Detecting subsurface interfaces with a physics-based level-set segmentation and additional geological constraints

Sharp interfaces often separate regions in the subsurface with distinctively different properties due to processes in geological evolution – and these   interfaces are relevant for a variety of scientific investigations, as well as practical applications.  The delineation of these layers with different properties is commonly attempted on the basis of geological and geophysical data, for example as picks in  prevalent seismic reflectors, interpreted from potential field measurements, and derived from observations in drillholes. 

We evaluate here a specific method to determine the position and shape of such an interface using measurements of state variables related to a physical flow field described with an elliptic PDE. A typical example is the measurement of temperatures related to heat flow through zones with distinctively different thermal conductivities. We use a level-set function to describe the interface and determine the optimal interface shape for a 2-D case. This type of shape inversion has been successfully attempted before, and we extend on this previous work by including additional shape constraints on orientation, interface, and observations of specific segmentation outcomes. These constrains are motivated by geological information that may be available, for example as derived and interpreted from additional geophysical measurements. 

We model this as an image segmentation problem, where we are looking for a segmentation of the image domain whose induced temperature minimizes the squared L² distance to temperature measurements on a lower dimensional set. From an optimal control perspective, the segmentation is the control and the temperature the state.  Numerically, the segmentation is represented by a level set and the minimization is done using a gradient flow, where the derivative with respect to the level set is computed using dualization. Moreover, we include additional geologically motivated constraints by adding soft penalties to the objective function.

We test our method with several conceptual examples to determine the feasibility and limitations, especially with regard to different interface shapes and the amount of available information and additional geological constraints, as well as the influence of noise on the detection  accuracy. Results show that these additional constraints help determining an interface. However, measurement noise and a non-homogeneous spatial distribution of physical properties reduces the accuracy of the derived interface.