Testing inverse kinematic models of paleocrustal thickness in extensional systems with high-resolution forward thermo-mechanical models

Authors


Abstract

[1] Reconstructing continental paleocrustal thickness is important for estimating tectonic accommodation, constraining three-dimensional basin geometry during early rifting phases of extensional margins and predicting the distribution of thick crustal sills that may block the global ocean and create restricted basins. We test an inverse kinematic method for modeling paleocrustal thickness by inverting the final bulk crustal structure produced from high-resolution thermo-mechanical models of lithospheric extension. The inverse kinematic method assumes pure shear, includes simple rules based on geodynamic models and field observations and requires displacement boundary conditions and the prescription of a transition from rigid to nonrigid deformation. The inverse pure-shear method produces a history of bulk crustal thickness that closely matches the forward models provided that the width of the rift zone is narrow during the later phases of continental extension when crust undergoes hyper-extension. We also observe that the width and surface trace of large-scale (LS) shear zones observed in the thermo-mechanical models coincide with inflection points and large gradients in inverted nonrigid velocity field. Our results demonstrate that if displacement boundary conditions can be constrained and the transition from rigid to nonrigid deformation defines a narrow rift zone during hyper-extension then relatively simple kinematic rules can be used to invert present-day bulk crustal structure for paleocrustal thickness, bulk lateral strain and aspects of upper crustal shear zone geometry from extensional systems with nonlinear rheology, structures dominated by simple shear in the upper crust, depth-dependent extension and asymmetric crustal thinning.

1. Introduction

[2] Reconstructing continental paleocrustal thickness is important for estimating tectonic accommodation, constraining three-dimensional basin geometry during early rifting phases of extensional margins and predicting the distribution of thick crustal sills that may block the global ocean and create restricted environments of deposition. Additionally, models of paleocrustal thickness and bulk lateral strain of the crust that are directly linked to displacement boundary conditions from global plate models can be used to test poorly constrained syn-rift plate kinematics against field observations that provide evidence for the timing and distribution of syn-rift deformation [Kneller et al., 2012]. Unfortunately, the paleocrustal thickness and bulk lateral strain history for a given continental margin cannot be accurately determined with purely forward thermo-mechanical models due to poorly constrained initial conditions and unknown initial rheological heterogeneity of the lithosphere. Furthermore, the application of high-resolution 3-D thermo-mechanical models in an inverse manner is presently not possible due to algorithmic and computational limitations. Because of these difficulties a forward thermo-kinematic approach is often used where crustal stretching history is prescribed and subsidence is calculated by solving equations governing vertical and horizontal heat conduction and flexural isostasy [e.g., McKenzie, 1978; Karner et al., 1997].

[3] Simplistic thermo-kinematic models use the ratio of an assumed initial crustal thickness over the postextension crustal thickness as a measure of the total bulk thinning of the lithosphere and apply this thinning to instantaneously advect material and heat [McKenzie, 1978]. More sophisticated thermo-kinematic models distribute crustal thinning over finite periods of time, prescribe the magnitude of crustal thinning using seismic observations that constrain the geometry of large-scale (LS) extension structures and have depth-dependent thinning in the crust and mantle lithosphere [e.g., Karner et al., 1997]. These more sophisticated approaches can be used to produce forward models of subsidence and predict the evolution of stratigraphic packages that can be directly compared to interpreted stratigraphy based on subsurface observations. However, the prescribed crustal stretching history required as input by thermo-kinematic models is difficult to define in three dimensions at plate-tectonic scales and in a manner that is directly linked to boundary conditions imposed by global plate motion.

[4] Thermo-kinematic models have also been applied to invert stratigraphic interpretations for mantle and crustal thinning history [White, 1994; Poplavskii et al., 2001; White and Bellingham, 2002; Bellingham and White, 2002; Ruepke et al., 2008; Hinsken et al., 2011]. These inverse thermo-kinematic methods search for crustal and mantle thinning parameters that yield subsidence histories that may in turn provide a good match to stratigraphic interpretations. However, these inverse thermo-kinematic methods require assumptions about initial crustal thickness, are difficult to link directly to displacement boundary conditions from global plate motion and do not necessarily produce crustal stretching histories consistent with present-day bulk crustal structure. Furthermore, these inverse thermo-kinematic methods are often more suited for modeling postrift thermal subsidence but are unlikely to predict accurately the bulk crustal structure during early syn-rift phases due to a limited syn-rift deposition or erosion of syn-rift sediments.

