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1. Introduction Methods Results Conclusion Quantifying uncertainties in national estimates of living biomass – a comparison of methods Johannes Breidenbach J. Heikkinen, G. Ståhl, H. Petersson, A. Ringvall, R. Astrup Norwegian Forest and Landscape Institute, Climate Center and National Forest Inventory P.O. Box 115, 1431 Ås Tel: +47 6494 8981; JOB@SkogogLandskap.no 1 / 18

2. Introduction Methods Results Conclusion Contents Introduction Background NFI Methods Analytic approach Parametric bootstrap Results Conclusion 2 / 18

3. Introduction Methods Results Conclusion Background • Sample surveys (NFI) Sampling error Measurement error • Biomass not measured – dbh and ht Model error • UNFCCC GPG require quantiﬁcation of uncertainties 3 / 18

4. Introduction Methods Results Conclusion Background • Sample surveys (NFI) Sampling error Measurement error • Biomass not measured – dbh and ht Model error ⇐ • UNFCCC GPG require quantiﬁcation of uncertainties 3 / 18

5. biometrics Sample-Based Estimation of Greenhouse Gas Emissions From Forests—A New Approach to Account for Both Sampling and Model Errors Go¨ran Sta˚hl, Juha Heikkinen, Hans Petersson, Jaakko Repola, and So¨ren Holm The Good Practice Guidance (GPG) for reporting emissions and removals of greenhouse gases from the land use, land-use change, and forestry (LULUCF) sector of the United Nation’s Framework Convention on Climate Change states that uncertainty estimates should always accompany the estimates of net emissions. Two basic procedures are suggested: simple error propagation and Monte-Carlo simulation. In this article, we argue that these methods are not very well-suited for uncertainty assessments in connection with sample-based surveys such as national forest inventories (NFIs), which provide a majority of the data for the LULUCF sector reporting in several countries. We suggest that a more straightforward approach would be to use standard sampling theory for assessing the sampling errors; however, it may be important to also include the error contribution from biomass and other models that are applied and this requires new methods for the variance estimation. In this article, a method for sample-based uncertainty assessment, including both model and sampling errors, is developed and applied using data from the NFIs of Finland and Sweden. The study revealed that the model error contribution to the combined sampling-model mean square error of ratio estimators of mean aboveground biomass on forestland amounted to about 10% in both countries. In estimating 5-year change of the corresponding biomass stocks, using permanent sampling units, the model error contribution was reduced to less than 1%. The smaller impact in the case of change estimation is due to the fact that any tendency of models to either over- or underestimate due to random parameter estimation errors will be the same both at the beginning and the end of a study period. The fairly small model error contributions in our study are due to the large number of sample trees used in the fitting of biomass models in Finland and Sweden; with less sample trees the model error contributions could be expected to be substantial. The proposed framework applies not only to greenhouse gas inventories but also to traditional NFI estimates of, e.g., growing stock in which uncertainties due to model errors typically are neglected in applications. Keywords: National forest inventory, model-dependent inference, uncertainty assessment, model error, UNFCCC, LULUCF sector, greenhouse gas inventory. T he Good Practice Guidance (GPG; IPCC 2003) for reporting emissions and removals of greenhouse gases for the land use, land-use change, and forestry (LULUCF) sector of the United Nation’s Framework Convention on Climate Change (UNFCCC) states that uncertainty estimates must accompany the annual estimates of greenhouse gas emissions and removals. Two different procedures are suggested: simple error propagation and use of Monte-Carlo simulation. The first procedure is based on standard approximation techniques for estimating the variance of the product of two stochastic variables, i.e., when the basic estimate is of the form C ϭ A ⅐ B, which is the standard assumption of the GPG as greenhouse gas emissions (C) are assumed to be estimated as an “activity estimate” (A) times an “emission factor” (B; IPCC 2003). With Monte-Carlo simulation the uncertainty estimate follows from detailed assumptions of probability densities linked to individ- ual random variables as well as the joint probability densities between different variables to capture dependencies. The step from simple error propagation to Monte-Carlo simulation is substantial in terms of required efforts to make the uncertainty analysis (e.g., Gertner 1987) and we argue that none of the methods suggested in the GPG are straightforward to apply in connection with LULUCF-sector related sample surveys, such as national forest inventories (NFIs). Monte-Carlo simulation requires large efforts combining sampling simulation with assumptions about model errors that are nonstandard in sample surveys. Further, sampling and model errors do not interact as straightforwardly as Manuscript received January 16, 2013; accepted May 3, 2013; published online August 29, 2013. Affiliations: Go¨ran Ståhl (goran.stahl@slu.se), Swedish University of Agricultural Sciences, Umeå, Sweden. Juha Heikkinen (juha.heikkinen@metla.fi), Finnish Forest Research Institute. Hans Petersson (hans.petersson@slu.se), Swedish University of Agricultural Sciences. Jaakko Repola (jaakko.repola@metla.fi), Finnish Forest Research Institute. So¨ren Holm (soren.holm@slu.se), Swedish University of Agricultural Sciences. FUNDAMENTAL RESEARCH For. Sci. 60(1):3–13 http://dx.doi.org/10.5849/forsci.13-005 Copyright © 2014 Society of American Foresters

