Rigid transformations for stabilized lower dimensional space to support subsurface uncertainty quantification and interpretation

Abstract

Subsurface datasets commonly are big data, i.e., they meet big data criteria, such as large data volume, significant feature variety, high sampling velocity, and limited data veracity. Large data volume is enhanced by the large number of necessary features derived from the imposition of various features derived from physical, engineering, and geological inputs, constraints that may invoke the curse of dimensionality. Existing dimensionality reduction (DR) methods are either linear or nonlinear; however, for subsurface datasets, nonlinear dimensionality reduction (NDR) methods are most applicable due to data complexity. Metric-multidimensional scaling (MDS) is a suitable NDR method that retains the data's intrinsic structure and could quantify uncertainty space. However, like other NDR methods, MDS is limited by its inability to achieve a stabilized unique solution of the low dimensional space (LDS) invariant to Euclidean transformations and has no extension for inclusions of out-of-sample points (OOSP). To support subsurface inferential workflows, it is imperative to transform these datasets into meaningful, stable representations of reduced dimensionality that permit OOSP without model recalculation.

We propose using rigid transformations to obtain a unique solution of stabilized Euclidean invariant representation for LDS. First, compute a dissimilarity matrix as the MDS input using a distance metric to obtain the LDS for $N$-samples and repeat for multiple realizations. Then, select the base case and perform a rigid transformation on further realizations to obtain rotation and translation matrices that enforce Euclidean transformation invariance under ensemble expectation. The expected stabilized solution identifies anchor positions using a convex hull algorithm compared to the $N+1$ case from prior matrices to obtain a stabilized representation consisting of the OOSP. Next, the loss function and normalized stress are computed via distances between samples in the high-dimensional space and LDS to quantify and visualize distortion in a 2-D registration problem. To test our proposed workflow, a different sample size experiment is conducted for Euclidean and Manhattan distance metrics as the MDS dissimilarity matrix inputs for a synthetic dataset.

The workflow is also demonstrated using wells from the Duvernay Formation and OOSP with different petrophysical properties typically found in unconventional reservoirs to track and understand its behavior in LDS. The results show that our method is effective for NDR methods to obtain unique, repeatable, stable representations of LDS invariant to Euclidean transformations. In addition, we propose a distortion-based metric, stress ratio (SR), that quantifies and visualizes the uncertainty space for samples in subsurface datasets, which is helpful for model updating and inferential analysis for OOSP. Therefore, we recommend the workflow's integration as an invariant transformation mitigation unit in LDS for unique solutions to ensure repeatability and rational comparison in NDR methods for subsurface energy resource engineering big data inferential workflows, e.g., analog data selection and sensitivity analysis.

Appendices

Appendix A

Based on Definition 5.8.5 in Lal and Pati [62], which states a map $T: {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ is said to be a rigid motion if $\Vert T\left(\mathbf{x}\right)- T\left(\mathbf{y}\right) \Vert = \Vert \mathbf{x}-\mathbf{y}\Vert , \forall \mathbf{x}, \mathbf{y}\in {\mathbb{R}}^{n}$, where $T$ is the rigid transformation operator such that when applied on set points, $\mathbf{x}, \mathbf{y}$ each with ${\text{dim}}N\times 3$, leads to an invariant matrix. Thus, if a rigid transformation operator is performed on an MDS realization that preserves pairwise dissimilarity in the lower dimensional space, the same properties from the prior definition are implied and an invariant dissimilarity matrix is obtained. The mathematical proof by rigor for rigid transformations is in Sorkine-Hornung et al. [63]. Hence, we propose Theorem A.1.

Theorem A.1: Let ${\mathbf{S}}_{E}$ be the ensemble expectation of rigid transformed solutions for multiple realizations in the workflow known as stabilized solutions, i.e.,${\mathbf{S}}_{E}={\mathbb{E}}\left[ T\left( {\mathbf{Z}}_{{\text{k}}}\right)\right]$, in a lower-dimensional space. If an out-of-sample-point (OOSP) that falls within a 95% confidence interval for each predictor feature of interest, $P \forall m=1,\dots , M$, and $M=\left\{ m \right|m\in {\mathbb{N}}\}$ is added, then the stabilized solution obtained by applying rigid transformation on the representation obtained from the low dimensional space, ${\mathbf{Z}}_{OOS{\text{P}}}$, is the same as the ensemble expectation of the stabilized solution for the $N$-sample case, which is Euclidean transformation invariant i.e., ${\mathbf{S}}_{E}\approx {\mathbf{S}}_{OOSP}$.

Proof: Suppose adding an OOSP within the 95% confidence interval causes the stabilized solution obtained by rigid transformation on ${{\varvec{Z}}}_{OOSP}$ to differ from ${\mathbf{S}}_{E}$. Let $T$ be the rigid transformation such that $T: {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$. This implies there is no applicable rigid transformation that ensures $\Vert T\left(\mathbf{x}\right)- T\left(\mathbf{y}\right) \Vert = \Vert \mathbf{x}-\mathbf{y}\Vert , \forall \mathbf{x}, \mathbf{y}\in {\mathbb{R}}^{n}$ when ${\text{dim}}\left\{\mathbf{x}\right\}\ne {\text{dim}}\left\{\mathbf{y}\right\}$. This contradicts the definition of a rigid transformation as stated in Lal and Pati [62] and Sorkine-Hornung et al. [63].

Therefore, assuming the shape and size of the sets of anchor points from the $N$-sample and $N+1$ sample cases are equal i.e., ${\text{dim}}\left\{{{\varvec{A}}}_{n}\right\}={\text{dim}}\{{{\varvec{A}}}_{OOSP}\}=n$, because the OOSP is within a 95% confidence interval for each predictor feature of interest. Then, a map $T: {\mathbb{R}}^{{\text{n}}}\to {\mathbb{R}}^{{\text{n}}}$ that enforces rigid motion in the low dimensional space is possible if $\Vert T\left({{\varvec{A}}}_{n}\right)- T\left({{\varvec{A}}}_{OOSP}\right) \Vert = \Vert {{\varvec{A}}}_{n}-{{\varvec{A}}}_{OOSP}\Vert , \forall {{\varvec{A}}}_{n}, {{\varvec{A}}}_{OOSP}\in {\mathbb{R}}^{n}$, provided that the OOSP does not reside in the tail distribution of each $P$. This constraint is imposed to restrict the dissimilarity matrix, ${\varvec{D}}$, from changing significantly. Therefore, Theorem A.1 holds.

Table 1 Criteria for evaluating normalized stress and its corresponding interpretation [57]

Credit author statement

Ademide O. Mabadeje: Data curation, Conceptualization, Methodology, Software, Validation, Visualization, Formal analysis, Writing – Original draft.

Michael Pyrcz: Data curation, Conceptualization, Supervision, Funding acquisition, Writing –Reviewing and Editing.