Impact of Observations on
Adjoint Sensitivity
S. Lakshmivarahan
School of Computer Science
University of Oklahoma
Norman, OK - 73019, USA
varahan@ou.edu

January 11, 2012

Acknowledgement: Joint work with John Lewis of NSSL/DRI

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Outline

Motivation
Computation of adjoint sensitivity
Forward Sensitivity
Relation between the two sensitivities
Impact of observations
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Motivation
Quantifying the impact of observations on forecast aspect - rainfall,
has a long and cherished history
Observation System Simulation Expriments (OSSE) are still being
used to assess the improvement in the analyses using various sets ofmodel generated observations
Targeted Observation approach on the otherhand, strives to identify
the impact of a particular subset of observations on a specified model
forecast aspect
This latter approach relies on expresing the sensitivity of the forcast
aspect w.r.to observation using the adjoint sensitivity
By expressing the adjoing sensitivity as a linear combination of thethe forward sensitivities, we uncover the fine structure of the adjoint
sensitivity using which we can identify the temporal zones where theobservations are likely to have little or no influence in the adjoint
sensitivity
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Literature on OSSE

E. Andersson, A. Hollingsworth, G. Kelly, J. Pailleux and Z. Zhang(1991) Global observation system experiments on operational
staistical retrivals of satellite data, MWR, Vol 119, 1851-1865
R. Atlas (1997) Atmospheric observations and experiments to assess
their usefulness in data assimilation, J. Meteor. Soc. Japan 75,111-130
C. Cardinali (2009) Monitoring the observations impact on the shortrange forecast, Q.J.R.M.S. Vol 135, 1469-1480

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Literature on Targeted observations

N.L. Barker and R. Daley (2000) Observations and backgroundadjoint sensitivity in adaptive observation-targeting probelm,QJR.M.S., Vol 126, pp1431-1451
R. Errico (2007) Interpretations of an adjoint-derived observational
impact measure, Tellus, Vol 59-A, 272-276
D. N. Daescu and R. Toddling (2009) Adjoint estimation of thevariation in model functional output due to assimilation of data,
MWR, Vol 137, 1705-1716
R. Gelaro, Y. Zhu, and R. Errico (2007) Examination of various-orderadjoint based approximation of observation impact, Meteor. Zeit. Vol16, 685-692

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Adjoint method - a resumex ∈ R n, a ∈ R p, M : R n × R p → R n
Model: x(k + 1) = M(x(k), a)x(0) Initial condition, a is parameterh : R n → R m is the forward operator
Observation: z(k) = h(x ∗ (k)) + v (k)x*(k) is the unknown but true state of the systemv(k) is the white Gaussain noise with known covariance (i.e.) N(0, R)

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6 / 35Forecast error and cost functional

Error e(k) = z(k) − h(x(k)) = [h(x ∗ (k)) − h(x(k))] + v (k)Cost functional: J : R n × R p → RP−1 e(k)J(x(0), a) = 21 N
k=1 e(k), Rhx, Ay i denotes the inner product of two vectors x and Ay
Goal is to minimize J w.r.to (x(0), a)

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Optimization problem

This is an off-line (batch) constrained minimization where the model
equations define the equality constraints
There are two ways to formulate it - storng or weak constrained
problem
4-D VAR first-order adjoint method is a result of the strong
constrained formulation
Solved by using the Lagrangian multiplier technique or can also besolved by using the method of first variation which is followed below
Weak constrained formulation is useful to account for the model
errors and uses a penalty functional

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First variation of J

δJ(x(0), a)) =PN

k=1 (f (k), δx(k))
Tf (k) = −Dh (k)R −1 e(k)Dh (x(k)) ∈ R m×n is the Jacobian of h(x) evaluated at x(k)f (k) is normalized forecast error viewed from model spaceδx(k) is the first variation of x(k) induced by the variation δx(0) inx(0) and δa in a propagated by the model dynamics

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Tangent linear model - propagation of perturbations

