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378 S. Petrovic et al.

hydrological mass redistributions performed here and may therefore be valuable for

further applications in other ﬁelds.

2 Input Data and Preprocessing

Our analysis is based on time series of spatial grids of surface mass anomalies from

GRACE gravity ﬁelds and from water storage ﬁelds of various global hydrology

models. These series are in both cases derived from time series of coefﬁcients of

spherical harmonics describing the gravity potential.

For GRACE such gravity potential coefﬁcients are provided by the groups

of the GRACE Science Data System (SDS) from the inter-satellite measure-

ments, which are related to the gravity variations along the satellites’ orbit. In

the following we use our own RL04 monthly model series generated at GFZ

(GFZ-RL04).

For hydrology four state-of-the-art global hydrology models are considered: the

WaterGAP Global Hydrology Model (WGHM, Döll et al., 2003), the H96 model

(Huang et al., 1996), the Land Dynamics (LaD) model (Milly and Shmakin, 2002),

and the Global Land Data Assimilation System (GLDAS, Rodell et al., 2004). In

order to be comparable with integral mass variations deduced from GRACE obser-

vations, a hydrological model should include all continental water existing in any

form. This holds for no available hydrology model. Furthermore, the considered

models differ with respect to several aspects of model design (Werth et al., 2009b).

Nevertheless, they give a realistic representation of global hydrology. All these

hydrology models are provided as total water storage variations on global grids.

All the monthly maps are then expanded into sets of spherical harmonics describ-

ing the gravity potential (e.g. Wahr et al., 1998) to allow for a uniﬁed preparation

of time series of grids of surface mass anomalies from GRACE and hydrological

data sets.

Each monthly spherical harmonic data set is referenced to the individual long

term mean to derive residual time-variable quantities. From these residual spherical

harmonic coefﬁcients time series of surface mass anomalies on 0.5

◦

×0.5

◦

grids are

computed applying the relations given by Wahr et al. (1998). In order to suppress the

correlated errors of the GRACE-based surface mass anomalies, known as striping,

the following ﬁlter versions are considered:

• DDK1: decorrelation ﬁlter with parameters a = 1 · 10

, p = 4 (Kusche, 2007),

• DDK2: decorrelation ﬁlter with parameters a = 1 · 10

, p = 4 (Kusche, 2007),

• DDK3: decorrelation ﬁlter with parameters a = 1 · 10

, p = 4 (Kusche, 2007),

• G500: Gaussian smoothing ﬁlter with R = 500 km (Jekeli, 1981).

For consistent treatment, the same ﬁlter is applied both to the GRACE and

the hydrological data sets. The focus in this study is on continental hydrology.

Therefore, we extract from the global grids the data points located over land and

data points located within 18 selected river basins worldwide. The size of the basins

Surface Mass Variability from GRACE and Hydrological Models 379

ranges from 5.9 · 10

for the Amazon to 85 · 10

for the Po basin, which

is at the limit of the resolution of the GRACE mission. Since the results with any

of the hydrological models are in general very similar, we present mainly results of

WGHM as a representative hydrological data set.

3 Methodology

The analysis consists of three main steps: (1) Principal Components Analysis

(PCA), (2) search for arbitrary periods in principal components, and (3) accuracy

assessment of the spectral parameters from GRACE.

Principal Components Analysis (PCA, also known as Empirical Orthogonal

Functions, EOF) allows for a proper pre-processing of the data in studying the

temporal characteristics of the spatial variability of surface mass anomalies from

GRACE and the hydrology models. We use conventional PCA (e.g. Preisendorfer,

1988, or Wilks, 1995), where decomposition into the mode-wise pairs of eigen-

vectors (evs) and associated principal components (pcs) is based on the signal

variance-covariance matrix of the input data. For each mode the eigenvector rep-

resents a pattern of the spatial variability. The associated principal component

describes the variation of this pattern with time.

