Are normal fault earthquakes due to elastic rebound or gravitational collapse?

We discuss two competing models for explaining the ground deformation associated with normal faulting earthquake in the brittle elastic upper crust. The classic elastic rebound theory is usually applied for all tectonic settings. In normal fault earthquakes, this model would predict a horizontal stretching eventually responsible for the elastic rebound at the earthquake. However, volumes mostly subside vertically during an extensional earthquake and the collapsed ground in the hanging wall is about one order of magnitude larger than the uplifted volumes of the surrounding hanging wall and footwall. The elastic rebound model would explain this asymmetry with a high horizontal elastic compressibility of the hanging wall and footwall absorbing the coseismic push. We rather suggest that the force activating normal fault earthquakes is mostly dictated by the sliding of the hanging wall, owing gravitational potential. The much larger coseismic subsidence with respect to the uplift can be explained by the closure at depth of a diffuse network of microfractures developed during the interseismic period. Since the horizontal stretching does not exist below ~1 km of depth, with the minimum horizontal stress tensor becoming positive below that depth, the development of a normal fault can be activated only by the vertical maximum stress tensor, i.e., the lithostatic load. The common fluids expulsion at the coseismic stage requires diffuse secondary permeability in the upper crust, in agreement with the presence of a diffuse network of microfractures.


Introduction
The brittle upper crust can be assumed as an elastic medium at the timescale of earthquakes, delivering elastic waves when perturbed instantaneously [e.g., Stein and Wysession, 2009]. Segall and Heimisson [2019] recently suggested that the elastic dislocation theory in extensional tectonic settings does not require a volume collapse at depth, as proposed by Bignami et al. [2019]. Segall and Heimisson [2019] indicate that the common theory of the horizontal elastic rebound reasonably reproduces the observed geodetic data, and infer how the extra coseismically subsided volume is accommodated by compressibility in the hanging wall. However, their analytical solution fails to answer two fundamental questions about the ground deformation associated with earthquakes in extensional tectonic settings. The first concerns the energy controlling the hanging wall motion; the second where the extra subsided volume would be adsorbed, considering the real mechanical constraints. Valerio et al. [2018] and Bignami et al. [2019] demonstrated that, for the Mw 6.5 central Italy, 2016 earthquake, the coseismically subsided volume is far greater than the uplifted crustal volume (Figure 1), posing the question of where this unbalance is compensated at depth, and what is the energy that can explain the geometry and kinematics of the half-graben (Figure 2).
We investigate here the energy source and kinematics of normal fault earthquakes, keeping in mind that the fault plane is only the passive discontinuity along which part of the energy accumulated by the crustal volume during the interseismic period is dissipated at the coseismic stage. Christian Bignami et al.
2 Figure 1. Four cross-sections of the ground motion generated by the Mw 6.5 Amatrice-Norcia 2016 (central Italy) seismic sequence [after Valerio et al., 2018]. For each section, the area of coseismic subsidence (red) is generally 10 times larger than the uplifted area (blue). The horizontal represents the pre-earthquake topography.
vertical movements that give the overall net difference between subsided and uplifted volumes, without any regard about the separated amount of these two volumes and their real (or approximate) ratio. The Segall and Heimisson [2019] formulation is therefore too simplified and much less precise than InSAR data that, on the contrary, have allowed calculating the ratio between the subsided and uplifted volumes. This is testified by the not negligible 20% 3 Graviquakes or normal fault earthquakes Figure 2. The elastic rebound and the gravitational collapse explain differently the asymmetry between the much larger coseismic subsidence associated to a normal fault earthquake, with respect to the uplifted volumes both in the hanging wall and footwall starting from the preparation phase during the interseismic period (above). In the middle panel are the observations, e.g., the vertical motion is larger than the coseismic horizontal shift recorded both by GPS and SAR. Below are the elastic rebound and the graviquake models at the coseismic stage. of discrepancy they found for the volume difference, which raises to 27% in our computation made with their technique. Hence, with this approach and numbers, they cannot confute the model proposed by Bignami et al. [2019]. Moreover, they compute the sum of the positive and negative uplift, without entering into the problem of where the ghost extra subsided volume disappears. Segall and Heimisson [2019] ignored the problem assuming high compressibility of the hanging wall or footwall, disregarding the real Poisson coefficient at crustal depth. Furthermore, they did not calculate in which volume this contraction should be accommodated in the hanging wall and at its bottom, close to the brittle-ductile transition. By the way, the compressibility should be allowed as well by a pre-existing dilation.
