IMAGING A SUBSURFACE GEOLOGICAL MODEL AT A PAST INTERMEDIATE RESTORATION TIME
A system and method is provided for restoring a 3D tomographic model of the Earth's subsurface geology from the presentday to a past restoration time. Whereas at the present time all faults represent active discontinuities, at a past restoration time some faults have not yet formed. Accordingly, the restored model divides the fault network into τactive faults (discontinuous surfaces for faults that intersect the layer deposited at the past restoration time) and τinactive faults (continuous surfaces for faults that do not intersect the layer deposited at the past restoration time). A new 3D restoration transformation is also provided that uses linear geological constraints to process the restoration model in less time and generate more accurate geological images.
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This application is a continuation of U.S. Ser. No. 16/244,544, filed on Jan. 10, 2019, which is incorporated in its entirety herein by reference.
FIELD OF THE INVENTIONEmbodiments of the invention relate to the field of geological tomography for generating an image of the interior subsurface of the Earth based on geological data collected by transmitting a series of incident waves and receiving reflections of those waves across discontinuities in the subsurface. The incident and reflected waves are reconstituted by a 3D model to generate an image of the reflecting surfaces interior to the Earth. Accordingly, geological tomography allows geophysicists to “see inside” the Earth.
Embodiments of the invention further relate to geological restoration in which the tomographic images of the present day geology are transformed into images of the past geology, as it was configured at an intermediate restoration time in the past τ before the present day and after the start of deposition of the oldest subsurface layer being imaged. New techniques are proposed herein to improve both the accuracy and computational speed of generating those images of the past restored geology. Improved images may aid geoscientists exploring the subsurface geology for applications such as predicting tectonic motion or earthquakes, or by engineers in the mining or oil and gas industries.
BACKGROUND OF THE INVENTIONThe accuracy of a geological model of the present day configuration of the subsurface of the Earth may be improved by “restoring” the model to a past intermediate time τ and checking model consistency at that time in the past. However, restoring geological models is a complex task and current methods are typically inefficient, requiring extensive processing resources and time, as well as inaccurate, relying on oversimplifications that induce errors to moderate the complexity of the task.
There is a longstanding need in the art to efficiently and accurately restore geological models from their present day geology to their past geology at restored past time τ.
SUMMARY OF EMBODIMENTS OF THE INVENTIONSome embodiments of the invention are directed to modeling restored geological models with τactive and τinactive faults. In an embodiment of the invention, a system and method is provided for restoring a 3D model of the subsurface geology of the Earth from a present day geometry measured at a present time to a predicted past geometry at a past restoration time. The 3D model of the present day measured geometry comprising a network of faults may be received, wherein a fault is a discontinuity that divides fault blocks that slide in opposite directions tangential to the surface of the fault as time approaches a modeled time. A past restoration time τ may be selected that is prior to the present time and after a time when an oldest horizon surface in the 3D model was originally deposited. The network of faults may be divided into a subset of τactive faults and a subset of τinactive faults, wherein a τactive fault is a fault that is active at the past restoration time τ and a τinactive fault is a fault that is inactive at the past restoration time τ. A fault may be determined to be τactive when the fault intersects a horizon H_{τ} that was originally deposited at the past restoration time τ and a fault may be determined to be τinactive when the fault does not intersect the horizon H_{τ}that was originally deposited at the past restoration time τ. The 3D model may be restored from the present day measured geometry to the predicted past geometry at the past restoration time τ by modeling each τactive and τinactive fault differently. Each τactive fault maybe modeled to join end points of a horizon H_{τ }separated on opposite sides of the fault in the present day model to merge into the same position in the restored model by sliding the end points towards each other in a direction tangential to the surface of the τactive fault. Each τinactive fault may be modeled to keep collocated points on opposite sides of the fault together.
Some embodiments of the invention are directed to modeling restored geological models with new restoration coordinates ρ_{τ}, v_{τ}, t_{τ}. In an embodiment of the invention, a system and method is provided for restoring a 3D model of the subsurface geology of the Earth from a present day measured geometry to a predicted past geometry at a restoration time in the past τ. The 3D model of the present day geometry of the subsurface may be received, including one or more folded geological horizon surfaces. A value may be selected of a restoration time in the past τ before the present day and after a time an oldest horizon surface in the 3D model of the subsurface was deposited. The 3D model may be restored from the present day measured geometry to the predicted past geometry at the restoration time in the past τ using a 3D transformation. The vertical component of the 3D transformation may restore the geometry to the vertical coordinate t_{τ} such that: points along a horizon surface H_{τ }modeling sediment that was deposited at the selected restoration time in the past τ have a substantially constant value for the restored vertical coordinate t_{τ*}; and at any location in the 3D model, the restored vertical coordinate t_{τ }is equal to a sum of a first approximation t′_{τ} of the vertical coordinate and an error correction term ϵ_{τ}, wherein the error correction term ϵ_{τ }is computed by solving a linear relationship in which a variation in the sum of the first approximation t′_{τ} of the vertical coordinate and the error correction term ϵ_{τ} between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation; and displaying an image of the restored 3D model of the subsurface geology of the Earth such that each point in the 3D model is positioned at the restored vertical coordinate t_{τ} as it was configured at the restoration time in the past τ.
The principles and operation of the system, apparatus, and method according to embodiments of the present invention may be better understood with reference to the drawings, and the following description, it being understood that these drawings are given for illustrative purposes only and are not meant to be limiting.
For simplicity and clarity of illustration, elements shown in the drawings have not necessarily been drawn to scale. For example, the dimensions of some of the elements may be exaggerated relative to other elements for clarity. Further, where considered appropriate, reference numerals may be repeated among the drawings to indicate corresponding or analogous elements throughout the serial views.
DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTIONEmbodiments of the invention improve conventional restoration techniques for imaging restored geological models as follows:
“τactive” faults vs. “τinactive” faults:
In conventional restoration models, all faults are active (as discontinuous surfaces) at all times. However, in reality, certain faults have not yet formed or activated at various intermediate restoration times τ. Accordingly, conventional restoration models generate false or “phantom” faults that erroneously divide geology that has not yet fractured, leading to geological inaccuracies in subsurface images.
Embodiments of the invention solve this problem by selectively activating and deactivating individual fault surfaces to be discontinuous or continuous, respectively, depending on the specific restoration geologicaltime τ. For each intermediate restoration time in the past τ, embodiments of the invention split faults into two complementary subsets of “τactive” faults and “τinactive” faults. τactive faults are activated at restoration time τ (e.g., a discontinuous fault surface along which fault blocks slide tangentially), whereas τinactive faults are deactivated at restoration time τ (e.g., a continuous surface that does not behave as a fault).
As faults form and evolve over time, they behave differently at different geological times in the past. For example, a fault that forms at an intermediate geologicaltime τ, where τ_{1}<τ<τ_{2}, is τactive in a restored model at later time τ_{2 }(after the fault has formed), but τinactive in a restored model at earlier time τ_{1 }(before the fault has formed). This fault classification allows faults to be modelled differently at each restoration time τ in a geologically consistent way, thereby preventing unrealistic deformations from being generated in the neighborhood of these faults.

