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Author's personal copy
Journal of Volcanology and Geothermal Research 178 (2008) 10–18
Contents lists available at ScienceDirect
Journal of Volcanology and Geothermal Research
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j vo l g e o r e s
Volcanic tremor during eruptions: Temporal characteristics, scaling and constraints
on conduit size and processes
Stephen R. McNutt a,⁎, Takeshi Nishimura b
a
b
Alaska Volcano Observatory, Geophysical Institute, University of Alaska, Fairbanks, Alaska, USA, 99775-7320
Department of Geophysics, Graduate School of Science, Tohoku University, Sendai 980-77, Japan
A R T I C L E
I N F O
Article history:
Received 1 June 2007
Accepted 7 March 2008
Available online 26 March 2008
Keywords:
volcanic tremor
eruptions
conduit radius
explosions
scaling
A B S T R A C T
We investigated characteristics of eruption tremor observed for 24 eruptions at 18 volcanoes based on
published reports. In particular, we computed reduced displacements (DR) to normalize the data and
examined tremor time histories. We observed: (a) maximum DR is approximately proportional to the square
root of the cross sectional area of the vent, however, with lower than expected slope; (b) about one half of the
cases show approximately exponential increases in DR at the beginnings of eruptions, on a scale of minutes to
hours; (c) one half of the cases show a sustained maximum level of tremor; (d) more than 90% of the cases
show approximately exponential decay at the ends of eruptions, also on a scale of minutes to hours; and (e)
exponential increases, if they occur, are commonly associated with the first large stage of eruptions. We
estimate the radii of the vents using several methods and reconcile the topographic estimates, which are
systematically too large, with those obtained from DR itself and theoretical considerations. We compare
scaling of tremor DR with that for explosions and find that explosions have large absolute pressures and scale
with vent radius squared, whereas tremor consists of pressure fluctuations that have lower amplitudes than
the absolute pressure of explosions, and the scaling is different. We explore several methods to determine
the appropriate scaling. This characteristic helps us to distinguish the type of eruptions: explosive (Vulcanian
or Strombolian) eruptions versus sustained or continuous ash (e.g. Plinian) eruptions. Average eruption
discharge, estimated from the total volume of tephra and the total duration of eruption tremor, is well
correlated with peak discharge calculated from cross sectional area of the vent and velocity of volcanic ejecta.
These results suggest similar scaling between different eruption types and the overall usefulness of
monitoring tremor for evaluating volcanic activity.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Volcanic tremor that occurs during eruptions (hereinafter termed
eruption tremor) is associated with the upward migration of magma
and gases through the vent, and includes much information on
eruption dynamics and kinematics of magma movement underground. Hence, source processes of eruption tremor are important to
understand eruption mechanisms, and monitoring tremor is useful to
determine eruption parameters quantitatively. Tremor in general,
including eruption tremor, has been documented at more than 160
volcanoes worldwide (McNutt, 1994a,b). In this paper we investigate
systematic relations between tremor reduced displacement, a normalized amplitude metric, and factors such as vent radius, erupted
volume, and tremor time history with the purpose of deducing general scaling relationships.
The source processes of volcanic tremor, which is not always
“eruption tremor”, have been investigated at many volcanoes around
the world for several decades (e.g., Aki and Koyanagi, 1981; McNutt,
⁎ Corresponding author.
E-mail address: [email protected] (S.R. McNutt).
0377-0273/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jvolgeores.2008.03.010
1986; Mori et al., 1989; Nishimura et al., 1990; Chouet, 1996; Neuberg
et al., 2000). As a result, many characteristics of tremor have been
identified and quantified. Examples include: ambiguous onset and
unclear phases with a predominant frequency of about 1–3 Hz, and an
overall frequency range of 0.5–10 Hz; various duration times from a
few tens of seconds (so called isolated tremor) to 10 days or more; and
hypocenter depths ranging from the surface down to 60 km
(summarized in McNutt, 1994b). To explain these features of tremor,
especially the predominant frequencies, numerous theoretical models
have been proposed. These include resonance of a magma body under
the ground (e.g., Crosson and Bame, 1985; Chouet, 1986, 1996), fluid
movement in volcanic conduits or channels (e.g., Ferrick et al., 1982;
Honda and Yomogida, 1993), bubble growth and collapse in magma or
water (e.g., Leet, 1988), and non-linear excitations caused by magma
flow (Julian, 1994). However, these proposed models do not always
explain well the characteristics of all volcanic tremor, because different types of tremor are observed associated with varying magma
properties, vent geometries, volcano structures, and eruption styles.
In the present study, we focus only on eruption tremor to avoid
difficulties arising from analysis of tremor from unknown sources. The
merits of this approach are as follows: (1) discrimination of eruption
Author's personal copy
S.R. McNutt, T. Nishimura / Journal of Volcanology and Geothermal Research 178 (2008) 10–18
tremor from other types is comparatively simple because eruption
tremor is associated with the surface phenomena of eruptions; (2)
data for many eruptions are available, with comparatively large seismic signal amplitudes, hence good signal-to-noise ratios; (3) the
general physical environments for generating eruption tremor are
similar, since all tremor accompanies the common phenomena of
eruptions; and (4) measured surface phenomena help us to constrain
realistic source processes.
Several detailed studies on eruption tremor have been reported.
For example, Eaton et al. (1987) showed that the amplitude of tremor
increases with heights of lava fountains at Kilauea Volcano. Yamasato
et al. (1988) investigated the temporal characteristics of eruption
tremor associated with the 1986 Oshima eruption. Nishimura et al.
