Finally, we have derived maps showing the relationship between the H/V -peak and trough frequency and key parameters of the model such as impedance contrast.. Because the relationships b
Trang 1Geophys J Int (2011) 184, 793–800 doi: 10.1111/j.1365-246X.2010.04863.x
Rayleigh waves and the transmission response of single layer over
half-space models
Tran Thanh Tuan,1 Frank Scherbaum2 and Peter G Malischewsky3
1Hanoi University of Science, VNU, Vietnam E-mail: tuantt@vnu.edu.vn
2University Potsdam, Germany E-mail: fs@geo.uni-potsdam.de
3Friedrich-Schiller University Jena, Germany E-mail: p.mali@uni-jena.de
Accepted 2010 October 21 Received 2010 July 22; in original form 2009 December 7
S U M M A R Y
One of the key challenges in the context of local site effect studies is the determination
of frequencies where the shakeability of the ground is enhanced In this context, the H/V technique has become increasingly popular and peak frequencies of H/V spectral ratio are
sometimes interpreted as resonance frequencies of the transmission response In the present
study, assuming that Rayleigh surface wave is dominant in H/V spectral ratio, we analyse
theoretically under which conditions this may be justified and when not We focus on ‘layer over half-space’ models which, although seemingly simple, capture many aspects of local site effects in real sedimentary structures Our starting point is the ellipticity of Rayleigh waves
We use the exact formula of the H/V -ratio presented by Malischewsky & Scherbaum (2004)
to investigate the main characteristics of peak and trough frequencies We present a simple
formula illustrating if and where H/V -ratio curves have sharp peaks in dependence of model
parameters In addition, we have constructed a map, which demonstrates the relation between
the H/V -peak frequency and the peak frequency of the transmission response in the domain
of the layer’s Poisson ratio and the impedance contrast Finally, we have derived maps showing
the relationship between the H/V -peak and trough frequency and key parameters of the model
such as impedance contrast These maps are seen as diagnostic tools, which can help to guide
the interpretation of H/V spectral ratio diagrams in the context of site effect studies.
Key words: Site effects; Theoretical seismology; Wave propagation.
1 I N T R O D U C T I O N
The analysis of ambient vibrations has become an increasingly
pop-ular tool for the estimation of local site effects and the
characteriza-tion of shallow site structure This can be seen for example in several
major recent research initiatives either being completely devoted to
ambient vibrations such as SESAME
(http://sesame-fp5.obs.ujf-grenoble.fr/index.htm) or having one or more subprojects dealing
with ambient vibration related issues, such as HADU and
NER-IES (http://www.geotechnologien.de/forschung/forsch2.11k.html;
http://www.neries-eu.org/) and the new monograph by Mucciarelli
et al (2009) just to name a few In particular, the H/V spectral ratio
technique, originally introduced by Nogoshi & Igarashi (1971), also
known as Nakamura’s method Nakamura (1989, 2000, 2009), has
become the primary tool of choice in many of the ambient vibration
related studies Considering that the most dominant contributions
to ambient vibrations are known to come from surface waves,
al-though the exact composition may change depending on the
partic-ular site (cf the publications from the SESAME project referenced
above), this means that it is the characteristics of the ellipticity of
Rayleigh waves which is actually analysed However, the
fundamen-tals of the H/ V-technique are controversial [the history and different opinions are discussed e.g by Bonnefoy-Claudet et al (2006) and
Petrosino (2006)] These different opinions even refer to the term
H/V -technique itself In this paper Rayleigh-wave ellipticity is
con-sidered an essential part of H/V -technique, without excluding the
important analysis of body waves Recently, Albarello & Lunedei (2009) found out that the body wave interpretation provides better results around the resonance frequency but not for higher
frequen-cies We aware of the fact that the trough of the H/V -curve may
be masked by higher modes, Love and body waves However, our intention in this paper is to theoretically analyse certain new rela-tionships between parameters of interest by using fundamental mode Rayleigh waves alone While the amount of applications of ambient vibrations analysis in recent years is impressive, on the theoretical side numerous challenges remain What for example is the
relation-ship between the H/V peak frequency and the peak frequency of
the transmission response of a medium where the shakeability of the site would be expected to be enhanced? Under what conditions
is it allowed to assume their approximate equivalence? This
ques-tion is important, especially when the H/V -ratio is obtained from
noise recordings only These questions turn out to be surprisingly
Geophysical Journal International
Trang 2794 T T Tuan, F Scherbaum and P G Malischewsky
challenging theoretically even for very simple model and they have
only rarely been addressed in the literature (e.g Malischewsky &
Scherbaum 2004; Malischewsky et al 2008 and Haghshenas et al.
