GEOLOGICA CARPATHICA, 53, 4, BRATISLAVA, AUGUST 2002
215—221
TOPOGRAPHY EFFECTS ON THE DISPLACEMENTS
AND GRAVITY CHANGES DUE TO MAGMA INTRUSION
MARIA CHARCO
1
, LADISLAV BRIMICH
2*
and JOSÉ FERNÁNDEZ
1
1
Instituto de Astronomía y Geodesia (CSIC-UCM), Facultad de Ciencias Matemáticas, Ciudad Universitaria, 28040 Madrid, Spain;
charco@mat.ucm.es, jft@iagmat1.mat.ucm.es
2
Geophysical Institute, Slovak Academy of Sciences, Dúbravská cesta 9, 842 28 Bratislava, Slovak Republic; geofbrim@savba.sk
*Corresponding author: Tel.: 421-7-59410603; Fax: 421-7-59410626
(Manuscript received June 13, 2001; accepted in revised form December 13, 2001)
Abstract: The paper describes a method for including topographic effects in a thermo-visco-elastic model. An approxi-
mate methodology for the consideration of topography in the computation of thermo-viscoelastic displacement and
gravity changes was used. It gave us a relatively general and simple solution useful for solving the inverse problem. Our
results show that for volcanic areas with an important relief, the perturbation of the thermo-viscoelastic solution (defor-
mation and total gravity anomaly) due to topography can be quite significant. In consequence, neglect of topographic
effects can give a rise to significant errors in the estimation of surface displacements and gravity changes, and therefore
in the estimation of the intrusion characteristics obtained solving the inverse problem.
Key words: ground deformation modelling, topographic effect.
Introduction
A magma intrusion in the Earth’s crust will cause effects (for
example deformation and gravity changes) related to its mass
as well to the pressurization of the chamber due to overfilling
or temperature changes. The deformation due to an expanding
or contracting magma chamber has frequently been modeled
as a dilatation source in an elastic halfspace. The simplest ex-
ample is the Mogi’s model (Mogi 1958). The source of defor-
mation is hypothesized as a hydrostatic pressure source, em-
bedded in a homogeneous elastic half-space. The pressure
source is regarded as a strain nucleus, that is a point like a
source with radial expansion, that is similar to the inflation of
a spherical cavity. A finite source can be satisfactorily approx-
imated by a point source, provided that the source dimensions
are small with respect to source depth. Other models have
been used to examine different types of deformation sources.
For instance, McTigue (1987) has derived an approximate ana-
lytical solution similar to the Mogi model that includes higher-
order terms to represent the finite shape of a spherical body;
Davis (1986) has developed approximate solutions for ellip-
soidal magma chambers.
The theory of the thermoelastic phenomena shows that ther-
moelastic stresses and deformations can arise in an elastic con-
tinuum if an inhomogeneous temperature field exists in the
media (see e.g. Nowacki 1962). Thus we can expect various
thermoelastic phenomena to occur in some regions like volca-
nic areas, with an anomalous behaviour of the heat flow. In
the geodynamic theory, it is well-known that thermo-elastic
strains and stresses play a considerable role in the stress state
of the lithosphere and its dynamics, specially in localities with
pronounced geothermal anomalies (Combs & Hadley 1977;
Teisseyre 1986). For this reason, Hvoždara & Rosa (1979,
1980) carried out a theoretical analysis of thermo-elastic de-
formations of a homogeneous half-space due to a point or lin-
ear source of heat, located at a particular depth in the half-
space. They proved that thermo-elastic stresses are expansive
in type and that they considerably disturb the normal lithostat-
ic stress, specially near the surface of the half-space. Hvoždara
& Brimich (1991) presented basic formulae and the results of
numerical calculations for the simplified mathematical models
of two important effects due to magmatic bodies in the Earth’s
lithosphere: a) static thermoelastic deformations, b) static elas-
tic deformations due to upward pressure. The magmatic body
is approximated by a finite volume source of heat in the first
model and by a concentrated vertical force in the second one.