11
(1995) inferred that the source mechanism of eruption tremor at Mt.
Tokachi can be represented as a counter force of the eruption (single
force). Nishimura (1998) extended this work to a suite of explosion
earthquakes at 11 volcanoes and determined scaling relationships.
From a comparison of tremor at different eruptions, McNutt (1994a,
2004) showed a linear relation between log10(reduced displacement)
(DR) of maximum sustained amplitude of eruption tremor and the
Volcanic Explosivity Index (VEI; Newhall and Self, 1982) as:
log10 DR ¼ 0:46 VEI þ 0:08
ð1Þ
In this paper, we examine systematic behavior of temporal variations
and scaling laws for the amplitude of eruption tremor. To describe the
Table 1
Observed parameters of volcanic eruption tremor. Notes: El Chichon total vol = 370; 200 assumed in Apr 4 eruption. Kilauea total dur Jan 83 = 99 h, we used Jan 5 part = 34 h, thus
vol 1/3. Usu total volume = 80, 20 per eruption. Nyiragongo Nov 94 Vt insignificant (lava lake). type: c (central vent); f (fissure);ph (phreatomagmatic explosion); lk (lava lake). Izu
Oshima (event 2) had a central vent at the surface but an inferred fissure at depth. Times for stages I, II, and III: h = hours, d = days, m = minutes. Under Vent, radius is given for circular
vents and length for fissures (see r or l column). ⁎Usu did not decay exp; concave downward
Event no., volcano
Date
Type
Event no., volcano
1 El Chichon
2 Izu Oshima
3 Izu Oshima
4 Kilauea
5 Kilauea
6 Kilauea
7 Miyake
8 Nyamuragira
9 Nyamuragira
10 Nyiragongo
11 Pavlof
12 Pavlof
13 Pinatubo
14 Piton de la Four.
15 Raoul Island
16 Redoubt
17 Shiveluch
18 Spurr
19 Spurr
20 St. Helens
21 Tokachi
22 Unzen
23 Usu
24 Veniaminof
82.Apr.4
86.Nov.15
86.Nov.21
83.Jan.5
83.Oct.2
83.Nov.5
83.Oct.3
81.Dec.25
86.Jul.16
94.Nov
80.Nov.12
83.Nov.14
91.Jun.15
85.Jun.14
64.Nov.20
89.Dec.15
64.Nov.12
92.Jun.27
92.Aug.18
80.May.18
62.Jun.29
90.Nov.17
77.Aug.7
83.Jun.4
Type
c
c
f
f
c
f
f
f
f
lk
c
c
c
f
ph
c
c
c
c
c
c
ph
c
c
Height
6
(start)
1 El Chichon
2 Izu Oshima
3 Izu Oshima
4 Kilauea
5 Kilauea
6 Kilauea
7 Miyake
8 Nyamuragira
9 Nyamuragira
10 Nyiragongo
11 Pavlof
12 Pavlof
13 Pinatubo
14 Piton de la Four.
15 Raoul Island
16 Redoubt
17 Shiveluch
18 Spurr
19 Spurr
20 St. Helens
21 Tokachi
22 Unzen
23 Usu
24 Veniaminof
Vt
c
c (f)
f
f
c
f
f
f
f
lk
c
c
c
f
ph
c
c
c
c
c
c
ph
c
c
× 10 m3
(km)
200
4.4
24
5
14
12
7.4
28
12
0.001
6
12.5
9000
1
–
13
300
44
52
400
71
–
20
9.8
17
5
16
2
2
2
10
6
5
1
11
7.5
30
2
1
12
15
14.5
18
24
12
1
12
8
VEI
5
2
3
2
2
2
3
3
2
1
3
3
6
1
1
3
4
3
3
5
3
1
3
3
Duration
DR
Freq.
2
(h)
(cm )
(Hz)
1.08
72
10
38
64
43
15.25
480
52
120
30
48
16.8
24
1.2
0.67
1
4.05
3.47
5.5
2
18
2.1
432
278
1230
2380
18
16
16
65
120
–
4
11
16
1070
8?
49
39
152
16
30
260
49
7.6
58
17
1
0.8
0.8
1–10
2
2
1.4
2
2
2
1.5
1.5
1
2?
1
1.6
2
2
2
1
3
2.5
2
1
Stage I
Stage II
Stage III
Vent
Area
τI
τII
τIII
(m)
(m2)
5m
1d
b1 h
8h
7h
none
none
1d?
none
none
12 h
14 h
20 m
none
none
20 m
52 m
3.3 h
16 m
3.6 h
47 m
none
63 m
1d
50 m
2d
5h
27 h
55 h
none
none
16d
12 h
none
16 h
20 h
2.7 h
none
none
none
none
none
3h
0.63 h
14 m
none
none
2d
15 m
none
4 h?
