Towards Three Component Seismograms From One Component DAS Records; Finite Frames in Geophysics

TL;DR

The paper addresses recovering full 3C seismic displacement vectors from one-component (1C) DAS measurements by employing finite-frame theory. It formulates the problem in a 3D Hilbert space using an overcomplete, non-orthogonal set of axis-vectors $\{\varphi_i\}$ and reconstructs $\mathbf{x}$ via the frame operator $S$, with $\mathbf{x} = \sum_i \langle \mathbf{x}, \varphi_i\rangle S^{-1}\varphi_i$, augmented by a boxcar window to produce local 3C estimates along the fiber. The Utah FORGE borehole example demonstrates successful recovery of horizontal components with high precision, validating the approach's practicality for creating virtual 3C DAS seismometers from 1C observations. The method broadens DAS applicability to arbitrary fiber deployments and re-surveys, enabling richer 3D seismic insights where only single-component data were previously available.

Abstract

Some geophysical observations commonly collect only one component (1C) of a three component (3C) vector field. For example, Distributed Acoustic Sensing (DAS) records seismograms derived from displacement differences along the axis of segments of a fiber optic cable. In practice, multiple observations from such 3C vector fields are available, but commonly along non-orthogonal directions -- i.e.\ desirable sets of observations along orthogonal basis vectors are not available. For DAS, the theory of (finite) frames allows the recovery of 3C vector observations as long as the set of measurements occur on axis-vectors that mathematically span 3D space. A reconstruction algorithm from finite frame theory is described and then applied to geometry data from a borehole at the Utah FORGE geothermal project. The results demonstrate recovery of high precision 3C vectors along the fiber optic cable from 1C input vector-projections.

Figures (3)

• Figure 1: Cartoon geometry of the problem at hand. Refer to the main text for the symbol definitions.
• Figure 2: A) A plot of the well survey at the correct aspect ratio. B) A horizontally exaggerated (vertically compressed) plot of the same data. Note the small deviations in the survey. The well is essentially vertical. Hence, the ability to recover horizontal vector components from measurements along its segments is much more problematic than recovering vertical components. The horizontal gray dotted arrow in A) shows the orientation of the test vector $(1\, 1\, 0)^T$ used in section \ref{['Example']}. That test vector and orientation is used throughout the reconstruction shown in Table \ref{['VectorRecovery']}.
• Figure 3: Angles (in degrees) between individual segments of the well survey shown in Figure \ref{['NoVertExag']}. The angles are calculated from $\cos^{-1}( <\varphi_{i+1},\varphi_i>)$ so orientation information in 3D space is not preserved. The standard histogram is plotted in dark blue against the left ordinate. Its CDF is plotted in a translucent tan against the right ordinate. A subdued red-tinted gray is where the two different kinds of histogram bins overlap. Note that 50% of the well survey segments have inter-segment angles of $\le 0.6^\circ$. The 3D nature of those angles is what is being exploited by the technique described in this paper.