The formulae for gravity anomaly due to non-uniform exten-
sion connected with thermo-elastic deformations were derived
in Hvoždara & Brimich (1995).
The elastic and thermoelastic models described above have
allowed an explanation of the measured geodetic data in many
volcanic regions, particularly when movements occur on rela-
tively short timescales. However, in many cases the elastic
models seem to be unable to reproduce the observed uplifts
unless unrealistic overpressures are considered and they are
often unable to explain the simultaneously observed displace-
ment and gravity changes (see e.g. Berrino et al. 1984; Jentz-
sch et al. 2001). The presence of incoherent materials and high
temperatures produce a lower effective viscosity of the Earth’s
crust, making it necessary to consider inelastic properties of
the media (Bonafede et al. 1986; Bonafede 1990; De Natale &
Pingue 1993). Bonafede et al. (1986) and Bonafede (1990)
worked out analytical solutions for the displacement and asso-
ciated stress fields induced by a pressure point source in a vis-
coelastic half-space and showed that the viscoelastic response
may reproduce the observed uplifts with plausible overpres-
216 CHARCO, BRIMICH
and FERNÁNDEZ
sure values. Several analytical models with inelastic properties
have been proposed by different authors. Hvoždara (1992)
considered a model of a viscoelastic half-space of the Kelvin
type, with a point source of heat. This paper presents basic for-
mulae for the stress and strain components of the deformation
field. The thermo-viscoelastic deformation field due to a
source of heat of prismatic shape embedded in a viscoelastic
half-space and the formulae for the gravity changes due to the
volume dilatation connected with the deformation field are de-
rived in Brimich (2000). Folch et al. (2000) obtained and com-
pared analytical and numerical solutions for ground displace-
ment caused by an overpressurized magma chamber placed in
a linear viscoelastic media composed by a layer over a half-
space. Different parameters such as the size, depth or shape of
the chamber, crustal rheologies or topography are considered
and discussed. The effect of the topography is also considered.
Fernández et al. (2001) presented a method extension of a de-
formation model previously developed to compute effects due
to volcanic loading in elastic-gravitational layered media
(Rundle 1982, 1983; Fernández & Rundle 1994; Fernández et
al. 1997), for the computation of time-dependent deformation,
potential and gravity changes due to magma intrusions in a
layered viscoelastic medium. They assumed a plane Earth ge-
ometry consisting of welded elastic and viscoelastic layers
overlying a viscoelastic half-space. They found that, in line
with prior results obtained by other authors (see e.g. Bonafede
et al. 1986), introducing viscoelastic properties in all of part of
the medium can extend the displacements and gravity changes
considerably, and therefore lower pressure increases are re-
quired to model given observed effects. In their results, the
viscoelastic effects seem to depend mainly on the rheological
properties of the layer (zone) where the intrusion is located,
rather than on the rheology of the whole medium. They ap-
plied the model to the 1982—1984 uplift episode at Campi
Flegrei modelling simultaneously observed vertical displace-
ment and gravity changes. Their results clearly showed that for
an appropriate interpretation of observed effects, it is neces-
sary to consider the gravitational field in the inelastic theoreti-
cal models. This consideration can change the value and pat-
tern of time-dependent deformation as well as the gravity
changes, allowing us to explain cases of displacement without
noticeable gravity changes or vice versa, cases with uplift and
increment of gravity values, and others.
Many of the models with inelastic properties considered so
far are analytical, assume point source of deformation and a
flat, horizontal free surface. Volcanoes are commonly associat-
ed with significant topographic relief. The approximation of
Earth’s surface as flat and use of half-space solutions can lead
to erroneous interpretation of the deformation data (see e.g.
Cayol & Cornet 1998; Williams & Wadge 1998, 2000; Folch
et al. 2000). Williams & Wadge (1998, 2000) and Cayol &
Cornet (1998) pointed out that topography has a significant ef-
fect on predicted surface deformation by elastic models in re-
gions of significant relief. Those authors pointed out that, in
the elastic case, the interpretation of ground-surface displace-
ments with half-space models can lead to erroneous estima-
tions. Cayol & Cornet (1998) found that the steeper the volca-
no, the flatter the vertical displacement field. Folch et al.