6. biometrics Sample-Based Estimation of Greenhouse Gas Emissions From Forests—A New Approach to Account for Both Sampling and Model Errors Go¨ran Sta˚hl, Juha Heikkinen, Hans Petersson, Jaakko Repola, and So¨ren Holm The Good Practice Guidance (GPG) for reporting emissions and removals of greenhouse gases from the land use, land-use change, and forestry (LULUCF) sector of the United Nation’s Framework Convention on Climate Change states that uncertainty estimates should always accompany the estimates of net emissions. Two basic procedures are suggested: simple error propagation and Monte-Carlo simulation. In this article, we argue that these methods are not very well-suited for uncertainty assessments in connection with sample-based surveys such as national forest inventories (NFIs), which provide a majority of the data for the LULUCF sector reporting in several countries. We suggest that a more straightforward approach would be to use standard sampling theory for assessing the sampling errors; however, it may be important to also include the error contribution from biomass and other models that are applied and this requires new methods for the variance estimation. In this article, a method for sample-based uncertainty assessment, including both model and sampling errors, is developed and applied using data from the NFIs of Finland and Sweden. The study revealed that the model error contribution to the combined sampling-model mean square error of ratio estimators of mean aboveground biomass on forestland amounted to about 10% in both countries. In estimating 5-year change of the corresponding biomass stocks, using permanent sampling units, the model error contribution was reduced to less than 1%. The smaller impact in the case of change estimation is due to the fact that any tendency of models to either over- or underestimate due to random parameter estimation errors will be the same both at the beginning and the end of a study period. The fairly small model error contributions in our study are due to the large number of sample trees used in the fitting of biomass models in Finland and Sweden; with less sample trees the model error contributions could be expected to be substantial. The proposed framework applies not only to greenhouse gas inventories but also to traditional NFI estimates of, e.g., growing stock in which uncertainties due to model errors typically are neglected in applications. Keywords: National forest inventory, model-dependent inference, uncertainty assessment, model error, UNFCCC, LULUCF sector, greenhouse gas inventory. T he Good Practice Guidance (GPG; IPCC 2003) for reporting emissions and removals of greenhouse gases for the land use, land-use change, and forestry (LULUCF) sector of the United Nation’s Framework Convention on Climate Change (UNFCCC) states that uncertainty estimates must accompany the annual estimates of greenhouse gas emissions and removals. Two different procedures are suggested: simple error propagation and use of Monte-Carlo simulation. The first procedure is based on standard approximation techniques for estimating the variance of the product of two stochastic variables, i.e., when the basic estimate is of the form C ϭ A ⅐ B, which is the standard assumption of the GPG as greenhouse gas emissions (C) are assumed to be estimated as an “activity estimate” (A) times an “emission factor” (B; IPCC 2003). With Monte-Carlo simulation the uncertainty estimate follows from detailed assumptions of probability densities linked to