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Uncertainty: Overview

Uncertainty: Overview

In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.

The Uncertainty Principle

The Uncertainty Principle

Werner Heisenberg considered the limits of how accurately one can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurate the measurement of the momentum of a particle is known, the less accurate the position at that time is known and vice versa. This is what is now called the Heisenberg uncertainty principle. He...

Propagation of Uncertainty from Systematic Error

Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this...

Propagation of Uncertainty from Random Error

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...

Estimation of the Physical Quantities

Estimation of the Physical Quantities

On many occasions, physicists, other scientists, and engineers need to make estimates of a particular quantity. These are sometimes referred to as guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. The physicist Enrico Fermi was famous for his ability to estimate various kinds of data with surprising precision. Estimating does not mean guessing a number or a formula at random. Instead, estimation means using prior experience and sound...

Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...

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Updated: Aug 24, 2025

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow

Published on: February 27, 2016

3D multi-physics uncertainty quantification using physics-based machine learning.

Denise Degen¹, Mauro Cacace², Florian Wellmann^3,4

¹RWTH Aachen University, Computational Geoscience, Geothermics and Reservoir Geophysics (CGGR), Mathieustraße 30, 52074, Aachen, Germany. denise.degen@cgre.rwth-aachen.de.

Scientific Reports

|October 19, 2022

Summary

This summary is machine-generated.

This study introduces a hybrid physics-based machine learning method to create accurate, scalable surrogate models for subsurface predictions. This approach significantly reduces computational cost for complex problems, enabling advanced uncertainty quantification.

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Published on: May 10, 2012

Area of Science:

Geophysics and Earth Sciences
Computational Science
Machine Learning Applications

Background:

Subsurface predictions rely on complex differential equations, leading to computationally intensive, high-dimensional problems.
Estimating uncertainties in material parameters further exacerbates computational challenges for existing models.

Purpose of the Study:

To develop a novel hybrid physics-based machine learning technique for creating efficient surrogate models.
To enable reliable, scalable, and interpretable predictions for complex subsurface physical processes.
To facilitate probabilistic analyses, including sensitivity studies and uncertainty quantification.

Main Methods:

Introduction of the non-intrusive reduced basis method, combining physical process models with data-driven machine learning.
Application to a thermo-hydro-mechanical reservoir simulation to demonstrate capabilities.
Development of surrogate models that overcome limitations of purely physics-based or data-driven approaches.

Main Results:

Achieved orders of magnitude computational gain compared to traditional methods.
Maintained accuracy exceeding measurement errors for complex, non-linearly coupled physical problems.
Enabled effective global sensitivity studies and uncertainty quantification.

Conclusions:

The hybrid physics-based machine learning approach offers a powerful solution for computationally prohibitive geoscientific problems.
The non-intrusive reduced basis method provides a scalable and interpretable alternative for subsurface modeling.
The technique is broadly applicable to various geoscientific challenges beyond the illustrated reservoir application.