Parameter Estimation in River Transport Models With Immobile Phase Exchange Using Dimensional Analysis and Reduced-Order Models

TL;DR

The paper tackles parameter estimation for river solute transport with immobile storage by introducing DSTE, a framework that uses dimensionless grouping to decouple advection velocity from other parameters and reduce forward-model evaluations. It combines a Laplace-domain analytical solution of the ADE with immobile exchange, KL-based reduced-order modeling, and Projected Barycentric Interpolation to map measurements to physical parameters, achieving accurate estimates across 295 field breakthrough curves. A reusable synthetic dataset and several interpolation strategies (notably PBI) enable robust, scalable inference, with DSTE-KL-PBI outperforming Laplace-fitting, moment-matching, and global optimization benchmarks. The approach yields actionable parameter-label datasets linking transport properties to hydraulic conditions, and offers a scalable path for applying dimensional analysis and reduced-order surrogates to river transport problems. Open data and code resources are provided to facilitate replication and extension to new tracer tests and higher-fidelity models.

Abstract

We propose a framework for parameter estimation in river transport models using breakthrough curve data, which we refer to as Dimensionless Synthetic Transport Estimation (DSTE). We utilize this framework to parameterize the one-dimensional advection-dispersion equation model, incorporating immobile phase exchange through a memory function. We solve the governing equation analytically in the Laplace domain and numerically invert it to generate synthetic breakthrough curves for different memory functions and boundary conditions. A dimensionless formulation enables decoupling the estimation of advection velocity from other parameters, significantly reducing the number of required forward solutions. To improve computational efficiency, we apply a Karhunen-Loeve (KL) expansion to transform the synthetic dataset into a reduced-order space. Given a measured breakthrough curve, we estimate the advection velocity by minimizing the distance from the measurement to the synthetic data in KL space, and infer the remaining dimensionless parameters by Projected Barycentric Interpolation (PBI). We benchmark our method against several alternatives, including Laplace domain fitting, moment matching, global random optimization, and variations of the DSTE framework using nearest-neighbor interpolation and neural network-based estimation. Applied to 295 breakthrough curves from 54 tracer tests in 25 rivers, DSTE delivers accurate parameter estimates. The resulting labeled dataset allows researchers to link transport parameters with hydraulic conditions, site characteristics, and measured concentrations. The synthetic dataset can be leveraged for the analysis of new breakthrough curves, eliminating the need for additional forward simulations.

Figures (8)

• Figure 1: Schematic of the Dimensionless Synthetic Transport Estimation (DSTE) framework with KL reduced-order modeling and Projected Barycentric Interpolation (PBI).
• Figure 2: Representations of the synthetic dataset ($N_{\text{synth}} =1,000$) for the ADE model with a first-order memory function in (a) parameter space, (b) breakthrough curves form, and (c) embedded in the space of KL coefficients, only first three coefficients being represented. The same synthetic datapoint has matching colors in the three plots. The colorscale is created to represent different combinations of parameters. The visualization of the KL embedding of the synthetic dataset and its association to the parameters (particularly when combined with measurements, as shown in Figure \ref{['fig:measured-dimensionless-and-embedding']}c) provides a way to evaluate how well the chosen distribution of parameters matches the measured data and which parameters could be modified to improve it.
• Figure 3: KL embedding process of a measured breakthrough curve. (a) Measured breakthrough curve. (b) Different dimensionless breakthrough curves corresponding to the same dimensional measurement but different advection velocities. (c) Embedding of the dimensionless breakthrough curves corresponding to a single measured breakthrough curve in the space of KL coefficients, only first three coefficients being represented. The definition of the KL space is the same as in Figure \ref{['fig:synthetic-construction-and-embedding']}c.
• Figure 4: Illustration of the PBI method for the example shown in Figures \ref{['fig:synthetic-construction-and-embedding']} and \ref{['fig:measured-dimensionless-and-embedding']}, but with rescaled color scale: (a) KL embedding of synthetic data forming the vertices of the simplex and different dimensionless forms of the field breakthrough curve corresponding to different advection velocities. Optimal and suboptimal projections with their corresponding simplices, and (b) Local coordinate representation of a simplex with the color scale corresponding to parameters interpolated by barycentric interpolation.
• Figure 5: Comparison of measured breakthrough curves to reconstructed breakthrough curves from the advection-dispersion model with immobile exchange (obtained with forward Laplace solver) using parameters estimated with different methods. Domain, boundary and initial conditions from column 4 in Table \ref{['tab:problems-comparison']}. (a) First-order exchange model. (b) Power-law memory function model. The title of each breakthrough curve shows the river, the date of the tracer test, and the number of the breakthrough curve in the tracer test (a letter before the number signifies there were multiple tracer tests on the same day).