 erroneous τtwin points (803,823) are transformed into a pair of collocated points 813=833,
 τtwin points (804,824) are transformed into collocated points 814=834.
 It is clear that, if d(a, b) denotes the distance between any arbitrary pair of points (a, b), then:
d(803,824)=d(833,835)≠d(833,834) (2)
This observation shows that erroneously considering fault 300 as a τactive fault inevitably generates unrealistic deformations.
This problem is solved according to embodiments of the invention, e.g., as shown in the bottom image of
Contrary to conventional methods, the use of τactive and τinactive faults produces more accurate results, e.g., even if there is no continuous path between (no way to continuously connect) a given fault block (e.g., 800) and the horizon H_{τ} (e.g., 210) deposited at geological time τ, which typically requires additional processing that may induce errors. By selectively activating and inactivating faults at the various restoration times according to when they form, embodiments of the invention eliminate erroneous phantom faults and more accurately represent the faulted geology.
Reference is made to
In operation 1610, a processor may receive a 3D model of the present day measured geometry comprising a network of faults (e.g., present day model 202). The present day model may be measured tomographically by scanning the Earth's subsurface e.g., as described in reference to
In operation 1620, a processor may select or receive a past restoration time τ that is “intermediate” or prior to the present time and after the start of the subsurface's deposition (the time period when an oldest horizon surface in the 3D model was originally deposited).
In operation 1630, a processor may divide the network of faults into a subset of τactive faults and a subset of τinactive faults. τactive faults may be faults that are active at the past restoration time τ and τinactive faults are faults that are inactive at the past restoration time τ. A fault is determined to be τactive when the fault intersects a horizon H_{τ} that was originally deposited at the past restoration time τ (e.g., see τactive faults 105 of
In operation 1640, a processor may restore the 3D model from the present day measured geometry to the predicted past geometry at the past restoration time τ. During restoration, the processor may flatten a horizon H_{τ} (e.g., 210 of
In operation 1650, for each τactive fault, a processor may model the τactive fault as an active discontinuous fault surface and restore the horizon surface by removing or omitting the fault surface at the time of restoration. The processor may eliminate the τactive fault during restoration by sliding its adjacent fault blocks together. This may join end points of a horizon H_{τ} separated on opposite sides of the fault in the present day model to merge into the same position in the restored model by sliding the end points towards each other in a direction tangential to the surface of the τactive fault.
In operation 1660, for each τinactive fault, a processor may model the τinactive fault, not as a discontinuous fault surface, but as a continuous nonfault surface in the restoration transformation. The τinactive fault may be modeled as a surface in which the discontinuity induced by the fault has been deactivated to prevent fault blocks from sliding in directions tangential to the surface of the fault as time approaches the restoration time τ. The processor may model the τinactive fault during restoration by keeping collocated points on opposite sides of the fault in the present day model together in the restored model.
After the geological model has been restored for a first past restoration time τ (operations 16201660), the process may repeat to restore the model for a second different past restoration time τ′. In some embodiments, the geological model may be sequentially restored to a sequence of multiple past restoration times τ_{1}, τ_{2}, . . . , τ_{n}. In multiple (all or not all) of the past restoration times τ_{1}τ_{2}, . . . , τ_{n}, the fault network may be divided into a different subset of τactive and τinactive faults, e.g., because different faults fracture the subsurface at different geological times. In some embodiments, a processor may play a moving image sequence in which the 3D model is iteratively restored in a forward or reverse order of the sequence of past restoration times τ_{1}, τ_{2}, , , , , τ_{n }to visualize changes in the subsurface geology over the passage of time.
In operation 1670, a processor may display a visualization of an image of the subsurface geology of the Earth overlaid with τactive faults and τinactive faults in the restored model at past restoration time τ. The processor may display the τactive faults and the τinactive faults with different visual identifiers, such as, different levels of translucency, different colors, different patterns, etc.
New Restoration Transformation u_{τ}, v_{τ}, and t_{τ}:
A restoration transformation may transform a geological image of the subsurface of the Earth from a present day space (e.g., x,y,z coordinates) to a restoration space (e.g., u_{τ}, v_{τ}, and t_{τ} coordinates) as it was formed at an intermediate restoration time in the past τ (before the presentday but after the start of the subsurface deposition). An ideal restoration should transform the vertical coordinate t_{τ} in a manner that strictly honors the thickness of layers, to preserve areas and volumes of the Earth, so that terrains are not stretched or squeezed over time in the vertical dimension. However, conventional restoration transformations typically deform the vertical coordinates, forcing terrains to stretch and squeeze, resulting in errors in the restoration model.
Embodiments of the invention improve the accuracy of the restoration model by establishing a vertical restoration coordinate t_{τ} that preserves layer thickness. This may be achieved by implementing a thicknesspreserving constraint that sets a variation in the vertical restoration coordinate t_{τ} between any two points separated by an infinitesimal difference in the direction of maximal variation of the vertical coordinate t_{τ} to be approximately equal to the distance between the points in the direction of maximal variation. An example of this constraint may be modeled by ∥grad t_{τ}(x,y,z)∥=1. This constraint, however, is nonlinear and highly complex and timeconsuming to solve. Due to its complexity, this constraint is rarely used in conventional restoration models, and instead replaced by oversimplifications, such as equations (33) and (34), that result in model errors as shown in histograms 501 and 502 of
Embodiments of the invention improve the accuracy of the restored model by establishing a new thicknesspreserving constraint that introduces an error correction term ϵ_{τ}. The new thicknesspreserving constraint sets the restored vertical coordinate t_{τ} to be equal to a sum of a first approximation t′_{τ} of the vertical coordinate and an error correction term ϵ_{τ}, wherein the error correction term ϵ_{τ} is computed by solving a relationship in which a variation in the sum of the first approximation t′_{τ} of the vertical coordinate and the error correction term ϵ_{τ} between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation. An example of this constraint may be modeled by ∥grad (t′_{τ}+ϵ_{τ})∥=1. The new thicknesspreserving constraint preserves layer thickness with greater accuracy as shown in histogram 503 of
Embodiments of the invention further improve the performance and computational speed of the computer generating the restored model by linearizing the new thicknesspreserving constraint. As an example, the new thicknesspreserving constraint may be linearized as follows. ∥grad (t′_{τ}+ϵ_{τ})∥=1 may be squared to obtain ∥grad t′_{τ}∥^{2}+∥grad ϵ_{τ}∥^{2}+∥2·grad t′_{τ}·grad ϵ_{τ}∥=1. The error correction term ϵ_{τ} may be generated such that the square of its spatial variation, ∥grad ϵ_{τ}∥^{2}, is negligible. Accordingly, the thicknesspreserving constraint simplifies to a new linear thicknesspreserving constraint of grad ϵ_{τ}·grad t′_{τ}≈½ {1−∥grad t′_{τ}∥^{2}} (eqn. (37)). This thicknesspreserving constraint is linear because t′_{τ} is already known, so the constraint is a relationship between the gradient of the error ϵ_{τ} and the gradient of the known first approximation of the vertical coordinate t′_{τ}. The computer may therefore compute the new thicknesspreserving constraint in linear time, which is significantly faster than computing the nonlinear constraints ∥grad t_{τ}∥=1 or ∥grad (t′_{τ}+ϵ_{τ})∥=1.
Contrary to conventional methods, the computational complexity for performing the restoration transformation according to embodiments of the invention is significantly reduced compared to classical methods that are based on the mechanics of continuous media. As a consequence, the modeling computer uses significantly less computational time and storage space to generate the inventive restoration model.
Contrary to conventional methods that allow variations of geological volumes and deformations, embodiments of the invention implement a new set of geometrical constraints and boundary conditions that preserve geological volumes and deformations while adhering to geological boundaries.
Contrary to conventional methods, embodiments of the invention restore faults along fault striae (e.g., see
An ideal restoration should also transform the horizontal coordinates u_{τ} and v_{τ} in a manner that strictly honors lateral spatial distribution, to preserve areas and volumes of the Earth, so that terrains are not stretched or squeezed over time in the horizontal dimensions. However, conventional restoration transformations based on depositional coordinates (e.g., paleogeographic coordinates u and v) typically deform the horizontal coordinates, forcing terrains to stretch and squeeze, resulting in errors in the restoration model.
Embodiments of the invention improve the accuracy of the restoration model at time τ by establishing horizontal restoration coordinates u_{τ} and v_{τ} that restore the horizon surface H_{τ} deposited at time τ consistently with horizontal depositional coordinates u and v whilst minimizing deformations. In one embodiment, on the horizon surface H_{τ} only, the horizontal restoration coordinates u_{τ} and v_{τ} are equal to the depositional coordinates u and v (see e.g., equation (20)) and the spatial variations of the horizontal restoration coordinates u_{τ} and v_{τ} are preserved with respect to the horizontal depositional coordinates u and v (see e.g., equation (21)). Thus, each restoration model at time τ, presents a horizon surface H_{τ}, as it was configured at that time τ when it was originally deposited. Additionally or alternatively, horizontal restoration coordinates u_{τ}0 and v_{τ} are modeled in a tectonic style (e.g., using constraints (22) or (23)) that is consistent with that of the horizontal coordinates u and v of the depositional model, which makes the restoration more accurate because the geological context is taken into account. Additionally or alternatively, horizontal restoration coordinates u_{τ} and v_{τ} are modeled to minimize deformations induced by the restoration of horizon H_{τ}, rather than minimizing deformations in the whole volume G. This may be achieved by implementing constraints (41) and (42) that only enforce orthogonality of gradients of u_{τ} and v_{τ} with local axes b_{τ} and a_{τ}, but which do not constrain the norm of grad u_{τ} and grad v_{τ}, as is typically constrained for horizontal depositional coordinates u and v consistent with the depositional time model. Horizontal restoration coordinates u_{τ} and v_{τ} may also be constrained only in G_{τ}, thereby only taking into account the part of the subsurface to be restored, not the entire model G. Additionally or alternatively, horizontal restoration coordinates u_{τ} and v_{τ} may be constrained to be equal on opposite sides of τactive faults at twin point locations, where the twin points are computed from fault striae, which also ensures consistency with the depositional model (see e.g., equation (43)). Additionally or alternatively, horizontal restoration coordinates u_{τ} and v_{τ} are constrained to be equal on opposite sides of τinactive faults at mate point locations to cancel the effect of inactive faults on the restoration model (see e.g., equation (43)).
