2. Deformable plate modelling methods and approaches

Rigid plate models can result in overlap at restored divergent margins, and underfit at restored convergent margins (Fig. 1, after Ady & Whittaker, Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019). To account for the internal deformation of plates, a variety of deformable plate modelling approaches have been developed. Recent works have provided highly insightful, yet variable, deformable plate models for a variety of tectonic settings (e.g. Cao et al. Reference Cao, Zahirovic, Li, Suo, Wang, Liu and Müller2020). However, this is not an entirely new approach and Dunbar and Sawyer (Reference Dunbar and Sawyer1987) were among the first to attempt a quantitative approach to deriving non-rigid plate reconstructions, followed by others (Srivastava & Verhoef, Reference Srivastava, Verhoef and Parnell1992; Williams et al. Reference Williams, Whittaker and Müller2011). A summary of the history of deformable plate modelling is provided in Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019). A recent advancement has been the inclusion of this approach in open source software such as GPlates (Gurnis et al. Reference Gurnis, Yang, Cannon, Turner, Williams, Flament and Müller2018; Müller et al. Reference Müller, Cannon, Qin, Watson, Gurnis, Williams, Pfaffelmoser, Seton, Russell and Zahirovic2018), resulting in deformable plate models of many regions (e.g. Cao et al. Reference Cao, Zahirovic, Li, Suo, Wang, Liu and Müller2020) as well as globally (Müller et al. Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019). In addition, other groups have also developed their own deformable modelling workflows (Smith et al. Reference Smith, Kurtz, Richards, Forster and Lister2007; Kneller et al. Reference Kneller, Johnson, Karner, Einhorn and Queffelec2012, Reference Kneller, Albertz, Karner and Johnson2013; Ady & Whittaker, Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019).

One of the most prevalent of the deformable plate modelling workflows is that of the Earthbyte Group that is built into the GPlates software (after version 2.0) described in Gurnis et al. (Reference Gurnis, Yang, Cannon, Turner, Williams, Flament and Müller2018) (Fig. 2). As a platform GPlates has been used extensively to produce both global (Matthews et al. Reference Matthews, Maloney, Zahirovic, Williams, Seton and Müller2016) and regional (Phethean et al. Reference Phethean, Kalnins, van Hunen, Biffi, Davies and McCaffrey2016; Gac et al. Reference Gac, Minakov, Shephard, Faleide and Planke2020; Romagny et al. Reference Romagny, Jolivet, Menant, Bessière, Maillard, Canva, Gorini and Augier2020) rigid-plate models, as well as deformable ones (e.g. Peace et al. Reference Peace, Welford, Ball and Nirrengarten2019; King et al. Reference King, Welford and Peace2020). When constructing deformable models in GPlates, the geometry and time-constrained existence (time of appearance plus disappearance) of plate deformation zones between portions of the plates considered rigid is defined (Gurnis et al. Reference Gurnis, Yang, Cannon, Turner, Williams, Flament and Müller2018). The deforming regions combine extension, compression and shearing that accommodate the relative motion between rigid blocks (Gurnis et al. Reference Gurnis, Yang, Cannon, Turner, Williams, Flament and Müller2018). This allows users to explore how strain rates, stretching and shortening factors, and crustal thickness evolve through space and time within deforming regions and interactively update the kinematics associated with deformation to see how these parameters are influenced by alternative scenarios (Müller et al. Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019). It could be argued that the main limitation of the GPlates deformable models is that deformation occurs in the context of rigid domains. However, previous studies applying this methodology claim to produce independently validated results (Welford et al. Reference Welford, Peace, Geng, Dehler and Dickie2018).

Fig. 2. Schematic depiction of the GPlates deformable plate tectonic modelling workflow followed by comparison with independent regional constraints modified after Peace et al. (Reference Peace, Welford, Ball and Nirrengarten2019) and King et al. (Reference King, Welford and Peace2020) based on methods described in Gurnis et al. (Reference Gurnis, Yang, Cannon, Turner, Williams, Flament and Müller2018). Panel 1 shows the model inputs (deformable boundaries and optional interior features) required to construct a topological network. ECC = edge of continental crust, and COB = continent–ocean boundary. Panel 2 demonstrates the resultant triangulation mesh created after a topological network is constructed in GPlates. Panel 3 displays a crustal thickness model created using the deformable mesh constructed in panel 2. Panel 4 demonstrates the need to assess the validity of crustal thickness models created in GPlates against independent observations such as estimates from gravity inversion and observations from seismic data as in Welford et al. (Reference Welford, Peace, Geng, Dehler and Dickie2018).

