Caldera Subsidence and Magma Chamber Depth of the …pmm/Oly_Caldera.pdf · Caldera Subsidence and Magma Chamber Depth ... radial ridges and circumferential graben that we hypothesize

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 97, NO. Ell, PAGES 18,295-18,307, NOVEMBER 25, 1992

Caldera Subsidence and Magma Chamber Depth of the Olympus Mons Volcano, Mars

M. T. ZUBER1

Geodynamics Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland

P. J. MOUGINIS-MARK

Planetary Geosciences Division, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, Honolulu

Observations of the distribution of tectonic features and their relationship to local surface topography indicate that the floor of the oldest and largest crater of the summit caldera of Mars' Olympus Mens volcano may have undergone subsidence in response to depressurizatien of a subsurface magma chamber. In this study, we construct an axisymmetric finite element model to calculate elastic stresses in a volcanic edifice to investigate the relationship between surface tectonism, caldera subsidence, and the physical characteristics of Olympus Mens' magmatic reservoir. Con- straints on the model are provided by the stress field within the crater indicated by the distribution of radial ridges and circumferential graben that we hypothesize formed due to a postcollapse and pestresurfacing phase of subsidence of the caldera floor. Model results show that the surface stress state is not strongly sensitive to the aspect ratio or pressure distribution of the magma chamber, or to the contrast in stiffness between the magma chamber and surroundings, but is strongly dependent on the width and depth of the chamber. For a range of plausible model parameters, we find the maximum depth to the top of the magma chamber to have been -<16 km, which indicates that the chamber, at the time of crater floor subsidence, was positioned within the Olympus Mens edifice, at a level much shallower than the estimated source depth of Martian magmas (--• 140-200 km). The allowable range of solutions indicates that the vertical position of the magma chamber may have been controlled by the neutral buoyancy level of ascending magma. Our results suggest a gross similarity between the configurations of the magmatic plumbing systems of Olympus Mens and several well-studied terrestrial volcanoes such as the Hawaiian shields.

INTRODUCTION

Studies of the eruptive characteristics of terrestrial basaltic volcanoes indicate that magma is generally transported from a deep, upper mantle source region to the surface via a shallow holding chamber [Eaton and Murata, 1960]. The configuration and depth of this chamber have implications for the nature of magmatic transport as well as for the processes which initiate and sustain eruptions. In closely monitored shields such as Hawaii's Kilauea volcano, the geometry and depth of the magma chamber have been constrained from seismic and gravity data [Crosson and Endo, 1981; Karpin and Thurber, 1987; Klein et at., 1987; Thurber, 1987;Ryan, 1988], as well as from geodetic leveling and triangulation measurements [Dieterich and Decker, 1975; Ryan et at., 1983; Delaney et at., 1990; Yang et at., 1992].

Shield volcanism is a common process on Mars, and the major shields have been the focus of numerous morpholog- ical studies [cf. Cart, 1973; Greeley and Spudis, 1981; Mouginis-Mark et al., 1992]. Unfortunately, due to the absence of seismic data, high spatial resolution gravity measurements, and measured ground displacements, analyses of magma chamber positions within Martian shields have been limited to gravitational scaling of terrestrial volcanoes

1 Also at Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland.

Copyright 1992 by the American Geophysical Union.

Paper number 92JE01770. 0148-0227/92/92JE-01770505.00

[Wilson and Head, 1988; Thomas et al., 1990]. In the vicinity of the summits of Martian volcanoes that exhibit a clear

record of tectonism, however, models of the stress field may represent an alternative and arguably more direct means of understanding the nature of magma accumulation and transport at shallow subsurface levels.

The summit caldera of the Olympus Mons volcano, shown in Figure 1, displays one of the clearest examples of tectonic processes associated with shield volcanism on Mars. Within the -90 x 60 km structure are six nested craters that

indicate that the summit has undergone multiple collapse episodes [Mouginis-Mark, 1981]. Observations of the caldera include high-resolution Viking images and local topography derived from stereophotogrammetry [Wu et at., 1984] and photoclinometry [Mouginis-Mark and Robinson, 1992]. Comparison of these data sets shows that the central portion of the largest and oldest crater (hereafter, crater 1), contains radial and circumferential ridges and represents a topographic low. In contrast, the outer part of the crater is characterized by circumferential graben which are found on a topographically elevated region along the perimeter of the crater floor. The total change in relief from the central ridges to peripheral graben is -•1300 m [Mouginis-Mark and Rob- inson, 1992]. Tectonic features within the interior of the caldera as a whole are most abundant in areas with the

greatest surface slopes [Watters and Chadwick, 1990]. Pho- togeologic evidence for basalt-like re surfacing of the caldera floor [Greeley and Spudis, 1981], in combination with the observed topography, has been interpreted to indicate that a large lava lake within the crater has subsided in its central region due to pressure reduction in the underlying magma

18,295

18,296 ZUBER AND MOUGINIS-MARK: OLYMPUS MONS CALDERA SUBSIDENCE AND MAGMA CHAMBER DEPTH

.;-";':¾?i

Fig. 1. Image of the Olympus Mons caldera showing six nested craters. The largest and oldest crater (referred to as crater 1 in text) has a radius of--•32 km. The tectonic features used in this study as constraints on magma chamber depth are located within the box. The resolution of the image is approximately 156 m pixel -• . North is at the top of the image. Illumination is from the right. (Viking Orbiter image 890A68.)

chamber [Mouginis-Mark, 1981; Mouginis-Mark and Robin- son, 1992].