Reference is made to
In operation 1710, a processor may receive a 3D model of the present day measured geometry (e.g., present day model 202) comprising one or more folded (e.g., curvilinear or nonplanar) geological horizon surfaces (e.g., 210). The present day model may be measured tomographically by scanning the Earth's subsurface e.g., as described in reference to
In operation 1720, a processor may select or receive a past restoration time τ that is “intermediate” or prior to the present time and after the start of the subsurface's deposition (the time period when an oldest horizon surface in the 3D model was originally deposited).
In operation 1730, a processor may restore the 3D model from the present day measured geometry (e.g., present day model G_{τ} 202 in xyzspace G 220) to the predicted past geometry at the restoration time in the past τ (e.g., restored model
The processor may restore the vertical coordinate t_{τ} such that points along a horizon surface H_{τ} (e.g., 210) modeling sediment that was deposited at the selected restoration time τ have a substantially constant value for the restored vertical coordinate t_{τ} (see e.g., eqn. (19)). Further, the processor may restore the vertical coordinate t_{τ} such that at any location in the 3D model, the restored vertical coordinate t_{τ} is equal to a sum of a first approximation t′_{τ} of the vertical coordinate and an error correction term ϵ_{τ}, wherein the error correction term ϵ_{τ} is computed by solving a relationship in which a variation in the sum of the first approximation t′_{τ} of the vertical coordinate and the error correction term ϵ_{τ} between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation. The error correction term ϵ_{τ} may correct errors in the first approximation t′_{τ} of the vertical coordinate. This constraint may be represented by a linear second order approximation (see e.g., eqn. (37)).
In some embodiments, the processor computes the first approximation t′_{τ} of the vertical coordinate by solving a relationship in which the spatial variation of the vertical coordinate t′_{τ} is locally approximately proportional to the spatial variation of a geological time of deposition t. In some embodiments, the coefficient of proportionality is locally equal to the inverse of the magnitude of the maximal spatial variation of the geological time of deposition (see e.g., eqn. (34)(1)). This relationship may give the vertical restoration coordinate t_{τ} the shape of the horizon H_{τ} because, on the horizon, the gradient of depositional time t is normal to the horizon surface. Thus, the ratio grad t/∥grad t∥ follows the shape of the horizon.
In some embodiments, the processor computes the first approximation t′_{τ} of the vertical coordinate by solving a relationship in which any infinitesimal displacement in the direction orthogonal to horizon surface H_{τ} results in a variation of the vertical coordinate t′_{τ} approximately equal to the length of the infinitesimal displacement for points on the horizon surface H_{τ} (see e.g., eqn. (33)(1)).
In some embodiments, the processor computes the restored vertical coordinate t_{τ} in parts of the subsurface which are older than restoration time τ such that isovalue surfaces of the restored vertical coordinate t_{τ} are parallel to the horizon surface H_{τ} and the difference in the restored vertical coordinate t_{τ} between two arbitrary isovalues is equal to the distance between the corresponding isosurfaces (see e.g., eqn. (31)). Parallel surfaces may be planar parallel in the restored model, and curved parallel (e.g., having parallel tangent surfaces) in present day model, such that the surfaces are nonintersecting at limits.
In some embodiments, the error correction term ϵ_{τ} is null at points along the horizon surface H_{τ} that was deposited at the selected restoration time in the past τ so that the restored horizon surface H_{τ} is flat (see e.g., eqn. (36)).
In some embodiments, the restored horizontal coordinates u_{τ} and v_{τ} are constrained such that for each point along the horizon surface H_{τ} that was deposited at the selected restoration time in the past τ: the restored horizontal coordinates u_{τ} and v_{τ} are equal to depositional horizontal coordinates u and v, respectively, and the spatial variations of the restored horizontal coordinates u_{τ} and v_{τ} are equal to the spatial variations of the depositional horizontal coordinates u and v, respectively (see e.g., eqns. (20)(21)). On average, globally over the entire model, the processor may compute ∥grad u∥=1 and ∥grad v∥=1. However, locally, this is not necessarily true e.g., on horizon Hτ. So, while the processor sets grad u_{τ}=grad u and grad v_{τ}=grad v on Hτ, the processor may not constrain ∥grad u_{τ}∥=1 and ∥grad v_{τ}∥=1 on Hτ. Moreover, the processor may not constrain grad u_{τ} to be orthogonal to grad t_{τ}. This results from the boundary condition on Hτ and propagation through its constant gradient.
In some embodiments, the restored horizontal coordinates u_{τ} and v_{τ} are constrained in parts of the subsurface which are older than restoration time τ such that directions of maximal change of the restored horizontal coordinates u_{τ} and v_{τ} are linearly constrained by a local coaxis vector b_{τ} and a local axis vector a_{τ}, respectively (see e.g., eqn. (41)).
In some embodiments, the local axis vector a_{τ} is oriented approximately in the direction of maximal change of depositional horizontal coordinate u and orthogonal to the direction of maximal change of the vertical restoration coordinate t_{τ}, and the local coaxis vector b_{τ} is oriented orthogonal to the direction of the local axis vector a_{τ} and orthogonal to the direction of maximal change of the vertical restoration coordinate t_{τ} (see e.g., eqn. (40)).
In some embodiments, if the tectonic style of the 3D model is minimal deformation, the restored horizontal coordinates u_{τ} and v_{τ} are computed over the part of the 3D model of the subsurface which is older than restoration time τ such that the directions of maximal change of u_{τ} and v_{τ} are approximately orthogonal to the local coaxis vector b_{τ} and the local axis vector a_{τ}, respectively. For example, equation (40) constrains the local axis vector a_{τ} to be parallel to the gradient of u and the local coaxis vector b_{τ} to be orthogonal to the local axis vector a_{τ}, which means that the gradient of u is orthogonal to the local coaxis vector b_{τ}. Equation (41) further constrains the gradient of u_{τ} to be approximately orthogonal to the local coaxis vector b_{τ}. Accordingly, the gradient of u_{τ} is approximately parallel to the gradient of u. The same logic implies the gradient of v_{τ} is approximately parallel to the gradient of v.
In some embodiments, if the tectonic style of the 3D model is flexural slip, the restored horizontal coordinates u_{τ} and v_{τ} are computed over the part of the 3D model of the subsurface which is older than restoration time τ such that projections of their directions of maximal change over the isovalue surfaces of the restored vertical coordinate t_{τ} are approximately orthogonal to local coaxis vector b_{τ} and the local axis vector a_{τ}, respectively (see e.g., eqn. (42)).
In some embodiments, the values of the restored horizontal coordinates u_{τ} and v_{τ} are constrained in parts of the subsurface which are older than the restoration time τ to be respectively equal on twin points on τactive faults, wherein twin points are points on opposite sides of a τactive fault that were collocated at the restoration time τ and are located on the same fault stria in the present day model, to merge the twin points into the same position in the restored model by sliding the twin points towards each other in a direction tangential to the surface of the τactive fault (see e.g., eqn. (43)).
In some embodiments, the values of the restored horizontal coordinates u_{τ} and y_{τ} are constrained in parts of the subsurface which are older than the restoration time τ to be respectively equal on mate points on τinactive faults, wherein mate points are points on opposite sides of a τinactive fault that are collocated at present day time, to move mate points together on opposite sides of τinactive faults (see e.g., eqn. (43)).
In operation 1740, a processor may display an image of the restored 3D model of the subsurface geology of the Earth such that each point in the 3D model is positioned at the restored coordinates u_{τ}, v_{τ}, t_{τ} defining the location that a piece of sediment represented by the point was located at the restoration time in the past τ.
In some embodiments, the processor may receive an increasing chronological sequence of past restoration times τ_{1}, τ_{2}, . . . , τ_{n}. For each restoration time τ_{i}, in sequence τ_{1}, τ_{2}, . . . , τ_{n}, the processor may repeat operations 17201730 to compute a corresponding 3D restoration transformation R_{τ}_{i}. 3D restoration transformation Rτ_{i }restores the part of the subsurface older than horizon H_{τ}_{i}to its predicted past geometry at time τ_{i}, e.g., to 3D restored coordinates u_{τ}_{i}, v_{τ}_{i}, and t_{τ}_{i}.
In operation 1750, in some embodiments, a processor may play a moving image sequence in which the 3D model is iteratively restored in a forward or reverse order of the sequence of past restoration times τ_{1}, τ_{2}, . . . , τ_{n }to visualize changes in the subsurface geology over the passage of time.
In some embodiments, the processor may edit the model in the restoration space and then reverse the restoration transformation to apply those edits in the present day space. For example, the processor may edit the depositional values u, v, and t associated with the restored 3D model, and then reverse transform the restored 3D model forward in time from the predicted past geometry at the restoration time in the past τ to the present day measured geometry using an inverse of the 3D restoration transformation 200 to incorporate the edits from the restored model into the present day model.
Operations of
In the past 30 years, many methods have been proposed to build geological models of sedimentary terrains having layers that are both folded and faulted. For any given geologicaltime τ, checking geological model consistency is considered both simpler and more accurate if terrains have previously been “restored” to their predeformational, unfolded and unfaulted state, as they were at geologicaltime τ.
Embodiments of the invention provide a new, purely geometrical 3D restoration method based on the input of a depositional (e.g., GeoChron model). Embodiments of the invention are able to handle depositional models of any degree of geometrical and topological complexity, with both small and large deformations, do not assume elastic mechanical behavior, and do not require any prior knowledge of geomechanical properties. Embodiments of the invention further reduce or eliminate gaps and overlaps along faults as part of the restoration transformation and do not resort to any postprocessing to minimize such gaps and overlaps. Compared to other conventional methods, embodiments of the invention minimize deformations and volume variations induced by geological restoration with a higher degree of precision, unequaled so far (see e.g.,
Referring to

