Digital Easter Egg Hunting: GeoPN-ProbabilityLab–>Covariance Geometry

May 3

3 min read

What once required considerable effort and imperfect data during my PhD work (1997-2001) —estimating acoustic attenuation in shallow marine sediments off the California coast—has now found a surprisingly elegant continuation.

This time, the setting is different: not a research vessel, but a classroom. Within a tutoring program for 10th–12th grade students, we revisited the same underlying concepts—now using high-quality NMR datasets. Over several weeks, we reconstructed and significantly improved the original numerical approach.

Back then (late 1990s to early 2000s), limited data quality—especially the lack of reliable porosity measurements—left many interpretations uncertain. Today, high-resolution NMR data provide an ideal foundation to revisit and refine these methods with clarity and confidence.

🧭 From First Steps to Probability Structure

What began as an introduction to discrete probability (Bernoulli chains) quickly evolved into continuous probability density functions (PDFs).

Instead of merely plotting data, the goal became clear:

👉 Quantify structure, variability, and meaning within the data

The chosen parameters:

Permeability (x)
Porosity (y)

📊 From Scatter Plot to Covariance Geometry

A simple scatter plot (permeability vs. porosity) offers only a first qualitative impression. The real insight emerges through covariance analysis:

σ-ellipses describe the distribution of the data
Eigenvectors define the principal directions
Eigenvalues quantify variance along these directions
Rotation follows directly from the covariance matrix

The 1σ ellipse contains ~68% of the data and represents the core structure of the distribution.

👉 A cloud of points becomes a geometrically interpretable system

📐 Why the Ellipses Matter

These ellipses are not decoration—they are interpretation:

They reveal correlation between parameters
They expose spread and orientation
They transform visualization into quantitative structure

A circular distribution would imply no correlation.The observed ellipsoidal shape clearly indicates a structured relationship.

"Covariance structure of permeability vs. porosity" as integrated visualization combining scatter plot with σ-ellipses (1σ–3σ) and principal axes, density contours, and a 3D probability density surface (“Gaussian hill”). Ellipses and contours represent probability levels and correlation structure, while the 3D surface expresses probability density as elevation. Data are peak-normalized (max = 1) for consistency across representations. — **IMAGE-PANEL** ***"Covariance structure of permeability vs. porosity"*** as integrated visualization combining scatter plot with σ-ellipses (1σ–3σ) and principal axes, density contours, and a 3D probability density surface (“Gaussian hill”). Ellipses and contours represent probability levels and correlation structure, while the 3D surface expresses probability density as elevation. Data are peak-normalized (max = 1) for consistency across the various representations according to the individual and preferred choice.

🌋 “Digital Easter Egg Hunting” & the "Loihi" Metaphor

The first complete results emerged over the Easter holidays—hence:

👉 “Digital Easter Egg Hunting” (aka: Data Mining)

However, inspired by earlier work near the Lōʻihi Seamount and research at the University of Hawaiʻi at Mānoa ( plus a pretty cool TV-documentary that we produced alongside :), the process earned a fitting name:

👉 “Loihi Seamount Gaussing” — from guessing to quantified probability

An intuitive analogy:

Scatter plot → volcanic vents (random distribution)
Ellipses → tectonic stress field (structure)
PDF → magma chamber (underlying system)

📈 Interpretation of Results

The dataset shows:

a clear ellipsoidal distribution
low variance around the mean
a stable correlation between porosity and permeability

In practical terms:

Higher permeability tends to coincide with lower porosity (coarser materials such as sands), while high porosity with low permeability indicates finer-grained materials (silts/clays).

This relationship may seem intuitive (to some perhaps even "trivial") — but the PDF confirms it quantitatively, highlighting the consistency and quality of the data.

🧊 From 2D to 3D

The 3D-PDF surface adds another layer of understanding:

visualization from multiple perspectives
clear identification of the probability peak
improved perception of uncertainty and spread

All visualizations are peak-normalized (max = 1) for comparability.

🧠 Conclusion

A scatter plot is a good start—but not the end.

Only by combining:

covariance
σ-ellipses
probability density functions

do we achieve a robust, interpretable model of the data.

👉 From point cloud → structure

👉 From correlation → geometry

👉 From guessing → Gaussing

🔭 Outlook

Next steps will focus on improving signal quality analysis for NMR and related datasets. The developed workflow acts as a vendor/client-independent analysis tool, applicable far beyond this specific use case. And perhaps the most practical take-away: