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Optimal array design:Application to the tropical Indian Ocean
Peter OkeNovember 2006
CSIRO Marine and Atmospheric Researchhttp://www.cmar.csiro.au/staff/oke/ email: peter.oke@csiro.au
Ensemble-based array designSakov and Oke
T
mAAP
1
1
Given a representative ensemble of anomalies:
maaaA ,,, 21
the ensemble covariance is given by:
Using Kalman filter theory, the covariances of the ensemble can be used to map (or grid, or analyse) a vector of observations:
1)(
)(
RHPHPHK
HwdKwwTT
bba
Giving an analysis error covariance of:
PHRHPHPHIP 1
TTa
Ensemble-based array design Sakov and Oke
aopt P
HH
min
TAA fa
HRHAHA
HAAAHA
HH
)1()(
))((trace
max
mT
TTopt
2/11
1
m
fTTf HARHAIT
An optimal array minimises some norm of : aP
We choose to minimise the analysis error variance:
Given an observation, or an array of observations, we update the ensemble to reflect the reduced variance:
Ensemble-based array design… in practice
Representative ensemble
calculate optimal
observation
update ensemble
Ensemble-based array designModel configurations
ACOM2 ACOM3 OFAM
Model code MOM2 MOM3 MOM4
Zonal res. 2o 0.5o 0.1-2o
Meridional res. 0.5-1.5o 0.33o 0.1-2o
# vert. levels 25 33 47
Wind forcing NCEP/NCAR + FSU ERS1/2 ERA40
Heat flux ABLM + FC ABLM + FC ERA40 + FC
Shortwave as above OLR + NCEP ERA40
Freshwater as above Monthly analyses Levitus
Time period 1982-1994 1992-2000 1992-2004
Ensemble-based array designModel-based – Intraseasonal
Ensemble-based array designModel-based – Intraseasonal
Table 1: The basin-averaged theoretical analysis error variance of IMLD (m2), and the percent reduction in parentheses
ACOM2 ACOM3 OFAM
Signal 15.2 22.7 49.3
Proposed array 12.3 (19%) 16.7 (26%) 36.6 (26%)
ACOM2 array 11.5 (24%) 16.2 (29%) 36.6 (28%)
ACOM3 array 11.9 (22%) 15.4 (32%) 35.8 (27%)
OFAM array 12.1 (20%) 16.3 (28%) 33.6 (32%)
… in practice, the proposed array looks pretty good, and will probably perform as well as any objectively defined optimal array.
Ensemble-based array designObservation-based – Intraseasonal to interannual
Ensemble-based array designObservation-based – Intraseasonal to interannual
Table 2: The basin-averaged theoretical analysis error variance of GSLA (m2), and the percent reduction in parentheses
Variance (m2)
Signal 81.0
Proposed array 34.9 (57%)
Unstructured array 27.1 (66%)
Structured array 29.5 (64%)
Ensemble-based array designCorrelation maps
Ensemble-based array designSummary
We demonstrate how simple it is to calculate optimal observations given the system
covariance.
In practice, if you have an ensemble that correctly represents the system covariance, it is
easy to calculate the corresponding optimal observations.
We show an example application to the tropical Indian Ocean, using a:
model-based ensemble
observation-based ensemble
Sakov, P., and Oke, P. R., 2006: Optimal array design: application to the tropical Indian Ocean. Monthly Weather Review, submitted.
Key questions in observing system design
What types of observations can be made?
temperature and salinity from a mooring or glider
sea-level from a tide gauge station
surface currents from a HF radar array
What are the observations intended to monitor?
area-averaged temperature
El Nino Southern Oscillation
location of a front
What are the practical constraints?
cost
maintenance
What is the best “complete” representation of the field being observed and monitored?
model temperature, sea-level, transport
observations satellite sea surface temperature
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