PALEOSTRIP graphical user interface (GUI) is composed of two different tabs. The first one is dedicated to input data, physical parameters, and choices of physical methods for each of the processes accounted for in backtracking (Fig. ). The second one is dedicated to plotting backtracked data and saving results (see Sect. ). The GUI can be launched either through the MATLAB main interface and double-clicking on the corresponding App Designer GUI file or by exporting it as a stand-alone application that can be executed without opening MATLAB. We do not provide a built-in PALEOSTRIP application, since the compilation of the applications depends on the operating system on which it is created, but we provide the PALEOSTRIP code instead. As such, the user is free to export it from the App Designer tool.

The total sediment thickness S accumulated at a given time can be obtained by reconstructing its compaction history. Eroded sediments are transported to the margin deposit and accumulate wherever the accommodation space allows for it. Accumulated sediments compact over time under loading, e.g. by overlying sediments. Therefore, to reconstruct the paleo-architecture of a continental margin at a given time, the sediment layers of a margin are stripped off sequentially, and the remaining underlying sediments need to be gradually decompacted. Decompaction is thus central to backstripping and backtracking.

Compaction or decompaction of sediments both imply a change in total volume Vtotal of deposited sediments; this is mostly caused by changes in the porosity of sediments and (to a lesser extent) changes in sediment compression. In submarine environments, sediments are saturated by water, and their compaction implies a decrease in pore fluid pressure via expelling of the water out of the sediment layers. Conversely, decompaction involves an increase in the pore fluid pressure by injecting water within the compacted sediments. The decompaction process consists of calculating the changes in porosity of the various sediment layers to determine the amount of water that was contained in the sediments at the time of their deposition based on their lithology. The depth of the decompacted sediment layers is recalculated on the basis of those porosity changes. Laboratory experiments have determined that for large depths, the evolution of porosity can be described by an exponential relationship rather than by a linear empirical equation: 4ϕ=ϕ0e-cz, where ϕ0 corresponds to deposition porosity at the seafloor (or at surface if emerged), c is the exponential slope decay coefficient (also referred to here as the compaction coefficient) depending on the lithology expressed in 1km-1, and z is depth in kilometres. This empirical equation implies that ϕ0 is decreased by the factor of 1/e at the depth of 1/c km.

In general, the mass of the total sediment column at a given time does not change: in submarine environments the water that is expelled from the sediments is implicitly added to the water column above the underlying sediments during the compaction process. Thus, only the volume of the sediment layers Vtotal changes due to water volume changes Vw, which allows for the integration of the porosity Eq. (). Considering a unit's cross-sectional area, the thickness of water WT to be added to the compacted sediment volume in order to decompact it is given by the following equation: 5WT=∫zizi+1ϕ0e-czdz, which upon integration gives 6WT(i)=ϕ0c{e-czi′-e-czi+1′}, where zi′ and zi+1′ are the newly decompacted depths of zi and zi+1. Because the volume of sediment grains Vsed remains unchanged, the integrated compacted sediment thickness Sdt between two given depths, considering a unit's cross-sectional area, can be written as follows: 7Sdt=zi+1-zi-ϕ0c{e-czi-e-czi+1}, Based on the fact that Vtotal=Vsed+Vw, the decompacted depths z′(i) for a unit's cross-sectional area can be inferred, and the final decompaction equation is given by the following equation: zi+1′-zi′=zi+1-zi-ϕ0c{e-czi-e-czi+1}8+ϕ0c{e-czi′-e-czi+1′}. PALEOSTRIP solves this equation iteratively (starting with zi′=0 and then the further iterations) until convergence is reached. Lithological parameters, i.e. the decompaction coefficient c and the depositional surface porosity ϕ0, are prescribed in the user-provided parameter files (see Sect. ).

