1 spp 1257 tutorial: analysis tools for grace time- variable harmonic coefficients jürgen kusche,...

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SPP 1257

Tutorial: Analysis Tools for GRACE time-variable harmonic coefficients

Jürgen Kusche, Annette Eicker, Ehsan ForootanBonn University, Germany

IGCP 565 WorkshopNovember 21-22, 2011

University of the Witwatersrand, Johannesburg, South Africa

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SPP 1257Jürgen Kusche

Analysis Tools

For everybody who has worked with GRACE data, it has become clear that GRACE solutions (say SH coefficients converted to TWS, total water storage) require some post-processing,

– to suppress correlated noise, remove ‚stripes‘ ( filtering)– to extract the dominating ‚modes‘ of temporal variability ( PCA, …)

Being in general use, these analysis tools always remove signal content together with ‚noise‘. For any comparison of GRACE data with geophysical modelling, it is imperative therefore that the same tool is applied to both. For getting ‚absolute‘ amplitudes, rates, etc., it is imperative to consider the ‚bias‘ of an analysis technique.

Note: If you download GRACE gridded products from GRACE Tellus website, GFZ ICGEM or others, some filtering has been applied already. Try to understand what has been done to the data, and what has to be done to be fully compatible with model output.

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Analysis Chain: Filtering

GRACE SHC

Treatment degree 1 & 2, temporal anomalies wrt epoch t, restore AOD

OR Filtering (in space domain)

Mapping to space domain or basin averaging

Filtering (in spectral domain)

SHA geophysical model SHC

Temporal alignement, possibly remove deg. 1 consistent to GRACE

OR Filtering (in space domain)



Multiplication in spectral domain

Convolution on the sphere

Globally defined?

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Analysis Chain: Filtering - quite often

GRACE SHC




Geophysical model


Filtering (in space domain)



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Analysis Tools: PCA

Computation of EOFs


Project gridded data onto EOFs: PCs for GRACE and model


Filtering (in spectral domain)Filtering (in spectral domain)

Truncated (filtered) reconstruction

Comparison

from GRACE or model

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Analysis Chain: PCA - also possible…

GRACE SHC



PCA & truncated reconstruction(in spectral domain)

SHA geophysical model SHC


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What is the purpose of filtering a GRACE solution?

Unfiltered GRACE solution

Part I: Filtering techniques and their application to GRACE data

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Filtering

– Attempts to suppress ‚noise‘ in data (here: SH coefficients)

– Requires that we have an a-priori knowledge of expected ‚noise‘ ( characterize spectral behaviour of noise)

– Filters take this into account either implicitly (‚deterministic filters‘ or explicitly ‚stochastic filters‘)

– Also regularizing or ‚constraining‘ GRACE normal equations corresponds to some kind of filtering (Kusche 2007, Klees et al 2008, Swenson & Wahr 2011)

Data sets e.g.

– GRACE-derived SH coefficients or maps of geoid, gravity anomalies,TWS

– SH coefficients or maps of TWS and other data from model output


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Boxcar-filtered GRACE solution: Remove stripes by averaging, convolution


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Gaussian-filtered GRACE solution: convolution with a smooth kernel


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Filtering a GRACE solution in spatial and in spectral domain


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Notations I use

– Positive and negative order

– Fully normalized

– All factors are put into the coefficientsPotential Surface Mass

(involves Love numbers)

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Isotropic filters

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Isotropic filters: Gaussian filter

harmonic degree n

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Destriping filters

Isotropic Non-isotropic

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Destriping filters

Swenson & Wahr 2006

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Destriping filters

Kusche 2007

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Destriping filters and WRMS reduction

GRACE TWS WRMS [cm], 3 filters

stronger lessfiltering

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…

t

Basin averaging

tOcean mass

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Basin averaging

(spatial domain)

(spectral domain)

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Basin averaging

Note: Equality requires• the integral is discretized at an error small enough• the SH truncation degree is big enough OR BOTH F and O are band-

limited

(spectral domain)

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Smoothed basin averaging and bias

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Leakage problem and filter bias: basin averaging, what happens?

Model GRACE (truncated SH)

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+ noise

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Model

Exact averaging Spectral leakage

GRACE (truncated SH)

Lost signal

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Lost signal

(leaking out)

Leaking outBias can be computed from model

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Lost signal (leaking out)

Signal added by surrounding region (leaking in)

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Part II: Principle component analysis and related ideas

Empirical orthogonal function analysis

Principle components

Related concepts

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PCA

– attempts to find a relatively small number of independent modes in a data set that convey as much as possible information without redundancy

– can be used to explore the structure of the data variability in an objective way, i.e. without assumptions on periodic behaviour etc.

– and EOF analysis are the same.

Data sets e.g.

– GRACE-derived maps of TWS, TWS and other maps from model output

– Sea level anomalies

– Other related spatial fields (SST, SLP, …)

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What does PCA do?

