Modeling radiocarbon in the Earth System. Radiocarbon summer school 2012

Transcription

1 Modeling radiocarbon in the Earth System Radiocarbon summer school 2012

2 Outline Brief introduc9on to mathema9cal modeling Single pool models Mul9ple pool models Model implementa9on Parameter es9ma9on

3 What is a model? A Belgium Netherlands B C MEIO DETRITUS MACRO N H Three examples of models. They have in common that they focus only on the object of interest, ignoring the irrelevant details. What is irrelevant depends on the aim of the model. Soetaert, K. and P.M.J. Herman A prac8cal guide to ecological modelling using R as a simula8on pla?orm. Springer.

4 The global carbon cycle as a conceptual model

5 Why we use models Synthesis and integra9on Predic9on and forecas9ng Observa9on and experimenta9on

6 Problem Conceptual model main components relationships Modelling steps and ingredients. Mathematical model general theory Parameterisation literature measurements Mathematical solution Calibration,sensitivity Verification,validation field data lab measurements OK? Prediction, Analysis Soetaert, K. and P.M.J. Herman A prac8cal guide to ecological modelling using R as a simula8on pla?orm. Springer.

7 Conceptual model: Radioac9ve decay Total number of 14 C atoms (N) Loss by radioac9ve decay

8 Mathema9cal model dc dt =Inputs Outputs

9 Mathema9cal model dn dt = λn Total number of 14 C atoms (N) Loss by radioac9ve decay

10 Mathema9cal solu9on? dn dt = λn

11 Mathema9cal solu9on dn dt = λn N(t) =N0 exp( λt)

12 Parameteriza9on Libby half- life = 5568 years t 1/2 = 5568 yr = ln 2 λ λ = yr 1

13 Predic9on How much radiocarbon would be available arer 10,000 years if the ini9al amount of 14 C atoms N 0 = 100 N(t = 10000) = 100 exp( λ10000) = 28.8

14 Single pool models Useful for integra9on and general understanding Spa9al and temporal averages of reservoirs Ignore heterogeneity and complexity

15 A simple reservoir model Inputs Stock Outputs dc dt = I O dc dt = I kc

16 Analy9cal solu9on dc dt = I kc C = I k I k C 0 e kt I = 10, k = 0.5, C 0 = 0

17 Effect of the ini9al condi9ons C 0 C C0=0 C0=20 C0=40 I = 10, k = 0.5 The system always reaches a steady- state independent on the ini8al condi8ons t

18 Steady State C = I k I k C 0 e kt C = I k

19 Steady State dc dt = I O =0 At steady- state the input flux is equal to the output flux. I = kc C = I k [mass]/[time] 1/[time] = [mass]

20 Turnover 9me How long would it take to empty the reservoir if we suddenly stop the input flux? τ 0 = C O = C I [mass] [mass/time] =[time] τ 0 = 1 k 1 1/[time] =[time]

21 Age and transit 9mes

22 Let s empty the reservoir dc dt = kc, C(t = 0) = C ss Start with a reservoir at steady state and reset time C = C SS exp( kt) Time evolution of the remaining C in reservoir O = kc SS exp( kt) Time evolution of the C leaving the reservoir

23 Let s empty the reservoir Carbon remaining in the reservoir C = C SS exp( kt) Carbon leaving the reservoir O = kc SS exp( kt) C Outputs t t

24 Let s empty the reservoir C = C SS exp( kt) O = kc SS exp( kt) C Mean age of C remaining in the reservoir Outputs Mean age of C leaving the reservoir t t

25 Age and transit 9me distribu9ons Proportion ψ(τ) =φ(τ) ψ(τ) = Age frequency distribution φ(τ) = Transit time frequency distribution E[ψ(τ)] = τ a E[φ(τ)] = τ r Mean age Mean transit time !