3h
2h
43 h
15 h
3d
40 h
5d
2h
14 h
13.8 h
24 h
1.2 h
20 m
8m
43 m
15 m
1.3 h
64 m
18 h
⁎ 62 m
14d
300
150
2000
1000
4
700
4500
1200
20
10
50
50
1000
1000
50
200
875
109
109
600
75
10
50
250
2.80E +05
7.00E +04
2.00E +03
1.00E +03
5.00E +01
7.00E +02
4.50E +03
1.20E +03
4.00E +02
3.10E +02
7.90E +03
7.90E +03
3.10E +06
1.00E +03
7.90E +03
1.30E +05
2.40E +06
3.70E +04
3.70E +04
1.10E + 06
1.77E +04
2.00E +01
7.90E +03
2.00E +05
r or l
r
r
l
l
r
l
l
l
l
r
r
r
r
l
r
r
r
r
r
r
r
l
r
r
References
(Sigurdsson et al. 1984; McClelland et al. 1989; Havskov et al. 1993)
Yamasato et al. (1988); Endo et al. (1988)
Yamasato et al. (1988); Endo et al. (1988)
Koyanagi et al. (1988); McClelland et al. (1989))
Koyanagi et al. (1988); McClelland et al. (1989)
Koyanagi et al. (1988); McClelland et al. (1989)
Uhira et al. (1984); McClelland et al. (1989)
Hamaguchi (1983); Ueki (1983); McClelland et al. (1989)
Kasahara et al. (1988)
Hamaguchi pers comm 1996; GVN 1994
McNutt (1987); McClelland et al. (1989); McNutt (1994a,b)
McNutt (1987); McClelland et al. (1989); McNutt (1994a,b)
Pin. Volc. Obs. Team (1991); White (1992); Wolfe (1992)
McClelland et al. (1989)
Adams and Dibble (1966)
Power et al. (1994); McNutt (1994a,b)
Gorshkov and Dubik (1970); Belousov (1995)
McNutt et al. (1995); Neal et al. (1995)
McNutt et al. (1995); Neal et al. (1995)
Scandone and Malone (1985); McClelland et al. (1989)
Yokoyama (1964); Katsui et al. (1978)
Shimizu et al. (1992)
Suzuki and Kasahara (1979); Niida et al. (1980)
McClelland et al. (1989); McNutt (1994a,b)
Velocity
Density
(m/s)
(g/cm3)
300
100
178
61
77
28
90
100
100
20
77
77
420
45
141
100
300
283
400
380
100
45
105
71
1.4
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
1.4
2.5
2.5
2.5
1.4
2.5
2.5
1.4
2.5
1.0
1.4
2.5
τb
τe
5m
8.1 h
–
4h
8h
–
–
–
–
–
7.5 h
7.1 h
20 m
–
–
10 m
12.5 m
2.2 h
16 m
54 m
16 m
–
19 m
–
15 m
–
4h
1.5 h
1h
7h
3.4 h
1.4d
24 h
4d
1h
4.8 h
3h
6h
16 m
15 m
4m
22 m
15 m
8m
10 m
5.5 h
–
9d
V0
(m3)
5.90E +10
–
4.00E +09
2.57E +08
1.08E +07
3.85E +08
3.87E +09
1.13E +10
2.70E + 09
1.67E +09
1.71E +09
8.20E +09
1.10E +13
7.58E +08
8.34E +08
9.13E + 09
1.35E +11
1.08E +10
1.04E + 10
1.56E +11
8.28E +08
1.39E +07
0.00E +00
8.61E +12
Author's personal copy
12
S.R. McNutt, T. Nishimura / Journal of Volcanology and Geothermal Research 178 (2008) 10–18
average features, we investigate eruption tremor from many eruptions
at different volcanoes around the world on the basis of published studies
and reports. First, we examine amplitude, estimates of vent size and
general characteristics of temporal variations of tremor amplitude.
Second, we compare the eruption tremor with explosion earthquakes
that accompanied Vulcanian or strong Strombolian eruptions. Subsequently, we examine the discharge rate of volcanic materials from the
vent, based on the data of eruption tremor and other geological evidence. Finally, we discuss the source mechanisms of eruption tremor in
light of our new interpretations.
2. Observed characteristics of eruption tremor
We investigate 24 examples of eruption tremor at 18 volcanoes on
the basis of published reports. In Table 1, parameters of eruption tremor
analyzed in the present paper are summarized with information on
eruption type, tephra volume, ash column height, and other parameters.
Height of ash column and volume of tephra are followed by references.
Most of the VEI values are determined by Simkin and Siebert (1994), but
some of the VEIs are estimated by us from heights of ash columns and
volumes of tephra. For descriptive purposes, we classified eruptions into
four types; (1) eruptions from a central circular vent, (2) fissure
eruptions, (3) lava lake activity, and (4) phreatic or phreatomagmatic
eruptions. Vents are characterized by either radius (r) or fissure length
(l), and the area (cross sectional area) of the vent is calculated from the
formula of either πr2 or l × 1 m (fissures are assumed to have a thickness
of 1 m; this may slightly underestimate some values). We use the
maximum fissure length and do not adjust for temporal variations in the
portion of the fissure erupting. Radii of vents and fissure lengths are
estimated from topographic maps, photographs of eruptions and
published
reports. Flow velocity v is estimated by using the relation
pffiffiffiffiffiffiffiffi
v 2gh, where h is the height of fountaining, ballistics, or plume, and g is
the gravitational acceleration. Because this formulation neglects the
momentum of entrained fluid flow, it may slightly underestimate the
velocity. We have used the best and highest resolution data available for
most of these estimates, however, as in any study using published data
(instead of original data), some prudence must be exercised in interpreting the measurements.