(2000) demonstrated that this result is dramatically empha-
sised in the viscoelastic case, where the topography changes in
a very important way both the magnitude and the pattern of the
displacement field. They also showed that neglect of the topo-
graphic effects may, in some cases, introduce an error greater
than that implicit in the point source hypothesis. These are the
reasons we want to study the effect of topography on the sur-
face displacements and gravity changes obtained by the ther-
mo-viscoelastic model described by Hvoždara (1992, 1998)
and Brimich (2000).
Thus we can quantify the error produced in the thermo-vis-
co-elastic solution. The effect of the topography is represented
allowing point source depth to vary with the relief, thus we re-
laxed the restriction of a flat free surface. If the topographic
perturbation is due primarily to the distance of the free surface
from the magma chamber rather than the local shape of the
free surface this type of solution should give a reasonable ap-
proximation (Williams & Wadge 1998).
If we use this methodology to introduce the topographic ef-
fects in the thermo-elastic and thermo-viscoelastic models we
still get analytical solutions. The advantage of this assumption
is something very clear, it allows us to obtain a relatively gen-
eral and simple solution useful for solving the inverse problem
(see e.g. Michalewicz 1994; Yu 1995; Yu et al. 1998; Tiampo
et al. 2000). Numerical methods, such as the finite element or
boundary element methods may be used to include the topo-
graphic effects when an accurate solution is desired for a par-
ticular deformation model, but such methods can be time-con-
suming in the length of time required to design a mesh and in
the actual computation time.
Thermo-viscoelastic deformation model
Elastic and thermoelastic models have allowed an explana-
tion of the measured geodetic data in many volcanic regions,
particularly when movements occur on relatively short time-
scales. The time evolution of heating of the halfspace (lithos-
phere) and associated deformation with it can be mathe-
matically calculated by means of the theory of thermo-
viscoelastic deformation. We consider a non-steady point
source of heat located at depth x in the viscoelastic halfspace
z > 0. For the uncoupled thermo-viscoelastic problem, the
temperature disturbance field T(x,y,z,t) due to this source
must obey the equation (Nowacki 1962):
where
λ
T
is heat conductivity, c
p
is specific heat under con-
stant pressure,
ρ
is the material density, w is the power of the
heat source,
δ
is the Dirac function, H(t) is Heavside’s unit
step function:
H (t) = 0 for t < 0,
H (t) = 1 for t > 0
.
If the surface of the half-space is kept at a constant tempera-
ture, which can be taken to be zero, then we have the boundary
condition on the surface z = 0:
0
)
(
0
=
=
z
x,y,z,t
T
Considering the initial temperature disturbance in all points of
the half-space as zero, we obtain the initial condition for t = 0:
(1)
(2)
t
T
ρ
c
t
H
ζ
z
δ
y
δ
x
wδ
T
λ
p
T
∂
∂
=
−
+
∇
)
(
)
(
)
(
)
(
2
TOPOGRAPHY EFFECTS ON THERMO-ELASTIC 217
0
)
(
0
=
=
t
x,y,z,t
T
(3)
Then, the solution of equation (1) under the boundary and
initial conditions, is obtained in the form (Carslaw & Jaeger
1959):
−
=
−
−
κt
R
erfc
R
κt
R
erfc
R
πλ
w
r,z,t
T
T
4
4
4
)
(
2
1
2
1
1
1
(4)
where R
1
= [r
2
+ (z —
ξ
)
2
]
½
, R
2
= [r
2
+ (z +
ξ
)
2
]
½
, with r = (x
2
+
+ y
2
)
½
being the horizontal distance from the polar axis z and
κ
=
λ
T
/(c
p
ρ
). The complementary error function erfc(s) is de-
fined by:
∫
−
−
=
s
u
du
e
π
s
erfc
0
2
2
1
)
(
(5)
The time and space variable temperature disturbance causes
variable stresses and displacements. Since the process of tem-
perature change is much slower in comparison with the propa-
gation time of elastic waves, it is sufficient to consider the stat-
ic equilibrium equation for a viscoelastic body:
3
2
1
0
3
1
,
,
i
x
σ
j
j
ij
=
=
∂
∂
∑
=
(6)
where
σ
ij
is the viscoelastic stress tensor. In the purely elastic
case the components
σ
ij
are given by the Duhamel-Neumann
relation, but in the viscoelastic case the stress-strain relations
are given by more complicated formulae (Nowacki 1962).