 deformations along faults 105 to be minimized,
 restoration to work correctly even though there are regions of G_{τ} not continuously connected to H_{τ},
 gaps and overlaps along faults and the geometry of fault striae 600 are minimized by the restoration transformation, so no postprocessing is needed to correct gaps or overlaps.
Embodiments of the invention input a 3D model of sedimentary terrains in the subsurface. In one example, the input model may be the GeoChron™ model generated by SKUA® software for use in mining and oil and gas industries. Embodiments of the invention may build a 3D restoration transformation of this model in such a way that, after transformation, the new model represents terrains as they were at a given intermediate restorationtime τ (where τ_{1}<τ<τ_{2}, before the present day τ_{2 }and after the time of the deposition of the oldest layer τ_{1}).
For example, G may represent the present day 3D geological domain of the region of the subsurface being modeled and G_{τ} 202 may represent the subset of G containing particles of sediment that were deposited at a time prior to or equal to τ. In some embodiments, for all points r ε G, a geologic restoration transformation may move a particle of sediment observed today at location r to a new restored location
where R_{τ}(r) represents a 3D field of restoration vectors, e.g., generated to minimize deformations in G_{τ}.
Depositional ModelA depositional model may be generated by inputting a tomographic model of the present day subsurface geology of the Earth and transforming that geology to a past depositional time as each particle was configured when originally deposited in the Earth. Sedimentary particles are deposited over time in layers from deepest to shallowest from the earliest to the most recent geological time periods. Since various layers of terrain are deposited at different geological times, a depositional model does not image the geology at any one particular time period, but across many times periods, each layer modeled at the geological time when the layer was deposited. Accordingly, the vertical axis or depth in the depositional model may be a time dimension representing the time period of deposition, progressing from oldest to newest geological time as the model progresses vertically from deepest to shallowest layers.
In one embodiment, the depositional model may be the GeoChron™ model, which is generated by SKUA™ software, that is routinely used by many oil & gas companies to build models of geologic reservoirs which help optimize hydrocarbon production and exploration. An example implementation of the GeoChron model is discussed in U.S. Pat. No. 8,600,708, which is incorporated by reference herein in its entirety. The depositional model is described in reference to the GeoChron model only for example, though any other depositional model may be used.
Reference is made to
In the example uvttransform 700 shown in
∥gradt(r)∥=1∀rεG (3)
Embodiments of the invention observe that when ∥grad t(r)∥ differs from “1,” replacing the depositional coordinates {u(r), v(r), t(r)} of the uvttransform 700 by new restoration coordinates {u_{τ}(r), v_{τ}(r), t_{τ}(r)} where ∥grad t_{τ}∥=1 allows the uvttransform to be replaced by a u_{τ} v_{τ} t_{τ}transform that generates a valid restoration model at restoration time τ.
In some embodiments, the depositional (e.g., GeoChron) model includes the following data structures stored in a memory (e.g., memory 150 of

 A network of geological faults 105 within the present day domain G 220.
 3D cornerpoint grid Γ 100 that fills the Gspace 220 with 3D polyhedral cells 108 (e.g., tetrahedra or hexahedra), without any gaps or overlaps in the studied domain, in such a way that no cell edge crosses any fault. The location of each node α 107 of grid Γ 100 in the Gspace is denoted r(α).
 For each geological fault F 105, two disconnected and independently meshed, collocated surfaces F^{+} 103 and F^{−} 104 on opposite sides of the fault 105. Surfaces F^{+} 103 and F^{−} 104 may be composed of 2D facets from the 3D polyhedral cells of grid Γ 100 bordering F 105 on either side of the fault 105. Fault surfaces F^{+} 103 and F^{−} 104 that are collocated in the present day model may, during the restoration process of transforming the model backwards in time, typically slide against one another, without generating gaps or overlaps between adjacent fault surfaces or fault blocks.
 Referring to
FIGS. 1, 3 and 6 , for each fault F 105, a set of pairs of points (r_{F}^{+},r_{F}^{−}) (101,102) called “twinpoints,” such that:  1. The two twin points in each pair are located on opposite sides of a corresponding pair of twin fault surfaces, r_{F}^{+} ε F^{+} and r_{F}^{−} ε F^{−}.
 2. At geological times before fault F 105 formed in the subsurface, particles of sediment were collocated which are observed today at locations r_{F}^{+} and r_{F}^{−}.
During the activation of fault F 105, particles of sediment initially located on F are assumed to slide along faultstriae (e.g., seeFIG. 12 ), which are the shortest paths, on F, between pairs of twin points (r_{F}^{+},r_{F}^{−}) (101,102).  A tectonic style which may be either a “minimal deformation” style or a “flexural slip” style. Choosing this tectonic style is a model decision assumed to have been made by a structural geologist.
 A triplet (e.g., {u(r), v(r), t(r)}) of discrete coordinates defined on a 3D grid Γ 100 of the depositional
G space, such that, for a particle of sediment observed today at location r, the coordinate values {u(r), v(r)} represent the paleogeographic coordinates of the particle at geologicaltime t(r) during the time period when it was deposited. According to the depositional (e.g., GeoChron) model, the paleogeographic coordinates {u(r), v(r)} may honor different differential equations depending on the tectonic style.
Moreover, referring to
Within the present day domain G, each geological horizon H_{τ} 210 may be defined by a set of particles of sediment which were deposited at geological time τ:
rεH_{τ}⇐⇒t(r)=τ (4)
In other words, each horizon H_{τ} 210 is a levelset (constant value) surface of the geologicaltime t.

