Freiwald, Grit

English

Electronic Publication Information Center

Ambiguity of signal--error estimates from normal equations

2 General procedureObservation equations: Aηd = ℓ+v, observation error covariance ΣGeneralized least squares:ATΣ−1Aηd = ATΣ−1ℓATΣ−1A︸ ︷︷ ︸Wηηd = ATΣ−1ℓ︸ ︷︷ ︸nWηηd = nUse ηd as data and Wη as weighting matrix in the ocean modeloptimization (equation (1)) =⇒ Result I.  75oW   50oW   25oW    0o    15oN   30oN   45oN   60oN   75oN   [m]−0.5 0 0.5Result I: The MDT is improved by the standard procedure.[m]   20oN  40oN  60oN  80oN    −5000−4000−3000−2000−10000[Sv]0 5 10 15 20Result I: The AMOC is improved, however overestimated.Ambiguity of signal–error estimates from normal equationsGrit Freiwald, Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany; Email: grit.freiwald@awi.de4 Revised procedureUsing the smoothing filter S, we derive a new interpretation of thegeodetic observation equations:Observation equations: A S−1S︸ ︷︷ ︸=Iηd = ℓ + v, error covariance ΣGeneralized least squares:(AS−1)TΣ−1AS−1Sηd = (AS−1)TΣ−1ℓS−T ATΣ−1A︸ ︷︷ ︸WηS−1Sηd = S−T ATΣ−1ℓ︸ ︷︷ ︸nS−TWηS−1︸ ︷︷ ︸DSηd = S−Tn Wη = STDSDSηd = S−Tn =⇒ Jˆη = (ηm− Sηd)TD(ηm− Sηd)Use Sηd as data and D as weighting matrix in the ocean modeloptimization =⇒ Result II.  75oW   50oW   25oW    0o    15oN   30oN   45oN   60oN   75oN   [m]−0.1 0 0.1 0.2 0.3 0.4∆ MDT difference: Result II - Result I. The Gulf Stream is shiftednorthwards. No further large changes occur.[m]   20oN  40oN  60oN  80oN    −5000−4000−3000−2000−10000[Sv]0 5 10 15 20Result II: The AMOC is improved: It is not as intense as in the firstapproach 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 thedata themselves.•The ocean model optimization with revised MDT data and error co-variance improves the AMOC.3 Smoothing filter SInverse error covariance matrix Wη is symmetric and positiv definite=⇒ A unique symmetric matrix square root exists:√Wη = VNormalize rows of V: V =√DS =⇒ Wη = STDSThe filter S is used to smooth the MDT data ηd.  75oW   50oW   25oW    0o    15oN   30oN   45oN   60oN   75oN   [m]−0.5 0 0.5Unsmoothed satellite MDT ηd  75oW   50oW   25oW    0o    15oN   30oN   45oN   60oN   75oN   [m]−0.5 0 0.5Smoothed satellite MDT SηdAdvantage: No arbitrary filter type, no arbitrary filter width!1 IntroductionThe inverse ocean model IFEOM assimilates Mean DynamicTopography (MDT) data ηd from satellite observations.Minimization of cost function: J = 12∑i Ji, i=T,S,v,η,...Jη =1α(ηm − ηd)TWη(ηm − ηd) (1)For a discussion of the weighting factor α, please see other poster.Wη = C−1η is the inverse MDT error covariance from the geodeticnormal 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, anew interpretation for MDT data and error covariance is derived.  75oW   50oW   25oW    0o    15oN   30oN   45oN   60oN   75oN   [m]−0.5 0 0.5First guess: MDT[m]   20oN  40oN  60oN  80oN    −5000−4000−3000−2000−10000[Sv]0 5 10 15 20First guess: Atlantic MeridionalOverturning Circulation (AMOC)The figures above show some oceanographic features of the first guessof the ocean model. The Gulf Stream and the AMOC are too weak(compare e.g. Johns et al., 1995, and Griffies et al.,2009).

Ambiguity of signal--error estimates from normal equations

Abstract

Similar works

Full text

Available Versions

Electronic Publication Information Center

Electronic Publication Information Center