Pergamon Computers & Geoscirnces Vol. 23. No. 3. pp. 231-249. 1997 0 1997 Elsevier Science Ltd PII: Soo98-3004(%)00066-0 All rights reserved. Printed in Great Britain 009%3004197 $17.00 + 0.00 PROGRAMS TO COMPUTE DEFORMATION DUE TO A MAGMA INTRUSION IN ELASTIC-GRAVITATIONAL LAYERED EARTH MODELS JOSE FERNANDEZ,’ JOHN B. RUNDLE,’ ROBERTO D. R. GRANELL*’ and TING-TO YU’ ‘Institute de Astronomia y Geodesia (CSIC-UCM), Fat. CC. Matematicas, Ciudad Universitaria. 28040, Madrid, Spain; ‘Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309, USA, and “Institute of Earth Science, Academia Sinica, P.O. Box l-55, Nankang, Taipei, Taiwan (e-mail: [email protected]) (Received 16 January 1996; revised 25 Jul_v 1996) Abstract-In previous papers from two of the authors, models, programs and results of programs developed to compute displacements due to volcanic loading with different assumptions have been pub- lished. In this paper, these previous models are extended in a set of programs written in FORTRAN 77. These codes allow us to calculate displacements (horizontal and vertical), tilt, vertical strain and po- tential and gravity changes on the surface of an elastic-gravitational layered Earth model. due to a magmatic intrusion. The intrusion is treated as a point source and the medium can consist of up to four layers overlying a homogeneous half-space. A topmost water layer can be included to calculate sea level changes. We describe the theoretical formulation of the deformation model together with the nu- merical programs developed. Examples of calculation for the different effects are presented. I?:# 1997 Elsevier Science Ltd Key Woralr: Displacement, Tilt, Strain, Sea level, Potential, Gravity, Elastic-gravitational layered Earth INTRODUCTION In recent years, the international scientific commu- nity has recognized the possibility of reducing and mitigating risks related to different natural hazards, using systematic study programs. Among these hazards are volcanic eruptions, earthquakes, floods and landslides (CEES, 1992). According to recent estimates, in the next 50 years more than one third of the world’s population will live in seismic and volcanic areas (Bilham, 1988). The fact that this decade has been declared the International Decade of Natural Disaster Reduction, by the International Council of Scientific Unions, UNESCO and the World Bank, has generated a variety of programs for research into problems related to the prediction and mitigation of these disasters, mainly in Third World countries. It should also be noted that one of the most promising ways of studying volcano- and earth- quake-related phenomena is the use of high-pre- cision crustal deformation data, which are obtained with modern space-borne geodetic methods, in par- ticular GPS. These data need physical deformation models to deduce features of the medium. Such *Also at Dpto. de Geofisica y Meteorologia, Fat. CC. Fisicas, Ciudad Universitaria. 28040. Madrid, Spain, models are also useful for estimating precursory phenomena, so that observation nets can be devel- oped in an optimum way. Normally, simple defor- mation models are used because of the simplicity of calculating the inverse problem. However, some- times it is better to use models that are as realistic as possible, specifically when the physics of the source process must be ascertained. We describe theoretical and computational methods for the calculation of the deformation, gravity, potential and sea level changes due to a point source of magma injection into a layered elas- tic-gravitational Earth model. In the calculations we consider a medium composed of four distinct layers overlying a half-space. The source can be located in any of the layers or the half-space. Rundle (1980, 1982) obtained and solved the equations that represent the coupled elastic-gravita- tional problem for a stratified half-space of homo- geneous layers, using the propagator matrix technique (Thomson, 1950; Haskell, 1953; Gilbert and Backus, 1966) to obtain the surface solutions (potential and gravity changes and deformation). Rundle (198 1) developed the numerical formulation needed to compute these perturbations for the situ- ation of a single layer in welded contact with an in- finite half-space. Expressions for the kernels in the situation of two layers may be seen in Fernandez 231
PII: Soo98-3004(%)00066-0 All rights reserved. Printed in Great Britain
009%3004197 $17.00 + 0.00
PROGRAMS TO COMPUTE DEFORMATION DUE TO A MAGMA INTRUSION IN ELASTIC-GRAVITATIONAL
LAYERED EARTH MODELS
JOSE FERNANDEZ,’ JOHN B. RUNDLE,’ ROBERTO D. R. GRANELL*’ and
TING-TO YU’
‘Institute de Astronomia y Geodesia (CSIC-UCM), Fat. CC. Matematicas, Ciudad Universitaria. 28040, Madrid, Spain; ‘Cooperative Institute for Research in Environmental Sciences, University of
Colorado, Boulder, CO 80309, USA, and “Institute of Earth Science, Academia Sinica, P.O. Box l-55, Nankang, Taipei, Taiwan
Abstract-In previous papers from two of the authors, models, programs and results of programs developed to compute displacements due to volcanic loading with different assumptions have been pub- lished. In this paper, these previous models are extended in a set of programs written in FORTRAN 77. These codes allow us to calculate displacements (horizontal and vertical), tilt, vertical strain and po- tential and gravity changes on the surface of an elastic-gravitational layered Earth model. due to a magmatic intrusion. The intrusion is treated as a point source and the medium can consist of up to four layers overlying a homogeneous half-space. A topmost water layer can be included to calculate sea level changes. We describe the theoretical formulation of the deformation model together with the nu- merical programs developed. Examples of calculation for the different effects are presented. I?:# 1997 Elsevier Science Ltd
In recent years, the international scientific commu-
nity has recognized the possibility of reducing and mitigating risks related to different natural hazards, using systematic study programs. Among these
hazards are volcanic eruptions, earthquakes, floods and landslides (CEES, 1992). According to recent
estimates, in the next 50 years more than one third of the world’s population will live in seismic and
volcanic areas (Bilham, 1988). The fact that this
decade has been declared the International Decade of Natural Disaster Reduction, by the International Council of Scientific Unions, UNESCO and the World Bank, has generated a variety of programs for research into problems related to the prediction and mitigation of these disasters, mainly in Third World countries.