2008) In the present paper we are focusing on some of these
is-sues, namely the properties of peaks and troughs of the ellipticity of
Rayleigh waves and their relationships to the transmission response
We derive a set of relationships, which might be used in practical
applications to guide the interpretation of H/V spectral ratios,
how-ever it is not yet applied to its full potential in the framework of this
paper
2 C H A R A C T E R O F H / V - R AT I O
O F R AY L E I G H WAV E S
Malischewsky & Scherbaum (2004) presented an analytically
ex-act formula of H/V for a 2-layer model of compressible media.
Later, Malischewsky et al (2008) used this formula together with
the secular equation to investigate the region of prograde Rayleigh
particle motion depending on material parameters These studies
form the basis of the present investigation of two special features
of the H/V -ratio: the singularity (or maximum) and the zero (or
minimum) point An older paper by Suzuki (1933) analyses the
sur-face amplitudes of Rayleigh waves in a stratified medium as well
However, the formulas are much more difficult than the ones from
the papers cited above and they are valid for Poissonian media only
Earlier studies of the singularity and the zero point [see e.g
SESAME H/V User Guidelines ] concluded that the singularity
occurs at a frequency which is close (i.e less than 5 per cent
dif-ference) to the fundamental resonance frequency for S-waves only
if the S-wave impedance contrast exceeds a value of 4
(Bonnefoy-Claudet et al 2008) For low contrast, the ellipticity ratio only
exhibits maxima and minima at certain frequencies and no zeros or
singularities In this case, the maxima occur at frequencies that may
range between 0.5 to 1.5 times the S-wave fundamental resonance
frequency It is also possible that H/V-curve of the fundamental
mode may exhibit a peak at the frequency f p and a trough at a
higher frequency f z Konno & Ohmachi (1998) reported a value of
f z /f pequalling two for a limited set of velocity profiles Stephenson
(2003) concludes that peak/trough structures with a frequency ratio
around two witness both a high Poisson’s ratio in the surface soil
and a high impedance contrast to the substrate
One question addressed here is under which conditions the H/V
-ratio derived from the Rayleigh-wave ellipticity exhibits a
singular-ity and a zero point, respectively, or if it only exhibits a maximum
and a minimum point Let us denote the shear-wave velocities of
the layer and the half-space by β1 and β2, respectively and the
corresponding densities of mass byρ1andρ2 The shear-wave
ra-tioβ1/β2is denoted by r s and the density ratioρ1/ρ2by r d One
can show numerically that the character of H/V is relatively stable
concerning changes of Poisson’s ratioν2of the half-space and of
the densities of massρ1andρ2 However, it changes dramatically
with Poisson’s ratio ν1 of the layer and the impedance contrast
Malischewsky et al (2008) prove for the simple model ‘layer with
fixed bottom’, which is a special case of the model ‘layer over
half-space’ when the impedance contrast is infinitive large or r s = 0,
thatν1= 0.2026 is the lower limit for the existence of a singularity
in H/V and ν1= 0.25 is the lower limit for existence of a zero point.
