4.1. Terrestrial Subglacial Eruptions
 Subglacial eruptions on the Earth are perhaps best recognized by the nature of table mountain structures such as those located in Antarctica [Smellie and Skilling, 1994; Skilling, 1994; Smellie, 2000, 2001], British Columbia [Mathews, 1947, 1951], Alaska [Hoare and Coonrad, 1978], and Iceland [Nielson, 1937; Van Bemmelen and Rutten, 1955; Einarsson, 1966] and by the rapid drainage of large volumes of meltwater through jokulhlaups [Major and Newhall, 1989], most prominent in Iceland [Bjornsson, 1974, 1975, 1977, 1992; Gudmundsson et al., 1995; Jonsson, 1982; Nye, 1976; Thorarinsson, 1957; Tomasson, 1974, 1996]. However, the shape of the construct developed from eruption into an ice sheet is not restricted to the low, flat-topped mounds of table mountains [Nielson, 1937; Einarsson, 1966; Hickson, 2000]. Conical features can also result from eruption into an ice sheet [Nielson, 1937; Mathews, 1947; Hickson, 2000]. These structures are known as subglacial mounds. As with most terrestrial analogs to Martian structures, table mountains (also referred to as tuyas) and subglacial mounds on Earth are significantly smaller than the Martian features we interpret to share a similar genetic origin. Terrestrial subglacial mounds are documented as reaching heights of up to 1 km and volumes of only a few cubic kilometers [Hickson, 2000].
 The morphological form of the structure built by subglacial eruptions is controlled by a number of factors [e.g., Wilson and Head, 2001b], such as the thickness of the ice sheet [Smellie and Skilling, 1994], type of ice sheet (polar versus temperate), the volume of meltwater produced and confined by the ice [Smellie, 2001], type of eruption (localized vent versus fissure eruption), size of cauldron formed on the ice surface, composition of the magma, rate of eruption, and volume of magma erupted [Hickson, 2000; Smellie, 2000]. Such differences in morphology are documented even for the same eruption, the 1934 eruption of Grimsvotn, Iceland, where two eruptive centers within a few kilometers of each other produced edifices of different morphologies [Nielson, 1937].
 Much of a subglacial eruption actually takes place subaqueously. A general sequence of events for a subglacial eruption is as follows (e.g., Figure 12). During the initial stages of eruption, large volumes of meltwater are produced by the increase in magmatic heat. This leads to formation of a depression in the surface of the overlying ice [Bjornsson, 1974; Gudmundsson et al., 1997]. Glacier hydrology is predominantly controlled by the topography of the ice surface [Nye, 1976; Benn and Evans, 1998]. The depression in the ice surface will cause the meltwater to drain toward the site of eruption and to collect around the forming edifice as an englacial lake [Bjornsson, 1975; Smellie, 2001]. The englacial lake is enclosed on all sides by the ice wall barriers.
 Further eruptions occur into the englacial lake, which quenches the lava, makes it less mobile, and causes it to either crumble or form pillows [Gudmundsson et al., 1997; Smellie and Skilling, 1994; Skilling, 1994; Smellie 2000, 2001]. The quenched lava acts as a good insulator, keeping further erupted lava out of contact with the ice. The easiest way for lava to flow in this englacial lake is sideways, where it is confined by the ice walls, thus causing the mound to have high width-to-height ratios and relatively steep slopes [Einarsson, 1966]. If the mound reaches a height sufficient to breach the level of the englacial lake, effusive subaerial capping flows will occur, leading to a relatively flat top, making the edifice less susceptible to erosion [Smellie and Skilling, 1994; Skilling, 1994; Smellie 2000, 2001]. If the eruption turns subaerial, pyroclastic activity can also occur, and a cone of material can be formed on top of the otherwise low, and often flat-topped, mound [Allen, 1979]. This pyroclastic activity could be due to either release of volatiles from the magma or phreatomagmatic interactions with the surrounding meltwater. Alternatively, it may date to a time after the edifice has been constructed when the ice sheet has retreated [Allen, 1979]. If the surface of the englacial lake is of sufficient height not to be breached, an entirely conical feature can be formed [Hickson, 2000]. Important in the constructional history of the volcano are the stability of the englacial lake and how the meltwater is drained.
