth- specific)
(Q10 Function, Energy of Activation)
From previous “incubation/decomposition” study (Ex 1)
Mass profiles
(substrate, microbes, roots)

59. (source- & depth- specific) (source-specific parameters)
The Deconvolution Problem
Objectives
Flux model
(source- & depth- specific)
Covariate data
What is i?
(source-specific parameters)
 Component fluxes
 Total soil flux
 Contributions
How to estimate i?

60. Bayesian Deconvolution
The Parameter Model (Priors)
Example: Lloyd & Taylor (1994) model
Informative priors for Eo and To:
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.

61. Implementation Markov chain Monte Carlo (MCMC) WinBUGS
Sample parameters (θi ) from posterior
Posteriors for: θi’s, ri(z,t)’s, pi(z,t)’s, etc.
Means, medians, uncertainty
WinBUGS

62. Results: Dynamic Source Contributions San Pedro Site – Monsoon Season
Zoom-in

63. Results: Root Respiration Responses
Zoom-in: July 27 – August 4
Date
Total root respiration
(umol m-2 s-1)
Soil water (v/v)
Rain (mm)
Mesquite (C3 shrub)
Sacaton (C4 grass)
Soil water
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.
Jul 27
Aug 4

64. Total root respiration
Results: Contributions Vary by Depth
Date
Total root respiration
(umol m-2 s-1)
Soil water (v/v)
Mesquite (C3 shrub)
Soil water
Sacaton (C4 grass)
Day 210
Day 213
Day 216
0-5
5-10
10-15
15-20
20-25
25-30
30-40
40-50
Depth (cm)
Relative contributions by depth
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.

65. 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

66. 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

67. 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)?

68. Questions? Photo by Travis Huxman
Monsoon flood, San Pedro River Basin; Sonoran desert

71. Results: Dynamic Source Contributions

72. Example WinBUGS OutputEO TO

73. Fractional contributions Substrate or root profiles
The Inverse Problem
Plant water uptake
Soil respiration
Isotope mixing model ??
Fractional contributions
Total flux
Flux
model
(Q10 Function, Energy of Activation)
Substrate or root profiles

74. The Inverse Problem
Isotope mixing model
(multiple sources & depths) ??
Contributions by source (i ) and depth (z )? Temporal variability?
Relative contributions
(by source & depth) ??
Source-specific respiration? Spatial & temporal variability?
Total flux
(at soil surface)
Flux model
(source- & depth- specific)
(Q10 Function, Energy of Activation)
Mass profiles
(substrate, microbes, roots)

75. The Deconvolution Problem
Data-Model Integration
Flux model
(source- & depth- specific)
Covariate data
What is i?
(source-specific parameters)
 Total soil flux
 Contributions
Likelihood of data
(isotopes & soil flux)
Depend on i
From isotope mixing model & flux models

