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