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

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Todays 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.

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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

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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

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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

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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 Jennifers talk … Observed data Latent (or true) process Data parameters Process parameters Unknown quantities Posterior LikelihoodProbabilistic process modelPrior(s)

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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

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Hierarchical Bayesian Model Observed data Latent (or true) processData parameters Process parameters Unknown quantities Posterior LikelihoodProbabilistic process modelPrior(s) Data model (likelihood) Probabilistic process model The process model

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

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Types of Process Models DeterministicStochastic Compartment models (differential or difference equns) Matrix models Reductionist models (include lots of details & components) Holistic models (use general principles) Static modelsDynamic models Distributed models (system depends on space & time) Lumped models Linear modelsNonlinear models Causal/mechanistic modelsBlack box models Analytical modelsNumerical/simulation models Jorgensen (1986) Fundamentals of Ecological Modelling. 389 pp. Elsevier, Amsterdam.

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Upcoming Example: Soil Carbon Cycle Model DeterministicStochastic Compartment models (differential or difference equns) Matrix models Reductionist models (include lots of details & components) Holistic models (use general principles) Static modelsDynamic models Distributed models (system depends on space & time) Lumped models Linear modelsNonlinear models Causal/mechanistic modelsBlack box models Analytical modelsNumerical/simulation models

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

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Source: Xu et al. (2006) Global Biogeochemical Cycles Vol. 20 GB2007. 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)

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

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How to Couple Data & Process Models? –Hierarchical Bayesian approach Observed data Latent (or true) process Data parameters Process parameters Unknown quantities Posterior LikelihoodProbabilistic process modelPrior(s) Data model (likelihood) Probabilistic process model

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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

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Ecosystem Processes Emphasis on aboveground What about belowground?

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N H20H20H20H20 H20H20H20H20 H20H20H20H20 C C N P Biogeochemical Cycles

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N H20H20H20H20 H20H20H20H20 H20H20H20H20C C N P Belowground system is critical Tightly linked to aboveground system

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Belowground Little info Difficult to measure Aboveground measurements (helpful but limited) Aboveground Lots of info Easy to measure Outstanding issues Partitioning above- & belowground Quantifying & partitioning belowground Implications for ecosystem function Examples: arid & semiarid systems Figure from Kieft et al. (1998) Ecology 79: Belowground Issues

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Motivating Questions: Soil Carbon Cycle From where in the soil is CO 2 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?

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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

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Examples Presented Today DeterministicStochastic Compartment models (differential or difference equns) Matrix models Reductionist models (include lots of details & components) Holistic models (use general principles) Static modelsDynamic models (implicit dependence on time) Distributed models (implicit dependence on space & time) Lumped models Linear modelsNonlinear models Causal/mechanistic modelsBlack box models Analytical modelsNumerical/simulation models

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Objectives: 1.Identify soil & microbial processes affecting decomposition 2.Learn how vegetation (i.e., microsite) controls these processes Ex 1: Soil organic matter decomposition

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Experimental Design Mesquite shrubland in southern Arizona Microsite types: 1.bare ground 2.grass 3.small mesquite 4.big mesquite Bare groundGrassSmall mesquiteBig mesquite 3 cores (reps)

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... 8 depths (layers)... Add water Add sugar + water Incubate at 25 o C CO 2 Measure CO 2 efflux (soil respiration rate) at 24 & 48 hours Experimental Design

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... 8 depths (layers)... Add water Add sugar + water Incubate at 25 o C CO 2 Measure CO 2 efflux (soil respiration rate) at 24 & 48 hours Measure: Microbial biomass Soil organic carbon Soil nitrogen Experimental Design

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Full-factorial design: Microsite 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) Design & Data Overview

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Some Data

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Soil depth microbes soil C CO 2 flux ? ? data Estimate microbial respiration (decomposition) parameters (i.e., process parameters) Carbon substrate Microbial biomass Respiration Analysis Objectives biomass & activity

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

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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 B C R N Things to consider: Multiple data types Nonlinear model Missing data Experimental design Data-Model Integration some data some data

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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) Observation precision (= 1/variance) Data Model (Likelihood)

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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 ObservedMean (truth) (latent process) Observation precision (= 1/variance) Data Model (Likelihood)

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Likelihood components Data parameters Latent processes Data Model (Likelihood)

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

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

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Process components Process parameters Probabilistic Process Model

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

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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) Parameter Model (Priors)

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

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The Posterior

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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 Posterior

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Model Implementation: WinBUGS

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Model Goodness-of-fit

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C* (total soil carbon, g C/m 2 )B* (microbial biomass, g dw/m 2 ) BareBig mesq. Med. Mesq. GrassBareBig mesq. Med. Mesq. Grass Example Results

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Bare groundBig mesquite Soil depth (or layer) SurfaceDeep Relative amount of microbial biomass

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Sensitivity to Data Sources

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From where in the soil is CO 2 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 Ex 2: Deconvolution of Soil Respiration data

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

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Stable Isotope Tracers CO 2 12 C 13 C Source isotope signatures Respired CO 2 signature Important data source: facilitates partitioning