It should also be noted that one of the most promising ways of studying volcano- and earth-
quake-related phenomena is the use of high-pre- cision crustal deformation data, which are obtained with modern space-borne geodetic methods, in par- ticular GPS. These data need physical deformation models to deduce features of the medium. Such
*Also at Dpto. de Geofisica y Meteorologia, Fat. CC. Fisicas, Ciudad Universitaria. 28040. Madrid, Spain,
models are also useful for estimating precursory phenomena, so that observation nets can be devel- oped in an optimum way. Normally, simple defor- mation models are used because of the simplicity of
calculating the inverse problem. However, some- times it is better to use models that are as realistic as possible, specifically when the physics of the source process must be ascertained.
We describe theoretical and computational
methods for the calculation of the deformation, gravity, potential and sea level changes due to a point source of magma injection into a layered elas- tic-gravitational Earth model. In the calculations we consider a medium composed of four distinct layers overlying a half-space. The source can be located in any of the layers or the half-space.
Rundle (1980, 1982) obtained and solved the
equations that represent the coupled elastic-gravita- tional problem for a stratified half-space of homo-
geneous layers, using the propagator matrix
technique (Thomson, 1950; Haskell, 1953; Gilbert and Backus, 1966) to obtain the surface solutions (potential and gravity changes and deformation). Rundle (198 1) developed the numerical formulation needed to compute these perturbations for the situ- ation of a single layer in welded contact with an in- finite half-space. Expressions for the kernels in the situation of two layers may be seen in Fernandez
231
232 J. Fernindez and others
(1992) and Fernandez and Rundle (1994a,b).
Solutions for media composed of more layers have
been obtained by Fernindez and Diez (1995).
In Fernandez and Rundle (1994b) the
FORTRAN program GRAVI used to compute the
solutions was described, presenting results obtained
for a point mass and a center of expansion located
at different depths in a medium composed of two
layers overlying a homogeneous half-space.
Fernrindez and Rundle (1994a) demonstrated that,
if the mantle is identified with the infinite half-
space, to use a homogeneous layer to represent the
earth’s crust may in some instances be too simple a
properties in the cortical models in volcanic areas
(e.g.. Banda and others, 1981; Zucca. Hill, and
Kovach, 1982; Abers, 1985; Elbring and Rundle,
1986; Rapolla. Fedi. and Fiume, 1989; Tilling and
Dvorak, 1993). Clearly it is better to approach this
change by using a stratified medium than by using
a homogeneous half-space. The changes in the med-
ium’s properties with depth may result in consider-
able changes to the magnitude and pattern of the
effects caused by a magmatic intrusion with regard
to those calculated for a homogenous half-space
(Roth, 1993; Fernlindez and Rundle, 1994a,b). As
an example we could consider the stratified cortical
model of the Island of Lanzarote, Canary Islands.
described by Fernindez and Rundle (1994a). and
also approach it by a homogenous medium with
elastic properties and density obtained as an aver-
age of the different values of the stratified medium.
Calculating the surface effects caused by a point
magmatic intrusion in both cortical models. the
differences for shallow intrusions are greater than
the present level of detection that can be reached
with the different geodetic techniques and instru-
ments used in volcano monitoring, and could be of
the order of metres in displacements and of various
mGal (IO-’ m s?) in gravity changes. Considering
these results. the model has been extended in differ-
ent aspects. The number of layers considered has
been increased from only one (Rundle, 1982) or
two (Fernindez and Rundle, 1994a,b), to three
(FernBndez and Diez, 1995) and four layers. New
effects associated with deformation. such as tilt and
vertical strain, are calculated. Also. a top water
layer can be considered, just as in the GRAVW
(Rundle, 1982) program, and in this manner the
medium would represent an oceanic crust, and sea
level changes produced by volcanic activity could
be calculated. All the developed codes compute the
same effects for the deformation model considered
(Rundle, 1980), but they differ in the number of layers to be considered overlying the half-space.
DEFORMATION MODEL
We use a model in which the Earth is represented as a stack of plane layers having both elastic prop-
erties and mass (Rundle, 1980, 1982), hence both
elasticity and self-gravitation are potentially import-
ant effects. The deformation, gravity, and potential
changes induced by a magmatic intrusion at (O,O,c)
are calculated. The equations that are satisfied by
the displacement vector u and perturbation poten-
tial 4 are given by Rundle (1980), who obtained a
general solution at : = 0 in the form
@=A4 -L J’ ~;(O)Jo(kr)k dk, and (2) 0
6g = -g = -M J’
mq;(0)JO(kr)k dk + pour, (3) 0
where M is the intrusion’s mass, and .x:(O), y:(O),
oh(O) and q&O) are kernel functions depending on
wave number k and the properties of the medium.