We first discuss the amplitude of eruption tremor. Fig. 1 shows a
relation between reduced displacement and cross sectional area of
the vent as measured at the ground surface. Reduced displacement
(DR) (Aki and Koyanagi, 1981; Fehler, 1983) represents a normalized
amplitude (DR is equal to rms amplitude times distance), which is
corrected for geometric spreading and instrument gain. Note that the
reduced displacement in the present study is estimated from the
maximum amplitude of eruption tremor; distances to stations are
given in maps or tables in the various papers cited. Most of the reduced
displacements were determined by us and the others were previously
determined by McNutt (1994a). We find that the reduced displacement
is roughly proportional to the cross sectional area of the vent or fissure,
although some of the tremor from fissures shows very large reduced
displacements (e.g., events 2, 3 and 8). The slope of the best fit
regression line is 0.3, with a regression coefficient of 0.52. This can be
written as log10(DR) = log10(0.29 × cross sectional area) + 0.52. We infer
that the maximum reduced displacement is approximately proportional to the square root of the area of vents, that is, the reduced
displacement linearly increases with crater radius when vents form a
circular crater. This correlation suggests that the area of the vent (crater
or conduit) plays an important role in controlling the amplitude of
eruption tremor, which is similar to the relation for volcanic explosion earthquakes (Nishimura and Hamaguchi, 1993; Nishimura, 1995,
1998).
Next, we examine temporal variations of eruption tremor amplitude.
Fig. 2(a) shows the observed variation of eruption tremor amplitude
during the October 1983 eruption of Mt. Miyake, which produced lava
fountaining (Uhira et al., 1984). We see that the amplitude increased
Fig. 1. Comparison of cross sectional area of the vent with the observed tremor reduced
displacement. Numbers in the figure correspond to each eruption in Table 1. Square
symbols represent fissure eruptions, and circles are eruptions from circular vents.
abruptly when the eruption started. After reaching a maximum level,
the tremor amplitude decreased approximately exponentially, and
eventually returned to the noise level. Fig. 2(b) is an example from the
November 1964 eruption of Raoul Island that produced a phreatomagmatic explosion (Adams and Dibble, 1966). Like Miyake, the tremor
amplitude at Raoul Island increased suddenly as the eruption began,
followed by approximately exponential decay (the amplitude also shows
a small perturbation in the middle of the decay sequence at about 18 h
30 m). Fig. 2(c) shows an example of tremor observed during the June 27,
1992 eruption of Mt. Spurr (McNutt et al., 1995). This case is slightly
different from the former two cases. Tremor amplitude showed an
approximately exponential increase for about 3.5 h after the eruption
started, and reached a maximum. Then, the amplitude decayed
approximately exponentially over about 43 m. Scandone and Malone
(1985) show the temporal variation of tremor accompanying the 1980
eruptions of Mt. St. Helens (Fig. 2(d)). On May 18, continuous tremor
started 3 h after the beginning of the initial gigantic explosion. The
amplitude increased approximately exponentially for about 4 h and
reached a maximum. After the tremor sustained this maximum level for
about 1 h, during which small fluctuations were observed, the amplitude
decayed. For the later eruptions at Mt. St. Helens (May 25, June 12, July
22, 1980 (Fig. 2(d)) and later), we find that for all the cases eruption
tremor increased abruptly, then decayed approximately exponentially.
Note that we have been careful to state the curve shapes as
“approximately exponential”. This reflects the fact that we have not
performed formal curve fitting to all the data. However, several cases for
which curve fitting have been done are indeed exponential (e.g. Pavlof –
1996 eruption; J. Benoit, writt. comm.; Shishaldin – 1999 eruption; G.
Thompson, writt. comm.). Benoit et al. (2003) showed that scaling
relationships between tremor amplitudes and durations for tremor at
nine volcanoes were exponential, further supporting this generalization.
The implications of this are explained below. In the remainder of the
paper, however, we use the terms “gradual increase” and “gradual
decrease” with the implicit understanding that these are approximations. In all cases but one the tremor time histories are concave upwards
for the increasing and decreasing segments.
Author's personal copy
S.R. McNutt, T. Nishimura / Journal of Volcanology and Geothermal Research 178 (2008) 10–18
13
Fig. 2. Examples of temporal variations of eruption tremor: (a) Miyake, October, 1983; (Uhira et al., 1984) (b) Raoul Island, November, 1964; (Adams and Dibble, 1966) (c) Spurr, June,
1992, (McNutt et al., 1995) (d) Mt. St. Helens, May, June, and July, 1980, (Scandone and Malone, 1985).
A detailed analysis of the events in Table 1 showed three main
characteristics in the temporal variation of tremor amplitude: (1) an
exponential increase [which we call stage I], (2) maintenance of a
maximum level [stage II], and (3) an exponential decrease [stage III].
Durations of the exponential increase and decrease usually differ,
with the increase having a longer time constant. We also observe
that small fluctuations and perturbations are common during these
stages and that some of the tremor has slightly more complicated
time histories. However, fluctuations are usually smaller than the
variations of the three main characteristics, and some of the
complicated shapes (time histories) can be explained by a superposition of several exponential increases and decreases (e.g., tremor
on February 4, 1989, at Mt. Tokachi; see Fig. 7 of Nishimura et al.,
1990). Therefore, we conclude that an exponential increase, the
maintenance of a maximum level, and an exponential decrease are
the three most basic stages of eruption tremor. The durations of the
three characteristics are displayed for each event in Table 1: 58% of
the events clearly show an exponential increase, 58% show the
maintenance of a maximum level, and 92% show an exponential
decrease.
The occurrence ratio for each stage in this analysis is approximate.
