Representation Error in Ocean Data Assimilation Robert N. Miller - - PowerPoint PPT Presentation

▶
representation error in ocean data assimilation
SMART_READER_LITE
LIVE PREVIEW

Representation Error in Ocean Data Assimilation Robert N. Miller - - PowerPoint PPT Presentation

Apr 13, 2023 198 likes •514 views

Representation Error in Ocean Data Assimilation Robert N. Miller James G. Richman College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, OR 97330 Representation Error in Ocean Data Assimilation p.1/30 Data

slide-1
SLIDE 1

Representation Error in Ocean Data Assimilation

Robert N. Miller James G. Richman College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, OR 97330

Representation Error in Ocean Data Assimilation – p.1/30

slide-2
SLIDE 2

Data Assimilation: Assumptions

Given

  • A model: ut − Lu = f
  • Chosen to mimic the “true” state u(t) which

evolve according to: u(t)

t − Lu(t) = f + b; b random

  • Observations z = Hu(t) + eobs; H defines the

relation between the state vector and the observed quantities

  • Question for Today: What, precisely, is u(t)?

Representation Error in Ocean Data Assimilation – p.2/30

slide-3
SLIDE 3

In Search of the True State

  • The ocean measured by instruments doesn’t

know about physical approximations, coarse resolution or their consequences

  • It is not subject to the limitations in computing

power that restrict models to coarse resolution

  • Measurements are not subject to the same

requirements for approximate physical parameterizations So ask: What quantity in nature is the “true” value of the model state? No specific answers today; Rather a suggestion for what to do while we are waiting.

Representation Error in Ocean Data Assimilation – p.3/30

slide-4
SLIDE 4

Representation Error

  • Data assimilation makes use of data misfits, aka

innovations: z − Hu(f)

  • u(f) is the forecast state
  • Let ˜

ut be the “true” ocean, as the instruments measure it.

Representation Error in Ocean Data Assimilation – p.4/30

slide-5
SLIDE 5

Representation Error

Write the innovation: z − Hu(f) = z − z(t) + z(t) − Hu(f) = ǫ0 + H(˜ u(t) − u(t)) + H(u(t) − u(f))

  • ǫ0 = z − z(t), the instrument error
  • H(˜

u(t) − u(t)) is the representation error

  • Estimates of its statistics must appear in the terms

reserved for instrument error

  • u(t) − u(f) is the forecast error

Representation Error in Ocean Data Assimilation – p.5/30

slide-6
SLIDE 6

Estimating Representation Er- ror

Our method for estimating the representation error for SST:

  • 1. Generate a long model run
  • 2. Calculate EOFs of the model run, considered as a

matrix whose (i, j) element is the value of state element j at time i

  • 3. Determine the number of meaningful degrees of

freedom

  • 4. Project the innovations on the meaningful EOFs
  • 5. Subtract the result from the innovations.
  • 6. The difference is an estimate of the representation

error

Representation Error in Ocean Data Assimilation – p.6/30

slide-7
SLIDE 7

Pacific Circulation Model

  • Parallel Ocean Program (POP)
  • Domain:
  • 105oE to 85oW
  • 30oS to 64oN
  • Resolution
  • 1o at the Equator, Mercator projection
  • 0.5o average resolution
  • 50 vertical levels, 25 in top 500m
  • 25 years (1978-2002), forced by NCEP/DOE

reanalysis

  • Initialized from Levitus, 30 year spinup

Representation Error in Ocean Data Assimilation – p.7/30

slide-8
SLIDE 8

EOFs of SST Anomalies

EO F Analy sis

  • fSea Sur

f ace Tem per at ur e Anom alie s

Representation Error in Ocean Data Assimilation – p.8/30

slide-9
SLIDE 9

Maps of Variance Described by SST EOFs

Representation Error in Ocean Data Assimilation – p.9/30

slide-10
SLIDE 10

EOF Analysis

  • f

HOT SST Anomalies

EO F Analy sis

  • fSea Sur

f ace Tem per at ur e Anom alie s

¥ The second EOF of the SST with 4% of the total variance described is dominated by variability in the strength of the subtropical gyre. In the subtropical gyre, this mode describes 30-50%

  • f the SST variance. The

SST anomaly at HOT (blue) is dominated by the second mode (red) with little contribution by the

  • ther two modes (black)

Representation Error in Ocean Data Assimilation – p.10/30

slide-11
SLIDE 11

Model and AVHRR Seasonal Anomalies: First EOF

Representation Error in Ocean Data Assimilation – p.11/30

slide-12
SLIDE 12

The Kalman Filter

Given a forecast state uf and observations z, correct according to: ua = uf + K(z − Huf) K = PfHT(HPfH

T + R)−1

  • Pf is the forecast error covariance matrix; K is

the Kalman Gain Matrix

  • HPfH

T + R is an estimate of the covariance of

the model-data misfit

  • In the Kalman filter, Pf evolves according to

model dynamics

Representation Error in Ocean Data Assimilation – p.12/30

slide-13
SLIDE 13

A Static Reduced State Space Filter

  • 1. Compute multivariate EOFs of the 23 year time

series

  • 2. Determine significant degrees of freedom: in this

case the Preisendorfer (1988) test indicates 35 significant modes, accounting for 59% of the total variance

  • 3. Estimate Pf by fitting the EOFs of the SST

misfits with the temperature portions of the multivariate model EOFs

  • 4. Estimate a static Kalman gain and assimilate SST

data

Representation Error in Ocean Data Assimilation – p.13/30

slide-14
SLIDE 14

Results of Data Assimilation

Model-data correlations, before and after assimilation:

Representation Error in Ocean Data Assimilation – p.14/30

slide-15
SLIDE 15

Why Not Better?