Since the input data sets are driven by climatological processes that show vari-

ability in the signal amplitude, phase and also period, both in space and time,

we estimate all three quantities simultaneously. Based on the principles described

in Mautz and Petrovic (2005), we estimate the amplitudes A

, phase lags φ

and

frequencies ω

of the model consisting of sinusoidal terms

(t) = A

sin (ω

t + φ

)(1)

from the principal components y(t) of individual dominant modes. For discrete data

this problem can be transformed into a least squares adjustment problem with the

observation equations:

= A

+ Dt



k=1

cos (ω

) +



k=1

sin (ω

) +ε

,(i = 1, ..., N)(2)

where in addition to the amplitudes A

and phase lags φ

(contained in the B

and

), also the unknown frequencies ω

= 2πf

= 2π/T

, and the trend parameter

D are to be determined. The latter is estimated to avoid aliasing of secular signals

into the periodic terms. The main advantage of the approach (2) is that the sig-

nal energy is mapped into few but representative frequencies, as it allows for the

estimation of arbitrary periods in addition to the amplitudes and phases. This con-

trasts the classical Fourier analysis where the signal energy is forced into a ﬁxed

basic period and multiples of the associated basic frequency, treating only the sig-

nal amplitudes and phases as unknowns. Equation (2) represents a highly non-linear

380 S. Petrovic et al.

least squares problem. In order to apply standard local optimization methods, very

accurate initial values for the unknown parameters are necessary. These can only

be provided by global optimization methods, which require high computational

effort (e.g. Horst and Pardalos, 1995). A reasonable strategy based on a sequential

algorithm is presented in Schmidt et al. (2008).

In contrast to the hydrological data sets, where only little is known about the

data accuracy, it is possible to assess the accuracy of the GRACE derived ampli-

tudes, phases and periods. This can be done based on the available GRACE gravity

ﬁeld model error estimates provided together with the gravity ﬁeld models by the

GRACE science data teams. We use a Monte Carlo method for error propagation

(Gundlich et al., 2003; Kalos and Whitlock, 1988), considering the spatial cor-

relation of the GRACE gravity ﬁeld errors. Based on a scaled version of the full

variance-covariance matrix of the monthly GRACE gravity model for August 2003

we create 200 spatial grids of noise in the surface mass anomalies for each of the

considered river basins by a rigorous error propagation of correlated model errors

by the Monte Carlo method. The noise grids are added to GRACE grids of surface

mass anomalies that are constructed from the strongest periods found in the pcs of

the PCA. To do this we replace the original principal components of the respec-

tive modes by the harmonic model from (2) evaluated at the epochs of the input

data, and perform the inverse PCA transformation for these modes. As a result we

obtain a time s eries of GRACE-based surface mass anomalies, containing a set of

well-deﬁned variations contaminated by authentic noise. Applying the PCA and the

frequency analysis to this data we derive the standard deviations of the empirical

distribution for the amplitudes, periods and phases.

4 Application to Continental Grids and River Basins

The described methodology was applied to continental grids and to 18 selected

river basins. The PCA reveals that in all cases 80% and more of the total vari-

ability observed in GRACE and water storage of hydrology models is contained

in the very ﬁrst three to ﬁve modes (Petrovic et al., 2007; Schmidt et al., 2008).

Hence, we limit the frequency analysis of the pcs and the accuracy assessment

for the GRACE data to these lower modes. Some plots of the evs patterns and

the associated pcs curves can be found in Petrovic et al. (2007) and Schmidt et al.

(2008). Here we illustrate the results taking as example the basins of Amazon and

Ganges.

Tables 1, 2, 3, and 4 summarize the periods detected in the pcs from the non-

standard frequency analysis outlined in previous section. In analogy to PCA, the

results are tabulated in descending order, i.e. from the strongest to the weakest peri-

odic component. This ranking is derived based on the percentage contribution of the

harmonic component to the total variability of the input grid data, i.e. from the ratio

(var

harm

/var

total

) · 100 [%], where var

total

denotes the variance of the original total

signal (which includes noise), and var

harm

is the signal variance of the spatial signal

reconstructed from the harmonic components only.