Because our data were selected from figures in previously published
reports, our classification depends on the resolution of the figures for
each eruption. For example, it is difficult to judge whether or not an
exponential increase occurred in cases of rapid increase as shown in
Fig. 2(a) and (b). The criteria of exponential decrease and maintenance
of a maximum level also have similar problems for other tremor
episodes. Hence, our classification in Table 1 is judged by which processes are dominant in the sequence of tremor. To evaluate each stage,
we measure total duration times for each stage, τI, τII, τIII, which are
the times from the start of eruption to peak amplitude, time of flat or
fluctuating part, and time from peak until end, respectively. If these
three stages are added together, we obtain the total duration of the
eruption. Table 1 also shows τb and τe, for an exponential increase at
the beginning and an exponential decrease at the end of eruption,
respectively, which are measured from the peak to 36% of the peak
(=1/e). These are not durations, but instead are characteristic times for
exponential increases or decreases in amplitude. We did not measure
the time constants τb and τe, for a few examples of tremor that were
not matched with the three stages (e.g., concave downward decay of
the 1977 eruption of Usu).
We find that the occurrence of an exponential increase (58% of
cases) is less likely than that of an exponential decrease (92% of cases),
although exponential increases are clearly observed at the May 18
eruption of Mt. St. Helens, at the first eruption of Mt. Spurr, June 27,
1992, and at the November 1964 event of Mt. Shiveluch. As in the 1980
eruption sequence of Mt. St. Helens (Fig. 2(d)), the exponential
increase often occurs accompanying the first main eruption, but
seems to be less frequently observed before the second and later
eruptions. Hence, we infer that the exponential increases are mainly
associated with the first paroxysmal phase, or in the early stages of an
eruption sequence at a volcano.
Author's personal copy
14
S.R. McNutt, T. Nishimura / Journal of Volcanology and Geothermal Research 178 (2008) 10–18
The main characteristics of eruption tremor are summarized as
follows; (a) the maximum reduced displacement is approximately
proportional to the square root of the cross sectional area of the vent,
(b) the eruption tremor shows three basic stages: a gradual increase,
maintenance of a maximum level, and a gradual decrease, and (c) the
gradual increase observed for the first big eruption of a volcano is
generally very clear.
3. Comparison of the eruption tremor to explosion earthquakes
Volcano seismologists have classified volcanic explosion earthquakes and eruption tremor mainly based on the following points.
Explosion earthquakes produce simple waveforms with short durations of less than a few tens of seconds, often accompanied by an airshock wave that is superimposed on the seismograms or is recorded
by a micro-phone or infrasound meter. On the other hand, eruption
tremor generally has a long duration (e.g. minutes to hours) with
emergent onsets, and is not accompanied by large air-shock waves.
These two types of signals are generally associated with different
types of eruptions, such as brief Strombolian or Vulcanian-type explosions and sustained ash or lava emissions, respectively. Hence
systematic differences between eruption tremor and explosion earthquakes reflect differences in the dynamics of eruption in these eruption styles.
In Fig. 3 we plotted explosions and tremor magnitudes as functions of the radius on the same axes. Here we covert DR to magnitude
(Tuboi, 1954; Watanabe, 1971; see Nishimura, 1998 for details) so we
can use the same base plot as Nishimura (1998). Explosions scale with
r2, so the slope is 2 in Fig. 3. Explosions require a high absolute
pressure; that is, pressure builds up under a sealed cap and rupture
occurs quickly when the pressure exceeds the strength of the cap. For
Vulcanian eruptions the cap is solid rock, whereas for Strombolian
eruptions the upper slug of magma serves the same purpose. This
Fig. 4. Comparison of different kinds of the volume discharge rate. a) discharge
estimated from column height versus average discharge rate. Average discharge is
obtained from dividing the total tephra volume by the total duration of eruption. Note
that units are volume per sec on the vertical axis and mass per sec on the horizontal
axis. b) Peak discharge rate versus average discharge rate. Peak or instantaneous
discharge is obtained from the product of cross sectional area of vent and flow velocity.
Numbers and symbols are the same as Fig. 3.
Fig. 3. Seismic magnitudes for explosions and tremor versus vent radius. Round black
symbols are explosions from Nishimura (1998). Numbers in the figure correspond to
each eruption in Table 1 for tremor and from Table 1 of Nishimura (1998) for explosions.
Square symbols represent fissure eruptions; open circles are eruptions from circular
vents. Fitted lines are for 10 m and larger radius for explosions, and for all data for
tremor.
pressure is estimated to be 1–10 MPa (Nishimura, 1998; Nishimura
and Uchida, 2005). Eruption tremor, by contrast, occurs under open
vent conditions. Tremor is generated by pressure fluctuations from
turbulence within the conduit, and the amplitude of these fluctuations
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must be less than the absolute pressure of explosions. Further, tremor
is a sustained signal, so the fluctuating pressures persist for periods of
minutes to hours or longer, in contrast to explosions, which have a
times scale measured in seconds to minutes. Even though we do not
know the exact source mechanics for eruption tremor, such a
difference as can be recognized in Fig. 3 is quite useful to empirically
distinguish the styles of eruption: explosive (Vulcanian or Strombolian) or continuous ash emissions (e.g. Plinian). One case of the
eruption tremor, for Izu Ohshima 1986 (square symbol labelled “3” in
Fig. 3), plots almost on the scaling relation for explosion earthquakes.