It is well established from the theory of mechanics of faulting [Anderson, 1951] that in extensional tectonic settings the maximum stress tensor (σ 1 ) is the lithostatic load (ρgz). Therefore, the main force moving the fault hanging wall is just gravity. Moreover, the pre-requisite for the Segall and Heimisson [2019] model is that, in extensional regimes, the crust is under horizontal tension, which is the case only above ~1 km depth [e.g., Twiss and Moores, 1992;Zoback, 2010] (Figure 3). This means that the differential stress has the minimum stress tensor (σ 3 ) positive below ~1 km depth, i.e., it is in compressive state. In these settings, the σ3 is horizontal and therefore the crust is not undergoing tensile stress below 1 km depth (Figure 4). The vertical load (σ 1 ) is roughly stable, apart oscillations due to variable fluid pressure, atmospheric load, snow, etc. Rupture and faulting occur because, due to plates stretching, the minimum stress (σ 3 ) decreases through time, thus increasing the differential stress during the interseismic period, and enlarging the Mohr circle to the left, hence intercepting the Coulomb critical failure envelope or the Byerlee's law, which governs the process of frictional sliding, since the seismogenic fault is usually not a newly-formed shear fracture ( Figure 5) [e.g., Carminati et al., 2020].
The other question is where does go the extra subsided volume. According to the general model supported by Segall and Heimisson [2019], this is accommodated by an elastic recovery associated with the horizontal rebound. Bignami et al. [2019] rather argued that the volume unbalance can be resolved only by a progressively dilated volume  In extensional tectonic settings, the minimum stress tensor (σ 3 ) generates horizontal traction only above 1 km of depth, where it is negative. Below that depth it becomes positive, i.e., the crust is in compressive state even in rifting areas due to the increase of the confining pressure generated by the vertical, lithostatic load. Here an average density of 2.5 g/cm 3 and an increase of the lithostatic load of 25 MPa/km is assumed. The area between the lithostatic load and the minimum horizontal stress represents the deviatoric stress and contains the stress data measured in wells. To the right, the compressional tectonic setting in which contraction in the crust occurs also at shallow depth.  compression test as it is for earthquakes. Therefore, values of high ν are highly plausible and when assuming a Poisson's coefficient ν = 0.4, the net volume decrease would be 0.03 km 3 , against the 0.10 km 3 of Bignami et al.
[2019] estimate. The unbalanced ratio of 7.5 between subsided versus uplifted volumes described by Bignami et al.
[2019] is only a minimum conservative value. Valerio et al. [2018] found a ratio up to 14 times ( Figure 1). A lower Poisson coefficient of those rocks can be obtained only in presence of a multitude pre-existing millimetric fractures, which are visible in outcrops of tectonized carbonates and in boreholes. Thus, the volume unbalance is far from being resolved via elastic recovery associated with the horizontal rebound, whereas the extra volume of the hanging wall has to be allocated somewhere else. Also, in the semi-infinite half-space of Okada-like models, this unbalance can be accommodated far away from the fault surface. In the real Earth, GPS and SAR data rather indicate that the coseismic motion is confined within few tens of km away from the fault and the extra subsided volume must be compensated within that upper crustal brittle volume.