individ- ual random variables as well as the joint probability densities between different variables to capture dependencies. The step from simple error propagation to Monte-Carlo simulation is substantial in terms of required efforts to make the uncertainty analysis (e.g., Gertner 1987) and we argue that none of the methods suggested in the GPG are straightforward to apply in connection with LULUCF-sector related sample surveys, such as national forest inventories (NFIs). Monte-Carlo simulation requires large efforts combining sampling simulation with assumptions about model errors that are nonstandard in sample surveys. Further, sampling and model errors do not interact as straightforwardly as Manuscript received January 16, 2013; accepted May 3, 2013; published online August 29, 2013. Affiliations: Go¨ran Ståhl (goran.stahl@slu.se), Swedish University of Agricultural Sciences, Umeå, Sweden. Juha Heikkinen (juha.heikkinen@metla.fi), Finnish Forest Research Institute. Hans Petersson (hans.petersson@slu.se), Swedish University of Agricultural Sciences. Jaakko Repola (jaakko.repola@metla.fi), Finnish Forest Research Institute. So¨ren Holm (soren.holm@slu.se), Swedish University of Agricultural Sciences. FUNDAMENTAL RESEARCH For. Sci. 60(1):3–13 http://dx.doi.org/10.5849/forsci.13-005 Copyright © 2014 Society of American Foresters biometrics Quantifying the Model-Related Variability of Biomass Stock and Change Estimates in the Norwegian National Forest Inventory Johannes Breidenbach, Clara Antoń-Fernańdez, Hans Petersson, Ronald E. McRoberts, and Rasmus Astrup National Forest Inventories (NFIs) provide estimates of forest parameters for national and regional scales. Many key variables of interest, such as biomass and timber volume, cannot be measured directly in the field. Instead, models are used to predict those variables from measurements of other field variables. Therefore, the uncertainty or variability of NFI estimates results not only from selecting a sample of the population but also from uncertainties in the models used to predict the variables of interest. The aim of this study was to quantify the model-related variability of Norway spruce (Picea abies [L.] Karst) biomass stock and change estimates for the Norwegian NFI. The model-related variability of the estimates stems from uncertainty in parameter estimates of biomass models as well as residual variability and was quantified using a Monte Carlo simulation technique. Uncertainties in model parameter estimates, which are often not available for published biomass models, had considerable influence on the model-related variability of biomass stock and change estimates. The assumption that the residual variability is larger than documented for the models and the correlation of within-plot model residuals influenced the model-related variability of biomass stock change estimates much more than estimates of the biomass stock. The larger influence on the stock change resulted from the large influence of harvests on the stock change, although harvests were observed rarely on the NFI sample plots in the 5-year period that was considered. In addition, the temporal correlation between model residuals due to changes in the allometry had considerable influence on the model-related variability of the biomass stock change estimate. The allometry may, however, be assumed to be rather stable over FUNDAMENTAL RESEARCH For. Sci. 60(1):25–33 http://dx.doi.org/10.5849/forsci.12-137 Copyright © 2014 Society of American Foresters

