Adjoint modeling and applications Consider a forward model y = F(x) with Jacobian matrix The adjoint of that forward model applies KT to vectors called adjoint forcings This is useful for Determining the sensitivity of model output to model variables going back in time; Solving the inverse problem numerically rather than analytically to accommodate large state vectors (4DVAR inversion)

Adjoint model as sensitivity analysis tool Forward model (CTM): State vector x(0) Concentrations y(0) y(1) y(p-1) y(p) Time t0 t1 tp-1 tp Jacobian expresses sensitivity of y(p) to x(0): Take transpose: Apply sequentially to unit vector v = (1,0…0)T: followed by etc… Single pass of the adjoint over [tp, t0] returns sensitivity of y(p),1 to all model variables at all previous times: y(p),1/y(p-1), y(p),1/y(p-2),… y(p),1/x(0)

Application to receptor modeling: sensitivity of smoke in Singapore to fire locations in Equatorial Asia MODIS fire observations, June 20-29 2013 Smoke in Singapore, June 20 Smoke transport simulated by GEOS-Chem Large fire emissions in Equatorial Asia from oil palm and timber plantations Adjoint can identify the sensitivity of smoke concentrations at a particular receptor site to fire emissions at all locations and previous times Smoke concentration in surface air Kim et al. [2015]

Using GEOS-Chem adjoint to compute sensitivity of smoke concentrations in Singapore to fire emissions Emissions E(x, t) from bottom-up inventory Concentrations y(x, t) =F(E(x, t)) computed with GEOS-Chem Sensitivities s (x, t,t’) = ySingapore(t)/E(x, t’) computed with GEOS-Chem adjoint Contributions to Singapore smoke Smoke concentrations in Singapore can be calculated for ANY emission inventory E and time t using archived sensitivities computed just ONCE from the adjoint:

Simple construction of a CTM adjoint Consider a CTM split into advection (A), chemistry (C), and emission (E) operators: Linearize each operator so that it operates as a matrix: This defines the Tangent Linear Model (TLM) and the model adjoint: TLM Adjoint Constructing the adjoint requires construction of the TLM by differentiating the model equations or the model code…and this represents most of the work

Adjoint of a linear advection operator transpose (adjoint) α 1 2 3 Courant number α = ut/ x α α Linear upstream advection scheme: Transpose applies reverse winds: Same applies in a linear Lagrangian model (reverse the winds to get the adjoint)

Adjoint of a linear chemistry operator A linear chemistry operator is self-adjoint; same operator can be used in forward and adjoint models. Sensitivity with respect to emissions is also self-adjoint:

Variational inversion xA Solve the inverse problem ° 1 2 ° x1 numerically rather than analytically ° x2 1. Starting from prior xA , calculate 2. Using a steepest-descent algorithm get next guess x1 3. Calculate 3 x3 ° , get next guess x2 4. Iterate until convergence Adjoint model computes by applying KT to adjoint forcings Minimum of cost function J

Adjoint method for calculating cost function gradient Time t0 t1 tp-2 tp-1 tp Observations y(0) y(1) y(p-2) y(p-1) y(p) Forward model F(xA) over[t0, tp] add add add add t0 t1 tp-2 tp-1 tp adjoint model

3DVAR and 4DVAR data assimilation State vector: 3-D time-dependent concentration field (very large) Observation vector: observations of state variables or related variables Example: assimilation of satellite observations of stratospheric ozone t0 t1 = t0 + h t2 = t0 + 2h 3DVAR 4DVAR Forecast model with assimilation at time increments h 3DVAR: same as Kalman filter but minimize cost function numerically 4DVAR: use adjoint to optimize field at t0 using observations spread over [t0, t0 + h]