Subsurface datasets inherently possess big data characteristics such as vast
volume, diverse features, and high sampling speeds, further compounded by the
curse of dimensionality from various physical, engineering, and geological
inputs. Among the existing dimensionality reduction (DR) methods, nonlinear
dimensionality reduction (NDR) methods, especially Metric-multidimensional
scaling (MDS), are preferred for subsurface datasets due to their inherent
complexity. While MDS retains intrinsic data structure and quantifies
uncertainty, its limitations include unstabilized unique solutions invariant to
Euclidean transformations and an absence of out-of-sample points (OOSP)
extension. To enhance subsurface inferential and machine learning workflows,
datasets must be transformed into stable, reduced-dimension representations
that accommodate OOSP.
Our solution employs rigid transformations for a stabilized Euclidean
invariant representation for LDS. By computing an MDS input dissimilarity
matrix, and applying rigid transformations on multiple realizations, we ensure
transformation invariance and integrate OOSP. This process leverages a convex
hull algorithm and incorporates loss function and normalized stress for
distortion quantification. We validate our approach with synthetic data,
varying distance metrics, and real-world wells from the Duvernay Formation.
Results confirm our method's efficacy in achieving consistent LDS
representations. Furthermore, our proposed "stress ratio" (SR) metric provides
insight into uncertainty, beneficial for model adjustments and inferential
analysis. Consequently, our workflow promises enhanced repeatability and
comparability in NDR for subsurface energy resource engineering and associated
big data workflows.

我们提出了一种稳定的欧几里德不变描绘方法，并通过计算输入的 MDS 差异矩阵，应用刚性变换和多个实现，确保变换不变并整合 OOSP，验证了我们的方法在达到一致的 LDS 表示方面的有效性。