In order to obtain the actual temporal behaviour of the
displacements and stresses we have to Laplace transform these
quantities. The detailed calculation was performed in
Hvoždara (1992).
The calculation was performed for a Kelvin body, for which
the generalized Duhamel-Neumann relation has the form of:
−
∂
∂
+
−
+
+
∂
∂
+
=
)
(
3
)
Θ(
1
2
3
3
1
)
(
1
2
)
(
,t
x
T
Kα
,t
x
t
t
µ
K
δ
,t
x
e
t
t
µ
,t
x
σ
r
T
r
*
ij
r
ij
*
r
ij
(7)
where e
ij
is the strain tensor,
µ
is the modulus of rigidity (the
Lamé’s constant), K =
λ
+ 2
µ
/3 is the bulk modulus, t* =
η
/
µ
is
decay time,
η
being the viscosity of material,
α
T
is the thermal
coefficient of the linear expansion and
Θ
(x
r
,t) is dilatation.
For the time dependence of displacements u and stresses
σ
on the surface of the viscoelastic half-space, we have the fol-
lowing formulae (Hvoždara 1992):
,
dτ
τ
r
S
τ
t
V
π
Qr
,t
r,
u
t
r
∫
−
=
0
1
)
,
(
)
(
)
0
(
,
)
(
)
(
)
4
(
)
(
2
)
(
2
)
,
0
,
(
0
2
0
4
2
3
1
3
0
2
0
−
−
−
−
+
−
=
∫
∫
−
−
−
−
t
t
κτ
R
z
dτ
r,t
S
τ
t
W
dτ
e
κτ
τ
τ
t
b
π
ζ
t
b
R
ζ
π
Q
t
r
u
,
dτ
τ
r
S
τ
t
U
dτ
τ
r
S
τ
t
B
π
Q
,t
r,
σ
t
t
rr
−
−
−
=
∫
∫
0
1
0
0
)
,
(
)
(
)
,
(
)
(
2
)
0
(
,
)
,
(
)
(
)
,
(
)
(
2
)
0
(
0
1
0
0
−
+
−
=
∫
∫
ϕϕ
t
t
dτ
τ
r
S
τ
t
U
dτ
τ
r
S
τ
t
N
π
Q
,t
r,
σ
where
,
ζ
r
R
,
λ
wκ
Q
/
T
2
1
2
2
0
)
(
+
=
=
,
t
N
t
B
t
U
)
(
)
(
)
(
−
=
,
t
M
t
B
t
V
)
(
)
(
)
(
−
=
,
µα
α
2
1
=
,
µ
K
/
Kα
α
T
)
4
3
(
9
2
+
=
,
µt
K/
α
*
)
2
(
9
3
=
,
µt
/
µ
K
κ
*
)
4
(
)
4
3
(
1
+
=
,
µt
/
µ
K
κ
*
)
(
)
4
3
(
3
+
=
,
t
κ
β
*
1
1
1
−
=
(
)
,
β
r
I
β
r
β
r
I
β
r
e
tβ
ζ
t
δ
R
ζ
R
r,t
S
β
ζ
R
+
−
−
−
=
+
−
−
−
2
2
1
2
2
2
2
0
2
2
2
4
2
0
2
3
0
0
2
2
1
2
)
(
1
3
)
(
2
2
2
0
,
β
r
I
β
r
I
e
tβ
ζ
t
δ
R
r,t
S
β
ζ
R
−
−
=
+
−
−
2
2
1
2
2
0
2
4
3
0
1
2
2
)
(
)
(
2
2
2
0
,
e
π
tβ
ζ
t
δ
ζR
r,t
S
β
R
−
−
−
=
2
2
0
3
3
0
2
2
)
(
)
(
ds
s
dI
I,
κt
β
)
(
4
0
1
=
=
is a modified Bessel function of the first
kind, order 1, I
0
is the same, order 0,
δ
(t) is the Dirac function.