76. (arid systems; total mass, heterotrophic activity)
Data Source Examples
stochastic data
Literature data
Pool Isotopes (δ13Ci)
(roots, soil, litter;
Keeling plots)
Soil Isotopes (δ13CTot)
(automated chambers
& Keeling plots)
Litter
(arid systems; total mass,
carbon, microbes)
Soil CO2 flux
(automated chambers)
Soil CO2 flux
(manual chambers)
Root mass
(arid systems;
total mass)
Soil samples
(carbon content,
C:N, root mass)
Microbial mass
(arid systems;
total mass)
Soil incubations
(root-free,
carbon substrate,
microbial mass,
heterotrophic activity)
Root respiration
(in situ gas exchange)
Soil temp & water
(automated,
multiple locations,
many depths)
covariate data
Root distributions
(arid systems,
different functional
types)
Soil carbon
(arid systems;
total C)
Root respiration
(arid systems,
different functional
types)
I am just starting to develop a hierarchical Bayesian melding scheme for merging these data sources and the individual-based growth model.
First, lets revisit the basic Bayesian framework. That is, the Bayesian framework used probability models or statements for everything, including data, parameters, and all other unknown quantities.
What I’m ultimately interested in is the posterior distribution for the unknown quantities, such as the growth parmeters (theta), variance terms (sigma), and model outputs or predictions (O). This posterior quantifies our uncertainty in these unknowns given our observed data. And, Bayes rule says that this posterior is proportional to the likelihood of all data TIMES the prior distribution for the unknowns. So, once I know the posterior, I can use it to make inferences about the growth parameters, how they vary amoung species, and what the implications are for tree growth and survival and forest dynamics.
Now, the likelihood function is really important because it explicitly links the model, data, and parameters.
First, let’s consider the FIA inventory, which gives measurements for tree i, of species j, at time t, growing in location s. We input to the model an initial diameter for this tree, a set of species and site-specific parameter values, and some covariates (X), and the model gives a slew of predictions for this particular tree, including height, trunk diameter, biomass totals for each structural compartment, leaf area, sapwood area, and bunch of other stuff.
Note though that some of these outputs can be matched-up with FIA data and go into the likelihood, E.g., the observed height and diameter are assumed to come from a normal distribution means that are given by the model outpus. But, the other quantities are not associated with data, but we have some information about these quantities based on previous studies or the literature, and we can use this information to come-up with prior for these unobservable outputs.
Now, what I really want are parameter estimates for each species, and I want to account for spatial random effects that might influence these estimates. So, I want to estimate the posterior distribution for theta_js, and I first assume that is can be broken-up into 2 independent pieces: one is the species effect, and one is the random spatial effect.
Recall that the literature database provides species-specific parameter estimates, and this also contributes to part of the likelihood.
For the spatial effects, I will begin be using a fairly simple model that allows for spatial autocorrelation and that can accommodate the huge, spatially extensive FIA dataset. I will use a conditionally autoregressive model, which assumes that the spatial effect at location s come from a normal distribution with a mean equal to a weighted average of its neighboring locations effects.
But, ultimately, what I am really interested in are the theta_tila_j’s – i.e., the species-specific parameter estimates.

77. Fractional contributions Substrate or root profiles
The Deconvolution Problem
Plant water uptake
Soil respiration
Isotope mixing model ??
Fractional contributions
Total flux
These water uptake, soil respiration, and isotopic mixing processes can be expressed as an inverse problem.
First, note that the stem isotope signatures at time t integrate over the soil isotope signatures at depth z TIMES the contribution of depth z to water uptake from that depth. We integrate over all depths from which roots are aquiring water, and this gives us the stem signature. Likewise, the C-13 signature of the respired CO2 sums over all heterotrophic and autotrophic sources times their fractional contributions. Note that that the fractional contribution of each source is determined by taking each source’s contribution at depth z and integrating it over all depths.These equations describe the isotope mixing model, and you should notice that they are very similar to the simple linear mixing model equation, which sums over potential sources.
Note that there are certain components of the mixing model that we can measure: the isotopic signatures of the stem water, soil water at different depths, the respired CO2, and the source signatures. But, what we don’t know, and what we really want to tease-apart are the fractional contributions. I.e., the contribution of each soil layer to plant water uptake (or – the plant water sources) and the contributions of each source and different soil layers to total soil respiration. Thus, this is an inverse problem because the reponse variable that we can measure is of lower dimension or contains less information that the thing inside the integral, and we essentially need to invert this integral in order to solve for the higher dimensional variables –i.e., the fractional contributions. Thus, just as in the simple linear mixing model, we wish to estimate the p’s and q’s, but we are going to do this using a mechanistic modeling framework.
First, note that the fractional contribution of different soil layers to plant water uptake is simply the amount of water taken-up from layer z divided by the total amount of water uptake. And, the fractional contribution of different sources and depths to soil respiration is simply the repiration rate of source i at depth z times the amount or mass of source i at depth z, divided by total respiration.
Note then that total water uptake is simply determined by integrating (or summing) water uptake over all depths from which water is acquired, and total respiration is determined by summing over all sources and integrating over all depths the total respiration at of each source at each depth. Note that we can measure components of the contributions and total fluxes. E.g., we can collect soil cores and get estimates of the mass distributions, and we can use soil respiration chambers to measure CO2 flux at the soil surface. But, we can’t directly measure mass-specific respiration rate of each source at each depth, and we can’t directly measure water acquisition at all depths.
But, I assume that we can use some sort of biophysical or semi-mechanistic model to describe both U and little r. For example, here’s a simplified version of a relativley simple water uptake model that I employed. This part of the uptake model describes a driving gradient for water uptake as given be differences between the water potentials and hydraulic conductivitis of the bulk soil and at the root surface. But, water uptake also depends on the active root area for water uptake, as shown by this term that includeds RA (active root area).
Similary, we can use some sort of model to describe mass-specific respiration, and examples of such models include the commonly used Q10 function or the Energy of activity model, both of which describe the dependency of respiration on temperature. I am currently exploring different respiration models, but in all cases, I’m incorporating the effects of soil water availability, and of course temperature on respiration rates. The function is defined by a set of parameters (theta) that capture differences between sources and that reflect the sensitivity of respiration to changes in temperature or soil water.
Note that once again, we can estimate components of the flux models with field data. For, example measurements of soil water content and soil texture can be used to estimate the driving gradient for water uptake, and soil water content in addition to soil temperature can also be measured as part of the respiration model. But, you should notice, that at least for the water uptake model, we still don’t know an important component, and that is the active root area profile. THis is something that we can’t directly measure in the field.
The analogy to the respiration model are the mass or substrate profiles, but as I mentioned earlier, we can estimate these from field data. So, what to do about the active root area profile? I assume that
But, again, we can assume some sort of model to describe RA, and I’m going to assume that the active root area profile can be described by a mixutre of gamma densities. I chose a gamma mixture model because it is extremely flexible and can capture an infinite number of shapes that are consistent with rooting profiles that have been measured in the field, including unimodal, bimodal and relatively uniform profiles. This gamma mixture has 5 unknown parameters (omega, alpha1, ....).
Flux
model
(Q10 Function, Energy of Activation)
Substrate or root profiles