 Paleogeographic coordinates {u(r), v(r)} and twinpoints (101,102) given as input are linked e.g. by the following equations:

 p Additionally or alternatively, each pair of twinpoints (r_{F}^{+},r_{F}^{−}) (101,102) may be the intersection of a level set 210 of vertical depositional coordinate t(r) with a “fault stria” σ(r_{F}^{−}) 600 comprising a curved surface passing through point r_{F}^{−} 102 whose geometry is defined by geological rules, e.g., defining fault blocks sliding against one another according to tectonic forces and geological context. As a consequence of constraints defined by equations (6), (7), and (8) above, faultstriae (e.g., see
FIG. 12 ) may characterize the paleogeographic coordinates {u(r), v(r)}, and vice versa.  Each point r ε G 214 may be characterized by its present day coordinates (e.g., {x(r), y(r), z(r)}) with respect to a present day coordinate system {r_{x}, r_{y}, r_{z}} 220 comprising three mutually orthogonal unit vectors, e.g., where r_{z }is assumed to be oriented upward.
 p Additionally or alternatively, each pair of twinpoints (r_{F}^{+},r_{F}^{−}) (101,102) may be the intersection of a level set 210 of vertical depositional coordinate t(r) with a “fault stria” σ(r_{F}^{−}) 600 comprising a curved surface passing through point r_{F}^{−} 102 whose geometry is defined by geological rules, e.g., defining fault blocks sliding against one another according to tectonic forces and geological context. As a consequence of constraints defined by equations (6), (7), and (8) above, faultstriae (e.g., see
It would be appreciated by a person of ordinary skill in the art that the GeoChron model and its features described herein are discussed only as an example of a depositional model and that these elements may differ in other models or implementations without changing the essence of the invention.
u_{τ} v_{τ} t_{τ}TransformationReferring to the volume deformation of
Accordingly, present day geological space G_{τ} 202 is transformed into a restored geological space

 t_{τ}(r) is a vertical spatial coordinate of the subsurface at the past restoration time τ, and is derived from, but different than, the geological time of deposition t(r). The vertical restoration coordinate t_{τ}(r) honors the following constraint:
∥gradt_{τ}(r)∥=1∀rεG_{τ} (10)

 {u_{τ}(r), v_{τ}(r)} are lateral restoration coordinates derived from, but different than, the paleogeographic coordinates {u(r), v(r)} of the depositional model.
 restoration coordinates {u_{τ}(r), v_{τ}(r), t_{τ}(r)} honor specific inventive constraints described below in such a way that, using the u_{τ} v_{τ} t_{τ}transform as a restoration operator minimizes deformations in the present day domain G_{τ}.
 for each point r ε G_{τ} 202, the restoration vector field R_{τ} may be defined e.g. by:
R_{τ}(r)=u_{τ}·r_{x}+v_{τ}(r)·r_{y}+t_{τ}(r)·r_{z}−r (11)
Compaction may be handled in pre and postrestoration stages, as is known in the art. Thus, the model may be restored without taking compaction into account.
Some embodiments of the invention provide an inventive volume deformation with a new set of inventive geometric constraints on the depositional model to allow geologic layers to be restored at a given geological time τ with a precision that has never before been reached. As shown in
As shown in

 a restored volume 203, denoted
G _{τ}, contains a (e.g., direct or “righthanded”) coordinate space 219 having orthogonal coordinate unit vectors {r _{u}_{τ},r _{v}_{τ},r _{t}_{τ}} and a family of horizontal planes {S _{τ}(d):d≥0} 207 parallel to horizontal coordinate vectors {r _{u}_{τ},r _{v}_{τ}};  a deformed version G_{τ} of
G _{τ} contains a (e.g., direct or “righthanded”) coordinate space 220 having orthogonal coordinate unit vectors {r_{x}, r_{y}, r_{z}} and a family of curved surfaces {S_{τ}(d):d≥0} 208 parallel to horizon {H_{τ}≡S_{τ}(0)} (210=208).
 a restored volume 203, denoted
For simplicity and without loss of generality, the coordinate frame unit vectors {
Referring to

 each point r ε G_{τ} 214 is transformed into point
r _{Υ} εG _{τ} 213 and vice versa:
 each point r ε G_{τ} 214 is transformed into point

 {u_{τ}(
r _{τ}), v_{τ}(r _{τ})} represent the horizontal restoration coordinates ofr _{τ} 213 with respect to {r _{u}_{τ},r _{v}_{τ}} 219 whilst t_{τ}(r ) represents the vertical restoration coordinate or altitude 204 ofr _{τ} 213 with respect tor _{t}, oriented upward;  for each point r ε G_{τ} 214:
 {x(r), y(r)} represent the horizontal present day coordinates of r with respect to {r_{x}, r_{y}} whilst z(r) represents the vertical present day coordinate or altitude of r with respect to the vertical unit frame vector r_{z }oriented upward; and
 {u_{τ}(r), v_{τ}(r), t_{τ}(r)} represent the restoration coordinates of
r _{τ} with respect to the restoration coordinate frame unit vectors {r _{u}_{τ},r _{v}_{τ},r _{t}_{τ}} 219 of the restored volumeG _{τ} 203.
 {u_{τ}(
Equivalently to equations (12) and in accordance with equation (1), during restoration of G_{τ}, a particle of sediment observed today at location r 214 is moved to a new location
with, in matrix notation:
Referring to
Referring to
t_{τ}(
such that:
Referring to

 each horizontal plane
S _{τ}(d) 207 is transformed into a curved surface S_{τ}(d) 208 “parallel” (e.g., this notion of “parallelism” may be characterized by equation (10)) to surface H_{τ} 210=208 and each surface S_{τ}(d) 208 is a level set of vertical restoration coordinate t_{τ}(r);  the images in G_{τ} 202 of the (u_{τ}), (v_{τ}) 205, 206 and (t_{τ}) 204 coordinate axes initially rectilinear and contained in volume
G _{τ} 203 now consist of curvilinear coordinate axes (223, 224) and 222.
 each horizontal plane
As shown in

 S_{τ}(0) is assumed to be identical to the horizon H_{τ} 210 to be restored:
S_{τ}(0)≡H_{τ} (19)
which is equivalent to defining that, on horizon H_{τ} 216, restored vertical coordinate t_{τ}(r) is equal to z_{τ}^{0};

 for any t<τ, the actual geologic horizon H_{t }216 is included (212) into the deformed volume G_{τ} 202; note that, contrary to surfaces {S_{τ}(d):d≥0} 208, horizons {H_{t}:t<τ} 216 may be nonparallel to {H_{τ}≡S_{τ}(0)} 210=208;
 after restoration of the volume G_{τ} 202 to its initial, unfolded state
G _{τ} 203:  all horizons, faults and geological objects included in G_{τ} 202 are dragged up by the embedding volume deformation,
H _{τ}≡S _{τ}(0)} 209=207 may be defined as the restored sea floor as it was at geological time τ.
With compaction handled separately in pre and post restoration steps, leaving aside the very particular case of clay and salt layers, tectonic forces generally induce no or negligible variations in volume. Therefore, restoration coordinates {u_{τ}(r), v_{τ}(r), t_{τ}(r)} may be chosen in such a way that the u_{τ} v_{τ} t_{τ}transform 201 of the presentday volume G_{τ} 202 into the restored volume

 Surfaces {S_{τ}(d):d≥0} 208 are level sets of the vertical restoration coordinate t_{τ}(r) and, for any infinitely small increment ϵ, the thickness of the thin slice of the volume bounded by S_{τ}(d) and S_{τ}(d+ϵ) are, as much as possible, constant and equal to ϵ. In other words, S_{τ}(d) and S_{τ}(d+ϵ) are as parallel as possible. This is equivalent to honoring equation (10) as precisely as possible.
 In the frame of this invention, the consistency between the depositional (e.g., GeoChron) model provided as input and its restored version at geological time τ is preserved. Such a consistency is preserved if, and only if, the uvttransform and the u_{τ} v_{τ} t_{τ}transform of H_{τ} are identical. This is achieved by honoring the following inventive boundary conditions, referred to as the (u_{τ}, v_{τ}) boundary constraints:

 Whilst taking the same given tectonic style into account (minimal deformation or flexural slip) as the one honored by paleogeographic coordinates {u(r), v(r)}, lateral restoration coordinates {u_{τ}(r), v_{τ}(r)} may be defined so that their associated restoration deformations are minimized. To preserve consistency with boundary conditions (20) and (21), this is achieved by honoring the following inventive constraints:
 in a minimal deformation tectonic style context, the following “minimal deformation constraints” may be honored by coordinates {u_{τ}, v_{τ}}, where t_{τ}(r) is given, e.g., as follows:

 in a flexural slip tectonic style context, the following “flexural slip constraint” is coupled (containing both lateral restoration coordinates u_{τ} and v_{τ}) and may be honored by coordinates {u_{τ}, v_{τ}}_{r}, e.g., as follows:
where subscript “S” refers to a projection of the directions of maximal change over isovalue surfaces of the restored vertical coordinate tτ.
So as not to conflict with equations (20) and (21), and contrary to conventional depositional coordinates u and v (e.g., in the GeoChron model), new constraints (22) and (23) do not constrain ∥grad u_{τ}∥, ∥grad v_{τ}∥, ∥grad_{s }u_{τ}∥, or ∥grad_{s }v_{τ}∥ to be equal to “1”.
RestorationReferring to

 restoring horizon H_{τ} 210 to its initial, unfaulted and unfolded state (e.g., mapping horizon H_{τ} onto the sea floor
S _{τ}(0)) 209 and  shifting all sedimentary terrains in such a way that, for each point r ε G:
 the particle of sediment currently located at point r moves to its former, “restored” location, where the particle was located at geological time τ,
 no overlaps or voids/gaps are created in the subsurface.
 restoring horizon H_{τ} 210 to its initial, unfaulted and unfolded state (e.g., mapping horizon H_{τ} onto the sea floor
At geological time τ, the sea floor
∀rεH_{τ}:z_{τ}^{0 }stands for z_{τ}^{0}(u(r),v(r)) (24)
Deformation of sedimentary terrains is typically induced both by tectonic forces and terrain compaction. In order to model separately the effects of these phenomena, the restoration process may proceed as follows:

 First, in a preprocessing phase, a total decompaction may be applied to the terrains to cancel the impact of compaction as it is observed today, at the present day or current geological time;
 Next, the effects of compaction being canceled, a depositionalbased restoration process taking only tectonic deformations into account (and not compaction) is applied to restore the geometry of the subsurface as it would have been observed at geological time τ;
 Finally, in a postprocessing phase, a partial recompaction is applied to the restored terrains in order to take compaction into account, as it could have been observed at geological time τ.
As an input to the restoration process, a given depositional (e.g., GeoChron) model may be received from storage in a digital device (e.g., from memory 150 of
Referring to
The region G_{τ} 202 may be retrieved as the part of the depositional model where geological time of deposition t(r) is less than or equal to τ (subsurface regions deposited in a layer deeper than or equal to the layer deposited at time τ).
The set of faults may be split into a subset of zactive faults cutting H_{τ} 210 and a subset of τinactive faults which do not cut H_{τ}.
A geologist or other user may decide to manually transfer some faults from the τinactive fault set to the τactive set, or vice versa, which typically causes greater restoration deformations. For example, manually altering the set of automatically computed τactive and τinactive faults typically makes the restoration less accurate.
Four new 3D piecewise continuous discrete functions {u_{τ}, v_{τ}, t_{τ}, ϵ_{τ}}_{r }may be created that are defined on grid Γ 100 whose discontinuities occur only across τactive faults;
Referring to
where {r_{F}^{⊕},r_{F}^{⊖}}_{τ} (304,306) represents a pair of “matepoints” collocated on both sides of F 300 and assigned to F^{+} 103 and F^{−} 104, respectively, and ϵ_{τ}(r) represents an error correction constraint. Constraints (25), (26), (27) and (28) may be referred to collectively as “fault transparency constraints.”
Assuming that TH_{min}>0 is a given threshold chosen by a geologist or other user, fault transparency constraints (25), (26), (27) and (28) may be locally installed at any point r_{F }on a τactive fault F where fault throw is lower than TH_{min}. As a nonlimitative example, TH_{min }may be equal to 1 meter.
Two new discrete vector fields r* and R_{τ} may be defined on 3D grid Γ 100.
For each node α ε Γ 107:

 r*(α) may be initialized as the initial location of α:
r*(α)=r(α) (29)

 a decompaction transformation C^{−}(r) known in the art may be used to move α vertically downward from its current compacted altitude z(α) to a new decompacted (e.g., deeper) altitude:
r(α)←C^{−}(r(α)) (30)
Referring to
∥gradt_{τ}(r)∥≃1∀rεG_{τ} (31)
In addition, to allow H_{τ} 210 to be restored on surface
t_{τ}(r_{H})=z_{τ}^{0}∀r_{H}εH_{τ} (32)
Due to its nonlinearity, thicknesspreserving equation (31) cannot be implemented as a DSI constraint, which must be linear. In order to incorporate the thicknesspreserving equation into the restoration model using the DSI method, various linear surrogates of equation (31) may be used to approximate t_{τ}(r) as follows:

 Referring to
FIG. 1 , to approximate thicknesspreserving equation (31), as a nonlimitative example, install the following DSI constraints on the grid Γ 100 where r_{T⋄} and r_{T* }are arbitrary points belonging to any pair (T^{⋄}, T*) of adjacent cells 108 of grid Γ 100 and where N(r_{h}) is the field of unit vectors defined on H_{τ}, orthogonal to H_{τ} and oriented in the direction of younger terrains:
 Referring to
1)gradt_{τ}(r_{H})=N(r_{H})∀r_{H}εH_{τ }2)gradt_{τ}(r_{T⋄})≃gradt_{τ}(r_{T*})∀(T^{⋄},T*) (33)

 Referring to
FIG. 1 , to approximate thicknesspreserving equation (31), alternatively as a nonlimitative example, install the inventive DSI constraints on grid Γ 100 as follows:
 Referring to
where r_{T⋄} and r_{T* }are arbitrary points belonging to any pair (T^{⋄}, T*) of adjacent cells of grid Γ 100 (e.g., the centers of T^{⋄} and T*, respectively).
Constraints (33) and (34) are only examples of possible surrogatethicknesspreserving constraints. Other examples of such surrogate thicknesspreserving constraints may be used.
Referring to
Assuming that constraints (32) and (33) or (34) are installed on grid Γ 100, a first approximation of vertical restoration coordinate t′_{τ}(r) may be computed by running the DSI method on grid Γ 100.
Honoring constraint (31) significantly increases the accuracy of the restoration model and a violation of this constraint not only degrades the accuracy of the vertical restoration coordinate t_{τ}(r) but also the horizontal restoration coordinates {u_{τ}(r), v_{τ}(r)} as they are linked to t_{τ}(r) (e.g., by equations (22) and (23)). Accordingly, there is a great need for validating any approximation technique used to compute t_{τ}(r).
To test the accuracy of the various approximations of t_{τ}(r), an example geological terrain is provided in
Similarly,
An approximation of the vertical restoration coordinate t′_{τ}(r) may be improved by a “t_{τ}incremental improvement” constraint, e.g., as follows:
t_{τ}(r)=t′_{τ}(r)+ϵ_{τ}(r0∀rεG_{τ} (35)
where ϵ_{τ}(r) is an error correction term, e.g., as characterized below.
Accordingly, assuming that an initial approximation t′_{τ}(r) has already been obtained, to compute an improved version of t_{τ}(r), the following inventive incremental procedure may be executed:

 For each point r_{H }ε H_{τ}, set the following equation as an inventive seafloorerror constraint e.g., using the DSI method:
ϵ_{τ}(r_{H})=0∀r_{H}εH_{τ} (36)
this constraint assumes that constraint (32) remains honored.

 For each cell T ε Γ ∩ G_{τ} 108, choose a point r_{τ} in the cell T (e.g., its center) and install the new linear thicknesspreserving constraint, e.g., using the DSI method as follows:
gradϵ_{τ}(r_{τ})·gradt′_{τ}(r_{τ})≃½{1−∥gradt′_{τ}(r_{τ})∥^{2}} (37)
This constraint is linear, deduced from a linear second order approximation of equation (31). Further, this constraint ensures that, after applying the t_{τ}incremental improvement correction constraint (35), the local value of ∥grad t_{τ}(r)∥ at any point r ε G_{τ} is as close as possible to “1.”

 For each sampling point r located on a τactive fault, install for ϵ_{τ}(r) the following inventive DSI constraint referred to as the “t_{τ}incremental boundary” constraint:
gradϵ_{τ}(r)×gradt_{τ}(r)≃0 (38)
This constraint specifies that, after applying correction constraint (35), in the close neighborhood of τactive faults, the shape of level sets of t_{τ}(r) remains roughly unchanged.

 To ensure piecewise continuity of the error correction ϵ_{τ}(r), install DSI gradient smoothness constraints, known in the art, for the error correction ϵ_{τ}(r).
 Assuming that constraints (36), (37) and (38) are installed on grid Γ 100, to interpolate the error correction ϵ_{τ}(r), run DSI on grid Γ 100.
 For each node α ε Γ 107, update the vertical restoration coordinate t_{τ}(α) as follows:
t_{τ}(α)=t′_{τ}(α)+ϵ_{τ}(α) (39)

 In the test case represented by
FIG. 4 andFIG. 5 , the histogram 503 of the distribution of ∥grad t_{τ}∥, where t_{τ} is approximated by constraints (37) over G_{τ} 202 is now considerably better than histograms 501 and 502 obtained with constraints (33) or (34), respectively. In particular:  As specified by equation (10), distribution 503 is now centered on value “1”. This condition is of paramount importance to minimize deformations during the restoration process generated by a u_{τ} v_{τ} t_{τ}transform.
 The standard deviation of distribution 503 is considerably reduced as compared to the relatively wider standard deviation of distributions 501 and 502.
 Moreover, in the test case represented in
FIG. 4 andFIG. 9 , the histogram 903 of the distribution of relative variations of volume ΔV/V induced by a restoration of H_{τ} 210 over G_{τ} 202, where t_{τ} is obtained using constraints (37) is significantly more accurate than histograms 901 and 902, where t_{τ} is obtained using constraints (33) or (34), respectively. The center of histogram 903 of volume variation ΔV/V is closer to the ideal value “0” than histograms 901 and 902, which indicates that variations of volume are better minimized after applying second order constraints (37) than constraints (33) or (34).
 In the test case represented by
Referring to

 install equations (20) and (21) as inventive boundary constraints.
 for all points r ε G_{τ} 214, define as follows inventive vectors fields a_{τ}(r) and b_{τ}(r) respectively, referred to as the “τaxe” and “τcoaxe” vector fields:
a_{τ}(r)=gradt_{τ}(r)×gradu(r)×gradt_{τ}(r)b_{τ}(r)=gradt_{τ}(r0×a_{τ}(r) (40)
The τaxe and τcoaxe vector fields a_{τ}(r) and b_{τ}(r) differ considerably from the local axe and coaxe vectors fields a(r) and b(r), e.g., as discussed in U.S. Pat. No. 8,711,140, which is incorporated by reference herein in its entirety. These new τaxe and τcoaxe vectors a_{τ}(r) and b_{τ}(r) strongly depend on the new vertical restoration coordinate t_{τ}(r) (e.g., already computed as above) and also take into account the gradient of the paleogeographic coordinate u(r) (e.g., associated to the depositional model provided as input).

 if the tectonic style is minimal deformation then, to approximate equations (22), install the following inventive “surrogate minimaldeformation” constraints e.g., using the DSI method:

 if the tectonic style is flexural slip then, to approximate equations (23), install the following inventive “surrogate flexuralslip” constraints e.g., using the DSI method:
where subscript “S” refers to a projection of the directions of maximal change over isovalue surfaces of the restored vertical coordinate tτ.