[5] Kneller et al. [2012] presented a new kinematic method for inverting present day crustal thickness and plate kinematics for the history of bulk lateral strain and paleocrustal thickness associated with lithospheric extension (Figure 1). With this kinematic method deformation is approximated as pure shear and simple rules, which are based on geodynamic models and field observations, control the timing and distribution of bulk vertical and lateral strain in the crust. A key advantage of this inverse method is that it simplifies the problem of predicting bulk crustal structure from a forward dynamic problem or an inverse thermo-kinematic problem, both of which involve many unknowns and poorly constrained initial and boundary conditions, to a kinematic inverse mass balance problem based on simple rules and only two boundary conditions: (1) the rate and direction of plate divergence and (2) the width of the rift zone. Additional advantages of the approach of Kneller et al. [2012] are (1) that it can easily be applied in 3-D at plate tectonic scales due to computationally inexpensive calculations and (2) that it can be directly linked to global plate motion and used to predict first order basin geometry associated with crustal thickness variations during syn-rift phases of extensional margin formation. Kneller et al. [2012] applied this method at plate-tectonic scales to test syn-rift to breakup plate reconstructions of the Central Atlantic and reproduced the general patterns of extension and subsidence associated with crustal thinning inferred from field observations.

Figure 1.

(a) Schematic example of the kinematic inversion method applied to an extensional system with two conjugate plates (present-day configuration 0 Ma) [modified from Kneller et al., 2012]. (b–d) With each time step a rigid reconstruction (inversion at 2, 4, 8 Ma) is performed followed by a nonrigid reconstruction (inversion at 2, 4, 8 Ma). The blue triangles represent the rigid nonrigid transition (RNT), which controls strain localization and the width of extension. Inverse displacement of particles occurs along evolving flow lines and is a function of incremental rigid overlap and time varying crustal thickness.

[6] Here we test the consistency of the kinematic inversion scheme of Kneller et al. [2012] with conservation laws by inverting crustal thickness produced from forward thermo-mechanical models of lithospheric extension. The thermo-mechanical models used in this work includes a high-resolution nested domain in the region of interest and frictional plastic (brittle) as well as temperature-dependent, viscous (power-law creep) behaviors of all rock types included in the model. The forward models exhibit a variety of processes that significantly affect the mass distribution of the crust during lithospheric extension such as strain accommodation along LS simple shear structures, depth-dependent thinning and asymmetric extension. We consider the kinematic pure-shear inversion successful if it provides an accurate approximation of bulk crustal thickness through time observed in the forward geodynamic model.

2. Kinematic Inversion Method

[7] With the inversion method of Kneller et al. [2012], conjugate continental plates are discretized using Lagrangian meshes composed of linear triangles in a manner similar to Smith et al. [2007]. Crustal thickness is defined in each particle of the Lagrangian mesh, and a pure shear approximation is applied to calculate past thicknesses in retro-deformed meshes. Important input parameters for the inversion are mobile boundaries on each conjugate plate called the rigid-nonrigid transitions (RNTs). The RNTs define the width of the rift zone and mark a transition from a region with lateral velocity equal to rigid plate velocity to a nonrigid region with lateral velocity that is a fraction of total plate velocity (Figure 1). The RNT must be defined by the modeler using surface and subsurface observations that provide evidence for the timing and distribution of syn-rift extension such as seismic reflection data or outcrops showing growth of sediment packages into fault planes and biostratigraphic and radiometric age control on syn-rift deposition [e.g., Olsen, 1997].

[8] The inversion algorithm involves incremental rigid and nonrigid reconstructions (Figure 1). With each rigid reconstruction, overlap is produced that is a measure of the total bulk lateral strain that must be restored at a particular time step. This overlap is denoted by OA,B,t where A and B refer to conjugate plates and t is the age of the rigid incremental reconstruction. Overlap is partitioned throughout conjugate Lagrangian meshes along planes defined by Euler pole flow lines using the following equation, which is shown here in expanded form:

display math(1)

where DA,i,t is the inverse displacement of particle i in plate A, EA,i,t is the total integrated crustal extension from the RNT of plate A to the location of the particle, EA,t is the total integrated crustal extension of plate A from the RNT of plate A to the distal edge of plate A and EB,t is the total integrated crustal extension of plate B from the RNT of plate B to the distal edge of plate B. Total crustal extension is calculated along Euler pole flow lines using the following equation

display math(2)

where E is the magnitude of total crustal extension, θ is the angle of rotation about an Euler pole, θo is the position along the flow line at the RNT, θi is the position either of particle i or the distal edge of a plate and β, the crustal thinning factor, is defined as follows:

display math(3)

where Lmax refers to the assumed maximum thickness for given plate and L(θ)t is the crustal thickness along the flow line at time t. We note that equation (1) is time-independent for cases where the RNTs are fixed to conjugate plates and extension directions do not change over time.

