Defining the search space of possible underground geological structures is a crucial aspect of the inversion process. Obsidian uses a layer-cake model, where the boundaries between each layer are modelled using a grid of offsets and control points.

The offset matrix is a high resolution 2D grid which represents the mean depths of the points in the grid. The control point grid is similar to the offset matrix, except the depths of the points are part of the model parameters. These points are varied by the inference engine to fit the sensor data, whereas the points in the offset matrix are fixed. The control point grids are interpolated to produce a smooth surface, and is added to the offset grid to produce the final layer boundary surface.

Both the offsets and control controls grid are defined by a X and Y resolution, and the grid points are evenly spaced along the bounds of each dimension. The resolution of the offset grid does not significantly impact the performance, but increasing the resolution of the control points grid increases the number of dimensions in the inversion problem.

Creating a World Specification

The WorldSpec class defines the basic structure of the world that do not vary during an inversion. It contains the x, y, and z bounds of the world, as well as a list of layer boundaries (each boundary is of type BoundarySpec. For instance, the following code creates a boundary with a 50x30 offset grid (a flat surface at 1500 depth) and a 5x3 control point grid:

worldSpec.boundaries.push_back({Eigen::MatrixXd::Ones(50, 30) * 1500, std::make_pair(5, 3), BoundaryClass::Normal});

An InterpolatorSpec wraps the information contained in a BoundarySpec and is used to perform interpolation for the layer boundaries.

Queries, transitions and height maps

Queries represent a 2D grid that is used to extract the depths of the layer boundaries. They can be thought of as a set of vertical lines parallel to the z (depth) axis. The points at which these vertical lines intersect with the layer boundaries are the layer transitions.

Queries are implemented in the Query class. The getTransitions() function takes a query, and returns, for each layer, the depth of each intersection point in the 2D grid defined by the query. Using this function, we can obtain a depth map of each layer boundary.

Voxelisation

Most of the forward models implemented by Obsidian cannot directly operate on these boundary height maps. The world model is first voxelised by approximating the boundaries and layers with a 3D mesh grid of voxels (3D pixels). Forward models can then treat each voxel as a rectangle prism with certain rock properties. Obviously, the results would not be identical to the real world model, but it serves as a good approximation when the resolution of the voxelisation is high. In Obsidian, voxelisation can be done using either the getVoxels() function or the voxelise() function. The former is a higher level wrapper around the latter to specifically extract a particular rock property.