On the basis of geologic mapping and numerical modeling of the stress regime of the Olympus Mons caldera complex, we suggest that certain tectonic features within crater 1 may preserve a record of the proposed subsidence event. If this is the case, then the stress field inferred from the features may contain information about the magma chamber at the time of subsidence. The objective of this study is to explore the

possible relationship between tectonic features within the Olympus Mons caldera and the nature of the magma chamber. To accomplish this, we begin by reviewing the geologic and structural context of the caldera, then describe a simple model of magma chamber depressurization and surface subsidence that can explain the distribution and timing of a specific assemblage of tectonic features within crater 1. We apply the model, with constraints provided by the observed spatial distribution of the tectonic features, to investigate the

ZUBER AND MOUGINIS-MARK: OLYMPUS MONS CALDERA SUBSIDENCE AND MAGMA CHAMBER DEPTH 18,297

;""• RADIAL WRINKLE RIDGE

-•- CIRCUMFERENTIAL RIDGE

• CIRCUMFERENTIAL GRABEN

5 km

N

Fig. 2a. Sketch map of tectonic features within the area of crater 1 outlined in Figure 1. The center of the crater is in the direction of the top of the image. This area was mapped from Viking Orbiter images 473S27-29 and 474S25-30.

extent to which the size, shape, depth, and pressurization state of the magmatic reservoir can be determined. We find that a wide range of magma chamber aspect ratios and pressure distributions can produce surface stresses consistent with the observed pattern of tectonism and conclude that neither the relative dimensions of the magma chamber nor the details of its pressurization can be confidently constrained from the tectonics alone. We also find that the

distribution of surface stress is a sensitive indicator of the

width and depth of the magma chamber. From our model results and observations, we find the depth to the top of the magma chamber at the time of subsidence to have been within the volcanic edifice, possibly at the level of neutral buoyancy, as is the case for several well-studied terrestrial volcanic shields.

CALDERA TECTONICS

Structural Context

Figure 2a shows a structural sketch map of crater 1. The area was mapped from high-resolution Viking Orbiter images, such as Figure 2b, which have a maximum spatial resolution of 15-20 m pixel -• [Mouginis-Mark and Robin- son, 1992]. The map shows three types of tectonic features. The first consists of prominent wrinkle ridges oriented radially from the center of the crater. These features are morphologically similar to wrinkle ridges that occur on spatially extensive plains units elsewhere on Mars and on the lunar maria [Greeley and Spudis, 1981; Plescia and Golombek, 1986; Watters, 1988]. The features have topographic amplitudes of---300 m, average widths of---2 km, and lengths of 5-15 km. The orientation of these features indicates that they formed as a consequence of circumferentially directed horizontal compression within the caldera. The second set of features consists of narrow, concentric ridges that are characterized by minor topographic expression (---20-40 m). These ridges are morphologically different from

the radial wrinkle ridges and are often superimposed upon the radial ridges. Unlike superficially similar features on the volcano Alba Patera, which were interpreted by Catterrnole [1986] to be spatter ridges, we see no evidence of comparable individual constructional cones or small flows within the

Olympus Mons caldera. We interpret the Olympus Mons circumferential ridges, which are distributed from the center of the crater out to approximately half its radius (r -• 0.53Rc

Fig. 2b. High-resolution (16.5 m pixel -•) image of the part of Figure 2a noted by the box. Illumination is from the left. (Viking Orbiter image 473S30.)

18,298 ZUBER AND MOUGINIS-MARK' OLYMPUS MONS CALDERA SUBSIDENCE AND MAGMA CHAMBER DEPTH

[Zuber and Mouginis-Mark, 1990]), to have formed as a consequence of radially directed compression. The third type of feature consists of prominent circumferential depres- sions that have previously been interpreted as graben [Mouginis-Mark, 1981; Watters and Chadwick, 1990]. These features have lengths of more than 25 km, widths of-420- 450 m, and depths of-180-200 m and are located in the outer part of the crater (r >- 0.53R c [Zuber and Mouginis- Mark, 1990]). The transition from a state of compression in the inner part of the crater, as indicated by the radial and concentric ridges, to extension in the outer part of the crater, as indicated by the circumferential graben, is sharp, occur- ring in some areas over distances comparable to the widths of individual features.

The assemblage of tectonic features discussed above can only be mapped at high spatial resolution (<50 m pixel-i) on the eastern side of the crater. However, lower-resolution images (>100 m pixel -1) of the western and northwestern segments of the crater floor display the same general trends. Specifically, concentric graben are observed around the perimeter of the crater floor, and there are hints of the narrow circumferential ridges closer to the crater center. Watters and Chadwick [1990] cite the topography and distribution of tectonic features within the caldera as evidence

for asymmetric subsidence, with greater downward displacement in the southern half of the caldera floor. However, this interpretation is based on observations of all of the craters that compose the caldera, rather than of crater 1 alone. There is no evidence for a significant deviation from axisym- metry in crater 1 from the observed distribution of tectonic features.

The relative sequence of formation of the tectonic features and their relationship to the formation of crater 1 can be derived from superposition relationships. Because all of the structures of interest are located within smooth materials on

the floor of the summit caldera, they must postdate both the collapse event responsible for the formation of crater 1 and an episode of subsequent (probably volcanic) resurfacing of the crater floor. The radial wrinkle ridges predate both the circumferential ridges and graben in all but possibly one instance shown by the arrow in Figure 2a. In this case, two radial ridges terminate against different parts of the same circumferential ridge. Here the timing of formation is ambig- uous, since the radial ridge may either predate or postdate the concentric ridge. The circumferential ridges and graben do not exhibit any superposition relationships and therefore may have formed either concurrently or at different times. However, both must have formed subsequent to many of the resolvable radial ridges.