[9] The first ratio on the right-hand side of equation (1) defines the asymmetry of extension by including total crustal extension from both plates and is used to partition overlap across conjugate plates. The second ratio defines a normalized crustal extension factor for a given Lagrangian particle and is used to distribute partitioned overlap throughout a given Lagrangian mesh. Equation (1) leads to a restoration where retro-deformation and thickening migrate from the distal parts of conjugate extensional systems to the proximal parts as the inversion proceeds. This pattern of extension is similar to the pattern associated with the field-based observations of Manatschal et al. [2007] that shows a transition from a wide rift zone with distributed strain during the early phases of extension to a focusing of extension during later phases.

[10] The pure shear assumption and inversion scheme presented in this work produce paleocrustal thickness and extension magnitudes that are identical to simple shear restorations of listric normal faults [Kneller et al., 2012]. Furthermore, the pure shear assumption used in this work produces nonrigid displacement with discontinuities and inflection points that mark key features of LS simple shear structures (Figure S1; supporting information).1

3. Dynamic Forward Models

3.1. Numerical Method

[11] The thermo-mechanical models of lithospheric extension were produced using the two-dimensional, plane-strain, thermo-mechanically coupled finite element method SOPALE (Simplified Optimized Arbitrary Lagrangian Eulerian), developed by the Geodynamics Group at Dalhousie University, Halifax, Canada [Fullsack, 1995; Willett, 1999]. The code solves the equilibrium force balance equations for incompressible viscous-plastic flows and the energy balance equation in two dimensions:

display math(4)
display math(5)
display math(6)

where σij is the deviatoric stress tensor, xi are the spatial coordinates, P = pressure, ρ = density, g = gravitational acceleration, vi is a component of velocity, Cp = specific heat, T = temperature, t = time, K = thermal conductivity, A = radioactive heat production per unit volume, and α = volumetric expansivity. The last term in equation (6) corrects temperature for adiabatic heating when material moves vertically. The system of equations generated by applying the finite element method to discretize equations (4), (5) is solved using a supernodal sparse Cholesky solver [Ng and Peyton, 1993; Gilbert et al., 1994].

[12] SOPALE employs an Eulerian grid that stretches vertically to accommodate the evolving material domain. A Lagrangian grid and a cloud of embedded particles are advected with the velocity field to update the distribution of properties on the Eulerian grid. We use a similar nested modeling approach as reported by Beaumont et al. [2009]: the calculations are made at two scales in order to improve model resolution in the region of interest; a small-scale (SS) subdomain is embedded within a LS model (Figure S2; supporting information). For each time step, the LS model solves the problem for the entire 600 (vertical) × 1200 (horizontal) km domain. The LS results are then used to define the velocity and temperature boundary conditions on the SS domain regions. The SS model solves the problem at higher resolution. The two model domains share a single cloud of Lagrangian tracking particles. The domain resolutions (Eulerian grid sizes and number of Lagrangian particles) are defined such that the LS domain has sufficient resolution to solve for the flow regime in all parts of the model domain. The SS domain is defined such that it overlaps significantly with the LS domain. This configuration ensures that the flows and stress states are the same in the outer part of the SS model and the adjacent LS. For example, the SS region includes the lithosphere where extension is actually occurring. Hence, both the SS and LS solutions are the same in the outer part of the SS model domain. Constant isostatic balance is achieved by maintaining an average pressure at the base of the model.

3.2. Material Properties

[13] The materials in our forward models have both plastic (brittle) and viscous (power law creep) properties. Frictional-plastic yielding is captured using the Drucker-Prager yield criterion:

display math(7)

where J2 is the second invariant of the deviatoric stress, P the mean stress, and C the cohesion. The effect of pore fluid pressure on yield strength is represented using an effective internal angle of friction, ϕeff such that

display math(8)

where ϕ=30° is the internal angle of friction for dry conditions when the pore fluid pressure, Pf = 0. In order to approximate hydrostatic fluid pressures, we define ϕeff = 15° for all model materials in the initial, undeformed configuration. Strain softening of the frictional-plastic materials is modeled using a linear decrease in the effective internal angle of friction, ϕeff, with accumulated plastic strain. This simplified method approximates strain softening occurring during progressive deformation.

[14] The effective viscosity η associated with power law creep is computed using

display math(9)

where B is the pre-exponential factor, converted to the tensor invariant form, I'2 is the second invariant of the deviatoric strain rate, n is the stress exponent, Q is the activation energy, V is the activation volume, and R is the universal gas constant. We use wet quartzite [Gleason and Tullis, 1995], wet feldspar [Rybacki et al., 2006], dry olivine [Karato and Wu, 1993], and wet olivine [Karato and Wu, 1993] to capture power law creep of upper crust, lower crust, lithospheric mantle, and asthenosphere, respectively (Table 1).

