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On the specification of the Background Error Covariance Matrix for Wave Data Assimilation Systems

• Jesús Portilla

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Introduction

• Motivation for Data Assimilation• Model and observations usually don’t match

• Users tend to trust more in observations

• Some situations are simply too difficult to model

• Some areas have dense monitoring networks that is a pity not to use to improve model results

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Introduction

• Background errors determine the extent and the magnitude in

which observations get introduced into the model wave field

your nameObjective DA Statistical DA

a b o bx x K x x

2

2 2b

o b

K

• Statistical DA concept

pdf

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1 2

,,

i

o mi j j

i j

J Q x x

Optimization problem

• The DA scheme

• Variational (3DVAR, 4DVAR)

• Optimal interpolation

• Kalman Filtering

• Adjoint modelling

• Neural Networks

• . . .

error covariance matrixerror covariance matrix

0J

2 0J

3DVAR

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• Background error covariance matrix (BECM)

Covariance (Target)

Variances (can be estimated, e.g., triple co-location)Correlation coefficient

(can be estimated, e.g., via the R2)

i j

iji i

w w

w w

Greenslade, D.J.M. and I.R. Young, 2005: The impact of Altimeter Sampling Patterns on Estimates of Background Errors in a Global Wave Model, J. Atmos. Oc. Tech., 22, No. 12 pp 1895 – 1917.

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• The North Sea

Voorrips A.C., V.K. Makin, and S. Hasselmann., 1997: Assimilation of wave spectra from pitch-and-roll buoys in a North Sea wave model, J. Geophys. Res., 102 (C3), 5829-5849

Parametric error correlation length

(using wave height)

expa

d

L

3 / 2

200( )

a

L km

• Background errors (parametric)

K13

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• Some remarks about Background errors● Our current knowledge about the structure (“shape and dimensions”) of Background Errors is very poor.

• For consistent DA, wave conditions must be homogeneous, isotropic, and ergodic over the assimilation domain.

• The computation of the Background Errors should consider the wave spectrum as the reference variable and not integral parameters like Hs.

• Background Errors depend on wave climate, which in turn might be characterized by the existence of different regimes. For a proper specification of the BECM each wave system has to be considered independently. • The wave climate and therefore the BECM is point specific and season dependent.

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• Wave climate

MODEL

BUOY

wind sea

wind sea

swell

swell

• Buoy Hs = 4.2 m

• Model Hs = 2.7 m

• Matching observations and model spectra

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• is the partition spectrum

• The truth is emulated from WWIII model output

• Computation of the BECM

i j

iji i

w w

w w

2

221

analyzed true

ij

true true

S SR

S S

,S S f

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• The spectral correction model

@ @, * * ,analysis remote o true obs o oS f S f

energy correctionfrequency correction

direction correction

• Each wave system is corrected individually

• No assumptions are made about the wind-sea or swell condition

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• Calculating for two main wave systems (e.g., North Sea)

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• Calculating for two main wave systems (e.g., North Sea)

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Northerly system K13 Southwesterly system K13 Parametric (general)

• Calculating for two main wave systems (e.g., North Sea)

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• Summary

• A method for the computation of the BECM has been developed

• This method considers explicitly:

a) The local spectral wave climate

b) The spectral correction model to be applied

• Assumptions about the wind-sea or swell condition are not used

• Conclusions

• The developed method allows calculating the BECM objectively on statistical bases

• The computed BECM’s implicitly define the spatial domain where the conditions of isotropy and homogeneity are fulfilled

• The condition of ergodicity can be included for instance by computing BECM’s for each season

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References

Voorrips A.C., V.K. Makin, and S. Hasselmann., 1997: Assimilation of wave spectra from pitch-and-roll buoys in a North Sea wave model, J. Geophys. Res., 102 (C3), 5829-5849

Greenslade, D.J.M. and I.R. Young, 2005: The impact of Altimeter Sampling Patterns on Estimates of Background Errors in a Global Wave Model, J. Atmos. Oc. Tech., 22, No. 12 pp 1895 – 1917.

Thanks for your attention!

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