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Results for “Inverse Problems”

Search indexed USGS publications on groundwater, aquifers, geologic maps, mineral resources and earthquakes. Explore source records by subject and place.

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USGS2009

Development of an objective‐oriented groundwater model for conjunctive‐use planning of surface water and groundwater

In this paper we construct an objective‐oriented model for conjunctive‐use planning of surface water and groundwater for the Warren groundwater basin in southern California. The goal of conjunctive‐use planning is to decrease high‐nitrate concentration while maintaining groundwater levels at desired elevations and meeting water demand. We formulate a management problem that minimizes the total cost over the proper choices of the time‐varying pumping and recharge rates at prespecified wells and surface ponds. To make the solution of the management problem reliable, we must have an accurate simulation model to predict groundwater level and nitrate concentration distributions under different management alternatives. The objective‐oriented model construction approach seeks a representative parameter that has the simplest structure and requires the minimum data for identification but can produce reliable results for a given model application. With the data from the Warren groundwater basin, we show how to incorporate management objectives into the construction of an objective‐oriented model, identify the parameter structure and its corresponding parameter values, solve the generalized inverse problem effectively by finding the worst‐case parameter (WCP), evaluate the sufficiency of existing data, and find a robust experiment design when the existing data are insufficient. Results of this case study show that the presented methodology is useful in practice because (1) data sufficiency can be judged before conducting actual field experiments and (2) the identified WCP drastically reduces the computation time for constructing an objective‐oriented model.

California↗

USGS2024

Seismic tomography 2023

Seismic tomography is the most abundant source of information about the internal structure of the Earth at scales ranging from a few meters to thousands of kilometers. It constrains the properties of active volcanoes, earthquake fault zones, deep reservoirs and storage sites, glaciers and ice sheets, or the entire globe. It contributes to outstanding societal problems related to natural hazards, resource exploration, underground storage, and many more. The recent advances in seismic tomography are being translated to nondestructive testing, medical ultrasound, and helioseismology. Nearly 50 yr after its first successful applications, this article offers a snapshot of modern seismic tomography. Focused on major challenges and particularly promising research directions, it is intended to guide both Earth science professionals and early‐career scientists. The individual contributions by the coauthors provide diverse perspectives on topics that may at first seem disconnected but are closely tied together by a few coherent threads: multiparameter inversion for properties related to dynamic processes, data quality, and geographic coverage, uncertainty quantification that is useful for geologic interpretation, new formulations of tomographic inverse problems that address concrete geologic questions more directly, and the presentation and quantitative comparison of tomographic models. It remains to be seen which of these problems will be considered solved, solved to some extent, or practically unsolvable over the next decade.

Bulletin of the Seismological Society of America↗

USGS2013

Uniform California earthquake rupture forecast, version 3 (UCERF3): the time-independent model

In this report we present the time-independent component of the Uniform California Earthquake Rupture Forecast, Version 3 (UCERF3), which provides authoritative estimates of the magnitude, location, and time-averaged frequency of potentially damaging earthquakes in California. The primary achievements have been to relax fault segmentation assumptions and to include multifault ruptures, both limitations of the previous model (UCERF2). The rates of all earthquakes are solved for simultaneously, and from a broader range of data, using a system-level "grand inversion" that is both conceptually simple and extensible. The inverse problem is large and underdetermined, so a range of models is sampled using an efficient simulated annealing algorithm. The approach is more derivative than prescriptive (for example, magnitude-frequency distributions are no longer assumed), so new analysis tools were developed for exploring solutions. Epistemic uncertainties were also accounted for using 1,440 alternative logic tree branches, necessitating access to supercomputers. The most influential uncertainties include alternative deformation models (fault slip rates), a new smoothed seismicity algorithm, alternative values for the total rate of M≥5 events, and different scaling relationships, virtually all of which are new. As a notable first, three deformation models are based on kinematically consistent inversions of geodetic and geologic data, also providing slip-rate constraints on faults previously excluded because of lack of geologic data. The grand inversion constitutes a system-level framework for testing hypotheses and balancing the influence of different experts. For example, we demonstrate serious challenges with the Gutenberg-Richter hypothesis for individual faults. UCERF3 is still an approximation of the system, however, and the range of models is limited (for example, constrained to stay close to UCERF2). Nevertheless, UCERF3 removes the apparent UCERF2 overprediction of M6.5–7 earthquake rates and also includes types of multifault ruptures seen in nature. Although UCERF3 fits the data better than UCERF2 overall, there may be areas that warrant further site-specific investigation. Supporting products may be of general interest, and we list key assumptions and avenues for future model improvements.