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stochastic dataLiterature data Data Source Examples Datasets: field/lab pubs Soil Isotopes ( δ 13 C Tot ) (automated chambers & Keeling plots) Soil CO 2 flux (manual chambers) Pool Isotopes ( δ 13 C i ) (roots, soil, litter; Keeling plots) Soil CO 2 flux (automated chambers) Root respiration (in situ gas exchange) Root distributions (arid systems, different functional types) 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

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Example Data Santa Rita pulse experiment Respiration ( mol / m 2 / s) San Pedro automated flux measurements San Pedro incubation experimentSanta Rita pulse experiment – 13 C

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Hierarchical Bayesian Model: Deconvolution Approach Integrate multiple sources of information Integrate multiple sources of information Diverse data sources Different temporal & spatial scales Literature information Lab & field studies Detailed flux models Detailed flux models Respiration rates by source type & soil depth Dynamic models Mechanistic isotope mixing models Mechanistic isotope mixing models Multiple sources

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stochastic dataLiterature data Data Source Examples Soil Isotopes ( δ 13 C Tot ) (automated chambers & Keeling plots) Soil CO 2 flux (manual chambers) Pool Isotopes ( δ 13 C i ) (roots, soil, litter; Keeling plots) Soil CO 2 flux (automated chambers) Root respiration (in situ gas exchange) Root distributions (arid systems, different functional types) 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)

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Bayesian Deconvolution The Hierarchical Bayesian Model Likelihood of data (isotopes & soil flux) Latent processes: from isotope mixing model & flux models Functions of parameters Some Likelihood Components Define process models… Observations (data)

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The 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) (Q 10 Function, Energy of Activation) ?? Contributions by source ( i ) and depth ( z )? Temporal variability? Source-specific respiration? Spatial & temporal variability? ?? ?? Theory & Process Models From previous incubation/decomposition study (Ex 1)

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

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Bayesian Deconvolution The Parameter Model (Priors) Example: Example: Lloyd & Taylor (1994) model E o T o Informative priors for E o and T o :

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Implementation Markov chain Monte Carlo (MCMC) Markov chain Monte Carlo (MCMC) Sample parameters ( θ i ) from posterior Posteriors for: θ i s, r i ( z,t ) s, p i ( z,t ) s, etc. Means, medians, uncertainty WinBUGS WinBUGS

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

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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 Jul 27Aug 4

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Results: Contributions Vary by Depth Date Total root respiration (umol m -2 s -1 ) Soil water (v/v) Day 210Day 213Day Depth (cm) Relative contributions by depth Mesquite (C3 shrub) Sacaton (C4 grass) Soil water

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Summary Sources of soil CO 2 efflux Sources of soil CO 2 efflux Mesquite (shrub): major contributor, stable source Sacton (grass): minor contributor, threshold response Microbes (bare): minor contributor, coupled to pulses Deconvolution & data-model integration Deconvolution & data-model integration Soil depth (including litter) By species or functional groups Quantify spatial & temporal variability Incorporate environmental drivers Implications & applications Implications & applications Identify mechanisms Predictions & forward modeling

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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

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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)?

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

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Results: Dynamic Source Contributions

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Example WinBUGS Output EOEO TOTO

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The Inverse Problem Plant water uptakeSoil respiration Isotope mixing model Fractional contributions Total flux Flux model Substrate or root profiles (Q 10 Function, Energy of Activation) ?? ??

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The 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) (Q 10 Function, Energy of Activation) ?? Contributions by source ( i ) and depth ( z )? Temporal variability? ?? Source-specific respiration? Spatial & temporal variability?

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

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stochastic dataLiterature data Data Source Examples Soil Isotopes ( δ 13 C Tot ) (automated chambers & Keeling plots) Soil CO 2 flux (manual chambers) Pool Isotopes ( δ 13 C i ) (roots, soil, litter; Keeling plots) Soil CO 2 flux (automated chambers) Root respiration (in situ gas exchange) Root distributions (arid systems, different functional types) 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)

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The Deconvolution Problem Plant water uptakeSoil respiration Isotope mixing model Fractional contributions Total flux Flux model Substrate or root profiles (Q 10 Function, Energy of Activation) ?? ??

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Plant water uptakeSoil respiration What are ω, 1, 1, 2, 2 ? What is i ? Likelihood of data From isotope mixing model & flux model The Deconvolution Problem

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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

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Process models Data P( | X ) Statistical tools data-model integration Key components

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The Process Model Conceptual models: –Systems diagrams –Graphical models Model formulation: –Explicit, mathematical eqns Systems equations State-space equations Compare Observations of real system Conceptual model Mathematical model Simulation model Analytical output Numerical/ simulation output Observational data

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Examples Presented Today DeterministicStochastic Compartment models (differential or difference equns) Matrix models Reductionist models (include lots of details & components) Holistic models (use general principles) Static modelsDynamic models (implicit dependence on time) Distributed models (implicit dependence on space & time) Lumped models Linear modelsNonlinear models Causal/mechanistic modelsBlack box models Analytical modelsNumerical/simulation models Jorgensen (1986) Fundamentals of Ecological Modelling. 389 pp. Elsevier, Amsterdam.

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

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