The value 0.2026 is the solution of equation
1− 2√γ sin√γ π
2
with γ = 1− 2ν1
For model ‘layer over half-space’, the H/V -ratio has a singularity if
and only if
r s < F(ν1, ν2, r d) (3) and it has a zero-point if and only if
r s < G(ν1, ν2, r d). (4)
The function F is given by
F( ν1, ν2, r d)= A(ν2, r d ) arctan [B( ν2, r d)(ν1− 0.2026)] (5)
and the auxiliary functions A and B are defined by
A(ν2, r d)= 0.297 + 0.061r d − 0.058r2+ 0.170ν2− 0.589r d ν2
+ 0.373r2ν2− 0.284ν2+ 0.817r d ν2− 0.551r2ν2 , B( ν2, r d)= 29.708 − 42.447r d + 23.852r2− 14.309ν2
+ 75.204r d ν2− 59.881r2ν2+ 121.370ν2
− 246.328r d ν2+ 170.027r2ν2 . (6) The formulas (5)–(6) are the result of numerical calculations and can be applied with good accuracy (the error is often less than 1–2 per cent) for the intervals 0< ν2 < 0.5 and 0.3 < r d < 0.9,
which cover the practically important cases For each pair of values (ν2, r d ), the equation r s = F(ν1, ν2, r d) represents a curve in the domain (ν1, r s ) on which the H/V -ratio curve changes its property
from having maximum points only to having singularities We are
not able to present a similar formula for the function G It has to be
determined numerically for each pair of values (ν2, r d) separately
by determining the critical value of r s, for each value ofν1, at which
the H/V -ratio curve changes its property from having a zero point
to having only a minimum point
A careful numerical analysis of function F shows that the leading
parameter isν1while there is almost no dependence onν2and only
a weak dependence on r d : the maximum difference of F( ν1) on
ν2 is only about 0.55 per cent and on r d about 3.3 per cent in the
whole range of r d from 0.3 to 0.9 and ofν2 from 0 to 0.5 By fixingν2 and r d (ν2 = 0.3449, r d = 0.7391) we can use F(ν1)
and G( ν1) to divide the region (ν1, r s ) into four parts R1, R2, R3, R4
with different character of H/V (see Fig 1) The blue curve AOB
is the graph of the function r s = F(ν1) and the green curve COD, which is plotted numerically in this case, is the graph of the function
r s = G(ν1)
To the right of the segments COB (R1) we observe one peak and
one zero point, which is the most interesting case The region R3
left from the segments AOD belongs to the case of one maximum
and one minimum The other regions R2and R4are smaller and less important: two peaks occur within the segments AOC and two zeros within BOD The special point O is common to all four regions, but the investigation of its meaning is beyond the scope of this paper
Because the relationships between the H/V peak and the trough
(or the maximum and the minimum, respectively) and the peak frequency of the transmission response of the medium are essential
in applying the H/V -method, we map them as contour lines in the
key parameters (ν1, r s)-plane in Figs 2(a) and (b), respectively We
have constructed contour lines for f p /f r es in Fig 2(a), where f pis
the position of the first peak (or maximum) and f r es is the S-wave resonance frequency of the medium and contour lines for f z /f p
in Fig 2(b), where f z is the position of the zero (or minimum) These maps show all possible values of these two ratios in the
key parameters domain for the fundamental mode H/V -curve of
Trang 3Figure 1 Illustration of the four regions R1, R2, R3, R4of H/V with different character in dependence on ν1 and r s The four figures in the lower panel
correspond to the four types of H/V -ratio curves as marked by the four stars in the upper panel The x-axis ¯f is non-dimensional frequency and is the ratio of thickness of the layer to wavelength of the S-wave in the layer.
Rayleigh surface waves In the region R4, in which two zero points
exist, the second one is represented The values forν2 and r s are
the same as in Fig 1 We have already proven (Malischewsky et al.
2008) that f p /f r es becomes 1 for r s= 0 (‘layer with fixed bottom’),
that is, when the shear-wave contrast is very high, the frequency of
the first peak is the S-wave resonance frequency of the layer.
The blue line in Fig 2(a) is the graph of the function r s = F(ν1)
which becomes with ourν2and r d
r s = 0.291 arctan [18.147 (ν1− 0.2026)] (7)
It separates regions of H/V with at least one singularity (on the
right—regions R1 and R2) from regions with a maximum (on the
left—regions R3and R4) The continuous red region is for the
do-main with at least one singularity of H/V while the dotted brown
region corresponds to a maximum For the region R2, with two
sin-gularities, the first one is plotted The value f p /f r es is close to 1
with 5 per cent deviation for high values ofν1and of the shear-wave
contrast This agrees well with other observations In the left (dotted
brown) region, where maxima and minima of H/V occur, this value
of f p /f r es is also observed We also note that the frequency of
the peak adopts its maximum on the blue curve r s = F(ν1)
This is in close connection with the remarkable curve in fig 6 of
Malischewsky & Scherbaum (2004), which presents the frequency
of the peak value of H/V in dependence on 1 /r s By using (7) with the valueν1 = 0.4375 from this paper, we obtain for the maximum
of the frequency of the peak valueβ2/β1 = 2.5637, which is very
close to 2.6 presented by Malischewsky & Scherbaum (2004) So this strange behaviour in Fig 6 finds a simple explanation The ratio of the position of the zero or trough to the position of
the peak or maximum (contour lines of f z /f p) in dependence onν1
and r s is presented in Fig 2(b) The green line r s = G(ν1)
sepa-rates the region of zero(s) right (solid red contour lines-regions R1
and R4) from the region of trough(s) (minima) left (dotted brown
contour lines-regions R2 and R3) For region R4 with two zero
points, the second one is plotted The function r s = G(ν1) starts
atν1 = 0.25 for r s = 0 with f z /f p = √3≈ 1.73 which is in conformity with the considerations of Malischewsky et al (2008).
