Variable seasonal and subseasonal oscillations in sea level anomaly data and

their impact on sea level prediction accuracy

W. Kosek 1,2, T. Niedzielski 3,2, W. Popiński 4, M. Zbylut 1 ,A. Wnęk 1

1) Environmental Engineering and Land Surveying Department, Agriculture University of Krakow, Poland2) Space Research Centre, Polish Academy of Sciences, Warsaw, Poland

3) Institute of Geography and Regional Development, University of Wrocław, Poland. 4) Central Statistical Office of Poland, Warsaw, Poland

VIII Hotine-Marussi Symposium, 17-21 VI 2013, Rome, Italy

DATAGridded Sea Level Anomaly (SLA) data 1o×1o for φ <-90o:90o>, Λ <1o,360o>

obtained from the Archiving, Validation and Interpretation of Satellite Oceanographic (AVISO) data in 1992-2013 with one week sampling interval. Data are produced from observations carried out by the satellites TOPEX/Poseidon, ERS, Jason-1, Jason-2 and Envisat. http://www.aviso.oceanobs.com/

Time-frequency analysis of sea level anomaly data

),()]([),(,

1,

AtxFTFTtu

where:

),(,

tu

)(,tx

- SLA time series

90;90 360;0

- broadband oscillation with central frequency ω

-geographic latitude and longitude

if

ifA

0

21),(

),( A1

- half of the bandwidth

- parabolic transmittance function

T

t t- sampling interval of the SLA data

T – mean period of broadband oscillation

Mean amplitude spectrum

kn

ktTtu

knTS

1

2),(

,2

2)(

,ˆ

where: n – number of SLA data, k – the number of points to be drop at the beginning and at the and of filtered time series due to filter errors,T=Δt/ω - the mean period of broadband oscillation.

Time variable amplitude spectrum

2/,...,22/,12/,2

),(,1

2),(

,ˆ

2/

2/mNmmtTktu

mtTS

m

mk

tTm /2

Mean amplitude spectrum of the whole ocean, northern and southern hemispheres

2

1

360

1

)(,

ˆ)(ˆ

TSTS

0 100 200 300 400 500 600 700 800period (days)

0

1

2

3

4

5c m

Northern Hem isphere

Southern Hem isphere

W hole ocean

ω

4ω3ω

2ω

Kuroshio Current Gulf Current

Antarctic Circumpolar Current

Time variable amplitude spectrum

PROGNOCEAN - Near real time system and service for sea level prediction Version

This is Prognocean 1.0.beta - solely experimental solution. Testing phase of Prognocean 1.0.beta is in progress. The system may be switched off without

publishing any notice. See Disclaimer below for terms of use.

ObjectiveThe near real time system and service for sea level prediction, known also as Prognocean is designed, implemented and based at the University of Wroclaw

(Poland). The initiative is supported by the Foundation for Polish Science through the European Regional Development Fund and the Innovative Economy

Programme and aims to compute predictions of Sea Level Anomaly (SLA) maps in near real time. Initially, daily data are predicted and lead time does not exceed two

weeks. Solutions for longer lead times will be available later, when time series sampled every 7 days are modelled. Along with predictions, Root Mean Squared Error (RMSE) of predictions is computed in near real time so that the users are

able to evaluate the performance of the system and service.

The mean prediction error of sea level anomaly data for 2 weeks in the future computed using the combination of the polynomial harmonic extrapolation model and 1) autoregressive prediction, 2) threshold autoregressive prediction.

Polynomial harmonic + autoregressive

Polynomial harmonic + threshold autoregressive

The mean prediction error of the SLA data for 2 weeks in the future and the mean amplitude of the annual oscillation

Computation of phase variations in real-valued time series

• Combination of complex demodulation and the Fourier Transform Low Pass Filter - CD+FTLPF

• Combination of the FTBPF and Hilbert transform - FTBPF+HT

Combination of complex demodulation and the Fourier transform low pass filter (CD+FTLPF)

)()),((),( 1 AtxFTFTtz oo

1. Multiplication of the time series by complex-valued harmonic with frequency :

2. Filtration of the transformed signal using FTLPF of complex-valued time series:

3. Computation of instantaneous phases:

.

)),(Re(

)),(Im(arctan),(

o

oo tz

tzt

o

- transmittance function,

)exp()(),( titxtx oo

,0

1 22

otherwise

ifAwhere

λ - window halfwidth

Combination of the FTBPF and Hilbert transform (FTBPF+HT)

),())((),( 1oo AtxFTFTtx

,0

)(1,

22

otherwise

ifAwhere oo

o

),(),(),( ooo txHitxtz

1. Computation of the oscillation with central frequency by the FTBPF:

λ - window

halfwidth

2. Forming of the complex-valued series using the Hilbert transform of the filtered oscillation:

or

3. Computation of instantaneous phases:

.

)),(Re(

)),(Im(arctan),(

o

oo tz

tzt

o

)1)((),())((),( 1 signAtxFTFTtz oo

- transmittance function

Conclusions

• The FTBPF analysis of sea level anomaly data reveals that the annual oscillation has a broadband character. This creates oscillations with frequencies being an integer multiplicity of the annual frequency. The amplitude maxima of all these shorter period oscillations are located almost in geographic regions of the annual oscillation amplitude maxima.

• The mean prediction errors for 2 weeks in the future of sea level anomaly data are usually big in geographic regions of annual oscillation amplitude maxima.

• The increase of the prediction errors of sea level anomaly data is mostly caused by variable phases and amplitudes of the broadband annual oscillation.