Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps

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Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps Lars Roggon a,b, Ralf Hetzel a,∗, Harald Hiesinger b, Jaclyn D. Clark b, Andrea Hampel c, Carolyn H. van der Bogert b a b c

Institut für Geologie und Paläontologie, Westfälische Wilhelms-Universität Münster, Corrensstr. 24, 48149 Münster, Germany Institut für Planetologie, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany Institut für Geologie, Leibniz Universität Hannover, Callinstr. 30, 30167 Hannover, Germany

a r t i c l e

i n f o

Article history: Received 9 April 2016 Revised 6 December 2016 Accepted 20 December 2016 Available online xxx Keywords: Lunar tectonics Lobate scarp Thrust faulting Displacement profile Fault growth

a b s t r a c t Fault populations on terrestrial planets exhibit a linear relationship between their length, L, and the maximum displacement, D, which implies a constant D/L ratio during fault growth. Although it is known that D/L ratios of faults are typically a few percent on Earth and 0.2–0.8% on Mars and Mercury, the D/L ratios of lunar faults are not well characterized. Quantifying the D/L ratios of faults on the Moon is, however, crucial for a better understanding of lunar tectonics, including for studies of the amount of global lunar contraction. Here, we use high-resolution digital terrain models to perform a topographic analysis of four lunar thrust faults – Simpelius-1, Morozov (S1), Fowler, and Racah X-1 – that range in length from 1.3 km to 15.4 km. First, we determine the along-strike variation of the vertical displacement from ≥ 20 topographic profiles across each fault. For measuring the vertical displacements, we use a method that is commonly applied to fault scarps on Earth and that does not require detrending of the profiles. The resulting profiles show that the displacement changes gradually along these faults’ strike, with maximum vertical displacements ranging from 17 ± 2 m for Simpelius-1 to 192 ± 30 m for Racah X-1. Assuming a fault dip of 30° yields maximum total displacements (D) that are twice as large as the vertical displacements. The linear relationship between D and L supports the inference that lunar faults gradually accumulate displacement as they propagate laterally. For the faults we investigated, the D/L ratio is ∼2.3%, an order of magnitude higher than theoretical predictions for the Moon, but a value similar for faults on Earth. We also employ finite-element modeling and a Mohr circle stress analysis to investigate why many lunar thrust faults, including three of those studied here, form uphill-facing scarps. Our analysis shows that fault slip is preferentially initiated on planes that dip in the same direction as the topography, because the reduced overburden increases the differential stress on prospective fault planes, and hence, promotes failure. Our findings highlight the need for quantifying vertical displacements of more lunar thrust-fault scarps with the methodology employed in this study, rather than relying only on measurements of local relief, which result in D/L ratios that tend to be too low. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Lobate scarps are straight to curvilinear positive-relief landforms that occur on all terrestrial bodies (e.g., Watters et al., 20 0 0; Williams et al., 2013; Massironi et al., 2015). Lobate scarps are the surface manifestation of thrust faults that cut through and offset the upper part of the crust. On Earth, the preservation potential of fault scarps, which are commonly developed in alluvial sediments, is low because they are rather quickly eroded, even in arid re-

∗

Corresponding author. E-mail address: [email protected] (R. Hetzel).

gions (e.g., Yeats et al., 1997). Therefore, well-preserved thrust-fault scarps on Earth for which displacements can be quantified from topographic profiles have ages of at most a few hundred thousand years (e.g., González et al., 2006; Cording et al., 2014). In contrast, fault scarps on Mars, Mercury, and the Moon can be preserved for hundreds of millions or even billions of years, because the absence of a substantial atmosphere (and its associated effects) results in extremely low erosion rates (e.g., Watters, 1993; Hauck et al., 2004; Grott et al., 2007). Hence, fault scarps on planetary surfaces provide the opportunity to study the growth of faults under a wide range of environmental conditions (e.g., gravity, temperature, pore pressure) (Schultz et al., 2006).

http://dx.doi.org/10.1016/j.icarus.2016.12.034 0019-1035/© 2017 Elsevier Inc. All rights reserved.

Please cite this article as: L. Roggon et al., Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2016.12.034

