Estimating observing system error thresholds using an asymptotic approach Gary Brassington and...

Estimating observing system error thresholds using an asymptotic approach

Gary Brassington and Prasanth Divakaran

Centre for Australian Weather and Climate Research

Bureau of Meteorology

AcknowledgementsWe gratefully acknowledge the valuable contributions from the BLUElink> science and technical team, Bureau of Meteorology, CSIRO the Royal Australian Navy, NASA, CNES, ESA, AVISO, NOAA, US Navy, the Argo Science team, Geoscience Australia, OSTST, GODAE and GHRSST.

Brassington, G. B. and P. Divakaran, 2009: The theoretical impact of remotely sensed sea surface salinity observations in a multi-variate assimilation system, Ocean Modelling, 27, 70-81 10.1016/j.ocemod.2008.12.005

Initial motivation

“What error is sufficiently large that it will have a negligible impact to the analysis system?”

Fact: SMOS and Aquarius has been estimated to have a large error

Can an error ever be regarded as too large?


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€

T a

η a

Sa

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

T f

η f

S f

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥+ P f HT (HP f HT + R)−1

T o − HT f

η o − Hη f

So − HS f

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

We can pose this formally asymptotically. Consider the simplest tri-variate problem

€

HP f HT =

σ TTf σ ST

f σ Tηf

σ STf σ SS

f σ Sηf

σ Tηf σ Sη

f σ ηηf

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

R =

σ TTo σ ST

o σ Tηo

σ STo σ SS

o σ Sηo

σ Tηo σ Sη

o σ ηηo

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥


The analysis for salinity can expressed as,

€

Sa = S f

+σ SS

f ˜ σ TT ˜ σ ηη − ˜ σ ηη σ STf ˜ σ ST − ˜ σ TTσ Sη

f ˜ σ Sη − σ SSf ˜ σ Tη

2+ ˜ σ Tη (σ ST

f ˜ σ Sη + ˜ σ STσ Sηf )

D

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥(So − S f )

+˜ σ ηη (σ ST

f σ SSo − σ ST

o σ SSf )+ ˜ σ Tη (σ SS

f σ Sηo − σ Sη

f σ SSo )+ ˜ σ Sη (σ ST

o σ Sηf − σ ST

f σ Sηo )

D

⎡

⎣ ⎢

⎤

⎦ ⎥(T o − T f )

+˜ σ TT(σ Sη

f σ SSo − σ Sη

o σ SSf )+ ˜ σ Tη (σ ST

o σ SSf − σ ST

f σ SSo )+ ˜ σ ST(σ ST

f σ Sηo − σ ST

o σ Sηf )

D

⎡

⎣ ⎢

⎤

⎦ ⎥(η o − η f )

(1)

where

€

D = ˜ σ TT ˜ σ ηη ˜ σ SS − ˜ σ ST2 ˜ σ ηη − ˜ σ Sη

2 ˜ σ TT − ˜ σ Tη2 ˜ σ SS + 2 ˜ σ ST ˜ σ Sη ˜ σ Tη

€

˜ σ xy = σ xyf +σ xy

o


Large error is when the analysis equation asymptotically converges to the same analysis in the absence of the observations (i.e., )

€

Sa → S f

+σ ST

f ˜ σ ηη − ˜ σ Sηf ˜ σ Tη

f

˜ σ TT ˜ σ ηη − ˜ σ Tηf 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥(T o − T f )

+˜ σ Sη

f ˜ σ TT − ˜ σ STf ˜ σ Tη

f

˜ σ TT ˜ σ ηη − ˜ σ Tηf 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥(η o −η f )

€

σSSo → ∞


This occurs when,

€

σ SSo ( ˜ σ TT ˜ σ ηη − ˜ σ Tη

2 ) >>σ SSf ( ˜ σ TT ˜ σ ηη − ˜ σ Tη

2 )

− ˜ σ ST2 ˜ σ ηη − ˜ σ Sη

2 ˜ σ TT + 2 ˜ σ ST ˜ σ Sη ˜ σ Tη

which is equivalent to the condition,

€

D = ˜ σ TT ˜ σ ηη ˜ σ SS − ˜ σ ST2 ˜ σ ηη − ˜ σ Sη

2 ˜ σ TT − ˜ σ Tη2 ˜ σ SS + 2 ˜ σ ST ˜ σ Sη ˜ σ Tη

→σ SSo ( ˜ σ TT ˜ σ ηη − ˜ σ Tη

2 )


