2 General procedure
Observation equations: Aηd = ℓ+v, observation error covariance Σ Generalized least squares:
AT Σ−1Aηd = AT Σ−1ℓ AT Σ−1A
| {z }
Wη
ηd = A| T Σ{z−1ℓ}
n
Wηηd = n
Use ηd as data and Wη as weighting matrix in the ocean model optimization (equation (1)) =⇒ Result I.
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Result I: The MDT is improved by the standard procedure.
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Result I: The AMOC is improved, however overestimated.
Ambiguity of signal–error estimates from normal equations
Grit Freiwald, Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany; Email: grit.freiwald@awi.de
4 Revised procedure
Using the smoothing filter S, we derive a new interpretation of the geodetic observation equations:
Observation equations: A S−1S
| {z }
=I
ηd = ℓ + v, error covariance Σ Generalized least squares:
(AS−1)T Σ−1AS−1Sηd = (AS−1)T Σ−1ℓ S−T A| T Σ{z−1A}
Wη
S−1Sηd = S−T A| T Σ{z−1ℓ}
n
S−T WηS−1
| {z }
D
Sηd = S−T n Wη = ST D S
D Sηd = S−T n =⇒ Jˆη = (ηm − Sηd)T D(ηm − Sηd)
Use Sηd as data and D as weighting matrix in the ocean model optimization =⇒ Result II.
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∆ MDT difference: Result II - Result I. The Gulf Stream is shifted northwards. No further large changes occur.
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Result II: The AMOC is improved: It is not as intense as in the first approach and a distinct maximum at about 40-45◦N is visible. ,
5 Results
• A filter for the MDT was generated directly from the geodetic nor- mal equations.
• No prior assumptions are made about the filter or the filter radius.
• The information content is shifted from the error covariance to the data themselves.
• The ocean model optimization with revised MDT data and error co- variance improves the AMOC.
3 Smoothing filter S
Inverse error covariance matrix Wη is symmetric and positiv definite
=⇒ A unique symmetric matrix square root exists: qWη = V Normalize rows of V: V = √
D S =⇒ Wη = ST D S The filter S is used to smooth the MDT data ηd.
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Unsmoothed satellite MDT ηd
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Smoothed satellite MDT Sηd
Advantage: No arbitrary filter type, no arbitrary filter width!
1 Introduction
The inverse ocean model IFEOM assimilates Mean Dynamic Topography (MDT) data ηd from satellite observations.
Minimization of cost function: J = 12 Pi Ji, i=T,S,v,η,...
Jη = α1(ηm − ηd)T Wη(ηm − ηd) (1) For a discussion of the weighting factor α, please see other poster.
Wη = C−1
η is the inverse MDT error covariance from the geodetic normal equations.
The matrix Wη is used to construct a filter S for the MDT data ηd. By inserting I = S−1S into the geodetic observation equations, a new interpretation for MDT data and error covariance is derived.
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First guess: MDT
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First guess: Atlantic Meridional Overturning Circulation (AMOC) The figures above show some oceanographic features of the first guess of the ocean model. The Gulf Stream and the AMOC are too weak (compare e.g. Johns et al., 1995, and Griffies et al.,2009).