Using the GPlates methodology, Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) produced a global Mesozoic–Cenozoic deforming plate motion model that captures the progressive extension of all continental margins since the initiation of Pangaea’s dispersal starting at ˜240 Ma. The Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) model includes major failed continental rifts and compressional deformation. Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) acknowledge that their global model will likely form the starting point for more regional models. The Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) model includes deforming zones for an area of ˜8 % of the Earth’s surface in which distributed deformation of the lithosphere occurs. However, Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) note that this value is significantly smaller than other estimates (e.g. the Kreemer et al. (Reference Kreemer, Blewitt and Klein2014) 14 % estimate). Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) state that the main reason for the discrepancy is that they solely focus on deformation of continental lithosphere, and therefore exclude large deforming regions in ocean basins. In addition, the model of Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) also excludes deforming edges of overriding plates along subduction zones, which form another significant portion of the Kreemer et al. (Reference Kreemer, Blewitt and Klein2014) geodetic strain-rate model. The Müller et al. (Reference Müller, Zahirovic, Williams, Cannon, Seton, Bower, Tetley, Heine, Le Breton, Liu, Russell, Yang, Leonard and Gurnis2019) model was applied by Ebbing et al. (Reference Ebbing, Dilixiati, Haas, Ferraccioli and Scheiber-Enslin2021) to provide new insights into Gondwana’s dispersal, demonstrating the applicability of such models.

At the regional scale, Peace et al. (Reference Peace, Welford, Ball and Nirrengarten2019) applied the GPlates deformable modelling workflow to the southern North Atlantic region by constructing a number of plate tectonic models and comparing the resultant crustal thicknesses with independently derived crustal thickness models from gravity inversion. This showed how resultant deformation from different plate models can be compared systematically. Similarly, Welford et al. (Reference Welford, Peace, Geng, Dehler and Dickie2018) applied the GPlates methodology to Baffin Bay showing that external model boundaries need to be a sufficient distance from the eventual location of break-up to prevent edge effects. King et al. (Reference King, Welford and Peace2020) studied the Galicia Bank on the Iberian margin and demonstrate that the GPlates methodology can be used to study structural inheritance. Gurnis et al. (Reference Gurnis, Van Avendonk, Gulick, Stock, Sutherland, Hightower, Shuck, Patel, Williams, Kardell, Herzig, Idini, Graham, Estep and Carrington2019) used the GPlates deformable plate modelling methodology to study the Puysegur Trench, New Zealand, and similarly Liu et al. (Reference Liu, Ma, Zhang and Gurnis2021) used the GPlates deformable model workflow to study the subduction of the Pacific plate in East Asia. Recently, Cao et al. (Reference Cao, Zahirovic, Li, Suo, Wang, Liu and Müller2020) applied this modelling workflow to the South China Block. The intraplate deformation in South China is closely related to the subduction of the Izanagi and Pacific plates. Thus, Cao et al. (Reference Cao, Zahirovic, Li, Suo, Wang, Liu and Müller2020) demonstrate the applicability at settings that include a compressional phase. Zhu et al. (Reference Zhu, Liu, Zhang, Gurnis and Ma2021) built a deformable plate model using GPlates of the Bohai Bay Basin in North China which is claimed to have implications for the entire East Asia region, demonstrating how insightful such reconstructions can be.

An alternative approach to deformable plate tectonic modelling has been developed and applied by Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) (Fig. 3). They produced a model for the entire North Atlantic region to study structural inheritance, and as an aid to exploration on the continental margins. Amongst other items, the inputs for the Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) model include a map of total beta (total beta = initial crustal thickness / final crustal thickness) in the model inputs. The map of total beta in the Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) workflow is calculated from crustal thickness obtained through gravity inversion and, where available, seismic data. A palinspastic deformation model guided by plate-derived Euler flowlines is then made in which deformation is iteratively restored. As with the GPlates models, the Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) models assume pure shear deformation and suggest that this is an adequate approximation at the scale of their study. Assumption of pure shear deformation in rift systems is not universally applicable, as many rifts have been shown not to obey symmetrical thinning (Becker et al. Reference Becker, Franke, Trumbull, Schnabel, Heyde, Schreckenberger, Koopmann, Bauer, Jokat and Krawczyk2014; Peace et al. Reference Peace, McCaffrey, Imber, Phethean, Nowell, Gerdes and Dempsey2016).

Fig. 3. Schematic demonstration of deformable restoration of a conjugate margin pair using an iterative palinspastic methodology as described in Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019). T1–T3 refer to the time-steps in such an iterative methodology. (a) Present-day (0 Ma) cross-section through a post-break-up hyperextended conjugate margin pair showing total beta trending to ∞ due to hyperextension (Hyp. Ext.). (b) Active rifting stage showing the restored beta factor increasing towards the centre of the rift. (c) Restored cross-section to pre-rift configuration with inferred pre-existing structures and total beta equal to 1. The concepts in this figure are modified from Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) who show the Norwegian – East Greenland conjugate margin pair (e.g. Peron-Pinvidic et al., Reference Peron-Pinvidic, Gernigon, Gaina and Ball2012), including the Jan Mayen Microcontinent (e.g. Schiffer et al. Reference Schiffer, Peace, Phethean, Gernigon, McCaffrey, Petersen, Foulger, Wilson, Houseman, McCaffrey, Doré and Butler2019). The example shown here, however, does not contain a microcontinent and shows relatively symmetrical rifting.

Both the GPlates method (Gurnis et al. Reference Gurnis, Yang, Cannon, Turner, Williams, Flament and Müller2018) and the Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) method require that continent–ocean boundaries (COBs) are accurately defined both spatially and temporally. Peace et al. (Reference Peace, Welford, Ball and Nirrengarten2019) found that the timing and location of break-up plays a significant role in the resultant deformation. However, challenges in defining COBs and break-up ages have been noted (Eagles et al. Reference Eagles, Pérez-Díaz and Scarselli2015), presenting a problem for plate reconstructions. Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) suggest that most of the issues with COBs pointed out by Eagles et al. (Reference Eagles, Pérez-Díaz and Scarselli2015) are unlikely to occur when COBs are picked consistently for the specific purpose of plate kinematic modelling. Nonetheless, it is clear this aspect of all plate models requires caution, particularly as the highly variable structure of passive margins is well documented (e.g. Biari et al. Reference Biari, Klingelhoefer, Franke, Funck, Loncke, Sibuet, Basile, Austin, Rigoti, Sahabi, Benabdellouahed and Roest2021).

Another major issue remains, in that plate movements for the Mesozoic are largely defined by reconstructions of the oceanic domain (Seton et al. Reference Seton, Müller, Zahirovic, Gaina, Torsvik, Shephard, Talsma, Gurnis, Turner, Maus and Chandler2012). As such, a common obstacle encountered during deformable plate tectonic modelling is availability of constraints. Deformation within continental domains often occurs during time periods when reliable plate kinematic constraints are scarce (King et al. Reference King, Welford and Peace2020). The constraints on deformation in the continental domains are harder to obtain and less reliable compared to those from the oceanic domains such as datable (and globally correlatable) oceanic magnetic anomalies and fracture zones (Peace et al. Reference Peace, Welford, Ball and Nirrengarten2019). Constraints on the deformation within continental domains can be derived from analysing rift styles, fault geometry, preserved continental fragments and the timing of deformation through stratigraphic evidence observed in seismic sections and well data (Peace et al. Reference Peace, Welford, Ball and Nirrengarten2019). The limiting factor on deformable plate models is thus not the computational or methodological restrictions; rather, it is the need to obtain adequate constraints on timing and kinematics of deformation events. As the constraints on continental deformation are much poorer, modelling multiple scenarios is suggested (Peace & Welford, Reference Peace and Welford2020).