Surface Mass Variability from GRACE and Hydrological Models 381

Table 1 Periods, phases and amplitudes of the most dominant periodic features for Amazon

deduced from GRACE time series 08/2002–09/2007. Applied decorrelation ﬁlter DDK1, ampli-

tudes in relative units, last column displays the percentage of total signal explained

Rank Mode Period T (d) Phase φ (d) Amplitude A Percentage of total

1 1 361.9 ±0.3 114.3 ± 0.7 7.63 ±0.06 73.9

2 2 361.5 ±0.4 201.3 ± 0.6 3.73 ±0.04 17.8

3 3 772.4 ±19.5 75.1 ± 13.7 0.50 ±0.06 1.1

4 1 459.6 ±6.6 469.7 ± 7.9 0.90 ±0.06 1.0

Table 2 Same as Table 1, but for the hydrology model WGHM

Rank Mode Period T (d) Phase φ (d) Amplitude A Percentage of total

1 1 360.3 101.9 4.80 73.3

2 2 361.0 197.0 2.09 13.8

3 1 1,072.7 455.0 1.28 4.0

4 1 467.0 453.3 0.77 1.3

Table 3 Periods, phases and amplitudes of the most dominant periodic features for Ganges

deduced from GRACE time series 08/2002–09/2007. Applied decorrelation ﬁlter DDK1, ampli-

tudes in relative units, last column displays the percentage of total signal explained

Rank Mode Period T (d) Phase φ (d) Amplitude A Percentage of total

1 1 365.2 ±0.6 264.3 ± 1.0 2.88 ±0.04 83.5

2 1 184.8 ±0.7 60.2 ± 1.7 0.76 ± 0.04 5.5

3 2 358.3 ±2.1 355.8 ± 3.8 0.47 ±0.03 2.3

Table 4 Same as Table 3, but for the hydrology model WGHM

Rank Mode Period T (d) Phase φ (d) Amplitude A Percentage of total

1 1 365.1 255.6 2.30 89.3

2 1 182.7 54.3 0.59 6.0

For the GRACE-based results the accuracy assessment according to Sect. 3 is

displayed in Tables 1 and 3. For the interpretation of the errors of the amplitudes

one has to note that the tabulated values have relative units, since absolute sig-

nal amplitudes are only given after the synthesis of the eigenvectors and principal

components. The mean errors displayed in Tables 1 and 3 show that the four most

dominant periodic variations found in Amazon and the three found in Ganges can

be regarded as signiﬁcant. However, the signiﬁcance of the 772.4-days wave in

Amazon is somewhat lower. In spite of the statistical signiﬁcance of the results,

it makes sense to check whether the resulting periods are independent of the ﬁlter

used. Table 5 illustrates the differences in the results for three different versions

of the decorrelation ﬁlter according to Kusche (2007), which represent a realistic

span of smoothing. The relative differences between the periods detected based on

382 S. Petrovic et al.

Table 5 Comparison of periods, phases and amplitudes estimated for data points inside the

Amazon and Ganges basin from GRACE for different versions (DDK1, DDK2, DDK3) of

decorrelation ﬁlter. Time span: 08/2002–09/2007

 period (d)  phase (d) Amplitude ratio

DDK2-1 DDK3-1 DDK2-1 DDK3-1 DDK2/1 DDK3/1

Amazon 0.05 −0.02 1.03 3.04 1.09 1.12

0.29 0.39 1.62 3.38 1.27 1.35

–5.01 −9.70 −15.31 −5.87 1.64 2.02

0.06 0.60 1.70 4.81 1.22 1.27

Ganges 0.06 0.04 −0.52 −3.14 1.12 1.16

–0.14 −0.56 −2.01 −4.59 1.11 1.09

0.46 3.56 −1.86 −1.74 1.85 2.55

the preprocessing by DDK1 and DDK2 vary between 0.01 and 0.13%, with the

sole exception for the 772.4-days wave (0.69%). For the comparison of DDK1 and

DDK3 these relative differences are mostly under 0.30% and reach in the worst case

1.26%. Note that the uncertainty of 0.01% corresponds to the periodic component

which explains 73.9% (respectively 83.5%) of the total signal, and the uncertainty of

1.26% to the periodic term explaining only 1.1% of the total variability. In short, the

determination of periods and phase lags is very stable, differences in the smoothing

strength affect, as expected, primarily amplitudes, cf. Kusche et al. (2009).