Note that this tremor is twice as strong as that from the 1991 eruption
of Pinatubo (tremor point labelled “13” in Fig. 3). Although the Izu
Oshima eruption was a basaltic type eruption, this tremor behaved
more like an explosion earthquake and not like typical eruption
tremor from the view point of seismic wave generation. This
determination is also supported by phenomenological observations
of a very high eruption column reaching 16 km a.s.l. and by the
generation of visible shock waves from the active vent. Additionally
the eruption may have broken fresh rock to form a new fissure, which
would contribute to stronger tremor by a geometric effect as shown in
Fig. 1.
4. Discharge rate of eruptions
Our systematic measurements in Table 1 permit an additional
comparison to be made. Fig. 4(a) shows a comparison of two kinds of
discharge rates of tephra. The first is the average discharge rate Qt
estimated by dividing the total tephra volume by the total duration
of the eruption estimated from tremor duration. The second is
discharge rate QH obtained from the column height (Morton et al.,
1956; see also below). In Fig. 4(b), the discharge rate in the vertical
axis is calculated as the product of the vent area and the estimated
peak flow velocity QSv; this is the peak or maximum instantaneous
discharge. Note that the units of the horizontal and vertical axes are
not the same, the former is kg/s and the latter is m3/s. We find that
the two peak discharge rates are highly correlated with the average
discharge rate. The flow velocity varies over about one order of
magnitude (20–420 m/s) and the vent area across five orders (5 × 101
to 3.1 × 106 m3), therefore, we conclude that cross sectional area is a
more important parameter in controlling the mass flux. Because
tremor amplitude is proportional to the square root of the cross
sectional area, we can in principle quantitatively evaluate the
discharge rate of eruption by monitoring tremor amplitude. It is
noteworthy to mention that Qt is strongly proportional to QSv, which
enables us to roughly evaluate an average discharge rate by
measuring only the cross sectional area of the vent and assuming a
representative flow velocity.
5. Discussion
We wish to compare quantitatively the differences between
tremor and explosion earthquakes, so that we can infer some of the
physical factors that govern tremor occurrence during eruptions. To
do so we present a straightforward model of tremor generation. First,
we suppose that the eruption tremor is generated by pressure
fluctuations in a cylindrical conduit due to volcanic flows. The conduit
shape is not critical here and a cylindrical conduit is mathematically
convenient. We envision that expanding gases and flow of magma
push against the wall rocks as magma moves towards the surface to
erupt. We then quantitatively represent a source of eruption tremor.
Eruption tremor consists mainly of surface waves (McNutt, 1994b), so
the source is presumed to be located at a shallow portion of the
volcanic conduit. We use a cylindrical conduit with a radius of R and a
length of Lc as a source configuration, and the tremor represents
radial oscillations of the conduit wall in and out from its neutral
position. In this case, the moment tensor of the source, M, is
15
expressed by e.g., Chouet (1996) in the xyz coordinate (z axis is the
vertical direction):
0
1
kþA
0
0
M¼@ 0
k þ A 0 ADV:
0
0
A
ð2Þ
where λ and μ are the Lame constants, ΔV the volume change of the
source. We use the far field expression for Rayleigh waves from this
seismic moment tensor (e.g., Aki and Richards, 1980; Eq. (7.149) on
page 316) as the displacement of eruption tremor:
juj ¼
r2 ð0Þ
8cUI1
rffiffiffiffiffiffiffiffi
2
dr2
jh M0
2kr1 ðhÞ þ i
pkr
dz
ð3Þ
where r2 (z) is the fundamental mode of the eigen function (we
neglect higher modes), c the phase velocity, U the group velocity, I1
the energy integral, k the wavenumber, and r the epicentral distance.
Here we assume λ = μ so that M0 = μΔV. For simplicity, we assume a
semi-infinite medium, so we obtain:
r1 ðhÞ ¼ e0:8475kh 0:5773e0:3933kh
r2 ðzÞ ¼ 0:8475e0:8475kz 1:4679e0:3933kz
I1 ¼ 1:2049q=k
c ¼ U ¼ 0:9194b
ð4Þ
where h is the source depth, ω is the angular frequency of tremor, β
the S-wave velocity, ρ the density of medium, and the units are MKS.
As a result, the reduced displacement is written as:
pffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
juj Lr
2r2 ð0Þ
DR ¼ pffiffiffi ¼
1:23e0:85kh 0:58e0:39kh M0 ;
8cUI1
2
ð5Þ
where L represents the wavelength.
The seismic moment M0 can be expressed by using the strain ε in
the radial direction at the conduit wall:
M0 ¼ 2pALc R2 e:
ð6Þ
Alternatively, we can express the M0 by the pressure disturbance
in the flow, ΔP:
M0 ¼ pLc R2 DP:
ð7Þ
These relations are quite important for estimating the conduit
radius R because the eruption dynamics and the magnitude of
eruptions are closely related to the cross sectional area of the
conduit. However, Eqs. (5) and (7) indicate that we cannot extract
the radius from DR without determining Lc and Δ P (or ε )
independently. We can use a fixed Lc of 500 m, and assume a
maximum value of ΔP at 1 MPa to determine a minimum conduit
radius under the assumption of cylindrical geometry. Assuming
β = 1.5 km/s, ω = 2π × 2 rad/s, h = 250 m, and ρ = 2500 kg/m3, we plot
these values versus the radius determined from surface topography
in Fig. 5 and list the values in Table 2. For comparison we also
determined the radius for spherical geometry (a point source) which
is the minimum possible radius that can be determined using
seismic data. This is of course physically unreasonable because there
is no way for the magma to move, however it provides some insight
into the limiting case. These values are quite small and are also
shown in Table 2.
