1. 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)

2. Adjoint model as sensitivity analysis tool
Forward model (CTM):
State vector x(0)
Concentrations y(0) y(1) y(p-1) y(p)
Time t t tp 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)

3. Application to receptor modeling: sensitivity of smoke in Singapore to fire locations in Equatorial Asia
MODIS fire observations, June
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]

4. 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:

5. 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

6. Adjoint of a linear advection operator
transpose (adjoint) α
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)

7. 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:

8. Variational inversionxA
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

9. Adjoint method for calculating cost function gradient
Time t t tp tp 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
t t tp tp tp
adjoint
model

10. 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
t 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]