Adjoint models: Theory ATM 562 - PowerPoint Presentation

Slide1

Adjoint models: Theory

ATM 562FovellFall 2018(See course notes, Chapter 16)

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Motivation

We often look at a forecast and wonder “where did this come from” and “how could this be changed”, especially if it involves model errorOften our strategy involves posing a hypothesis that explains the error’s origin, changing the model initial conditions and/or model parameters to compensate, and running the model… usually over and over and over

Ideally, we want an

active, dynamic tracer

that shows how the error came about in the model – which fields, where, and whenThe adjoint model is a simplified version of the forward model that runs backwardsThe forward model predicts temperature, pressure, winds, etc., forward in time“The future evolves from the past”The adjoint model predicts sensitivity to temperature, pressure, winds, etc.., backwards in time“Traces error back to its roots” (in theory)

2Slide3

Background: simple problem

1D wave equation: M real (interior) grid points, 1 prognostic variable, 2 time level schemeRewrite in explicit form

c

> 03Slide4

Background: matrix form

Write in matrix form (interior points j = 1, 2, …, M)

M

x1

M

x

M

M

x1

4

For this simple problem, same

C

every time step.

Not true for more realistic problems.

Boundary condition needs to be applied

(currently presuming u=0 at left side)Slide5

Background: integrate model

Initial condition is u0. Integrate for N time steps.

We can relate the

final forecast

to the

initial condition

through the

transition matrix

P

N

5Slide6

Background: generalize model

x is p prognostic fields x M interior gridpoints

p

≥ 4 (i.e., u, w,

q’, p’,…) M = MX x MY x MZ

6Slide7

Tangent linear model (TLM) #1

A simple model equationExamples:

u

= prognostic variable

= model parameter

1D wave equation (parameter = -

c

)

Exponential decay (parameter = -

a

)

7Slide8

Tangent linear model (TLM) #2

Run the model twice, using two different initial conditions and/or two different values for parameter a.

Control solution

u

C(x,z,t)Alternative solution uA(x,z,t) [example: more warm vs. less warm thermal]Control parameter aCAlternative parameter

a

A

[example: faster vs. slower sound speed]

Difference between simulations and their parameters

8Slide9

Tangent linear model (TLM) #3

You can always subtract two simulations.However, the TLM is a model that attempts to estimate the difference between the control and alternative runs, based on the control run

run

run

calculate

TLM run

run

estimate

Instead of…

…do this

Why?

We’ll see…

9Slide10

Tangent linear model (TLM) - example

10

Example:

Model task 0B with

slightly different

c

or slightly shifted

initial conditions

Keep in mind:

differences have to

be smallSlide11

Tangent linear model (TLM) #4

TLM can be formed via

perturbation analysis

, and as usual presumes the perturbations are (& remain)

small so higher order terms absent1Uses Taylor series to approximate u”if

then the TLM is

11

1

See course notes for qualifications and disclaimersSlide12

Tangent linear model (TLM) #5

The perturbation model has been linearized (no u’’a

’’ term) and is constrained to (“tangent to”) the control run (

u

C, aC).“Tangent linear model”Discretize TLM and write in matrix form

12

C

n

based on

control

model run

- Run control simulation

- Archive

C

n

“every time step” (ideally)

- Initialize and run TLM

Ignore for simplicitySlide13

Tangent linear model (TLM) #6

Integrate the TLM. Initial condition is u’’0

13

C

n

based on

control

model run

- Run control simulation

- Archive

C

n

“every time step” (ideally)

- Initialize and run TLMSlide14

Tangent linear model (TLM) #7

Generic form. x

” is

p

variables by M points. Initial condition is x’’014Slide15

Forecast aspect

JThe forecast aspect

J

is something about the control run we want to examine

How did some feature appear?Why did some error occur?A J at time N is a scalar function of the control run at that timeJN = J(xN)JN can be changed by perturbing

the control run (ignoring higher order terms)

15Slide16

Change of forecast aspect ∆

JN

16

Change of

J at time N

Sensitivity

of

J

to

x

l

at time

N

Perturbation applied

to variable/location

x

l

at time

N

p

variables

M

locations

Perturb a variable/location

It only changes

J

if

J

is

sensitive

to it!!!!Slide17

Change of forecast aspect ∆

JN

17

Let

JN be surface pressure at one point, say 30˚N 60˚W, at time N∆JN is the change in surface pressure at that place and timeAt time N, ∆

J

N

is sensitive to only

one

variable and location – pressure at that location

Therefore, of the

p

x

M

terms in the sum,

only one sensitivity is nonzero

, and it is

equal to 1

Thus ∆

J

N is KNOWN information and is TRIVIALSlide18

Rewrite as an inner product

Postulate the adjoint model, a prediction model for sensitivity x*

Making this less trivial

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Take the TLM model and

Replace

x

’’ by

x

*

Transpose

C

n

Operate it

backwards

Note

C

n

T

≠ C

n

-1

, so we are NOT running the TLM backwards

From control run

Renaming sensitivity for convenience

TLM

AdjointSlide19

TLM vs. adjoint

19

The control run “information” used to step

perturbations

forward in time...

...is transposed and used to step

sensitivity

backwards

in timeSlide20

TLM and adjoint

Relating initial and final times for TLM and adjoint models

Next, we will make use of the

adjoint

propertyThis is how the adjoint model got its name…20

a

,

b

are vectors

M

x 1

L

is a matrix

M

x

M

x”

,

x*

are vectors

M

x 1

P

is a matrix

M

x

MSlide21

The recipe

21

trivial

by definition

relate final to initial time

invoke

adjoint

property

relate final to initial time

Note therefore that Slide22

The recipe

22

KNOWN

TRIVIAL

KNOWN

NOT TRIVIAL!

x* identifies what perturbations

x

’’ have

to be at the

initial time

to get that

desired change to J at time

NSlide23

Integrating the adjoint

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(1) Run the control model forwards to time

N

and archive Cn every time step (ideally)(2) Initialize adjoint model at time N(3) Integrate adjoint model backwards,

reading in

C

n

from archive

You DON’T need to integrate the TLM at all!!!Slide24

To summarize

The control simulation is made by integrating a (likely nonlinear) model forward in time, producing forecasts of temperature, pressure, winds, etc..The tangent linear model is a linearized version of the forward model, producing forecasts of perturbations (deviations) from the control forecast

The inescapable assumption is the deviations are small (truncated Taylor series)

The

adjoint model is a transposed version of the TLMThe adjoint model runs backwards in timeThe adjoint propagates sensitivity to temperature, pressure, winds, etc., backwardsIt represents an active, dynamical “tracer” that shows how model got to final stateIt also must assume that deviations are small

Our simple examples involved differentiating the model differential equation to create the TLM. In practice, we differentiate the model

code

.

For code with a lot of complicated physics, especially on/off switches (microphysics!), this is very difficult

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