Table 1. Material Properties and Thermal Parameters
Earth domainUpper CrustLower CrustLithospheric MantleAsthenosphereUnit
  1. a

    Initial value.

  2. b

    Value for strain softening.

  3. c

    inline image

Rock typeWet quartziteWet feldsparDry olivineWet olivineN/A
Angle of internal frictiona, ϕ15151515°
Angle of internal frictionb, ϕ5555°
Cohesion, C2 × 1072 × 1072 × 1072 × 107Pa
Density, ρ2700290033003300kg m−3
Uniaxial pre-exponent factor, A1.1 × 10281.59 × 10112.42 × 10163.9 × 1015Pa−n/s
Tensor-invarient pre-exponent factorc, B8.57 × 10287.13 × 10111.43 × 10151.76 × 1014Pa s1/n
Stress exponent, n433.53 
Activation energy, Q223345540430kJmol−1
Activation volume, V03.8 × 10−51.5 × 10−51.5 × 10−5m3mol−1
Heat capacity, CP750750750750m2 s−2 K−1
Thermal conductivity, K5.8582.253.5 → 2.8852.885 → 50W m−1 K−1
Temperature range for change in KN/AN/A600 → 13201320 → 1330°C
Radioactive heat production, A2.5 × 10−61.85 × 10−700W m−3
Volumetric expansivity, α3.1 × 10−53.1 × 10−53.1 × 10−53.1 × 10−5K−1
Basal heat fluxN/AN/AN/A0.025W m−2
Geotherm21118.670.5°C km−1

3.3. Initial Configuration and Boundary Conditions

[15] The geodynamic models of lithospheric extension are generic and utilize an overall similar design as previously published [e.g., Huismans and Beaumont, 2007, 2011], except that we use a SS model domain to achieve higher resolution in the region of interest. The total model domain is 1200 km (horizontal) by 600 km (vertical) large and it comprises 15 km thick upper crust, 20 km thick lower crust, 90 km thick lithospheric mantle, and 475 km thick asthenosphere (Figure S2). The resolutions of the LS and SS domains are (horizontal by vertical) ca. 3000 m by 1250 m and 750 m by 333 m, respectively. In contrast to up to 200 km thick cratons, which we do not seek to represent in this study, this configuration is consistent with relatively young continental lithosphere.

[16] In order to trigger extensional deformation in the model, a constant, outgoing velocity of 0.25 cm/yr is applied to either side of the model lithosphere (i.e., full extension velocity = 0.5 cm/yr). The ingoing velocity of asthenosphere flowing back into the model is computed dynamically to achieve mass and isostatic balance. The initial temperature at the base of upper crust, Moho, and base of lithosphere are 326°C, 550°C, and 1330°C, respectively (Figure S3; supporting information). The thermal model parameters and associated thermal gradients are reported in Table 1. This combination of material and thermal properties yields a lithosphere with depth-dependent strength, such that upper strong crust exists above weak lower crust, which together is superimposed onto strong upper mantle (Figure S3).

[17] Two thermo-mechanical models are used to test the inversion scheme presented in this work and are referred to as TM1 and TM2. With model TM1 a 10 km by 10 km large weak seed (light gray box on Figure S2) is placed in the center of model, in the uppermost lithospheric mantle. The purpose of the seed is to nucleate the extension at the desired location. Thermo-mechanical model TM1 produces relatively symmetric and narrow crustal thinning. The distribution of material particles and strain rate at several time steps of model TM1 are shown in Figures 2 and 3. The evolution of crustal thickness from TM1 is shown in the supporting information (Animation 1).

Figure 2.

Distribution of material particles from thermo-mechanical model 1 (TM1) with a single weak seed at (a) 0 Ma. (b) 5 Ma. (c) 10 Ma. (d) 15 Ma. (e) 20 Ma. (f) 26 Ma. The boundary between original mantle lithosphere and accreted lithosphere is denoted by the black dashed line. Isotherms are denoted by thin blue lines. The region of hyper-extended crust is denoted in (Figure 2a). The bulk crustal structure shown in this figure at 0 Ma is inverted with inversion cases 1–3 using different interpretations for the RNT.

Figure 3.

Snapshots of strain localization from thermo-mechanical model TM1 at (a) 0 Ma. (b) 5 Ma. (c) 10 Ma. (d) 15 Ma. (e) 20 Ma. (f) 26 Ma. Isotherms are denoted by thin black lines. Arrows are velocity vectors. Inverted blue triangles show the location of the RNT used for inversion case 3. The location of the RNT through time was qualitatively estimated based on the distribution of high-strain rate calculated from computed velocity.