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In general, fault growth is governed by factors such as rock strength, the prevailing stress field, fluid pressure, and the presence and orientation of pre-existing discontinuities (Muraoka and Kamata, 1983; Barnett et al., 1987; Pollard and Segall, 1987; Scholz and Cowie, 1990; Cowie and Scholz, 1992; Bürgmann et al., 1994; Peacock, 2002; Kim and Sanderson, 2005; Torabi and Berg, 2011). Several studies of faults on Earth, as well as theoretical considerations, indicate that the maximum fault displacement, D, is linearly related to fault length, L (i.e., D = c . L; with the constant c depending mainly on rock properties) (Walsh and Watterson, 1987; Cowie and Scholz, 1992; Dawers et al., 1993; Mansfield and Cartwright, 1996; Clark and Cox, 1996; Schlische et al., 1996; Hetzel et al., 2004). This fault-scaling law implies that isolated faults transecting a rock with uniform properties (i.e., with a constant value of c), should maintain a constant D/L ratio as they grow (e.g., Cowie and Scholz, 1992; Dawers et al., 1993). D/L values for faults on Earth lie mainly between 0.005 and 0.05 (i.e., 0.5–5%), with slightly higher values of 0.02 to 0.1 for thrust faults (e.g., Elliott, 1976; Kim and Sanderson, 2005). For faults on Mars and Mercury, D/L values are at the lower end of values for Earth and cluster around 0.0 02–0.0 08 (i.e., 0.2–0.8%) (Schultz, 1997; Watters et al., 20 0 0; Wilkins et al., 20 02; Schultz et al., 20 06; Polit et al., 20 09; Nahm and Schultz, 2011; Byrne et al., 2014), although large normal faults bounding the Thaumasia graben on Mars were reported to have D/L values of up to 0.02 (Hauber and Kronberg, 2005). For the Moon, theoretical considerations suggest a low D/L ratio of ∼0.001 (i.e., 0.1%) owing to the relatively low lunar gravity (Schultz et al., 2006). On the other hand, Watters and Johnson (2010) inferred a D/L value of ∼0.012 for lunar thrust faults, but this estimate was based only on relief estimates for lobate scarps derived from shadow measurements on Apollo Panoramic images. Detailed investigations of displacement profiles and length–displacement scaling relationships of lunar thrust faults have only recently begun (Banks et al., 2013). Such investigations are important, for example, to derive estimates for the Moons´ global contraction by elastic deformation and thrust faulting, as was recently performed for Mercury (Byrne et al., 2014; Klimczak, 2015). Lobate scarps on the Moon are morphologically simple landforms that occur most commonly in the highlands (Schultz, 1976; Binder, 1982; Binder and Gunga, 1985). Early studies of lobate scarps were conducted with photographs from the Apollo Panoramic Camera, which only covered a limited area of the lunar equatorial zone. The Lunar Reconnaissance Orbiter (LRO) spacecraft has greatly improved our ability to study small-scale topographic features by providing high-resolution images of the lunar surface. Since the launch of the LRO in 2009, its camera systems have provided almost two million images of the lunar surface (e.g., Robinson et al., 2010; Burns et al., 2012; Keller et al., 2016). In particular, the LRO Narrow Angle Cameras (NAC) have provided highresolution, 5-km-wide panchromatic swaths, ideal for the detection and investigation of small-scale landforms (Chin et al., 2007; Robinson et al., 2010; Burns et al., 2012). So far, over 3200 lobate scarps have been identified, which range in length from ∼0.6 km to ∼22 km and have relief of between ∼5 m and ∼150 m (Banks et al., 2012; Watters et al., 2015). In general, scarps related to thrust faulting are characterized by an asymmetric shape in cross-sectional view, with the slope changing at the toe and the crest of the scarp (Fig. 1). If the topography adjacent to a scarp is not level, the scarp may either face downslope or upslope. We refer to such scarps as downhill-facing and uphill-facing, respectively (Fig. 1a, b). The topographic asymmetry of scarps can be used to derive the dip direction of the underlying faults (e.g., Yeats et al., 1997; Burbank and Anderson, 2011). Beneath a downhill-facing scarp, the underlying thrust fault dips in the opposite direction than the adjacent topography, whereas the

thrust fault dips in the same direction as the topography beneath an uphill-facing scarp (Fig. 1a, b). With respect to lobate scarps on the Moon, several previous studies have noted that many such scarps face uphill (Schultz, 1976; Binder and Gunga, 1985; Banks et al., 2012; Williams et al., 2013) although these works did not provide exact numbers. Our review of the 97 scarps listed by Binder and Gunga (1985) and Banks et al. (2012) shows that 57 (i.e., ∼60%) are uphill-facing. This proportion is different from Earth, where thrust faults in accretionary wedges and foreland fold-and-thrust belts commonly dip in the opposite direction to the surface slope and so form downhill-facing scarps (e.g., Davis et al., 1983; Avouac et al., 1993; Palumbo et al., 2009). On Earth, the formation of uphill-facing scarps requires special conditions such as rapid loading by sediments during glacial times (Adam et al., 2004) or postglacial unloading due to the melting of glaciers (e.g., Ustaszewski et al., 2008). To the best of our knowledge, the reasons for the relative abundance of uphill-facing scarps on the Moon remain unclear. On the basis of high-resolution NAC-derived digital terrain models (DTMs), this study provides a detailed investigation of the topographic signature of four lunar thrust-fault scarps that is complemented by numerical modeling. The four scarps were selected because (1) they have a relatively simple morphology and are nearly straight, (2) they cover a length scale of one order of magnitude, (3) DTMs of the entirety of each scarp and the adjacent region are available, and (4) maximum-relief values, commonly interpreted as vertical fault offsets, were reported for two of the scarps in a previous study and can be compared with the results of our different methodology to quantify fault offsets (see Section 3). In particular, we address the following questions: How does the displacement on individual faults vary along strike? What are the D/L ratios of lunar thrust faults and are they different from previous estimates? And why are uphill-facing scarps so common on the Moon?