Model:x(k + 1) = M(x(k), a)δx(0) and δa are given initial perturbations
δx(k + 1) = A(k)δx(k) + B(k)δaA(k) = DM (x(k)) ∈ R n×n is the Jacobian of M w.r.to the state x
B(k) = DM (a) ∈ R n×p is the the Jacobian of M w.r.to parameter a
The above recurrence relation for δx(k) can be solvedPδx(k) = A(k − 1 : 0)δx(0) + k−1
j=0 A(k − 1 : j + 1)B(j)δaA(i : j) = A(i)A(i + 1)...A(j) is the product of the Jacobian along thetrajectory from time i to j when i ≤ jS. Lakshmivarahan (School of CS)

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Two properties of inner products

Adjoint: hx, Ay i = AT x, ySummation: : hx, Azi + hy , Azi = hx + y , Azi
Linear operators in finite dimensional vector spaces are represented bymatrices. For these operators, transpose is the adjoint

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Gradient of J w.r.to IC X(0)Substituting δx(k) in the expression for δJ and simplifying:
DPENT (k − 1 : 0)f (k), δx(0)δJ =A
k=0EDP PNk−1 TT (k − 1 : j + 1)f (k), δa
B(j)A+k=0
j=0
From first principles:δJ = ∇x(0) J, δx(0) + h∇a J, δaiComparing term by term we get the expressions for the adjoint
sensitivities
PT∇x(0) J = N
k=1 A (k − 1 : 0)f (k)S. Lakshmivarahan (School of CS)

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Adjoint dynamics - initial conditon

Computation of the adjoint sensitivity using the above summation isvery inefficientAn efficient method is to compute it recursively using the adjointdynamics
λ(k) = AT (k)λ(k + 1) + f (k)Final condition: λ(N) = f (N)Iterating backward from N to 0, we get
Adjoint sensitivity: λ(0) = ∇x(0) J
Complexity is O[N(n2 + n)]

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Gradient of J w.r.to a

Recall ∇a J =PN Pk−1
k=0

j=0 B

T (j)AT (k − 1 : j + 1)f (k)This expression, since it involves double summation, takes longer time
to compute
we can likewise rewrite it as a recursive computation using a doubly
nested loop
Complexity is O[N 2 (n2 + n) + N(n2 )]Computation of the adjoint sensitivity w.r.to parameters is
considerably more expensive compared to that w.r.to the initialcondition

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Adjoint dynamics - parameters
for k = 1 : N
λ(N) = f (N)for j = N-1:kλ(j) = AT (j)λ(j + 1) + f (j)end - j loopλ(k) = B(k − 1)λ(k)sum = 0.0
for i = 1:N
sum = sum + λ(i)end - i loop
end - k loop

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January 11, 201215 / 35Reference on adjoint methods

E. Kalnay (2003) Atmospheric Modeling, Data Assimilation, andPredictability, Cambridge University Press, 341 pages
J. Lewis, S. Lakshmivarahan and S. K. Dhall (2006) Dynamic Data
Assimilation: a least squares approach, Cambridge University Press,654 pages

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Forward Sensitivity Method - a summary

Recall that error e(k) = z(k) − h(x(k)) = [h(x ∗ (k)) − h(x(k))] + v (k)This is contingent on the IC, x(0) and parameter a
Goal is to find corrections δx(0) and δa :The solution from (x(0) + δx(0)) and (a + δa) will annihilate theerror e(k)S. Lakshmivarahan (School of CS)

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A pictorial view-single observation

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First variation in h(x)δh(x(k)) = Dh (x)δx(k)Dh (x) ∈ R m×n is the Jacobian of h(x)Next step is to compute δx(k)S. Lakshmivarahan (School of CS)

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Computation of δx(k)Let δx(0) and δa be the perturbations in x(0) and aLet δx(k) be the induced perturbation in x(k)From the Model: x(k + 1) = M(x(k), a)
we get δx(k) = V (k)δx(0) + U(k)δaV (k) is the (forward) sensitivity of the solution x(k) w.r.to x(0)
i (k)
n×n
V (k) = [Vij (k)] = [ ∂x∂xj (0) ] ∈ R