We suggest that the spatial and temporal distribution of this assemblage of tectonic features can be explained by subsidence of the floor of crater 1. Subsidence is a common

event in the evolution of calderas [cf. Williams, 1941; Smith and Bailey, 1968] and is often associated with depressurization of the underlying magma chamber. On Olympus Mons, depressurization could be attributable to flank eruptions [Mouginis-Mark, 1981] or simply to withdrawal of magma to lower levels in the volcano. In the case of crater 1, the amount of subsidence was probably a maximum in the center of the crater, since the (presumably) originally horizontal central region of the floor is now 600-1000 m lower than the perimeter of the crater floor [Mouginis-Mark and Robinson, 1992].

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Fig. 3. Elastic surface stresses arising due to an instantaneous pressure drop in an elliptical subsurface source vs. distance from crater center (r/Rc, where Rc is crater radius). The source has a width a = Rc, height c = 0.5a, depth beneath the surface d = 0.25R c, and maximum force vector within the source Fmc = 1 x 10 9 kg m s -2. The Young's modulus contrast between the source and surroundings (Emc/Es) = 0.01. Negative stresses are compressional and positive stresses are extensional.

Stresses Due to Subsidence

If the tectonic features within crater 1 formed as a conse-

quence of surface subsidence related to magma chamber deflation, then the distribution of the features should be consistent with the pattern of deformation implied from stresses arising from this process. Figure 3 shows the theoretical distribution of elastic surface stresses associated

with an instantaneous pressure drop within an ellipsoidally shaped source situated within a simple axisymmetric volcanic edifice. (Details of how the stresses were calculated are discussed later.) The source is intended to approximate a subsurface magma chamber. The distribution of surface deformation associated with a given stress field can be deduced if the signs and relative magnitudes of principal stresses are known [Anderson, 1951]. At the surface where the tectonic features are observed, shear stresses vanish, and the principal stresses are identical to the normal stresses plotted in Figure 3. Near the crater center (r/R c _< 0.5), the maximum compressive stress is the circumferential or hoop

stress (c r000, where the subscript 0 indicates stress at the surface (z - 0)). Interpretation in the context of the Anderson [1951] criteria for faulting and a critical value of stress necessary to overcome the frictional resistance of slip surfaces [Byerlee, 1978] predicts the formation of radial ridges in this region. In the outer part of the crater, where

Crrr ø is the least compressive and or00 ø the most compressive principal stresses, circumferential graben are predicted by interpretation of Anderson's [1951] criteria. The possibility that observed concentric graben are alternatively controlled by a tensile failure criterion for rock, in which the least

compressive principal stress (in this case cry0) exceeds some limit at the inner and outer limits of faulting, has also been explored [Comer et al., 1979; Solomon and Head, 1980; Comer et al., 1985; Hall et al., 1986], as has the assumption that surficial graben form where the maximum principal stress difference cr 1 - cr 3 (equivalent to Cr•r 0 - or000 in our model) exceeds a critical value [Gephart, 1987]. These criteria


predict surface stresses consistent with the observed transition of inner radial ridges to outer concentric graben. Thus, on the basis of the surface stress field associated with

deflation, we suggest that radial ridges near the center of the crater and circumferential graben at larger radii may be indicative of subsidence of the crater floor associated with

deflation of a subsurface magma chamber. Faulting, which presumably accompanies wrinkle ridge

formation [Plescia and Golombek, 1986; Watters, 1988; Golombek et al., 1991], will be characterized by a decrease in the magnitude of the stress in the vicinity of the faults, with radial ridges preferentially releasing the circumferentially directed component of the stress. Fracture mechanics models incorporating realistic material properties would ultimately be required to ascertain the magnitude and distribution of the stress drop. However, dislocation models of mode III cracks [Pollard and Segall, 1987] show that areas within approximately three crack lengths both parallel and perpendicular to the crack would be influenced by the crack stress field. This distance is larger than the spacings of radial ridges in crater 1. A reduction of stress along the radial reverse fault presumably associated with the ridge would thus be accompanied by a reduction in the hoop stress of the surroundings. A significant stress drop associated with the formation of the radial ridges would modify the pattern of

stress shown in Figure 3. For sufficient decreases in rr000, rrrr0 would become the maximum compressive stress in the inner part of the crater. Thus, circumferential ridges might be expected to form subsequent to radial ridges in the inner part of the crater, as is observed within crater 1. Solomon and Head [1980], though they did not specifically invoke the argument offered here, used the radial component of stress associated with surface loading to explain the formation of concentric wrinkle ridges within some lunar maria. In this scenario, concentric ridges that postdate the radial ridges in the interior of the crater may be explained by subsidence as well, though this scenario is admittedly based upon assump- tions regarding the nature of how rocks will respond to stress after initial failure, which is difficult to either observe or model.

We recognize the possibility that the tectonic features we identified were produced by a mechanism other than subsidence related to deflation, such as simple cooling of the proposed lava lake that comprises the smooth matedhals in which the tectonic structures formed. However, we note that the pattern of observed surface deformation within crater 1 is not consistent with the theoretically predicted distribution of thermal stresses induced in a uniformly cooling circular plate [Boley and Wiener, 1966]. In addition, this pattern of features is not observed in recently cooled lava lakes in Hawaii [Wright and Okamura, 1977; Peck, 1978]. Neither is it likely that convective shear stresses at the base of a cooling lava lake would produce stresses that can be consistent with the observed pattern of deformation.

If the tectonic features in crater 1 are indeed a conse-

quence of floor subsidence associated with magma chamber deflation, then their spatial distribution provides a constraint on the geometry and depth of the deflational source, i.e., the subsurface magmatic reservoir, at the time of the subsidence event. We adopt this working hypothesis and develop models of magma chamber deflation constrained by the transition from compressional radial ridges to extensional concentric graben within the caldera. The association of the circumfer-

r,u r=Rc

w=O

Fig. 4. Axisymmetric finite element grid used in the analysis. The grid contains 1827 nodes connecting 1878 quadrilateral elements. The left boundary is a symmetry axis on which the horizontal displacement (u) vanishes. On the bottom boundary the vertical displacement (w) vanishes, and on the right-hand boundary, both horizontal and vertical displacements vanish. The crater perimeter is located at r = R c. The grid is sufficiently large so that the boundary conditions do not influence the stress field in the vicinity of the summit.

ential ridges with the proposed subsidence event, while supported by superposition relationships, is not a require- ment of our model.

MECHANICAL MODEL

Method of Solution

To quantify the edifice stress field, we employed a finite element approach in which the volcano and its surroundings were approximated by a grid composed of a specified number of structural elements that are connected at nodal points. We used the program TECTON [Melosh and Raefsky, 1980] to develop a general linear elastic model that includes the effects of gravity and self-compression. For simplicity, we used an axisymmetric model.

In a finite element analysis, the nodal displacements, which we represent by a vector U, are the principal un- knowns and define the number of degrees of freedom in the problem [cf. Bathe, 1982]. This vector can be related to another vector F that contains the components of force acting at each node and a global stiffness matrix K by

F = KU (1)

where K was constructed from the elastic constants, element dimensions, and local force-displacement relationships. Equation (1) constitutes a set of simultaneous equations that was solved for a specified set of boundary conditions using a Gaussian elimination routine [Desai and Abel, 1972]. A quadratic interpolator was used in the computation of the nodal displacements.

Boundary Conditions

Figure 4 illustrates the boundary conditions on one of the grids used in the analysis. The grid represents the Olympus Mons volcano, with a radius of 300 km and height of 30 km and the underlying lithosphere down to a depth of 150 km beneath the base of the volcano. The following conditions were imposed: vanishing horizontal displacements (u) at the center of symmetry of the volcano (left boundary), vanishing vertical displacements (w) at depths much greater than the crater radius (bottom boundary), and vanishing horizontal and vertical displacements at radial distances far from the


Fig. 5a. Geometry of the magma chamber. Parameters of interest include the horizontal (a) and vertical (c) axial half lengths, and the depth beneath the center of the crater to the top of the chamber (d).

crater rim (right boundary). Numerical analyses were performed to assure that the calculated stress fields in the

vicinity of the caldera were not sensitive to any of the far field conditions. The boundary condition on the magmatic reservoir, which is discussed in more detail in the next section, was taken to be a force per length simulating a uniform change of pressure on the walls.

Magma Chamber Geometry

Constraints on the shapes of terrestrial magma chambers are based on observations of exposed plutonic complexes and models based on observed tectonics, structure, and eruptive characteristics [Chevallier and Verwoerd, 1988]. Structures range from simple ovoidal plutons of the Ker- guelen Archipeligo [Giret and Lameyre, 1983] to a complex network of interconnected dikes and conduits for the

Kilauea volcano [Ryan, 1988]. Due to the absence of subsurface information for Olympus Mons, the magma chamber was modeled as a simple ellipsoidally shaped source (Figure 5a). Inward directed point forces on nodes within an elliptical cross section were imposed to represent an instantaneous pressure drop. The cross section was characterized by a horizontal half axis of length a, vertical half axis of length c, and depth d to the top of the chamber at the center of the crater. Models with both constant and variable vector magnitudes were considered. In most of the numerical experiments it was assumed that each force vector within the

magma chamber was directed normal to the boundary of the chamber, as illustrated in Figure 5b. However, force distributions in which vectors were directed radially toward the center of the chamber, and those in which all forces were directed downward (parallel to the z direction) were also analyzed. Because of the enhanced temperatures in active terrestrial magma chambers indicated from the attenuation of seismic waves, it was assumed that elements within the model magma chamber were characterized by a lower value of Young's modulus than elements composing the surrounding volcano. The magma chamber geometry was chosen so

Fig. 5b. Distribution of point forces on nodes within the magma chamber. In this example, forces are directed normal to the boundary of the chamber.


TABLE 1. Typical Model Parameters

Symbol Parameter Value Units

Emc Young's modulus of magma 1 x 10 9 Pa chamber

Es Young's modulus of surroundings 1 x 10 TM Pa v Poisson's ratio 0.25

p density 2700 kg m -3 -2

g acceleration of gravity 3.71 m s Fmc magma chamber force vector 1 x 10 9 kg m s -2

correlates with the inner limit of circumferential graben. We then compared the stress patterns to the observed radial positions of concentric graben in crater 1 of Olympus Mons. The approach of using the radial component of stress to explain the positions of circumferential graben has previously been employed in analyses which estimated the effec- tive elastic thickness of the lithosphere in the vicinity of lunar maria and Martian volcanoes [Comer et al., 1979; Solomon and Head, 1980; Comer et al., 1985; Hall et al., 1986].

that its size, shape, depth, orientation, stiffness, density, and pressure distribution could be varied without modifying the grid, which allowed numerical experiments to be performed over a broad parameter space by simply varying input values. In addition, it was possible to construct solutions for even simpler magma chamber geometries than shown in Figure 5 (e.g., point and line sources). These solutions were directly compared to existing analytical [cf. Mogi, 1958] and finite element [Dieterich and Decker, 1975; Decker, 1987] solutions to test the accuracy of the finite element code and the reproducibility of numerical solutions from different codes. Table 1 lists typical material properties used in the analysis, though all parameters were varied to characterize the model sensitivity.