We now need to link the conduit radii determined from seismic
data to those determined from topography and to use these data to
bring together the constraints from both explosions and eruption
tremor. Three of our cases have data for both explosions and tremor:
Pavlof, St. Helens and Tokachi. These cases are especially useful to
determine how much lower the pressure fluctuations are for tremor
compared to the absolute pressure of the explosions. In each case the
seismic magnitude of the explosions is larger than the tremor by 1–4
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orders of magnitude as seen in Fig. 3. Thus the relative sizes of explosions versus tremor appear to be correct in terms of the pressure
arguments in Section 3 above; high absolute pressure for explosions
and lower pressure for tremor.
We next attempt to reconcile the systematic errors in measuring
the vent/conduit radius. We observe in Fig. 3 that the explosion and
tremor data have different slopes, which imply fundamentally different scaling relations and different underlying processes. But we
must first consider whether the measurements of vent radius (seen
at the surface) and conduit radius (what we infer acts to produce
tremor at depth) can be improved or corrected. We explored three
correction schemes. First, we used Mount St. Helens as a reference
value of 50 m because it has constraints based on several different
methods (Carey and Sigurdsson, 1985; Chadwick et al., 1988), and
shifted tremor data to the left to agree with the same slope as
explosions and the observed offset of Mount St. Helens. This
scheme is the most restrictive and assumes that the scaling of
explosions is correct. Second, we used the conduit radius
determined from DR (cylindrical geometry in Table 2) to rotate
the data, again using Mount St. Helens at 50 m as a reference value.
This gives the same slope as explosions but preserves the scatter of
the data; we presume the scatter has physical meaning. A third
scheme, sliding the data to the left but retaining the slope of the
tremor data, was rejected because it moves the leftmost points
(smallest R) unrealistically too far to the left. Of the three methods
we prefer method 2 as being best grounded in the observations and
also agreeing reasonably well with the theory. However, none of
the various correction methods we considered can satisfactorily
explain the offset and scatter of tremor data with respect to
explosion data without invoking parameters that cannot be directly
measured.
We considered other factors that contribute to our understanding and assessment of appropriate values for the vent/conduit radii.
For example, plume rise theory based on Morton et al. (1956)
suggests that H = 1.67Q0.259 where H is height and Q is discharge.
Fig. 5. Vent size estimated from DR versus vent size estimated from topography.
Numbers and symbols are the same as Fig. 3. Note that radius estimated from DR is
effective radius. See text and Table 2 for details.
Table 2
Estimates of vent radii using several methods. Numbers in brackets are equivalent radii
for fissures
Volcano
Date
1 El Chichon
2 Izu Oshima
3 Izu Oshima
4 Kilauea
5 Kilauea
6 Kilauea
7 Miyake
8 Nyamuragira
9 Nyamuragira
10 Nyiragongo
11 Pavlof
12 Pavlof
13 Pinatubo
14 Piton de la Four.
15 Raoul Is
16 Redoubt
17 Shiveluch
18 Spurr
19 Spurr
20 St. Helens
21 Tokachi
22 Unzen
23 Usu
24 Veniaminof
82.Apr.4
86.Nov.15
86.Nov.21
83.Jan.5
83.Oct.2
83.Nov.5
83.Oct.3
81.Dec.25
86.Jul.16
94.Nov
80.Nov.12
83.Nov.14
91.Jun.15
85.Jun.14
64.Nov.20
89.Dec.15
64.Nov.12
92.Jun.27
92.Aug.18
80.May.18
62.Jun.29
90.Nov.17
77.Aug.7
83.Jun.4
DR
radius
(spherical)
radius
(cylindrical)
radius
(topo)
(cm2)
(m)
(m)
(m)
278
1230
2380
18
16
16
65
120
8.3
17.5
24.4
2.1
2
2
4
5.5
4
11
16
1070
8
49
39
152
16
30
260
49
7.6
58
17
1
1.7
2
16.4
1.4
3.5
3.1
6.2
2
2.7
8.1
3.5
1.4
3.8
2.1
46.9
98.6
137.1
11.9
11.2
11.2
22.7
30.8
0.0
5.6
9.3
11.2
91.9
8.0
19.7
17.6
34.7
11.2
15.4
45.3
19.7
7.7
21.4
11.6
300
150
[25.2]
[17.8]
4
[14.9]
[37.8]
[19.5]
[11.3]
10
50
50
1000
[17.8]
50
200
875
109
109
600
75
[2.5]
50
250
Since Q = velocity × cross sectional area, for circular conduits this
implies H is proportional to R0.5. In Fig. 6 we plot R determined
from H (using data from Table 1) versus R from DR. We assumed
velocity in Table 1 and we used the R determined from DR with
cylindrical geometry (Table 2). We observe that the slope is
approximately 1.0 for values of R determined from H greater than
10 0 m hence we consider this to be the range where observations
are most reliable. Smaller values give extremely low R estimates
(lower part of Fig. 6). We also note that all the values of R based on
H are very small. In fact they agree better with the radii determined
from DR using a point source (Table 2; spherical geometry), which
is the minimum possible size using seismic data. This suggests
again that the main part of the flow during eruptions is concentrated near the center of the conduits so that the effective radius is
indeed quite small. The choice of R for various modeling schemes
depends very strongly on the conditions of the eruptions and the
constraints allowed by the data.