Table 6 displays the results obtained for Ganges basin earlier using a simple

Gaussian smoothing ﬁlter and a shorter time span (Schmidt et al., 2008). There is

a very good agreement with Table 3, conﬁrming that the detected periods are not

merely an artefact of the applied ﬁlter or time interval.

The results for Amazon and Ganges presented here are representative for all

investigated basins and can be summarized as follows. A major part of the total

variability, usually 80% and more, is contained in a few ﬁrst modes resulting from

PCA. In each basin a few (2–4) signiﬁcant (according to the accuracy assessment)

periodic terms are detected, which explain 70–95% of the entire variability. The

signiﬁcance of the long-periodic terms (periods of 2 years and more) is somewhat

lower due to the limited time span of the available data. Preprocessing GRACE data

with different decorrelation or smoothing ﬁlters affects the determination of periods

and phase lags only slightly.

Table 6 Periods, phases and amplitudes of the most dominant periodic features for Ganges

deduced from GRACE time series 02/2003–12/2006. Applied Gaussian smoothing ﬁlter G500,

amplitudes in relative units, last column displays the percentage of total signal explained

Rank Mode Period T (d) Phase φ (d) Amplitude A Percentage of total

1 1 362.7 ±1.0 265.1 ± 1.4 2.69 ±0.06 79.0

2 1 183.4 ±1.0 57.8 ± 2.3 0.71 ± 0.05 4.6

3 2 352.7 ±5.2 358.6 ± 13.2 0.60 ±0.06 4.6

Surface Mass Variability from GRACE and Hydrological Models 383

In Tables 1, 2, 3, and 4 we see a good agreement between the periods detected

from GRACE observations and from the hydrology model WGHM. The most

dominant features observed are annual periodic variations (with almost identical

results for the three other hydrology models considered here). They are found in

all investigated basins; in most of them 70–80% of the total variability seem to

be represented by annual waves. The accuracy assessment of the GRACE-based

results indicates a very good resolution of the annual terms; for the Amazon and

the Ganges basin the standard deviations of the derived annual periods and their

phases are at the level of or even below 1 day. The detection of several annual

signals in different modes, i.e. associated to different spatial patterns, is not sur-

prising. This is caused by the s patial variability of the signal amplitudes, phases

and periods inside the region of interest. For the Amazon these variations are well

explicable by the distribution of the rainfall inside this basin, cf. Zeng (1999) and

Güntner et al. (2007). Hence, ﬁnding several annual periods associated to different

spatial patterns inside a basin is a consequence of the processes governing the mass

redistributions.

Semiannual variations are found only in few investigated river basins. The contri-

bution to the total variability is small, but can be determined signiﬁcantly, as shown

by standard deviations for the Ganges (Table 3). This semiannual term is found in

the principal component of mode 1; hence, it has the same spatial variability pat-

tern as the annual term. Therefore, it might be just a correction of the dominating

annual term, and it is not easy to attribute the semiannual signal to some uniquely

deﬁned physical cause. However, there are several physical causes that may explain

semiannual oscillations indeed, cf. Güntner et al. (2007).

The results of the frequency analysis (Tables 1 and 2) also show long-periodic

variations in the range of 2 years and more. As for semiannual signals, these rep-

resent only small contributions to the total variability, too. The standard deviations

for the GRACE-based estimates are much larger than for the annual and semiannual

terms. This holds e.g. for the 2.1-yearly oscillation in the Amazon (Table 1), and

illustrates the difﬁculties in clearly identifying such terms, in particular in view of

the rather short period covered by GRACE so far. However, for several reasons such

terms may be plausible nevertheless, cf. Schmidt et al. (2008).

