1. Tutorial: Analysis Tools for GRACE time-variable harmonic coefficients Jürgen Kusche, Annette Eicker, Ehsan Forootan Bonn University, Germany IGCP 565 Workshop November 21-22, 2011 University of the Witwatersrand, Johannesburg, South Africa

2. 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.

3. Analysis Chain: Filtering
GRACE SHC
SHA geophysical model  SHC
Globally defined?
Treatment degree 1 & 2, temporal anomalies wrt epoch t, restore AOD
Temporal alignement, possibly remove deg. 1 consistent to GRACE
Filtering (in spectral domain)
Filtering (in spectral domain)
Multiplication in spectral domain
Mapping to space domain or basin averaging
Mapping to space domain or basin averaging
OR Filtering (in space domain)
OR Filtering (in space domain)
Convolution on the sphere 3

4. Analysis Chain: Filtering - quite often
GRACE SHC
Geophysical model
Treatment degree 1 & 2, temporal anomalies wrt epoch t, restore AOD
Temporal alignement, possibly remove deg. 1 consistent to GRACE
Filtering (in spectral domain)
Filtering (in spectral domain)
Mapping to space domain or basin averaging
Mapping to space domain or basin averaging
Filtering (in space domain) 4

5. Analysis Tools: PCA Filtering (in spectral domain)
Mapping to space domain or basin averaging
Mapping to space domain or basin averaging
Computation of EOFs
from GRACE or model
Project gridded data onto EOFs: PCs for GRACE and model
Comparison
Truncated (filtered) reconstruction 5

6. Analysis Chain: PCA - also possible…
GRACE SHC
SHA geophysical model  SHC
Treatment degree 1 & 2, temporal anomalies wrt epoch t, restore AOD
Temporal alignement, possibly remove deg. 1 consistent to GRACE
PCA & truncated reconstruction
(in spectral domain)
Mapping to space domain or basin averaging 6

7. Part I: Filtering techniques and their application to GRACE data
What is the purpose of filtering a GRACE solution?
Unfiltered GRACE solution 7

8. Part I: Filtering techniques and their application to GRACE data
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 8

9. Part I: Filtering techniques and their application to GRACE data
What is the purpose of filtering a GRACE solution?
Boxcar-filtered GRACE solution: Remove stripes by averaging, convolution 9

10. Part I: Filtering techniques and their application to GRACE data
What is the purpose of filtering a GRACE solution?
Boxcar-filtered GRACE solution: Remove stripes by averaging, convolution 10

11. Part I: Filtering techniques and their application to GRACE data
What is the purpose of filtering a GRACE solution?
Boxcar-filtered GRACE solution: Remove stripes by averaging, convolution 11

12. Part I: Filtering techniques and their application to GRACE data
What is the purpose of filtering a GRACE solution?
Gaussian-filtered GRACE solution: convolution with a smooth kernel 12

13. Part I: Filtering techniques and their application to GRACE data
Filtering a GRACE solution in spatial and in spectral domain 13

14. Part I: Filtering techniques and their application to GRACE data
Notations I use
Positive and negative order
Fully normalized
All factors are put into the coefficients
Potential  Surface Mass
(involves Love numbers) 14

15. Part I: Filtering techniques and their application to GRACE data
Isotropic filters 15

16. Part I: Filtering techniques and their application to GRACE data
Isotropic filters: Gaussian filter
harmonic degree n 16

17. Part I: Filtering techniques and their application to GRACE data
Destriping filters
Isotropic
Non-isotropic 17

18. Part I: Filtering techniques and their application to GRACE data
Destriping filters
Swenson & Wahr
2006 18

19. Part I: Filtering techniques and their application to GRACE data
Destriping filters
Kusche 2007 19

20. Part I: Filtering techniques and their application to GRACE data
Destriping filters and WRMS reduction
GRACE TWS WRMS [cm], 3 filters
stronger
filtering
less 20

21. Part I: Filtering techniques and their application to GRACE data
Basin averaging …
Ocean mass t t 21

22. Part I: Filtering techniques and their application to GRACE data
Basin averaging
(spatial domain)
(spectral domain) 22

23. Part I: Filtering techniques and their application to GRACE data
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) 23