Stresses
σ
zz
and
σ
rz
are zero according to the known bound-
ary condition of the theory of elasticity. The heat flow anoma-
ly due to the temperature field is:
−
+
=
=
∂
∂
=
−
=
κt
R
πκt
R
κt
R
erfc
πR
wζ
z
T
λ
,t
r,
q
z
T
z
4
exp
)
(
4
2
)
0
(
2
0
1
2
0
2
0
3
0
0
In addition we can calculate the perturbation of gravity due
to a point source of heat. There are two principal reasons for
the gravity changes. The first one is the change of density
ρ
0
by the increment of
∆ρ
due to volumetric dilatation. The den-
(9)
(8)
,
t
M
t
B
t
W
)
(
2
)
(
)
(
−
=
(
)
[
]
b t
t
e
t
( )
,
=
−
−
−
−
α
κ
κ
2
1
1
1
1
(
)(
)
(
)
[
]
N t
t
e
e
e
t
t
t
( )
/
(
)
,
=
−
+
−
+
−
−
−
−
−
−
−
−
α α κ
κ
β κ
βκ κ κ κ
κ
κ
κ
1 3 3
1
3
1
1
1
1 3
1 3
1
2
1
1
3
1
3
(
)
[
]
B t
t
e
t
( )
,
=
−
−
−
α
βκ
κ
1
1 1
1
1
1
(
)(
)
(
)
[
]
M t
t
e
e
e
t
t
t
( )
(
)
,
=
−
+
−
+
−
−
−
−
−
−
−
−
−
α α κ
κ
κ
κ κ κ κ
κ
κ
κ
2 3 3
1
1
1
3
1
3
1 3
1
2
1
1
3
1
3
218 CHARCO, BRIMICH
and FERNÁNDEZ
sity change
∆ρ
generates perturbation of the gravity potential,
which for z
≥
0 obeys the Poisson equation. For z < 0 this po-
tential satisfies the Laplace equation, that is, it is a harmonic
function. Brimich (1998) obtained the following formula for
the gravity anomaly:
,
dτ
πκτ
κτ
e
τ
t
W
ζ
R
ζ
t
W
Q
Gρ
,t
r,
∆g
t
κτ
/
R
TVE
−
+
−
=
∫
−
0
3
)
4
(
2
3
0
0
4
)
2
(
)
(
2
)
(
2
1
)
0
(
2
0
where G = 6.67
×
10
11
kg
—1
m
3
s
—2
is the gravity constant and
[
]
.
-
e
-
κ
t
µ
K
Kα
t
W
t
κ
T
)
1
(
2
6
9
)
(
3
1
3
2
−
−
+
=
The formula given by (10) determines the integral effect of
the volumetric dilatation due to thermal expansion in a vis-
coelastic half-space.
The second reason for the gravity changes is the free-air
change and Bouguer correction as an effect of vertical uplift of
the surface above the source of heat. The gravity effect due to
the upward doming of the surface of the Earth, that originally
was the plane z = 0, is given by the sum of the free-air change
of gravity and the Bouguer correction:
[
]
,
r
h
πGρ
γ
∆g
FA
FAB
)
(
2
–
0
+
=
where h(r) = —u
z
(0,r,t) is the doming, that is the vertical uplift,
γ
FA
= 3.086
×
10
—6
ms
—2
/m is the vertical gradient of normal
gravity and 2
π
G
ρ
0
is the Bouguer correction.
The effect of topography
In this chapter the effect of the topography on the surface
displacements and gravity changes obtained by the thermo-
viscoelastic model described above is investigated. We pro-
pose a simple method of evaluating the topographic effects in
a three-dimensional deformation model which consists of as-
suming a different source depth at each point for which a solu-
tion is desired. This methodology was introduced by Will-
iams & Wadge (1998) and permits that we still have analytical
solutions even if we relax the restriction of a free flat surface.