5 Reconstruction of Filtered GRACE Time Series of Surface

Mass Anomalies

Phase lags in Tables 1, 2, 3, and 4 show that the variations of surface mass anomalies

from WGHM are ahead of the GRACE estimate for Amazon and Ganges. This holds

also in general. Taking into account all investigated basins and different hydrology

models, an average systematic offset of about 1 month can be detected. In order to

validate and calibrate hydrology models, the GRACE time series is reconstructed

taking into account only the part of the signal which can be regarded as reliably

detected and representative for the hydrological variations. In order to achieve this,

384 S. Petrovic et al.

280°

285°

290°

295°

300°

305°

310°

–10°

–5°

0° 0°

5° 5°

-15

-1

-10

-5

–20° –20°

–15° –15°

–10°

–5°

0° 0°

–20 –15 –10

–5

10 15 20

280°

285°

290°

295°

300°

305°

310°

–15°

0° 0°

5° 5°

–20 –20°

° –15°

–10° –10°

–5° –5°

0° 0°

5° 5°

–20 –15 –10

–5

10 15 20

(b)(a)

Fig. 1 (a) Synthesis of GRACE-based surface mass anomalies in the Amazon derived from the

ﬁrst four harmonic terms from Table 1 at epoch 15 August 2006. (b) Residual of total surface mass

anomaly map minus the map shown in (a). Units: cm water column

we construct ﬁltered GRACE time series of surface mass anomalies using in the

PCA synthesis the harmonic components of the pcs that

• are signiﬁcant according to the accuracy assessment, and

• are found also in the hydrology data.

As an example, Fig. 1(a) shows the ﬁltered surface mass anomaly signal for

the Amazon basin for the month of August 2006. This reconstruction is based on

the four dominant periodic terms from Table 1. Figure 1(b) displays the residual

signal obtained by subtracting the map in Fig. 1(a) from the total GRACE-based

signal at that epoch. The anomalies in Fig. 1(a) represent about 94% of the total

signal (Table 1), while the residual signal in Fig. 1(b) represents only about 6%

of the variability. Although both maps still contain GRACE data errors, the spa-

tial distribution of the residual signal, dominated by North-South oriented striping

features well comparable to the correlated GRACE gravity model error, indi-

cates a clear signal-noise separation obtained by the applied approach. For the

Ganges, the three dominating periodic terms (Table 3) explain 91% of the total

variability.

The residual signal may still contain traces of hydrological signal and cannot

be interpreted as pure errors. Since the amplitude of the residual signal is rela-

tively small (in the Amazon only 6% of the total variability, in the Ganges 9%), the

potentially neglected hydrological signal should be very s mall, too. Conversely, the

reconstructed signal from the selected harmonic terms may also contain some small

portions of the error signals. However, the accuracy assessment shows that the rela-

tive contributions of errors in the derived dominating terms are much smaller than in

the residual part. The percentage of the total signal explained by this reconstruction

is comparably high in most of the considered basins. In this way the construction of

ﬁltered surface mass anomalies from the dominating terms is useful for the basin-

speciﬁc calibration of hydrological models with respect to the dominant periodic

variations of water storage, as demonstrated in Werth et al. (2009a).

Surface Mass Variability from GRACE and Hydrological Models 385

6 Conclusions

We present the methodology for deriving characteristic spectra of periodic sur-

face mass variations induced by continental hydrology observable in time series

of GRACE gravity and global hydrology models. To this end we apply a combined

PCA and frequency analysis method, where the latter allows for the determination

of arbitrary periods from the temporal patterns of the derived PCA modes. This is

an interesting alternative to conventional, Fourier-based frequency analysis of data,

as it may allow for a more adequate description of time variability contained in geo-

physical data. For the GRACE-based spectra we perform an accuracy assessment

using available GRACE error estimates and a Monte-Carlo simulation.

