ISSN 1028-334X, Doklady Earth Sciences, 2007, Vol. 417, No. 8, pp. 1247–1251. © Pleiades Publishing, Ltd., 2007.
Original Russian Text © O.V. Pavlenko, 2007, published in Doklady Akademii Nauk, 2007, Vol. 417, No. 1, pp. 94–98.

GEOPHYSICS

Quasi-Stationary Form of the Spectrum of Intense Seismic Waves
at the Surface of the Ground
O. V. Pavlenko
Presented by Academician V.N. Strakhov May 19, 2005
Received November 5, 2006

DOI: 10.1134/S1028334X07080260

Transformations of spectra of intense seismic waves
propagating in ground columns are studied in numerical experiments. It was found that transformation ofspectra to the form E( f ) ~ f –k in the ground occurs due
to nonlinear (hysteresis) energy dissipation and redistribution of the energy of oscillations in the spectrum
by means of interaction between waves. Emphasizing
the low-frequency components agrees with the Manley–Row relations of decay instability, according towhich energy transport of low-frequency oscillations tohigh-frequency oscillations is hampered, but decay ofhigh-frequency oscillations and transport of theirenergy to low-frequency oscillations is facilitated.
The authors of papers [1, 2] devoted to accelogramsof earthquakes in 1995 in Kobe and 2000 in Tottori(Japan), which were recorded by borehole seismic
groups, developed numerical models of the behavior ofground layers in situ; i.e., they estimated the stresses
and strains excited by strong motions in the upper~100 m of the ground. The numerical models describe
the behavior of ground layers in ten 1.5-s time intervals
during 15 strong motions. The authors of [3] used thesemodels in numerical experiments to study the propagation of intense noise (transverse) waves in the groundlayers. The obtained results characterize the statisticalphenomena observed during the propagation of intense
seismic waves in nonlinear media (the near-surfaceground). The interactions between individual spectral
components lead to multiplication of spectral lines,
nonlinear widening of spectra, and the appearance of aconstant component of the seismic field.
Figure 1 illustrates transformation of the spectra ofintense seismic waves propagating in the ground layers:
the results of testing the numerical models are presented for the behavior of the ground in Port Island andthe SJK site (a conventional name of the location ofInstitute of Physics of the Earth, Russian Academy of Sciences,
Bol’shaya Gruzinskaya ul. 10, Moscow, 123995 Russia;
e-mail: olga@ifz.ru

vertical groups) during the Kobe earthquake 1995 by
the Gaussian white noise (the duration of test signals
was chosen to be sufficiently large (~40 min) to obtainsmooth spectra; their intensity was chosen so that thestresses and strains excited in the ground coincided on
average with the stresses and strains generated during
earthquakes to estimate the degree of the response ofthe ground to earthquakes). The results are presented
for five 1.5-s intervals (first, third, …) among ten intervals studied. It is seen in the figure that the spectra ofoscillations at the surface take a form close to E( f ) ~ f –k.
In the layers at intermediate depths (16 and 32 m in Port
Island; 24.9 m in SJK), the forms of spectra demonstrate a resonance intensification of individual components. Interaction between spectral components takes
place along with the resonance intensification. Highand low-frequency combination harmonics are formed.
The energy of the propagating seismic waves is redistributed over the spectral band. The spectral peaks aresmoothed, and the resulting spectrum of the signal at
the surface of the ground tends to obtain the limiting
form E( f ) ~ f –k. Relative deformations in the groundat both points reached ~10–3 in the surface layers and~10–5 at depths.
Table 1 presents the estimates of coefficient k
obtained in the numerical modeling for the spectrum ofoscillation velocity V(t) in five 1.5-s intervals (corresponding to the intervals in Fig. 1), as well as estimatesof k based on velocigrams for Kobe earthquakes in1995. In Port Island at the station closest to the fracture
plane (2 km from the fracture plane), dilution of theground was observed during the earthquake. The nonlinear distortions of the propagating signals are large.
All spectra of signals at the surface obtain a smoothed
form close to the limiting one E( f ) ~ f –k (Fig. 1). As
seen from the table, coefficient k changes only slightly
in time with the development of ground dilution (intervals 7 and 9), but remains within 1.13–1.44. In SJK(6 km from the fracture plane), the nonlinearity ofground behavior is slightly smaller than in Port Islandand variations in k are more notable. If signal spectra at1247

1248

PAVLENKO
V, m/s
(a)
0.1
10–2
0m
10–3
10–4
10–5
0.1
10–2
16 m
10–3
–4
10
10–5
0.1
10–2
32 m
10–3
–4
10
10–5
0.1
10–2
83 m
10–3
–4
10
10–5
0 20 40 0 20 40 0 20 40 0 20 40 0 20 40 f, Hz
5:47:00
5:47:03
5:47:06
5:47:09
5:47:12 t, s
V, m/s
(b)
0.1
10–2
0m
10–3
10–4
10–50.1
10–2
24.9 m
10–3
10–4
10–5
0.1
10–2
97 m
10–3
–4
10
10–5
0 20 40 0 20 40 0 20 40 0 20 40 0 20 40 f, Hz
5:47:00
5:47:03
5:47:06
5:47:09
5:47:12 t, sFig. 1. Spectra of test signals of the Gaussian white noise in ground layers at the depths of the location of recording instruments ofborehole groups in Port Island (a) and SJK (b) at five 1.5-s intervals during strong motions. Component E–W (east–west).

the surface can be approximated sufficiently exactly by
the dependence E( f ) ~ f –k, the values of k are close to
1.3–1.4. In SJK, intervals with a sufficiently high (thirdinterval) level of the oscillations are characterized by
k ~ 1.26–1.51, which is close to the values of k in Port
Island. The estimates of k calculated from thevelocigrams for the Kobe earthquake are presented in
the last column of the table. In general, they are close to
the values obtained by numerical modeling.
The analysis of real accelograms shows an increasein the low-frequency components and smoothing ofspectral peaks in the oscillations at the surface. This isclearly seen in the examples of accelogram spectra during the earthquake in 1995 in Kobe, Port Island, andSJK (Fig. 2). The spectra signals at the surface and at~100 m differ notably, which evidences significant nonlinearity in the response of the ground. Thus, the input

signals of any spectral composition are transformed at
the surface to the signals with spectra E( f ) ~ f –k; i.e.,
the information about the spectral composition ofwaves propagating from the earthquake source is either
completely or partly lost as a result of nonlinear distortions of seismic waves in the near-surface ground.
In acoustics, the description of the transformation ofbroadband spectra is related to the problem of acousticturbulence. For simple acoustic waves, the measured
statistical characteristics of nonlinear perturbations canbe calculated exactly. The general relation for the correlation function of a random process was found for anarbitrary section of a nonlinear medium as a function of
the 2D characteristic function of the input signal. Relations were found for a process with normal distribution
between the spectral density in the medium and the correlation function at the input [4, 5]. If the input of theDOKLADY EARTH SCIENCES Vol. 417 No. 8 2007

QUASI-STATIONARY FORM OF THE SPECTRUM OF INTENSE SEISMIC WAVES

system is broadband noise, the anomalously low attenuation of high-frequency components evidences energy
transport upward in the spectrum to lower frequencies[6–8]. Energy transport also takes place to the low-frequency part of the spectrum. The spectrum of broadband acoustic noise widens to both the high-frequency
and low-frequency parts of the spectrum due to nonlinear interactions [9].
The equilibrium form of the spectral distribution or
the universal law of decay at high frequencies is mostinteresting. According to the data of many researchers,
the spectrum decays as ~ω–2 in the case of strong nonlinearity in the region of the existence of developed
shock fronts, because the amplitudes of the harmonics,
which make up the rupture, decrease as ~ω–1 [4]. Thisdependence was obtained theoretically in the approximation of chaotic phases for the medium without dispersion in the inertial frequency ranges (without
sources and sinks of energy) [10].
In the case of the existence of spatially distributed
sources, which generate waves in a nonlinear medium,the propagation of these waves is described by nonuniform nonlinear equations, the right part of which is arandom function, e.g., Burgers equations or equationsof simple waves [9]. By calculating the correlation
functions for noise waves for approximate solutions ofthese equations and spectra of the wave intensity basedon these correlation functions, we also obtain that lowfrequency components are emphasized in the quasi-stationary spectrum. This is usually explained by magnification of the field scale due to the merging of moving‡, m/s20.50
E–W

Coefficient k estimated for five 1.5-s intervals in numericalexperiments and on the basis of Kobe earthquake (EQ)
records in 1995 for two horizontal components: east–west(E–W) and north–south (N–S)
Location

Direction

Port Island

E–W

1.33 1.35 1.24 1.31 1.28 1.21

N–S

1.31 1.35 1.28 1.25 1.22 1.39

E–W

0.31 1.51 1.43 1.3

N–S

0.25 1.26 1.48 1.45 1.42 1.39

SJK1357

9EQ1.08 1.46

ruptures. High-frequency asymptotics formed by nonlinearity is related to the structure of the front of longlived shock waves, which has the form ω–2. Sometimes,
at small inverse Reynolds numbers Γ (Γ = xp/x‰, i.e., atsmall ratios of the characteristic nonlinear xp and dissipative x‰ wavelengths), the power asymptotics is transferred to the exponential form exp(–βω) [9].
As shown in review [9], emphasizing the low-frequency components is a general feature and is physically related to the fact that high-frequency dissipation
cannot prevent energy accumulation in the low-frequency range. According to the Manley–Row relations,low-frequency modes do not transfer energy to highfrequency modes, whereas high-frequency modes
can decay and transfer the energy to low-frequency

(a)

0.25

N–S0m(b)‡, m/s2

00.50

0.6

0.25

16 m

00.50

E–W

N–S
0m

0.3
00.6

0.25

32 m

00.50

24.9 m

0.3
00.6

0.25
01249

5 10 15 083 m

0.3

5 10 15f, Hz

097 m
1020 01020f, HzFig. 2. Spectra of accelograms of the Kobe earthquake in 1995 recorded by borehole groups in Port Island (a) and SJK (b). Components E–W (east–west) and N–S (north–south).
DOKLADY EARTH SCIENCES Vol. 417 No. 8 2007

1250

PAVLENKO
(a)

τ0.6
0.4
0S(f)
0.1
10–2
10–3
10–4
10–5
00.8 1.6

00.8 1.6

00.8 1.6

00.8 1.6
γ

40

040

04060

040

0.8 1.6

00.8 1.660

60

60
f, Hz(b)

τ0.6
0.4
0S(f)
0.1

0.8 1.6

00.8 1.60γ

10–2
10–3
10–450 750 20 50 75

0 20 50 75

0 20 50 75
f, HzFig. 3. Linear and nonlinear responses of the simple ground section to (a) Gaussian white noise and (b) narrowband noise of variable
intensity. At the top, operation intervals of nonlinear stress–strain dependences are shown (stresses and strains are normalized as
described in [12]). At the base, spectra of oscillations at the surface are shown.

modes [11]. Similar regularities are also found for seismic waves.
The mechanisms of spectra transformation for
intense seismic waves in the near-surface ground areimportant for engineering seismology because they arerelated to the problems of intensification of seismic
waves in the ground columns. Intensification depends
on the level of seismic forcing, which is related, in particular, to the dependence of absorption on the intensityof seismic waves. According to the theory, nonlinear
mechanisms of absorption, which do not depend on thedissipative properties of the medium [9], begin to act
when intense waves propagate in nonlinear ground. In
seismology, increase in the absorption coefficient has
been recorded in sedimentary rocks with increasing
intensity of seismic waves [12].
In the subsequent numerical experiments, we studied the transformation of spectra of seismic waves in
the near-surface ground. We studied the propagation ofbroad- and narrow-band noise signals in grounds of different compositions and water saturation levels.

Figures 3a and 3b demonstrate the spectra of signals at
the surface of linear and nonlinear responses of a simple ground section (200-m of alluvium) with gradualincrease of seismic velocities with depth to the Gaussian white noise (a) and narrow-band noise (b) of different intensities. The figures show that, in the case of nonlinear response of the ground with increasing intensityof the input signal, the spectrum of oscillations at thesurface gradually decays in the high-frequency direction and takes the form E( f ) ~ f –k. If the intensity of theinput (narrowband) noise signal increases, the spectraof oscillations at the surface also tend to transform as
E( f ) ~ f –k. This form of the spectrum is achieved atstrong manifestations of nonlinear response of theground. The energy from the frequency band of theinput signal is transferred to both the high-frequency
and the low-frequency spectral ranges. In the case ofthe linear response, the form of the signal spectrum atthe surface does not change (the spectrum of the linear
response has been smoothed for better visualization;
DOKLADY EARTH SCIENCES Vol. 417 No. 8 2007

QUASI-STATIONARY FORM OF THE SPECTRUM OF INTENSE SEISMIC WAVES

periodicity of the spectra is related to reverberation in
the ground layer).
The results of modeling of the responses of otherground sections to the Gaussian white noise demonstrate that the amplitudes of low-frequency oscillations
during nonlinear response of the ground practically do
not change as compared to the linear case, while theamplitudes of the oscillations in the medium- and highfrequency ranges decrease due to nonlinear decay. Thisagrees with the results in [13], in which the authors
used the methods of numerical modeling of accelograms of strong earthquakes and demonstrated that thenonlinear response of the ground decreases oscillationsin the medium-frequency range. Low-frequency components are not attenuated, while the high-frequency
components slightly increase.
Thus, transformation of spectra to E( f ) ~ f –k in thenear-surface ground occurs as a result of nonlinear(hysteresis) dissipation and redistribution of the energyof oscillations in the spectra due to the interaction
between waves. Emphasizing the low-frequency components agrees with the relations of the Manley–Row
decay instability, according to which energy transportof low-frequency oscillations to high-frequency oscillations is hampered, while the decay of high-frequency
oscillations and transport of their energy to low-frequency oscillations is facilitated [11].

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