Isostasy in Move Defining the former elevation and shape of the lithosphere, in particular the elevation of the Earth s surface, is important in the restoration of a model as it aids in reducing uncertainty in palaeo-water depth predictions or erosional potential. This has implications for sediment delivery, burial and decompaction (e.g. Reynolds et al. 1991). For a given thickness, density and temperature, and discounting the dynamic effects of mantle flow, the absolute elevation of the lithosphere depends on its gravitational equilibrium with the underlying mantle (e.g. Molnar et al. 2015). This is a process known as isostasy (e.g. Watts 2001). By changing the thickness, density and temperature of the lithosphere, tectonic processes such as rifting or mountain-building change the surface elevation, as do changes in mass distribution by sedimentary loading or erosion (e.g. Tucker & Slingerland 1994). During sequential restoration of a regional cross-section or a 3D model, it is possible to adjust for elevation changes caused by the isostatic response of the lithosphere to, for example, variation in sedimentary loading or tectonically driven thickening or thinning. In Move TM, the isostatic response of the lithosphere can be modelled for changes in load using two methods: Airy Isostasy and Flexural Isostasy. Airy Isostasy treats loads that are fully compensated by local buoyancy forces; Flexural Isostasy treats loads that are not compensated by local buoyancy and are regionally supported by the strength of the lithosphere, which bends in response. With Airy Isostasy in either 2D or 3D, the entire model behaves as a rigid block with a uniform density floating as a fully compensated load on a fluid mantle of greater density. Changes in thickness or load will cause vertical movement of the whole model. Thickening of the crust will sink the lithosphere deeper into the mantle, but will increase surface elevation. Thinning will cause the lithosphere to rise, but will reduce surface elevation. Airy Isostasy should generally be used for models with maximum dimensions less than ~10 km, although this depends on the strength of the lithosphere (see below) and there is no effective upper limit. It can also be applied in the restoration workflow for models containing salt. For models in rifted areas with maximum dimensions greater than ~10 km, Flexural Isostasy is recommended. For this approach, the lithosphere is characterised as having an inherent strength and rigidity defined by a parameter called Effective Elastic Thickness (T e ). This is the thickness of a perfectly elastic layer with the same flexural strength as the lithosphere (Roberts et al. 1998). In conjunction with the Young s Modulus (E), this defines how easily and on what length scale (flexural wavelength) the lithosphere will bend when loaded or unloaded. The flexural wavelength determines the length scale over which loading is distributed for an uncompensated load and defines the magnitude of bending. T e is a function of the lithospheric thickness and temperature. T e is large for orogenic collision zones or cratonic areas (>60 km) and small for oceanic or rifted continental crust (<15 km), although in these areas, lithospheric cooling means that it increases with time. A T e of 0 km implies a load fully compensated by local buoyancy forces,
which can be modelled with Airy Isostasy. It is not always necessary to account for isostatically driven variations in lithosphere elevation using these methods. For local lines, or where isostasy parameters are not available, a first order approximation to changes resulting from Airy Isostasy can be made where sediment facies information provides estimates of depth of deposition. In these cases, the whole section can be treated as a single block and its vertical position can be adjusted for depth of deposition at a given restoration step using the Transform tools in Move. Modelling Isostasy in Move The Isostatic Relief sheet is part of the Decompaction tool in the 2D or 3D Kinematic Modelling modules (Figure 1). The user can choose between Airy isostatic compensation or Flexural isostatic compensation depending on the workflow, the maximum dimensions of the model and other considerations such as T e. Isostatic response is strongly dependent on density. Archimedes Principle states that where sediments are immersed in water, part of the load will be compensated by the weight of the displaced water and density will be less (Hamilton 1976). As a result, isostatic adjustment is greater for loads that are emergent and less for those in, for example, submerged sedimentary basins. In the Isostatic Relief sheet it is essential that the load applied is set to either Sub Marine or Sub Aerial. A sub-marine load is equivalent to the sediment load minus the load of the displaced water. This assumes that when a sedimentary load is removed, the void will be infilled with water, partially loading the lithosphere and consequently reducing the vertical displacement. If a sub-aerial load is selected, it is assumed that water is absent and the net density of the load will be greater, resulting in more vertical displacement than in a sub-marine scenario. There may be instances where a model lacks a Rock Properties database or it is not appropriate for the workflow being carried out. In these cases an average density value for the total load can be used and where Calculate Isostasy Only option is selected, a bulk density for the crust can be entered (Load Bulk Density).
