Data Assimilation by the Adjoint Method in Coastal and River Models

by Graham Copeland and Igor Gejadze

Department of Civil Engineering

Workshop on Coastal Observatories.17 – 19 October 2006, Proudman Oceanographic Laboratory.

Content

Background to DA using Variational Methods:-

Direct and Inverse Models

The Variational Approach

The Adjoint Method

Examples of DA for the SWE and fsNSE

Propagation of Adjoint Sensitivity Information

Effective use of Data

Model Coupling and Data Assimilation

1D St.Venant - 2D SWE

This begs the questions:-

What data are we likely to have?

What data do we really need?

How does the data control the solution?

How do we make best use of data?

PROBLEM STATEMENT

We require to either Reconstruct or Predict

state variables (e.g. flow fields) based on:-

OBSERVATIONS and GOVERNING EQUATIONS.

Required Data – Model Controls

Topography Depth(x,y)

Friction coeff (x,y)

Current v(L,y)

Control Data

Depth(x,y)

Friction(x,y)

Initial conditions: water levels & currents.

Boundary conditions: currents or water levels .

These values are needed to completely determine the solution

but the boundary values are usually not all known from observations.

Current v(0,y)

Limited Area Model

An initial or boundary value problem, known as a direct model, becomes inverse when:- some or all of its boundary or initial data is missing and isreplaced by data located inside the domain.

data

data

data

datadata

data

ill-posed ? Inversealso ill-posed

Direct and Inverse Models

Direct and Inverse Models

So, since it most unlikely for all control data to beavailable at the boundaries, all models are inverse.

However, in practice, reasonable assumptions are made about the boundary controls and distributed coefficients and so good solutions can be found.

But there are methods available that use available data either

to recover the values of boundary controls to recover the initial conditions to recover values of distributed model parameters to recover information about sources

and so lead to:- assimilation of data into a model solution.

Variational Data Assimilation Methods

Variational Methods have the following features:-

They define an objective function J based on differencesbetween the solution of the equations and observationsi.e. a measure of ‘goodness of fit’.

Using a variational method, they calculate a gradient measure that determines how to adjust the controls

to minimise J and so improve the ‘goodness-of-fit’.

They use an iterative procedure to move step by step towards a solution that minimises J.

The Adjoint Sensitivity Method

This finds the gradient sensitivities of the type

This sensitivity information is propagated away from thedata locations back to the controls (e.g. at the boundary).

The controls are then adjusted systematically and the direct model is recomputed.

Eventually the values of controls are recovered that produce a solution that agrees with measured data.

It carries out an Objective Calibration of the model.

Useful for hindcasts and estimating initial conditions for forecasts

q

J

Integrate direct model forward

boundary conditions

xt

initial conditio

ns

Integrate adjoint model backwards

Adjust initial and boundary conditions untilSolution matches data

THE ADJOINT SENSITIVITY METHODShallow Water EquationsExample 1- dimensional(x), unsteady(t)

DATADATA

0

x

q

t

H

The Nonlinear shallow water equations:

x

z

t

H (x,t)

z(x,t)Channel bed

Free surface

0)(

/2

2

H

qqk

x

z

x

HgH

x

Hq

t

q

q

Define the Objective function & the Lagrangian:

Measure sensitivity to water level H(x0bs ,t0bs ) ≠ H0bs(x0bs ,t0bs)

Measure data mismatch : by quadratic measuring function r)( 0bsHH

)()()(5.0 002

0 bsbsbs ttxxHHr

T L

dtdxH

qqk

x

z

x

HgH

x

Hq

t

q

x

q

t

HrJ

0 0 2

2

)(/

Or, if we have flow data qobs as well as Hobs

r = 0.5 ( - obs)2 (x – xobs) (t – tobs) + 0.5 (q - qobs)2 (x – xobs) (t – tobs)

Take the first variation of J to get the adjoint equations in the Lagrange multipliers and (adjoint variables)

Source terms

02/3

22

H

r

H

qqK

x

zg

x

Hg

xHq

02

2/2

q

r

H

qK

xHq

x

Sensitivities at Control Boundaries by Adjoint Method

• Following the variational approach, the adjoint variables and give us the sensitivities :

the sensitivities of water depths at x=xobs being different from an observed depth H0bs In terms of changes in water depth “H” downstream.

),(

),(),(

),( 2

2

p

p

p

p tLH

tLqtLgH

tLH

J

( m s2 )

( m2 s )

),0( ptq

J

the sensitivities of calculated water depths at x=xobs being different from an observed depth H0bs in terms of changes in inflow discharge “q” upstream.

