The approach to construction and discretization of complex 3D models of geological media in software for airborne geophysical surveys maintenance

The paper presents a new approach to construction of complex geological models and generation of finite element meshes in order to perform 3D modeling and multidimensional processing of airborne geophysical data. The main principles of geological models construction are considered. They are based on defining geological models in terms of the subject field and take into account their main specifics: complex background media, curvilinear Earth surface relief and borders between the geological layers, complex shape of target objects, etc. We consider the necessary data structures and specifics of finite element mesh generation. The example of generation of the finite element mesh for a geological model typical of airborne geophysical surveys is shown.


Introduction
Nowadays, the need for applying 3D approaches to modeling and data processing in airborne geophysical surveys is undeniable [1][2][3]. In this regard, on the one hand, the requirements for accuracy of representing real geological media grow steadily, and, on the other hand, the requirements for automation of their discretization and modeling accuracy increase.
Together with that, some software applications, which implement 3D modeling and are characterized by sufficiently high degree automation, make it possible to describe a medium only by the objects of simple rectangular shape and use the parallelepiped meshes for discretization. Other software applications allow the description of a medium with complex borders [4][5][6], but they do not take into account the specifics of the problems, and, correspondingly, do not provide sufficient instruments of automation both for model construction and for optimization of computational cost to obtain the numerical solutions of the required accuracy.
In this paper, we consider the approach, which, on the one hand, allows convenient construction of "continuous" geological media with curvilinear borders and local 3D objects of complex shape, and, on the other hand, makes it possible to automatically generate the optimized non-conforming meshes with hexahedral cells, which ensure both quite an accurate representation of complex borders and the required accuracy of numerical solution approximation and low computational cost. • computational modules including modules of solution of forward and inverse airborne electromagnetic and magnetic problems; • graphical user interface (GUI); • control program for parallel solutions; • information system for keeping the observed data, geological media recovered, and corresponding numerical data.
In this paper, we consider the principles of construction and discretization of geological models, which are implemented in GUI and corresponding computational modules.

Geological model construction
In the software developed, the geological model is represented by the basis layers the borders between which (including the Earth surface relief) are defined by "heights" Z m (xp,yp) in the sets of points {xp,yp}, p=1,…,Pm. Using these sets, the bicubic Hermite splines Zm(x,y) are constructed. They pass through or as close as possible to these points and ensure the smoothing of corresponding surfaces.
Besides, for each border between the geological model layers, the so-called master coordinate is preset. These coordinates are determined by some middle depth of the corresponding border.
At the next stage of the geological model construction, 3D inhomogeneities are formed within each of the layers.
The 3D objects of parallelepiped shape are represented by six coordinates (minimal and maximal values in each of the coordinate axes). We note that the values of the coordinate in the Z-axis are preset in master coordinates, i.e., in fact, the object location along the vertical axis is determined relative to the top and bottom surfaces of the corresponding layer and its top and bottom borders are curved in accordance with the curvilinear borders of the layer within which this object is contained.
The 3D objects of complex shape in plan are represented with the use of polygons for which we preset the value d defining the accuracy of their borders representation. The positions in the Z-axis of these complex objects are defined in the same manner as for the objects of parallelepiped shape.
The dipped geological structures with possible change of their dimensions at different levels in the Z are defined with the use of the hexahedrons-stencil-plates the top and bottom bases of which are rectangulars with possible different dimensions and locations in plan. These hexahedrons, in fact, define the deformation of 3D objects (structures) located within them.
We note that for 3D objects, the priority can be defined, which enables us to place some objects "over" others (i.e. at the places of their intersections, the properties of the medium are defined according to the object with a higher priority). This is convenient when there is a need for constructing a "continuous" but laterally inhomogeneous medium with curvilinear borders in plan, which, in its turn, includes 3D target objects of complex shape. In this case, the priority of such target 3D objects is set higher than the priority of the objects which represent a laterally inhomogeneous background medium.
For all structural parts considered above, the electrophysical characteristics are set: the values of conductivity (it may be anisotropic) and parameters of induced polarization (chargeability, relaxation time, and the coefficient describing the form of drop).

