Geology topics

S. Christensen

Publications and source records attributed to S. Christensen.

6 recordsLinked to original sources

USGS2011

Evaluation of MODFLOW-LGR in connection with a synthetic regional-scale model

This work studies costs and benefits of utilizing local‐grid refinement (LGR) as implemented in MODFLOW‐LGR to simulate groundwater flow in a buried tunnel valley interacting with a regional aquifer. Two alternative LGR methods were used: the shared‐node (SN) method and the ghost‐node (GN) method. To conserve flows the SN method requires correction of sources and sinks in cells at the refined/coarse‐grid interface. We found that the optimal correction method is case dependent and difficult to identify in practice. However, the results showed little difference and suggest that identifying the optimal method was of minor importance in our case. The GN method does not require corrections at the models' interface, and it uses a simpler head interpolation scheme than the SN method. The simpler scheme is faster but less accurate so that more iterations may be necessary. However, the GN method solved our flow problem more efficiently than the SN method. The MODFLOW‐LGR results were compared with the results obtained using a globally coarse (GC) grid. The LGR simulations required one to two orders of magnitude longer run times than the GC model. However, the improvements of the numerical resolution around the buried valley substantially increased the accuracy of simulated heads and flows compared with the GC simulation. Accuracy further increased locally around the valley flanks when improving the geological resolution using the refined grid. Finally, comparing MODFLOW‐LGR simulation with a globally refined (GR) grid showed that the refinement proportion of the model should not exceed 10% to 15% in order to secure method efficiency.

Ground Water↗

USGS2006

Bias and uncertainty in regression-calibrated models of groundwater flow in heterogeneous media

Groundwater models need to account for detailed but generally unknown spatial variability (heterogeneity) of the hydrogeologic model inputs. To address this problem we replace the large, m-dimensional stochastic vector ?? that reflects both small and large scales of heterogeneity in the inputs by a lumped or smoothed m-dimensional approximation ????*, where ?? is an interpolation matrix and ??* is a stochastic vector of parameters. Vector ??* has small enough dimension to allow its estimation with the available data. The consequence of the replacement is that model function f(????*) written in terms of the approximate inputs is in error with respect to the same model function written in terms of ??, ??,f(??), which is assumed to be nearly exact. The difference f(??) - f(????*), termed model error, is spatially correlated, generates prediction biases, and causes standard confidence and prediction intervals to be too small. Model error is accounted for in the weighted nonlinear regression methodology developed to estimate ??* and assess model uncertainties by incorporating the second-moment matrix of the model errors into the weight matrix. Techniques developed by statisticians to analyze classical nonlinear regression methods are extended to analyze the revised method. The analysis develops analytical expressions for bias terms reflecting the interaction of model nonlinearity and model error, for correction factors needed to adjust the sizes of confidence and prediction intervals for this interaction, and for correction factors needed to adjust the sizes of confidence and prediction intervals for possible use of a diagonal weight matrix in place of the correct one. If terms expressing the degree of intrinsic nonlinearity for f(??) and f(????*) are small, then most of the biases are small and the correction factors are reduced in magnitude. Biases, correction factors, and confidence and prediction intervals were obtained for a test problem for which model error is large to test robustness of the methodology. Numerical results conform with the theoretical analysis. ?? 2005 Elsevier Ltd. All rights reserved.

Advances in Water Resources↗

USGS2002

Experience gained in testing a theory for modelling groundwater flow in heterogeneous media

Usually, small-scale model error is present in groundwater modelling because the model only represents average system characteristics having the same form as the drift, and small-scale variability is neglected. These errors cause the true errors of a regression model to be correlated. Theory and an example show that the errors also contribute to bias in the estimates of model parameters. This bias originates from model nonlinearity. In spite of this bias, predictions of hydraulic head are nearly unbiased if the model intrinsic nonlinearity is small. Individual confidence and prediction intervals are accurate if the t-statistic is multiplied by a correction factor. The correction factor can be computed from the true error second moment matrix, which can be determined when the stochastic properties of the system characteristics are known.