Sediments load the basement of the continental margin and cause a local and regional subsidence during accumulation. Water also loads the basement due to eustatic sea level variations, leading to changes in the water depth WD that are also caused by ocean bottom subsidence and/or uplift through time. During the decompaction, the newly computed decompacted depths of the remaining sediment layers also need to be adjusted to account for the changes in their porosity and thus their density due to the presence of water filling the pores. For each sediment layer, the porosity ρds(i) of the decompacted thickness Sd(i) of the ith sediment layer is 9ϕds(i)=ϕ0ce-cz′(i+1)-e-cz′(i)Sd(i), and the decompacted sediment bulk density ρds(i) of the ith sediment layer is given by 10ρds(i)=ϕds(i)ρw+(1-ϕds(i))ρS(i). At each time step, the removal of the top layer causes the decompaction of the underlying sediment layers.

Removed sediment layers are substituted with seawater. Unloading causes an isostatic compensation that can be calculated by using different isostatic methods. In PALEOSTRIP, two methods are implemented.

Thermal subsidence corresponds to a vertical contraction of the lithosphere. During a rifting phase, as a first step, the lithosphere stretches apart and thins, which causes a net increase in heat outflow towards the surface due to the upwelling of the underlying asthenosphere. The stretching is generally not uniform, but reconstructions of past stretching processes require knowledge about plate tectonic strain rates e.g., which is beyond the scope of our software. The first step is called “initial” subsidence. In the second step, the “thermal subsidence” occurs; i.e. the stretched lithosphere constricts due to cooling. To account for those effects during backstripping, PALEOSTRIP adopts the 1D-thermal subsidence model from that assumes an instantaneous stretching (syn-rift) and a single rifting phase. The initial subsidence Zinit is given by 16Zinit=TIL(ρm-ρc)TICTIL1-αvTmTIC2TIL-ρmαvTm21-1βρm(1-αvTa)-ρw, where TIL and TIC are the initial lithospheric and crustal thicknesses at the beginning of rifting, β is the stretching factor, αv is the coefficient of thermal expansion, ρc is the crustal density, and ρm is the mantle density. In this equation, the subsidence is isostatically compensated by water (ρw) using the Airy local compensation. This 1D instantaneous model can be improved by adopting a time-evolving approach to the extension, for example that of . However, comparison between and revealed that the two models show no or only little discrepancy if the duration of extension, given the time required to extend it by a factor of β, is about 60/β2 Myr if β≤2 or 60(1-1/β)2 Myr if β≥2 . During the second step, the thermal subsidence occurs after the end of rifting (post-rift) and accounts for the vertical thermal conduction, which takes the shape of an exponential function decaying with time: 17Zthermal(t)=E0βπsin⁡πβ1-exp⁡-tτ, where t corresponds to the time elapsed since the end of rifting expressed in seconds (e.g. time of backtracked horizon = 14 Ma; age of the end of rifting = 76 Ma; and t=(76-14)×3600×24×365×106). 18E0=4ρmαvTmTILπ2(ρm-ρw), and τ=TIL2π2κ, where κ is the thermal diffusivity. The total thermal subsidence is given by 19Ztot_thermal=Zinit+Zthermal.

In PALEOSTRIP, during the backtracking procedure, paleo-water depths are corrected by removing the increment of thermal subsidence between each time step and present-day. This is because backtracking is an iterative process departing from a known state, i.e. present day. Because Zinit is constant in time it cancels out, and the thermal subsidence correction is given by 20ΔZtot_thermal(t)=(Zinit+Zthermal(t))-(Zinit+Zthermal(0)), and thus, 21ΔZtot_thermal(t)=Zthermal(t)-Zthermal(0), where t varies from present to past in this equation following backstripping procedure, i.e. t is an age older than present day (0). ΔZtot_thermal is thus positive, with Zthermal being larger for younger times, i.e. a larger time is elapsed since the end of rifting.

Note that by applying the 1D model from to 2D transects and 3D maps, it is assumed that no horizontal heat advection occurs. Almost all parameters involved in the thermal subsidence can be modified (age of end of rifting, TIL and TIC, αv, κ) from the GUI tab interface (Fig. b). The stretching factor β can be formulated through several methods. (1)

By prescribing a constant and uniform β factor that will be used at each time step of backtracking to calculate the thermal subsidence.