Rocks under compression may undergo pressure solution with related cleavage development, but this process does not occur instantaneously as required by the earthquake duration. Although the crust is under contraction even in extensional settings, fractures may form due to the lithostatic load. These fractures tend to parallel σ 1 and they may form as the positive σ 3 decreases in time due to the extensional tectonics, or because of compaction. Rocks under compression have much higher strength with respect to dilation, and once broken, they lose most of their elasticity. Valerio et al. [2018] demonstrated that a possible interpretation of the observed volumetric asymmetry related to the Mw 6.5 central Italy 2016 earthquake can be given through a numerical analysis showing the role of the hanging wall gravitational sliding. They proposed a 2D numerical model of the 2016 earthquake developed in a Finite Element (FE) environment under a structural-mechanic physic context. Specifically, this approach takes into account the linear elasticity behavior of the medium by considering the plain strain approximation mode. This aims to reduce the differences among the modelled vertical movements and those computed with the available ALOS-2 DInSAR measurements. In addition, in the numerical modelling, the tectonics and distribution of the seismicity in the study area are included, in order to account for the mechanical anisotropies and the active faults in the Apennines chain. We also specify that the Mw 6.5 2016 earthquake is not the first case study in which this approach has been applied, because it has already been successfully exploited for modeling other seismic sequences recorded in Central Italy [Tizzani et al. 2013;Lavecchia et al., 2016;Castaldo et al., 2018].

Results
For sake of brevity, here we do not provide an extensive discussion on the applied numerical approach, which can be found in Tizzani et al. [2013]. We prefer to focus on the results of the simulated scenario accounting for the structural frameworks of the investigated area, which allows to retrieve the best fit coseismic deformation pattern.
This consists of a SW-dipping normal fault and a conjugate antithetic dilated zone that may evolve into normal faults. See for example the ENE-dipping conjugate fault of the 2016 Amatrice-Norcia events that has been imagined both by hypocentral distribution of aftershocks, and it was mapped in the field [Galli et al., 2019a;2019b;Improta et al., 2019;Galderisi and Galli, 2020]. Both structures lay above a low-angle ENE-dipping decollement (Figure 6a).
The lithological setting is characterized by the following geological units having isotropic mechanical properties and constant densities [Valerio et al., 2018] surface deformation observed by the ALOS-2 DInSAR measurements (Figure 6c), [Valerio et al., 2018]. In addition, the crustal distribution of the modeled vertical displacement, derived by the optimization procedure (Figure 6d), highlights that the considered crustal wedge produced a significant modulation effect of the crustal displacement field that is coherent with the observed ground deformation. This analysis allows us to interpret the subsidence and uplift phenomena caused by the 2016, Mw 6.5 earthquake as the result of the gravitational motion of the hanging wall sliding along the normal fault and the frictional force acting in the opposite direction. This is coherent with the generation of the double couple along the fault plane recorded in the moment tensor, although its component may often be only partial [e.g., Miller et al., 1998]. The data analysis and the performed numerical models demonstrate that: (i) the coseismic uplift in the hanging wall block is about 1/14 of the subsidence, suggesting a strong volumetric asymmetry and unbalance between the subsided and the uplifted volumes within the upper crust, (ii) our 2D modelling highlights the presence of an antithetic zone as a prerequisite to reach the best fit between measured and simulated coseismic surface motions.

Discussion
The discussion in this article is not whether the brittle upper crust is elastic or not, but where the energy that generates normal fault earthquakes comes from and which is the main mechanical framework. These events dissipate energy, obviously in part by means of elastic waves in an elastic medium. However, normal fault earthquakes appear to be governed by different energy with respect to the other types of earthquakes. There are several differences between the seismicity of extensional tectonic settings with respect to strike-slip and compressive earthquakes that support a different origin for normal fault earthquakes with respect to the other tectonic settings, where the volume moves without vertical shift (transcurrent) or either against gravity (thrust).
Here is a list of some relevant differences among normal faults and thrusts: 1. In extensional tectonic settings the σ 1 is the lithostatic load (gravity), whereas in compressive settings the σ 1 is the horizontal tectonic stress (elasticity); 2. At the global scale, the Gutenberg-Richter b-value is around 1.1 for normal faults, whereas is about 0.9 for thrusts [Schorlemmer et al., 2005], according to the lower maximum magnitude generated by normal fault earthquakes; 3. Normal fault-related earthquakes in crustal rifting areas have the maximum magnitude around 7.5, whereas thrusts may reach Mw 9.5; 4. Within the crust, the volumes involved by extensional faulting have a length about 3 times larger than the seismogenic thickness, which is usually the brittle-ductile transition or shallower decoupling surfaces, as in the Amatrice-Norcia 2016 seismic sequence. An example is Wasatch Fault in the central US, which is about 350-400 km long, but only segments in average 40-50 long ruptured [Machette et al., 1991]. Along compressive belts with high convergence rate (>6-7 cm/yr) this ratio may reach over 25 times (e.g., the 2004 Banda Aceh or the 2011 Tohoku Mw>9 events). In regions where the convergence rate is slower (few mm/yr), this ratio decreases to 2-4 times as well as the magnitude that can be up to Mw 6-7 (e.g., Italy accretionary wedges, ); 5. With a given magnitude, the aftershocks of normal faults last longer than those of seismic sequences associated with thrusts [Valerio et al., 2017]. This is consistent with the contrast between the volume motion in favor or against gravity, respectively; 6. With a given magnitude, the stress drop of extensional earthquakes tends to be lower than that of compressive earthquakes [Cocco and Rovelli, 1989;Rovelli and Calderoni, 2014], which also supports the notion that rocks need more energy to move against gravity; 10. Earthquake magnitude increases with fault dip for normal faults, whereas it tends to be larger for a lower dip along thrust faults. The shallower the normal fault dip, the smaller the vertical gravitational movement and the shorter and slower the coseismic slip. The opposite can be observed for steeper normal faults. 11. Fluids react oppositely when comparing extensional and compressive tectonic settings [Muir Wood and King, 1993;Doglioni et al., 2014;Barberio et al., 2017]; fluids are contained in fractures or primary porosity and their squeezing or recall requires contraction and dilation respectively as expected during the coseismic inversion in the two opposite settings; the pre-coseismic Vp/Vs ratio increase [Lucente et al., 2010] in extensional tectonic settings supports this interpretation. InSAR data provide an extremely precise coseismic amount and extent of subsidence, showing that it is the entire hanging wall volume that collapsed vertically [Bignami et al., 2019]; the macroseismic larger damages occurred in the area involved by the coseismic subsidence (near field), providing evidence of the influence of the vertical component of peak ground acceleration in weakening the strength of masonry buildings [Liberatore et al., 2019].
For all these reasons it has been suggested to classify earthquakes as a function of their source of energy, which is controlling the different phenomenology associated to normal faulting in favor of gravity (graviquake) and thrusting against gravity dominated by elastic energy (elastoquake) [Doglioni et al., 2015;Petricca et al., 2015;. The extensional seismicity of the Apennines is distributed along faults activated by volumes that, episodically, move brittle crustal prisms downward. The deformation gradually migrates over the whole seismogenic area undergoing tectonic extension. For example, the Norcia fault responsible for the 1703 sequence [Galli et al., 2018] was affecting a different volume to the west of the last 2016 seismic sequence. Segall and Heimisson [2019] stated that the horizontal stretching is responsible for an elastic rebound during the earthquake and that the larger coseismically volume is elastically adsorbed in a compressible hanging wall.

Conclusions
In turn, Bignami et al. [2019] suggest that the force is rather dictated by the gravitational potential, whereas the much larger coseismic subsidence with respect to the uplift can be explained by the abrupt closure at depth of a diffuse network of fractures, which progressively developed during the interseismic period.