[18] The initial configuration of model TM2 is identical to that of model TM1 except that multiple seeds are placed in the upper crust (5 seeds) and uppermost mantle (four seeds). The seeds are evenly distributed over a ca. 400 km wide region (Figure 4) to account for the effect of multiple inherited weaknesses and thus more realistic incipient extension during the early model evolution. In order to prevent a lasting impact of the seeds on the long-term evolution in both models, the seeds are progressively changed to ambient crust or mantle when the pressure at the top and bottom of a given seed changes from the initial configuration. Therefore, the seeds localize extension only in the very early model evolution and the bulk of the model dynamics are controlled by the inherent behavior of the extension system. The net effect of embedding multiple seeds in the numerical model is the development of a wider rift zone as well as the emergence of an abandoned rift and more asymmetric crustal thinning.

Figure 4.

Distribution of material particles from thermo-mechanical model 2 (TM2) with multiple weak seeds and a wider at (a) 0 Ma. (b) 5 Ma. (c) 10 Ma. (d) 15 Ma. (e) 20 Ma. (f) 26 Ma. See figure 5 for a descriptions of features. The bulk crustal structure shown in this figure at 0 Ma is inverted with case 4.

4. Inversion of Forward Models

[19] We define two conjugate plates at 26 Myr by splitting the forward models with a vertical line centered in the region where strain is localized into a narrow 10 km zone and crustal thickness is less than 10 km (Figure S4; supporting information). Crustal thickness at this time step is interpolated onto conjugate inversion meshes that are 111 km wide and have element spacing of 5 km. For each inversion, we use an Lmax from equation (3) equal to 35 km, an inversion time step equal to 1 Myr and a full spreading velocity of 0.5 cm/yr.

[20] Figures 6-8 show inversions of thermo-mechanical model TM1 for cases 1–3, which involve different assumptions about how the RNT on each plate moved throughout time. Figure 9 shows case 4, which involves the inversion of model TM2. Animated output from these inversions is shown in the supporting information (Animations 2, 3, 4, 5, and 6). In order to simplify the comparison between inverted crustal structure and crustal structure produced by the forward model we plot the surface and Moho from both forward and kinematic models using local isostasy with assumed constant densities in the crust and mantle.

[21] Case 1 involves an inversion of model TM1 and assumes that the RNTs are fixed to tectonic plates and have large separation throughout rift evolution (Figure 6). Case 2 involves an inversion of model TM1 but includes mobile RNTs that become progressively narrower over time (Figure 7). For case 2, the RNTs on both conjugate plates approximately follow the boundaries of the region with geologically significant strain rates in the forward model. Case 3 is similar to case 2 but involves an alternative scenario for RNTs that is also based on the distribution of regions with high-strain rate observed in the forward model (Figure 3). This alternative model for RNTs has a narrower zone of nonrigid deformation in the region of hyper-extended crust during the initial inversion steps and a rift zone that is shifted toward plate B relative to case 2 from 5 Ma to 10 Ma (Figure 8). Case 4 involves an inversion of model TM2 (Figures 4 and 5) and mobile RNTs that are defined with an approach similar to that used in cases 2 and 3 (Figure 9). RNT locations are shown at each time step of the inversion in animations in the supporting information (Animations 2, 3, 4, and 6).

Figure 5.

Snapshots of strain localization from thermo-mechanical model TM2 at (a) 0 Ma. (b) 5 Ma. (c) 10 Ma. (d) 15 Ma. (e) 20 Ma. (f) 26 Ma. Inverted blue triangles show the location of the RNT used for inversion case 4. The location of the RNT through time was qualitatively estimated based on the distribution of high-strain rate calculated from computed velocity. See Figure 6 for a descriptions of features.

Figure 6.

Comparison of bulk crustal structure between the forward dynamic model TM1 (black lines) and kinematic inversion (green lines) for case 1 at different time steps: (a) Age = 0.0 Ma. (b) Age = 5.0 Ma. (c) Age = 10.0 Ma. (d) Age = 15.0 Ma. (e) Age = 20.0 Ma. (f) Age = 26.0 Ma. This case involves a wide rift zone and RNTs that are fixed to their associated plate. The surface of the crust and Moho from both the forward model and the inverse model are displayed using local isostasy for direct comparison.

Figure 7.

Comparison of bulk crustal structure between the forward dynamic model TM1 (black lines) and kinematic inversion (green lines) for case 2 at different time steps: (a) Age = 0.0 Ma. (b) Age = 5.0 Ma. (c) Age = 10.0 Ma. (d) Age = 15.0 Ma. (e) Age = 20.0 Ma. (f) Age = 26.0 Ma. This case involves mobile RNTs that define a narrow rift zone during the latter stages of extension and follow the boundaries of the region with geologically significant strain rates in the forward model. The surface of the crust and Moho from both the forward model and the inverse model are displayed using local isostasy for direct comparison.

Figure 8.