Likewise U(k) is the (forward) sensitivity of x(k) w.r.to theparameters ai (k)U(k) = [Uij (k)] = [ ∂x∂a] ∈ R n×p
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Basic Equation in FSM

Substitutingδx(k) in δh, we gete(k) = δh = Dh [V δx(0) + Uδa]e(k) = [H1 , H2 ][δx(0), δa]T
where H1 = Dh V ∈ R m×n , H2 = Dh U ∈ R m×p ,Define H = [H1 , H2 ] ∈ R m×(n+p)
We get the standard linear least squares problem:e(k) = Hζ where ζ = [δx(0), δa]TS. Lakshmivarahan (School of CS)

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Multiple Observations

Let there be N observations z(k) for k = 1 to n
Let e = (e T (1), e T (2), ...e T (N))T ∈ R Nm be the collection of all theerrors
Let H = [H1 , H2 , ...HN ] ∈ R Nm×(n+p) matrix products of Jacobian ofh (x) and the forward senstivities
Solve the resulting linear least squares problem:
Hζ = eS. Lakshmivarahan (School of CS)

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Evolution of forward sensitivities w.r.to x(0)Model: x(k + 1) = M(x(k), a)
Differentiate this model equation w.r.to x(0):Forward Sensitivity dynamics w.r.to x(0) in component form:
P ∂Mi (k) ∂xq (k)∂xi (k+1)∂xj (0) =∂xq (k) ∂xj (0)Thus we get V (k + 1) = A(k)V (k) with I.C. V (0) = I
Iterating V (k) = A(k − 1)A(k − 2)...A(1)A(0)Denote V (k) = A(k − 1 : 0)

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Evolution of forward sensitivities w.r.to a

Model: x(k + 1) = M(x(k), a)
Differenciate the above model equations w.r.to a in component form:
Forward Sensitivity dynamics w.r.to a:
P ∂Mi (k) ∂xq (k) ∂Mi∂xi (k)∂aj =∂xq∂aj + ∂aj
We get U(k + 1) = A(k)U(k) + B(k) with U(0) = 0
P
Iterating: U(k) = k−1
j=0 A(k − 1 + j)B(j)S. Lakshmivarahan (School of CS)

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S. Lakshmivarahan and J. Lewis (2010) Forward sensitivity approachto dynamic data assimilation”’ Advances in Meteorology, doi:
10.1155/2010/375615
S. Lakshmivarahan and J. Lewis (2011) Impact of observation onadjoint sensitivity, J. Meteor. Soc. Japan (under review)

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Fine Structure of adjoint sensitivity w.r.to x(0)∇x(0) J =PN

T
k=1 A (k − 1 : 0)f (k)Using the defintion of V(k):PT∇x(0) J = Nk=1 V (k)f (k)In words:
adjoint sensitivity w.r.to I.C. is the sum of the products of thetranspose of the forward sensitivity w.r.to the IC, V (k) and thenormalized forecast error, f (k) = −DhT (k)R −1 e(k) viewed from themodel space where e(k) = z(k) − h(x(k))

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Fine Structure of adjoint sensitivity w.r.to a∇a J =PN

Pk−1

k=0 [j=0 BT (j)AT (k − 1 : j + 1)]f (k)Using the defition of U(k): ∇a J =PN

k=0 UT (k)f (k)In words:
adjoint sensitivity w.r.to parameters is the sum of the products of thetranspose of the forward sensitivity w.r.to the IC, U(k) and thenormalized forecast error, f (k) = −DhT (k)R −1 e(k) viewed from themodel space where e(k) = z(k) − h(x(k))