 The general structure of an ice sheet is that of an impermeable ice layer overlain by a permeable layer composed of snow and firn. The permeable layer is referred to as the passage zone [Smellie and Skilling, 1994]. On the Earth the passage zone is located within ∼150 m of the ice sheet surface [Smellie, 2001]. There exist two main mechanisms by which the englacial lake can drain, discussed in detail by Smellie and Skilling  and Smellie  and briefly summarized here. The first mechanism involves floating of the surrounding ice sheet. If the hydrostatic pressure at the base of the englacial lake reaches the magnitude at the base of the surrounding ice, the ice sheet will be lifted off its bed, and the meltwater will drain subglacially, underneath the ice along the basement topography, and will carve out channels. This is presumably the mechanism by which the majority of meltwater drains from underneath the Vatnajokull ice cap in Iceland [Bjornsson, 1974, 1975, 1992]. In order for this mechanism to occur, the height of the englacial lake required to reach the critical hydrostatic pressure must be less than the height required to reach the passage zone. If the passage zone is reached before the critical hydrostatic pressure is reached, the second mechanism of drainage results. Meltwater accumulates within the ice chamber, increasing the height of the englacial lake. When the level of the passage zone is reached, the overflow of meltwater drains supraglacially, through the permeable firn and snow. This leads to a more stable lake level than if drainage were to occur subglacially. Some subglacial drainage may still occur occasionally if lava melts the ice at the base of the ice sheet [Bjornsson, 1975; Bjornsson and Kristmannsdottir, 1984].
 Supraglacial drainage through the passage zone requires a relatively thick permeable layer and is thus thought to be the predominant mechanism of drainage in thick glaciers and ice sheets. Estimates of the thickness of the Hesperian-aged ice sheet performed by Head and Pratt  suggest that in places it was as much as 2500 m thick. Also, the lower gravity of Mars relative to that of Earth would suggest that there would be less compaction of snow into ice in a Martian ice sheet, and thus a Martian ice sheet would be expected to have a thicker passage zone relative to a terrestrial one. Together, the estimates of the ice sheet thickness, as well as the lower gravity of Mars, suggest that the predominant mechanism of drainage of meltwater produced from eruption into the Hesperian ice sheet would have been supraglacial drainage through the passage zone.
 After the subglacial eruptions have ceased, meltwater can still accumulate from the increase in geothermal heat near the erupted structure [Bjornsson and Gudmundsson, 1993; Gudmundsson et al., 1995]. Thus jokulhlaups can continue to form, leading to continued drainage of water along the basement topography and the carving out of channels. Once the geothermal heat level returns to pre-eruption conditions, the ice is able to reform around and upon the volcano. Such glacial activity could lead to erosion of the flanks of the edifice [Van Bemmelen and Rutten, 1955]. As pointed out by Chapman and Tanaka , the postconstructional environment provides ample opportunity for degradation of the ideal tuya form. The identification of well-preserved examples may be indicative of an ice sheet that was melted and did not reform.
 The heights of the mountains can be used as an estimate of the thickness of the ice sheet at the time of construction of the volcanoes. The conically shaped mountains would have erupted entirely within the englacial lake, while the flat-topped varieties would have just breached the surface of the lake. Since the lake surface corresponds to the level of the passage zone, which is below the surface of the ice sheet, estimates made from the heights of the volcanoes would be minimum estimates. These observations provide a basis for comparison to the characteristics of the Martian mountains interpreted to be volcanoes erupted in the proximity of the Hesperian-aged ice sheet [Head and Pratt, 2001].
4.2. The Mountains
 Many of the morphological differences observed between the mountains and Martian subaerial volcanoes of similar size could be explained by the processes involved in subglacial eruptions, as described above (Figure 12). The high width-to-height ratios of the mountains (Figure 11), along with their relatively steep slopes, and in some cases flat tops, are very similar to those predicted by the sequence of events outlined for a subglacial eruption (Figure 12).