Comparison of bulk crustal structure between the forward dynamic model TM1 (black lines) and kinematic inversion (green lines) for case 3 at different time steps: (a) Age = 0.0 Ma. (b) Age = 5.0 Ma. (c) Age = 10.0 Ma. (d) Age = 15.0 Ma. (e) Age = 20.0 Ma. (f) Age = 26.0 Ma. This case involves mobile RNTs, a narrower rift zone relative to case 2 during the latter stages of extension and a shift in the rift zone toward plate B from 5 to 10 Ma. The surface of the crust and Moho from both the forward model and the inverse model are displayed using local isostasy for direct comparison.

Figure 9.

Comparison of bulk crustal structure between the forward dynamic model TM2 (black lines) and kinematic inversion (green lines) for case 4 at different time steps: (a) Age = 0.0 Ma. (b) Age = 5.0 Ma. (c) Age = 10.0 Ma. (d) Age = 15.0 Ma. (e) Age = 20.0 Ma. (f) Age = 26.0 Ma. This case involves asymmetric crustal thinning and a wider distribution of extensional deformation. The surface of the crust and Moho from both the forward model and the inverse model are displayed using local isostasy for direct comparison.

[22] For inversion case 1 where RNTs are fixed to plates and the region of nonrigid deformation is wide equation (1) leads to crustal thinning that is too rapid in distal regions of the extensional system where the crust thinned to less than 15 km (Figure 6). This mismatch between forward and inverse models is greatest in hyper-extended regions where final crustal thickness is less than 10 km. However, with case 1 the timing and magnitude of crustal thinning in proximal basins is reproduced reasonably well. This indicates that equation (1) is capturing the general pattern of strain localization calculated in the thermo-mechanical model but is under-estimating the magnitude of strain localization in the distal parts of the model.

[23] For cases 2 and 3 where the RNTs track the edges of regions in the thermo-mechanical model undergoing geologically significant strain rates the match between the inverse model and forward model significantly improves in the hyperextended domain (Figures 7 and 8). The match between the inverted paleocrustal thickness and the results of the forward model are improved further as the separation between RNTs becomes narrower during the later phases of extension (Figures 7 and 8). This result is consistent with field studies of exhumed extensional systems that suggest progressive localization in the distal parts of extensional systems [Manatschal et al., 2007]. The inversion of thermo-mechanical model TM2, which involves wider more asymmetric extension relative to TM1, also shows a good match between inverted crustal thickness and crustal thickness from the forward model. Case 4 demonstrates that equation (1) can accurately mimic lateral bulk strain and crustal thickness distributions from the forward model in wide regions (> 175 km) undergoing asymmetric crustal thinning permitted that minimum crustal thicknesses are greater than 15 km.

[24] The magnitude of inverted lateral velocity also closely follows the lateral velocity of the forward model, and inflection points and large gradients in the nonrigid inverted velocity approximately mark the surface trace of LS shear zones and the intersection of shear zones with zones of ductile deformation in the lower crust (Figures 10 and S5; supporting information). The nonrigid inverted velocity is calculated in reference to the associated plate and is defined with the following equation:

display math(10)

where vx,i,nr is the nonrigid inverted velocity at particle i, vx,i is the inverted velocity at particle i and vmax,plate is the maximum lateral velocity of the plate associated with particle i.

Figure 10.

A comparison between inverted nonrigid velocity from case 2 and strain rate from the thermo-mechanical model at (a) 15 Ma and (b) 20 Ma. Nonrigid velocity magnitude is calculated in a reference frame where the associated plate is held fixed (equation (10)). Inflection points (vertical black arrows) and large gradients in the nonrigid velocity approximately mark the surface trace of LS shear zones and the intersection of shear zones with zones of ductile deformation in the lower crust.

5. Discussion

[25] In the proximal environment of the extensional system the simple kinematic rule described in equation (1) produced a history of paleocrustal thickness that closely matches the general bulk crustal thickness produced by the forward model (Figures 6-9) for both widely separated and progressively narrower RNTs. For cases 2–4 where the RNTs from conjugate plates were placed in close proximity during the later phases of extension (Figures 7-9), the inverse pure-shear kinematic method presented in this work provided a more accurate reproduction of bulk crustal thickness produced by the forward model in both the proximal and distal parts of the extensional system.

[26] The accurate reproduction of crustal thickness by the simple kinematic scheme presented in this work may be unexpected given that the forward models display LS simple shear deformation, depth-dependent thinning, large asymmetry and deformation of the lithospheric mantle. For cases 2–4 the relative accuracy of the inverted crustal thickness ranges from 80% to 98% and the absolute error in inverted crustal thickness ranges from 1 to 5 km in these regions (Figures 8 and 9), errors that are comparable to the magnitude of uncertainty associated with the best available seismic constraints on crustal thickness in continental settings [e.g., ±4.5 km, Li et al., 2002].