78. From isotope mixing model & flux model
The Deconvolution Problem
Plant water uptake
Soil respiration
What are ω, a1, m1, a2, m2?
What is
i?
Thus, the goal in both of these problems is to estimate the unknown quantities and parameters including the gamma mixture parameters omega, alpha1, mu1, alpha2, mu2, and the source- or substrate-specific respiration parameters as depicted by theta i.
Once we know these parameters, they will tell us the active root area profile, with will give up the water uptake profile, which will ultimate tell us the fractional contributions of the different soil depths to plant water uptake.
Likewise, once we know theta, then we have an estimate of the mass-specific repiration rates, which gives us total soil respiration, both of which tell use the fractional contribution of different sources and soil layers to total soil respiration.
We get estimates of these parameter by fitting field data to the isotope mixing and flux models that I described in the previous slides. In particular, a simple example is to assume that the observed stem signatures, soil CO2 signature and total soil respiration data come from normal distributions whose means are defined by the isotope mixing models and the flux models.
From isotope mixing model & flux model
Likelihood of data

79. 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

80. data-model integration
Key components
Data
I am employing an integrative approach in developing the mechanistic framework and for addressing important and interesting ecological questions.
A key element of my appraoch is the strong integration of data and models.
I incorporate as much information as possible by conducting targeted field experiments and using large databases and existing data from the literature.
I develop and employ mathematic and simulation models that explicitly translate what I see are the key ecological processes associated with a particular problem into a series of equations.
Importantly, I use the models as a way to analyze the disparate data sources, and I couple the models and data via statitical modeling methods.
I tend to use Bayesian approaches because they are particularly appealing for merging large datasets from diverse sources with process-based models because they are relatively straightforward to implement and can easily accommodate prior information such as that from the literature.
This integrative approach greatly helps to tease apart complicated plant-environment interactions and facilitates linking processes operating at different scales.
P(q | X ) q
Process models
Statistical tools
data-model integration

81. The Process Model Conceptual models: Model formulation:
Systems diagrams
Graphical models
Model formulation:
Explicit, mathematical eqn’s
Systems equations
State-space equations
Observations
of real system
Conceptual
model
Mathematical
model
Analytical
output
“Compare”
Observational
data
Simulation model
Numerical/
simulation
output

82. 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.

83. Data Model (Likelihood)
Likelihood components
Assuming conditional independence,
likelihood of all data is: