ISSN 1028-334X, Doklady Earth Sciences, 2007, Vol. 415, No. 5, pp. 727–733. © Pleiades Publishing, Ltd., 2007.
Original Russian Text © O.V. Pavlenko, 2007, published in Doklady Akademii Nauk, 2007, Vol. 414, No. 5, pp. 676–682.

GEOPHYSICS

Self-Modulation of Seismic Waves
in the Subsurface Soil
O. V. Pavlenko
Presented by Academician V.N. Strakhov May 19, 2005
Received August 28, 2006

DOI: 10.1134/S1028334X07050157

Nonlinear effects of modulation instability of seismic waves during their propagation in subsurface soils
are studied in numerical experiments. The results showthat self-modulation and generation of subharmonics
take place at high amplitudes of seismic waves in casesof their resonance intensification in soil layers and precede the chaotic stage in the case of further increase inamplitudes. Generation of these effects is related to dispersion of seismic wave propagation velocities in subsurface soils due to their nonlinear response.
The nonlinearity of soils during strong earthquakes
remains a debatable issue in seismology. Interactions
and self-effects of seismic waves (see, for instance,[1, 2] and others), attenuation of resonance frequency
of soil layer oscillations, an increase in nonlinear
absorption, and a decrease in elastic moduli (see [3, 4]and others) were studied in field, laboratory, andnumerical experiments and based on earthquake
records. This article presents results of the numericalsimulation of modulation instability observed during
intense seismic wave propagation in soil layers. We
used numerical models of soil behavior in fault zones ofstrong earthquakes based on the records of verticalgroups [5].
In particular, models of soil behavior in fault zonesof the 2000 Tottori (Japan) earthquake (MW = 6.7) wereconstructed; i.e., the stress and strain excited by strong
motions in the upper ~100-m soil layer were determined [6]. The obtained vertical distribution of stressand strain in layers at station TTRH02, the nearest to
the fault plane (~2 km), are presented in [6, 7]. The profile at TTRH02 consists of a sandy–clayey sequence(VS ~ 210 m/s) in the upper 9-m layer. The lower layer(down to ~100 m) is composed of denser layers with VS
gradually increasing with depth to ~790 m/s per 100 m.

Joint Institute of Physics of the Earth, Russian Academyof Sciences, Bol’shaya Gruzinskaya ul. 10, Moscow,
123810 Russia; e-mail: olga@ifz.ru

Subsurface soils are partially water-saturated (VP ~860 m/s), resulting in a noticeable intensification ofoscillations on the surface. Stress and strain wereassessed in 10 consecutive 1.5-s intervals corresponding to 15 s of strong motion [6].
To assess the degree of nonlinearity of soil behavior,
numerical models were tested by harmonic signals witha 3-Hz frequency [8]. Stress–strain relationships
obtained for each 1.5-s interval were used for calculating the propagation of testing signals in soil layers. The
results of testing (Fig. 1) demonstrate that high harmonics of basic frequency are generated in the courseof propagation of harmonic signals in soils. As a rule,the harmonics are of an odd order (3rd, 5th, and others).
Harmonics of an even order (2nd, 4th, and others) aregenerated along with harmonics of the odd order only
during the highest intensity of the input signal (intervals 4, 6, 7). Similar results were obtained when testingother soil profiles [9]. However, subharmonics of thebasic frequency (intervals 3, 5) are also generated inintervals with the highest intensity of oscillations when
testing the soil profile at station TTRH02.
To elucidate conditions of subharmonic generation,
the propagation of harmonic signals in the profile at station TTRH02 was studied in greater detail. Figure 2
illustrates the results of testing the TTRH02 profile bysignals with a 3-Hz frequency. The input signal ofamplitude V0 = 0.01 m/s yielded a 3-Hz signal with several even and odd high harmonics. Increase in amplitude
V0 to 0.011 m/s also fostered an increase in the working
interval of the stress–strain relationship and appearanceof an inflection point. The output signal became amplitude-modulated, and its spectrum demonstrated subharnfmonics of type ----- : (1, 5, 7, 11, 13, 17, 19 Hz, and oth3ers). Upon further increasing V0, modulation wasretained initially at V0 = 0.012 m/s. Subsequently, theinput signal resembled noise at V0 = 0.013 m/s (Fig. 2).
The experiment was repeated with a 6-Hz frequency of
the input signal, and a slight modulation effect was727

103f

5f

20 0f

f

103f
5f

20 0

f

f

103f
5f

20 0f

f
2f
4f

103f
5f

20 0

6f

f

f

103f
5f

20 0f

f
2f

20 0

4f 5f 6f

103f

f

f
2f
4f

103f
5f

20 0

6f

f

f

103f
5f

20 0f

f

103f
5f

20 0f

f

5f

20F, Hz104 m

103f0mFig. 1. Results of testing the soil profile at station TTRH02 by harmonic signals (f = 3 Hz): spectra of testing signals (below) and signals on the surface (above) in 10 successive
1.5-s intervals (15 s of strong motions).

V, m/s
0.1 f
10–2
10–3
10–4
10–5
0.1
10–2 f
10–3
10–4
10–5
0728
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010200:00:35 0:00:30

06108200:00:35

10 1200:00:30

–0.2

00.2

6108200:00:35

10 1200:00:30

–0.2

00.2

6108209m0m

95 m
0:00:35
Timef, Hz

10 12Time, sFig. 2. Self-modulation and chaotic character of the harmonic signal with 3-Hz frequency and amplitudes V0 = 0.010, 0.011, 0.012, and 0.013 m/s in the soil profile at station
TTRH02. From top to bottom: calculated oscillations on the surface, working intervals of stress–strain relationships in the upper 9-m interval, spectra of signals on the surface,schematic view of oscillations in soil layers (upper and lower rows are related to surface and 100-m depth, respectively; oscillations in upper layers are presented in more detail;maximal amplitudes of oscillations correspond to the depth of ~9 m).

0:00:3040
0–40

V, m/s
0.1
10–2
10–3
10–4
10–5

–0.2

10 12–0.2
800

60.2

V, m/s
0.2

SELF-MODULATION OF SEISMIC WAVES IN THE SUBSURFACE SOIL729

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PAVLENKO

revealed at the input signal amplitude of ~0.03–0.05 m/s.
Oscillations became chaotic when the amplitude wasincreased to 0.1 m/s.
Figure 3a demonstrates stress and strain in soil layers in the TTRH02 profile at different depths excited bypropagating harmonic signals (V0 = 0.01 m/s, frequency
1–21 Hz). The 3-Hz frequency is resonance frequency
fres of this soil profile, and it shows resonance amplification of oscillations in the upper ~9 m (fres decreases
due to nonlinearity of the soil response, while fres isequal to ~210/36 = 5.8 Hz in the linear case). The nonlinearity of stress–strain relationship leads to noticeable
decrease in the effective propagation velocity of wavesof the 3-Hz resonance frequency compared to waves ofother frequencies, i.e., velocity dispersion. Conditions
for the development of self-modulation appear for harmonics with resonance frequencies.
The phenomenon of harmonic wave self-modulation in nonlinear media with dispersion has been studied sufficiently [10]. Owing to the dispersion, high harmonics generated due to nonlinearity are asynchronouswith the wave of basic frequency and, therefore, do notgrow, whereas oscillations at the output appear to be
amplitude-modulated. Modulation is caused by parametric instability leading to the appearance of ghostreflections with frequencies close to the carrier frequency. Modulation instability is possible only at a certain relation between nonlinearity signs and dispersionof the group velocity (Lighthill condition), owing towhich nonlinear distortion caused by synchronism
compensates the linear dispersion asynchronism [10].
Figure 3b demonstrates stresses and strains excited
in soil layers in profile TTRH02 by testing signals withdifferent V0 values at f = 3 Hz. Hysteresis characteristics in the upper 9 m of soil, where deformations are thegreatest, mainly determine the mode and spectrum composition of oscillations on the surface: at V0 = 0.012 m/s,
their spectrum becomes more complicated. Upon further
increase in the amplitude, the oscillations become chaotic. The chaotic character corresponds to large working intervals of stress–strain relationships and sets ofclosed loops of hysteresis curves at different depths. It
is likely that extension of the working intervals ofstress–strain relationships beyond the inflection point
also contributes to fast complication and chaotizationof oscillations. Changes in the shape of hysteresis
curves correspond to variations in the trajectories ofoscillations of medium particles in the phase space, i.e.,
to variations in phase portraits of the system (bifurcations). Transition to stochasticity via a series of bifurcations of period doubling is typical of dissipative nonlinear systems [10]. The described experiments exhibit
period trebling, especially prominent in the layer withthe maximal oscillation amplitude of ~9 m (Fig. 2),
which is likely to be related to the prevalence of cubic(instead of quadratic) nonlinearity in the soil response.
Transition to stochastic oscillations takes place in thecase of increase in V0, i.e., increase in the number ofnonlinear interactions and formation of mixed harmonics. A sufficiently high intensity of the input signal
leads to increase in the number of harmonics and chaotization in an avalanche manner, and the output signal
spectrum becomes continuous.
Phenomena of self-modulation and recovery of
Fermi–Pasta–Ulam waveforms in seismic fields werestudied by Dimitriu in field experiments [11]. The spatial evolution of harmonic signals emitted by a vibrator
within the frequency band 7–14.65 Hz was investigatedat a distance of 10–70 m from the vibrator. The V0 valueon the surface near the vibrator was 0.00015–0.006 m/s.
The recovery of Fermi–Pasta–Ulam waveforms wasobserved at low V0 values. We noted a complete recovery of initial waveforms (both envelope and carrierwaves) over a short distance, probably due to strong
nonlinearity of subsurface soils. We also recorded a
substantial dependence of observed effects on the signal frequency and a slight dependence on the amplitude. For the highest frequency of 14.65 Hz, only weaksigns of instability were observed and the most remarkable signs were recorded for f = 10 Hz [11].
The observed phenomena can be explained from theviewpoint of the nonlinear theory of wave propagation.
The Shredinger nonlinear equation [12]iAt + bAxx + c|A|2A + dAx = 0
describes three main classes of the phenomena
observed during quasi-harmonic wave propagation innonlinear media with dispersion: modulation instability(self-modulation), stationary waves of envelopes(including solitons), and periodic (in space and time)
recovery of a slightly modulated wave (approaching aperiodic sequence of solitons). In this equation, Ä is thecomplex amplitude of an envelope of propagating wave
2d ωof type exp[i(kx – ωt)]; real constants b = --------2- , c, anddk
d determine the value and sign of dispersion, cubic nonlinearity, and absorption, respectively. The sign of bc
determines the process evolution: if bc < 0, the wave isstable and solitons do not form, and instability develops
if bc > 0.
In experiments carried out by Dimitriu, the minimum of the group velocity of wave propagation corresponds to 10–12 Hz and the explanation given is confirmed by the experimental results. In our numericalexperiments (Figs. 1–3), the appearance of the velocity
dispersion in the medium is related to the nonlinearityof the soil response; in this case, the minimum of wave
propagation velocity also corresponds to the range ofresonance frequencies and the condition of modulation
instability bc > 0 is fulfilled. As in Dimitriu’s experiments, the effects of modulation and subharmonic generation show a direct correlation with the amplitude ofthe testing signal.
The experiments described here were also carriedout with other soil profiles. Self-modulation and generDOKLADY EARTH SCIENCES Vol. 415 No. 5 2007

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100.4

Depth, m

096.6

96.6

15
45.1

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–0.4 0 0.4

86.8

45.1

158

6.8

0.4

135125112.5–0.4–0.8 0 0.8

00.4

V0, mm/s

2.50.4

Depth, m(b)Fig. 3. Stress and strain excited in soil layers: (a) at different frequencies of the input signal, V0 = 0.010 m/s, and (b) at different amplitudes of the input signal, f = 3 Hz.