PO and B. are vector functions that are given in
terms of the Bessel function of the first kind of
order zero Jo(kr), and PO = 4nGp0, with p. being
the density of layer 1. This solution can be shown
to be unique (Rundle, 1982). From Equation (1)
the expression for tilt changes can be obtained as
T(r) = 2 = -A4 I ‘m.y,!,(0)Jl(kr)k’ dk. (4)
.O
and, also from Equation (l), we obtain the ex-
pression for the value of the vertical strain at 2 = 0
at any arbitrary radial distance.
Al cc s,(r) = %(r) = ~
Al +%I J’ _&O)Jo(kr)k2 dk. (5)
0
Tilt and vertical strain values for this model and
point volcanic loading sources have not been obtained previously.
The integration kernels ~$0). y:(O), q;(O) and
w&O) are given as a function of the characteristics
of the medium’s layers, through matrix [_!?I and a
vector [fi of the form
&O) = EllAOp + ‘%BOp + ~15Qp - FI.
.&O) = ~ZIAOI, + &3Bop + E25Dop - F2.
dJux = ESIAO,, +E53BOp +EssDop - F5,
dj(o) = E41'4Op + E63BO,, + E65DO,, - F6. (6)
The matrix [a, a product of propagator matrices (layer matrices), is obtained using the solutions of the infinite space problem (Rundle, 1980). Vector [1;1 is a product of layer matrices with the jump dis- continuity vector [D]. Matrix [El is given by
Computing deformation due to magmatic intrusions 233
[E] = [a’][d] ” . [d][Z”f’(Hj]. (7)
where [a”] is the layer matrix (propagator matrix)
defined by Rundle (1980)
[N”] = [Z”(-c&)J[z”(o)]-I. (8)
The elements of the matrix [Zn(z)] are functions of k given by Rundle (1980, 1982) and Fernindez (1992). The vector [E;‘1 is defined by the product
[F] = [a’][a”] [a”‘][D], (9)
where a new layer boundary is introduced at depth c corresponding to the depth of the intrusion zrl = L’, and si corresponds to the division of the layer where the intrusion is located. The jump dis- continuity vector [O] for : = c is given in equations
(16)) (19) in Fernindez and Rundle (1994b). The
coefficients Ao,, BoP and Do,, are also given by their
equations ( I O)-( 15).
EXPLICIT SOLUTIONS FOR A GENERAL LAYERED
HALF-SPACE
The numerical formulation required to obtain the integration kernels for a layer overlying a half-space is described in Rundle (1981), though it differs slightly from the formulation described here and used in the GRAVW 1 program. The situation of
two layers overlying a half-space appears in Fernindez (1992) and Fernindez and Rundle (1994a,b). Since the numerical formulation required to obtain the explicit solutions in the situation of more than two layers overlying the half-space has not been published, we will explain this briefly and in a general way for a layered half-space formed by p layers, specifying the results in the situations of p = 3 and 4.
As in Rundle (197s) and Fernandez and Rundle (1994b), matrix [Z”(z)] can be expressed as
[Z”(=)] = ([A”(z)] + [B’(c)]ee2”‘)e” (IO)
being the elements of matrices [A”(z)] and [B”(c)] given by Fernandez (1992). Consequently matrix [a”] is given by
[all] = ([M !‘] + [N U]e-?k~l!‘)ek&, (11)
For a p-layered half-space, layer p + I being the bottom homogeneous half-space, matrix [a has the
expression
[El = (f-p’l)Lzwl~ (12)
which, considering Equations (10) and (1 I ), gives
[E] = fPH(jJE~‘]ez~). (13)
where nP = 3, 7, 15, 31 for p = 1, 2, 3, 4 respect- ively, and in general, n,, = 2”+’ - 1. For p = 1, Z, are ZI = 0, Zl = -2kH, 23 = -4H. For p = 2
they are given by Fernindez and Rundle (1994a,b). For p = 3 and 4 they are given in Table I.
The expression of vector [fl will depend on the layer where the intrusion is located. For an intru-
sion in layer k, [Fj is given by
[F] = F[F;]eDk,. (14)
where mr = 2”. Dk,$ for k = 1, 2 and 3 is given by
Fernindez and Rundie (1994a,b). The correspon- ding coefficients for k = 4 and 5 are given in rou- tines CK10L4 and CKlOLS, in arrays DF. of GRAVW4 listing code. It is also necessary to know E$, and E&, defined in Fern;indez and Rundle
(1994a) to calculate the kernel functions. Following
Table I. Z, exponents used in Equation (13) for p = 3 and 4. p being number of layers considered In layered medium. rl,
represents thickness of layer i (i = I. 2.3.4) and H = 2 d, ,=I
Rundle (1981) _t$ can be expressed as a sum of products of k functions multiplied by exponentials in the form
p again being the number of layers of the half-space and R, = 5. The functions EL,,,, and exponents G,?
are computed in the subroutines that calculate the
integration kernels (CKIOLI, CKlOL2, CK10L3, CK10L4 or CKlOLS, depending on the layer where
the intrusion is located). El:k can be expressed as a function of the layer where the magmatic intrusion
is located and the number of layers of the medium in a similar way,
,y’/ = e2hH FA (16)
where NMk = nhmk.The coefficients Tk,, are also
obtained in the subroutines that compute the inte- gration kernels. Ekk,i, functions are calculated in the
routines EFA, EFB, EFC, EFD and EFE for the situations of intrusion in layer one, two, three, four and five, respectively, in each code.