26 Mean age and transit 9mes are not always equal Human popula9on Uranium Water vapor, soils Bolin & Rhode (1972).

27 Mul9ple pool models No. Pools Parallel Series Feedback

28 Models as systems of differen9al equa9ons γ I 1 γ C 1 C 2 dc 1 dt dc 2 dt = γi k 1 C 1 =(1 γ)i k 2 C 2

29 Models as systems of differen9al equa9ons I α C 1 C 2 dc 1 dt dc 2 dt = I k 1 C 1 = αk 1 C 1 k 2 C 2

30 Models as systems of differen9al equa9ons I α 1 C 1 C 2 α 2 dc 1 dt dc 2 dt = I + α 2 k 2 C 2 k 1 C 1 = α 1 k 1 C 1 k 2 C 2

31 Turnover time is calculated as the inventory (here, the mass of carbon in soil organic matter) divided by the total flux of C out of the reservoir. Transit (residence) time is the time spent in the reservoir by an individual C atom, which is equivalent to the average age of a molecule leaving the reservoir (assuming its age was 0 when it entered the reservoir). Average age of C atoms in the soil organic matter reservoir is the average time since the atoms entered (whether they are leaving or not). A homogeneous reservoir (one for which the probability of every atom leaving is equal), at steady state, the turnover time, residence time and average age of organic matter in the reservoir are equal.

32 Faster 4 grams τ = 1 years Slower 20 grams τ = 10 years Flux = 4 g/yr Flux = 2 g/yr One Pool (wrong) 24 grams τ = 4 years Flux = 6 g/yr Mean age of C in Reservoir Turnover Time Mass/Flux Mean Age of respired C [ 20 g *10yr + 4g *1yr] 24 g 24g 6g / yr = 4 year [ 4 g/ yr *1yr + 2g / yr *10 yr] 6g/ yr = 8.5 year = 4year [ 24 g * 4yr] 24 g = 4 year 24 g 6g/ yr = 4year 6g/ yr * 4yr = 4year 6g/ yr

33 Model structure, age, and transit 9mes Manzoni et al. (2009). J. Geophys. Res. 114, G04025.

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35 Modeling radiocarbon Single pool case Inputs Stock Outputs Radioac9ve decay

36 Modeling radiocarbon Single pool case d 14 C dt = df C dt = IF ex kfc λfc = IF ex (k + λ)fc

37 Example: incorpora9on of bomb carbon in the terrestrial biosphere! 14 C ( ) ! 14 C in Atmosphere! 14 C in Terrestrial biosphere df C dt = IF atm (k + λ)fc I = GP P = 120 k = I C = GP P C = Year

38 Steady- state solu9on df C dt = IF ex (k + λ)fc Differential equation IF ex =(k + λ)fc Steady-state solution FC = IF ex k + λ Solve for FC F = k k + λ Assuming F ex = 1 (pre 1950), and I = C*k

39 Model implementa9on Analy9cal vs. numerical solu9ons Spreadsheets vs programing language

40 Analy9cal solu9on dc dt = I kc C = I k I k C 0 e kt Gives the exact solution to the differential equation Can give insights into the properties and long-term behavior of the system Not always possible to compute

41 Numerical solu9on df C dt = IF atm (k + λ)fc Most practical approach to model radiocarbon! 14 C ( ) ! 14 C in Atmosphere! 14 C in Terrestrial biosphere The calibration curve is not a function but a set of numbers Is prone to approximation (numerical) errors Year

42 Numerical approxima9on of an ODE The idea is to break the problem in small pieces and approximate a solution to the ODE at each interval

43 A Explicit euler method B Runge-Kutta method 4 f(t2) df f(t1) +! t (t1) dt f(t1) C df f(t1) +! t (t2) dt f(t2) f(t1) tangent line! t t1 t2 Implicit euler method true value estimated D estimated 3 2 1! t 2! t t1 t2 Semi-Implicit euler method A. The explicit Euler integration method. The function value at time t2 is estimated based on the value at time t1 and the rate of change (as denoted by the tangent line). It is assumed that the rate of change remains constant in the time step between t1 and t2. As this is not the case, the estimated value considerably underestimates the true value. B. The 4- th order Runge-Kutta method. One function evaluation at the start point (1), two evaluations at the mid-point (2,3), and one evaluation at the end point (4) are combined to provide the final projected value (grey dot). C. The implicit Euler integration method uses the estimate of the rate of change at the next time step, t2. D. The semi-implicit method combines both the explicit and implicit approach t1 t2 t1 t2 Soetaert, K. and P.M.J. Herman A prac8cal guide to ecological modelling using R as a simula8on pla?orm. Springer.