The characteristic features of hydrologically induced mass redistributions are

dominated by annual variations which explain about 60% to more than 90% of

the total data variability, and are well determined in the GRACE data. The abso-

lute errors of the annual periods and phases are small and seem to be at the level

of about ± 1 day for most of the basins. For the phases of the annual terms we

observe a systematic delay between GRACE-based surface mass anomalies and

the hydrology models (depending on the model and the region the phases from the

hydrology models are about 1–6 weeks ahead of the GRACE estimates), which may

point to systematic deﬁciencies in hydrological modelling. Semiannual terms do

not play a signiﬁcant role, at least on the global scale. For river basins only in a few

regions, e.g., the Ganges, the Nile, the Congo, and the Ob, semiannual variations

are traceable. Moreover long-periodic signals with periods of 2 years and more are

observed. Due to the short data period covered by GRACE so far, a separability of

such features may be limited.

Finally, it is possible to derive ﬁltered GRACE surface mass anomaly data. These

are reconstructed by the inverse PCA transformation where the original principal

components are replaced by harmonic terms that (1) are signiﬁcant according to the

error estimates, and (2) are found consistently also in hydro-climatological mod-

els. Such anomaly data, revealing a distinct signal-noise separation, are of beneﬁt

for calibration and validation of hydroclimatological models. The method is not

restricted to the presented study case of hydrology, and it may be of general inter-

est for a wide set of geophysical applications in the context of an assimilation of

GRACE-based surface mass data.

Acknowledgments The German Ministry of Education and Research (BMBF) supported these

investigations within the geoscientiﬁc R+D programme GEOTECHNOLOGIEN “Erfassung des

Systems Erde aus dem Weltraum” under grant 03F0424A. We thank P.C.D. Milly, Y. Fan and H.

van den Dool, M. R odell as well as P. Döll and J. Alcamo for providing the LaD, H96, GLDAS

and WGHM model data, respectively. Thanks also go to D.W. Pierce for his Empirical Orthogonal

Functions (EOF) software.

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Time-Space Multiscale Analysis and Its

Application to GRACE and Hydrology Data

Willi Freeden, Helga Nutz, and Kerstin Wolf

1 Introduction

The satellite gravity mission GRACE (Gravity Recovery and Climate Experiment)

(Tapley et al., 2004a, b) provides a huge amount of data which allows to quantify

both temporal and spatial variations of the Earth’s gravity ﬁeld caused by mass trans-

port and mass distribution (Swenson and Wahr, 2006; Swenson et al., 2003). For

modelling the gravitational ﬁeld of the Earth, usually, truncated Fourier series based

on spherical harmonics are used where the accuracy of the approximation is given

by the maximum degree. A fundamental disadvantage of this method is the localiza-

tion of the spherical harmonics in the frequency domain, which leads to a smearing

of the spatial detail information over the whole globe. The need of a method which

allows local analysis of the gravitational potential led to the development of spher-

ical wavelets in the Geomathematics Group of TU Kaiserslautern (Freeden, 1999;

Freeden and Schneider, 1998; Freeden and Schreiner, 2009; Freeden et al., 1998).

The spherical wavelets are kernel functions, which are built up using clusters of a

ﬁnite number of polynomials, and, thus, guarantee a good localization in the space

domain. Note that the uncertainty principle states that localization in both frequency

and space domain are mutually exclusive.

Based on the theory of spherical wavelets we propose two different methods

which allow a temporal and spatial wavelet analysis of monthly data sets given

by the satellite mission GRACE or resulting from WGHM (WaterGAP Global

Hydrology Model) (Döll et al., 2003) for a period of several years (some ﬁrst results

are presented in, e.g., Fengler et al., 2007). With the help of the wavelet analysis

we are able to locally compare the GRACE and hydrology data in both temporal

and spatial sense and, ﬁnally, to give a ﬁrst idea of how to construct an improved

hydrological model on the basis of a time-space multiresolution.

The layout is as follows: in Sect. 2 the combination of the spatial analysis and

the temporal analysis is performed with the help of two different methods: First

W. Freeden (B)

Geomathematics Group, Department of Mathematics, TU Kaiserslautern,

67653 Kaiserslautern, Germany

e-mail: freeden@mathematik.uni-kl.de

387

F. Flechtner et al. (eds.), System Earth via Geodetic-Geophysical Space Techniques,

Advanced Technologies in Earth Sciences, DOI 10.1007/978-3-642-10228-8_33,

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Springer-Verlag Berlin Heidelberg 2010