California↗

USGS2017

A probabilistic approach to remote compositional analysis of planetary surfaces

Reflected light from planetary surfaces provides information, including mineral/ice compositions and grain sizes, by study of albedo and absorption features as a function of wavelength. However, deconvolving the compositional signal in spectra is complicated by the nonuniqueness of the inverse problem. Trade-offs between mineral abundances and grain sizes in setting reflectance, instrument noise, and systematic errors in the forward model are potential sources of uncertainty, which are often unquantified. Here we adopt a Bayesian implementation of the Hapke model to determine sets of acceptable-fit mineral assemblages, as opposed to single best fit solutions. We quantify errors and uncertainties in mineral abundances and grain sizes that arise from instrument noise, compositional end members, optical constants, and systematic forward model errors for two suites of ternary mixtures (olivine-enstatite-anorthite and olivine-nontronite-basaltic glass) in a series of six experiments in the visible-shortwave infrared (VSWIR) wavelength range. We show that grain sizes are generally poorly constrained from VSWIR spectroscopy. Abundance and grain size trade-offs lead to typical abundance errors of ≤1 wt % (occasionally up to ~5 wt %), while ~3% noise in the data increases errors by up to ~2 wt %. Systematic errors further increase inaccuracies by a factor of 4. Finally, phases with low spectral contrast or inaccurate optical constants can further increase errors. Overall, typical errors in abundance are <10%, but sometimes significantly increase for specific mixtures, prone to abundance/grain-size trade-offs that lead to high unmixing uncertainties. These results highlight the need for probabilistic approaches to remote determination of planetary surface composition.

Journal of Geophysical Research E: Planets↗

USGS2019

Estimating sand concentrations using ADCP‐based acoustic inversion in a large fluvial system characterized by bi‐modal suspended‐sediment distributions

Quantifying sediment flux within rivers is a challenge for many disciplines due, mainly, to difficulties inherent to traditional sediment sampling methods. These methods are operationally complex, high cost, and high risk. Additionally, the resulting data provide a low spatial and temporal resolution estimate of the total sediment flux, which has impeded advances in the understanding of the hydro‐geomorphic characteristics of rivers. Acoustic technologies have been recognized as a leading tool for increasing the resolution of sediment data by relating their echo intensity level measurements to suspended sediment. Further effort is required to robustly test and develop these techniques across a wide range of conditions found in natural river systems. This article aims to evaluate the application of acoustic inversion techniques using commercially available, down‐looking acoustic Doppler current profilers (ADCPs) in quantifying suspended sediment in a large sand bed river with varying bi‐modal particle size distributions, wash load and suspended‐sand ratios, and water stages. To achieve this objective, suspended sediment was physically sampled along the Paraná River, Argentina, under various hydro‐sedimentological regimes. Two ADCPs emitting different sound frequencies were used to simultaneously profile echo intensity level within the water column. Using the sonar equation, calibrations were determined between suspended‐sand concentrations and acoustic backscatter to solve the inverse problem. The study also analyzed the roles played by each term of the sonar equation, such as ADCP frequency, power supply, instrument constants, and particle size distributions typically found in sand bed rivers, on sediment attenuation and backscatter. Calibrations were successfully developed between corrected backscatter and suspended‐sand concentrations for all sites and ADCP frequencies, resulting in mean suspended‐sand concentration estimates within about 40% of the mean sampled concentrations. Noise values, calculated using the sonar equation and sediment sample characteristics, were fairly constant across evaluations, suggesting that they could be applied to other sand bed rivers.

