Estimating emissions from observed atmospheric concentrations: A primer on top-down inverse methods Daniel J. Jacob

The inverse modeling problem Quantify selected variables driving a physical system (state vector x, dim n) by using: –The observable manifestations of the system (observation vector y, dim m ) –a physical model relating x to y (forward model F) –consideration of errors (error covariance matrix S). prior estimate x A ± S A observations y ± S O forward model y = F(x) ± S M optimal estimate statistical fit Forward model gives y = F(x), but solving x = F -1 (y) is not possible due to errors Need instead a statistical fit: “inversion” “data assimilation”

Inverting for emissions: a conceptual view 2-D gridded emission field x Observed atmospheric concentration y i Sensitivity Fit x to y Emission footprint Incude prior information x A ± S A to: target the fit to most relevant components prevent unphysical solutions weigh observations vs. prior knowledge quantify information content of observations from chemical transport model (forward model) transport Adjoint of forward model (reverse flow)

Chemical transport model (CTM): forward model for inverting emissions from atmospheric observations Solve 3-D continuity equation for chemical concentrations in the atmosphere: change in concentration with time grid-resolved transport (advection) subgrid transport (turbulence) chemical production and loss emission, deposition solve for fixed gridboxes over  t utut toto to+tto+t Eulerian framework solve for point masses moving with the flow Lagrangian framework Resolution of current models: ~100 km (global) ~ 10 km (continental) ~ 1 km (regional)

Backward Lagrangian particle dispersion model (LPDM) Run Lagrangian model backward from receptor location, with points released at receptor location only Single simulation defines footprint of a given observation Adjoint of forward Lagrangian model Efficient with limited number of local observations flow backward in time

CTM relating methane emissions to concentrations is linear simulated concentrations are proportional to emissions This is not an obvious result because methane and OH are sinks for each other: Emission ↑ [CH 4 ] ↑ [OH] ↓ [CH 4 ] ↑ ↑ However, we know [CH 4 ] from global background observations to within a few % so we should not let model changes in [CH 4 ] affect [OH] The linear CTM is solely defined by its Jacobian matrix K: The CTM adjoint is the transpose K T (reverse flow)

Bayes’ theorem: foundation for inverse modeling P(x) = probability density function (PDF) of x P(x,y) = PDF of (x,y) P(y | x) = PDF of y given x prior PDF observation PDF normalizing factor (unimportant) posterior PDF Optimal estimate solution for x given y is max[P(x | y)] solve for  P(x,y)dxdy  Bayes’ theorem

General Bayesian solution to the inverse problem State vector Observation vector Forward model Prior x A with prior error covariance matrix S A Observations y with observation error covariance matrix S O prior PDF P(x)Observation PDF P(y | x) solve

Error covariance matrices and PDFs for vectors Consider vector x with expected value E[x] and error Error covariance S : Gaussian error PDF:

Optimal estimate solution assuming Gaussian errors prior PDF observation PDF Minimize cost function: solve

Linear forward model allows analytical solution y = F(x) = Kx where Is the Jacobian matrix Solution:with gain matrix posterior error covariance matrix Relate solution to true value: with A = GK averaging kernel matrix smoothing error observational error truth

A little more on the averaging kernel matrix A describes the sensitivity of the optimized estimate to the true state and hence the smoothing of the solution: Analytical inversion gives A as part of the solution: The trace of A gives the # of independent pieces of information in the inversion – also called the degrees of freedom for signal (DOFS) smoothing error observational error truth

Pros and cons of analytical solution to inverse problem PRO: Complete error characterization as part of the solution CON: Calculating Jacobian matrix can be expensive Assumption of Gaussian errors may not be best, allows negative solutions Prior error PDFs Lognormal error PDF may be more physical and enforces positivity - but does not allow analytical solution

Adjoint-based numerical solution ° ° ° ° AA 22 11 33 x2x2 x1x1 x3x3 xAxA Minimum of cost function J Solve the inverse problem by applying adjoint K T to 1. Starting from prior x A, calculate 2. Using a steepest-descent algorithm get next guess x 1 3. Calculate, get next guess x 2 4. Iterate until convergence

Pros and cons of adjoint-based solution to inverse problem PRO: No computational limitation on size of state vector Non-Gaussian error PDFs can be accommodated CON: No error characterization as part of the solution Requires availability of CTM adjoint

Monte Carlo Markov Chain (MCMC) solution to inverse problem Brute-force calculation of posterior PDF: Start from x A, calculate P(x A ) and P(y|x A ) and from there P(x A |y) Now apply a random displacement x 1 = x A + ∆x, calculate P(x 1 |y) Apply next displacement ∆x to either x A or x 1, calculate P(x 2 |y) Apply next displacement ∆x to either x 1 or x 2, calculate P(x 3 |y) … eventually the full PDF P(x|y) is constructed xAxA x1x1 x2x2 x3x3 sample population PDF isolines PRO: Use any form of prior PDF Full posterior PDF is calculated Positivity of solution can be enforced CON: Computationally expensive

Summary flow diagram for inverse methods

Selection of state vector If state vector is too large, cost function is dominated by prior: smoothing error Correct this by aggregating state vector elements, but this incurs aggregation error There is an optimal state vector dimension for fitting observations: # state vector elements aggregation smoothing As dim(x) increases, the importance of the prior terms increases Prior Observations native grid aggregated grid

Optimal clustering of 1/2 o x2/3 o gridsquares Correction factor to bottom-up emissions Native CTM resolution (n = 7096) 1000 clusters SCIAMACHY data cannot constrain emissions at 1/2 o x2/3 o resolution; reduce to 1000 clusters Wecht et al. [2014a] Selection of emission state vector in methane source inversions

Quantifying stratospheric contribution to methane columns solar occultation satellite data Stratosphere contributes 5% of methane column in tropics, 20% at high latitudes Stratospheric meridional transport in models tends to be too fast, causing overestimate of methane columns at high latitudes Not correcting for this bias can cause ~5% bias in inversion results at northern mid- latitudes, 40% in Arctic Patra et al. [2011], Ostler et al. [2015] transport observations model

Observing System Simulation Experiments (OSSEs) to evaluate future satellite missions