2. Location of the studied faults and database for topographic analysis The smallest of the fault scarps selected for this study, Simpelius-1, is located on the nearside of the Moon, whereas the other three – Morozov (S1), Fowler, and Racah X-1 – are located in the lunar highlands on the farside of the Moon (Fig. 2). The position of the latter scarps ensures that the underlying faults have likely formed in rocks with relatively similar mechanical properties (i.e., the lunar megaregolith with a thickness of several kilometers; Hiesinger and Head (2006); Jaumann et al. (2012)). Two of the scarps, Morozov S1 and Racah X-1, occur close to the equator, whereas Fowler and Simpelius-1 are located on the northern and southern hemisphere, respectively. The exact location of each scarp is provided in Table 1. Using LRO NAC stereo pairs, which provide different viewing angles of the regions including the scarps, DTMs for each landform were derived with the SOCET–SET software by correlating the images and edge-matching each pixel (Tran et al., 2010; Burns et al., 2012; Henriksen et al., 2015). The horizontal resolution of these terrain models (i.e., their pixel size) ranges from 0.5 to 1.2 m. The vertical precision of the terrain models varies with the convergence angle for each stereo pair (Tran et al., 2010), which lies between 18 ° and 31 ° for the images we used here. These convergence angles translate into a relative vertical precision between 0.5 m and 0.7 m for the terrain models (cf. Tran et al. (2010)). Their absolute position error is < 10 m in longitude/latitude and about 1 m in elevation (Henriksen et al., 2015).

Please cite this article as: L. Roggon et al., Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2016.12.034

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Fig. 1. Sketch of an idealized thrust-fault scarp with relevant fault-scarp parameters noted (modified from Hanks and Andrews (1989)). The motion of the thrust fault leads to a displacement of the surface, which has a far-field slope angle of θ f . Depending on the dip of the surface the resulting fault scarp faces downhill (a) or uphill (b). In both cases, the vertical fault displacement, V, can be found by extrapolating the far-field slope from the footwall and hanging wall to the scarp (dashed grey lines) (c, d). Importantly, the local relief, R, at the scarp is not the same as the vertical displacement, V. Moreover, R is greater than V for downhill-facing scarps, whereas for uphill-facing scarps R is smaller than V.

Fig. 2. Global image of the Moon showing the location of the four thrust-fault scarps investigated in this study. Table 1 Location, length and displacements of the four studied thrust fault scarps. Name of fault scarp

Simpelius-1

Morozov (S1)

Fowler

Racah X-1

Location

73.55°S 13.11°E yes 1.31 17 ± 2 34 ± 4 (26–50) 0.026

6.56°N 129.95°E yes 4.97 70 ± 15 140 ± 30 (110–210) 0.028

43.28°N 147.06°W yes 10.8 94 ± 15 188 ± 30 (150–280) 0.017

10.06°S 178.10°E No 15.4 192 ± 30 384 ± 60 (300–560) 0.025

Uphill-facing scarp Fault length L (km) Maximum vertical displacement (m) Maximum total displacement D (m)a D/L ratio

a Values were calculated under the assumption of a fault dip of 30°. The displacement range given in parentheses was determined for a dip range of 20° to 40° for the thrust faults (note that the calculated total displacement D decreases with increasing fault dip).

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3. Method for quantifying vertical fault displacements from topographic profiles For analyzing the slip distribution along the lobate scarps, we extracted between 20 and 27 topographic profiles along each of the four fault scarps with the DTMs described above. The profiles along each scarp were chosen with a regular spacing; in areas where the endpoints of the scarps were not well defined or where two fault segments appeared to overlap, however, we extracted additional profiles and reduced the profile spacing. Each profile is oriented perpendicular to the average strike of the respective scarp (Fig. 3). To quantify the vertical fault displacement (or fault throw) on each profile, we applied the approach of Hanks and Andrews (1989), which is commonly employed for the analysis of both normal- and thrust-fault scarps on Earth (e.g., Wallace, 1977; Nash, 1980; Avouac et al., 1993; Hetzel et al., 2002). In this approach, which can be applied to both uphill and downhill-facing scarps (Fig. 1a, b), the vertical displacement of the scarp-forming fault is determined by extrapolating the regional (i.e., far-field) slope to the fault scarp (Fig. 1c, d). Ideally, the far-field slope, θ f , on both sides of the fault scarp should be the same; in reality, the slopes in the footwall and the hanging wall often differ by a few degrees. Hence, to determine the vertical displacement for each fault as accurately as possible, the topographic profiles must be long enough to reliably quantify the far-field slope in both the footwall and hanging wall. For measuring the far-field slope and quantifying the vertical displacement, we fit straight lines to the topographic data in both the footwall and hanging wall on a visual basis. The uncertainty of the fault offset was estimated from the range of dips permitted by the data. Hence, a topographic profile with more variation in elevation values along its length leads to a larger range of dip angles (i.e., θ f ) and, therefore, to a larger uncertainty in the fault offsets, whereas data that plot close to a straight line yield a smaller uncertainty. For most of the profiles, the uncertainty of the vertical displacement is of the range 15–30%, although for small offsets near the fault tips, the estimated uncertainty is up to 50%. It is important to note that, in general, the local relief, R, at a scarp (i.e., the elevation difference between the crest and the base of the scarp) is not the same as the vertical fault displacement: for downhill-facing scarps the local relief is greater than the vertical displacement, whereas for uphill-facing scarps it is smaller than the vertical displacement (Fig. 1c, d). Only in the special case of a horizontal surface will the scarp relief be equal to the vertical displacement. 4. Results of the topographic analysis The shortest of the investigated fault scarps, Simpelius-1, is an uphill-facing scarp at the southern rim of Simpelius crater (Banks et al., 2012). The 1.31 km-long scarp faces south–southeast and the underlying thrust fault dips to the north–northwest (Fig. 3a). The DTM shows that the fault is composed of two primary overlapping segments. As revealed by the analysis of 20 scarp profiles, the main segment in the west reaches a maximum vertical displacement of 17 ± 2 m close to the western end of the fault (Fig. 4). The displacement along this main segment decreases gradually to the east, where the fault throw on the shorter eastern segment reaches 5.0 ± 1.5 m (Fig. 5a). The Morozov (S1) scarp, located close to the equator, constitutes a 4.97-km-long uphill-facing scarp. Based on its morphology, we interpret the scarp to be formed by a west-dipping thrust fault composed of two principal segments (Fig. 3b). East of the fault scarp, the far-field slope dips more or less uniformly to the west at angles of 12 ° to 18 °, whereas the slope angles west of the scarp are lower and more variable owing to natural undula-

