⚡ electrical engineering

Do P-Wave Velocity-UCS Equations Transfer Across Studies? How Many Local Tests Are Needed for Intact-Rock Calibration

This study demonstrates that generalized P-wave velocity to UCS equations lack transferability across different rock types and laboratories, necessitating local calibration and direct UCS testing for reliable engineering design, particularly for weak rocks.

Original authors: Ali Mehrabifard¹˒², Zahra Yousefabadi³

Published 2026-09-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ali Mehrabifard¹˒², Zahra Yousefabadi³

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep underground, engineers face a critical question before they can build a tunnel, a dam, or a mine: how strong is the rock? The most reliable way to answer this is to take a cylindrical core of the rock, crush it in a machine, and measure the exact force required to break it. This test, known as the uniaxial compressive strength test, is the gold standard for safety. However, it is destructive, expensive, and slow. It requires a perfect piece of rock and hours of laboratory time. Because of these constraints, geologists and engineers have long hoped for a shortcut. They wanted to use sound waves to guess the strength. By sending a pulse of sound through a rock sample and measuring how fast it travels, they could theoretically estimate how hard the rock is without breaking it. This speed, known as P-wave velocity, is easy to measure and non-destructive. For decades, researchers have tried to write simple mathematical rules that link this sound speed directly to rock strength, hoping to apply these rules to any rock, anywhere.

A new study challenges the idea that these universal rules actually work. The researchers gathered a massive collection of data, bringing together nearly 2,340 pairs of measurements from 59 different scientific studies. Each pair consisted of a rock sample where both the sound speed and the crushing strength had been measured in a laboratory. They then took the most popular equations found in existing literature—rules that claim to predict strength based on sound speed—and tested them against this new, independent data. The results were stark. When these published equations were applied to rocks they had never seen before, they failed almost everywhere. Out of eighteen tested equations, fourteen performed no better than simply guessing the average strength of all rocks and ignoring the sound speed entirely. The sound waves, it turns out, do not carry a universal message that can be read the same way in every laboratory or for every type of rock.

The reason for this failure lies in the hidden differences between studies. While the sound speed and rock strength are related, the relationship shifts depending on who is doing the measuring and what specific rock they are testing. Factors like the mineral makeup of the rock, how wet it is, the shape of the sample, and even the specific equipment used in the lab create a unique "offset" for every study. The researchers found that these study-to-study shifts account for more than 60 percent of the uncertainty when trying to predict strength. It is as if every laboratory speaks a slightly different dialect of the same language; a rule written in one lab does not translate correctly to another. This means that a formula that worked perfectly for a specific set of granite samples in one country might be completely wrong for a different set of granite samples in another, or for a completely different type of rock like sandstone or limestone.

The study also looked at whether a small amount of local testing could fix these broken rules. Engineers often wonder if they can measure the sound speed on a new site, use a published equation, and then just run a few local crushing tests to "calibrate" or adjust the result. The researchers tested this by simulating what happens when a team runs a small number of local strength tests—say, four samples—and uses those results to tweak the prediction. They found that this approach does help, but only to a point. With just four local tests, the ability to correctly sort rocks into broad strength categories improved significantly, rising from a 45 percent success rate to over 63 percent. This is useful for screening: it helps engineers decide if a rock is generally weak, medium, or strong. However, the uncertainty remained too high to determine a precise number for engineering design. Even after calibration, the predicted strength could still be off by nearly half its value. For critical decisions involving weak rocks, which often govern the safety of slopes and underground excavations, this level of guesswork is not enough.

The situation is even more precarious for weak rocks, those with a strength below 25 megapascals. This is the category of rock that often causes the most trouble in construction, yet it is the least represented in the data. The study found that no single published equation could reliably predict the strength of these weak materials without local testing. In fact, for rocks in this weak range, the existing equations did not outperform a simple average-based benchmark. While some patterns suggested overestimation, the data was too sparse to definitively prove a universal bias, and the observed patterns could be statistical artifacts rather than a consistent flaw in the equations. The data was simply too sparse and the variations too complex to create a general rule for weak rock. The researchers concluded that for weak materials, or whenever a precise numerical strength is needed for a design, there is no substitute for direct measurement. The sound speed can be a helpful tool for sorting and organizing rock samples, but it cannot replace the crushing test.

Ultimately, this research does not say that sound speed is useless. Instead, it redefines how it should be used. The sound wave is a good tool for preliminary classification, helping engineers quickly screen large volumes of core to identify which parts need more attention. But it cannot be used as a standalone calculator for strength. The study proposes a practical workflow: use the sound speed to guide where to test, run a small number of local crushing tests to calibrate the specific conditions of that site, and then use those local results to make decisions. If the rock is weak or the design requires high precision, the local testing must continue until the uncertainty is low enough to be safe. The dream of a single, universal equation that works for every rock on Earth has been shown to be an illusion. The reality is that every rock formation and every laboratory has its own unique signature, and understanding that signature requires listening to the rock itself, not just applying a rule from a book.