 Some of the mountains display a conical feature on top of an otherwise flat-topped summit (mountains 6 and 11) (Figure 5). Such a conical feature could be accounted for by the transition in the eruption from a subaqueous phase to a subaerial phase, or more simply from subaerial eruptions at a time when the ice sheet had retreated from the area [Allen, 1979]. Most of the mountains do display a conical shape, although some are irregular in outline. How is this morphology accounted for among the others attributed to subglacial origin? As documented by Mathews , there are several conical volcanoes interspersed among table mountains in northern British Columbia. These conical features likely erupted at a time when the overlying ice was sufficiently thick that breach of the englacial lake was not possible by the erupting structure. In such a situation the ice walls would have confined accumulation of the pillow material and hyaloclastites. A conical feature would result and is a likely scenario for the origin of the more conical and irregularly conical mountains under discussion here. The cone-shaped mountains all fall within the center of the regional deposits of the Dorsa Argentea Formation (Figure 7). Such locations are consistent with the origin just outlined, as the formerly large Hesperian-aged ice sheet [Head and Pratt, 2001] would be expected to be have been thicker toward the pole and thinner farther away. The only exception is mountain 17, which, as discussed earlier, is a likely candidate for an impact crater central peak.
 An interesting terrestrial analog for the spacing and alignment of the mountains is a trio of volcanic edifices erupted subglacially in Iceland (Figure 13). This trio consists of Kverkfjoll in the northeast, Grimsvotn in the middle, and Thordarhyrna and the Laki craters in the southwest [Grondvold and Johannesson, 1983; Bjornsson and Einarsson, 1990]. The alignment spans ∼120–130 km, with spacing between the different centers of 30–70 km. A similar trio can be found within the alignment of the Martian mountains, specifically, mountains 5, 6, and 7, which span ∼123 km (Figure 2b). As indicated by the Icelandic features, eruptions at one center do not require simultaneous eruption at one of the other centers. Additionally, the morphologies among the Icelandic structures in this trio are not constant, nor are those exhibited by mountains 5, 6, and 7. It should be noted that while this Icelandic analog serves to show that different morphologies can be constructed underneath the same ice sheet, the conditions at Iceland are certainly different from the Martian volcanoes. Specifically, there is no indication of a rift system, as is the case with Iceland, other than the alignment of the mountains.
Figure 13. Annotated sketch map of the western portion of the Vatnajokull ice cap, Iceland, showing the spacing and alignment of the volcanic centers erupted beneath the ice. Inset shows outline of Iceland, with the location of its four ice caps, and a box around the enlarged view of western Vatnajokull.
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 The relatively large size of the summit calderas found on some of the mountains may also have a relationship to a subglacial origin. During times between eruptions, meltwater may have collected within the summit craters in a fashion similar to that occurring with the Grimsvotn caldera in Iceland today [Bjornsson, 1983]. In Grimsvotn the meltwaer lake has served to both erode away at the banks of the summit crater as well as fill in the crater with eroded material [Gudmundsson, 1989]. The summit craters found on at least four of the south polar mountains may share a similar evolutionary history to that of the Grimsvotn cladera.
 The low-domed mountains could likely have erupted entirely subaerially. Together they account for the four shallowest slopes and lowest heights among the mountains (Table 1). Also, all four of these mountains are located outside the boundary of the Dorsa Argentea Formation (Figure 7).
 Finally, the interpretation of the mountains as subglacial volcanoes is also supported by the surrounding features discussed earlier. The channels found extending away from some of the mountains may be indicative of some basal meltwater drainage (Figure 2b). Such meltwater could have accumulated during the constructional history of the mountains and been released as jokulhlaups by melting of the surrounding ice walls along the basement. As mentioned in section 4.1, the main mechanism of meltwater drainage is inferred to be supraglacial drainage through the passage zone [Smellie, 2001].
 The circular depressions located around several of the mountains are remarkably similar to depressions observed around some volcanoes in the north polar region of Mars [Grosfils and Sakimoto, 2001]. These volcanoes are slightly smaller than the south polar mountains, but their corresponding circular depressions extend outward for about one diameter of the volcano and are at most ∼100 m deep, similar to the observations made of the circular depressions around the south polar mountains. Grosfils and Sakimoto  interpreted the north polar depressions as resulting from volatile loss from the surrounding plains and ground subsidence due to emplacement of the volcanoes, and inferred underlying sills. While there is no evidence for underlying sills beneath the south polar mountains, the approximate central location of the mountains within the circular depressions suggests that the two are related and that melting and loss of volatiles could have resulted from heat conduction from the magma reservoir as well as sill emplacement [Head and Wilson, 2002]. A volatile loss origin for the circular depressions is consistent with the volatile-rich nature of the Dorsa Argentea Formation and the former presence of an ice sheet [Head and Pratt, 2001].