The condition on for observation error for this simplified problem is then,

€

σ SSo >>σ SS

f −˜ σ ST

2 ˜ σ ηη + ˜ σ Sη2 ˜ σ TT − 2 ˜ σ ST ˜ σ Sη ˜ σ Tη

˜ σ TT ˜ σ ηη − ˜ σ Tη2


€

σ SSo >>σ SS

f −˜ σ ST

2 ˜ σ ηη + ˜ σ Sη2 ˜ σ TT − 2σ ST

f σ Sηf σ Tη

f

˜ σ TT ˜ σ ηη −σ Tηf 2

R is uncorrelated

Assume

€

˜ σ ST2 ˜ σ ηη + ˜ σ Sη

2 ˜ σ TT >> 2σ STf σ Sη

f σ Tηf

€

˜ σ ηη ˜ σ TT >> σ Tηf 2

and

€

σ SSo >>σ SS

f −˜ σ ST

2

˜ σ TT

−˜ σ Sη

2

˜ σ ηη


Lets consider four conditions

€

σ SSo >>σ SS

f

€

σ SSo >>σ SS

f −σ ST

2

˜ σ TT

€

σ SSo >>σ SS

f −σ ST

2

˜ σ TT

−σ Sη

2

˜ σ ηη

Univariate

Bivariate

Tri-variate, simplified

€

σ SSo >>σ SS

f −˜ σ ST

2 ˜ σ ηη + ˜ σ Sη2 ˜ σ TT − 2σ ST

f σ Sηf σ Tη

f

˜ σ TT ˜ σ ηη −σ Tηf 2

Tri-variateR is uncorrelated


Lets corrupt illustrate these conditions with an analysis scheme, BODAS

(Up to this point the analysis is system independent)


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€

σ SSo ~ 1psu2

€

σ SSo ~ 0.44( )

2psu2 ~ 0.19psu2

Interpretation

€

σ SSo >>σ SS

f

>10σ SSf

or an impact will occur when

€

σ SSf > σ SS

o /10


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€

σ SSo /10 ~ 0.1psu2

€

σ SSo /10 ~ 0.019psu2

Alternatively observations will have an impact when

€

σ SSf > σ SS

o /10 Univariate

€

σ SSf −

˜ σ ST2 ˜ σ ηη + ˜ σ Sη

2 ˜ σ TT − 2σ STf σ Sη

f σ Tηf

˜ σ TT ˜ σ ηη −σ Tηf 2 > σ SS

o Tri-variateR is uncorrelated

Aquarius

SMOS


Lets corrupt illustrate these conditions with an analysis scheme, BODAS

(Up to this point the analysis is system independent)


Tri-variate assimilationUni-variate assimilation


Tri-variate assimilation Bi-variate assimilation


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Perf

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ceHow does this compare to the increment variance?


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Multi-variate is always a tougher constraint for observation impact(i.e., GODAE type systems cannot tell you it is more valuable than a univariate objective analysis)

Asymptotic conditions are system independent, applicable to hypothesis testing and diagnosing impact to any system

Results will be assumption and system dependent and open to interpretation, although the “good” systems will always pose tougher constraints.

For BLUElink, Aquarius will impact the tropics and equatorial bandLess impact at mid-latitude with good coverage from SST

SMOS will have considerably less impact for BLUElink


Conclusions

The asymptotic conditions can be reposed for other variables

The conditions are extendible to larger matrices

It can tell you where to expect impact

It can tell the relative impact


Perf

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C=


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€

DC → σ Sop

DN

€

α0σ Sop

>> α1σ Sop−1

+α2σ Sop−2

+L +αp

€

σSo >> α1 /α0

>> aiaiT

i =1, p

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟+

i =1, p

∑ (−1)2k(−1)1+ j −i+1 (−1)1+ 2

k =i+1, p

∏k =1, i −1

∏ ⎛

⎝ ⎜

⎞

⎠ ⎟aiaj

TDN, ij / DNj =p+1, p+ q+ r

∑


Computing – Cramer’s rule

€

ANx i = gi

€

xij = DN, ij / DN

Cost of one inversion of the reduced observation space

€

x =AN−1G


What about Argo?


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Reposed for etaTri-variate assimilation

Estimating observing system error thresholds using an asymptotic approach Gary Brassington and...

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