The analytical solutions are useful for solving the inverse
problem and avoiding inclusion of numerical models that can
be time consuming. Therefore, we allow magma chamber
depth to vary with topography, thus in the equations (8), (10)
and (11)
ζ
is replaced by
ζ
’=
ζ
+ H, where H is the point ele-
vation, we want to obtain the viscoelastic deformation and
gravity changes. If the topographic effect is due primarily to
the distance of the free surface from magma chamber rather
than the local shape of the free surface, this methodology
comes near the actual case (Williams & Wadge 1998, 2000).
To study the effect of the topography the relief of an area can
be represented by a volcanic cone with height H and average
slope of the flanks
α
. We consider the surface displacements
and respective gravity changes caused by a point source of
heat located beneath an axis-symmetric volcano with average
slopes of their flanks of 0°, 15°, 20°, and 30°. The volcano
models with slopes of 15° and 20° are representative of basal-
Table 1: Properties of the homogeneous half-space considered in
Figs. 2 to 4. Keys:
λ
and
µ
Lamé parameters, K Bulk modulus,
ν
Poisson ratio,
ρ
density,
λ
T
coefficient of heat conductivity, c
p
spe-
cific heat under constant pressure,
α
T
coefficient of linear thermal
expansion and
κ
coefficient of thermal conductivity.
λ
=
7.05
× 10
10
Pa
µ
=
6.075
× 10
10
Pa
K
=
1.11
× 10
10
Pa
ν
=
0.26857
ρ
=
3 000 kg m
–3
λ
T
=
3 Wm
–1
K
–1
c
p
=
840 Jkg
–1
K
–1
α
T
=
10
–6
K
–1
κ
=
1.1905
× 10
–6
m
2
s
–1
Fig. 1. Characteristics of the model used to determine the influ-
ence of the topography on the surface displacements.
tic shield volcanoes, whereas the volcano models with slopes
of 30° are representative of andesitic volcanoes (Cayol & Cor-
net 1998). Schematic illustration of the problem is given in
Fig. 1. The effect of the topography is neglected when
α
= 0
(H = 0). The rheological behaviour of the crust is represented
by a homogeneous half-space of Kelvin’s type with Lamé pa-
rameters
λ
and
µ
with the topography characterized by the
same parameters. As a reference model we have used a point
source of heat at the depth
ζ
= 2 km, its intensity (power) w =
2.6384
×
10
7
W in order to achieve the epicentral heat flow
anomaly q
z
(0) = 42 mW/m
2
, since q
z
(0) = w(2
π ζ
2
)
—1
. The den-
sity and elastic parameters of the half-space and the thermal
parameters of the medium are shown in Table 1.
We set the decay time for the Kelvin’s type of the viscoelas-
tic body as t* = 3.3
×
10
12
s, where t* =
η/µ
and
η
is the mean
viscosity of the crustal rocks. The time evolution of the hori-
zontal and vertical displacements u
r
, u
v
and terms of total grav-
(11)
(10)
TOPOGRAPHY EFFECTS ON THERMO-ELASTIC 219
Fig. 3. Same as Fig. 2 but for radial displacements.
Fig. 4. Same as Fig. 2 but for total gravity changes. Units are m/s
2
.
Fig. 2. Thermo-viscoelastic vertical displacement in meters com-
puted for different time values, the source described in Table 1
and considering (a) a flat surface, and (b)—(d) axis-symmetric vol-
canic cone with an average slope of the flanks of 15°, 20° and 30°
respectively. t
κ
is the decay time defined in the text.
ity anomaly
∆
g
sum
=
∆
g
TVE
+
∆
g
FAB
was calculated for various
times using multiples of the characteristic heat disturbance
time t
κ
=
ζ
2
(4
κτ
)
—1
, which corresponds to the value e
—1
of the
known heat propagation factor exp(—
ζ
2
/4
κτ
) in the epicentre of
the heat source. The results for the depth
ζ
= 2 km (t
κ
= 8.37
×
10
9
s) are presented in Figs. 2 to 4. The results are compared
with the flat-surface solution given by the analytical method.