The value 2 of this ratio is observed in the down-left corner of the figure with high value of Poisson’s ratio and impedance con-trast This is consistent with conclusion of Stephenson (2003) However, this value of ratio is also observed in the region
with low Poisson’s ratio value where the H/V -ratio curve has
only maximum and minimum points and it is almost unchanged with the impedance contrast Since the minimum points are generally not well distinct, they have not a great practical significance
Trang 4796 T T Tuan, F Scherbaum and P G Malischewsky
Figure 2 (a) Contours of f p /f r esas a function ofν1and r s (b) Contours of f z /f pas a function ofν1and r s In regions with red continuous lines, the H/V -ratio
curve has at least one singularity (Fig.2a) or at least one zero point (Fig 2b) In regions with brown dotted lines, it has only a maximum point (Fig 2a) and a minimum point (Fig 2b) In this figure,ν2= 0.3449, r d = 0.7391.
3 P O T E N T I A L P R A C T I C A L
A P P L I C AT I O N S
H/V -measurements yield positions of peaks and troughs, i.e f p
and f z Let us assume that for a certain region the less important
parameters ν2 and r d are known (ν2 = 0.3449, r d = 0.7391 in
our case) Since we get only two data points from the H/V -ratio
curve, we are able to derive two unknown parameters maximum
The first example of the potential application is to get the resonance
frequency of the medium from the map in Fig 2(a) if we know the
two key parameters (Poisson’s ratio of the layerν1and the impedance
contrast r s) In this case, the thickness of the layer does not have an effect We can get the resonance frequency without knowing it Let us demonstrate this application by using a synthetic data set for a model with two layers over half-space of Lieg´e, Belgium (Wathelet 2005; see Table 1) and taking into account that this more complex model can be replaced with reasonable accuracy by the simple model one layer over half-space Note thatν2and r d used for constructing Fig 2 are different from the values of Table 1 But usually these values are not known in advance and we have avoided
Trang 5Table 1 Parameters for the model two layers over half-space used for the
synthetic data set.
Thickness (m) P-wave (m/s) S-wave (m/s) Density (g cm–3)
a recalculation of Fig 2 The error in doing this is very small (see
Section 2)
The use of synthetic H/V -ratio data is quite common F¨ah et al.
(2003) used them to constrain the velocity solutions from the H/V
-ratio inversion and they found a good agreement with the observed
data especially in the frequency range between the peak and the
first trough in the H/V -ratio curve Fig 3 shows how H/V -ratios’
dependence on frequency extracted from our synthetic data set The
computation has been performed with GEOPSY (www.geopsy.org)
and follows the SESAME H/V User Guidelines (2005) It is based
on modal summation with all modes of Rayleigh and Love waves
(i.e in practice 20 modes) by using Bob Hermann’s code The
source distribution is spatially homogeneous and located on the
surface (i.e no deep sources) The source orientations are randomly
distributed The window length is 60 s with 50 per cent overlap, i.e
20 windows are used for the whole time interval of about 10 min
The frequencies of the peak and trough of H/V -curve observed in
Fig 3 are f p ≈ 5.1 Hz and f z ≈ 9.8 Hz To apply the method,
we make the approximation that these peak and trough frequencies
are those of Rayleigh waves
This model is ‘two layers over half-space’ but we will treat it as a
simple model ‘layer over half-space’ whose the new parameters of
the layer are the average of those from the original model, as it is
necessary in applying our methods As a starting point, we assume
that we do not know anything about this model except the synthetic
H/V -ratio curve We now calculate the average parameters to be used
in the method First, the new thickness is d = d1 + d2 = 27.8 m
and the new shear velocity is
¯
β = β1d1+ β2d2
Here we have denoted the thicknesses of the two layers by d1and
d2 and the shear-wave velocities byβ1 andβ2, respectively Note
that the average shear-wave velocity can be alternatively taken by
conserving the traveltime in two layers as
¯
β = d
d1/β1+ d2/β2
but our investigation for approximating the model ‘two layers over half-space’ by the simple model ‘layer over half-space’ shows that the first averaging method gives more accurate results in peak and trough frequencies The new Poisson’s ratio is calculated to be 0.4579 in the same manner
We now assume that we know nothing about this model ex-cept the Poisson’s ratio ¯ν1 = 0.4579 and the impedance ratio
¯r s[= 553.43/2086] = 0.2653 From the map in Fig 2(a), by
pro-jecting, we can derive the ratio f p /f r esto be about 1.035 And from
the H/V -ratio curve in Fig 3, we observe a clear peak at a fre-quency about f p ≈ 5.1 Hz in the H/V-ratio curve This implies that f r es = 4.9275 Hz.
Since the resonance frequency is calculated when the ratio of the thickness of the layer to the wavelength of the shear wave in the layer is one fourth (i.e d
λ¯ β1 = d f r es
¯
4), we can infer either the thickness or the shear wave velocity if the other is known already
For instance, if we assume d = 27.8 m is also known in advance,
we can calculate the shear wave velocity as
¯
β1= 4d f r es = 547.94 m/s which is very close to the calculated
average value of the model
It should be mentioned that the theoretically obtained possible
deviation of the H/V -peak from the S-wave resonance (up to 40 per cent, see Section 2) still has to be demonstrated with simulated H/V
-curves for simple models similar to the ones in Fig 3 However, this
is beyond the scope of the present paper
A second potential application is to get the key parameters (Pois-son’s ratio of the layer and the impedance contrast) from the other
parameters and the peak and trough frequency in the H/V -ratio
curve We assume that somehow we have the resonance frequency
of the layer (in this example, it is calculated from the total thickness
and the average shear velocity) as f r es[= β1¯
4d] = 4.9769 Hz The
other parameters are assumed to be unknown
We can then map the peak frequency ratio f p /f r es as a function
of r s and f z /f p(Fig 4a) and ofν1and f z /f p(Fig 4b), respectively From these maps one can infer the average Poisson’s ratio of the layersν1and the average shear-wave contrast r sof any several-layer
model from the H/V -measurements at one single station when it is
treated as a simple model ‘layer over half-space’ These two maps
Figure 3 Synthetic H/V-curves versus frequency Each coloured curve corresponds to the result from one individual time window The black solid line is the
average H/V (geometric mean of individual H/V curves) Dashed lines correspond to confidence intervals (±1 sigma for log normally distributed amplitude ratios; courtesy of M Ohrnberger).
Trang 6798 T T Tuan, F Scherbaum and P G Malischewsky
Figure 4 (a) Contours of f p /f r es as a function of r s and f z /f p in region R1(b) Contours of f p /f r esas a function ofν1and f z /f p in region R1
Trang 7refer to the region R1of Fig 1(a) We omit here maps for the
re-gions R2, R3and R4as they are less important For example, if we
know from the data that the peak occurs at the S-wave resonance
fre-quency and the ratio between the trough and peak frefre-quency is about
2, we obtain from Figs 4(a) and (b) that r s = 0.25 and ν1 = 0.46,
respectively
In our example, the ratio of trough to peak frequency and the
ratio of the peak frequency to the resonance frequency of the layer
can be obtained from the H/V -ratio curve as
f z
f p
= 9.8 Hz
5.1 Hz = 1.9216 and
f p
f r es
= 5.1 Hz
4.9769 Hz = 1.0247 (10)
By using the contour line of f p /f r es = 1.0247 in Fig 4(a)
to-gether with the value of f z /f p = 1.9216 we obtain the shear-wave
contrast of our simple model as 0.269; similarly, from Fig 4(b) we
obtain Poisson’s ratio in the layer as 0.485 From the parameters of
the actual model we have the average of the shear-wave contrast and
of Poisson’s ratio as 0.2653 and 0.4579, respectively These values
are close to the results obtained from the maps Figs 4(a) and (b)
with a relative error of 6 per cent for the shear-wave contrast and of
1.3 per cent for Poisson’s ratio
One thing that should be noted here is the relative error of the
results with the uncertainty of the data (frequencies of peak and
trough) While the real data provides the reliable indication of
the fundamental resonance frequency of the soils (see
Bonnefoy-Claudet et al (2008)), the trough frequency is not reliable because
the experimental H/V -ratio is related not only to the ellipticity
of the fundamental mode of Rayleigh waves, but also to higher
modes of Rayleigh waves, Love waves and body waves From the
maps in Fig 4, we observe that the contour lines can be
consid-ered parallel in most cases with the inclined angle of α1 in the
first map and ofα2 in the second map where tanα1 ≈ 1.37 and
tanα2≈ 1
The relative errors of the results can be calculated as
δr s= f z
f p r stanα1
δ f z and δν1= f z
f p ν1tanα2
δ f z (11)
whereδr s = s
f z In this example, the relative errors of results caused by the uncertainty of data are