 Referring to
FIG. 1 andFIG. 6 , to prevent the u_{τ} v_{τ} t_{τ}transform used as a restoration operator from generating gaps and overlaps along τactive faults, specific constraints may be added along fault striae induced by twinpoints of the depositional model provided as input. For that purpose, for each pair of twinpoints (r_{F}^{+},r_{F}^{−}) (101,102) located on faces F^{+} 103 and F^{−} 104 of a τactive fault F 105, respectively, a process may proceed according to as follows:  retrieve the fault stria σ(r_{F}^{−}) 600 passing through twin points (r_{F}^{+},r_{F}^{−}) (101,102), and
 on curve σ(r_{F}^{−}) 600, retrieve a point {tilde over (r)}_{F}^{+} 601 located on F^{+} 103 and such that t_{τ}({tilde over (r)}_{F}^{+}) is approximately equal to t_{τ}(r_{F}^{−}).
 install the following inventive “τfaultstriae” constraints e.g., using the DSI method:
 Referring to

 To ensure piecewise continuity of horizontal restoration coordinates u_{τ}(r) and v_{τ}(r), install gradient smoothness constraints e.g., using the DSI method.
 To compute the pair of horizontal restoration coordinates {u_{τ}(r), v_{τ}(r)} honoring constraints (20), (41 or 42) and (43), run DSI on grid Γ 100.
The restoration vector field R_{τ}(r) represents the field of deformation vectors from the present day (e.g., xyz) space to the restoration (e.g., u_{τ} v_{τ} t_{τ}) space, e.g., computed from the u_{τ} v_{τ} t_{τ}transform.
Referring to
For each node α 107 of 3D grid Γ 100:

 if, to compute vertical restoration coordinate t_{τ}(r), compaction was taken into account, then, using a recompaction operator C^{+}(r) known in the art, move α vertically upward from its current decompacted altitude z(α) to a new recompacted (shallower) altitude:
r(α)←C^{+}(r(α)) (45)
save the restoration vector Rτ(α) on a digital device:
Rτ(α)=r(α)−R*(α) (46)
where r*(α) is defined e.g., in equation (29).

 reset location r(α) of α to its initial location before restoration:
r(α)ƒr*(α) (47)
stop.
Scanning the Subsurface through TimeConsider a series of geological restoration times {τ_{1}<τ_{2}< . . . <τ_{n}} associated with reference horizons H_{τ}_{1}, H_{τ}_{2}, . . . , H_{τ}_{n}, respectively. Using the restoration method described herein, for each (τ_{1}=τ), build and store on a digital device a restoration vector field R_{τ}_{i}(r)=R_{τ}(r), e.g., as:
In addition to these reference restoration times, an additional restoration time τ_{n+1 }may be added to be associated with the horizontal plane H_{t}_{n+1 }located at a constant altitude z_{τ}_{n+1}^{0 }of the sea level. Time τ_{n+1 }may be the present day geological time and, provided that τ_{n+1 }is greater than τ_{n}, any arbitrary value may be chosen for τ_{n+1}. As a nonlimitative example, τ_{n+1 }may be defined as:
τ_{n+1}=τ_{n}+1 (49)
Because τ_{n+1 }is the present day, applying the restoration vector field Rτ_{n+1}(r) to the present day horizon H_{t}_{n+1 }should leave H_{t}_{n+1 }unchanged e.g., as follows:
Rτ_{n+1}(r)=0∀rεG (50)
To explore subsurface evolution throughout geological times, a process may proceed as follows:

 as input, read a depositional (e.g., GeoChron) model and associated series of restoration vector fields {R_{τ}_{1}, R_{τ}_{2}, . . . , R_{τ}_{n+1}} stored on a digital device;
 using an input device such as, in a nonlimitative example, the keyboard of a computer or the wheel of a computer mouse, select a geological time τ_{i }in the given list of geological times {τ_{1}<τ_{2}< . . . <τ_{n+1}};
 for each vertex α ε Γ 107, save a copy r*(α) of the location of this node in the depositional model given as input;
 apply the restoration vector field R_{τ}_{i}(r) to the depositional model given as input;
 display the restored model on a device such as, in a nonlimitative example, a display (e.g., display 180 of
FIG. 15 ), such as, a screen, a hologram or a 3D printer;  repeat the previous operations as long as desired.
 optionally, to modify the geometry of the horizons at geological time τ_{i}, use a computerized tool known in the art to edit the geological time of deposition t(r);
 for each vertex α 107 of 3D grid Γ 100, use copy r*(α) to restore r(α) to its present day location:
r(α)←r*(α)∀αεΓ (51)
such an operation implicitly and automatically propagates the modifications of the geometry of horizons optionally performed above;

 return to the first step above.
Geological models are generated using geological or seismic tomography technology. Geological tomography generates an image of the interior subsurface of the Earth based on geological data collected by transmitting a series of incident waves and receiving reflections of those waves across discontinuities in the subsurface. A transmitter may transmit signals, for example, acoustic waves, compression waves or other energy rays or waves, that may travel through subsurface structures. The transmitted signals may become incident signals that are incident to subsurface structures. The incident signals may reflect at various transition zones or geological discontinuities throughout the subsurface structures, such as, faults or horizons. The reflected signals may include seismic events. A receiver may collect data, for example, reflected seismic events. The data may be sent to a modeling mechanism that may include, for example, a data processing mechanism and an imaging mechanism.
Reference is made to
One or more transmitter(s) (e.g., 190 of
One or more receiver(s) (e.g., 140 of
One or more processor(s) (e.g., 140 of
The processor(s) may compose all of the reflection points 50 to generate an image or model of the present day underground subsurface of the Earth 30. The processor(s) may execute a restoration transformation (e.g., u_{τ} v_{τ} t_{τ}transform 201) to transform the present day model of subsurface 30 to a restored subsurface image 203 at a restoration time τ. One or more display(s) (e.g., 180 of
Reference is made to
System 1505 may include one or more transmitter(s) 190, one or more receiver(s) 120, a computing system 130, and a display 180. The aforementioned data, e.g., seismic data used to form intermediate data and finally to model subsurface regions, may be ascertained by processing data generated by transmitter 190 and received by receiver 120. Intermediate data may be stored in memory 150 or other storage units. The aforementioned processes described herein may be performed by software 160 being executed by processor 140 manipulating the data.
Transmitter 190 may transmit signals, for example, acoustic waves, compression waves or other energy rays or waves, that may travel through subsurface (e.g., below land or sea level) structures. The transmitted signals may become incident signals that are incident to subsurface structures. The incident signals may reflect at various transition zones or geological discontinuities throughout the subsurface structures. The reflected signals may include seismic data.
Receiver 120 may accept reflected signal(s) that correspond or relate to incident signals, sent by transmitter 190. Transmitter 190 may transmit output signals. The output of the seismic signals by transmitter 190 may be controlled by a computing system, e.g., computing system 130 or another computing system separate from or internal to transmitter 190. An instruction or command in a computing system may cause transmitter 190 to transmit output signals. The instruction may include directions for signal properties of the transmitted output signals (e.g., such as wavelength and intensity). The instruction to control the output of the seismic signals may be programmed in an external device or program, for example, a computing system, or into transmitter 190 itself.
Computing system 130 may include, for example, any suitable processing system, computing system, computing device, processing device, computer, processor, or the like, and may be implemented using any suitable combination of hardware and/or software. Computing system 130 may include for example one or more processor(s) 140, memory 150 and software 160. Data 155 generated by reflected signals, received by receiver 120, may be transferred, for example, to computing system 130. The data may be stored in the receiver 120 as for example digital information and transferred to computing system 130 by uploading, copying or transmitting the digital information. Processor 140 may communicate with computing system 130 via wired or wireless command and execution signals.
Memory 150 may include cache memory, long term memory such as a hard drive, and/or external memory, for example, including random access memory (RAM), read only memory (ROM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), flash memory, volatile memory, nonvolatile memory, cache memory, buffer, short term memory unit, long term memory unit, or other suitable memory units or storage units. Memory 150 may store instructions (e.g., software 160) and data 155 to execute embodiments of the aforementioned methods, steps and functionality (e.g., in long term memory, such as a hard drive). Data 155 may include, for example, raw seismic data collected by receiver 120, instructions for building a mesh (e.g., 100), instructions for partitioning a mesh, and instructions for processing the collected data to generate a model, or other instructions or data. Memory 150 may also store instructions to divide and model τactive faults and τinactive faults. Memory 150 may generate and store the aforementioned constraints, restoration transformation (e.g., u_{τ} v_{τ} t_{τ}transform 201), restoration coordinates (e.g., u_{τ}, v_{τ}, t_{τ}), a geologicaltime and paleogeographic coordinates (e.g., u, v, t), a model representing a structure when it was originally deposited (e.g., in uvtspace), a model representing a structure at an intermediate restoration time (e.g., in u_{τ}, v_{τ}, t_{τ}space), and/or a model representing the corresponding present day structure in a current time period (e.g., in xyzspace). Memory 150 may store cells, nodes, voxels, etc., associated with the model and the model mesh. Memory 150 may also store forward and/or reverse u_{τ}, v_{τ}, t_{τ}transformations to restore present day models (e.g., in xyzspace) to restored models (e.g., in u_{τ}, v_{τ}, t_{τ}space), and vice versa. Memory 150 may also store the threedimensional restoration vector fields, which when applied to the nodes of the initial present day model, move the nodes of the initial model to generate one of the plurality of restored models. Applying a restoration vector field to corresponding nodes of the present day model may cause the nodes to “move”, “slide”, or “rotate”, thereby transforming modeled geological features represented by nodes and cells of the initial model. Data 155 may also include intermediate data generated by these processes and data to be visualized, such as data representing graphical models to be displayed to a user. Memory 150 may store intermediate data. System 130 may include cache memory which may include data duplicating original values stored elsewhere or computed earlier, where the original data may be relatively more expensive to fetch (e.g., due to longer access time) or to compute, compared to the cost of reading the cache memory. Cache memory may include pages, memory lines, or other suitable structures. Additional or other suitable memory may be used.
Computing system 130 may include a computing module having machineexecutable instructions. The instructions may include, for example, a data processing mechanism (including, for example, embodiments of methods described herein) and a modeling mechanism. These instructions may be used to cause processor 140 using associated software 160 modules programmed with the instructions to perform the operations described. Alternatively, the operations may be performed by specific hardware that may contain hardwired logic for performing the operations, or by any combination of programmed computer components and custom hardware components.
Embodiments of the invention may include an article such as a nontransitory computer or processor readable medium, or a computer or processor storage medium, such as for example a memory, a disk drive, or a USB flash memory, encoding, including or storing instructions, e.g., computerexecutable instructions, which when executed by a processor or controller, carry out methods disclosed herein.
Display 180 may display data from transmitter 190, receiver 120, or computing system 130 or any other suitable systems, devices, or programs, for example, an imaging program or a transmitter or receiver tracking device. Display 180 may include one or more inputs or outputs for displaying data from multiple data sources or to multiple displays. For example, display 180 may display visualizations of subsurface models including subsurface features, such as faults, horizons and unconformities, as a present day subsurface image (e.g., 202), a restored subsurface image (e.g., 203) and/or a depositional model (e.g., 703). Display 180 may display one or more present day model(s), depositional model(s), restoration model(s), as well as a series of chronologically sequential restoration models associated with a sequence of respective restoration times (e.g., τ_{1}<τ_{2}<τ_{3}<τ_{4}, as shown in
Input device(s) 165 may include a keyboard, pointing device (e.g., mouse, trackball, pen, touch screen), or cursor direction keys, for communicating information and command selections to processor 140. Input device 165 may communicate user direction information and command selections to the processor 140. For example, a user may use input device 165 to select one or more preferred models from among the plurality of perturbed models, recategorize faults as τactive faults and τinactive, or edit, add or delete subsurface structures.
Processor 140 may include, for example, one or more processors, controllers or central processing units (“CPUs”). Software 160 may be stored, for example, in memory 150. Software 160 may include any suitable software, for example, DSI software.
Processor 140 may generate a present day subsurface image (e.g., 202), a restored subsurface image (e.g., 203) and/or a depositional model (e.g., 703), for example, using data 155 from memory 150. In one embodiment, a model may simulate structural, spatial or geological properties of a subsurface region, such as, porosity or permeability through geological terrains.
Processor 140 may initially generate a three dimensional mesh, lattice, grid or collection of nodes (e.g., 100) that spans or covers a domain of interest. The domain may cover a portion or entirety of the threedimensional subsurface region being modeled. Processor 140 may automatically compute the domain to be modeled and the corresponding mesh based on the collected seismic data so that the mesh covers a portion or the entirety of the threedimensional subsurface region from which geological data is collected (e.g., the studied subsurface region). Alternatively or additionally, the domain or mesh may be selected or modified by a user, for example, entering coordinates or highlighting regions of a simulated optional domain or mesh. For example, the user may select a domain or mesh to model a region of the Earth that is greater than a userselected subsurface distance (e.g., 100 meters) below the Earth's surface, a domain that occurs relative to geological features (e.g., to one side of a known fault or riverbed), or a domain that occurs relative to modeled structures (e.g., between modeled horizons H(t_{1}) and H(t_{100})). Processor 140 may execute software 160 to partition the mesh or domain into a plurality of threedimensional (3D) cells, columns, or other modeled data (e.g., represented by voxels, pixels, data points, bits and bytes, computer code or functions stored in memory 150). The cells or voxels may have hexahedral, tetrahedral, or any other polygonal shapes, and preferably threedimensional shapes. Alternatively, data may include zerodimensional nodes, onedimensional segments, twodimensional facet and threedimensional elements of volume, staggered in a threedimensional space to form threedimensional data structures, such as cells, columns or voxels. The cells preferably conform to and approximate the orientation of faults and unconformities. Each cell may include faces, edges and/or vertices. Each cell or node may correspond to one or more particles of sediment in the Earth (e.g., a node may include many cubic meters of earth, and thus many particles).
Data collected by receiver 120 after the time of deposition in a current or present time period, include faults and unconformities that have developed since the original time of deposition, e.g., based on tectonic motion, erosion, or other environmental factors, may disrupt the regular structure of the geological domain. Accordingly, an irregular mesh may be used to model current geological structures, for example, so that at least some faces, edges, or surfaces of cells are oriented parallel to faults and unconformities, and are not intersected thereby. In one embodiment, a mesh may be generated based on data collected by receiver 120, alternatively, a generic mesh may be generated to span the domain and the data collected by receiver 120 may be used to modify the structure thereof. For example, the data collected may be used to generate a set of point values at “sampling point”. The values at these points may reorient the nodes or cells of the mesh to generate a model that spatially or otherwise represents the geological data collected from the Earth. Other or different structures, data points, or sequences of steps may be used to process collected geological data to generate a model. The various processes described herein (e.g., restoring a geological model using τactive and τinactive faults, or restoring a geological model using a new thicknesspreserving constraint) may be performed by manipulating such modeling data.
Restoration coordinates may be defined at a finite number of nodes or sampling points based on real data corresponding to a subsurface structure, e.g., one or more particles or a volume of particles of Earth. Restoration coordinates may be approximated between nodes to continuously represent the subsurface structure, or alternatively, depending on the resolution in which the data is modeled may represent discrete or periodic subsurface structures, e.g., particles or volumes of Earth that are spaced from each other.
The computing system of
“Restoration” or “intermediate” time τ may refer to a time in the past before the present day and after a time when an oldest or deepest horizon surface in the 3D model was deposited. “Restoration” or “intermediate” transformation or model may refer to a model or image of the surface as it was configured at the “intermediate” time in the past τ. An intermediate horizon may refer to a horizon that was deposited at the “intermediate” time τ, which is located above the deepest horizon and below the shallowest horizon.
“Time” including the presentday, current or present time, the past restoration time τ, and/or the depositional time τ, may refer to geological time periods that span a duration of time, such as, periods of thousands or millions of years.
“Geologicaltime” t(r) may refer to the time of deposition when a particle of sediment represented by point r was originally deposited in the Earth. In some embodiments, the geologicaltime of the deposition may be replaced, e.g., by any arbitrary monotonic increasing function of the actual geologicaltime. It is a convention to use an monotonically increasing function, but similarly an arbitrary monotonic decreasing function may be used. The monotonic function may be referred to as the “pseudogeologicaltime”.
The geologicaltime of the deposition and restoration time of particles are predicted approximate positions since past configurations can not typically be verified.
“Current” or “present day” location for a particle (or data structure representing one or more particles) or subsurface feature may refer to the location of the item in the present time, as it is measured.
In stratified terrain, layers, horizons, faults and unconformities may be curvilinear surfaces which may be for example characterized as follows.

 A horizon, Hτ, may be a surface corresponding to a plurality of particles of sediment which were deposited approximately at substantially the same geologicaltime, τ.
 A fault may be a surface of discontinuity of the horizons that may have been induced by a relative displacement of terrains on both sides of such surfaces. In other words, the geologicaltime of deposition of the sediments is discontinuous across each fault. Faults may cut horizons and may also cut other faults.
 An unconformity may be a surface of discontinuity of the horizons that may have been induced by for example an erosion of old terrains replaced by new ones. In other words, similarly to faults, the geologicaltime of deposition of the sediments is discontinuous across each unconformity.
Terrain deformed in the neighborhood of a point r in the Gspace may occur according to a “minimal deformation” tectonic style when, in this neighborhood:

 the strain tensor is approximately equal to the null tensor.
Terrain deformed in the neighborhood of a point r in the Gspace may occur according to a “flexural slip” tectonic style when, in this neighborhood:

 the length of any small increment of distance d(r) on the horizon passing through point r is preserved, e.g., in any direction, and,
 the volume of the terrains in the neighborhood of point r does not vary.
DiscreteSmoothInterpolation (DSI) is a method for interpolating or approximating values of a function f(x,y,z) at nodes of a 3D grid or mesh Γ (e.g., 100), while honoring a given set of constraints. The DSI method allows properties of structures to be modeled by embedding data associated therewith in a (e.g., 3D Euclidean) modeled space. The function f(x,y,z) may be defined by values at the nodes of the mesh, Γ. The DSI method allows the values of f(x,y,z) to be computed at the nodes of the mesh, Γ, so that a set of one or more (e.g., linear) constraints are satisfied. DSI generally only applies linear constraints on the model.
In some embodiments, bold symbols represent vectors or multidimensional (e.g., 3D) functions or data structures.
In some embodiments, the “simeq” symbol “≃” or “≅” may mean approximately equal to, e.g., within 110% of, or in a least squares sense. In some embodiments, the symbol “≡” may mean identical to, or defined to be equal to.
While embodiments of the invention describe the input depositional model as the GeoChron model, any other depositional model visualizing the predicted configuration of each particle, region or layer at its respective time of depositional may be used.
While embodiments of the invention describe the present day coordinates as xyz, the restoration coordinates as u_{τ}v_{τ}t_{τ}, the depositional coordinates as uvt, the restoration transformation as a u_{τ}v_{τ}t_{τ}transform, and the depositional transformation as a uvttransform, any other coordinates or transformations may be used.
In the foregoing description, various aspects of the present invention have been described. For purposes of explanation, specific configurations and details have been set forth in order to provide a thorough understanding of the present invention. However, it will also be apparent to one skilled in the art that the present invention may be practiced without the specific details presented herein. Furthermore, well known features may have been omitted or simplified in order not to obscure the present invention. Unless specifically stated otherwise, as apparent from the following discussions, it is appreciated that throughout the specification discussions utilizing terms such as “processing,” “computing,” “calculating,” “determining,” or the like, refer to the action and/or processes of a computer or computing system, or similar electronic computing device, that manipulates and/or transforms data represented as physical, such as electronic, quantities within the computing system's registers and/or memories into other data similarly represented as physical quantities within the computing system's memories, registers or other such information storage, transmission or display devices. In addition, the term “plurality” may be used throughout the specification to describe two or more components, devices, elements, parameters and the like.
Embodiments of the invention may manipulate data representations of realworld objects and entities such as underground geological features, including faults and other features. The data may be generated by tomographic scanning, as discussed in reference to
When used herein, a subsurface image or model may refer to a computerrepresentation or visualization of actual geological features such as horizons and faults that exist in the real world. Some features when represented in a computing device may be approximations or estimates of a real world feature, or a virtual or idealized feature, such as an idealized horizon as produced in a u_{τ} v_{τ} t_{τ}transform. A model, or a model representing subsurface features or the location of those features, is typically an estimate or a “model”, which may approximate or estimate the physical subsurface structure being modeled with more or less accuracy.
It will thus be seen that the objects set forth above, among those made apparent from the preceding description, are efficiently attained and, because certain changes may be made in carrying out the above method and in the construction(s) set forth without departing from the spirit and scope of the invention, it is intended that all matter contained in the above description and shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.
It is also to be understood that the following claims are intended to cover all of the generic and specific features of the invention herein described and all statements of the scope of the invention which, as a matter of language, might be said to fall therebetween.
Claims
1. A method of restoring a 3D model of the subsurface geology of the Earth from a present day measured geometry to a predicted past geometry at a restoration time in the past τ, the method comprising:
 receiving the 3D model of the present day geometry of the subsurface, including one or more folded geological horizon surfaces;
 selecting a value of a restoration time in the past τ before the present day and after a time an oldest horizon surface in the 3D model of the subsurface was deposited;
 restoring the 3D model from the present day measured geometry to the predicted past geometry at the restoration time in the past τ using a 3D transformation, wherein the vertical component of the 3D transformation restores the geometry to the vertical coordinate tτ such that: points along a horizon surface Hτ modeling sediment that was deposited at the selected restoration time in the past τ have a substantially constant value for the restored vertical coordinate tτ′; and
 at any location in the 3D model, the restored vertical coordinate tτ is equal to a sum of a first approximation t′τ of the vertical coordinate and an error correction term ϵτ, wherein the error correction term ϵτ is computed by solving a relationship in which a variation in the sum of the first approximation t′τ of the vertical coordinate and the error correction term ϵτ between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation; and
 displaying an image of the restored 3D model of the subsurface geology of the Earth such that each point in the 3D model is positioned at the restored vertical coordinate tτ as it was configured at the restoration time in the past τ.
2. The method of claim 1, wherein the direction and magnitude of maximum variation of ϵτ linearly depend on the direction and magnitude of maximum variation of t′τ.
3. The method of claim 1, wherein the first approximation t′τ of the vertical coordinate is computed by solving a relationship in which the spatial variation of the vertical coordinate t′τ is locally approximately proportional to the spatial variation of a geological time of deposition, wherein the coefficient of proportionality is locally equal to the inverse of the magnitude of the maximal spatial variation of the geological time of deposition.
4. The method of claim 1, wherein the first approximation t′τ of the vertical coordinate is computed by solving a relationship in which any infinitesimal displacement in the direction orthogonal to horizon surface Hτ results in a variation of the vertical coordinate t′τ approximately equal to the length of the infinitesimal displacement for points on the horizon surface Hτ.
5. The method of claim 1 comprising computing the restored horizontal coordinates uτ and vτ at the restoration time in the past τ in parts of the subsurface deposited at a time prior to restoration time τ, wherein the restored horizontal coordinates uτ and vτ represent the predicted horizontal positions where particles in the subsurface were located in the Earth at the restoration time in the past τ.
6. The method of claim 5, wherein the restored horizontal coordinates uτ and vτ are constrained such that for each point along the horizon surface Hτ that was deposited at the selected restoration time in the past τ: the restored horizontal coordinates uτ and vτ are equal to depositional horizontal coordinates u and v, respectively, and the spatial variations of the restored horizontal coordinates uτ and vτ are equal to the spatial variations of the depositional horizontal coordinates u and v, respectively.
7. The method of claim 5, wherein the restored horizontal coordinates uτ and vτ are constrained in parts of the subsurface which were deposited at a time prior to restoration time τ such that directions of maximal change of the restored horizontal coordinates uτ and vτ are linearly constrained by a local coaxis vector bτ and a local axis vector aτ; respectively, wherein the local axis vector aτ is oriented approximately in the direction of maximal variation of depositional horizontal coordinate u and orthogonal to the direction of maximal variation of the vertical restoration coordinate tτ, and wherein the local coaxis vector bτ is oriented orthogonal to the direction of the local axis vector aτ and orthogonal to the direction of maximal variation of the vertical restoration coordinate tτ.
8. The method of claim 7, wherein if the tectonic style of the 3D model is minimal deformation, the restored horizontal coordinates uτ and vτ are computed over the part of the 3D model of the subsurface deposited at a time prior to restoration time τ such that the directions of maximal change of uτ and vτ are approximately orthogonal to the local coaxis vector bτ and the local axis vector aτ; respectively.
9. The method of claim 7, wherein if the tectonic style of the 3D model is flexural slip, the restored horizontal coordinates uτ and vτ are computed over the part of the 3D model of the subsurface deposited at a time prior to restoration time τ such that projections of their directions of maximal change over the isovalue surfaces of the restored vertical coordinate tτ are approximately orthogonal to local coaxis vector bτ and the local axis vector aτ, respectively.
10. The method of claim 5, wherein the values of the restored horizontal coordinates uτ and vτ are constrained in parts of the subsurface which are older than the restoration time τ to be respectively equal on twin points on τactive faults, wherein twin points are points on opposite sides of a τactive fault that were collocated at the restoration time τ and are located on the same fault stria in the present day model, to merge the twin points into the same position in the restored model by sliding the twin points towards each other in a direction tangential to the surface of the τactive fault.
11. The method of claim 5, wherein the values of the restored horizontal coordinates uτ and vτ are constrained in parts of the subsurface which are older than the restoration time τ to be respectively equal on mate points on τinactive faults, wherein mate points are points on opposite sides of a τinactive fault that are collocated at present day time, to move mate points together on opposite sides of τinactive faults.
12. The method of claim 5 comprising:
 editing depositional values u, v, and t associated with the restored 3D model; and
 reverse transforming the restored 3D model forward in time from the predicted past geometry at the restoration time in the past τ to the present day measured geometry using an inverse of the 3D restoration transformation to incorporate the edits from the restored model into the present day model.
13. The method of claim 1 comprising: receiving an increasing chronological sequence of past restoration times τ1, τ2,..., τn; and for each restoration time τi in sequence τ1, τ2,..., τn, computing the corresponding 3D restoration transformation Rτi which restores the part of the subsurface deposited at a time prior to horizon Hτi to its predicted past geometry at time τi, wherein each 3D restoration transformation Rτi restores the part of the subsurface deposited at a time prior to horizon Hτi to 3D restored coordinates uτi, vτi, and tτi.
14. The method of claim 13 comprising playing a moving image sequence in which the 3D model is iteratively restored in a forward or reverse order of the sequence of past restoration times τ1, τ2,..., τn to visualize changes in the subsurface geology over the passage of time.
15. A system for restoring a 3D model of the subsurface geology of the Earth from a present day measured geometry to a predicted past geometry at a restoration time in the past τ, the system comprising:
 one or more processors configured to: receive the 3D model of the present day geometry of the subsurface, including one or more folded geological horizon surfaces, select a value of a restoration time in the past τ before the present day and after a time an oldest horizon surface in the 3D model of the subsurface was deposited, restore the 3D model from the present day measured geometry to the predicted past geometry at the restoration time in the past τ using a 3D transformation, wherein the vertical component of the 3D transformation restores the geometry to the vertical coordinate tτ such that: points along a horizon surface Hτ modeling sediment that was deposited at the selected restoration time in the past τ have a substantially constant value for the restored vertical coordinate tτ, and at any location in the 3D model, the restored vertical coordinate tτ is equal to a sum of a first approximation t′τ of the vertical coordinate and an error correction term ϵτ, wherein the error correction term ϵτ is computed by solving a relationship in which a variation in the sum of the first approximation t′τ of the vertical coordinate and the error correction term ϵτ between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation, and display an image of the restored 3D model of the subsurface geology of the Earth such that each point in the 3D model is positioned at the restored vertical coordinate tτ as it was configured at the restoration time in the past τ.
16. The system of claim 15, wherein the direction and magnitude of maximum variation of ϵτ linearly depend on the direction and magnitude of maximum variation of t′τ.
17. The system of claim 15, wherein the one or more processors are configured to compute the first approximation t′τ of the vertical coordinate by solving a relationship in which the spatial variation of the vertical coordinate t′τ is locally approximately proportional to the spatial variation of a geological time of deposition, wherein the one or more processors are configured to compute the coefficient of proportionality to be locally equal to the inverse of the magnitude of the maximal spatial variation of the geological time of deposition.
18. The system of claim 15, wherein the one or more processors are configured to compute the first approximation t′τ of the vertical coordinate by solving a relationship in which any infinitesimal displacement in the direction orthogonal to horizon surface Hτ results in a variation of the vertical coordinate t′τ approximately equal to the length of the infinitesimal displacement for points on the horizon surface Hτ.
19. The system of claim 15, wherein the one or more processors are configured to compute the restored horizontal coordinates uτ and vτ at the restoration time in the past τ in parts of the subsurface deposited at a time prior to restoration time τ, wherein the restored horizontal coordinates uτ and vτ represent the predicted horizontal positions where particles in the subsurface were located in the Earth at the restoration time in the past τ.
20. The system of claim 19, wherein the one or more processors are configured to constrain the restored horizontal coordinates uτ and vτ such that for each point along the horizon surface Hτ that was deposited at the selected restoration time in the past τ: the restored horizontal coordinates uτ and vτ are equal to depositional horizontal coordinates u and v, respectively, and the spatial variations of the restored horizontal coordinates uτ and vτ are equal to the spatial variations of the depositional horizontal coordinates u and v, respectively.
21. The system of claim 19, wherein the one or more processors are configured to constrain the restored horizontal coordinates uτ and vτ in parts of the subsurface which were deposited at a time prior to restoration time τ such that directions of maximal change of the restored horizontal coordinates uτ and vτ are linearly constrained by a local coaxis vector bτ and a local axis vector aτ, respectively, wherein the one or more processors are configured to generate the local axis vector aτ to be oriented approximately in the direction of maximal variation of depositional horizontal coordinate u and orthogonal to the direction of maximal variation of the vertical restoration coordinate tτ, and to generate the local coaxis vector bτ to be oriented orthogonal to the direction of the local axis vector aτ and orthogonal to the direction of maximal variation of the vertical restoration coordinate tτ.
22. The system of claim 21, wherein if the tectonic style of the 3D model is minimal deformation, the one or more processors are configured to compute the restored horizontal coordinates uτ and vτ over the part of the 3D model of the subsurface deposited at a time prior to restoration time τ such that the directions of maximal change of uτ and vτ are approximately orthogonal to the local coaxis vector bτ and the local axis vector aτ, respectively.
23. The system of claim 21, wherein if the tectonic style of the 3D model is flexural slip, the one or more processors are configured to compute the restored horizontal coordinates uτ and vτ over the part of the 3D model of the subsurface deposited at a time prior to restoration time τ such that projections of their directions of maximal change over the isovalue surfaces of the restored vertical coordinate tτ are approximately orthogonal to local coaxis vector bτ and the local axis vector aτ, respectively.
24. The system of claim 19, wherein the one or more processors are configured to constrain the values of the restored horizontal coordinates uτ and vτ in parts of the subsurface which are older than the restoration time τ to be respectively equal on twin points on τactive faults, wherein twin points are points on opposite sides of a τactive fault that were collocated at the restoration time τ and are located on the same fault stria in the present day model, and to merge the twin points into the same position in the restored model by sliding the twin points towards each other in a direction tangential to the surface of the τactive fault.
25. The system of claim 19, wherein the one or more processors are configured to constrain the values of the restored horizontal coordinates uτ and vτ in parts of the subsurface which are older than the restoration time τ to be respectively equal on mate points on τinactive faults, wherein mate points are points on opposite sides of a τinactive fault that are collocated at present day time, to move mate points together on opposite sides of τinactive faults.
26. The system of claim 15, wherein the one or more processors are configured to: receive an increasing chronological sequence of past restoration times τ1, τ2,..., τn, and for each restoration time τi in sequence τ1, τ2,..., τn, compute the corresponding 3D restoration transformation Rτi which restores the part of the subsurface deposited at a time prior to horizon Hτi to its predicted past geometry at time τi, wherein the one or more processors are configured to generate each 3D restoration transformation Rτi to restore the part of the subsurface deposited at a time prior to horizon Hτi to 3D restored coordinates uτi, vτi, and tτi.
27. The system of claim 26, wherein the one or more processors are configured to play a moving image sequence in which the 3D model is iteratively restored in a forward or reverse order of the sequence of past restoration times τ1, τ2,..., τn to visualize changes in the subsurface geology over the passage of time.
Type: Application
Filed: Nov 12, 2019
Publication Date: Jul 16, 2020
Applicants: Emerson Paradigm Holding LLC (Houston, TX), (LuxembourgBeggen)
Inventors: JeanLaurent MALLET (LuxembourgBeggen), AnneLaure TERTOIS (Saint Cyr la Riviere)
Application Number: 16/681,061