1. Case studies in Gaussian process modelling of computer codes for carbon accounting Marc Kennedy, Clive Anderson, Stefano Conti, Tony OHagan

2. Talk Outline Centre for Terrestrial Carbon Dynamics Computer Models in CTCD Bayesian emulators Case Study 1: SPA Case Study 2: SDGVM

3. Centre for Terrestrial Carbon Dynamics The CTCD… is a NERC centre of excellence for Earth Observation made up of groups from Sheffield, York, Edinburgh, UCL, Forest Research brings together experts in vegetation modelling, soil science, earth observation, carbon flux measurement and statistics

4. Net Ecosystem Production Plant respiration Photosynthesis Gain Loss Soil respiration Loss – Terrestrial carbon source if NEP is negative – Terrestrial carbon sink if NEP is positive

5. Computer Models in CTCD SPA – Simulates plant processes at 30-minute time intervals ForestETP – Stand scale – Localised modelling SDGVM – Global scale – Coarse resolution

6. Statistical objectives within CTCD Contribute to the development of these models – through model testing using sensitivity analysis Identify the greatest sources of uncertainty Correctly reflect the uncertainty in predictions – Uncertainty analysis: propagating the parameter uncertainty through the model

7. Bayesian Emulation of Models Model output is an unknown function of its inputs – Convenient prior is a Gaussian process – Run code at set of well chosen input points – Obtain posterior distribution The emulator is the posterior distribution of the output – Fast approximation – Measure of uncertainty – Nice analytical form for further analysis

8. Case study 1: Soil Plant Atmosphere (SPA) Model SPA is a fine scale model created by Mat Williams – Aggregated SPA outputs were used to create the simpler up-scaled model (ACM: the Aggregated Canopy Model) by fitting a set of simple equations with 9 parameters Can an emulator do any better than ACM as an approximation to SPA?

9. ACM vs. Emulator for predicting SPA Bayesian emulator created using only 150 of the total 6561 points used to create ACM Predicted remaining 6411 SPA points using emulator and ACM – Compare Root Mean Square Errors (RMSE)

10. SPA Predictions Emulator Predictions RMSE = 0.314 using emulator ACM Predictions RMSE = 0.726 using ACM

11. Case Study 2: Sheffield Dynamic Global Vegetation Model SDGVM is a point model – each pixel represents an area, with an associated vegetation type / land use Vegetation type is described using 14 plant functional type parameters SDGVM is constantly being developed – To improve process modelling – To incorporate more detailed driving data

12. Plant Functional Type inputs Examples: Leaf life span Leaf area Temperature when bud bursts Temperature when leaf falls Wood density Maximum carbon storage Xylem conductivity Emulator will allow small groups of inputs to vary, others fixed at original default values

13. Soil inputs Soil clay % Soil sand % Soil depth Bulk density

14. Emulator for SDGVM Built an emulator for the NEP output of SDGVM – 80 runs in the 5-dimensional input space were used as training data – A maximin Latin hypercube design was used to ensure even coverage of the input space. Plant scientists specified the ranges 24.259 14.24 18.384 36.204 -3.214 1.774 254.0 6.304346 7.913044 20.28985 6.521775 330.0 8.739128 8.173912 13.4058 19.56525 326.0 8.30435 5.56522 7.971025 50.000023 145.0 5.521742 5.043478 0.72465 33.695625 236.0 9.43478 8.782606 1.08695 75.0 123.0 9.608696 9.478258 21.0145 71.739151 Run code … …

15. Model testing: Sensitivity analysis We use sensitivity analysis for model checking and for model interpretation Calculate main effects of each code input – How does output change if we vary the input, averaged over other inputs? Building the emulator has uncovered bugs – simply by trying different combinations of input values

16. Main Effect: Leaf life span

17. Main Effect: Leaf life span (updated)

18. Main Effect: Senescence Temperature

19. Main Effects: Soil inputs Soil inputs had been fixed in SDGVM Output sensitive to sand content, but not clay content, over these ranges More detailed soil input data are now used

21. Error discovered in the soil module NEP -20 0 20 40 60 80 050000010000001500000 Before…After… Bulk density

23. SDGVM: new sensitivity analysis We initially analysed uncertainty in the NEP output at a single test site, using rough ranges for the 14 plant functional type parameters Assumed default (uniform) probability distributions for the parameters The aim here is to identify the greatest potential sources of uncertainty

24. NEP (g/m 2 /y)

25. Leaf life span 69.1% Minimum growth rate 14.2% Water potential 3.4% Maximum age 1.0%

26. Plant Functional Type parameters Uncertainty is driven by just a few key parameters – Maximum age – Leaf life span – Water potential – Minimum growth rate The next step was to refine the rough probability distributions for these parameters

27. Elicitation We elicited formal probability distributions for the key parameters – based on discussion with Ian Woodward – representing his uncertainty about their values within the UK – noting that each really applies as an average over the species actually present in a given pixel

28. Leaf life span (days)Minimum growth rate (m) Maximum age (years)Water potential (M Pa)

29. Leaf life span 69.1% Minimum growth rate 14.2% Water potential 3.4% Maximum age 1.0% Mean NEP = 174 gCm -2 Std deviation = 14.32 gCm -2 Mean NEP = 163 gCm -2 Std deviation = 12.65 gCm -2 Uniform probability distributionsRefined probability distributions

30. Uncertainty analysis at sample sites We computed uncertainty analyses on NEP outputs from SDGVM for 9 sites/pixels NEP Stockten on the Forest (Nr York) Milton Keynes Barnstaple (Devon) Keswick (Lake District) Lowland (Scotland) Dartmoor New Forest (Hampshire) Kielder S. Ballater (Scotland) 2070120170220270

31. Uncertainty is clearly substantial, even when we only take account of uncertainty in these parameters The most important parameter is minimum growth rate, which accounts for typically at least 60% of overall NEP uncertainty – This suggests targeting this parameter for research Seeding density?

32. Ongoing work We need to estimate uncertainty in the overall UK carbon budget – Developing new theory for aggregating uncertainty over many pixels Windows software will be made available later this year www.shef.ac.uk/st1mck