tions of the topography (Fig. 4). The five topographic profiles depicted in Fig. 4 illustrate that the scarp face has a less-steep slope than the regional topography, which is why the scarp appears as a narrow band of lower surface slope values (∼150–300 m wide) in the slope map (Fig. 3b). The maximum vertical displacement of the main fault segment is 70 ± 15 m, whereas the smaller fault segment in the north only possesses a displacement of 20 ± 4 m (Fig. 5b). The Fowler scarp consists of one segment with a length of 10.8 km (Fig. 3c). The controlling thrust fault dips to the east (i.e., it has the same dip direction as the regional topography). An oblique 3D view along strike of the fault scarp (Fig. 3c) shows that this scarp is also facing uphill (towards the west). The scarp profiles reveal a maximum vertical offset of 94 ± 15 m (at the location of profile 5, Fig. 4). The displacement profile constructed from 27 scarp profiles shows that the fault offset decreases gradually towards the northern fault tip over a length of about three kilometers (Fig. 5c). In contrast, the displacement gradient is steeper at the southern end of the fault, where the terrain is more irregular with several craters that can be seen on the DTM and the oblique-view image (Fig. 3c). With a length of 15.4 km, Racah X-1 is the longest of the studied scarps and underlain by a west-dipping thrust fault (Fig. 3d). The topography in the central part of the fault is essentially flat and the relatively steep scarp faces to the east (see slope map in Fig. 3d). In this region, the vertical displacement values derived from several scarp profiles are highest, with a maximum of 192 ± 30 m (Fig. 5d). Farther to the north and south, the fault transects two large craters and in the vicinity of these craters, the vertical offset drops markedly by ∼100 m over a distance of 1–2 km. The terrain near the craters is irregular with an undulating topography and slopes that are locally steeper than 20° As a consequence, it was impossible to determine vertical fault offsets for four of the scarp profiles (shown as white lines in Fig. 3d). To the north and south of the two craters, the vertical fault displacement is lower and decreases smoothly to zero at the fault tips. In order to estimate the maximum total fault displacement from the maximum vertical displacement recorded at each fault, we assume that the thrust faults have a dip of 30 ° (cf., Anderson, 1951). This is the same value that was used in previous studies to obtain total displacement values for lunar thrust faults (Watters and Johnson, 2010; Watters et al., 2010). This assumption leads to total displacements that range from 34 ± 4 m to 384 ± 60 m (Table 1). As the 30 ° dip angle is uncertain, we also report a range of total displacement for dip angles between 20 ° and 40 ° in Table 1, following Banks et al., (2012). This range encompasses the dip values obtained recently for ten lunar thrust faults by fault dislocation modeling (Williams et al., 2013; Byrne et al., 2015). Only one fault has an inferred dip that is lower (i.e., 15 °; Byrne et al., 2015). A plot of maximum fault slip, D, versus fault length, L, reveals a linear relationship between D and L and indicates that the maximum displacement is about 2.3% of the fault length for the investigated thrust faults (Fig. 6). We discuss this finding in detail in Section 5.1. Our topographic analysis shows that three of the studied thrust fault scarps are uphill-facing scarps generated by slip on faults that dip in the same direction as the topography adjacent to the scarps. We hypothesize that the reason for the formation of thrust faults with such a geometry may arise from the lower vertical stress acting on the fault plane, which in turn results from a lower gravitational load, as compared with thrust faults that dip in a direction opposite to that of the regional topography (Fig. 1a, b). In Section 5.2, we evaluate this hypothesis by means of finite-element modeling and a Mohr-circle analysis for stress.