24. Part I: Filtering techniques and their application to GRACE data
Smoothed basin averaging and bias 24

25. Part I: Filtering techniques and their application to GRACE data
Leakage problem and filter bias: basin averaging, what happens?
Model
GRACE (truncated SH) 25

26. Part I: Filtering techniques and their application to GRACE data
Leakage problem and filter bias: basin averaging, what happens?
Model
GRACE (truncated SH)
+ noise 26

27. Part I: Filtering techniques and their application to GRACE data
Leakage problem and filter bias: basin averaging, what happens?
Model
GRACE (truncated SH)
Exact averaging
Spectral leakage
Lost signal 27

28. Part I: Filtering techniques and their application to GRACE data
Leakage problem and filter bias: basin averaging, what happens?
Model
GRACE (truncated SH)
Lost signal
(leaking out)
Bias can be computed from model
Leaking out 28

29. Part I: Filtering techniques and their application to GRACE data
Leakage problem and filter bias: basin averaging, what happens?
Lost signal (leaking out)
Signal added by surrounding region (leaking in) 29

30. Part II: Principle component analysis and related ideas
Empirical orthogonal function analysis
Principle components
Related concepts
From IPCC Rep. 2007, Working Group I: The Physical Science Basis, Steric Sea Level Changes
EOF1 and (normalized) PC1 from Ishii 2006, thermosteric sea level 700m from aRGO, red dotted = (negative) Southern Oscillation Index
(i.e. from air pressure difference Tahiti – Darwin (Australia) 30

31. Part II: Principle component analysis and related ideas
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, …) 31

32. Part II: Principle component analysis and related ideas
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.
epochs (time)
PCs (expansion coefficients)
grid points
EOFs (new basis) 32

33. Part II: Principle component analysis and related ideas
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 33

34. Part II: Principle component analysis and related ideas
PC1
What do we get from PCA? t
62%
EOF1
PC2 t
24%
EOF2 …
PCn t
<1% t
EOFn 34

35. Part II: Principle component analysis and related ideas
Example (I)
GRACE TWS 400km Gaussian filtered (note: EOF  PC has unit of data) 35

36. Part II: Principle component analysis and related ideas
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 36

37. Part II: Principle component analysis and related ideas
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) 37

38. Part II: Principle component analysis and related ideas
Representing data in an eigenvector basis 38

39. Part II: Principle component analysis and related ideas
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. 39

40. Part II: Principle component analysis and related ideas
The EOFs are found as the eigenvectors of the data covariance matrix.
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 40

41. Part II: Principle component analysis and related ideas
Example
GRACE global analysis (left: EV, right: cumulative percentage of TV)
(courtesy E. Forootan) 41

42. Part II: Principle component analysis and related ideas
The EOFs are found as the eigenvectors of the data covariance matrix.
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 42

43. Part II: Principle component analysis and related ideas
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. 43

44. Part II: Principle component analysis and related ideas
Example (I) revisited
Trend (decrease)+ annual
Semi-annual (modulation of
annual)
Annual signals out of phase
Trend ?
Trend (increase)+ annual
2007: unusually strong rainfall in Congo 44

45. Part II: Principle component analysis and related ideas
Examples (II)
Wouters & Schrama (2007) Direct EOF filtering of GRACE SH coefficients
Top: Unfiltered/Gaussian, Middle: EOF filtered, Bottom: difference. Unit [cm] 45

46. Part II: Principle component analysis and related ideas
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. 46

47. Part II: Principle component analysis and related ideas
Synthetisches Beispiel aus North et al Eigenwerte liegen alle zwischen 14 und 10, nicht unbedingt schlecht konditioniert.
Zwei jeweils benachbarte Paare von Eigenwerten.
Mit 300 (3000) Realiserungen (p=300) sind die geschätzten delta-lambda im Bereich 1 (0.6)
Daraus lassen sich relevante Störungen der EOFs vermuten für alle (300) bzw. für 3 und 4 (3000).
North et al (1982), reprinted in Von Storch & Zwiers 47

48. Part II: Principle component analysis and related ideas
Related concepts: (Orthogonal) EOF rotation
REOFs are still orthogonal - RPCs are correlated now.
How can we use this degree of freedom? 48

49. Part II: Principle component analysis and related ideas
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 49

50. Part II: Principle component analysis and related ideas
simulated
PCA
VARIMAX
Dommenget & Latif (2001) 50

51. 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. 51