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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
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Objective DA Statistical DA
( )a b o bx x K x x= + −
( )
2
2 2
b
o b
K
σ
σ σ
=
+
• Statistical DA concept
pdf
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( )
1 2
,
,
i
o m
i 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
ij
i i
w w
w w
ρ
σ σ
=
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)
exp
a
d
L
ρ
= − ÷
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
ij
i i
w w
w w
ρ
σ σ
=
2
2
2
1
analyzed true
ij
true true
S S
R
S S
ρ
− ≅ = −
−
∑
∑
( ),S S f θ=
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• The spectral correction model
( ) ( )@ @, * * ,analysis remote o true obs o oS f S fθ α β θ δ= +
energy correction
frequency correction
direction correction
• Each wave system is corrected individually
• No assumptions are made about the wind-sea or swell condition
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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!