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Fig. 3. LROC NAC images of the four studied fault scarps, showing the topographic profiles (thin black lines) used to derive along-strike vertical displacements. The white numbers mark the locations of scarp profiles shown in Fig. 4. To further illustrate the topography in the vicinity of the scarps, we also show color-coded slope maps for Morozov S1 and Racah X-1 and an oblique along-strike view of the Fowler scarp. (a) The two-segment (see inset) Simpelius-1 scarp (LROC NAC images M139804021, M139817589; horizontal pixel scale of 0.5 m). (b) The Morozov (S1) scarp (LROC NACs M1113318442, M1113332648; horizontal pixel scale of 1.2 m). (c) The Fowler scarp (images M143221393, M143228175; pixel scale of 0.7 m). (d) The Racah X-1 scarp. In the vicinity of two craters, indicated on the slope map for this scarp, the topography is too irregular to allow reliable topographic measurements (i.e., the four profiles indicated by white lines) (images M143452995, M143459779; pixel scale of 0.5 m). The geographic coordinates of the four scarps are given in Table 1.

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Fig. 4. Topographic profiles across three of the four studied fault scarps, Simpelius-1, Morozov (S1), and Fowler. The vertical displacement for each profile was derived with the approach illustrated in Fig. 1. The locations of the profiles are shown in Fig. 3 (i.e., those profiles marked by white numbers).

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Fig. 4. Continued

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Fig. 5. Variations in vertical displacement along strike for the four studied thrust faults. The scarp profiles from which the displacement values were derived are shown in Fig. 3. The bold numbers next to some displacement values indicate the respective scarp profiles shown in Fig. 4. The topography in each displacement profile is exaggerated by twenty times.

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Fig. 6. The relationship between fault length, L, and maximum displacement, D, for the studied faults, together with two additional fault data sets from the Moon. A linear regression through our data yields a D/L ratio of 0.023 (or 2.3%). The second data set yields a D/L ratio of 0.012 (Watters and Johnson, 2010). The third data set is from Banks et al., (2012). Note that scarps not entirely covered by NAC images were excluded from this data set. The total displacements for the faults from all three data sets were calculated assuming a fault dip of 30°. See Section 5.1 for discussion.

5. Discussion 5.1. Fault displacements and displacement-length scaling We calculated vertical displacements along four lunar thrustfault scarps ranging in length from 1.3 km to 15.4 km (Table 1) by extrapolating the far-field slope to the scarp. This approach is commonly used for fault scarps on Earth and can be applied to any fault scarp, regardless of whether it faces uphill, downhill, or occurs on a horizontal surface (Fig. 1). The gradual lateral variations in displacement along strike of the four fault scarps suggest that the methodology is generally well suited to study lunar faults (Fig. 5). Nonetheless, in a few regions with irregular topography and craters, such as parts of the Racah X-1 scarp, we were unable to measure reliable fault offsets. Other studies of lunar scarps used the maximum relief determined from detrended topographic profiles as a proxy for vertical fault displacement (e.g., Watters et al., 20 0 0; Banks et al., 2012). With respect to this approach, it should be noted that the local relief along uphill-facing scarps is consistently lower than the vertical fault offset on the underlying fault, whereas it is larger than the fault offset for downhill-facing scarps (Fig. 1c, d). For two of the scarps studied here (Simpelius-1 and Racah X-1), it is possible to compare published values for their maximum relief with the results of our methodology. We obtain a vertical displacement of 17 ± 2 m at the Simpelius-1 scarp (Fig. 4), whereas Banks et al., (2012) reported a maximum relief of 13 ± 5 m in the same portion of the scarp. Although the two values agree within their uncertainties, our nominal value is somewhat higher. The lower value derived by Banks et al., (2012, see their Fig. 1c) is consistent with the geometric inevitability that the local relief at this uphill-facing scarp is smaller than the vertical displacement. An additional reason for the discrepancy may be that the detrending of topographic profiles can lead to measurement errors when

the regional slopes in the fault footwall and hanging wall differ by a few degrees, as is the case for Simpelius-1 and Morozov (S1) (see slope angles given in Fig. 4) and other lunar scarps. For the central portion of the Racah X-1 scarp, Banks et al., (2012) calculated a maximum relief of 150 ± 5 m and noted that pre-existing topography complicates the definitive assessment of the structural relief. Our two profiles of the Racah X-1 scarp, which are closest to the profile of Banks et al., (2012), yield fault throws of 138 ± 25 m and 144 ± 25 m, respectively (Fig. 5), which are (within error) identical to the previous estimate. The good agreement between the two approaches is not surprising, because for fault scarps in flat terrain, the maximum local relief should be the same as the vertical offset derived from extrapolating the (negligible) far-field slope to the scarp. However, one of our profiles gave a considerably larger vertical displacement of 192 ± 30 m (Fig. 5), which indicates that – in order to determine the maximum displacement of faults – it is important to quantify displacements on many profiles, rather than to rely on a limited number of measurements. As shown in Fig. 6, the maximum total displacements of the four studied thrust faults follow a linear increase with length over one order of magnitude. Following previous interpretations of fault displacement data sets on Earth (e.g., Cowie and Scholz, 1992; Dawers et al., 1993; Schlische et al., 1996; Hetzel et al., 2004; Kim and Sanderson, 2005), we interpret this relationship to indicate that during the progressive accumulation of slip, lunar faults propagate laterally and increase in length. For the investigated faults, the ratio of maximum displacement, D, to fault length, L, ranges from 0.017 to 0.028 (Table 1) with a mean value of ∼0.023 (or 2.3%) (Fig. 6). This is an order of magnitude higher than the value of ∼0.1% derived by theoretical considerations that took the influence of lunar gravity into account (Schultz et al., 2006), and about twice as large as the value of ∼0.012 estimated by Watters and Johnson (2010) (Fig. 6). The latter estimate was based on Apollo Panoramic images of nine fault segments and the assumption that