Earth Surface Processes and Landforms↗

USGS2017

Heat as a groundwater tracer in shallow and deep heterogeneous media: Analytical solution, spreadsheet tool, and field applications

Groundwater flow advects heat, and thus, the deviation of subsurface temperatures from an expected conduction‐dominated regime can be analysed to estimate vertical water fluxes. A number of analytical approaches have been proposed for using heat as a groundwater tracer, and these have typically assumed a homogeneous medium. However, heterogeneous thermal properties are ubiquitous in subsurface environments, both at the scale of geologic strata and at finer scales in streambeds. Herein, we apply the analytical solution of Shan and Bodvarsson (2004), developed for estimating vertical water fluxes in layered systems, in 2 new environments distinct from previous vadose zone applications. The utility of the solution for studying groundwater‐surface water exchange is demonstrated using temperature data collected from an upwelling streambed with sediment layers, and a simple sensitivity analysis using these data indicates the solution is relatively robust. Also, a deeper temperature profile recorded in a borehole in South Australia is analysed to estimate deeper water fluxes. The analytical solution is able to match observed thermal gradients, including the change in slope at sediment interfaces. Results indicate that not accounting for layering can yield errors in the magnitude and even direction of the inferred Darcy fluxes. A simple automated spreadsheet tool (Flux‐LM) is presented to allow users to input temperature and layer data and solve the inverse problem to estimate groundwater flux rates from shallow (e.g., <1 m) or deep (e.g., up to 100 m) profiles. The solution is not transient, and thus, it should be cautiously applied where diel signals propagate or in deeper zones where multi‐decadal surface signals have disturbed subsurface thermal regimes.

Hydrological Processes↗

USGS1985

Spatial analysis of extension fracture systems: A process modeling approach

Little consensus exists on how best to analyze natural fracture spacings and their sequences. Field measurements and analyses published in geotechnical literature imply fracture processes radically different from those assumed by theoretical structural geologists. The approach adopted in this paper recognizes that disruption of rock layers by layer-parallel extension results in two spacing distributions, one representing layer-fragment lengths and another separation distances between fragments. These two distributions and their sequences reflect mechanics and history of fracture and separation. Such distributions and sequences, represented by a 2 ?? n matrix of lengthsL, can be analyzed using a method that is history sensitive and which yields also a scalar estimate of bulk extension, e (L). The method is illustrated by a series of Monte Carlo experiments representing a variety of fracture-and-separation processes, each with distinct implications for extension history. Resulting distributions of e (L)are process-specific, suggesting that the inverse problem of deducing fracture-and-separation history from final structure may be tractable. ?? 1985 Plenum Publishing Corporation.

Journal of the International Association for Mathe↗

USGS1992

Simultaneous parameter estimation and contaminant source characterization for coupled groundwater flow and contaminant transport modelling

Parameter estimation and contaminant source characterization are key steps in the development of a coupled groundwater flow and contaminant transport simulation model. Here a methodologyfor simultaneous model parameter estimation and source characterization is presented. The parameter estimation/source characterization inverse model combines groundwater flow and contaminant transport simulation with non-linear maximum likelihood estimation to determine optimal estimates of the unknown model parameters and source characteristics based on measurements of hydraulic head and contaminant concentration. First-order uncertainty analysis provides a means for assessing the reliability of the maximum likelihood estimates and evaluating the accuracy and reliability of the flow and transport model predictions. A series of hypothetical examples is presented to demonstrate the ability of the inverse model to solve the combined parameter estimation/source characterization inverse problem. Hydraulic conductivities, effective porosity, longitudinal and transverse dispersivities, boundary flux, and contaminant flux at the source are estimated for a two-dimensional groundwater system. In addition, characterization of the history of contaminant disposal or location of the contaminant source is demonstrated. Finally, the problem of estimating the statistical parameters that describe the errors associated with the head and concentration data is addressed. A stage-wise estimation procedure is used to jointly estimate these statistical parameters along with the unknown model parameters and source characteristics.

Journal of Hydrology↗

USGS1984

A simple model of a phreatic surface through an earth dam

A simple numerical model for estimating a phreatic surface in an earthen dam is presented. The numerical approach is based upon the Complex Variable Boundary Element Method (CVBEM). By expanding the CVBEM approximation geometric functions into a first order Taylor series, the unknown phreatic surface location geometrics can be approximated without iteration by solving a single matrix system. The developed technique provides for the numerical solution of the inverse problem of locating the phreatic surface coordinates. A comparison of results produced from this simple approach to results produced from a finite element analog and an iterative CVBEM analog for an example problem is presented.