[27] The kinematic analysis presented in this work also sheds light on the type of information that is necessary to accurately model bulk crustal thinning by inverting present-day bulk crustal structure. Required inputs are: (1) present-day crustal thickness grids from both conjugate margins that accurately represent the volume of extended crust and the topography of the basement and Moho, (2) relative displacement between diverging plates that are undergoing rifting and breakup and (3) the location of the RNT on conjugate plates through time.

[28] The methods presented in this work invert present-day crustal thickness using equation (1), which suggests that thickness variations in extended crust record a bulk lateral strain history that can be used to estimate past crustal thickness. Therefore, accurate models of present-day crustal thickness are critical for the application of this inversion scheme. Sufficiently accurate models of bulk crustal thickness for the plate-scale application of this inversion scheme can be obtained by inverting sediment thickness and bathymetry using a model of local isostasy provided that Receiver functions and/or refraction data is available to constrain an appropriate reference crustal thickness and depth of compensation [e.g., Kneller et al., 2012]. Furthermore, seismic reflection and refraction data are necessary to estimate the volume of igneous material that was added to the crust during extension and breakup and must be removed from the crust prior to performing the inversion [Kneller et al., 2012].

[29] The RNT can be constrained by mapping the spatial and temporal distribution of syn-kinematic sediments using surface and subsurface observations. For example, regional deep reflection lines can be used to identify syn-kinematic sediment packages with growth into faults. If the age of syn-kinematic sediments can be constrained using field observations, well data and regional seismic correlations and there is sufficient coverage across the extensional system then accurate estimates of the location of RNTs through time may be determined [Manatschal et al., 2007].

[30] Geologic constraints on the location RNTs are sparse in most systems due to limited age control on deeply buried syn-rift sediments. Nonetheless, as case 1 demonstrates even when RNTs are wide and fixed to tectonic plates useful approximations of the past distribution of crustal thickness in proximal environments of extensional systems can be obtained (Figure 6). Furthermore, cases 2–4 (Figures 7-9) demonstrate that more accurate approximation for the past distribution of crustal thickness in hyper-extended regions (crustal thickness ≤ 10 km) can be obtained by using closely spaced RNTs during the later phases of extension [Kneller et al., 2012]. For cases investigated in this work inverted paleocrustal thickness has an accuracy of 80–90% provided that the RNTs are first placed along the perimeter of the hyper-extended domain and then shifted beyond major basin bounding shear zones in the proximal environment after crustal thickness in the hyper-extended domain has been restored to 15–20 km. This suggests that if extensional systems deform in a manner similar to the forward models used in this work accurate paleocrustal thickness estimates can be obtained using this simple model for the RNT. Finally, even sparse age control on syn-rift sediments that provides an indication of how deformation localized in extensional systems can be used to better define RNTs and improve approximations of past crustal thickness [Kneller et al., 2012].

[31] In extensional systems that have undergone a diachronous onset of sea floor spreading displacement boundary conditions can be inferred from geophysical observations if calibrated magnetic anomalies in the oceanic crust are available that intersect the edge of extended continental crust [e.g., South China Sea, Briais et al., 1993]. However, relative plate motion is often poorly constrained during the syn-rift and early breakup phases of extensional margin formation. One approach that can be used to estimate displacement boundary conditions in these poorly constrained regions involves defining the prerift configuration of conjugate extended plates by fitting palinspastically restored crustal thickness grids. This requires an estimate of the prerift crustal thickness based on available seismic constraints located around the perimeter of the extensional system. The direction of relative plate velocity is estimated using available constraints on the motion along syn-rift normal faults or by extrapolating the oldest observable fracture zones along continental margins. The rate of divergence from prerift phases to breakup is then estimated using and the best available age control on the earliest syn-rift sediments and constraints on the timing of continental breakup and the onset of sea floor spreading [Kneller et al., 2012].

[32] One advantage of the kinematic inverse approach presented in this work is that a range of scenarios can quickly be explored at plate-tectonic scales with computational time on the order of hours as opposed to the days, weeks or months required to execute a single high-resolution thermo-mechanical model. The computational efficiency of the kinematic approach permits the generation of multiple inversion scenarios and facilitates the analysis of uncertainty in plate kinematics, crustal structure and the displacement history of the RNT.

[33] The inversion scheme of Kneller et al. [2012] approximates crustal extension as a mass-balance problem and is designed to approximate LS lateral displacement gradients and changes in bulk crustal thickness. The user of this scheme should be aware that the pure shear assumption used in this work is inadequate for modeling internal crustal structure since large stretching in the upper crust is accommodated by simple shear deformation. However, displacement gradients can be used to make predictions about the general location and width of LS shear zones. Furthermore, this inversion scheme will not capture the effects of several processes that may partly control crustal mass distribution in extensional systems such as orogenic collapse, lower crustal flow associated with pressure gradients, and erosion. Although these processes may play a secondary role in controlling mass distribution of the crust during extension, caution should be used when applying the kinematic inversion method presented in this work.