IAHS-AISH Publication↗

USGS2002

Experiences from the testing of a theory for modelling groundwater flow in heterogeneous media

Usually, small-scale model error is present in groundwater modelling because the model only represents average system characteristics having the same form as the drift and small-scale variability is neglected. These errors cause the true errors of a regression model to be correlated. Theory and an example show that the errors also contribute to bias in the estimates of model parameters. This bias originates from model nonlinearity. In spite of this bias, predictions of hydraulic head are nearly unbiased if the model intrinsic nonlinearity is small. Individual confidence and prediction intervals are accurate if the t-statistic is multiplied by a correction factor. The correction factor can be computed from the true error second moment matrix, which can be determined when the stochastic properties of the system characteristics are known.

Acta Universitatis Carolinae, Geologica↗

USGS1999

Evaluation of confidence intervals for a steady-state leaky aquifer model

The fact that dependent variables of groundwater models are generally nonlinear functions of model parameters is shown to be a potentially significant factor in calculating accurate confidence intervals for both model parameters and functions of the parameters, such as the values of dependent variables calculated by the model. The Lagrangian method of Vecchia and Cooley [Vecchia, A.V. and Cooley, R.L., Water Resources Research, 1987, 23(7), 1237-1250] was used to calculate nonlinear Scheffe-type confidence intervals for the parameters and the simulated heads of a steady-state groundwater flow model covering 450 km2 of a leaky aquifer. The nonlinear confidence intervals are compared to corresponding linear intervals. As suggested by the significant nonlinearity of the regression model, linear confidence intervals are often not accurate. The commonly made assumption that widths of linear confidence intervals always underestimate the actual (nonlinear) widths was not correct. Results show that nonlinear effects can cause the nonlinear intervals to be asymmetric and either larger or smaller than the linear approximations. Prior information on transmissivities helps reduce the size of the confidence intervals, with the most notable effects occurring for the parameters on which there is prior information and for head values in parameter zones for which there is prior information on the parameters.The fact that dependent variables of groundwater models are generally nonlinear functions of model parameters is shown to be a potentially significant factor in calculating accurate confidence intervals for both model parameters and functions of the parameters, such as the values of dependent variables calculated by the model. The Lagrangian method of Vecchia and Cooley was used to calculate nonlinear Scheffe-type confidence intervals for the parameters and the simulated heads of a steady-state groundwater flow model covering 450 km2 of a leaky aquifer. The nonlinear confidence intervals are compared to corresponding linear intervals. As suggested by the significant nonlinearity of the regression model, linear confidence intervals are often not accurate. The commonly made assumption that widths of linear confidence intervals always underestimate the actual (nonlinear) widths was not correct. Results show that nonlinear effects can cause the nonlinear intervals to be asymmetric and either larger or smaller than the linear approximations. Prior information on transmissivities helps reduce the size of the confidence intervals, with the most notable effects occurring for the parameters on which there is prior information and for head values in parameter zones for which there is prior information on the parameters.

Advances in Water Resources↗

USGS1996

Simultaneous confidence intervals for a steady-state leaky aquifer groundwater flow model

Using the optimization method of Vecchia & Cooley (1987), nonlinear Scheffe??-type confidence intervals were calculated tor the parameters and the simulated heads of a steady-state groundwater flow model covering 450 km2 of a leaky aquifer. The nonlinear confidence intervals are compared to corresponding linear intervals. As suggested by the significant nonlinearity of the regression model, linear confidence intervals are often not accurate. The commonly made assumption that widths of linear confidence intervals always underestimate the actual (nonlinear widths was not correct for the head intervals. Results show that nonlinear effects can cause the nonlinear intervals to be offset from, and either larger or smaller than, the linear approximations. Prior information on some transmissivities helps reduce and stabilize the confidence intervals, with the most notable effects occurring for the parameters on which there is prior information and for head values in parameter zones for which there is prior information on the parameters.

IAHS-AISH Publication↗