Advances in Water Resources↗

USGS2016

A python framework for environmental model uncertainty analysis

We have developed pyEMU, a python framework for Environmental Modeling Uncertainty analyses, open-source tool that is non-intrusive, easy-to-use, computationally efficient, and scalable to highly-parameterized inverse problems. The framework implements several types of linear (first-order, second-moment (FOSM)) and non-linear uncertainty analyses. The FOSM-based analyses can also be completed prior to parameter estimation to help inform important modeling decisions, such as parameterization and objective function formulation. Complete workflows for several types of FOSM-based and non-linear analyses are documented in example notebooks implemented using Jupyter that are available in the online pyEMU repository. Example workflows include basic parameter and forecast analyses, data worth analyses, and error-variance analyses, as well as usage of parameter ensemble generation and management capabilities. These workflows document the necessary steps and provides insights into the results, with the goal of educating users not only in how to apply pyEMU, but also in the underlying theory of applied uncertainty quantification.

Environmental Modelling and Software↗

USGS2016

Bayesian estimation of magma supply, storage, and eruption rates using a multiphysical volcano model: Kīlauea Volcano, 2000–2012

Estimating rates of magma supply to the world's volcanoes remains one of the most fundamental aims of volcanology. Yet, supply rates can be difficult to estimate even at well-monitored volcanoes, in part because observations are noisy and are usually considered independently rather than as part of a holistic system. In this work we demonstrate a technique for probabilistically estimating time-variable rates of magma supply to a volcano through probabilistic constraint on storage and eruption rates. This approach utilizes Bayesian joint inversion of diverse datasets using predictions from a multiphysical volcano model, and independent prior information derived from previous geophysical, geochemical, and geological studies. The solution to the inverse problem takes the form of a probability density function which takes into account uncertainties in observations and prior information, and which we sample using a Markov chain Monte Carlo algorithm. Applying the technique to Kīlauea Volcano, we develop a model which relates magma flow rates with deformation of the volcano's surface, sulfur dioxide emission rates, lava flow field volumes, and composition of the volcano's basaltic magma. This model accounts for effects and processes mostly neglected in previous supply rate estimates at Kīlauea, including magma compressibility, loss of sulfur to the hydrothermal system, and potential magma storage in the volcano's deep rift zones. We jointly invert data and prior information to estimate rates of supply, storage, and eruption during three recent quasi-steady-state periods at the volcano. Results shed new light on the time-variability of magma supply to Kīlauea, which we find to have increased by 35–100% between 2001 and 2006 (from 0.11–0.17 to 0.18–0.28 km 3 /yr), before subsequently decreasing to 0.08–0.12 km 3 /yr by 2012. Changes in supply rate directly impact hazard at the volcano, and were largely responsible for an increase in eruption rate of 60–150% between 2001 and 2006, and subsequent decline by as much as 60% by 2012. We also demonstrate the occurrence of temporal changes in the proportion of Kīlauea's magma supply that is stored versus erupted, with the supply “surge” in 2006 associated with increased accumulation of magma at the summit. Finally, we are able to place some constraints on sulfur concentrations in Kīlauea magma and the scrubbing of sulfur by the volcano's hydrothermal system. Multiphysical, Bayesian constraint on magma flow rates may be used to monitor evolving volcanic hazard not just at Kīlauea but at other volcanoes around the world.

Hawai'i↗

USGS2021

New insights on faulting and intrusion processes during the June 2007, East Rift Zone eruption of Kilauea volcano, Hawai'i

The East Rift Zone (ERZ) of Kīlauea Volcano, Hawai'i, represents one of the most volcanically active regions in the world. The 2007 Father's Day (FD) dike intrusion, eruption, and accompanying slow-slip event (SSE) has been previously modeled using geodetic data to constrain the geometry of the intrusion and the timing and magnitude of the SSE. Here, we perform inversions of three interferometric synthetic aperture radar (InSAR) datasets and a new intensity offset tracking dataset to assess the effect of integrating intensity cross-correlation offsets into inversion problems and explore additional potential models for the intrusion geometry of the FD event based on this additional data. The overall lowest misfit single Okada model for all datasets opens 2.3 m, strikes 73 degrees while dipping sub-vertically at 83 degrees, and extends approximately 2.9 km to the ENE and 2.4 km downdip. The differences are minor between complex en-echelon distributed Okada and decollement model of (Montgomery-Brown et al., 2010) or 3D-MBEM breaching models including multiple surface breaches and free-slipping decollement movement. Finally, we examine the static Coulomb stress changes for the proposed decollement fault created by our preferred model and a representative model of deep rift opening and find that deep rift zones dilation, not shallow ERZ intrusions, are likely modulating slip on the decollement.