The curves for t/t
κ
= 0.5, 1.0, 2.0, 3.0, 5.0, 7.0 gradually ap-
proach the curves that were calculated by means of the formu-
lae for the stationary thermo-elastic problem (Hvoždara &
Brimich 1991). We can see that the displacements and gravita-
tional anomalies approach their static values slowly, because
of the viscoelastic behaviour of the half-space, which is math-
ematically expressed by the convolution integrals in the previ-
ous chapter. As is pointed out by other authors, the principal
effect of topography is a reduction of vertical displacement
and total gravity anomaly magnitudes due to the greater dis-
tance from the source to the free surface (the steeper the volca-
no, the flatter the displacement field and the gravity change).
In Fig. 2 the changes of the vertical displacements caused by
the topography are presented. Fig. 4 shows that the topography
effect changes the pattern of the total gravity anomaly, too.
Vertical displacements and total gravity changes are mostly in-
fluenced by the topographic effect, thus neglecting the topog-
220 CHARCO, BRIMICH
and FERNÁNDEZ
raphy may lead to a miss-interpretation of the volume change
of the source. We observe in our results that, as Folch et al.
(2000), the effects of the topography are dramatically empha-
sized in the viscoelastic case.
It is not the case for radial displacements (Fig. 3), where the
effects of the considered topography obtained with the used
approximate method are not very important. This result should
be tested by comparing with numerical methods as pointed out
by Williams & Wadge (1998) or similar to that used by Cayol
& Cornet (1998) and Folch et al. (2000).
We observe in our results, like Folch et al. (2000) for a pure-
ly viscoelastic medium, that neglecting the topography gener-
ates distortions in a viscoelastic half-space of Kelvin type,
which would lead to inaccuracies in the predicted displace-
ments and gravity changes.
Conclusions
The thermo-elastic models used to interpret the anomalous
behaviour of the heat flow in some volcanic regions, can be
used particularly when movements associated with volcanic
activity occur on relatively short timescales. However, the
presence of incoherent materials and high temperatures pro-
duce a lower effective viscosity of the Earth’s crust, making it
necessary to consider inelastic properties of the media. That is
the reason why Hvoždara (1992, 1998) and Brimich (2000)
considered a thermo-viscoelastic half-space with a point
source of heat to model displacements and gravity changes
caused by a magma intrusion. The results show that the ther-
mo-viscoelastic solution gradually approachs the solution ob-
tained for the stationary problem (thermo-elastic solution).
The models used to interpret the geodetic data measured in
volcanic areas, typically compute the deformation field and
gravity changes at the surface of an elastic half-space due to a
point source at depth and assume that topography does not sig-
nificantly affect the results. Considering previous results ob-
tained by other authors for elastic (Williams & Wadge 1998,
2000; Cayol & Cornet 1998) and viscoelastic media (Folch et
al. 2000), we have included topographic effects in the thermo-
viscoelastic model. We have used an approximate methodolo-
gy. This methodology permits us to have an analytical solution
which allows us to solve the inverse problem. With the meth-
odology described above, we can observe the reduction of ver-
tical displacements in regions with higher topography due to
the greater distance from the source of heat to the free surface.
In volcanic areas of greater relief the perturbation of the ther-
mo-viscoelastic solution (deformation and total gravity anom-
aly) due to topography can be quite significant. Therefore we
have demonstrated that the topography may significantly af-
fect the surface displacements and gravity changes computed
for a magma chamber represented by a heat point source. Thus
we can conclude that any model which neglects the topograph-
ic effect could cause a significant error in the estimation of
surface displacements and gravity changes, or in the determi-
nation of the characteristics of the intrusion if we use the mod-
el to solve the inverse problem.
Acknowledgments: This research was mainly supported by
funds from the Collaboration Project 98SK0003 and Grant No.
2/7059 of VEGA, the Slovak Grant Agency. The research by
María Charco and José Fernández was also supported with
funds from research project AMB99-1015-C02.
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