3. Inclusion or omission of microcontinental fragments in plate models

It is well known that in rifts a number of different types of relatively undeformed regions of continental material may prevail despite surrounding deformation (Peron-Pinvidic & Manatschal, Reference Peron-Pinvidic and Manatschal2010; Foulger et al. Reference Foulger, Doré, Emeleus, Franke, Geoffroy, Gernigon, Hey, Holdsworth, Hole, Höskuldsson, Julian, Kusznir, Martinez, McCaffrey, Natland, Peace, Petersen, Schiffer, Stephenson and Stoker2020; Schiffer et al. Reference Schiffer, Peace, Phethean, Gernigon, McCaffrey, Petersen, Foulger, Wilson, Houseman, McCaffrey, Doré and Butler2019; Neuharth et al. Reference Neuharth, Brune, Glerum, Heine and Welford2021). The genetic origin of such entities may vary (Whittaker et al. Reference Whittaker, Williams, Halpin, Wild, Stilwell, Jourdan and Daczko2016), as well as their size and degree of internal deformation, and they may be referred to as microplates, continental fragments, terranes, blocks, or by combinations of these or other terms (Peron-Pinvidic & Manatschal, Reference Peron-Pinvidic and Manatschal2010). Here the term ‘continental fragments’ is used for consistency, but no preference over other comparable terms is implied. Previous work shows how the inclusion or omission of continental fragments in a model can profoundly influence plate models (Peace & Welford, Reference Peace and Welford2020). As such, another theme in recent plate tectonic models is the defining of smaller regions that move independently. For example, in the Southern North Atlantic, Nirrengarten et al. (Reference Nirrengarten, Manatschal, Tugend, Kusznir and Sauter2018) defined a number of microcontinental fragments that, like the major plates, also have kinematics described by Euler poles. Likewise, Gion et al. (Reference Gion, Williams and Muller2017) reconstructed the Eurekan orogeny as a number of independent entities, each with its own Euler pole. These types of reconstructions address some of the same challenges that deformable models are attempting to solve. Moreover, models have been developed that include both independent smaller plates and deformable regions (Peace et al. Reference Peace, Welford, Ball and Nirrengarten2019).

However, questions remain regarding how small it is possible to go when defining individual plates. For example, could it ever be feasible that individual fault blocks are defined in plate models with their own Euler pole? The reality is that everything in an actively deforming region can undergo variable degrees of deformation. Even the microcontinental fragments in models such as that of Nirrengarten et al. (Reference Nirrengarten, Manatschal, Tugend, Kusznir and Sauter2018) have likely undergone some amount of internal deformation (e.g. rift-related deformation on the Flemish Cap), and as such the unresolved question of where to draw the line between deforming and undeforming regions remains.

The necessity of including microcontinental fragments to produce a better fit has been questioned by Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) who argue that their quantitative restoration methodology provides an alternative to their inclusion. In the Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) model, only five independent plates for the entire North Atlantic region are defined, demonstrating that adequate deformable models can be produced without smaller plates. However, it has also been suggested that models that include both microcontinental fragments and a deformable modelling workflow are likely to give the most realistic reconstruction of deformation (Peace et al. Reference Peace, Welford, Ball and Nirrengarten2019). It is thus clear that further investigation into the inclusion/omission of microcontinental fragments is necessary.

Another substantial and outstanding issue is how the constraints on the kinematics of continental fragments are derived. It has been suggested that through detailed mapping of structures in deformed regions the kinematics of continental fragments may be elucidated (King et al. Reference King, Welford and Peace2020). However, even with the most detailed mapping, resolving deformation histories is problematic and many scenarios must be considered to give any level of validation to a model (Peace et al. Reference Peace, Welford, Ball and Nirrengarten2019).

4. A classification system for deformable plate models: two fundamentally different types?

It is clear that not all deformable plate models are the same in terms of their capabilities, inputs and outputs. Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) propose a classification system with rigid, dynamic and deformable plate models, as well as subcategories. The Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) classification provides a good overview of the types of models. However, an alternative method of categorizing deformable models is through the required inputs, and specifically whether a model uses a beta/thinning factor as a model input or as a result. For example the Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) deformable model uses gravity inversion to derive crustal thickness which is then used to calculate beta factors. An iterative approach is then used to restore deformation. On the other hand, the GPlates methodology calculates a beta or stretching factor based on the relative movements of plates. Each of these approaches has different uses. The Ady & Whittaker (Reference Ady, Whittaker, Wilson, Houseman, McCaffrey, Doré and Butler2019) method likely more accurately reproduces deformation, but as deformation is not purely driven by the plate motions this method cannot be used to compare different plate models as easily as the GPlates method. The GPlates method, on the other hand, can be used as a validation tool for rigid plate models, as the fact that deformation is only driven by rigid plates allows kinematic models to be evaluated. Specifically, comparison of the resultant crustal geometries produced through different models with independent estimates can be made (Welford et al. Reference Welford, Peace, Geng, Dehler and Dickie2018). The question, however, remains of how best to validate the reconstructions depicted in deformable plate models.