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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Part II: Principle component analysis and related ideas Empirical orthogonal function analysis Principle components Related concepts http://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch5s5-5-4-1.html From IPCC Rep. 2007, Working Group I: The Physical Science Basis, 5.5.4.1 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Part II: Principle component analysis and related ideas Synthetisches Beispiel aus North et al 1982. 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

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

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

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

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