7. Introduction Methods Results Conclusion Aims of the study • Comparison of methods to quantify model-related error Analytical approach (Ståhl et al., 2014) Parametric bootstrap (Breidenbach et al., 2014) 5 / 18

8. Introduction Methods Results Conclusion The Norwegian National Forest Inventory • Interpenetrating panel design – 1/5th of plots measured every year • 22.008 sample plots for land-use classiﬁcation • Circular sample plots, 250 m2 • Species, diameter, location, height (subsample) • Swedish biomass functions 6 / 18

9. Introduction Methods Results Conclusion The Norwegian National Forest Inventory 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 NFI 5 NFI 6 NFI 7 NFI 8 NFI 9 NFI 10 Rotating inventory Yearly change estimates possible Inventory of mountain forests Inventory of Finnmark Last year for KP rep. 7 / 18

10. Introduction Methods Results Conclusion Basic idea • Biomass estimate based on 2 stage sample • Stage 1 (S1): NFI • Stage 2 (S2): Independent sample to estimate model parameters 8 / 18

11. Introduction Methods Results Conclusion Biomass models Species speciﬁc, ﬁtted on independent stage 2 (S2) sample: AGBS2i = f(β, xi, ei), xi = x1i, ..., xpi 9 / 18

12. Introduction Methods Results Conclusion Biomass models Species speciﬁc, ﬁtted on independent stage 2 (S2) sample: AGBS2i = f(β, xi, ei), xi = x1i, ..., xpi • 2 sources of uncertainty Parameter estimates ˆβ → ˆΣ, p × p Residual error ei → ˆσ 9 / 18

13. Introduction Methods Results Conclusion Sampling error Stage 1 (S1) sample = NFI yj = i ˆf(xij) 10 / 18

14. Introduction Methods Results Conclusion Sampling error Stage 1 (S1) sample = NFI yj = i ˆf(xij) ˆy = 1/n j yj VarS(ˆy) = 1 n s2 10 / 18

15. Introduction Methods Results Conclusion Sampling error Stage 1 (S1) sample = NFI yj = i ˆf(xij) ˆy = 1/n j yj VarS(ˆy) = 1 n s2 ˆY = Nˆy VarS( ˆY) = N2 VarS(ˆy) 10 / 18

16. Introduction Methods Results Conclusion Analytic approach VarC(ˆy) = VarS(ˆy) + VarM(ˆy) 11 / 18

17. Introduction Methods Results Conclusion Analytic approach VarC(ˆy) = VarS(ˆy) + VarM(ˆy) VarM(ˆy) = p j=1 p k=1 CovS2(ˆβj, ˆβk ) · ˆ¯fj · ˆ¯fk = ˆ¯f T · ˆΣ · ˆ¯f 11 / 18

18. Introduction Methods Results Conclusion Analytic approach VarC(ˆy) = VarS(ˆy) + VarM(ˆy) VarM(ˆy) = p j=1 p k=1 CovS2(ˆβj, ˆβk ) · ˆ¯fj · ˆ¯fk = ˆ¯f T · ˆΣ · ˆ¯f ˆ¯fj = avg S1 (NFI) 1st partial derivatives 11 / 18

19. Introduction Methods Results Conclusion Analytic approach VarC(ˆy) = VarS(ˆy) + VarM(ˆy) VarM(ˆy) = p j=1 p k=1 CovS2(ˆβj, ˆβk ) · ˆ¯fj · ˆ¯fk = ˆ¯f T · ˆΣ · ˆ¯f ˆ¯fj = avg S1 (NFI) 1st partial derivatives Residual error ei ignored Assumption: reasonably accurate parameter estimates Ståhl et al. (2014); Cunia (1986) 11 / 18

20. Introduction Methods Results Conclusion Parametric bootstrap Repeat B times: ˆβ∗ = N( ˆβ, Σ) AGB∗ i = f( ˆβ∗, di, hi) 12 / 18

21. Introduction Methods Results Conclusion Parametric bootstrap Repeat B times: ˆβ∗ = N( ˆβ, Σ) AGB∗ i = f( ˆβ∗, di, hi) B Estimates of ˆy∗ or ˆY∗ 12 / 18

22. Introduction Methods Results Conclusion Parametric bootstrap Repeat B times: ˆβ∗ = N( ˆβ, Σ) AGB∗ i = f( ˆβ∗, di, hi) B Estimates of ˆy∗ or ˆY∗ VarMB(ˆy) = Var(ˆy∗) and VarMB( ˆY) = Var( ˆY∗) 12 / 18

23. Introduction Methods Results Conclusion Parametric bootstrap – residual error – meas. error Repeat B times: ˆβ∗ = N( ˆβ, Σ) AGB∗ i = f( ˆβ∗, di, hi) 13 / 18

24. Introduction Methods Results Conclusion Parametric bootstrap – residual error – meas. error Repeat B times: ˆβ∗ = N( ˆβ, Σ) AGB∗ i = f( ˆβ∗, di, hi) e∗ = N(0, σ) → same e∗ i for remeasured trees AGB∗∗ i = f( ˆβ∗, e∗ i , di, hi) 13 / 18

25. Introduction Methods Results Conclusion Parametric bootstrap – residual error – meas. error Repeat B times: ˆβ∗ = N( ˆβ, Σ) AGB∗ i = f( ˆβ∗, di, hi) e∗ = N(0, σ) → same e∗ i for remeasured trees AGB∗∗ i = f( ˆβ∗, e∗ i , di, hi) B Estimates of ˆy∗∗ and ˆY∗∗ 13 / 18