Technical Summary: Transferability of P-Wave Velocity–UCS Equations

Problem Statement
P-wave velocity ( $V_p$ ) is widely used to estimate the intact-rock uniaxial compressive strength (UCS) via empirical power-law relations ( $UCS = aV_p^b$ ). However, equations calibrated in one study frequently fail when applied to new rock types or laboratories. While high coefficients of determination ( $R^2$ ) are often reported within source studies, these do not guarantee predictive accuracy in independent datasets. The core difficulty lies in "transfer": factors such as porosity, mineralogy, fabric, saturation, weathering, and test procedures can shift the velocity-strength relationship without altering its overall slope. Previous compilations have noted this site dependency, but a quantitative assessment of how much predictive information is retained when applying published equations to independent studies has been lacking. Furthermore, the performance of these equations in weak rock (below 25 MPa), which is critical for excavation and slope design, remains poorly understood due to sparse data in that range.

Methodology
This study assembled a corpus of 2,337 paired laboratory measurements of $V_p$ and UCS from 59 distinct studies. The dataset includes sedimentary, igneous, and metamorphic rocks, though it is heavily weighted toward strong rocks (ISRM classes R4 and R5). The analysis employed the following methodological framework:

Leave-One-Study-Out Validation: To evaluate transferability, 19 published velocity-strength equations were tested against the corpus. For each equation, the source study was withheld, and the model was validated on the remaining studies. This prevents the model from "seeing" the specific study offset it is supposed to predict.
Hierarchical Modeling: A hierarchical Bayesian regression was used to separate the common velocity dependence from systematic differences among studies. The model is defined as $\log UCS_{ij} = \beta_0 + \beta_1 \log V_{p,ij} + u_j + \epsilon_{ij}$ , where $u_j$ represents the study-specific offset and $\epsilon_{ij}$ the within-study residual. This allows for the quantification of "transformation uncertainty" distinct from inherent variability.
Benchmarking: Each equation was compared against two baselines: a "mean-only" benchmark (ignoring velocity) and a "genetic class" benchmark (conditioning on rock type but ignoring velocity).
Local Calibration Experiments: The study simulated local calibration by withholding a study, drawing $m$ specimens (where $m = 1, 2, 4, 10, 20$ ) as a calibration set, and correcting the pooled relation using the mean residual of these specimens. Different specimen selection rules (random, velocity-spanning, slowest-first) were tested.
Weak Rock Analysis: Specific attention was paid to records below 25 MPa (ISRM R2 and below) to determine if published equations exhibit systematic bias or if the data is too sparse to draw conclusions.

Key Results

Limited Transferability of Published Equations: Of the 19 equations evaluated, 14 failed to outperform the mean-only benchmark when restricted to their declared lithology and velocity ranges. Only four equations (Tuğrul and Zarif, 1999; Karakul and Ulusay, 2013; Moradian and Behnia, 2009; Kahraman, 2001) demonstrated statistically significant skill over the benchmark after family-wise error correction. Even for these successful equations, the transfer is imprecise; the best equation achieved a study-balanced RMSE of ~39.6 MPa against a benchmark of ~53.4 MPa.
Dominance of Study Offsets: The hierarchical model revealed that study-level offsets dominate predictive uncertainty. Approximately 62% of the observation-level predictive variance for a new study is attributable to the study offset, while only ~38% is due to within-study residual variation. Reported study characteristics (lithology, velocity range, density) explained very little of this offset variance (only ~14% combined), indicating that unreported factors (e.g., test procedure, saturation, specimen geometry) drive the transfer failure.
Value of Local Calibration: Integrating a small number of local UCS tests significantly improves performance, but with diminishing returns.
- Classification: With zero local tests, the correct strength-class prediction rate was 45.0%. With four local tests, this increased to 63.2%.
- Precision: While four tests narrowed the 95% predictive interval for a single specimen from a factor of 13.34 to 5.82, the relative margin of error for the locally calibrated conditional strength level remained high at 45%.
- Sample Size: Reaching a 25% relative margin requires approximately 13 local tests; reaching 10% requires 70 tests.
Weak Rock Limitations: No published equation outperformed the mean-only benchmark for weak rocks (below 25 MPa). While outcome-binned plots suggested overprediction in weak rock, simulation showed this pattern could arise from an unbiased relation due to binning artifacts. However, the lack of transferable equations and the scarcity of data below 5 MPa (only 86 records) mean that weak-rock strength cannot be reliably estimated without local testing.
Selection Rules: Specimen selection matters. Calibration sets concentrated entirely on the slowest (lowest velocity) specimens resulted in significant under-coverage of predictive intervals. However, no single selection rule (random vs. velocity-spanning) was universally superior for small sample sizes (e.g., $m=4$ ).

Significance and Claims
The paper concludes that published P-wave velocity–UCS equations are not generally transferable estimates of intact strength. The primary barrier to transfer is not the functional form of the equation but the systematic, unexplained offsets between studies.

The authors claim that $V_p$ should be restricted to preliminary core classification and screening rather than providing precise numerical strength values. While a small local dataset (e.g., four tests) can improve strength classification and guide further testing, it is insufficient to replace direct UCS measurements where weak materials or precise design strengths are required.

The study's contribution is a quantified local-calibration workflow: use velocity to stratify and conditionally classify intact core within a local population, estimate the study-level offset with direct tests, and size the remaining UCS program based on the uncertainty the engineering decision can tolerate. The paper emphasizes that future studies must report saturation, specimen geometry, transducer frequency, and test standards to improve the transferability of velocity-strength relations.

Technical Summary: Transferability of P-Wave Velocity–UCS Equations

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