The Estimation of the Net CO2 Flux for England & Wales and its Uncertainty using Emulation Marc Kennedy, Tony O’Hagan, Clive Anderson, Mark Lomas, John Paul Gosling and Ian Woodward (University of Sheffield) Andreas Heinemeyer (University of York)

Carbon flux Carbon dioxide (CO2) is one of the principal greenhouse gases that drives global warming To what extent can vegetation reduce the quantity of CO2 going into the atmosphere? Source or sink? Kyoto agreement signatories are required each year to account for carbon (C) emissions How to estimate this? Inventories Models

Computer models In almost all fields of science, technology, industry and policy making, people use mechanistic models to describe complex real-world processes For understanding, prediction, control Growing realisation of importance of uncertainty in model predictions Can we trust them? Without any quantification of output uncertainty, it’s easy to dismiss them

Uncertainty analysis Consider just one source of uncertainty We have a computer model that produces output y = f (x) when given input x But for a particular application we do not know x precisely So X is a random variable, and so therefore is Y = f (X ) We are interested in the uncertainty distribution of Y How can we compute it?

Monte Carlo The usual approach is Monte Carlo Sample values of x from its distribution Run the model for all these values to produce sample values yi = f (xi) These are a sample from the uncertainty distribution of Y Neat but impractical if it takes minutes or hours to run the model We can then only make a small number of runs

Emulation A computer model encodes a function, that takes inputs and produces outputs An emulator is a statistical approximation of that function Estimates what outputs would be obtained from given inputs With statistical measure of estimation error Given enough training data, estimation error variance can be made small

So what? A good emulator estimates the model output accurately with small uncertainty and runs “instantly” So we can do uncertainty analysis etc fast and efficiently Conceptually, we use model runs to train the emulator then derive any desired properties of model

Gaussian process We use Gaussian process (GP) emulation Nonparametric, so can fit any function Error measures can be validated Analytically tractable, so can often do uncertainty analysis etc analytically Highly efficient for up to 100 inputs The method uses Bayesian theory Formally, the posterior distribution of the function is a GP This posterior distribution is the emulator

BACCO This has led to a wide ranging body of tools for inference about all kinds of uncertainties in computer models All based on building the GP emulator of the model from a set of training runs This area is known as BACCO Bayesian Analysis of Computer Code Output Includes not just uncertainty analysis Sensitivity analysis, calibration, data assimilation, validation, optimisation …

CTCD and MUCM Centre for Terrestrial Carbon Dynamics (CTCD) http://ctcd.group.shef.ac.uk Mission: To understand C fluxes from vegetation Managing Uncertainty in Complex Models (MUCM) http://mucm.group.shef.ac.uk To develop robust and widely applicable BACCO methods

The England & Wales carbon flux in 2000 Recent application of these methods Dynamic vegetation model (SDGVMd) Predicts carbon sequestration and release from vegetation and soils NBP (net biosphere production) GPP (gross primary production) Over 700 pixels across E&W 4 plant functional types separately modelled Deciduous broadleaf (DcBl), evergreen needleleaf (EvNl), C3 grasses and crops

SDGVMd outputs for 2000 Plug-in maps

Outline of analysis Build emulators for each PFT at a sample of sites Identify most important inputs Define distributions to describe uncertainty in important inputs Analysis of soils data Elicitation of uncertainty in PFT parameters Need to consider correlations

Carry out uncertainty analysis in each sampled site Interpolate across all sites Mean corrections and standard deviations Aggregate across sites and PFTs Allowing for correlations

Sensitivity analysis for one pixel/PFT Main effects at site (54.417, -0.75)

Elicitation Beliefs of expert (developer of SDGVMd) regarding plausible values of PFT parameters Important to allow for uncertainty about mix of species in a pixel and role of parameter in the model In the case of leaf life span for evergreens, this was more complex

EvNl leaf life span

Correlations PFT parameter in one pixel may differ from in another Because of variation in species mix Common uncertainty about average over all species induces correlation Elicit beliefs about average over whole UK EvNl joint distributions are mixtures of 25 components, with correlation both between and within years

Mean NBP corrections Interpolated maps of E(NBP) – plugins, assuming 100% coverage of each PFT

NBP standard deviations Standard deviations of NBP, assuming 100% coverage of each PFT

Land cover (from LCM2000) Land cover map

Aggregate across 4 PFTs Corrected map for NBP, with standard deviation, after weighted aggregation over different PFTs

Sensitivity analysis Map shows proportion of overall uncertainty in each pixel that is due to uncertainty in the parameters of PFTs As opposed to soil parameters Contribution of PFT uncertainty largest in grasslands/moorlands

Aggregate over England & Wales PFT Plug-in estimate (Mt C) Mean (Mt C) Variance (Mt C2) Grass 5.279 4.639 0.269 Crop 0.853 0.445 0.034 Deciduous 2.132 1.683 0.013 Evergreen 0.798 0.781 0.001 Covariances Total 9.061 7.548 0.321 Breakdown of variance in total E&W variance of NBP, and comparison with plug-in totals

Conclusions BACCO methods offer a powerful basis for computation of uncertainties in model predictions Analysis of E&W aggregate NBP in 2000 Good case study for uncertainty and sensitivity analyses Involved several technical extensions Has important implications for our understanding of C fluxes Policy implications