Figure 1 3D Decompaction toolbox in Move showing Isostatic Relief sheet with parameters set up for flexural isostatic compensation of a sub-marine load Airy Isostasy To compensate for isostatic effects using the Airy method (Figure 2) the user must enter the mantle density for the area of interest (crust density will be calculated from the Rock Properties database, from values defined in the Parameters sheet or, if the Calculate Isostasy Only option has been selected it will use the Load Bulk Density value). Where not provided, the default mantle value of 3300 kg/m³, based on the North Sea, will be used. Isostatic adjustment of elevation is calculated using the formula:
Airy Isostasy Where: D1 = Water depth before sediment load Z = Amount of subsidence relative to a basement reference D2 = Water depth after sediment load Pc= Crust density S = Thickness of sediment loaded or unloaded Pw= Water density H1 = Crustal thickness before sediment load Pm= Mantle Density H2 = Crustal thickness after sediment load Figure 2 Changes in elevation for loads fully compensated by local buoyancy using Airy Isostasy.
Flexural Isostasy To compensate for isostatic effects using the Flexural method the user must understand how T e (Effective Elastic Thickness), the average Young s Modulus (E) of the lithosphere, and the difference in density between the asthenosphere and lithosphere affect the response to loading, particularly in defining the rigidity and flexural wavelength of the lithosphere (e.g. Watts 2001). The rigidity of the lithosphere is defined by the formula (Landau & Lifshitz 1986): 3 ETe D 2 12(1 ) Where: E = Young s Modulus T e = Effective Elastic Thickness Poisson s Ratio Young Modulus (E) is defined as the stiffness of an elastic material in this case an average for the lithosphere being loaded in the model, including the load being added/removed. It is assumed to be constant throughout the model. Flexural Rigidity (D) is a key parameter used to determine the flexural wavelength. How it connects to the other parameters can be seen by varying the parameter for E (Young's Modulus), or T e (Elastic Thickness) in the toolbox and observing how they change the Flexural Wavelength. As indicated by examining the formula above, reducing T e or E will reduce D, and consequently reduce the flexural wavelength. In Move, it is recommended to keep E constant and vary T e. A table of representative T e values is provided in the Move help pages for regions around the globe, with relevant references. When modelling Flexural Isostasy, the vertical displacement can be approximated as being distributed sinusoidally (Einsele 2000). The amount of flexure decays away from the load with the rate of decrease determined by the flexural wavelength.
Flexural Isostasy Load Extension For some models, loads that lie outside the area of interest will have an influence on the vertical displacement of the model. Where this is the case, the load being added can be extended to take account of regional loading using the Load Extension setting. Care must be taken if the total length of the load including the extension, exceeds the flexural wavelength for the model because no flexure will occur and changes in load will be treated as being fully compensated. In this case, Airy Isostatic compensation will prevail. Sensitivity testing to constrain the appropriate load extension is advised. Adding the average structural trend to the toolbox will constrain the optimum direction of load extension (Figure 3). Figure 3 3D model with main structural trend displayed and load extension part of the toolbox highlighted in red. The basin deepens to the west and load extension has been set up to estimate flexural isostasy along the basin margin.
Although isostasy is a process that operates in 3D there are strong benefits to applying it in 2D. This allows the user to assess isostatic response along individual basin profiles and gain a more sophisticated understanding of the mechanisms involved in the response of the section to changes in loading and thickness and will help reduce uncertainty in any restoration. References Einsele, G. 2000. Sedimentary basins: evolution, facies, and sediment budget. Springer Science & Business Media. Hamilton, E.L. 1976. Variations of density and porosity with depth in deep-sea sediments. Journal of Sedimentary Research, 46 (2). Landau, L.D. & Lifshitz, E. 1986. Theory of Elasticity, vol. 7. Butterworth-Heinemann. Molnar, P., England, P.C. & Jones, C.H. 2015. Mantle dynamics, isostasy, and the support of high terrain. Journal of Geophysical Research: Solid Earth, 120 (3), 1932-1957. Reynolds, D.J., Steckler, M.S. & Coakley, B.J. 1991. The role of the sediment load in sequence stratigraphy: The influence of flexural isostasy and compaction. Journal of Geophysical Research: Solid Earth, 96 (B4), 6931-6949. Roberts, A.M., Kusznir, N.J., Yielding, G. & Styles, P. 1998. 2D flexural backstripping of extensional basins: the need for a sideways glance. Petroleum Geoscience, 4 (4), 327-338. Tucker, G.E. & Slingerland, R.L. 1994. Erosional dynamics, flexural isostasy, and long-lived escarpments: A numerical modeling study. Journal of Geophysical Research: Solid Earth, 99 (B6), 12229-12243. Watts, A.B. 2001. Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge.