5.3. Numerical Modeling of Withdrawn Magma Volume
 The magnitude of deflation that induced surface collapse may be numerically investigated by comparing the volume of the deflated magma and the volume of the void generated at surface. Figure 24 shows two end-members volume transfer mechanisms that are accounted for in the models. In a closed system, the whole magma body is the deflated lens, and there is no magma replenishment. In an open system, the magma lens is connected with a reservoir that begins replenishing as soon as significant magma underpressurization occurs (Figure 24). In the first step, limited crustal stretching traps magma at depth, which in turn leads to overpressure in the magma lens and helps focus tectonic strain at surface (step 1). Dike emplacement (not shown on Figure 24) occurs between the magma lens and the surface in such a way that the total dilation remains vertically constant. Gradually magma overpressure decreases (step 2). As long as magma pressure is positive, the erupted magma volume does not decrease the magma lens volume, and there is no surface collapse. When the magma pressure falls below zero, the differential stress in the host rock increases to a critical value σc equal to rock strength (step 3). Then the whole host rock body above σc fails and collapses into the deflated lens (step 4). The volume of the unstable host rock may be equal to the volume of the collapse trough that eventually forms at surface. In the closed system considered, the volume of destabilized host rock material as deduced from the volume of the surface trough is thus a direct measure of the deflation magnitude. If the magma body is being replenished, the volume of destabilized host rock, as deduced from the volume of surface trough, is an underestimate of the magnitude of deflation.
 In order to extrapolate surface trough volume to the volume of destabilized materials above the magma lens, the fraction of the measured trough volume that formed initially during the early tectonic stretching episode should be removed. However, the contribution of tectonic stretching is thought to be small/negligible compared with the volume created by volcanic collapse, both because strain estimates at unmodified narrow grabens shows very low strain [Plescia, 1991; Golombek et al., 1994], and because part of the void created this way should have been filled by early lava flows (Ve1 on Figure 24).
 In the case of a single deflation event, the expected collapse mechanism is rapid stoping over the reservoir [Roche et al., 2000]. However, in some cases the volume of the surface trough may result from successive withdrawal events, and the sum of the pressure drops equals the pressure drop in case of a single larger collapse event.
 In order to constrain the magnitude of underpressure Pm we use the trough volume data obtained from morphometric analysis of the DEMs (Table 1). We assume that the volume of the magma lens roof remains constant during surface collapse, and compute the magma underpressures using a boundary element code.
 First the extent (width B and length L) of the deflating part of the reservoir is defined assuming that it is not larger than the extent of the bottom of the trough above (Table 2). Its shape is flat bottomed lens with a convex top aligned with the trough. This geometry is consistent with a buoyant magma generated by adiabatic decompression, or laterally flowing from a remote source region, and trapped at a rheological boundary below a slightly thinned brittle crust (Figure 25). In most cases its maximum thickness t is taken to be 1 km. This value is usually close to the minimum thickness required, for the volume that can be made available in the lens, to exceed the volume of the collapsing hot rock material. However, if magma reservoir replenishment occurs before the end of the collapse event then the volume of the reservoir may not be correlated with surface trough volume. In the Valles Marineris region, some reservoir thicknesses values, t, were taken to more than 1 km when taking 1 km resulted in magma underpressures that were so large that the magmatic processes involved might need to significantly differ from those in the other areas. The magnitude of the pressure drop is obtained at various depths e. For the most voluminous collapse troughs, thought to correspond to very deep reservoirs, e may be at the level of neutral buoyancy, 10 km [Wilson and Head, 1994]. Compute3D (Rocscience, Inc.) was selected for its ability at measuring the volume of host rock material below or beyond any stress.
Table 2. Reservoir Parameters Consistent With Observed Trough Volumesa
|N°||L, km||B, km||e, km||t, km||σc, MPa||Pp/Pl, MPa||ΣPm, MPa||Vt, km3|
 For a given reservoir size (L, B, t, e), the magma pressure Pm is initially set to the vertical stress. Then its magnitude is increased until the volume of the host rock, beyond the in situ rock strength, approximates the volume of the trough. The model consists of an elastic half-space subject to gravitational forces. Remote tectonic stress is set to 0 in order to visualize the structural effect of deflation alone. We investigate two extreme vertical stress conditions. In some experiments pore pressure Pp is assumed to be null. The vertical stress is thus ρge, where 3.72 m.s−2, is taken to 2900 kg.m−3, g = 3.72 m.s−2, and e the depth of the top of the magma body. In other experiments we conducted, the pore pressure was considered to be as high as ρge/2 (Table 2). Based upon reasonable assumptions about the strength envelope of the Martian crust [Banerdt et al., 1992], and with inferences of elastic lithosphere thickness on Mars from MOLA data [Zuber et al., 2000], the ductile level that possibly exists in the deep crust is assumed to be deeper than the modeled part of the crust. Stress amplification induced by the possible existence of a ductile crust [Kusznir and Bott, 1977] is not taken into account. Young's modulus is taken to be 40 GPa and the friction angle is 30°. The computations are performed in a grid that covers one quarter of the model and extrapolated to the whole model. Node spacing is not a critical issue in the models because we are not interested in the details of the tectonic processes during subsidence, but on large-scale rock volumes. We found that > 70,000 nodes are usually enough to obtain reasonably accurate results.
 We determined that the cumulated underpressure ΣPm required to explain the trough volumes ranges between tens of MPa and 350 MPa, depending on pore pressure and reservoir size and depth. Figure 26 gives an example of a trough from the Calydon Fossae, SW of Valles Marineris, whose volume may be explained by this range of underpressures in a reservoir located at 5 km depth. For the smallest troughs the reservoir thickness needs to be > 2 km for the volume of the host rock above failure not to exceed the volume of the reservoir and hence allow trough formation in a single collapse event with negligible reservoir replenishment. This results from the lower width/depth ratio for small troughs than for more voluminous troughs such as chasmata. The reservoir below the smallest measured trough (12 on Table 2) needs to be at >3–4 km thick. This is based upon the assumption that the reservoir width (top 1 km at the top, 2 km at the bottom) is on the order of the trough width (1 km). It would still require a reservoir at least 2 to 3 km thick if reservoir width 2 km (top) to 5 km (bottom).
Figure 26. Example of deflation computation results for one of the Calydon Fossae troughs, SW of Valles Marineris (trough 10 in Tables 2 and 3). The gray levels give σ1–σ3 for a model in which reservoir depth is 5 km and extends as long as the chasma extent. The grayscale does not apply to the stress arrows, which are almost everywhere between 0 and 30 MPa except close to the reservoir. Major arrow axis plane follows σ1, and σ3 is perpendicular to the arrow plane. The curved plane is differential stress isocontour 26 MPa, the stress level at which the host rock located near the top of the reservoir exceeds the strength envelope selected by Banerdt et al. .
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