Estimating observing system error thresholds using an asymptotic approach Gary Brassington and Prasanth Divakaran Centre for Australian Weather and Climate.
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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?
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.
Perf
orm
an
ce
€
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
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
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.
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
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.
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 → ∞
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.
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 )
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.
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
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.
€
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
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.
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
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.
Lets corrupt illustrate these conditions with an analysis scheme, BODAS
(Up to this point the analysis is system independent)
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.
Perf
orm
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ce
€
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
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.
Perf
orm
an
ce
€
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
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.
Lets corrupt illustrate these conditions with an analysis scheme, BODAS
(Up to this point the analysis is system independent)
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.
Tri-variate assimilationUni-variate assimilation
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.
Tri-variate assimilation Bi-variate assimilation
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.
Perf
orm
an
ce
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.
Perf
orm
an
ceHow does this compare to the increment variance?
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.
Perf
orm
an
ceConclusions
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
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.
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
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.
Perf
orm
an
ceExtendable
C=
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.
Perf
orm
an
ceExtendable
€
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
∑
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.
Computing – Cramer’s rule
€
ANx i = gi
€
xij = DN, ij / DN
Cost of one inversion of the reduced observation space
€
x =AN−1G
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.
What about Argo?
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.
Perf
orm
an
ce
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.
Reposed for etaTri-variate assimilation
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.
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