The expressions required for (k&, - E6i) and (kFs - Fb) are
kEs, - Eh, = e2iH($E&iez,), (17)
with
E ;6i = kE;, - I?;,; (18)
element of row i, column j of matrix [E”] where the
is denoted by I?$, and
kF5 - F6 = xFf,,eD”,‘. (.=I
(19)
being
F;,, = k(F;), - (F;)6. (20)
Using Equations (lo)-(20) it is possible to calcu-
late A03, B03, 003, with which the kernels can be obtained. It is obvious that the expressions for Aol,
Bos, 003, depend on the location of the intrusion. By way of example, the general form for the kernel
.&O) in the situation of an intrusion located in layer k is
where element F, (i = 1, 2, 5 and 6) appears multi- plied and divided by r given by (Fernandez and Rundle, 1994a,b),
r = A/e6’H, (22)
with A given by equation (90d) of Rundle (1980). The sum of functions that multiply ek” must be zero for convergence in Equation (21) and advantage is taken of this by setting them equal to zero (for more details see Rundle (1981) and Fernandez and Rundle (1994b)).
EXISTENCE OF A WATER LAYER
The presence of a top water layer in these calcu-
lations necessarily modifies the boundary conditions taken into account at the surface (Rundle, 1982). Instead of requiring null elastic traction for z = 0 (Rundle, 1980) the change of tractions must be con- sidered to be caused by the redistribution of water as a consequence of the deformation. Thus the ver- tical component of traction in surface z = 0. T,(O),
has the following expression (Rundle. 1982)
T,(O) = -&Vgu;(O) + P,$(-D,). (23)
The first term, -pwgu,(0), is the traction caused by the additional weight of the water column of alti- tude u,(O). The minus sign is due to the fact that z is positive downwards in the medium, while traction
is defined positive along the normal to the surface outwards. The second term, p,+(-D,), is the weight of the water column that is originated by sea potential changes in z = -D,. Consequently, the traction condition is
2kx-A(O) + p,gxA(O) - pwwi(0)e-hD” = 0 (24)
[this equation must replace equation (28) of Rundle (1982) where an equal sign appears instead of the minus sign between the second and third terms on the left side of equation]. This affects the definition of the integration kernels in the equations, see Rundle (1982). The effects of the existence of a water layer on the integration kernels through matrix [a and vector [fl are considered in the CALKRN routine after calling the subroutines that compute both arrays.
NUMERICAL INTEGRATION OF THE KERNELS
The main problem found in the integration pro-
cess of the kernels function is that the existence of a pole or poles on the real axis depends on the values of the elastic moduli adopted. If a pole is present, it always occurs with a gravitational wave number and close to the associated k,, given by equation (2.16) of Rundle (1981). The technique used to solve this problem was described by Rundle (1981) and Fernindez and Rundle (1994b). Fernandez, Yu and Rundle (I 996) and Yu, Rundle, and Fernindez (1996) demonstrate that the integration around the poles does not contribute an important part to the final solution, therefore it is possible to determine
Computing deformation due to magmatic intrusions 235
0
-so
loo
40
20
0
0
-0.1
-0.2
-0.3
I I I I I I
-3 - (XAVI
----I (iKAVW2
. _ I I I I I I
#( . I I l I 1 0 10 20 30 40 SO
Radial Distance (km)
Figure 1. Comparison of output of GRAVl program that performs poles computation and integration around them if they exist, and GRAVW2 program that set kernel equal to zero when they go to infin- ity. Both outputs are obtained for center of expansion located at 5 km depth in layered medium com- posed of two layers each 10 km thick. Medium properties are given in text. U_ is vertical displacement
U,. is radial displacement, and g, is surface gravity change.
236 J. Fernindez and others
1
0.S
0
0. 1
0.05
0
0.2
0.1
0
- MUVI
- - - - GRAVW2
O 10 20 30 40 50
0 10 20 30 40 50
Radial Distance (km)
Figure 2. Comparison of output of GRAVI program that performs poles computation and integration around them if they exist, and GRAVW2 program that sets kernel equal to zero when they go to infin- ity. Both outputs are obtained for point mass located at 5 km depth in layered medium composed of two layers each 10 km thick. Medium properties are given in text. Uz is vertical displacement, U,. is
radial displacement, and gs is surface gravity change.
Computing deformation due to magmatic intrusions 237
Table 2. List of codes forming GRAVW PROGRAMS SET describing number of layers considered in each one, possible locations of source, if integration around poles is done or not if they exist, and computed effects. Codes are compared
with other previous codes, GRAVW (Rundle, 1982) and GRAVI (Fernandez and Rundle, 1994a.b)
GRAVW GRAVWl GRAVI GRAVWZ GRAVW3 GRAVW4
Numer of layers over the half-space 1 1 2 2 3 4 Places where the intrusion can be located Layer Anywhere Anywhere Anywhere Anywhere Anywhere Poles computation Yes No Yes No No
Computed effects 11, tl:, u,, T, E: u,, u, u;, 11,. T. 6: u,, u,. T, E: I,,. ,;(;: I:,
&. RET RFA R\. gB3 gFA SW gs. ‘?FA & c?B, i?FA g\, c!?B, PFA Rv ,?H. PF4
dd dd dd d4 W dd
SL SL SL SL SL
the existence of poles and to compute more than
one integral in k. The way to solve the problem is
to make the kernels equal to zero when these tend
to infinity. This can be done by using the same
technique as Yu. Rundle, and Fernindez (1996).
which we apply in the subroutine AKNINT of the
programs.