Finite element meshes generation
The accuracy of the receiver signals obtained as a result of finite element calculation of the 3D electromagnetic field significantly depends on the mesh refinement in small vicinity around the receivers and abnormal 3D inhomogeneties placed quite close to them. Therefore, in order to reduce the computational cost required to obtain the signal with the acceptable accuracy, the non-conforming meshes with local refinements are used [7]. These meshes are generated taking into account the following principles: • The regular nonuniform mesh with quite a small equal step in the small subdomain defined by transmitter-receiver position (or positions joined into a group) is used as the base mesh. Further, this subdomain is named the uniform regularity subdomain.
• The additional local refinements in the vicinities of transmitters and receivers are performed in order to increase the accuracy of the calculated signals.
• The "superfluous" nodes located outside the uniform regularity subdomain are deleted, and the corresponding "elongated" cells are joined, which allows decreasing in the number of nodes in the obtained irregular non-conforming mesh approximately by an order.
• 3D objects are embedded with the local refinements of near-border cells of the mesh (in order to take into account the borders of these objects).
In order to implement these principles, before finite element mesh generation, the following specific data structures are formed on the basis of the geological model data considered above.
• The first structure is the structure describing the computational domain with the use of control points, which are put in a regular macro mesh. The control points are put on the plates, which will be transformed into curvilinear borders between the layers, and their coordinates in the Z are defined using master coordinates of the corresponding borders. In plan (at each border between the layers), the control points are put in the following way. Firstly, the control points are put on the remote lateral boundaries of the computational domain, which are defined depending on the time range in which the modeling is carried out. Secondly, the control points are put on the boundaries of the uniform regularity subdomain. And finally, the control points are put at the place of vertexes of the hexahedrons-stencil-plates defining the deformation of geological structures.
• The second structure is the structure describing 3D objects locations relative to the control points positions. In order to form this structure, the 3D objects of complex shape are preliminarily approximated in plan (in lateral) using rectangulars taking into account the values d, which defines the required accuracy of representation of such objects borders. The approximation is performed in such a way that, on the one hand, the required accuracy would be ensured, and, on the other hand, the number of rectangulars obtained (which are further interpreted as independent 3D objects with the same properties) would be minimized.
• The third structure is the structure containing the information on splines, which represent the curvilinear borders between the geological layers.
On the basis of these structures, the finite element mesh is generated in the following way.
• At the first stage, the base mesh is generated. This mesh takes into account control points positions, the step-size in the uniform regularity subdomain, and the factor of increase of the cell size when moving to the remote boundaries of a computational domain. During the mesh generation process, the "superfluous" nodes are deleted from the mesh, and "elongated" cells are joined into more isometrical ones.
• At the second stage, local refinements are performed in the vicinities of transmitter and receiver positions.
• At the third stage, 3D objects are embedded in accordance with their priorities (the objects with higher priorities are embedded over the objects with lower priorities).
• At the fourth stage, the finite element mesh is deformed in accordance with the real coordinates of the control points and splines representing the curvilinear borders between the layers.

Examples
We consider a three-layered geoelectrical model the layers of which contain the inhomogeneities of different types. The master coordinates of the borders between the layers are taken equal to -40 m and -240 m. In the top layer of the geoelectrical model, two non-overlapping objects were constructed IOP Publishing doi:10.1088/1757-899X/1019/1/012075 4 using the polygons. These objects are shown in Figure 1a. Figure 1b shows 3D inhomogeneities in the form of polygons in the second layer. The target objects are shown by red color, inhomogeneities simulating geological noise-objects are shown by other colors. The left target object has higher priority, it therefore "slots" noise-objects. The noise-objects also have different priorities (because they overlap one another). So, the objects whose contours are shown by green color have the highest priority, and the objects the contours of which are shown by dark blue color have the lowest priority. Besides, the right target object narrows downward, and this narrowing is defined by the corresponding hexahedron-stencil-plate.
For the Earth surface and surfaces between the layers, the sets of points {xp,yp}, p=1,…,Pm representing the shape of these surfaces were set.
The measurements system is shown by light points, and the points of the current group (for which the calculations are carried out) around which the uniform regularity subdomain is constructed are shown by black color.  As we noted above, before the finite element mesh generation, the objects-polygons are approximated in the plan taking into account the value d, which defines the accuracy of the borders representations. The result of this approximation is shown in Figure 2 for d=20 m (Figure 2a) and for d=10 m (Figure 2b).
As previously mentioned, the mesh generation consists of several stages one of which (the third) includes the embedding of 3D objects into the mesh. In order to show the base mesh with local refinements and deformations, we exclude this stage. The finite element mesh without the embedded 3D objects is shown in Figure 3.  In Figures 4a and 4b, the 3D inhomogeneities in the top and second layers are visible, and in Figure 4c, we can see how the sizes of the right target object are reduced in the plan.  Figure 5 shows one of the slices of the finite element mesh, which was generated for the case when the value d was taken equal to 10 m for approximating the polygons. IOP Publishing doi:10.1088/1757-899X/1019/1/012075 6 We can see that the mesh refinements are only in the vicinities of 3D objects borders. This increases the accuracy of borders representation, but the number of nodes in this mesh is bigger by half. Figure 5. The finite element mesh slice inside the second layer (approximation of the objects of complex shape is performed for d=10 m).

Conclusion
The approach developed enables us to construct complex geological models and generate the corresponding finite element meshes. These meshes, on the one hand, include all borders of the structural parts of a geological model and make it possible to obtain the finite element solution with the required accuracy, and, on the other hand, they are optimized and do not contain the "superfluous" nodes, which finally provides the computational effectiveness of 3D modeling and the inversions based on it.