It is worth noting that the elastic rebound model neglects either the strain partitioning between the upper brittle and the lower ductile portions of the crust, or the variations in competence of any further interlayered formations. Segall and Heimisson [2019] concluded the criticism to the Bignami et al. [2019] model claiming that geodetic data are consistent with elastic dislocation theory and do not require a volume collapse at depth.
Conversely, in a number of papers [Doglioni et al., 2011;2015;Bignami et al., 2019] it was emphasized the role of the brittle-ductile transition (BDT), or shallower decollements as the zone separating two different styles of interseismic deformation during the preparatory phase of an earthquake. A dilated wedge is kinematically required in the upper cold crust because normal faults are locked, whereas the lower crust is permanently shearing. This is a prerequisite in accommodating the collapsing volume during a normal fault earthquake.
Moreover, the dilated wedge should be populated by thousands of millimetric fractures filled by fluids as suggested by magnetotelluric surveys and higher Vp/Vs ratio. The search of those fractured volumes may illuminate where will be the next major seismic sequences.
The claimed Occam razor is in favor of the simplest explanation for the activation of normal fault earthquakes, which in this case is the force of gravity (Figure 7). The gravity force is absent in the Segall and  Comparison between the coseismic model of elastic rebound and graviquake for normal faulting (BDT, brittleductile transition). In the elastic rebound, the larger subsided volume is compensated by the diffuse horizontal contraction in the hanging wall or by the dissipation of the volume outside the section, being required a decoupling at the brittle upper crust base where the differential stress for rupture is maximum. In the graviquake model, the extra volume is rather adsorbed by an interseismic pre-dilated wedge. The hanging wall moves primarily downward, as dictated by the lithostatic load σ 1 . Since the horizontal pull is missing below 1 km, the coseismic dominant motion is not horizontal but rather vertical, and the maximum stress tensor is given by the lithostatic load; therefore, we consider the graviquake model more realistic.

Figure 8.
In extensional tectonic settings, the ratio between the seismogenic layer depth and the hanging wall length appears to be, in average, 3(±1). As the thickness increases, the volume enlarges, the fault is longer, and the hanging wall collapse determines a greater fault slip and a consequent higher earthquake magnitude. Note that doubling the seismogenic layer from 5 to 10 km, the volume increases about 1 order of magnitude, as the maximum earthquake size, from M5 to M6. BDT, brittle-ductile transition. The red wedges in the section above suggest the inferred dilated crustal volume permeated by thousands of millimetric microfractures.  fault aftershocks are much more numerous and last longer. This can be interpreted as related to the motion in favor of gravity and the sequence will last when the final gravitational equilibrium is reached. Along thrust faults, the motion against gravity inhibits and buffers most of the further motion after the mainshock. Therefore the two sequences can be inferred as earthquakes delivering gravitational and elastic energy, respectively, and for this reason they should be distinguished into graviquakes and elastoquakes (below).
In spite of a smaller maximum magnitude, normal faults earthquakes are followed by a large number of aftershocks with respect to thrust related events [Valerio et al., 2017], this being compatible with the motion in favor of gravity ( Figure 10). Normal faults rupture as soon as the hanging wall can slide down when the lithostatic load overcomes the fault friction and the conjugate interseismically dilated wedge is present to accommodate the extra volume at depth. The gravitational energy can explain the lower volumes and magnitude of earthquakes associated to normal faults with respect to those moving against gravity [Doglioni et al., 2015] as shown in Figure 11. The hanging wall does not fall into a giant cave below, but it is obviously re-closing an intensely fractured rock volume like closing an accordion. The fundamental process governing stick-slip behavior holds and is frictional sliding, but the energy comes from the lithostatic load and can occur because of the deep presence of the pre-stretched upper crust.
This can explain why magnitude of normal fault earthquakes increases with dip, whereas it decreases for thrustrelated events as shown in Figure 12. Last but not least, the search for the fractured volume in the antithetic zone of forthcoming normal fault activation may represent a relevant target of investigation by tomography and magnetotelluric analysis, providing a clue for the areas more prone to seismic activation.