Figure I shows different effects caused by a cen-
ter of expansion of strength pa”. where p is the
increment of pressure in 0.1 MPa and a is the
radius in km. Figure 2 shows the same effects com-
puted for a point mass of 1 MU. The effects dis-
played in both figures have been computed using
the GRAVI program (Fernandez and Rundle,
1994b). with poles calculation. and GRAVWZ,
without poles calculation. The source is supposed
to be located at 5 km depth in the first layer of an
Earth model composed by two layers overlying the
mantle. Both layers are 10 km thick. Densities are
2600, 2900 and 3300 kg m-s for layer one. two and
half-space respectively. Both Lame parameters for
layers and half-space are considered equal to
3 x IO’? Pa.
We see clearly that suppressing poles compu-
tation and the integration around them, if any exist,
causes no important afteration in the computed
effects, as can be observed in the example. The
differences are of the order of the numerical accu- racy or even smaller, I mm in displacements and
1 pGal for gravity changes (Fernindez and Rundle.
Table 3. Program utilities of GRAVW4 program and computations performed in each one
PROGRAM GRAVW4: Reads data and computes deformation, potential and gravity change using HLFSPC and THFLTT subroutines. SUBROUTINE ALAYER: Determines the layer where the intrusion is located. SUBROUTINE HLFSPC: Computes the effects due to a magma intrusion located in a homogeneous halfspace with the same characteristics as the layer where the intrusion is. SUBROUTINE THFLTT: Integrates the Green functions. SUBROUTINE CALKRN: Computes the integration kernel differences. SUBROUTINE EMA’T: Computes [&‘I matrix given by Equation 12. SUBROUTINE AMAT: Computes layer matrix given by Equation 1 I. SUBROUTINE ZCOMPC: Computes [Z”(z)] matrix for k<min(k,,, i = 1.2,3). SUBROUTINE ZOINVC: Computes [Z,,(O)]-’ matrix for k<min(k,<,, i = I, 2,3]. SUBROUTINE ZCPC: Computes [Z”(z)] matrix for min(k,,, i = 1, 2, 3)<k<max(k,~,, i := 1. 2. 3). SUBROUTINE ZCPIN: Computes [Z”(O)]-’ matrix for min(k,,, i = 1, 2. 3)<k<max(k,,. i = 1.2.3). SUBROUTINE ZCOMPR: Computes [Z ‘(;)j, matrix for k > max[k,,, i = 1, 2. 3). SUBROUTINE ZOINVR: Computes [Z”(O)]- matrix for k > max(k,,, i = 1, 2. 3). SUBROUTINE FMI: Computes [4 vector for an intrusion located in layer I. SUBROUTINE FMZ: Computes [fi vector for an intrusion located in layer 2. SUBROUTINE FM3: Computes [4 vector for an intrusion located in layer 3. SUBROUTINE FM4: Computes [Fj vector for an intrusion located in layer 4. SUBROUTINE FM5: Computes [4 vector for an intrusion located in the half space. SUBROUTINE GFINT: Performs the integration on the Green function. SUBROUTINE AKNINT: Integrates the Green function along the considered contour. SUBROUTINE El: Computes the differences given by Equation 18. SUBROUTINE EBAR: Computes the E& functions given in Equation 15. SUBROUTINE EFA: Computes the EF, functions given by Equation 16 for an intrusion in layer one. SUBROUTINE EFB: Computes the E% functions given by Equation 16 for an intrusion m layer two. SUBROUTINE EFC: Computes the Ej., functions given by Equation 16 for an intrusion in layer three. SUBROUTINE EFD: Computes the i?& functions given by Equation 16 for an intrusion in layer four. SUBROUTINE EFE: Computes the Ex.1 functions given by Equation t 6 for an intrusion in layer five (half-space). SUBROUTINE CKlOLl: Computes the kernels for a source located in layer one. SUBROUTINE CKlOLZ: Computes the kernels for a source located in layer two. SUBROUTINE CKIOL3: Computes the kernels for a source located in layer three. SUBROUTINE CKIOL4: Computes the kernels for a source located in layer four. SUBROUTINE CKlOLS: Computes the kernels for a source located in the half-space.
238 J. Fernindez and others
+
THFLlT
AKNINT
ZCOMPC 7
F i
f
FM4 CK10L4 EFD
I F
Figure 3. Flow diagram of GRAVW4 FORTRAN program.
Computing deformation due to magmatic intrusions 239
Table 4. Input files for each program of GRAVW set. For explanations see text
1994a). By eliminating these calculations we can shorten the code, removing the subroutines POLCAL, POLDEC, POLINT, FKERN and ZBRENT, which are now no longer necessary and which were used in program GRAVl to carry out these processes. A large part of the subroutine GFINT has also been removed. This is due to the fact that in GRAVI program it could have been necessary to perform more than one integral, depending on the number of poles; now, with these modifications, a single integral will always be calcu- lated from k = IO-’ (~0) to k,,, which is given in equation (49) in Fernandez and Rundle (1994b). By computing this integral, we use a logarithmic par- tition of the kernel which is given in expressions (45)-(46) of the same paper.
GRAVW PROGRAMS SET
Using the expressions described, different FORTRAN-77 programs have been developed for the situations of one, two, three and four layers over a homogeneous half-space. The programs are named in general GRAVW PROGRAMS SET. This set is formed by GRAVWI, GRAVWZ, GRAVW3 and GRAVW4 programs. The number included in the name of the program indicates the number of layers considered in the program. We have used previous existing codes, GRAVW (Rundle, 1982) and GRAVl (Ferncindez and Rundle, 1994a,b) for the situations of one and two layers, respectively. Some improvements, apart from the aforementioned changes in the integration
POINT MAGMA SOURCE: DEPTH(KM) = 10.00000 PRESSURE (BAR) = .lOOOOE + 05 RADIUS(KM) = I .OOOOO MASS(MU) = 1.00000
THE SOURCE IS IN LAYERI
x = 0. Y = 0.