Why would explosions scale differently than volcanic tremor? We
suggest that explosions actually form the craters, so the size
necessarily scales with the strength of the explosion, assuming that
the strength of the rock is constant (Sato and Taniguchi, 1997;
Nishimura, 1998). Tremor, on the other hand, apparently occurs
associated with sustained eruptions that use only a portion of the
available conduit, generally the central part, and does not modify the
conduit significantly during the course of the eruption, except perhaps
at the vent.
We were surprised to see so little obvious difference between
basalt and andesite/dacite composition (different symbols in Fig. 3).
We had anticipated an effect because basalt is less viscous and may
have a lower gas content, whereas andesite/dacite is more viscous and
has higher gas content. The lack of a difference suggests that the
physics of sustained explosive eruptions are not very sensitive to
magma composition, but depend more strongly on parameters such as
conduit size, ascent velocity, etc.
The vent sizes estimated from topographic maps, photographs, etc.
are known to be too large, for several reasons. First, the radius is
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measured at the ground surface, whereas the eruption tremor signal
originates at depths of a few hundred meters or more. Geological
investigations generally show that volcanic vents are flared at the
tops, hence surface values are always maxima. Second, geologic
evidence at Mule Creek (Stasiuk et al., 1996) shows breccia, vitrophyre,
and degassed magma near the wall rocks, suggesting that the part of
the magma that moves, or the effective size, is smaller than the full
size measured to the wall rocks. Third, flow models for viscous fluids
such as magma (Poiseuille type flow or plug flow) show that the flow
velocity is high near the center of a conduit and very slow near the
edges, hence the part of the conduit involved in the main magma flow
is again smaller than the full conduit dimensions. This implies that
only a small part of the flow processes contribute to volcanic tremor
generation. This would be the parts near the walls, suggesting that the
central maximum flow part is decoupled or does not transmit its
energy very efficiently to the adjacent material that is closer to the
wall. All these factors likely contribute to the scatter in plots of
vent size versus seismic amplitude, and also affect the slope of scaling
relations.
Based on the characteristics of eruption tremor, we infer the
behavior of the vent and flow as follows. In stage I, either the effective
size of vent is enlarged by the flow of volcanic ejecta, or the fluctuation
of pressure in the volcanic flow systematically increases, or both. The
distribution of gasses in the magma may contribute to the exponential
increase, with low gas at first and more gas later, because the upper
portion would be partially degassed to the surroundings during slow
ascent (e.g. Jaupart, 1998). New magma may gradually be supplied
from deeper regions. These factors may increase amplitude of tremor
exponentially. In stage II, the cross sectional area of the vent has
reached a maximum size and remains roughly constant. The velocity
and density of flow do not change because magma supplied to the
reservoir and magma withdrawn through the vent are almost equal.
Hence, the amplitude of tremor keeps a maximum level. In stage III,
the cross sectional area of the vent remains almost at the maximum
level, but the velocity of flow decreases as the pressure of the reservoir
becomes lower due to cessation of magma supply from deeper
reservoirs. As a result, tremor eventually ceases at the end of eruption.
17
Table 3
Variations of parameters for each stage of eruption tremor
Stage
Area size
of vent
Velocity and
density of flow
Magma
supply
Tremor
amplitude
I
Increase
Yes or no
II
Constant
III
Constant
Constant with
fluctuation
Constant with
fluctuation
Decrease with
fluctuation
Exponential
Increase
Maintenance of a
Maximum Level
Exponential
Decrease
Yes
No
These changes of the radius, density, flow velocity, etc. for each stage
are summarized in Table 3.
6. Conclusions
We investigated characteristics of 24 cases of eruption tremor at 18
volcanoes based on published reports. Detailed analyses of reduced
displacements and temporal variations of eruption tremor with geological data such as areas of vents, flow velocities of ejecta, and tephra
volumes, reveal the following basic characteristics of the tremor: (1)
reduced displacement is approximately proportional to the square root
of the cross sectional area of the vent; (2) temporal variation of tremor
amplitude shows an exponential increase at the beginning of eruption,
followed by maintenance of a maximum level, and exponential decay
at the end; (3) exponential increase is often observed at the first big
eruption of a sequence.
To explain these characteristics, we investigated scaling relations
between tremor amplitude, cross sectional area of the vent (conduit),
velocity and density of volcanic flow, and other parameters. From the
comparison of these parameters with the characteristics of the eruption tremor, we find that cross sectional area of the conduit or vent is
an important parameter controlling the amplitude of tremor and its
temporal variation, despite measurement problems. Explosions are
stronger than tremor at the same volcano, reflecting the fact that
explosions require a high absolute pressure to break the cap rock,
whereas tremor consists of lower magnitude fluctuations of pressure
in a sustained eruption from an open vent. The observed features of
eruption tremor may help us to better understand temporal changes
in the magnitude of volcanic eruptions.
Acknowledgments
We are grateful to H. Hamaguchi for providing us the unpublished
data of Volcano Nyiragongo and to K. Uhira for giving us information
on tremor data of Mt. Miyake. An earlier draft of the paper was
reviewed by J. Benoit, M. Garces, H. Shimozuru and B. Sturtevant.
Matthias Hort and Silvio de Angelis kindly provided comments on the
current draft. This study was partly supported by the foreign scientist
invitation program of the Japan Society for Promoting Science, by the
U.S. National Science Foundation under grant EAR-9418219, by the
21COE Program of Tohoku University, and by the U.S. Geological
Survey as part of the Volcano Hazards Program, and by additional
funds from the State of Alaska to the Alaska Volcano Observatory.
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