Hydrograph amplitude = 2 m

characteristic

Example: - Solution to forward equations

Discharge (q)

Calculated water depth (H)

Control (inflow) boundary

Driving Hydrograph

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0 0.5 1.0 1.5 2.0

Time

Dis

char

gh

(q

)

x

time

‘Source’ for Adjoint (H-H0bs)

Trial Inflow hydrograph

‘water level mismatch’ informationcreates a source thatpropagates backwards awayfrom data towards boundaries

‘sensitivity’ informationhow control boundariesaffect levels at xobs

q

Jxobs

Source term & Adjoint Solution

t

x

Source

0

200

400

600

800

1000

1200

0 100 200 300 400

time step

se

ns

itiv

itie

s m

2 s

0

10

2030

40

50

6070

80

90

0 50 100 150 200 250 300 350

time step

sen

siti

viti

es m

s2

PhiPsi

),( ptLH

J

),0( ptq

J

Solution to adjoint equations & Sensitivities on Control Boundaries

Results courtesy of H. Elhanafy, U of Strathclyde

-1 .00

0 .00

1 .00

Wave of amplitude = 1 mdriven by data at x = 0 at all times

xtime

characteristic

Data time series

Data amplitude = 0.8 m

Control boundary

Sinusoidal solution to forward equations

xt

Final

data

data

Remainder of control not recovered

Control recovered

u(0,y,t)

y = 0 x

y

bed

Surface elevation data

Current dataUnknown boundary condition

),(ˆ txS k

),,(ˆ 1 tYxu l

),,(ˆ 2 tYxu l

flu id

L/2

channel bed 2/3

Z

L /2

passive boundary:

open

'open D irichet 'contro l boundary

X

Y

f(0 )=0

Z

free surfaceS(0,t)

L /4

e levation andve locity sensors

u

v

Adjoint Method2D section fsNSE

Example problem

definition

2D vertical section

p(x,y,t)

Adjoint variables

S*, u*, v*, p*

flu id

L/2

channel bed 2/3

Z

L /2

passive boundary:

open

'open D irichet 'contro l boundary

X

Y

f(0 )=0

Z

free surfaceS(0,t)

L /4

e levation andve locity sensors

u

v

Adjoint pressure p* at mid-depth based on velocity measurements

velocitysensors

step

Antiphase.Discontinuity in p*

Adjoint solution showing propagation of p* through model

S*

p*, u*,v*

c = (g d)0.5 20 ms-1

Wave transport Advectionu = 1 ms-1

800 m data‘source’

u*,v*

con

tro

l b

ou

nd

ary

y=0

y=40 m

Transport mechanisms in adjoint model•Wave transport affects all depths•Advective transport preserves profile but only within advective range

location of s ingulardriv ing source

(u - ve locity sensor)

Non-uniform profile in the vertical due to convertive transport

u - sensitivity at the control boundary.Wave disturbance arrives firstas p*followed by convective termsas uu*

Wave celerity = 20x convective speed

Vertical spread due relative vertical motion of source, not diffusion

u sensitivity through vertical at x = 0 due to a single velocity source at mid-depth at x = 800 m

Infl

ow

co

ntr

ol

bo

un

dar

y

Recovered by advection at speed u

Rec

over

ed b

y

plan

e SW

wav

e at

spee

d c=

(g h

)0.5

Observed

Current u

Separate influences on boundary control

by advection and by wave propagation in

2D SWE

Recovery of Boundary

Controls in presence

of standing waves

inflow open

Recovery of Boundary

Controls in presence

of standing waves.

Data from nodes does not recover controls at all.

Only H or only u does not recover controls very well

Co-located H and u recovers controls best.

Must retain phase info’

inflow closed

Recovery of distributed parameters by Adjoint DA

)()()(5.0 002

0 bsbsbs ttxxHHr

),()(

fnxz

J

T L

dtdxH

qqk

x

z

x

HgH

x

Hq

t

q

x

q

t

HrJ

0 0 2

2

)(/

Fom observations of water level H and velocities q/H we can recover bathymetry z(x).

This assumes that the boundary controls are known

For example recover y of bathymetry in SWE model

2 D example. Reference bathymetry used in an identical twin experiment

Flow field computed over this bathymetry and used to provide ‘observations’ of H and velocity at sparse locations

DA proceeds from an initially plane bed to recover bathymetry from sparse observations of H

Recovered z(x)

Recovery of bathymetry by DA of co-located observations of velocities u,v and H

Demonstrates the importance of having both u,v and H data

Initial z(x)

Recovered z(x)

Results courtesy of M. Honnorat, INRIA Rhone-Alpes, Grenoble

Flow modelFlow model - 1 - 1G lobal ocean (PE or SW E)

Estuarian/ m iddle stream(2D SW E)

F luvia l (S t.VE)

Precip ita tion, evaporation, w ind stress, heat flux,ground w aters, sedim ents- from other m odels

Tidal zone

C oasta l (PE or fsN S)Surface water flow modelSurface water flow model – essential part of the full physical model used for eco-modelling.

