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Many astrophysical analyses depend on estimates of redshifts (a proxy for distance) determined from photometric (i.e., imaging) data alone. Inaccurate estimates of photometric redshift uncertainties can result in large systematic errors. However, probability distribution outputs from many photometric redshift methods do not follow the frequentist definition of a Probability Density Function (PDF) for redshift -- i.e., the fraction of times the true redshift falls between two limits z1 and z2 should be equal to the integral of the PDF between these limits. Previous works have used the global distribution of Probability Integral Transform (PIT) values to re-calibrate PDFs, but offsetting inaccuracies in different regions of feature space can conspire to limit the efficacy of the method. We leverage a recently developed regression technique that characterizes the local PIT distribution at any location in feature space to perform a local re-calibration of photometric redshift PDFs. Though we focus on an example from astrophysics, our method can produce PDFs which are calibrated at all locations in feature space for any use case.
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A conditioned backward fractional-derivative model for identifying super-diffusive pollutant source in aquatic systems with multiple observation data
Backward models for super-diffusion have been developed to identify pollutant source locations, but they are limited to a single observation and disregard field-measured concentrations. To overcome these limitations, this study derives the adjoint of the space-fractional advection–dispersion equation, incorporating measured concentrations from multiple observation data. Backward probabilities, such as the backward location probability density function (PDF), describe the likely source location(s) at a fixed time before sampling, offering a comprehensive modeling approach for source identification. By applying Bayes’ theorem, the individual PDFs from each observation and its corresponding concentration are combined into a joint PDF, enhancing both the information and reliability compared to the previous single PDF. Field applications show that the improved model enhances accuracy (with PDF peak locations closer to the actual source) and precision (with reduced variance) of backward PDFs for identifying point sources in a natural river and aquifer. The model’s performance is affected by observation count and measurement errors, with double peaks in the backward location PDF possible due to source mass uncertainty. Future refinements, such as incorporating backward travel time analysis and extending applications to reactive pollutants, could further enhance the utility of the conditioned backward fractional-derivative model developed in this study.
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- Award ID(s):
- 2412673
- PAR ID:
- 10655837
- Publisher / Repository:
- Advances in Water Resources
- Date Published:
- Journal Name:
- Advances in Water Resources
- Volume:
- 203
- Issue:
- C
- ISSN:
- 0309-1708
- Page Range / eLocation ID:
- 105039
- Subject(s) / Keyword(s):
- Backward probability method Super-diffusion Bayes’ theorem Space fractional advection–dispersion equation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1016/j.advwatres.2025.105039