To constrain the characteristics of the magma chamber, we calculated surface stresses as a function of radius in the

crater that would be expected due to deflation of a magma chamber for a range of specified model parameters. We calculated the entire stress tensor but in order to most simply compare various model configurations, we plot only the radial component of stress. Our examination of the three normal stress components indicates that the crossover of trr•0 from compression to extension is indicative of the radial position of the onset of surface extension that presumably

RESULTS

Baseline Solution

We first consider the case of a magma chamber with an aspect ratio of c/a - 0.5 and Young's modulus contrast between the magma chamber and surroundings of Emc/E s -- 0.01. The chamber width was assumed to equal the width of the crater (a - Rc- 32 km), as is generally observed for terrestrial volcanoes [e.g., Ryan, 1988]. Figure 6 plots the

distribution of the radial surface stress, trr•0, as a function of radius, r/R c, from the center of the crater (where r/R c - 1 is the crater perimeter) for a range of d. At shallow depths

(e.g., d - 0.125Rc), trrr 0 is compressional and large in magnitude close to the center of the crater. With increasing

distance from the center, the magnitude of trrr ø decreases, and at about halfway to the crater perimeter it becomes extensional. At progressively increasing depths, the dy- namic range of surface stress magnitudes decreases, and the transition from compression to extension shifts to greater distances from the crater center. At shallow depths, a specified change in the depth of the magma chamber corresponds to a much smaller shift in the compression to extension crossover point than at greater depths. Conse-

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I I I I

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-7oo I I I I I I I I 0 0.2 0.4 0.6 0.8

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Fig. 6. Radial surface stress (Crrr 0) due to magma chamber deflation as a function of distance from the crater center (r/Rc, where R c is the crater radius of 32 km). Stress patterns are shown for a range of depths to the top of the chamber (d). Assumes a chamber with width a = Rc, height c = 0.5a, maximum magma chamber force vector magnitude Fmc = 1 x 10 9 kg m s -2, and Young's modulus contrast between chamber and surroundings Emc/E s = 0.01. For these parameters, the theoretical distribution of stresses that best fits the observed pattern of tectonic features (transition from radial ridges to circumferential graben at r --• 0.53Rc) occurs for d = 0.25Rc = 8 km.

18,302 ZUBER AND MOUGINIS-MARK.' OLYMPUS MONS CALDERA SUBSIDENCE AND MAGMA CHAMBER DEPTH

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Fig. 7. Plot of rrrr0 versus r/Rc for a range of magma chamber aspect ratios (c/a) assuming a = Rc, d = 0.5Rc, Fmc = 1 x 10 9 kg m s -2 and Emc/E s = 0.01.

quently, the model is not very sensitive to the observational constraints for very shallow depths (--• 1 km).

The magnitudes of the stresses are most strongly a function of the assumed Young's moduli (Es and Emc ) and force magnitude of the deflating magma chamber (Fmc). For the parameter values assumed in Figure 6 (Es = 1 x 10 • Pa, Emc = 1 x 109 Pa, and Fmc = 1 x 109 kg m s-2), both the compressional and extensional radial stress magnitudes at- tain values sufficient to cause failure in intact rock through- out much of the radial extent of the crater. In crater 1 of

Olympus Mons, the distribution of ridges and graben indicate that the transition from radial compression to extension occurs at a radial distance from center (r) of approximately 0.53R c (17 km [Zuber and Mouginis-Mark, 1990]). For the parameters assumed in Figure 6, the magma chamber depth that most closely corresponds to the observations is d = 0.25Rc (8 km), though solutions with d up to 0.5R c are allowable when model uncertainties are taken into account.

We next examine the sensitivity of the model to various parameters and demonstrate that the solution shown in Figure 6 likely represents a maximum depth to the top of the magma chamber.

lipsoidally shaped chamber, in which the half axes of the magma chamber were assumed to be tilted with respect to the r and z coordinate axes. While this shape is idealized and not physically justifiable, it was easily implemented and served to illustrate the effect of varying magma chamber shape on the best-fit depth. For this solution, a positive tilt angle 0 corresponds to a counterclockwise rotation of the chamber axes and a negative angle 0 corresponds to a clockwise rotation. Figure 8 shows that positive and negative tilts cause the transition from compression to extension to shift slightly farther from and closer to the center, respectively. However, for the range of tilt angles examined the change in the crossover distance is insignificant. Unless the magma chamber was highly inclined with respect to the horizontal, chamber orientation is not an important factor in constraining the best fit depth.

Stiffness Contrast

The sensitivity of the solution to the contrast in Young's modulus between the magma chamber and surroundings (Emc/E s) is shown in Figure 9. Figure 9 demonstrates that if Emc < Es, the depth of the magma chamber is essentially insensitive to this ratio. For Emc/E s -- 1, the transition from compression to extension shifts to a greater distance from the center and would require a shallower magma chamber than determined in Figure 6 to explain the observed pattern of tectonism. The case in which Emc/E s > 1, which corresponds to a magma chamber that is more stiff than its surroundings, is characterized by an even greater transition distance. However, this situation is not physically realistic.