44 Explicit or forward Euler method

45 Explicit or forward Euler method

46 Applica9on radiocarbon model: spreadsheet implementa9on df C dt = IF atm (k + λ)fc Differen9al equa9on (FC) t+1 =(FC) t + (FC) t =(FC) t + h((if atm ) t (k + λ)(fc) t ) Numerical solu9on F t+1 = IF atm +(FC) t (1 k λ) C t+1 Simplifica9on for annual 9me step

47 ODE solvers Accurate solu9ons to ODEs Requires some level of programming Excel can do it, but befer try something different Matlab, Mathema9ca, and R have really good and easy to implement ODE solvers For large computa9ons C, C++, or Fortran are recommended

48 How to implement models in R Use the desolve package to solve systems of ODEs. It can handle linear and nonlinear problems. For linear models in soil applica9ons, use SoilR!

49 desolve Reading the first 5 pages of package vignefe Solving Ini8al Value Differen8al Equa8ons in R is enough to implement linear and nonlinear systems in R. You can get it here: hfp://cran.r- project.org/ web/packages/desolve//vignefes/ desolve.pdf

50 SoilR Can be used to implement linear soil organic mafer decomposi9on models Includes a radiocarbon module and facilitates parameter es9ma9on To install simply type inside R: install.packages("soilr", repos= c("h9p://r- Forge.R- project.org", getop@on("repos"))

51 Stocks and fluxes of radiocarbon in bulk soil and respired CO 2

52 How to do it in SoilR? example( ThreepSeriesModel, package= SoilR " See also: example( ThreepParallelModel, package= SoilR " example( ThreepFeedbackModel, package= SoilR "

53 Model structure and mean transit 9mes

54 Parameter es9ma9on Numerical op9miza9on Bayesian parameter es9ma9on

55 Numerical op9miza9on Minimize or maximize an objective function Set some constraints Find optimal solution

56 Numerical op9miza9on Objective or cost function Constraints

57 Numerical op9miza9on Commonly, Residual sum of squares RSS = n (y i f(x i )) 2 i=1

58 Local versus global minima In radiocarbon problems we ooen find two local minima

59 Example! 14 C ( ) We are interested in finding the turnover 8me that minimizes the difference between the observa8on and the predic8on of a one pool model min f(τ) =(O τ R n t=2004 FC t=2004 ) 2 df C dt = IF atm (1/τ + λ)fc Year Δ 14 C = 200 (per mil) in 1998

60 Difference between observed and possible values of 14C in Single pool Squared Residual Turnover time

61 14C trajectories for the two possible solutions! 14 C ( ) ! 14 C in Atmosphere 13 yrs turnover time 36 yrs turnover time In radiocarbon we usually get two optimal solutions without constraints Year

62 Possible solutions for the two pool model, with τ 1 < τ 2

63 Two pool solution Addi8onal informa8on such as the Δ 14 C of different pools or fluxes are helpful constraints in the op8miza8on problem

64 Bayesian parameter es9ma9on Powerful method for parameter es9ma9on Parameters are assumed to be random variables instead of fixed values Bayesian methods seek to es9mate a posterior probability distribu9on given some prior knowledge and some observed data.

65 Bayesian inference Likelihood P (θ D) =P (θ) P (D θ) P (D) Posterior Prior Posterior distribu8on provides knowledge about the mean and variance of the parameters. Can be used to es8mate uncertainty of the predic8ons

66 How to implement Bayesian techniques? R is a powerful environment for finng Bayesian models Package FME provides func9ons to fit models to data using Bayesian techniques SoilR provides an interface to FME for Bayesian parameter es9ma9on

67 14 CO 2 efflux from soils at Harvard Forest

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69 In summary Models are helpful tools to understand system behavior They can help you es9ma9ng mean ages, transit 9mes, and uncertain9es in radiocarbon applica9ons However, beware of the assump9ons of the modeling techniques you use (e.g. steady- state, negligible numerical errors)