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Bayesian reversal of the liquid level trajectory in a draining tank for pollution forensics

T0 review · 0 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Bayesian inversion recovers the initial liquid level in a drained tank from the final level and drainage duration estimate.

desk verdict Bayesian inversion with a discrepancy term recovers initial tank levels from final level and drainage time in lab tests, but the approach stays narrow and leans on a fixed duration input. read the letter →

arxiv 2605.29193 v1 pith:U2YHWSOJ submitted 2026-05-28 stat.AP

Kyla D. Jones , Gbenga Fabusola , Alexander W. Dowling , Cory M. Simon This is my paper
classification stat.AP
keywords BayesianinferenceinverseproblemtankdrainingTorricelli'slawpollutionforensicsmodeldiscrepancyliquidlevelstatisticalinversion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out a Bayesian framework to solve an inverse problem in pollution forensics: recovering the unknown starting liquid level in a storage tank after partial drainage through a leak. It fuses a physics model based on Torricelli's law, an empirical discrepancy term for model shortcomings, prior knowledge, and the observed final level while taking drainage duration as a fixed input. Experiments with water tanks show the inferred initial level is accurate, though uncertainty grows as drainage time lengthens. This matters because the discharged volume can then be estimated for regulatory assessment and remediation when the original inventory is unknown.

What carries the argument

Bayesian statistical inversion that combines a physics-based drainage model, an empirical discrepancy function, and experimental time series data to quantify uncertainty in the inferred initial liquid level.

What would settle it

Run a controlled tank-draining experiment with a measured initial level and known duration, observe the final level, apply the Bayesian procedure, and check whether the true initial level falls outside the reported uncertainty bounds.

Watch

Extended reading notes

Core claim

The central claim is that Bayesian statistical inversion, applied to a Torricelli's law model augmented with an empirical discrepancy function, can recover the initial liquid level from the final observed level and an estimate of drainage duration, with the inference accurate in experiments but with increasing uncertainty for longer drainage times.

Load-bearing premise

The drainage duration is known or can be estimated with sufficient accuracy to serve as a fixed input to the inversion.

Editorial extensions

If this is right

  • The discharged volume can be estimated after a pollution incident when the original inventory is unknown.
  • Uncertainty in the initial level prediction is quantified and grows with longer drainage duration.
  • The discrepancy function accounts for missing or imperfectly modeled physics in the drainage dynamics.
  • The framework provides a classroom example of dynamic modeling, model discrepancy, and Bayesian inference.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same inversion structure could be tested on other fluid systems where duration is only approximately known.
  • Real-time sensor streams could be used to update the initial-level posterior sequentially rather than after the fact.
  • Sensitivity of the posterior to small errors in the supplied drainage duration estimate remains unexamined in the reported experiments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a Bayesian inversion framework to recover the unknown initial liquid level in a storage tank after partial drainage, given the final observed level and an estimate of drainage duration. It employs a physics-based model from Torricelli's law augmented by an empirical discrepancy function, places priors on uncertain parameters, and validates the approach through laboratory experiments on water tanks, reporting that the inferred initial levels are accurate while posterior uncertainty grows with longer drainage times.

Significance. If the experimental results hold with the quantitative detail expected in the full manuscript, the work provides a practical, uncertainty-quantified method for pollution forensics applications where initial inventory is unknown. The combination of mechanistic modeling, data-driven discrepancy correction, and Bayesian updating is a standard and defensible approach for this inverse problem; the classroom-project angle is a minor but positive additional contribution.

minor comments (2)
  1. Abstract: the statement that the inferred initial level 'was accurate' should be accompanied by at least one quantitative error metric (e.g., mean absolute error or coverage of credible intervals) rather than a qualitative claim; this is needed to substantiate the central experimental result.
  2. The manuscript should clarify whether drainage duration is treated as a known fixed input in the laboratory experiments or as an uncertain estimate, and include a brief sensitivity check on duration error even if the central claim remains intact.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring response or manuscript changes at this time.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper's derivation uses a standard physics model (Torricelli's law) augmented by an empirical discrepancy function, combined with Bayesian inversion and priors to solve an inverse problem for initial liquid level given final level and drainage duration. Experimental results report accuracy of the inferred initial level under controlled lab conditions. No step reduces a prediction to a fitted quantity by construction, invokes self-citation as load-bearing justification for a uniqueness claim, or renames a known result; the framework is self-contained against external benchmarks and the reported accuracy is an independent empirical outcome rather than a definitional tautology.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Abstract-only review; ledger populated from stated elements only. Torricelli's law is treated as standard physics input. No explicit free parameters, invented entities, or ad-hoc axioms listed beyond the general Bayesian framework and empirical discrepancy term.