On long timescales, sea level has been varying with time due to plate tectonics changing the dimensions of ocean basins e.g. and, on shorter timescales, due to continental ice storage within ice sheets, ice caps, and glaciers during cold periods, as was the case, e.g. during the second half of the Cenozoic the last 34 Myr, e.g.. Classically, only eustatic sea level changes have been considered for correcting the subsidence or the paleo-water depth. For the last 34 Myr, eustatic sea level is usually defined relative to the present-day total ocean area (362.15×106 km2) because it is assumed that this area evolved only a little and not enough to significantly alter this number. For time periods older than that, eustatic sea level changes have to be expressed according to the ocean area of the time. Considering only eustatic sea level changes results in highly approximated ice sheet growth and decay, leading to sea level variations of growing amplitude over the past 34 Myr e.g.. The variations induced by glaciations have much shorter timescales, i.e. 10–105 years, compared with those induced by plate tectonics (105–106 years). Associated sea level changes are not spatially uniform and induce changes in the Earth's gravity field e.g., and regional self-gravitating sea level changes can substantially vary from eustasy e.g..

In PALEOSTRIP, sea level (SL, in m) is corrected the same way as thermal subsidence: 22ΔSL(t)=SL(t)-SL(0), where t varies from present to past in this equation following a backtracking procedure, i.e., t is an age older than present day (0). ΔSL is expressed relative to present, and it can therefore assume positive or negative values, i.e. induce an uplift or a subsidence. Most of the sea level variation time series found in the literature already correspond to variations relative to present, i.e. are already in the form of Eq. (). Thus, in order to avoid confusion for the user, three different ways of correcting water depth with sea level changes have been implemented within PALEOSTRIP and can be managed through the GUI (Fig. c): (1)

by prescribing constant and uniform sea level correction ΔSL applied at each time step of the backtracking (e.g. correction at a time where sea level is lower by 100 m relative to present is prescribed as -100 in the GUI),

The last option allows us to prescribe sea level changes calculated with glacio-isostatic adjustment models e.g. SELEN4 to account for regional self-gravitating effects of ice sheet growth and decay. ΔSL is used to compute the water load correction to adjust the water depth WD. In Eq. (), water load due to sea level change is described based on Airy local compensation. In PALEOSTRIP, the water load due to sea level change is computed using the isostatic method previously selected to carry out decompaction (see Sect. ). Note that if isostasy is deactivated, no water load correction is computed.

Mantle dynamics generate flows that cause time-varying surface topography and bathymetry deformations, which are called dynamic topography. The timescale at which the mantle flow produces dynamic topography that occurs at long wavelengths ≈5000 to 10 000 km. The current mantle-driven dynamic topography was revealed by estimating the residual topography, obtained after removing the isostatic topography, which is generated by thickness and density contrasts within the lithosphere and the isostatic effect of the lithosphere calculated using seismic data compilations e.g.. Stratigraphic observations of continental margins revealed that the dynamic topography evolves quickly with time, potentially impacting long-term climate evolution . showed that the magnitude of past interglacial sea level proxies partly results from dynamic topography, and they suggested correcting sea level proxies before inferring the relative contribution of past ice sheet to the sea level changes at that time. Furthermore, simulated Pliocene dynamic topography changes accounted for in Antarctic ice sheet simulations revealed that ice sheet stability is highly influenced by mantle dynamics that create or cancel pinning areas at the surface . Thus past reconstructions of continental morphology and shallow margins should account for past evolution of dynamic topography. recently provided time slices of reconstructed dynamic topography over the past 240 Myr, which constitutes a strong basis to calculate a dynamic topography correction.

PALEOSTRIP provides three ways to correct for dynamic topography changes through its GUI (Fig. d):

by prescribing a uniform and constant dynamic topography change relative to present day that will be applied to each time step of the backtracking procedure;

The correction is calculated as for sea level changes (Eq. ) as follows: 23ΔDynT(t)=DynT(t)-DynT(0), where t varies from present to past in this equation following backstripping procedure, i.e., t is an age older than present (0). ΔDynT is expressed relative to present, and since dynamic topography is not spatially uniform at a global level, it can therefore assume positive or negative values, i.e. induce an uplift or a subsidence.