PCA uses a set of orthogonal functions (EOFs) to represent a spatio-temporal data field in the following way

EOF ej =e (, ) show spatial patterns of the major factors (‚modes‘) that account for temporal variations.

PC dj;i = d(t) tells how the amplitude of EOF varies with time.

PCs (expansion coefficients)

EOFs (new basis)

epochs (time)

grid points

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How do we obtain the PCs?

Lets assume the EOFs are orthogonal (why are they called EOF, after all) and normalized.

The PCs are found as an orthogonal projection of the data onto the new basis functions (the EOFs). We can try to reconstruct the original data using only the ‚major‘ EOFs

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What do we get from PCA?

…

t

62%t

24%t

<1%t

PC1

PC2

PCn

EOF1

EOF2

EOFn

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Example (I)GRACE TWS 400km Gaussian filtered (note: EOF PC has unit of data)

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How do we choose the EOFs?

Total variance

Then, the maximum variance is concentrated in a single EOF if

I.e., maximize subject to

Or, solve an eigenvalue problem

max. variance

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How do we choose the EOFs?The EOFs are found as the eigenvectors of the data covariance matrix.

Some remarks (I)

– The covariance as above is temporal, i.e. it considers auto- and covariances of time series per grid point (for n grid points)

(we‘ll come back to this point)

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Representing data in an eigenvector basis

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The EOFs are found as the eigenvectors of the data covariance matrix.

Some remarks (II)

– It is an empirical realization. Should we know the true covariance, we might better use this one instead of the empirical one

– All data has been considered as (perfectly) centered

– Eigenvectors require normalization and a further convention for uniqueness, e.g.

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Some remarks (III)

– Alternatively, we could use SVD applied to the data matrix

– Each EOF explains a fraction of the total variance, given by the ratio of the EV vs. TV (total variance)

– It is common to choose the number of EOFs as to ‚explain‘’ 90% (or …) of the TV

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Example

GRACE global analysis (left: EV, right: cumulative percentage of TV)

(courtesy E. Forootan)

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Some remarks (IV)

– Instead of computing the temporal data covariance, we may compute the spatial covariance (spatial variance and covariances for the p epochs)

– Requires less memory for p < n– Temporal and spatial covariance matrices share the same p EVs– k-th EOF (spatial) ~ k-th PC (temporal) and vice versa

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Some remarks (V)

– Linear transformations of the data lead to new eigenvectors and –values

Therefore:

– EOFs and PCs computed on a regional grid look different from EOFs/PCs computed on a global grid

– GRACE: EOFs of geoid change look different from EOF of TWS change– PCA applied to GRACE SH coefficients (EOF filter of Schrama & Wouters)

looks different from PCA applied to grids.

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Example (I) revisited Trend (decrease)+ annual

Trend (increase)+ annual

Annual signals out of phase Semi-annual (modulation ofannual)

Trend ?

2007: unusually strong rainfall in Congo

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Examples (II) Wouters & Schrama (2007) Direct EOF filtering of GRACE SH coefficients

Top: Unfiltered/Gaussian, Middle: EOF filtered, Bottom: difference. Unit [cm]

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How many modes (EOFs) should we retain? In other words, how many % of the data TV should we reconstruct?

North et al. 1982, Month. Weath. Rev.: Consider the spatio-temporal data as stochastic, i.e. perturbed by e.g. Gaussian noise. Then, the covariance

and the eigenvalues / -vectors will be stochastic as well. In first order…

If the sampling error in the eigenvalue is comparable to the spacing of the eigenvalues, then the sampling error of the EOF will be comparable to the nearby EOF.

And then it is time to truncate.

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North et al (1982), reprinted in Von Storch & Zwiers

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Related concepts: (Orthogonal) EOF rotation

REOFs are still orthogonal - RPCs are correlated now.

How can we use this degree of freedom?

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Related concepts: REOF, how can we use this degree of freedom? I.e. how do we find the rotation matrix?

E.g. VARIMAX, maximize the variance of the square of the REOFs (i.e. the spreading of the total variability of the modes)

E.g. ICA, minimize 3th / 4th statistical moments of the REOFs to maximize the independence of the RPCs

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Dommenget & Latif (2001)

simulated

PCA

VARIMAX

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Take-home message

Filtering is a necessary tool for interpreting GRACE data correctly.

– What we have discussed here are the options that the user of Level-2 data products has. Some filtering has been applied in the Level-1 processing as well. There is simply no point in using ‚unfiltered‘ data.

PCA is a useful tool for interpreting GRACE and other geophysical data and model outputs. It is either applied as a kind of filtering (see above) or as a tool to explore the major directions of data variability.

– It is easy to construct counterexamples where PCA fails to isolate physical modes!

– „PCA may help you to find the needle in the haystack. But once you found it, you should be able to recognize it as a needle“ [v. Storch]. I.e. you should be able to assign some physics to it, otherwise it might be just an artefact.

1 spp 1257 tutorial: analysis tools for grace time- variable harmonic coefficients jürgen kusche,...

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