Hawaii↗

USGS2007

A simple model for calculating tsunami flow speed from tsunami deposits

This paper presents a simple model for tsunami sedimentation that can be applied to calculate tsunami flow speed from the thickness and grain size of a tsunami deposit (the inverse problem). For sandy tsunami deposits where grain size and thickness vary gradually in the direction of transport, tsunami sediment transport is modeled as a steady, spatially uniform process. The amount of sediment in suspension is assumed to be in equilibrium with the steady portion of the long period, slowing varying uprush portion of the tsunami. Spatial flow deceleration is assumed to be small and not to contribute significantly to the tsunami deposit. Tsunami deposits are formed from sediment settling from the water column when flow speeds on land go to zero everywhere at the time of maximum tsunami inundation. There is little erosion of the deposit by return flow because it is a slow flow and is concentrated in topographic lows. Variations in grain size of the deposit are found to have more effect on calculated tsunami flow speed than deposit thickness. The model is tested using field data collected at Arop, Papua New Guinea soon after the 1998 tsunami. Speed estimates of 14 m/s at 200 m inland from the shoreline compare favorably with those from a 1-D inundation model and from application of Bernoulli's principle to water levels on buildings left standing after the tsunami. As evidence that the model is applicable to some sandy tsunami deposits, the model reproduces the observed normal grading and vertical variation in sorting and skewness of a deposit formed by the 1998 tsunami.

Sedimentary Geology↗

USGS2001

Resolution analysis of finite fault source inversion using one- and three-dimensional Green's functions 1. Strong motions

We develop a methodology to perform finite fault source inversions from strong motion data using Green's functions (GFs) calculated for a three-dimensional (3-D) velocity structure. The 3-D GFs are calculated numerically by inserting body forces at each of the strong motion sites and then recording the resulting strains along the target fault surface. Using reciprocity, these GFs can be recombined to represent the ground motion at each site for any (heterogeneous) slip distribution on the fault. The reciprocal formulation significantly reduces the required number of 3-D finite difference computations to at most 3NS, where NS is the number of strong motion sites used in the inversion. Using controlled numerical resolution tests, we have examined the relative importance of accurate GFs for finite fault source inversions which rely on near-source ground motions. These experiments use both 1-D and 3-D GFs in inversions for hypothetical rupture models in order (1) to analyze the ability of the 3-D methodology to resolve trade-offs between complex source phenomena and 3-D path effects, (2) to address the sensitivity of the inversion results to uncertainties in the 3-D velocity structure, and (3) to test the adequacy of the 1-D GF method when propagation effects are known to be three-dimensional. We find that given "data" from a prescribed 3-D Earth structure, the use of well-calibrated 3-D GFs in the inversion provides very good resolution of the assumed slip distribution, thus adequately separating source and 3-D propagation effects. In contrast, using a set of inexact 3-D GFs or a set of hybrid 1-D GFs allows only partial recovery of the slip distribution. These findings suggest that in regions of complex geology the use of well-calibrated 3-D GFs has the potential for increased resolution of the rupture process relative to 1-D GFs. However, realizing this full potential requires that the 3-D velocity model and associated GFs should be carefully validated against the true 3-D Earth structure before performing the inverse problem with actual data.

Journal of Geophysical Research B: Solid Earth↗

USGS2007

A field assessment of the value of steady shape hydraulic tomography for characterization of aquifer heterogeneities

Hydraulic tomography is a promising approach for obtaining information on variations in hydraulic conductivity on the scale of relevance for contaminant transport investigations. This approach involves performing a series of pumping tests in a format similar to tomography. We present a field‐scale assessment of hydraulic tomography in a porous aquifer, with an emphasis on the steady shape analysis methodology. The hydraulic conductivity (K) estimates from steady shape and transient analyses of the tomographic data compare well with those from a tracer test and direct‐push permeameter tests, providing a field validation of the method. Zonations based on equal‐thickness layers and cross‐hole radar surveys are used to regularize the inverse problem. The results indicate that the radar surveys provide some useful information regarding the geometry of the K field. The steady shape analysis provides results similar to the transient analysis at a fraction of the computational burden. This study clearly demonstrates the advantages of hydraulic tomography over conventional pumping tests, which provide only large‐scale averages, and small‐scale hydraulic tests (e.g., slug tests), which cannot assess strata connectivity and may fail to sample the most important pathways or barriers to flow.