UZ (CM) = - 29.5143 RELATIVE UZ (CM) = - 29.4631 UZ OF SEA (CM) = - 0.05125725 GRAVITY CHANGE FROM DENSITY CHANGE (MGAL) = 0.07990089 SURFACE GRAVITY CHANGE (MGAL) = - 0.01115084 FREE AIR GRAVITY CHANGE (MGAL) = 0.07990089 BOUGUER GRAVITY CHANGE (MGAL) = 0.034135 15 RADIAL DISPLACEMENT (CM) = 0. TILT (MSECA) = 0. VERTICAL STRAIN (MICROSTRAIN) = - 0.20024461E - 04
of the kernels, have been made. They are described form [Z”(z)] matrix inversion, instead of the LEQ2C
in Table 2. The programs run on a workstation routine (IMSL, 1982). By way of example, the pro-
using a UNIX operating system. gram utilities of GRAVW4 program are listed in
Some modifications have been also made to the Table 3, indicating the computations performed in
subroutines ZOINVC, ZOINVR and ZCPCIN listed each one. Figure 3 presents a flow diagram.
in Fernandez and Rundle (1994b). They now use the Table 4 shows input files for all the four pro-
mathematical routines ZGECO and ZGESL to per- grams (the numerical values are those used to test
Table 6. Part of output file for plotting. r is radial distance, u, and Us are vertical and radial displacement respectively, T is tilt. E, is vertical strain, g,, gra, and ga indicate surface, free air, and Bouguer gravity change respectively, and SL
Computing deformation due to magmatic intrusions 241
-0.1
-0.12
-0.14
-0.16
- CiKAVW
MB__ CiKAVWl
- - (iKAVW2
- - (iKAVW3
0 1 2 3 4 5
Radial Distance (km)
Figure 4. Test of GRAVW PROGRAMS SET comparing results to those from GRAVW (Rundle. 1982) program. Vertical displacement (U,), free air gravity (grJ and sea level (Sea-Level) changes pro- duced by point mass located in medium composed of 1, 2, 3, and 4 layers overlying half-space are rep- resented. Top water layer 3 km thick is also assumed. Source is located at 1 km depth for every medium. All layers have same properties but different to those of the half-space. See text for specific
values. Success of each program is denoted by its name.
-40
,- &
a 0
-60
E
2 -8
0
-100
I I
I I
I I
I I
0 1
2 3
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0 1
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Rad
ial
Dis
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m)
Rad
ial
Dis
tanc
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40
20 0
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5
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- H
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Tes
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pari
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effe
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prod
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on
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5 km
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ogen
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.)
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ing
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.
Computing deformation due to magmatic intrusions 243
Ep8 1
I
I
I I I’ I
I
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,
d’ _** _ . --
__-- __--
-- --------- _-
244 J. Fernandez and others
Table 7. Properties of media 0, 1. 8, 9, 10. 11 and 12, considered in Figure 7
the program in next section). They are similar to the files described in Fernindez and Rundle (1994b) for GRAVl program but now also include the characteristic of a possible top water layer.
Therefore the data are as follows: the characteristics of the crustal model (layer thicknesses. Lame con-
stants and densities), magma intrusion character- istics (radius, pressure and depth), thickness and density of the water layer, and the X, Y coordi- nates, or radial distance r, of the points where we
want to compute the effects. The units used for the program for data input are km (IO” m) for dis- tances, 10s kg me3 for densities, lOI Pa for elastic
parameters, lOI kg (MU) for mass and 0.1 MPa for pressure values.
We use the NAMELIST statement for the data input. In Table 4, H, Dl, D2, D3 and D4 are layer thicknesses. AMU, ALAM, AMUZ, ALAM2, AMU3, ALAM3, and AMU4, ALAM are the
Lame constants for layers 1, 2, 3 and 4 respectively, ALAMl, AMUl are the same quantities for the half-space, YSTART is the first value of Y coordi-
nate, DYOBS the increment on Y, X the considered value for this coordinate, NNY is the number of
different Y coordinates. If INDIC is greater than zero the programs read individual points. RHOL, RHOLZ, RHOL3 and RHOL4 correspond to the layer densities and RHOH to the density of the half-space. We can suppress the gravity effect by setting RHOFCT equal to a small value (for example, lo-‘). NSTEP is the number of integration intervals for Green function integration. DWATER is the thickness of the possible water layer and RHOW is the water density. If they are equal to zero the water layer is not considered. PMAGMA, RMAGMA and AMASS are pressure increment,
radius and mass of the intrusion, respectively. The output of the program consists of four files,
three of which are similar to the files described in Fernindez and Rundle (1994b) for the program GRAVl. In one file the characteristics of the crustal model and magma intrusion are written, together with the various effects computed for each point, shown in Table 5 (the numerical values that appear in the table corresponds to the output for the data shown in Table 4). Another file, useful for plotting the effects, gives coordinates of radial distances, ver- tical displacement. radial displacement, tilt, vertical
strain, surface gravity change, free-air gravity
change, Bouguer gravity change and sea level change in columns, see Table 6. The third output file contains the values of the kernel differences, X,, - X,, as shown in figures l-2 by Fernindez and Rundle (1994b). The fourth output file compares the results with the results corresponding to a homogeneous half-space with the same character- istics as the layer where the intrusion is located. This file is similar to the file displayed in Table 6. The programs use different subprograms from the IMSL library (IMSL, 1982).