Magma Chamber Pressure Distribution

In addition to assuming magma chamber point force vectors directed normal to the wall of the magma chamber (Figure 5b), we also calculated solutions in which vectors

5O

Magma Chamber Aspect Ratio

Figure 7 shows that the best-fit depth is not very sensitive to the aspect ratio of the chamber for c/a < 1, which corresponds to an oblate ellipsoid (semimajor axis in the x direction). For c/a > 1, corresponding to a prolate ellipsoid (semimajor axis in the z direction), the transition from radial compression to extension shifts to greater distances. There- fore, if the magma chamber is prolate, it must have been either shallower or narrower than determined in Figure 6 at the time of crater floor subsidence to explain the observed transition from radial compression to extension. Since it is unlikely that the chamber is significantly narrower than the collapse crater [cf. Marsh, 1984], a shallower magma chamber would be implied.

Magma Chamber Orientation

In order to investigate the stress field associated with a tilted magma chamber, we examined a model with a nonel-

o:1 -5o Q_

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Fig. 8. Plot of rrrr0 versus r/Rc for a range of magma chamber orientations (0) assuming a = R c, d = 0.125R c, c/a = 0.5, F c = 1 x 10 9 kg m s -2 Emc/E s = 0.01 and Fmc = 1 x 10 9 kg m s-n•. The orientation 0 ='0 ø corresponds •o a magma chamber with axes orientated along the x and z coordinate axes. Positive 0 corresponds to a counterclockwise rotation of the magma chamber axes and negative 0 corresponds to a clockwise rotation of the axes.

ZUBER AND MOUGINIS-MARK' OLYMPUS MONS CALDERA SUBSIDENCE AND MAGMA CHAMBER DEPTH 18,303

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-500 I I I I I I I I 0 0.2 0.4 0.6 0.8

r/R c

Fig. 9. Plot of •rrr0 versus r/Rc for a range of Young's modulus contrasts between the magma chamber and surroundings (Emc/Es) assuming a = R c, d = 0.125Rc, and c/a = 0.5. In this solution, E s is held fixed. Notice that the transition from compression (•rrr negative) to extension (•rrr positive) is not sensitive to this ratio except for the unrealistic case of Emc/E s • 1.

were oriented radially from center and in which all vectors were directed downward along the z axis. The crossover points of the radial stress for these cases were found to be virtually indistinguishable from the case in which the vectors were oriented normal to the chamber wall. We also consid-

ered the potential importance of the magnitude of magma chamber pressurization. This was done by varying all force vector lengths by the same amount. Figure 10 shows that this affects the magnitude of the radial stress. However, the distance from center of the transition of radial compression to extension, and the depth of the magma chamber implied by the transition, remain unchanged. We also investigated a variety of distributions of point force magnitudes. In addition to the case of uniform magnitudes at all nodes within the chamber shown (Figure 5 b), we also calculated solutions for models in which magnitudes were dependent on the distance from the center of the chamber. For some cases, changing

lOO

-lOO

-200

-4oo x 1,,• Fmc = 1 X 109 kg ms '2 -5oo •1 I I I I

0 0.2 0.4

r/R c

I I I I

0.6 0.8

Fig. 10. Plot of %r0 versus r/Rc for a range of magma chamber force vector magnitudes (Fmc) assuming a = R c, d = 0.125Rc, c/a = 0.5, and Emc/Es = 0.01. The plot shows that the transition from compression (Crrr0 negative) to extension (•rrr0 positive) is insensitive to this parameter.

Fig. 11. Plot of •rrr0 versus r/R c for a range of magma chamber widths (a) assuming c = d = 0.25Rc and Emc/Es = 0.01. Note that a wide magma chamber undergoes the transition from compression (negative) to extension (positive) at greater distances from the crater center than a narrow magma chamber. Therefore a chamber that is wider than the crater (a/Rc • 1) must be shallower than a chamber that is less than or equal to the crater width (a/Rc -• 1) to explain the transition from compressional to extensional deformation at r • 0.53Rc in crater 1 of the Olympus Mons caldera.

the distributions of magnitudes of the point forces modified the magnitudes of the surface stresses and some details of the shape of the radial stress distribution, but in no cases did

these changes significantly alter the transition of %r0 from compression to extension. We thus conclude that the best fit depth range is not strongly sensitive to the details of the imposed pressure distribution as long as the chamber is depressurizing.

If the magma chamber were instead overpressured, i.e., inflating instead of deflating, then the signs of the stresses in Figure 3 would be reversed. The center of the crater would be in extension rather than compression and the crater periphery would be in compression. Such a state of stress is not consistent with the observed distribution of tectonic

features within the Olympus Mons caldera, indicating that the magma chamber could not have been significantly overpressured at the time the tectonic features formed.

Magma Chamber Width

Figure 11 demonstrates that the depth of the magma chamber is highly sensitive to chamber width. Again, the magma chamber is unlikely to have been markedly narrower than the crater [Marsh, 1984]. If it was wider, then the transition from compression to extension would have oc- curred at a greater radius from the crater center than for the scenario in Figure 6 where the magma chamber radius equals the crater radius. For a wider magma chamber, a shallower maximum depth than determined for the parameters assumed in Figure 6 is implied.

DISCUSSION

Influence of Far-Field Stresses

In all of the solutions discussed above we assumed a

continuity of stresses across the crater edge. We did not explicitly include in the model a steeply-dipping bounding


fault, which is commonly observed in terrestrial calderas [Williams and McBirney, 1968; Smith and Bailey, 1968]. It is not clear to what extent stress would be transmitted across

such a structure, so we consider the limiting case of a perfectly frictionless fault which effectively isolates the crater floor from the surrounding volcanic edifice. Analytical solutions for loading of circular plates [Timoshenko and Woinowsky-Krieger, 1959] show that the radial stress for a plate that is welded to its surroundings will be modified from that for a freely floating plate if a remote stress is transmitted across the boundary. However, for remote stresses to be important, they must be similar in magnitude to those for magma chamber depressurization and accumulate on comparable timescales.