δr s = 5.21 δ f zandδν1= 3.96 δ f z, which are relatively big The third potential application of the method is for the case when only the shear wave velocity of the basin and the thickness of the layer are already known We will determine the shear wave and the Poisson’s ratio of the layer Since the resonance frequency is not known in this case, we have to construct another map to apply the method Fig 5 is a modified map of the first map in Fig 4 This map consists of contour lines of the peak frequency in the domain
of the ratio f z /f pandβ1 The procedure of the method in this case
is as follows: first, from the new map we can infer the value ofβ1
to be 554 m/s by projecting the first peak as 5.1 Hz and the ratio
of trough to peak frequency as 1.9216 This value ofβ1is almost consistent with the average value of the model To get the value ofν1
we calculate the resonance frequency from the derivedβ1and apply the second maps in Fig 4 The relative error of the resultingβ1in this case due to the uncertainty of the trough frequency is calculated
in the same manner as in example two and isδβ1= 0.49 δ f z Finally, it should be noted that the method applied above is sim-ple and easy to use but it is based on the relationship between peak/trough frequencies of the fundamental mode of the ellipticity
of Rayleigh surface waves and the parameters for the simple model
‘layer over half-space’ For models of several layers over half-space, the average values of parameters in layers have to be used Hence, the method achieves accurate results only for models without big
jumps in parameters, especially in S-wave velocity The method also requires at least the thickness or the S-wave velocity of the layer to be known a priori For some sites where both of them are
not available, for example, when the sedimentary thickness cover is large, the method cannot be applied
4 C O N C L U S I O N S
The H/V -method has become increasingly popular over the last
few decades as a convenient, practical and low cost tool used to
Figure 5 Contours of f pas a function ofβ1and f z /f p in region R1 withβ2 = 2086 m/s.
Trang 8800 T T Tuan, F Scherbaum and P G Malischewsky
determine subsurface site characteristics in urbanized areas.The
physics behind the H/V -peak are not yet completely understood.
However, Bonnefoy-Claudet et al (2008) have shown on simulated
microtremors that the H/V peak frequency astonishingly provides
the resonance frequency of the site regardless of the underlying
physics of the H/V peak is (Rayleigh waves for high impedance
contrast, S-wave resonance and/or Love waves for moderate to low
contrast) We studied Rayleigh waves and have presented a map,
which shows the relationship between the H/V -peak frequency of
the Rayleigh fundamental mode and the S-wave resonant frequency
of the layer with dependence on other model parameters In addition,
a general formula (function F) is presented showing under which
conditions the H/V -ratio curve has a sharp peak or only a broad
maximum It turns out that when Poisson’s ratio of the layer and the
impedance contrast, which are the key parameters of the site, are
located on or in the vicinity of the graph of F, the difference between
the H/V -peak frequency and the S-wave resonance frequency of the
layer is often very high (up to 40 per cent) Finally, we have presented
some maps indicating values of the model parameters (e.g Poisson’s
ratio of the layer and the impedance contrast) by using information
about the H/V -peak and trough frequency These maps are meant
as tools to aid in the interpretation of site characteristics from H/V
measurements
A C K N O W L E D G M E N T S
We kindly acknowledge the support of Matthias Ohrnberger from
the University of Potsdam in providing us with the synthetic data
set and its processing This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) under Grant No MA 1520/6-2
Further, P.G.M gratefully acknowledges the support of
Bundesmin-isterium f¨ur Bildung und Forschung (BMBF) in the framework
of the joint project ‘WTZ Germany-Israel: System Earth’ under
Grant No 03F0448A This work was also supported by Vietnam’s
National Foundation for Science and Technology Development
(NAFOSTED) in a project under Grant No 107.02-2010.07
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