 Polygonal patterns are found primarily in the circular depressions and marginal troughs surrounding many of the mountains. If these date from the Hesperian period, the polygons may have formed in the process of formation of the circular depressions, when volatiles were removed or when the ground subsided. Other possible origins for the cracking are from either mud desiccation or ice-wedge cracking [Seibert and Kargel, 2001] following pooling of meltwater in the depressions around the mountains or in some other low-lying areas acting as basins for meltwater drainage. Alternatively, the polygonal terrain could be due to more recent cycling of a volatile-rich upper layer [Lane and Christensen, 2000].
 On the basis of the above detailed analysis of the mountains and surrounding features found in association with them, we interpret several of the mountains as having been constructed into the Hesperian-aged ice sheet. The well-preserved state of the mountains indicates that little postconstructional erosion has occurred. This suggests that once the ice was melted and removed from the vicinity of the mountains, the ice sheet did not reform [Chapman and Tanaka, 2001]. Therefore the mountains could have likely played a significant role in local meltback of the Hesperian-aged ice sheet. This is consistent with the work of Milkovich et al. , which suggests that meltback of the Hesperian ice sheet likely resulted from bottom-up, rather than the top-down, melting.
 The Dorsa Argentea Formation, the deposits of the Hesperian ice sheet, dates to the middle part of the Hesperian Period [Tanaka and Scott, 1987], suggesting that the larger ice sheet had melted back and retreated by that time. Therefore the sources of the melting must have operated prior to that time. Among the most prominent candidates for bottom-up melting in this region is the emplacement of the Hesperian-aged ridged plains (Hr), which have been mapped by Tanaka and Scott  throughout much of the south polar region. The ridged plains material forms numerous patches of “broad, planar, moderately cratered surfaces marked by long, linear to sinuous ridges., embays all adjacent units.,” and is interpreted as “extensive flows of low-viscosity lava erupted at high rates from local sources.” [Tanaka and Scott, 1987]. According to Tanaka and Scott , Hr lies largely stratigraphically below the Dorsa Argentea Formation and was emplaced during the Early Hesperian. Thus, if the Dorsa Argentea Formation represents the remnants of a previously larger volatile-rich ice sheet unit, then the emplacement of volcanic deposits of the ridged plains (Hr) during the early part of the Hesperian could have served as a heat source to melt portions of the ice sheet.
 Indeed, such melting could have occurred in several different modes. Recent mapping of the northern lowlands has shown evidence for the widespread occurrence of Hr, lying below the Vastitas Borealis Formation (Late Hesperian) and resurfacing the northern lowlands [Head et al., 2002]. This new northern lowlands occurrence brings the total area of Hr to ∼30% of the surface of Mars, strengthening arguments that this phase of volcanism and tectonism may have represented a significant global event on the planet [e.g., Watters, 1993]. Such important global events could have involved a period of enhanced global heat flow, potentially leading to enhanced basal melting of thick volatile-rich deposits. Second, local intrusion of Hr into volatile-rich deposits or ice sheets could obviously lead to the equivalent of terrestrial supraglacial, intraglacial, and subglacial melting, in which features such as tuyas might have formed. Thus the widespread nature and style of occurrence of the Early Hesperian ridged plains, together with their geographic and temporal proximity, mean that the emplacement of Hr is a clear candidate for a source of bottom-up melting of the unit that became the Dorsa Argentea Formation. Furthermore, the individual occurrences of the mountain structures could well mark the manifestations of an Hr source vent erupted under an ice sheet, producing these unusual morphologic features, rather than laterally extensive volcanic plains.