One possible source of remote stress is flexure of the underlying lithosphere due to the load of the volcanic edifice. At times shortly (compared to the Maxwell time of the asthenosphere) after an episode of significant shield building, the near-summit stress field would not be significantly al- tered by flexure [McGovern and Solomon, 1990]. However, at later times, the most compressive stress axis near the summit becomes nearly horizontal [McGovern and So- lomon, 1990], which would cause the radial stress calculated in our models to become more compressional. Superposition of stresses due to depressurization and flexure would shift the transition from compression to extension farther from the center of the crater, which would require a shallower magma chamber than indicated by Figure 6 to explain the observed transition from compression to extension. How- ever, the time scale for the accumulation of flexural stresses is likely to have been much greater than that for magma chamber depressurization. For flexure to have contributed to modifying the stress field examined in this study, depressurization would have had to occur at a presently indeter- minate time after flexural stresses had begun accumulating.

Implications of Model Results

Solutions for ellipsoidal chambers with a variety of force distributions as well as those invoking simple line sources do not yield significantly different stress crossover points. This result, in combination with the insensitivity of the model to a wide range of chamber aspect ratios (Figure 7), unfortunately indicates that the state of stress implied by the surface tectonics does not provide useful constraints on the relative geometry, volume, or degree of depressurization of the magma chamber. Determination of these quantities would require detailed models of eruption mechanics [e.g., Spera, 1984; Wilson and Head, 1988] and geodetic measurements of surface displacements [e.g., Dvorak et al., 1983], neither of which are available for Martian volcanoes.

For the range of models that we have examined, parameter uncertainties tend to shift the calculated depth of the top of the magma chamber to shallower levels than indicated by our baseline model (Figure 6). This trend holds for plausible ranges of values of density and Poisson' s ratio. We therefore consider a value of 0.5Rc (16 km) to be an upper limit of the depth of the magma chamber. Because of the nature of the observational constraints and the model behavior, it is not possible to reliably define a lower limit, so we conservatively assume that all depths -< 16 km are allowable. The summit of Olympus Mons rises 27 km above its surroundings [Wu et al., 1986]. Thus if concentric ridges and graben within crater

1 of the Olympus Mons caldera complex formed as a consequence of subsidence related to magma chamber withdrawal, then at the time of subsidence the magma chamber must have been located within the volcanic edifice.

Analyses of magma chamber depth for a number of terrestrial intraplate volcanic islands indicate a shallow reservoir between 2 and 4 km beneath the summit [Chevalier and Verwoerd, 1988]. In addition, magma chamber depths for Iceland's Krafla [Tryggvason, 1986], and Hawaii's Mauna Loa [Zucca et al., 1982] and Kilauea [Ryan et al., 1983; Decker, 1987; Klein et al., 1987; Thurber, 1987; Ryan, 1988; Delaney et al., 1990; Yang et al., 1992] volcanoes are approximately 2-5 km. In a volcano, magma ascending from partial melt zones at depth decreases in density due to decompression and, at shallow depths, volatile exsolution. The density of volcanic country rock increases with depth due to compaction associated with the weight of the over- burden. The depth of terrestrial magma chambers has been interpreted to represent a state of neutral buoyancy at which the density of ascending magma equals the density of the surroundings [Rubin and Pollard, 1987; Ryan, 1987; Walker, 1988]. Since the distribution of compaction with depth depends on the lithostatic pressure, the depth to a given compaction state will scale inversely with gravity [Wilson and Head, 1990]. With appropriate scaling for the difference in gravity between Mars and Earth, the 2-5 km range of terrestrial depths would correspond to an approximate range of 5-13 km for Martian magma chambers. This gravity- scaled depth range is in good agreement with our calculated depth range of d -< 16 km, which suggests that if the densities of melt and country rock on Mars are generally similar to those on Earth, then the depth of the Olympus Mons magma chamber may have been controlled by neutral buoyancy as well. The allowable range of magma chamber depths is much shallower than the estimated depth of Mar- tian primary melts of 140-200 km [Carr, 1973; Blasius and Cutts, 1976; Thurber and Toksoz, 1978; Bertka and Hollo- way, 1987]. This result, in combination with photogeologic evidence for multiple eruptions and summit collapse episodes [Greeley and Spudis, 1981; Mouginis-Mark, 1981; Mouginis-Mark et al., 1992], suggests that magma which formed in a partial melt zone at depth was transported to a shallow subsurface reservoir where it fed intermittent eruptions. If this scenario is viable, then the structure and magmatic transport process of Olympus Mons may be fun- damentally similar to those of terrestrial volcanoes such as the Hawaiian shields.

Terrestrial Subsidence Studies

One of the best-known examples of tectonic features associated with deflational subsidence of a subsurface magmatic reservoir on Earth are the "ring dikes" that have been mapped within the Cullin intrusive complex in Skye, Scot- land [Anderson, 1936]. Ring dikes consist of narrow, out- ward sloping, concentric intrusions that surround individual plutonic complexes. These structures were hypothesized to have formed along zones of maximum shear stress associated with deflation of a subsurface magmatic reservoir. The peripheral graben within crater 1 of Olympus Mons are characterized by a sense, orientation, and spatial position that could plausibly correspond to a surficial manifestation of the ring dikes described by Anderson. However, the

ZUBER AND MOUGINIS=MARK: OLYMPUS MONS CALDERA SUBSIDENCE AND MAGMA CHAMBER DEPTH 18,305

ground surface on which the ring dikes are exposed at Skye is erosional and thus does not correspond to the surface that existed at the time of the deflation event. Anderson [1936] expressed uncertainty concerning whether the ring fractures in the Skye region extended to the ground surface at the time of formation. In contrast, the floor of crater 1 at the summit of Olympus Mons shows little evidence of erosion, from which we infer that features currently at the surface formed in its vicinity. Consequently, it is not clear whether or not the features within crater 1 formed in a manner analogous to those at Skye.