Flow model – 2Flow model – 2

G lobal ocean (PE or SW E)

Estuarian/ m iddle stream(2D SW E)

F luvia l (S t.VE)

Precip ita tion, evaporation, w ind stress, heat flux,ground w aters, sedim ents- from other m odels

Tidal zone

C oasta l (PE or fsN S)

It does not look always possible to create a single (integral) surface flow model based on most complete flow eqns (3D fsNSE). It does not look elegant either.

Can we use the existing models (blocks) to Can we use the existing models (blocks) to simulate integral phenomena?simulate integral phenomena?

Assuming all blocks are based on different eqns, implemented using different methods, run by different users in different places, runs are not necessarily synchronized, etc. No essential modifications in existing software are allowed!

Can we assimilate data collected in different parts of the flowCan we assimilate data collected in different parts of the flowto model integral phenomena without creating the single (integral)to model integral phenomena without creating the single (integral)model? model?

NO DEFINITE ANSWER !

Paradoxical, but more definitely YES !

Simple flow model: 1D St.Venant-2D SWE

shallow water flow m odel by 1D S t.Venan

2D SW E m odel

.

This simple set contains most properties of the general case.

Interface between models is ‘liquid’ or ‘open’ boundary.

1) What must be specified at the model interface to provide correct transfer ofinformation (in both directions!)?Exact answer exists only for 1D SWE

3) How to arrange information exchange in time? Time step is very different ! Different models could be run in different places!

2) Keep in mind: models are different. 1D St.V model cannot provide sufficient information to specify well-posed 2D SWE model by definition.

Flow example with propagating dry/wet front: reference solutionFlow example with propagating dry/wet front: reference solution

a) b)

c) d)

Difficulties in coupling flow models - 1

Interface between models is ‘liquid’ or ‘open’ boundary.

Therefore, all difficulties related to ‘open’ boundaries are applied here.What information should be transferred between models?How to synchronize information transfer?

Trivial case: two overlapping sub-domains, information exchange every time step, explicit numerical scheme

1

2

12

1'

2'

Derichlet or Neumann

Why does it work? Because both Derichlet and Neumann specify well-posed semi-discrete (frozen time) problem in sub-domain.However, not a global time problem!!!

General rule to follow:Boundary conditions applied at the ‘open boundary’ model interface must specify well-posed flow problem for each model (sub-domain).

Difficulties in coupling flow models - 2

Can we afford information exchange every time step?Do we use explicit solvers for all models?Are the models perfectly synchronized?

NO

Waveform relaxation methodWaveform relaxation method, orGlobal Time Schwarz methodGlobal Time Schwarz method(E. Lelarasmee, L.Halpern, M. Gander, …)

Need for ‘global time’ boundary conditions‘global time’ boundary conditions at the model interface, which specify a well-posed flow problem for each model.

Characteristic analysis Characteristic analysis : systematic (but still approximate) way

Characteristics and higher order invariants – 1

x,u

y,v

'W est boundary'

w

w

w

1

2

3

Characteristic analysis of the 2D SWE:Eigenvalues:

Invariants:

cuucu ,,

hucqw

vhpw

hucqw

)(

,

,)(

3

2

1

Characteristic analysis is approximate (even assuming small bathymetry and friction slopes) since x is considered as dominant incidence and flow direction.This results to the partial reflection (when using w asbc), which increases when incidence direction deviates from normal.Problem: flow may support many wave packages coming by different incidence angle.

Incoming invariants to be prescribed !,outgoing – extrapolated from the interior

Characteristics and higher order invariants – 2

1

'

2

21

Using characteristic invariants at the interface between sub-domains iseventually equivalent to the waveform relaxation method.The solution can be obtained by successively solving problems insub-domains (global time Schwarz ).In theory (for the 1D SWE), the process should converge in 2 iterations!

Outgoing quantities from = Incoming quantities to1 2Vice versa

0,

,1

22

2

11

)2(1

unww

ww

0,

,2

21

2

23

)1(3

unww

ww

Characteristics and higher order invariants – 3

More complex invariants can be build based on the theory of absorbing BC(B. Engquist, A. Majda)Higher order invariants used as bc allow reflections of waves coming to the boundary by angles deviating from the normal to be essentially reduced.

Example: second order invariants (E. Mazauric-Nourtier, build for

linearized SWE )

33

33222

211

,

2,

,4

2),(

wtyW

c

fw

y

wuc

y

wv

t

wtyW

y

wuc

t

wtyW

(in)

(in)

(out)Remarks:1. Still (x) remains the dominant flow and incidence direction. Improves treatment of waves in a certain cone around the normal.2. Numerical implementation of higher order invariants (more than 2nd)could be unstable.

