1

Approximate decorrelation and non-isotropic smoothing of time-variable

GRACE gravity field models

Jürgen Kusche, Roland Schmidt

with input fromSusanna Werth, Roelof Rietbroek

GFZ Potsdam

IUGG 2007, Perugia, GS002

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Outline of the talk

• GRACE fields exhibit artefacts (“stripes”) which may be seen as a realization of spatially correlated noise - smoothing and/or “de-striping” is required

• Theory: Discussion of ways to decorrelate (“de-stripe”) the noise in GRACE solutions (including method from Swenson-Wahr 2006 (SW06) and a new method)

• Theory: The scaling (bias) problem

• Results: De-striped GFZ GRACE RL4 fields, surface mass grids, and a time series of basin-averaged GRACE-derived OBP ( talk in JGS001)

• Conclusions

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“Stripes” in GRACE solutions

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Stripes in GRACE solutions

NS-oriented artefacts

gravity field determination =ill-posed problem

Stochastic (noise) and deterministic (background model) errors cause unphysical oscillations

RMS variability of 40 GFZ RL04 monthly solutions in

2/03-12/06 relative to their mean (7-10/04 and

12/06 excluded), Gaussian 550km

Surface Mass

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Decorrelation, “de-striping”

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• degree-dependent (isotropic)• Gauss (Jekeli 1981, Wahr & al 1998), Gauss-Weierstrass (Freeden 1998), Hanning (Jekeli 1981), Blackman (Schmidt & al 2006), CuP (Fengler & al 2006)

• degree- and order-dependent• modified Gauss (Han 2005)• removing single coefficients based on hypothesis testing (Sasgen & al 2005)

• full non-isotropic (general two-point kernel)• constrained fields (Tikhonov)• empirical signal decorrelation combined with Gaussian (Swenson & Wahr 2006)• empirical error decorrelation and Tikhonov smoothing (Kusche 2007)

Issues • de-striping property• amplitude damping (bias) and phase lags• interpretability• optimality criteria, multiresolution properties

Filter Methods for GRACE-L2 Products

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• combine approximate error decorrelation and Tikhonov smoothing (Kusche 2007)• scaled dense synthetic, “smooth” normal matrix for 1 month• synthetic, smooth signal variance model from Hydrology + Ocean circulation• damping “on normal equation level”

This work

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Construction of E and S GRACE orbits (coverage)

Hydrology Model +Ocean Circulation Model

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LAT=60oCross-sections

N-S direction (o)

W-E direction (*)

Impulse response

Filter Properties

This work

LAT=0o

Distance from kernel center

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This work Swenson and Wahr (2006)

black circle = Gaussian 500km

Impulse response

Filter Properties

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can be approximated as block-diagonal

Decorrelation/Smooth. Filter W for L = 70

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C-Block (m+1)

odd/even degrees

Asymmetric order/parity weighting

degree

C-Block (m)

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Scaling (bias) problem

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• ratio between filtered and exact basin average

• depends on • filter• shape of basin• signal within and outside basin

All smoothed GRACE-based functionals, global maps or basin averages, are systematically biased low

Scaling (bias) problem

• damping of the global rms

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Relative bias from true and filtered signal, including hydrology

apparent phase lag

Scaling (bias) problem

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Relative bias from true and filtered signal, hydrology removed

400km: 56%

Scaling (bias) problem

year

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Comparison of filters based upon variance and standard scaling bias

Comparison Gaussian – This Work

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Results

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Gaussian Filter

RMS variability of 40 GFZ RL04 monthly solutions in 2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded)

Further “de-striping” reduced amplitude (biased towards zero)

Left: Gaussian 500km, Right: Gaussian 550km

wrms=3.85cm

Surface MassGeoid

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Empirical signal decorrelation according to Swenson and Wahr

(2006)

Filter > l=10, Gaussian 400km

RMS variability of 40 GFZ RL04 monthlySolutions in 2/03-12/06 relative to theirMean (7-10/04 and 12/06 excluded)

wrms=3.76cm

Decorrelation – Swenson and Wahr 2006

Surface Mass

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RMS variability of 40 GFZ RL04 monthly solutions in2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded)

Left and right: approx. decorrelated using 8/03 orbits and LaD+ECCOfor W-matrix (up to deg/ord = 70), a = 10E+14

wrms=3.83cm

Decorrelation – This Work

Surface MassGeoid

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BOTH are decorrelated/smoothed using the SAME operator, i.e.8/03 orbits and LaD+ECCO for W-matrix (up to deg/ord = 70), a = 10E+14

Approx. (GRACE) decorrelation does not distort hydrology model

wrms=3.85cmwrms=2.30cm

Decorrelation – This Work

Surface Mass – GRACESurface Mass - WGHM

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DFG-Mass Transport project STREMP See talk by L. Fenoglio et al in JSG001

Regional Averaging

GRACE “raw” time series of mass change over the Mediterraneanby different methods

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• Stripes in GRACE solutions still visible; although RL04 improvement over earlier releases

• Best strategy: remove during processing (but perfect de-aliasing impossible)

• Second-best strategy: post-processing using error correlation model (here: from an arbitrary GRACE- or GRACE-type orbit + a-priori model information)

• Proposed technique removed stripes much more effectively compared to Gaussian; simultaneously smoothing (“amplitude bias”) is comparable to Gaussian

• Use for mass transport studies (hydrology, ocean); higher resolution at comparable damping

Conclusions

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Thank you

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Decorrelation – This Work

Full W-matrixOrder/parity only

RMS variability of 40 GFZ RL04 monthly solutions in2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded)

Approximately decorrelated using 8/03 orbits and LaD+ECCOfor W-matrix (up to deg/ord = 70), a = 10E+14

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Decorrelation – This Work

W-matrix based on syntheticnormals from orbit 8/03

W-matrix based on covariancematrix for 8/03 GFZ-RL04

RMS variability of 40 GFZ RL04 monthly solutions in

2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded)

Apriori model information for W-matrix (70,70) from LaD+ECCO, a = 10E+14

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Decorrelation – This Work

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Filtering Regularization

eq.

interpretation

estimate of x estimate of x

reduces variance

yes yes

introduces bias

yes (“scaling factor”)

yes

alternative interpretatio

n

unbiased estimate of averaged x

no

required processing

L2 data L1 data (but…)

LSx̂Wx̂

LS

1T11T x̂WyA)RAA(x̂

Decorrelation – This Work

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Filtering Regularization

parameter tuning

S/N geophysical signals /GRACE

(Wiener/”optimal”)

S/N geophysical signals/ GRACE (LSC)

ordata-driven (GCV,VCE)

latitude-dependence

no (but…) automatically

anisotropic decorrelatio

nno (but…) automatically

geometric interpretatio

nyes no

Decorrelation – This Work

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Spherical disc signal + Gaussian(can be analytically treated)

Disc radius [km]

Gauss

ian

sm

ooth

ing r

adiu

s [k

m]

amplitude scaling error (relative bias)Mediterranean

Amplitude Scaling Error - Gaussian