[34] The kinematic scheme presented in this work inverts present-day bulk crustal structure and plate kinematics to estimate paleocrustal thickness whereas previous work inverts interpreted stratigraphy for crustal and mantle thinning factors and requires an assumed initial crustal structure. Both types of inversions have strengths and weaknesses depending on the problems being addressed. We think that the inverse kinematic method described in this work is well suited for estimating plate-scale distributions of crustal mass during the early syn-rift phases of extensional systems when limited sedimentation or erosion may present challenges for other inversion methods that require stratigraphic information.

[35] Since the kinematic inversion presented in this work can be easily linked to plate reconstructions, it has potential to make global predictions of the locations of crustal barriers to the global ocean that may produce restricted environments of deposition throughout Mesozoic extensional systems that developed during the breakup of Pangaea [Kneller et al., 2012]. Identifying these restricted conditions is important for hydrocarbon exploration because restricted environments are conducive for the deposition of organic-rich source rocks. A uniform stretching model can be applied using the crustal thinning parameters predicted by the approach presented in this work. Such models will not be applicable to model subsidence in distal environments where depth-dependent mantle deformation is likely to occur. However, uniform stretching model linked to a plate-scale application our kinematic inversion may provide a useful global reference model of tectonic subsidence that could be used to make first order predictions about basin geometry and identify where more complicated nonuniform mantle thinning models are required to model subsidence.

[36] Future thermo-mechanical benchmarks should be applied to fully explore the limitations of this kinematic approach by inverting crustal thickness produced by thermo-mechanical models that include processes not considered in the present study (e.g., SS convection in the mantle lithosphere). The kinematic inversion scheme presented in this work assumes that deformation is dominated by simple shear in the upper crust and pure shear in the lower crust. If deformation becomes heterogeneous in the lower crust due to significant lateral flow, the simple rules used with the inversion scheme presented in this work will provide a poor approximation to paleocrustal thickness. Although more geodynamic modeling is necessary, we think that the primary effect of sedimentation on lithospheric extension is to control the location and magnitude of strain localization. Therefore, the distribution of sedimentation may help constrain the location of RNTs and improve the predictions of the kinematic inversion scheme presented in this work.

6. Conclusions

[37] The inverse pure-shear kinematic method presented in this work produces an accurate reproduction of Moho topography and bulk crustal thickness from forward thermo-mechanical models that display LS simple shear deformation and depth-dependent thinning. The relative accuracy of the inverted crustal thickness ranges from 80% to 98% and the absolute error in topography is less than 1 km (Figures 8 and 9). Our results demonstrate that if displacement boundary conditions and the width of the rift zone can be constrained, then the pure-shear approximation and simple kinematic rules can be used to invert present-day bulk crustal structure and accurately predict paleocrustal thickness evolution from extensional systems with complex nonlinear rheology and asymmetric crustal thinning, provided that there is limited lateral flow in the lower crust.

[38] The inverse pure shear kinematic approach also produced lateral displacements that mimicked the lateral component of velocity from the geodynamic model and accurately predicted geometric aspects of LS simple shear structures. Inflection points and large-gradients in inverted lateral nonrigid velocity correlated with the surface trace and intersection of shear zones with lower crustal detachments observed in the forward thermo-mechanical model (Figures 10 and S5).

[39] The kinematic inversion method presented in this work produced results with accuracy adequate for modeling plate-scale paleocrustal thickness in extensional systems dominated by LS simple shear and pure shear thinning in the lower crust, to provide input for thermo-kinematic models of lithospheric extension and to define lateral nonrigid deformation in plate reconstructions. If boundary conditions can be constrained with observations then this approach also provides a way of linking plate kinematics to lateral strain and paleocrustal thickness in specific extensional systems and can be used to better understand how the evolution crustal mass distribution controls basin geometry, the destruction of crustal barriers to the global ocean and the potential of restricted environments of deposition during the syn-rift phases of basin evolution.

Acknowledgments

[40] We thank ExxonMobil Upstream Company for permission to publish this work. We also thank Chris Beaumont, Steven Ings, and Douglas Guptill for discussions concerning the use of SOPALE for modeling lithospheric extension. Furthermore, we thank Gianreto Manatschal for discussions of how field observations from extensional systems can be used to constrain boundary conditions in kinematic models. Finally, we thank an anonymous reviewer and Boris Kaus for comments and suggestions that significantly improved the presentation of this work.

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