TESTING OF THE PROGRAMS
The results obtained using GRAVWl,
GRAVWZ. GRAVW3 and GRAVW4 programs have been tested by comparing the results obtained in two limiting situations, in the same way as GRAVl program was tested by Fernandez and Rundle (1994a). The first situation is one where all
the layers have the same physical characteristics but are different from those of the underlying half- space, the results being compared with those
obtained using the GRAVW program used by Rundle (1982). The second situation, where both layers and the half-space have the same physical characteristics, produced the same results as in the situation of a homogeneous half-space.
The results obtained for the first situation are shown in Figure 4, where the results obtained using the GRAVW PROGRAMS SET are compared with those obtained using the GRAVW code (Rundle, 1982). The figure shows vertical displace- ment (ui_), free-air gravity (grJ and sea level (SL) changes due to a point mass located in media com- posed of one, two, three, and four layers overlying a homogeneous half-space, with a 3 km thick water layer above. A 1000 kg m-j density is considered. Source is located in the first layer at 1 km depth. All layers are considered to have the same proper- ties, with the density being pi = 2670 kg rnd3 and Lame parameters j+_ = pL = 3 x 10’” N rn-‘. A density value of pH = 3300 kg rne3, and elastic par- ameters & = 3 x lOI N me2 and &i = IO9 Nrn-’ are assumed for the half-space. All these calcu- lations would represent a stage some time after the active intrusion process, where the fluid properties
-750
0
-150
-15
Computing deformation due to magmatic intrusions 245
c=5 km
-- 8
- 9 -
-_- 10
--- 11
- 12 --
0 10 20 30 40 50
c=15 km
I I I I I ----__ I_ . - _ _ - - - - i _- -- .
/’ /
\ / \ /
’ 1’ rl I I I I II
0 JO 20 30 40 50
c=25 km
I I I I I I_
0 10 20 30 40 50
Radial Distance (km)
Figure 7. Tilt values in milliseconds of arc (mseca) for point mass of 1 MU (IO” kg) located in media described in Table 8 at 5, 15 and 2.5 km depth.
CAGE0 23/3--B
246 .I, Fernindez and others
-7
-8
-9
I so0
z s 2
1000
-
n ;:
so0
0
-1.25
-z 0,
-1.5
-’
m -1.75
J I I I I
0 1 2 3 3 5
1 I I I I
‘I
0 1 2 3 4 5
1 I I I I I-
0 I 2 3 4 5
Radial Distance (km) Radial Distance (km)
‘w’
s v
W
-Me-06
-4e-06
0.0047
0.0046
I I I I I I
J I I I I I .
0 I 2 1 4 5
I I I I r
1 I I I I 1
3 1 2 3 4 5
1 I I I I
Figure 8. Vertical (C,) and radial (U,j dispkicements, tilt, vertical strain (E;). free air grdWty fgr.,) and sea level (SL) changes produced by center of expansion located at 13 km depth (third layer) in medmm
described in Table 9 and considering a top water layer 3 km thick.
Computing deformation due to magmatic intrusions 247
of the half-space (asthenosphere) have been taken into account in the considered shear modulus.
The results for the second situation are rep- resented in Figure 5, showing vertical and horizon- tal displacements, tilt and vertical strain produced by a center of expansion of strength pa3 located at 5 km depth in a homogeneous half-space and false layered half-spaces having the same reference den- sity and elastic properties. These characteristics are p = 3300 kg m-3 , i = p = 3.1012 N mv2. In this situation, no water layer is considered. The compu- tations shown in Fig. 5 represent the effects pro- duced during the active intrusion process.
Once again, it will be noticed for both tests that the differences between the effects computed by each code, assuming identical sources and medium, is of negligible order, smaller than the accuracies mentioned previously. A brief comment must be made on the difference found in Fig. 3 between the sea level changes computed by GRAVWQ code and the rest of the programs. Although it stands out in the graph, this difference is of the order of 0.1 mm or less, caused probably by the accumulation of rounding and numerical errors in computations. These errors are much bigger if this code is used, because the number of calculations carried out is also larger, since this increases exponentially with the quantity of layers considered for the medium. Figures 4 and 5 show clearly that this difference is only detectable in this situation, where sea level changes are of the order of accuracy for displace- ments.
RESULTS AND DISCUSSION
The importance of including gravity can be seen from the example in Fig. 6. This figure shows verti- cal and horizontal deformation, tilt and gravity changes due to a center of expansion (ce) and a point mass (mp), both located at 6 km depth in the Lanzarote crustal model composed of two layers over the mantle. See Fernandez and Rundle (1994a) for model properties. It can be seen that gravita- tional effects are not significant for displacements and tilt at the surface. The contribution to the total effect (in the situations of displacement and tilt) of the intrusion mass (mp in Fig. 6) is almost null compared to the contribution made by the pressure changes (ce in Fig. 6). If we suppress the effect of the gravitational field, giving RHOFCT a small value, for example 10-3, we get gravity change and deformation values similar to those displayed in Figure 6 for a center of expansion. This is consist- ent with the results obtained by Rundle (1980), who shows that gravitational effects in ground defor- mation become significant over wavelengths greater than 1000 km, but have little effect on deformation in the near source region. As shown in Figure 6, this is not the situation for surface gravity change,
Table 8. Properties of the medium used in Figure 8
P (103 Thickness Layer kgm-‘) E. (lO’*Pa) p (10”Pa) (km)
where the mass of the intrusion is significant and the effect may be important. Therefore it is seen that gravity field is important for modeling and understanding the observed gravity change data. This effect is also important if we consider visco- elastic properties for the medium. Hofton, Rundle, and Foulger (1995) show that the effects of gravity produce important differences in the displacement fields when long intervals of time are considered. Their model predicts the presence of a long wave- length component in the post-diking deformation field following a single event.
The effect of vertical structure is discussed by Fernindez and Rundle (1994a), who show that de- viations of the displacements in a layered medium from those in a homogeneous half-space are affected more heavily by variations in elastic moduli than by changes in layer reference densities. Thus, variations in strain contribute more to gravity changes than do variations in density. Taking into account the same media, they attempted to test the influences of changes in elastic properties on the computed effects (see Table 7), and by way of example we have carried out the computations for tilt at the same depths as in their figures 2. 3 and 4. The results obtained are shown in Figure 7. They confirm the results obtained by Fernandez and Rundle (1994a). Similar results are obtained for ver- tical strain.
Figure 8 presents displacements, tilt, vertical strain, free air gravity and sea level changes occur- ring a long time after the active intrusion process in the layered medium described in Table 8. We con- sider a center of expansion located in the third layer of the medium, at 13 km depth. The con- sidered pressure increment is 10’ MPa, and the radius is 1 km. The considered value for the shear modulus of the bottom half-space (asthenosphere), lo6 Pa, would indicate the relaxation produced a long time after the intrusion process.
Moreover, using deformation models and knowl- edge of previous volcanic activity, we can predict changes in the deformation and gravity patterns that precede an eruption. Prediction may allow early detection of eruptive activity and thus possible mitigation of risk (Fernandez and Diez, 1995). To carry this out, inversion of the model will be necess- ary.
248 J. Fernindez and others
SUMMARY AND CONCLUSIONS
This paper deals with the numerical implemen- tation of a mathematical technique for the calcu- lation of ground deformation, gravity and potential changes caused by a point source magma intrusion into an elastic-gravitational multilayered earth model using the model described by Rundle (1980). Rundle (1982) and Fernindez and Rundle (1994a,b). We have developed FORTRAN-77 pro-
grams, named GRAVWl, GRAVWZ, GRVW3 and GRVW4, that solve the problem for a medium
formed by one, two, three and four layers overlying a half-space, respectively.
As regards numerical calculation, one of the
main problems found in previous works (Rundle, 198 1; Fernindez and Rundle, 1994b) has been eliminated by removing poles computation and inte-
gration around them, if any. This has reduced the number of lines of code considerably. as well as the execution time. This reduction is important in the situation of codes that consider more than two layers overlying the half-space, which are larger,
since all different situations are taken into account, and they need a longer execution time as a result of an exponential increase of the number of calcu-
lations in relation to the number of layers. To
attain a higher level of accuracy in the numerical computations, the subroutines that calculate the
inverse matrices have been changed from those of the GRAVI code. Again. the precision of the calcu- lated results is of the same order as in Fernindez
and Rundle (1994a,b).
All the codes forming GRAVW PROGRAMS SET are similar in the sense that they perform the
same computations using the same numerical pro- cedures. New effects are now computed (tilt and
vertical strain), others have been extended to all codes and situations (sea level changes) and the number of source locations has increased (see
Table 1). A maximum of four layers has been con- sidered. In Ferm’mdez and Rundle (1994a) the model was tested in relation to the changes in the
elastic parameters and densities. In this paper the effect of the gravity field has been taken into account, and it has been found that it can be funda- mental to adjust and to explain properly the gravity changes measured in active zones. Also. consider-
ation of the gravity field is necessary for compu- tation of altitude changes with regard to an equipotential surface if we wish to obtain ortho- metric and geometric gravity gradients that are sen- sitive to the dynamics of the intrusion processes (Rundle. 1982).
An example where this model has been applied to adjust and interpret real data, with regard to Kilauea volcano, may be seen in Rundle (1982). considering a stratified medium of a single layer overlying the half-space. In future works we shall look at the adjustment of gravity and deformation
data, and how to solve the inverse problem for the parameters of the model considering the cortical structure is known, in active areas such as Long Valley Caldera, California (see for example Jachens and Roberts, 1985; Dixon, Farina, and Mao, 1995; Langbein and others, 1995).
It is also known that ground deformation is an
important precursor for long-dormant volcanoes. The use of deformation models can thus be a powerful tool for the design of techniques for moni- toring volcanic deformation, especially when there is no information on prior episodes of ground de- formation. For this application. the most realistic model would be necessary, and the point source considered here is only a first step (Delaney and McTigue, 1994). The programs described are avail- able to the scientific community via anonymous FTP from the server IAMG.ORG.
Acknon/e~ggment,s~-The research of J. F. cjft(u iagmatl. mat.ucm.es) and R. d. R. G. was supported with funds from contract EVSV-CT93-0283 from the Environment Program of the European Union. The research of J. B. R. (rundle(a~fractal.colorado.edu) and T. T. Y. (yutt(a earth.sinica.edu.tw) was sup- ported by NASA grant NAG52353 to the University of Colorado. Part of the research of J. F. was car- ried out during his stay at CIRES, University of Colorado at Boulder, supported by the Consqjrricl t/c Etk~mciorz j’ C’ulturtr of 1/w C’ommidut/ tk MLILirid. We thank Mark Jesse1 and Richard Williams for giv- ing us valuable suggestions and comments to improve the manuscript.
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