Geology Reports⌕ Search
Home/Archive/USGS

USGS · 70015091

A comparison of several methods for the solution of the inverse problem in two-dimensional steady state groundwater flow modeling

Logan K. Kuiper
View original source ↗Export metadata ↓

Abstract

Two geostatistical approaches for the estimation of hydraulic conductivity and hydraulic head from hydraulic conductivity and hydraulic head measurements are developed for two-dimensional steady flow with sinks. For both approaches the field of the logarithm of hydraulic conductivity (log-conductivity) is represented as a random field with mean θ 1 +θ 2 x +θ 3 y where x and y denote Cartesian coordinates, variance σ 2 , and covariance σ 2 , exp (−α R ), σ 2 , exp (−α R 2 ), or, σ 2 Δ R r where α, Δ, and r are constants and R is separation distance. The first approach uses linearization of the discretized flow equations to allow the construction of the joint covariance matrix of the hydraulic head and log-conductivity measurements as functions of the parameters θ, θ 2 , θ 3 , σ 2 , and α or Δ. It then uses maximum likelihood estimation to obtain these parameters and also a parameter associated with log-conductivity measurement error. Having found values for the parameters, it then uses kriging to form predictors for log-conductivity and hydraulic head from measured values of hydraulic conductivity and hydraulic head. The second approach uses kriging to form a parameter-dependent predictor for log-conductivity from measured hydraulic conductivity, and then uses this predicted log-conductivity placed into the discretized flow equations to compute hydraulic head. The parameters are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads. A third approach simply allows the hydraulic conductivity field to be a step function with a different value for hydraulic conductivity assigned to each of several chosen regions in the two-dimensional aquifer. The assigned hydraulic conductivities are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads. The three approaches are tested for hydraulic head prediction accuracy on two generated test problems, one of which is statistically generated, and also on a field problem. The third approach, despite its simplicity, performs as well or better than the other approaches.

Explore related subjects

Water Resources ResearchArticle

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Logan K. Kuiper. 2010-07-09. A comparison of several methods for the solution of the inverse problem in two-dimensional steady state groundwater flow modeling. https://doi.org/10.1029/wr022i005p00705

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related USGS reports

USGS2026

Big data for small streams: Leveraging a coarse national model to enhance fine-resolution stream temperature predictions

Predictive stream temperature models are commonly used to address spatial and temporal gaps in observational data because stream temperature information is essential for environmental management decisions. National models capture broad patterns driven by climate, topography, and seasonality, but often lack the spatial resolution to simulate fine-scale variation in groundwater discharge, riparian shading, or tributary inflows. In contrast, watershed models may simulate finer-resolution patterns but miss regional drivers or be limited by scarcity of observations. To generate watershed-extent, fine-resolution stream temperature predictions, we developed 18 multiscale machine-learning models that leverage (a) a broad-extent, coarse-resolution stream temperature model and (b) finer-resolution observations and predictor data. The coarse-resolution model simulates daily stream temperature across the conterminous United States on the National Hydrologic Geospatial Fabric (13.7 km median reach length; 865,364 km total). The fine-resolution models simulate daily stream temperature on the National Hydrography Dataset Plus version 2 (1.5 km median; 5,192,808 km total). We compared multiscale models to seven purely broad- (national) or watershed-extent models, testing them in eight watersheds (436–985 km 2 ) with differing hydrologic settings and under realistic data availability scenarios. Especially when observations within a watershed were scarce, multiscale models generated more accurate predictions than national or watershed-specific models. A recurrent graph convolution neural network pretrained on coarse predictions and fine-tuned on watershed-specific observations was most accurate across watersheds and data scenarios. These results highlight that leveraging both coarse, national results and watershed-specific data can increase model accuracies, particularly when observations within a watershed are spatially or temporally limited.

Water Resources Research↗
USGS2026

Landscape controls river continuum dissolved organic carbon in northeastern Alaska

Rivers draining continuous permafrost landscapes on the North Slope of Alaska span from the mountains to the coast, integrating gradients in vegetation, soil organic matter, and hydrologic flow. We investigated how these landscape transitions influence dissolved organic carbon (DOC) concentrations, radiocarbon ages, and export along the river continuum over seasonal timeframes in northeastern Alaska. Water samples were collected from the Canning River and its tributaries from the spring freshet through late summer via multiple campaigns during 2011–2024. A simulation from the Permafrost Water Balance Model (PWBM) estimated terrain-specific contributions to freshwater and DOC export. River DOC concentrations peaked during the spring and declined in summer, with episodic increases linked to summer rainfall. Summer river DOC concentrations from the Brooks Range remained low (0.41–0.64 mg C L −1 ) but increased downstream (0.5–3.0 mg C L −1 ), becoming younger from the mountains (805 yBP) to the river delta (525 yBP) as flow from DOC-enriched streams draining the foothills (6–18 mg C L −1 ; Modern yBP) entered the Canning. The PWBM simulation revealed that while the mountains dominate the Canning's annual (90%) and spring-summer (88%–93%) freshwater export, the foothills disproportionately influence its annual (56%) and spring-summer (44%–58%) DOC export. DOC yields from the coastal plain were also high but had little influence due to its small contributing area. Our findings highlight the outsized role of the foothills in modulating river-integrated DOC export to coastal ecosystems. Considering the impact of these landscapes on carbon source and runoff will aid in anticipating future DOC fluxes from Arctic rivers.

Alaska↗
USGS2026

Predicting Minnesota lake ice phenology with deep learning, explainable methods, and a physically based benchmark, 1980-2018

Globally, lakes are losing ice cover, but our understanding of the rates and variability of change are biased toward few lakes with long-term records. Recent works have used remote sensing and predictive modeling to supplement the observational record, but these approaches produce large errors when estimating ice phenology for individual lakes. Accurate, lake-specific estimates of ice formation and breakup through time are important for estimating variability in ice loss across heterogenous lakes and for understanding broader implications. This work explores machine learning (ML) approaches for hindcasting ice phenology using observations from 1980 to 2018 across 625 Minnesota lakes, covering 4359 lake-years of record. We used daily weather and static lake attributes to develop 60 neural networks for hindcasting lake ice time series. We considered LSTMs and attention-based transformers of varying size and initial parameters. We found that the largest LSTM performed most accurately on withheld data, and on a test set of unseen years and lakes, it outperformed a state-of-the-art physically based model in daily accuracy (97% vs. 95%), year-level metrics (RMSE for ice formation = 6.9 vs. 13.2 days; ice breakup = 7.5 vs. 12.2 days; ice duration = 9.6 vs. 13.7 days), and estimating loss of ice cover from 1980 to 2018 (observed = 8.4 days, LSTM = 7.2 days, GLM = 3.4 days). This demonstrates that ML can estimate historical lake ice formation and breakup dates within a week of observed phenology across heterogeneous lakes and recreate broad scale trends in ice cover phenology with limited long-term ice records.

Minnesota↗