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their maximum relief equals their vertical displacement. We argue that this assumption is not correct and yields D/L ratios that are too small. The inference that scarp relief should not be used as a proxy for fault offset is further supported by measurements of fault length and greatest relief based on NAC mosaics and Lunar Orbiter Laser Altimeter (LOLA) data, respectively (Banks et al., 2012). As shown in Fig. 6, most data points of Banks et al., (2012, their Tables 1 and 2) plot below the regression line obtained in this study. Our D/L ratio of ∼0.023 for lunar thrust faults is based on the analysis of but four faults, and could be confirmed in future studies by analyzing more scarps with the methodology presented in Section 3. Nonetheless, our finding has important implications for studies attempting to quantify the global contraction of the Moon from fault scaling relations, because a higher D/L ratio used as input in the calculation will increase the strain calculated by the same degree (cf., Scholz and Cowie, 1990). Our results, in addition to recently published findings for other lunar scarps (Banks et al., 2013; Williams et al., 2013), indicate that the displacement– length ratios of lunar thrust faults are similar to faults on Mercury and Mars (e.g., Watters et al., 20 0 0; Hauber and Kronberg, 2005; Hauber et al., 2013; Byrne et al., 2014), and almost as high as the average displacement–length ratio of ∼3% for faults on Earth (Schlische et al., 1996; Torabi and Berg, 2011). Our estimate for the maximum total displacement on each of the four investigated faults (Table 1) remains somewhat uncertain because, like the other displacement data shown in Fig. 6, it is based on the assumption of a fault dip of 30°. In favorable circumstances, fault dips can be quantified directly from the surface geometry of displaced craters, as was recently shown for thrust faults on Mercury with large displacements of ∼1 to ∼5 km (Galluzzi et al., 2015). However, owing to the relatively small displacements of the studied faults and the absence of tectonically displaced craters, it was not possible to employ the method of Galluzzi et al., (2015) in our study. One of the most reliable approaches to determine the dip angle and the depth extent of faults in a more indirect way is dislocation modeling (e.g., Grott et al., 2007; Williams et al., 2013; Byrne et al., 2015). In this approach, fault dip, penetration depth and displacement are varied until the difference between the deformed surface in the model and the observed topography is minimized according to some metric. With this method, Williams et al., (2013) derived dip angles of 35 ± 3° to 40 ± 4° and a depth extent of 250 ± 50 m to 900 ± 50 m for six lunar thrust faults, including Simpelius-1 and Racah X-1. When these dip values are applied to all four investigated faults, their total calculated displacements fall in the lower part of the displacement ranges given in Table 1. The fault data presented by Williams et al., (2013, their Table 1) are also consistent with the finding that the depth extent of lunar faults increases with fault length. Given that the conditions for the onset of contraction-induced thrust faulting are most favorable at or near the surface (Klimczak, 2015), we argue that faulting has likely started at or near the surface, if these scarps formed as a result of global lunar contraction from interior cooling, and then propagated downwards as the faults grew and increased in length. With respect to the depth extent of ∼900 m estimated for Racah X-1 by Williams et al., (2013), we note that this value was derived from a scarp profile located near the most northern of two craters, where the inferred total slip is only 45 ± 2 m. At the center of the Racah X-1 scarp, our minimum estimate for the total fault displacement is much larger (i.e., 300 m for an assumed fault dip of 40 °: Table 1). Here, the thrust fault has presumably propagated to a substantially greater depth. At the surface, the rapid decrease of the measured vertical fault offset towards the two large impact craters (Fig. 5d) may be caused by the presence of mechanically stronger rocks (cf. Schlische et al., 1996). More competent rocks, such as solidified impact melts or impact melt breccias