Sediment erosion and depositions are the main processes involved in the building of continental margins. Present-day identified sedimentary units may not reflect the real number of sedimentary units at a given time in the past because some of the layers might have been eroded in the meantime. Erosion is identified within sediment cores as hiatus and as high-amplitude (sometimes truncational) reflectors within marine seismic profiles. During backstripping, one should move back eroded sediments to their presumed original locations e.g., which can be either within the backstripped area or outside the backstripped area. If eroded sediments are moved back in the backstripped area, then the total number of layers should be modified during the backward modelling process. The current version of PALEOSTRIP does not treat erosion and does not allow for the variation of the total number of sedimentary layers once they are input at the beginning of the procedure. A good description of how to handle erosion during backstripping is provided in Sect. 5.4 of .

Vertical transects imply that input data are provided along an horizontal direction X and a vertical direction (depth) Z. Due to the needs of flexure calculations, the original sediment loads array is extended (duplication of last values at both edges) to about 90 % of its length from both edges (Fig. a). The sediment loads are then placed at the centre of an expanded domain (3 times larger than the original array length) to avoid edge effects on flexure correction. After flexure calculation, the correction is extracted and relocated on the original array domain. All the other backtracking calculations occur on the original input array. Note that spatially variable lithospheric elastic thickness is also interpolated and extrapolated following this procedure.

Maps imply that data are provided along the two horizontal directions, X and Y, and along the vertical direction, Z. Input data are read as scattered data and not as gridded data. This means that points are unstructured and are not written in a file following a classical NX×NY structure but are instead treated as independent single points. When data are treated as scattered this takes more computational time, but this also allows one to input either structured gridded data or irregular polygon data. This facilitates all computations and avoids unnecessary duplication of routines for 1D, 2D, or 3D cases. For the need of flexure, 3D data are interpolated (preserving their original horizontal resolution) on a regular rectangular grid. The original input grid is expanded (extrapolation of the last values from the edges) to about 30 % from all sides of the grid. The sediment loads are then placed at the centre of an expanded domain (twice as large as the original grid dimension) to avoid edge effects on flexure correction (Fig. b). After flexure calculation, the correction is extracted and relocated on the original grid domain. All the other backtracking calculations occur on the original input grid. Note that spatially variable lithospheric elastic thickness is also interpolated and extrapolated following this procedure.

In this example, we decompact the drilling site DSDP273 from the western Ross Sea (Fig. ). It has two main identified seismic unconformities above the basement. Lithological parameters are taken from , and the lithological parameters file (see Sect. ) is as follows:

Backtracking is performed using Airy local isostasy since flexure cannot be applied to 1D drilling sites. We also add thermal subsidence in order to illustrate the difference in backtracked depths of those unconformities (Fig. ). The layout of Fig.  is not the layout of PALEOSTRIP, but the results have been assembled to highlight the impact of thermal subsidence on backtracked depths. As for 2D transects, all intermediate variables and input conditions can be plotted and saved.

In this example we backtrack the depth of the mid-Miocene unconformity (14 Ma) across the Ross Sea from ANTOSTRAT atlas . The initial grid is an irregular polygon. The set of initial data is composed of three files: the actual bathymetry of the Ross Sea, the present-day depth of mid-Miocene unconformity, and the presumed present-day depth of the basement below (Fig. ).

The fifth file contains the lithological parameters of the layers to be decompacted (see Sect. ) and are taken from . The lithological parameters file is as follows:

PALEOSTRIP is run several times to add one of the following components at a time: isostasy, thermal subsidence, sea level correction, and dynamic topography correction. Thanks to this approach, the user can perform ensembles of simulations to retrieve sound statistics of the model parameter space. Two series are shown, one using the Airy isostatic correction (Fig. ) and the other using the flexural isostatic correction (Fig. ). Paleo-bathymetry retrieved with flexural isostasy produces a quite different morphology from the one computed using Airy isostasy, as already observed by . Sea level and dynamic topography (Fig. b) corrections are not big enough to produce a significant change of the overall morphology. However, they matter for the bathymetric highs (especially the shallowest one), as a few tens of metres can uplift those highs above sea level. PALEOSTRIP allows the user to plot different variables amongst initial input data and computed quantities, such as isopach, density, porosity, or isostatic correction, allowing the user to check and separate the various processes required to perform a detailed analysis of their impact on the paleo-bathymetric reconstruction (Fig. ).