Most other terrestrial deflational subsidence studies focus

on surface displacements rather than surface stresses [e.g., Mogi, 1958], as the former observation is more diagnostic of a particular subsurface source geometry. Unfortunately, ground displacement data is not available for volcanic structures on Mars.

Summit Tectonics of Other Major Martian Shields

The summit area of the volcano Alba Patera is perhaps the second most interesting locus of volcanic edifice tectonism on Mars. Despite the low elevation of this structure compared to other major shields, the appearance of the caldera complex is essentially shield-like [Wood, 1984]. Multiple collapse events were associated with the formation of two discrete calderas [Mouginis-Mark et al., 1988]. Large wrinkle ridges are found both on the caldera floor and radial to the summit region. Circumferential graben also bound the approximate border of the summit as defined by the two calderas. Without good topographic information, however, it is not possible to infer the locations and/or magnitudes of subsidence events for the volcano. Such a study should be possible with topographic information to be obtained in the upcoming Mars Observer Mission [Zuber et al., 1992]. Pavonis Mons also displays radial ridges within its caldera and circumferential graben outside the rim [Crumpier and Aubele, 1978]. The positions and orientations of some of the ridges and graben of these shields can be consistent with a stage of postcollapse subsidence comparable to that which we postulate for Olympus Mons. The summit caldera of Ascraeus Mons consists of eight separate craters. The floor of the largest and youngest crater contains very small ridges of probable tectonic origin, while the edifice just outside the crater rim exhibits mare-type wrinkle ridges and circumferential graben [Mouginis-Mark, 1981]. The summit caldera of Arsia Mons is infilled with flows of probable volcanic origin and no tectonic features are visible on the floor, though circumferential graben occur outside the rim [Crumpier and Aubele, 1978]. Calderas of Hadriaca Patera, Amphitrites Patera, and Syrtis Major exhibit similar structures to Arsia Mons, although it is not clear whether they have been infilled by lava flows or other materials [Mouginis-Mark et al., 1992]. However, none of these caldera complexes exhibits a clear transition from compression to extension that would permit magma chamber depth to be constrained using the model employed in this analysis. Since many calderas have undergone subsidence during their evolution, the question arises as to why Olympus Mons uniquely exhibits tectonic evidence for this process. At least part of the answer may be related to the relative timing of initial collapse, resurfacing of the crater floor, and subsidence. In addition, the mechanical properties of the summit region and the geometry, mechan-

ical properties, and manner of depressurization of the magma chamber may have contributed.

SUMMARY

If certain ridges and graben, shown in Figure 2, formed as a consequence of subsidence of the floor of the Olympus Mons caldera in response to deflation of the underlying magma chamber, then the stress field implied by these features can be used as a constraint on models to investigate the gross characteristics of the chamber. Application of a linearly elastic, axisymmetric finite element model to deter- mine elastic stresses in the vicinity of a volcanic summit demonstrates that a broad range of magma chamber aspect ratios and pressure distributions can produce surface stress fields consistent with the observed pattern of tectonic features. Therefore the relative geometry, volume, and details of the pressurization of the magma chamber cannot be confidently constrained on the basis of stresses alone. How- ever, the surface stress field is sensitive to the width and depth of the chamber; the width of the chamber trades off with its depth such that a wider chamber implies a shallower depth. The constraint that the chamber cannot have been significantly narrower than the crater, and observations of the distribution of tectonic features within the crater, constrain the depth to the top of the Olympus Mons magma chamber at the time of subsidence to have been _< 16 km.

This depth range suggests that if the densities of melt and country rock associated with terrestrial and Martian shield volcanoes, respectively, are similar, then the chamber formed at the level where ascending magma and surrounding rocks were gravitationally stable. A magma chamber at this depth would have been located within the Olympus Mons edifice, at a level much shallower than the probable source depth of Martian primary magmas. Thus the general model for the internal magmatic plumbing system of many terrestrial volcanoes [Eaton and Murata, 1960], i.e., a deep source of partial melt which ascends to a shallow magma chamber that fuels episodic eruptions, may describe Olympus Mons as well.

Acknowledgments. We thank Lionel Wilson and Bruce Marsh for helpful discussions, Mark Robinson for the photoclinometric measurements of caldera topography, and George Wyatt for writing the graphics display software. We also thank Jay Melosh for making available the TECTON finite element code and for answering numerous questions concerning its use. MTZ acknowledges support from the NASA Planetary Geology and Geophysics Program and The Johns Hopkins University Space Grant Consortium. P.M.-M. was supported by grant NAGW-437 from the NASA Planetary Geology and Geophysics Program.

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P. J. Mouginis-Mark, Planetary Geosciences Division, Depart- ment of Geology and Geophysics, School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, HI 96822.

M. T. Zuber, Geodynamics Branch, Code 921, NASA Goddard Space Flight Center, Greenbelt, MD 20771.

(Received February 5, 1992; revised July 14, 1992;

accepted July 15, 1992.)

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