–0.4–0.4 0 0.4

00.4

–0.4
–0.8 0 0.8

00.4f, Hz(a)SELF-MODULATION OF SEISMIC WAVES IN THE SUBSURFACE SOIL731

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V, m/s0.1

0.0005

0.002

0.004

0.006

0.01

0.014

0.016

0.018

0.02

0.04

0.06

0.008

0.1

0.12

0.14

0.16

0.18

0.2

10–210–3
10–4
0.1
10–2
10–3
10–4
0.1
10–2
10–3
10–4
0

5 10 15 20 0

5 10 15 20 0

5 10 15 20 0

5 10 15 20 0

5 10 15 20 0

5 10 15 20f, HzFig. 4. Spectra of oscillations on the surface of the soil profile in Port Island at different amplitudes of the input harmonic signal(f = 3 Hz). Amplitudes of input signals are shown above (m/s).

ation of harmonics were observed in each case for harmonics characterized by resonance intensification ofoscillations. We investigated profiles without seismic
boundaries with velocities VS gradually increasing withdepth. We also studied profiles composed of the upper
loose layer and the lower (more dense) layer with VSincreasing with depth. The soil profile in Port Island is
an example of a complicated section (systems withmany degrees of freedom). A model of its behavior in
the 1995 in Kobe (Japan) earthquake was reported in[5]. The profile includes five groups of layers with different properties: artificial soil above the groundwaterlevel (0–13 m), the same below the groundwater level(13–17 m), alluvial clay (17–28 m), alluvial sand (28–34 m), and dense layers of gravel and clay (deeper than34 m). Figure 4 illustrates the results of testing this soil
profile by the harmonic signal with a 3-Hz frequency of
the resonance frequency band (1–4 Hz) of the upper liquefied layer. The soil response is essentially nonlinear
even at low V0. Upon increasing V0, its nonlinear distortions are intensified. This is followed by the successive
appearance of high harmonics of odd orders (3rd, 5th, …),
high harmonics of even orders (2nd, 4th, 6th, …), andnfsubharmonics of type ----- in the soil response spectra.
3Finally, the oscillations become chaotic (Fig. 4, upperrow, 0.01 m/s). Upon further increasing V0, we observea phenomenon similar to the recovery state: the system
periodically passes from the state of subharmonic generation with subsequent chaotization to the state of generation of high odd and even harmonics until theachievement of the limiting amplitude of V0 ~ 0.525 m/scorresponding to the ultimate soil rigidity.
The increase in V0 augments stress and strain in theupper layers. Liquefied soils of the upper group aremore susceptible to changes in amplitudes. Growth ofstress and strain in one of the layer groups entails variations in the whole vertical distribution of stress andstrain in layers, and, consequently, variations in oscillation spectra on the surface (Fig. 4). At V0 ~ 0.006 m/s,subharmonics appear for the first time in the signal
spectrum on the surface. At higher V0 values, different
soil layers become resonant and the oscillation spectrum on the surface changes qualitatively. Oscillationsare modulated at resonance frequencies, and signal
spectra on the surface give an idea of the complicatedbehavior of the system with many degrees of freedomnf(Fig. 4). One can see subharmonics of types ----- (V0 ~3
nf0.004, 0.006, …) or ----- (V0 ~ 0.018, 0.06, …) and both2types in some cases. Types of both subharmonics andhigh harmonics are likely to be determined by forms ofstress–strain relationships in the soil layers: liquefied
soils in Port Island exhibit nonlinearity of the evennforder [9]. Subharmonics of type ----- appear in spectra of
2output signals only in this case. Such subharmonics areabsent in spectra of output signals of other studied soil
profiles, which lack nonlinearity of the even order.
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