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Adjoint vs. forward sensitivity
Recall adjoint sensitivity is computed using one forward run of themodel, compute the normalized forecast errors f (k) and using it as
the forcing, run the backward adjoint model which is always linear
In FSM on the otherhand, the forward sensitivity V (k) is also also
computed using the model and the sinsitivity equations both forward.
There is no need to compute the adjoint model.
The adjoint model related the evolution of the Lagrangian multiplier
which is a vector, but the forward sensitivity is a matrix which has asystem of n-vector equations to be solved simultaneously.
When done serially, FSM will require more time, but V (k)computation can be readily implemented in parallel - it isembarassingly parallel
Despite its computational demands, it provides very useful
information on the impact of observations on adjoint sensitivity
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Impact of observations
Despite its computational demands, FSM provides very useful
information using which we can in addition also use it to evaluate theimpact of the observation.
For example if the qth column of V (k) is very small, that is, when theqth compnent xq (k) is insenitive to the elements of xj (0) for all j= 1to n, then the qth row of V (k) is zero and the qth element of theproduct V T (k)f (k) will have its qth component also very close tozero.
That is, the observation at time k does not materially affect theadjoint sensitivity
A similar commnets apply to the adjoint sensitivity w.r.to theparameter aHere in lies the advantage FSM based approach. By running themodel, FSM for V (k) and U(k) off-line in parallel, we should be able
to assess the impact of the location of the observation on the adjoint
sensitivity
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Example

Scalar dynamics: ∂x∂t = CT (θ − x)
Scalar dynamics: x(k + 1) = (1 − c)x(k) + cθx(0), c, θ are three control variables
Solution: x(k) = (1 − c)k (x(0) − θ) + θ
Sensitivities of the solution:∂x(k)k∂x(0) = (1 − c) - exponentially decreases to zero with k∂x(k)k∂θ = [1 − (1 − c) ] - exponentially increases to 1 with k∂x(k)k−1 [x(0) − θ] - decreases to a minimum and then∂c = −k(1 − c)increases to zero

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Solution and its sensitivities

Solution saturates, two sensitivities go to zero and the third one tends to θ

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Fine structure of adjoint sensitivity - Well placed
observations

Example 1 : Generate observation from x(0) = 1, θ = 11, c=0.25
Observations are chosen at times k =1, 2, 17, 18
Forecast started from x(0) = 2.0, θ = 10, c = 0.30
∇x(0) J = 0.971, ∇θ J = −1.267 and ∇c J = 1.436
Notice that the gradients are of the correct sign

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Fine structure of adjoint sensitivity - Ill placed observations
Example 2: Generate observation from x(0) = 1, θ = 11, c=0.25
Observations are chosen at times k =15, 16, 17, 18
Forecast started from x(0) = 2.0, θ = 10, c = 0.30
∇x(0) J = −0.011, ∇θ J = −3.721 and ∇c J = −0.203
Notice that the gradient of w.r.to θ dominates and the othergradients are small
The small values of the two gradients is a direct consequence of therelative insensitivity of the solution w.r.t. x(0) and c when thesolution reaches a saturation stage
Also the sign of the gradients ∇x(0) J and ∇c J are inconsistent
Smallness of the gradients further indicate that inherent flatness ofthe J function the the θ and c directions

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Conclusions
Forward sensitivities of the solution can be used in two ways
First, it can be used to isolate the effective from the ineffective
temporal regions for placing the observations in order that they havea good impact on the adjoint gradient with respect to the chosenforecast aspect
Second, once the sensitivities are made available they can be readily
used to compute the adjoint sensitivity for use in an optimization
procedure
FSM requires more computations compared to the calssical adjoint
method, but it is amenable to embarassingly parallel computation.
This extra cost is well worth the effort since it helps to identify the”best” rigions to place the observations. It is a great off-line tool to
do diagnostic studies

S. Lakshmivarahan (School of CS)

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January 11, 201234 / 35Other applocations of FSM

E. Tromble, S. Lakshmivarahan, R. Kollar and K.M. Drescback(2011) ”Application of Forward Sensitivity Mtehod to a GWCE-based
Shallow Water Model”, Computational Geosciences, (Under Review)
D. Phan, S. Lakshmivarahan. E. Weng and Y. Luo (2011) ”ForecastError Correction in Ecosystem Carbon Pool Model using Forward
Sensitivity Method”, Under Preparation

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