assumptions (2)
  • domain assumption Torricelli's law governs the draining dynamics
    Abstract states 'physics-based model based on Torricelli's law'
  • standard math Bayesian inference combines prior physical knowledge with data to produce posterior over initial level
    Core method described in abstract

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Cite this review

Pith. "Pith review of Bayesian reversal of the liquid level trajectory in a draining tank for pollution forensics." pith.science (2026). https://pith.science/paper/U2YHWSOJ

@misc{pith2026260529193,
  author       = {Pith},
  title        = {Pith review of: Bayesian reversal of the liquid level trajectory in a draining tank for pollution forensics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2YHWSOJ}},
  note         = {Machine review of arXiv:2605.29193}
}
read the original abstract

Storage tanks for hazardous liquids are common in industry and agriculture. During a pollution incident, liquid may drain from a storage tank through a small hole, crack, or pipe. After containing the leak, estimating the discharged volume of liquid is essential for public safety, regulatory assessment, and remediation. When the original inventory of liquid is unknown, this constitutes an inverse problem. In this work, we present a framework for inferring the initial liquid level in a partially drained tank from the observed final liquid level after a pollution incident and an estimate of the drainage duration. Because the drainage dynamics, model parameters, and observations are uncertain, we employ Bayesian statistical inversion to combine prior physical knowledge with experimental liquid level time series data to predict the initial liquid level with quantified uncertainty. We use a physics-based model based on Torricelli's law to describe the tank-draining dynamics and augment it with an empirical discrepancy function to account for missing or imperfectly modeled physics. In our experiments with a tank draining of water, we found that our inferred initial liquid level was accurate, although uncertainty increased with drainage duration. Beyond its application to pollution forensics, this work may also serve as a hands-on classroom project illustrating dynamic modeling, model discrepancy, and Bayesian inference.

Figures

Figures reproduced from arXiv: 2605.29193 by the authors.

Figure 1
Figure 1. summarizes the problem setup. An above-ground storage tank holds a hazardous liquid. The initial height of the liquid in the tank, h0 [cm], is unknown. At time t = t0 [s], the tank suddenly begins draining through a small orifice near its bottom due to the hydrostatic pressure generated by the liquid column caused by gravity, contaminating the surrounding area. Owing to a vent, the headspace in the tank remains at a… view at source ↗
Figure 2
Figure 2. Experimental setup and liquid level time series data. (a) Water drains out of a small hole in the bottom side of a tank. (b) The measured height of water in the tank over time during three separate tank-draining experiments. The first experiment mimics a pollution event, where only the final condition is observed. The next two experiments are for model calibration. Beginning with a filled tank, the liquid level time… view at source ↗
Figure 3
Figure 3. The prior and posterior distribution over water level trajectories in the tank over the three experiments. Each curve shows the forward model for the liquid level dy￾namics in the draining tank, hθ(t;t0, h0) + δa(hθ(t;t0, h0)), with parameters θ and a and associated initial condition (t0, h0) sampled from the (a) prior and (b) posterior. The liquid level time series data for obtaining the posterior are the hollow po… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Calibrated parameters of the forward model. (a) Marginal prior and posterior densities, approximated by kernel density estimation on the MCMC samples, of each parameter of the forward model. Titles show equal-tailed 90% credible interval. (b) Samples of the model discr…
Figure 5
Figure 5. Figure 5: The inferred initial condition of the draining tank for the pollution scenario. The joint (bottom right) and marginal (top right and bottom left) prior and posterior distri￾butions of the initial condition of the water level in the tank during the pollution scenario. T…
Figure 6
Figure 6. Figure 6: Two additional instances of the inverse problem with a shorter (top) and longer (bottom) drainage duration. Left: Prior and posterior distribution over the initial water level. Right: Water-level trajectories sampled from the posterior compared with the observed final …

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