⚡ electrical engineering

Generation, Annihilation and Flow of Structural Information in Ultrasonic Nondestructive Evaluation

This paper introduces and quantifies a physical concept of structural information in ultrasonic nondestructive evaluation by deriving a balance equation analogous to Poynting's theorem, thereby mapping the generation, annihilation, and flow of information from defects to sensors to enhance testing, monitoring, simulation, and machine learning applications.

Original authors: Frank Schubert

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Frank Schubert

Original paper licensed under CC BY 4.0 (http://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

Imagine you are trying to figure out what's inside a sealed, opaque box without opening it. You tap on it with a hammer (ultrasound) and listen to the echo. In the world of Non-Destructive Testing (NDT), engineers have been doing this for decades. But until now, they've been guessing how much "information" they are actually getting from those echoes. Is the echo telling them about a crack's size? Its location? Its material? It's been a bit like trying to measure the "weight" of a thought.

Dr. Frank Schubert's paper introduces a revolutionary new way to think about this. He proposes a physics-based theory of information specifically for sound waves. Instead of treating information as a vague concept, he treats it like energy.

Here is the breakdown of his ideas using simple analogies:

1. The "Difference" is the Key

Imagine you have a perfect, pristine guitar string. You pluck it, and it makes a specific sound. Now, imagine you slightly bend the string (a "parameter change"). If you pluck it again, the sound is almost the same, but with a tiny, almost invisible difference.

In traditional testing, engineers look at the whole sound. Dr. Schubert says: "Ignore the whole sound. Only look at the tiny difference between the two sounds."

He calls this the "Difference Wave Field."

Analogy: Think of the original sound as a loud, noisy crowd. The "information" about the defect isn't in the crowd's noise; it's in the change in the noise when someone coughs. By mathematically subtracting the "perfect" sound from the "defective" sound, you isolate the pure "message" about the defect.

2. Information is Like a River (The Balance Equation)

Dr. Schubert realized that information behaves very much like water flowing in a river or energy moving through a wire.

Information Density: This is how much "information water" is in a specific spot at a specific time.
Information Flow: This is the current carrying that water.
Sources and Sinks: This is the most exciting part.
- Sources: When the sound wave hits a defect (like a crack or a layer of different material), it creates new information. It's like a faucet turning on, pouring fresh information into the river.
- Sinks: Sometimes, information disappears! If a wave hits a boundary and reflects in a way that cancels out the "difference," that specific piece of information is "annihilated." It's like a drain in the river where the water vanishes.

The Big Discovery: Unlike energy (which is conserved and never disappears), information is not conserved. It can be created and destroyed depending on how the wave interacts with the object.

3. The "Change Bit" (Cbit)

In computer science, we measure data in "bits." Dr. Schubert introduces a new unit called the "Change Bit" (Cbit).

Analogy: Imagine you are a detective. You have a suspect (the defect).
- If you move the suspect 1 inch, and the witness (the sensor) notices a huge change in the story, that's a high Cbit value. You learned a lot.
- If you move the suspect 1 inch, and the witness notices almost nothing, that's a low Cbit value. You learned very little.
- This allows engineers to calculate exactly how much they can learn about a specific defect (like its depth vs. its thickness) before they even build the testing equipment.

4. Why This Matters (The "So What?")

This theory changes how we design tests and use Artificial Intelligence (AI).

Better AI Training: AI needs to learn to spot defects. Usually, AI is fed thousands of images of "good" and "bad" parts. Dr. Schubert suggests feeding the AI the "Difference" instead.
- Analogy: Instead of showing a student a picture of a car and a picture of a flat tire, you show them a picture of the difference between the two. The student (AI) learns the specific feature of the flat tire much faster and more accurately.
Optimizing Sensors: Engineers can now calculate exactly where to place a sensor to get the maximum "Cbits" for a specific type of defect. If you want to find a crack's depth, put the sensor in the reflection path. If you want to find its speed of sound, put it in the transmission path.
Universal Application: While this paper uses sound waves, the math could apply to X-rays, heat imaging, or even financial markets. Any system where a "signal" changes because of a "parameter" can use this logic.

Summary

Dr. Schubert has built a map for information.

Before: We knew information existed in sound waves, but we didn't know how to measure it, where it came from, or where it went.
Now: We have a formula. We know information is generated at interfaces (like a wall), flows with the wave, and can be destroyed. We can measure it in "Change Bits."

It's like going from guessing how much rain is falling by looking at the clouds, to having a precise rain gauge that tells you exactly how many drops hit the ground, where they came from, and how hard they hit. This allows us to build better, smarter, and more efficient ways to inspect the world around us.

1. Problem Statement

In Non-Destructive Evaluation (NDE) and Structural Health Monitoring (SHM), the term "information" is used colloquially but lacks a physically formalized and quantifiable theory. While Shannon's information theory (probabilistic) and Fisher information (statistical) exist, they do not directly address the deterministic interaction between physical fields (ultrasound) and structural defects.

Current NDE relies on extracting data from scattered wave fields, but there is no rigorous framework to:

Quantify how much "structural information" is generated by specific defects.
Track the flow of this information from the defect to the sensor.
Distinguish between information sources (where data is created) and sinks (where data is lost/annihilated).
Optimize testing setups based on information yield rather than just signal amplitude.