4.3. Estimates of the Thickness of the Ice Sheet
 If these features are correctly interpreted to be volcanic eruptions beneath and in the vicinity of an ancient ice sheet, then the thickness of the ice sheet can be estimated from the heights and morphologies of the mountains. Specifically, the heights of the flat-topped mountains and cone-shaped mountains can be used to develop constraints on the minimum thickness of the ice sheet into which the volcanoes erupted. The flat-topped mountains are interpreted to have erupted up to about the level of the englacial lake. Different authors have proposed various connections between the level of the lake and the thickness of the cap [e.g., Smellie and Skilling, 1994]. However, it can be agreed that the maximum level of the lake is less than the thickness of the overlying ice. Therefore an estimate of the meltwater lake level can provide a minimum value for the thickness of the ice. This is especially true for Mars, where the passage zone, corresponding to the lake level, would be expected to be farther from the surface of the ice sheet. The heights of the flat-topped south polar mountains provide an estimate of between 700 and 1300 m.
 As discussed in section 4.2, the mountains that are more cone-shaped could have erupted entirely within the englacial lake. Discounting mountain 17, which may well be a central peak, the remaining cone-shaped mountains provide ice sheet thickness estimates ranging from 1200 to 2200 m. Interestingly, the tallest of the cone-shaped mountains (13) is found in the middle of the regional deposit of the Dorsa Argentea Formation. The remaining cone-shaped mountains decrease in height closer toward the edge of the deposit (Figure 7, Table 1).
 The above estimates combine to give a minimum estimate of the thickness of the overlying ice sheet, ranging from 700 m to 2.2 km and averaging to ∼1.4 km. Thicker estimates generally come from mountains closer toward the middle of the regional deposit of Hd, and smaller estimates come from mountains found closer toward the boundary of the deposit. This trend is consistent with the expectation that the ice sheet would have thinned toward its edges. The estimates are also consistent with the thickness of the formerly large Hesperian-aged south polar ice sheet as estimated by Head and Pratt .
 An alternative explanation for the different ice sheet thicknesses obtained from the heights of the mountains is a variation in ice sheet thickness with time rather than simply in location. If the mountains were constructed at somewhat different times, then the heights of the mountains may be indicating the thickness of the ice sheet at these times, implying that it was not constant. However, there does not appear to be any good evidence to suggest a significant age variation among the mountains. The simplest explanation for the varying heights seems to be a variation in ice sheet thickness with location.
4.4. Other Candidate Martian Subglacial Volcanoes
 To date, other features on Mars interpreted as subglacial eruptions have been low, elongated, steep-sided mounds, considered to be analogous to terrestrial table mountains [Hodges and Moore, 1978; Allen, 1979; Chapman, 1994; Chapman and Tanaka, 2001]. To some degree, this has been due to the limited resolution of Viking images. However, planetwide surveys in search of volcano-ice interaction have also been restricted to features resembling Icelandic table mountains and moberg ridges. Such features are not the sole type of subglacial volcanoes found on Earth and therefore should not be considered the sole type that might be found on Mars.
 The majority of the table mountains found on Mars are located in the northern lowlands. The structures are generally smaller than the mountains found near the south pole and longer than they are wide, with widths ranging from 1 to 16 km in diameter and heights reaching up to 1 km (Figure 11). An exception to this is a feature in Hebes Chasma which stands 4 km above the chasma floor [Chapman and Tanaka, 2001]. They have steep slopes, and many are flat-topped [Hodges and Moore, 1978; Allen, 1979; Chapman, 1994; Chapman and Tanaka, 2001]. What reason is likely to explain the difference in morphology and morphometry between the south polar features and the table mountains of the northern lowlands?
 A possible explanation might be found with the thickness and type of ice layer into which the volcanoes erupted [Smellie, 2000]. Allen  proposes that the table mountains he describes erupted into a subsurface ground ice layer. Chapman  proposes that the table mountains found in Utopia erupted into a frozen paleolake, of thickness estimated to be ∼180 m. The south polar mountains would have erupted into a formerly large south polar ice sheet. The heights of the flat-topped and conical features have been used to arrive at a minimum value of 1.4 km for the average thickness of the ice sheet. Clearly, a different set of constraining variables would have been involved in the formation of the south polar mountains and the table mountains found elsewhere. Additionally, the state of the base of the ice sheet (wet or dry) would have played a role in the eruption [Smellie and Skilling, 1994]. Unfortunately, there is no method by which to determine such conditions on Mars at this time.