  • The model does pretty well at what it can do.
  • Most of the signal variability outside of the

tropics comes from eddies, and this can’t be usefully assimilated

Representation Error in Ocean Data Assimilation – p.15/30

slide-16
SLIDE 16

Representation Error

  • Project model-data misfits on multivariate EOFs.

This is the portion of the data that is compatible with the model

  • Subtract the result from the model-data misfits.

This is an estimate of the error of representation

  • Calculate the EOFs of the error of representation

Representation Error in Ocean Data Assimilation – p.16/30

slide-17
SLIDE 17

EOFs of SST Error of Represen- tation

Representation Error in Ocean Data Assimilation – p.17/30

slide-18
SLIDE 18

Next: The NCEP Climate Fore- cast System

  • Climate models are often run past their forecast

horizons

  • Climate models can only produce forecasts

consistent with their internal physics

  • Representation errors could take on crucial

importance

  • Maybe a statistical characterization of

representation errors can be used to devise a stochastic model to simulate them

Representation Error in Ocean Data Assimilation – p.18/30

slide-19
SLIDE 19

Ocean Component, NCEP CFS

  • Basically global MOM/POP
  • Resolution 1o over most of the ocean, tapering to

0.33o from 30o N/S to 8.5o

  • 24 years (1982-2005), 9-month forecasts, from

the restart files

Representation Error in Ocean Data Assimilation – p.19/30

slide-20
SLIDE 20

Spectral Analysis of the CFS Restart Records

Temperature EOFs and Time Series of Amplitudes

100 200 300 −50 50 1980 1990 2000 2010 −4 −2 2 4 100 200 300 −50 50 100 200 300 −50 50 1980 1990 2000 2010 −4 −2 2 4 1980 1990 2000 2010 −5 5 11% 4.3% 3.5%

Representation Error in Ocean Data Assimilation – p.20/30

slide-21
SLIDE 21

Lead EOF, SSH

Lead EOF, SSH 50 100 150 200 250 300 350 −60 −40 −20 20 40 60 −12 −10 −8 −6 −4 −2 2 4 6 8 10 Representation Error in Ocean Data Assimilation – p.21/30

slide-22
SLIDE 22

Second EOF, SSH

EOF No. 2, SSH 50 100 150 200 250 300 350 −60 −40 −20 20 40 60 −25 −20 −15 −10 −5 5 10 15 20 25 Representation Error in Ocean Data Assimilation – p.22/30

slide-23
SLIDE 23

Third EOF, SSH

EOF No. 3, SSH 50 100 150 200 250 300 350 −60 −40 −20 20 40 60 −15 −10 −5 5 10 Representation Error in Ocean Data Assimilation – p.23/30

slide-24
SLIDE 24

Amplitude of the Lead EOF

1985 1990 1995 2000 2005 −8 −6 −4 −2 2 4 Year SOI and First PC

Blue=lead PC; Red=SOI. Correlation ≈ 0.4

Representation Error in Ocean Data Assimilation – p.24/30

slide-25
SLIDE 25

The Preisendorfer Test

50 100 150 200 250 300 0.02 0.04 0.06 0.08 0.1 0.12 Eigenvalue Number Proportion of Total Variance Spectrum of Model Covariance Matrix

61 modes ≈ 60% of the total variance

Representation Error in Ocean Data Assimilation – p.25/30

slide-26
SLIDE 26

EOFs of SST Observations

First EOF of Pathfinder SST

50 100 150 200 250 300 350

  • 60
  • 40
  • 20

20 40 60

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2

Representation Error in Ocean Data Assimilation – p.26/30

slide-27
SLIDE 27

Significance Test for SSTA EOFs

50 100 150 200 250 −5 5 10 15 20 25 30 35 Preisendorfer Test, AVHRR SSTA Representation Error in Ocean Data Assimilation – p.27/30

slide-28
SLIDE 28

Amplitude

  • f

Lead EOF

  • f

SSTA

1980 1985 1990 1995 2000 2005 2010

  • 0.25
  • 0.2
  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.2

Amplitude of Pathfinder SST EOF

Representation Error in Ocean Data Assimilation – p.28/30

slide-29
SLIDE 29

Lead EOF of Residuals

First EOF of SST Residuals

50 100 150 200 250 300 350

  • 60
  • 40
  • 20

20 40 60

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2

Representation Error in Ocean Data Assimilation – p.29/30

slide-30
SLIDE 30

Amplitude

  • f

Lead EOF

  • f

Residuals

1980 1985 1990 1995 2000 2005 2010

  • 0.25
  • 0.2
  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.2 0.25

Amplitude of SST Residuals EOF

...This is going to be harder than we thought.

Representation Error in Ocean Data Assimilation – p.30/30