1. Atmospheric Inventories for Regional Biogeochemistry Scott Denning Colorado State University

2. The Global Carbon Cycle Atmosphere 750 + 3/yr Ocean 38,000 Plants and Soils 2000 Fossil Fuel ~90 ~120 6 ~90

4. Global trend known very accurately Provides an integral constraint on the total carbon budget Interannual variability is of the same order as anthropogenic emissions!

6. Meridional Gradients (Tans et al, 1990) Simulated gradient way too steep for most emission scenarios

7. Diurnal Variations Diurnal cycle of 70 ppm over an active forest is 5x as strong as seasonal cycle Vertical gradient at sunrise is 15x as strong as pole-to-pole gradient Vertical gradient reverses in daytime, much weaker

8. Diurnal Variations

9. Advection: –Stuff that was “upstream” moves here with the wind: –To predict the future amount of q, we simply need to keep track of gradients and wind speed in each direction Turbulence –Tracer transport by “unresolved” eddies –Behaves like down-gradient diffusion or “mixing” Moist (cumulus) convection –Vertical transport by updrafts and downdrafts in clouds Atmospheric Transport

10. Cumulus Transport (example) Cloud “types” defined by lateral entrainment Separate in-cloud CO2 soundings for each type Detrainment back into environment at cloud top

11. Inverse Modeling of Atmospheric CO 2 Air Parcel Sources Sinks transport (model) (solve for) concentration transport sources and sinks (observe) Sample

12. Synthesis Inversion Procedure (“Divide and Conquer”) 1.Divide carbon fluxes into subsets based on processes, geographic regions, or some combination 1.Spatial patterns of fluxes within regions? 2.Temporal phasing (e.g., seasonal, diurnal, interannual?) 2.Prescribe emissions of unit strength from each “basis function” as lower boundary forcing to a global tracer transport model 3.Integrate the model for three years (“spin-up”) from initially uniform conditions to obtain equilibrium with sources and sinks 4.Each resulting simulated concentration field shows the “influence” of the particular emissions pattern 5.Combine these fields to “synthesize” a concentration field that agrees with observations

13. Consider Linear Regression Given N measurements y i for different values of the independent variable x i, find a slope ( m ) and intercept ( b ) that describe the “best” line through the observations –Why a line? –What do we mean by “best” –How do we find m and b ? Compare predicted values to observations, and find m and b that fit best Define a total error (difference between model and observations) and minimize it!

14. Our model is Error at any point is just Could just add the error up: Problem: positive and negative errors cancel … we need to penalize signs equally Define the total error as the sum of the Euclidean distance between the model and observations (sum of square of the errors): Defining the Total Error

15. Minimizing Total Error (Least Squares) Take partial derivs Set to zero Solve for m and b

16. Geometric View of Linear Regression Any vector can be written as a linear combination of the orthonormal basis set This is accomplished by taking the dot product (or inner product) of the vector with each basis vector to determine the components in each basis direction Linear regression involves a 2D mapping of an observation vector into a different vector space More generally, this can involve an arbitrary number of basis vectors (dimensions)

17. Generalized Least Squares The G's can be thought of as partial derivatives and the whole matrix as a Jacobian.

18. “Determinedness” Even-determined –Exact solution, zero prediction error Over-determined –Single "best" solution, finite prediction error Underdetermined –Infinite solutions, zero prediction error Mixed-determined –Many solutions, finite prediction error

19. TransCom Inversion Intercomparison Discretize world into 11 land regions (by vegetation type) and 11 ocean regions (by circulation features) Release tracer with unit flux from each region during each month into a set of 16 different transport models Produce timeseries of tracer concentrations at each observing station for 3 simulated years

20. Synthesis Inversion Decompose total emissions into M “basis functions” Use atmospheric transport model to generate G –Fill by columns using forward model, or –Fill by rows using backward (adjoint) model Observe d at N locations Invert G to find m datatransport fluxes d 1 = G 11 m 1 + G 12 m 2 + … + G 1N m N CO 2 sampled at location 1Strength of emissions of type 2 partial derivative of CO 2 at location 1 with respect to emissions of type 2

21. Rubber Bands Inversion seeks a compromise between detailed reproduction of the data and fidelity to what we think we know about fluxes The elasticity of these two “rubber bands” is adjustable Prior estimates of regional fluxes Observational Data Solution: Fluxes, Concentrations

22. MODEL a priori emission a priori emission uncertainty concentration data concentration data uncertainty emission estimate emission estimate uncertainty Bayesian Inversion Technique

23. MODEL a priori emission a priori emission uncertainty reduction concentration data concentration data uncertainty emission estimate emission estimate uncertainty Bayesian Inversion Technique

24. Our problem is ill-conditioned. Apply prior constraints and minimize a more generalized cost function: Bayesian Inversion Formalism Solution is given by Inferred flux Prior Guess Data Uncertainty Flux Uncertainty observations data constraintprior constraint

25. Uncertainty in Flux Estimates A posteriori estimate of uncertainty in the estimated fluxes Depends on transport (G) and a priori uncertainties in fluxes (C m ) and data (C d ) Does not depend on the observations per se!

26. Covariance Matrices Inverse of a diagonal matrix is obtained by taking reciprocal of diagonal elements How do we set these? (It’s an art!)

27. Accounting for “Error” Sampling, contamination, analytical accuracy (small) Representativeness error (large in some areas, small in others) Model-data mismatch (large and variable) Transport simulation error (large for specific cases, smaller for “climatological” transport) All of these require a “looser” fit to the data

28. Regional Problem is Harder! Higher density of observations over central US or Europe allows more detailed analysis High-resolution transport modeling typically requires specifying inflow! How to treat tracer variations on lateral boundaries? Can “nest” models of various scales, but how to treat errors? The smaller the region, the less trace gas variations are determined inside it!

31. Ingredients for Inversions Observations! Choices of domain, spatial & temporal structure of solution to seek Transport model to specify winds, turbulence, cloud motions Source-receptor matrix linking fluxes to observable concentrations An optimization scheme Error covariance statistics for expected model- data mismatch Numerical matrix algebra