This document proposes a method for calculating the background error covariance matrix (BECM) for wave data assimilation systems. The method considers the local spectral wave climate and the spectral correction model applied. It calculates separate BECMs for different wave systems, such as the northerly and southwesterly systems in the North Sea, without assumptions about wind-sea or swell conditions. The developed method allows objectively computing BECMs based on statistics and implicitly defines the spatial domains where isotropy and homogeneity are fulfilled.

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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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• 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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