2. Methodology

The author proposes a new deterministic information theory based on difference wave fields. The methodology involves:

Difference Wave Fields ( $W_{diff}$ ): Instead of analyzing absolute wave fields, the study calculates the difference between wave fields generated by a baseline system ( $S_0$ $S_{0}$ ) and systems with small variations in a specific structural parameter ( $P$ $P$ ), such as defect position, thickness, density, or sound velocity.
- $I_P \propto \frac{\partial W}{\partial P}$ (The partial derivative of the wave field with respect to the parameter).
Analogy to Poynting's Theorem: The author derives a balance equation for structural information analogous to the conservation of energy (Poynting's theorem) in elastodynamics.
- Information Density ( $I_P$ ): Defined as the sum of kinetic and potential components of the differential wave field.
- Information Flux ( $F_P$ ): Analogous to the Poynting vector, representing the flow of information.
- Source/Sink Terms ( $Q$ ): Unlike energy, which is conserved in lossless media, the information balance equation includes source and sink terms. Information is generated at interfaces (scattering) and can be annihilated during subsequent interactions.
Normalization and Scaling: To make the quantity dimensionless and independent of excitation amplitude, the information density is normalized by the energy of the excitation signal ( $E_f$ ) and scaled by the square of the parameter ( $P^2$ ).
Numerical Simulation: The theory is verified using 1D and 2D Elastodynamic Finite Integration Technique (EFIT) simulations involving a layered medium with varying geometric and material parameters.

3. Key Contributions

A. Formalization of "Change Information"

The paper introduces "Change Bits" (Cbits) as a new absolute unit of structural information. This allows for the quantitative comparison of different parameters (e.g., position vs. density) and different measurement setups (reflection vs. transmission).

B. The Information Balance Equation

A major theoretical breakthrough is the derivation of the balance equation showing that structural information is not a globally conserved quantity.

Generation: Information is generated primarily at scattering interfaces (boundaries between materials) and within volumes where material properties change (e.g., sound velocity gradients).
Annihilation: Information can be annihilated (lost) when waves interact with subsequent boundaries, even in a lossless system. This explains why certain parameters (like layer position) may not be detectable in transmission but are in reflection.
Non-Conservation: Unlike total energy, total information increases or decreases with every interaction event.

C. Separation of Parameters

The methodology demonstrates that difference wave fields allow for the separate investigation of individual structural parameters. While absolute wave fields for small changes look nearly identical, the difference fields for position, thickness, density, and velocity are distinct and orthogonal, making them superior for classification and parameter estimation.

D. Relationship to Machine Learning

The paper establishes a link between the magnitude of the difference signal in "feature space" and the information content. Larger distances between signals in the feature space (caused by parameter changes) correspond to higher information content, providing a theoretical basis for optimizing training data generation and feature selection in ML for NDE.

4. Key Results (Simulation Findings)

The simulations on a 1D layered model yielded the following quantitative insights:

Parameter Sensitivity (Information Yield):
- Sound Velocity Changes: Generated the highest information content (approx. 1750x more than density changes). Information is generated volumetrically as the wave traverses the layer.
- Position Changes: Generated very high information content (approx. 1000x density), primarily detectable in reflection.
- Thickness Changes: Moderate information content (approx. 60x density), detectable in both reflection and transmission.
- Density Changes: Generated the lowest information content.
Measurement Setup Efficiency:
- Reflection Setup: Optimal for detecting position and density changes.
- Transmission Setup: Optimal for detecting sound velocity and thickness changes.
Source/Sink Dynamics:
- In the "Layer Position" case, information generated at the first boundary was partially annihilated at the second boundary in the forward direction (explaining why position info is lost in transmission).
- In the "Layer Thickness" case, the second boundary acted as a strong source for forward-propagating information.
Kinetic vs. Potential: Kinetic and potential information components oscillate in opposite phases (self-interference) but sum to a stable total information curve that increases stepwise with each interaction.

5. Significance and Implications

Optimization of NDE Systems: Engineers can now calculate the "information yield" of a specific setup before building it, allowing for the design of inspection systems that maximize the extraction of specific defect parameters.
Advanced Imaging: The theory suggests that imaging algorithms should be based on difference wave fields rather than absolute fields to achieve higher resolution and parameter specificity.
Machine Learning: Provides a quantitative metric to weight training data. Configurations that generate high "Cbits" are more valuable for training ML models to detect specific defects.
Fundamental Physics: Establishes a deterministic, physics-based information theory that is distinct from Shannon's probabilistic theory. It bridges the gap between physical field interactions and information quantification without relying on statistical noise assumptions.
Generalizability: While derived for ultrasonics, the concept of "change information" (proportional to the square of the differential field) is applicable to other NDE methods (eddy current, thermography) and potentially non-technical systems (biological, financial) where system states change in response to parameters.

In conclusion, Dr. Schubert's work transforms "information" from a vague concept in NDE into a rigorously defined, quantifiable physical quantity with its own conservation laws (or lack thereof), offering a powerful new tool for the next generation of structural health monitoring and inspection technologies.