1. 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)

2. 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

3. 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

4. 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?

5. 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

6. 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

7. 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

8. 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

9. 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 …

10. CTCD and MUCM Centre for Terrestrial Carbon Dynamics (CTCD)
Mission: To understand C fluxes from vegetation
Managing Uncertainty in Complex Models (MUCM)
To develop robust and widely applicable BACCO methods

11. 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

12. SDGVMd outputs for 2000
Plug-in maps

13. 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

14. 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

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

16. 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

17. EvNl leaf life span

18. 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

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

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

21. Land cover (from LCM2000)
Land cover map

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

23. 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

24. 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

25. 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