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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...
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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.
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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...
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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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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...
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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...
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Uncertainty Quantification in Inverse Scattering Problems.

Carolina Abugattas1,2, Ana Carpio2, Elena Cebrián3

  • 1Facultad de Ingeniería, Universidad Alberto Hurtado, Santiago de Chile 8340575, Chile.

Entropy (Basel, Switzerland)
|May 4, 2026
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Summary
This summary is machine-generated.

This study introduces a Bayesian framework for inverse scattering, progressively adapting from low to high dimensions. It enhances anomaly detection in complex media by quantifying uncertainty and reducing computational costs using prior information.

Keywords:
Bayesian inversionKarhunen–Loève expansionsMarkov chain Monte Carlogeophysical imagingmedical imaginguncertainty quantification

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Area of Science:

  • Computational physics and geophysics
  • Bayesian inference and uncertainty quantification
  • Wave propagation and inverse problems

Background:

  • Inverse scattering problems aim to identify subsurface anomalies from wave interaction data.
  • Noise in measurements necessitates uncertainty estimates for anomaly predictions.
  • Existing methods struggle with complex scenarios like overlapping anomalies or limited prior information.

Purpose of the Study:

  • To develop a progressive Bayesian framework for inverse scattering problems.
  • To reduce computational costs by leveraging prior information and adapting dimensionality.
  • To provide uncertainty estimates for detected anomalies in various geological and biological settings.

Main Methods:

  • Progressive Bayesian formulation from low- to high-dimensional models.
  • Low-dimensional parameterization for well-separated anomalies (e.g., tissue anomalies, subsoil deposits).
  • High-dimensional parameterization using Karhunen-Loève (KL) or Fourier expansions for complex scenarios (e.g., oil/gas reservoirs).

Main Results:

  • Demonstrated effectiveness in detecting anomalies in tissues and subsoil deposits using low-dimensional models.
  • Successfully characterized oil and gas reservoirs in salt dome configurations using high-dimensional KL expansions.
  • Utilized MCMC samplers to characterize posterior probability, providing insights into high-probability configurations and uncertainty reduction.

Conclusions:

  • The proposed progressive Bayesian framework effectively handles inverse scattering problems of varying complexity.
  • High-dimensional KL-based methods show superior stability and effectiveness for complex subsurface characterization.
  • Educated choices for a priori covariance are crucial; further research could incorporate them as additional parameters.