(e.g., Grieve and Therriault (2013), and references therein) associated with the craters may be responsible for the high displacement gradient on the fault plane adjacent to the craters, although linkage of fault segments (see next paragraph) could also explain this pattern. The high displacement gradients at the western tip of Simpelius-1 and the southern tip of Fowler could be explained by the presence of relatively strong rocks, too (Fig. 5a, c). As neighboring faults grow and increase in length, they start to interact and eventually link with each other. This process leads to the formation of segmented fault zones and relay zones at the junctions of neighboring faults (e.g., Cartwright et al., 1995; Willemse, 1997; Peacock, 20 02; Densmore et al., 20 03). Relay zones appear to have formed at two of the studied faults, Simpelius-1 and Morozov (S1), where they occur between the main fault segment and adjacent smaller segments (Fig. 3a, b). The displacement profiles are consistent with this interpretation, and show that the two segments of these faults overlap with each other and are therefore linked (Fig. 5a, b). Near the center of the Fowler scarp (i.e., 4–5 km from its northern end), the relatively low vertical displacements recorded by three profiles may also indicate the presence of a relay zone (Fig. 5c). With respect to segmented fault zones and relay zones, it is important to note that in rocks of the same mechanical strength, the D/L ratios of both individual faults and segmented fault zones are expected to be similar (Dawers and Anders, 1995). 5.2. Formation of uphill-facing scarps: insights from finite-element modeling Our analysis of topographic profiles indicates that three of the studied thrust fault scarps are uphill-facing scarps generated by slip on faults that dip in the same direction as the local topography. As mentioned above, thrust faults with such a geometry are quite common on the Moon (e.g., Banks et al., 2012; Williams et al., 2013). To test our hypothesis that the surface topography controls the vertical load on a fault plane and, hence, plays an important role in the formation of uphill-facing fault scarps, we simulated thrust faulting and its relation to topography with two-dimensional finite-element models using the commercial code ABAQUS (version 6.14). The models consist of an elastic crust, which is shortened by a velocity boundary condition applied at both model sides to simulate horizontal shortening induced by long-term cooling of the lunar interior (Fig. 7a). Based on the relatively young age estimates for lunar lobate scarps (i.e., < 1 Gyr; Binder and Gunga, 1985), and vertical offsets of a few tens of meters, we chose a total shortening rate of 2 mm/kyr (note, however, that the results presented below do not depend on the actual value of the shortening rate). Gravity was included as a body force (acceleration due to gravity is 1.6 m/s2 : Vaniman et al., 1991). In different model setups, we varied both the topography of the model surface (i.e., as horizontal, or with a slope of 15°) and the position of a predefined thrust fault (i.e., either dipping in the same or the opposite direction to surface slope; Fig. 7a). The predefined fault plane had a dip of 30 ° (e.g., Anderson, 1951; Watters et al., 2010) and was embedded in the uppermost part of the 50 km-thick model crust (Fig. 7a). The fault plane was implemented as a frictional contact interface between the footwall and hanging wall (cf. Hampel and Hetzel, (2006); Turpeinen et al., (2008)). Slip initiation was controlled by the Mohr-Coulomb failure criterion with a friction coefficient of 0.8, which is appropriate for normal stresses < 200 MPa (Byerlee (1978); note that a normal stress of ∼200 MPa is equivalent to a depth of ∼40 km on the Moon). Our model results indicate that the onset of faulting in the 200 km-long model is a function of the surface topography (Fig. 7b). Thrust fault 1, which dips in the same general direction as the topography (and which forms an uphill-facing scarp), starts to slip

Please cite this article as: L. Roggon et al., Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2016.12.034

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Fig. 7. Model setup and results of our finite-element analysis. (a) Geometry and parameters of the model (ρ density, E Young’s modulus, μ friction coefficient; v velocity, g acceleration due to gravity). A velocity boundary condition was applied at both sides to shorten the model in the x-direction. The model sides were free to move in the z-direction; the model bottom was free to move in the x-direction but was fixed in the z-direction. (b) Fault slip histories derived from the models. In the case of a horizontal model surface, both faults show the same intermediate slip evolution (grey and dashed black curves). See Section 5.2 for discussion.

4.2 Myr after the onset of shortening and reaches a total slip (i.e., the vector sum of the throw and the heave) of 5.8 m after 70 Myr (Fig. 7b). In contrast, slip on fault 2 (which leads to the generation of a downhill-facing scarp) initiates much later (i.e., after 20 Myr of elapsed model time) and attains a total slip of only 1.8 m in 70 Myr (Fig. 7b). If the surface of the model is horizontal, faulting on both structures starts after 4.4 Myr, but faulting proceeds at a lower rate than for fault 1, which generated the uphill-facing scarp. We emphasize that, although the absolute ages for fault initiation (as well as the total fault slip) depend on the arbitrarily chosen shortening rate (as well as on the size of the model and the elastic parameters), this relative timing of fault activation was consistently observed irrespective of the chosen shortening rate. The model results demonstrate that, for all other factors being equal, the differing weight of the hanging wall above the two modeled faults is responsible for the different timing of fault initiation, as well as the difference in total slip. For faults dipping in the same direction as the regional topography, the work needed to uplift the hanging wall is less than that required for a fault that dips in the opposite direction (i.e., against regional topography).

The principle of work minimization has been used to analyze the development of fault systems on Earth in diverse tectonic settings (see e.g., Cooke and Madden (2014) for a review) and was, for example, applied to explain the growth of fold-and-thrust belts as critically tapered wedges (Hardy et al., 1998). In general, the development of thrust faults in convergent settings on Earth is associated with the underthrusting of oceanic or continental crust beneath an upper plate. This geometric configuration leads to the development of wedge-shaped accretionary prisms and fold-andthrust belts in the upper plate (e.g., Davis et al., 1983; Gutscher et al., 1998). As the lower plate is commonly covered by weak sedimentary rocks saturated with fluids, plate convergence leads to the development of thrust faults, which generally form at the front of the upper plate and dip toward the respective accretionary wedge or fold-and-thrust belt, thus forming downhill-facing scarps (e.g., Kumar et al., 2010; Ran et al., 2010; Schmidt et al., 2011). Owing to the absence of plate tectonics on the Moon, similar tectonic settings are not present there. However, strains expected to result from contraction of the lunar crust due to secular cooling, in addition to diurnal tidal and orbital recession stresses, appear to be

Please cite this article as: L. Roggon et al., Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2016.12.034

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Fig. 8. Schematic sketches and Mohr diagrams for stress illustrating the state of stress at point P on a potential thrust-fault plane beneath horizontal (a) and dipping (b, c) surfaces. σ 1 and σ 3 are the maximum and minimum principal stresses, respectively. In the Mohr diagram, the differential stress (σ 1 – σ 3 ) is the diameter of the Mohr circle. (a) A notional thrust fault beneath a horizontal surface. (b) A notional fault plane that dips in the same direction as the surface. (c) A notional fault plane that dips in the opposite direction as the surface. See Section 5.2 for discussion.

consistent with the distribution and orientations of lunar scarps (Watters et al., 2015). Our model results indicate that uphill-facing scarps on the Moon preferentially form in areas where the topography is not flat. To illustrate the state of stress that controls the initiation of faulting for different topographic slopes, we present a Mohr circle analysis of the problem (Fig. 8). The Mohr diagram for stress (Mohr (1900); see, for example, Twiss and Moores (1992)), where the normal stress, σ N, is plotted on the horizontal axis and the shear stress, τ , is shown on the vertical axis, illustrates the state of stress at a point (Fig. 8). The Mohr circle is always centered on the horizontal axis and its two intersections with this axis define the maximum and minimum principal stresses, σ 1 and σ 3 , respectively, such that the diameter of the Mohr circle equals the differential stress (i.e., σ 1 – σ 3 ). Each point on the Mohr circle represents the orientation of a plane in physical space, and the normal and shear stresses acting on this plane can be read from the two axes. To initiate faulting on a potential failure plane, the Mohr circle (i.e., the differential stress) needs to be large enough to touch the Mohr-Coulomb failure envelope (Fig. 8). This is illustrated for a point, P, located on a potential thrust fault below a horizontal (Fig. 8a) and two dipping surfaces (Fig. 8b and c), respectively. Note that for thrust faulting to occur, σ 1 must be horizontal and σ 3 vertical (e.g., Anderson, 1951). Let us now assume that the differential stress at point P is not high enough to cause failure beneath a horizontal surface for rocks with a given set of mechanical properties (Fig. 8a). However, for a surface that dips in the same direction as the potential fault plane, the overburden and hence the vertical principal stress σ 3 at point P is reduced, which increases the Mohr circle diameter, possibly to the point of fault initiation in rocks with identical mechanical properties (Fig. 8b). Beneath a surface that dips in the opposite direction as the potential fault plane, the minimum principal stress σ 3 is larger and the Mohr circle is therefore smaller and moves away from the failure envelope, which suppresses faulting (Fig. 8c). This analysis provides a dynamical basis for how thrust faults with uphill-facing scarps form on the Moon.

6. Conclusions We quantified the maximum displacement of four thrust-fault scarps on the Moon with a series of topographic profiles, from which we evaluated the along-strike variations in fault slip. Such measurements were not possible prior to the LRO mission, which has provided NAC-derived terrain models of unprecedented spatial resolution. Our investigation of these lunar scarps indicates that the ratio of maximum displacement to fault length is ∼0.023, and is thus similar to terrestrial faults. This result needs to be taken into account in lunar tectonic studies including, for example, when attempting to quantify the magnitude of the Moons´ global contraction using fault scaling relations. The formation of uphill-facing scarps requires less energy than do downhill-facing scarps. Our numerical model indicates that uphill-facing scarps form earlier and grow faster than downhill-facing scarps under otherwise similar conditions. Acknowledgments We gratefully acknowledge the work and help of the LRO and LROC teams to obtain and process the new data that made this study possible. HH, JDC, and CvdB were supported by the German Space Agency (Projects 50OW0901 and 50OW1504). We thank Paul Byrne and Matteo Massironi for their detailed and thoughtful reviews, which greatly improved the manuscript. References Adam, J., Klaeschen, D., Kukowski, N., Flueh, E., 2004. Upward delamination of Cascadia Basin sediment infill with landward frontal accretion thrusting caused by rapid glacial age material flux. Tectonics 23, TC3009. doi:10.1029/ 20 02TC0 01475. Anderson, E.M., 1951. The Dynamics of Faulting and Dyke Formation with Applications to Britain. Oliver and Boyd, Edinburgh, p. 206. Avouac, J.P., Tapponnier, P., Bai, M., You, H., Wang, G., 1993. Active thrusting and folding along the northern Tien Shan and Late Cenozoic rotation of the Tarim relative to Dzungaria and Kazakhstan. J. Geophys. Res. 98, 6755–6804.

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Please cite this article as: L. Roggon et al., Length-displacement scaling of thrust faults on the Moon and the formation of uphill-facing scarps, Icarus (2017), http://dx.doi.org/10.1016/j.icarus.2016.12.034