Difficulties in coupling flow models - 3

The main problem is 1D->2D information transfer.The 1D model cannot provide full set of boundary conditions for the 2D

model :a) For the 2D model all invariants are distributed in the tangential direction.b) There is no tangential invariant in the 1D formulation exists.

x,u

y,v

1D SW E 2D SW E

1 2

'

1D->2D

2D->1D

11

)2(1 bwdw How to distribute along interface?

Need distribution rule! Uniform??

)2(1w

dwbw )2(3

13

x,u

y,v

'W est boundary'

w

w

w

1

2

3

No problem with it. No point in controlling

0,22 unw No information available! Trivial??

13w

Model coupling: steady-state, non-homogenous, optimal control - 1

Virtual control methods: J-L. Lions, O. Pironneau, A. Quarteroni

1

2

12

1'

2'

211211

11111

111

,

)(,)(

),(,

ugu

yxfuL

122122

22222

222

,

)(,)(

),(,

ugu

yxfuL

01 )( bbdivL 02 )( bbdivkL

convection convection-diffusion

2,1 - virtual controls

21

2221121 2

1,

dxuuJ

21, ,inf21

JCoupling is considered as the minimization problem:

Model coupling: steady-state, non-homogenous, optimal control – 2

1

'

2

21

A disjoint partition of or non-overlappingnon-overlapping case

dsdsuuJ 221

22121 2

1

2

1),(

11 unb

222 unbunk

The objective function reflects our a-priori ‘natural’ guess about solutions at the coupling interface.

Model coupling: global time, non-homogenous, optimal control – 3

x,u

y,v

1D SW E 2D SW E

1 2

'

Coupling is considered as the minimization problem for the objective functional:

T

vuuhh

T

ddddt

bwdwdtwJ

0

22212212

0

21

12

1)2(

1

2

1

2

1

vuh ,, - fluxes

dwbw )2(3

13Open bc for 1D:

Incoming invariant for 2D problem is the unknown control. 21w

Solution of the control problem gives result which tends to the solution bySchwarz iterations under trivial assumptions. Thus, by itself optimal control approach is not enough to get better results.

Its advantage is that it may allow additional information to be used.

Model coupling & data assimilation

x,u

y,v

1D SW E 2D SW E

1 2

'

sensors

If data is available and has to be assimilated,then coupling and assimilation can be performed in a single optimization loopsingle optimization loop. .

ddddt

bwdwdtwJwJ

vuuhh

T

T

22212212

0

2

0

21

12

11)2(

10)2(

1

2

2,,

Extended objective functional:

Could contain other controls, i.e. 0,11 xtw

This approach should be preferred for the non-homogenous modelbecause the ‘rigid’ coupling at the interface could be eventually harmful.Otherwise, solution in 2D area is dominated by data, which fills in gaps in our knowledge about bc at the coupling interface.

Common DA objectiveCommon DA objective

Numerical results on WFR coupling

(t)

1

2

51

2

3 4

6(t)

x, x '

y, y '

L

sensor A

sensor B

Generalized objective functionGeneralized objective function

21 JJJJ

)ˆ(,

2

1

2

2

UUmlar

dtdyyxxrJ

h

l mml

DA component

T

dtIWdwJ0

2111 )|)(

2

1

33

For the inlet coupling interface

For the outlet coupling interface

For the interfaces we use exactly what we know !

T

dtIWdwJ0

2222 )|)(

2

1

44

Joint DA-coupling algorithm – 1 (consistent 1D/2D meshes)Joint DA-coupling algorithm – 1 (consistent 1D/2D meshes)

Results of data assimilation: recovering the unknown inlet BC for the 1D model (together with BC for the 2D local model, which are not presented). Both elevation and discharge data are used.

The convergence history: for the generalized objective function (in bold line) and for its components

Joint DA-coupling algorithm – 2 (consistent 1D/2D meshes)Joint DA-coupling algorithm – 2 (consistent 1D/2D meshes)

Results of data assimilation: recovering the unknown inlet BC for the 1D model (together with BC for the 2D local model, which are not presented). Only elevation data are used.

The convergence history: for the generalized objective function (in bold line) and for its components

Conclusions

1. Coupling of flow models can be achieved together with DA within the same control loop.

2. The models and their adjoints run independently exchanging information between global time runs. No need to create single integrated model or generate its single integrated adjoint.

3. This arrangement should be preferred when incompatible (due to different reasons) models are involved into joint consideration. Measured data fills gaps in a-priori data needed in ‘coarse-to-fine’ information transfer. Thus, coupling together with DA is simpler task than without.

4. Boundary conditions applied on the model interfaces should be based on characteristics or higher order characteristic invariants.

5. In the overlapping areas the ‘finer’ model should adjust ‘coarser’ model via ‘defect correction’