Water Resources Research↗

USGS2008

An interactive Bayesian geostatistical inverse protocol for hydraulic tomography

Hydraulic tomography is a powerful technique for characterizing heterogeneous hydrogeologic parameters. An explicit trade-off between characterization based on measurement misfit and subjective characterization using prior information is presented. We apply a Bayesian geostatistical inverse approach that is well suited to accommodate a flexible model with the level of complexity driven by the data and explicitly considering uncertainty. Prior information is incorporated through the selection of a parameter covariance model characterizing continuity and providing stability. Often, discontinuities in the parameter field, typically caused by geologic contacts between contrasting lithologic units, necessitate subdivision into zones across which there is no correlation among hydraulic parameters. We propose an interactive protocol in which zonation candidates are implied from the data and are evaluated using cross validation and expert knowledge. Uncertainty introduced by limited knowledge of dynamic regional conditions is mitigated by using drawdown rather than native head values. An adjoint state formulation of MODFLOW-2000 is used to calculate sensitivities which are used both for the solution to the inverse problem and to guide protocol decisions. The protocol is tested using synthetic two-dimensional steady state examples in which the wells are located at the edge of the region of interest.

Water Resources Research↗

USGS2023

Can hydrological models benefit from using global soil moisture, evapotranspiration, and runoff products as calibration targets?

Hydrological models are usually calibrated to in-situ streamflow observations with reasonably long and uninterrupted records. This is challenging for poorly gage or ungaged basins where such information is not available. Even for gaged basins, the single-objective calibration to gaged streamflow cannot guarantee reliable forecasts because, as has been documented elsewhere, the inverse problem is mathematically ill-posed. Therefore, the inclusion of other observations, and the reproduction of other hydrological variables beyond streamflow, become critical components of accurate hydrological forecasting. In this study, six single- and multi-objective model calibration schemes based on different combinations of gaged streamflow, global-scale gridded soil moisture, actual evapotranspiration (ET), and runoff products are used for the calibration of a process-based hydrological model for 20 catchments located within the Lake Michigan watershed, of the Laurentian Great Lakes. Results show that the addition of gridded soil moisture to gaged streamflow in model calibration improves the ET simulation performance for most of the catchments, leading to the overall best-performing models. The monthly streamflow simulation performance for the experiments using gridded runoff products to inform the model is outperformed by those using the gaged streamflow, but the discrepancy is mitigated with increasing catchment scale. A new visualization method that effectively synthesizes model performance for the simulations of streamflow, soil moisture, and ET was also proposed. Based on the method, it is revealed that the streamflow simulation performance is relatively weak for baseflow-dominated catchments; overall, the 20 catchment models simulate streamflow and ET better than soil moisture.

Lake Michigan↗

USGS2008

Evaluation of unsaturated zone air permeability through pneumatic tests

Predicting the steady state distribution of air pressure in the unsaturated zone resulting from a pneumatic test provides a method for determining air-phase permeability. This technique is analogous to the inverse problem of well hydraulics; however, air flow is more complicated than ground water flow because of air compressibility, the Klinkenberg effect, variations in air density and viscosity that result from temperature fluctuations in the unsaturated zone and the possibility of inducing water movement during the pneumatic test. An analysis of these complicating factors reveals that, when induced water movement can be neglected, a linear version of the airflow equation can provide an appropriate approximation for the purpose of determining air-phase permeability. Two analytical solutions for steady state, two-dimensional, axisymmetric airflow to a single well partially screened in the unsaturated zone are developed. One solution applies where there is a stratum of relatively low air permeability, separating the stratum in which the well is completed, from the atmosphere. The other solution applies where there is no separating stratum between the domain and atmosphere. In both situations the water table forms the lower horizontal boundary. Applications of both solutions to determine air permeability from data collected during pneumatic tests are presented.

Water Resources Research↗