26. Introduction Methods Results Conclusion Biomass stock 500000 510000 520000 530000 540000 0e+002e−054e−056e−05 NFI 9 AGB (kt) Density 14 / 18

27. Introduction Methods Results Conclusion Biomass stock 500000 510000 520000 530000 540000 0e+002e−054e−056e−05 NFI 9 AGB (kt) Density 14 / 18

28. Introduction Methods Results Conclusion Biomass stock 500000 510000 520000 530000 540000 0e+002e−054e−056e−05 NFI 9 AGB (kt) Density RMSE %: Analytical Boot Sampling 1.055 1.057 0.967 14 / 18

29. Introduction Methods Results Conclusion Biomass stock change 45000 46000 47000 48000 49000 50000 0e+001e−042e−043e−044e−045e−046e−047e−04 Change NFI 9 − NFI 8 AGB (kt) Density 15 / 18

30. Introduction Methods Results Conclusion Biomass stock change 45000 46000 47000 48000 49000 50000 0e+001e−042e−043e−044e−045e−046e−047e−04 Change NFI 9 − NFI 8 AGB (kt) Density 15 / 18

31. Introduction Methods Results Conclusion Biomass stock change 45000 46000 47000 48000 49000 50000 0e+001e−042e−043e−044e−045e−046e−047e−04 Change NFI 9 − NFI 8 AGB (kt) Density RMSE %: Analytical Boot Sampling 1.185 1.187 3.540 15 / 18

32. Introduction Methods Results Conclusion Biomass stock change - including residual error 45000 46000 47000 48000 49000 0e+001e−042e−043e−044e−045e−046e−047e−04 Change NFI 9 − NFI 8 AGB (kt) Density RMSE %: Boot no res err Boot res err Sampling 1.18 1.20 3.54 1.7% difference 16 / 18

33. Introduction Methods Results Conclusion Summary – comparison of methods • Analytic approach & param. bootstrap were equal • Both methods require the same information - parameter covariance matrix – Many repititions needed in boostraping to obtain stable result + Bootstrap simpler in theory - e.g., extension for residual error or measurement error 17 / 18

34. Introduction Methods Results Conclusion Summary of results • Model error = sampling error in stock estimates • Sampling error dominates over model error in change estimates • Biomass residual error = marginal inﬂuence • Dependent on S1 and S2 sample size! 18 / 18

35. Additional Slides Biomass models Species speciﬁc, ﬁtted on independent stage 2 (S2) sample from Finland (Ståhl et al. 2014, Repola 2009): AGB = f(d, h) = exp(ˆβ1 + ˆβ2g(d) + ˆβ3g(h) + ei) g(d) = d d + kd g(h) = h h + kh 18 / 18

36. Additional Slides Biomass models Species speciﬁc, ﬁtted on independent stage 2 (S2) sample from Finland (Ståhl et al. 2014, Repola 2009): AGB = f(d, h) = exp(ˆβ1 + ˆβ2g(d) + ˆβ3g(h) + ei) g(d) = d d + kd g(h) = h h + kh f(d, h) d = d · exp(ˆβ1 + ˆβ2g(d) + ˆβ3g(h)) 18 / 18

37. Additional Slides Biomass models Species speciﬁc, ﬁtted on independent stage 2 (S2) sample from Finland (Ståhl et al. 2014, Repola 2009): AGB = f(d, h) = exp(ˆβ1 + ˆβ2g(d) + ˆβ3g(h) + ei) g(d) = d d + kd g(h) = h h + kh f(d, h) d = d · exp(ˆβ1 + ˆβ2g(d) + ˆβ3g(h)) ¯f(d, h) d = 1 nS1 i di · f(di, hi) 18 / 18

38. Additional Slides Parameter error and tree height measurement error 45000 46000 47000 48000 49000 0e+001e−042e−043e−044e−045e−046e−047e−04 Change NFI 9 − NFI 8 AGB (kt) Density Ht err: 2.5% and 7.5% RMSE %: Boot no ht err Boot ht err Sampling 1.18 1.20 3.54 1.7% difference Ht err: 5% and 20% RMSE %: Boot no ht err Boot ht err Sampling 1.18 1.27 3.54 5.0% difference 18 / 18

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