Geology topics

Richard L. Cooley

Publications and source records attributed to Richard L. Cooley.

13 recordsLinked to original sources

USGS2010

Comment on RamaRao et al. [1995] and LaVenue et al. [1995]

A method for stochastic modeling of groundwater flow systems using a combination of pilot point parameterization and conditional simulation was presented by RamaRao et. al [1995] and LaVenue et. al [1995]. (We will collectively term these two papers RLMM and term the method developed in RLMM the CS method here.) RLMM (pp. 478-479) state that the CS method is intended to provide a frequency distribution of possible alternative spatial transmissivity T distributions that (1) are statistically similar to the observed T distribution, (2) are equally likely given the calibration data, and (3) closely reproduce the measured pressures. The frequency distribution of transmissivities is then used to form frequency distribution of derived functions, such as travel times, which are summarized in the form of uncertainty measures, such as (RLMM, p. 512) "confidence (or tolerance) intervals," on the actual values, which in the case of travels times, are values that could occur sometime in the future (RLMM, p.513). Cooley [2000] analyzes the RLMM method using linearization and bootstrap theory and concludes that that the method can yield accurate uncertainty estimates but only under some limited circumstances. In this comment we use Cooley's analysis to critique the method. We also identify and discuss some statements made my by RLMM about model calibration and their methodology that appear to be misleading. It is unusual to comment on a paper so long after publication. Subsequent work has expanded the method of RLMM, and the method was used advantageously in the testing documented by Zimmerman et. al [1998], so it is clear the method has significant strengths. We go back to the 1995 papers for this comment, however, because they display most clearly the methodological difficulties with which we are concerned.

Water Resources Research↗

USGS2010

Exact Scheffé-type confidence intervals for output from groundwater flow models: 2. Combined use of hydrogeologic information and calibration data

Calibration data (observed values corresponding to model-computed values of dependent variables) are incorporated into a general method of computing exact Scheffé-type confidence intervals analogous to the confidence intervals developed in part 1 (Cooley, this issue) for a function of parameters derived from a groundwater flow model. Parameter uncertainty is specified by a distribution of parameters conditioned on the calibration data. This distribution was obtained as a posterior distribution by applying Bayes' theorem to the hydrogeologically derived prior distribution of parameters from part 1 and a distribution of differences between the calibration data and corresponding model-computed dependent variables. Tests show that the new confidence intervals can be much smaller than the intervals of part 1 because the prior parameter variance-covariance structure is altered so that combinations of parameters that give poor model fit to the data are unlikely. The confidence intervals of part 1 and the new confidence intervals can be effectively employed in a sequential method of model construction whereby new information is used to reduce confidence interval widths at each stage.

Water Resources Research↗

USGS2010

Comment on "Horizontal aquifer movement in a theis-theim confined system" by Donald C. Helm

In a recent paper, Helm [1994] presents an analysis of horizontal aquifer movement induced by groundwater withdrawal from a confined aquifer in which fluid and grains are incompressible. The analysis considers the aquifer in isolation (ignoring overlying and underlying strata) and assumes that the aquifer deforms purely in the horizontal direction (with no vertical movement). Helm's solution for grain displacement is obtained through introduction of a quantity known as bulk flux, q b , defined as q b = n v w + (1 - n )v s where n is porosity, v w is velocity of water, and v s is the velocity of the solid grains. On the basis of the bulk flux concept, Helm develops an explanation for the driving force on the bulk material. It is our view that Helm's analysis is subject to four limitations. First, Helm's assumption of zero vertical displacement is not supported by field observations and could result in over- estimation of radial displacement. Second, in ignoring the role of overlying and underlying strata, Helm's solution does not yield reliable estimates of aquifer deformation. Third, Helm's solution method works only for problems that involve one spatial coordinate (for example, x or r) but does not generally work for problems involving three-dimensional flow and de- formation. Fourth, Helm's explanation of the driving force on the bulk material is faulty for general three-dimensional problems. The purpose of our comment is to discuss these four issues.

Water Resources Research↗

USGS2010

Incorporation of prior information on parameters into nonlinear regression groundwater flow models: 2. Applications

This paper investigates factors influencing the degree of improvement in estimates of parameters of a nonlinear regression groundwater flow model by incorporating prior information of unknown reliability. Consideration of expected behavior of the regression solutions and results of a hypothetical modeling problem lead to several general conclusions. First, if the parameters are properly scaled, linearized expressions for the mean square error (MSE) in parameter estimates of a nonlinear model will often behave very nearly as if the model were linear. Second, by using prior information, the MSE in properly scaled parameters can be reduced greatly over the MSE of ordinary least squares estimates of parameters. Third, plots of estimated MSE and the estimated standard deviation of MSE versus an auxiliary parameter (the ridge parameter) specifying the degree of influence of the prior information on regression results can help determine the potential for improvement of parameter estimates. Fourth, proposed criteria can be used to make appropriate choices for the ridge parameter and another parameter expressing degree of overall bias in the prior information. Results of a case study of Truckee Meadows, Reno-Sparks area, Washoe County, Nevada, conform closely to the results of the hypothetical problem. In the Truckee Meadows case, incorporation of prior information did not greatly change the parameter estimates from those obtained by ordinary least squares. However, the analysis showed that both sets of estimates are more reliable than suggested by the standard errors from ordinary least squares.

Water Resources Research↗

USGS2008

A comparison of several methods of solving nonlinear regression groundwater flow problems

Computational efficiency and computer memory requirements for four methods of minimizing functions were compared for four test nonlinear-regression steady state groundwater flow problems. The fastest methods were the Marquardt and quasi-linearization methods, which required almost identical computer times and numbers of iterations; the next fastest was the quasi-Newton method, and last was the Fletcher-Reeves method, which did not converge in 100 iterations for two of the problems. The fastest method per iteration was the Fletcher-Reeves method, and this was followed closely by the quasi-Newton method. The Marquardt and quasi-linearization methods were slower. For all four methods the speed per iteration was directly related to the number of parameters in the model. However, this effect was much more pronounced for the Marquardt and quasi-linearization methods than for the other two. Hence the quasi-Newton (and perhaps Fletcher-Reeves) method might be more efficient than either the Marquardt or quasi-linearization methods if the number of parameters in a particular model were large, although this remains to be proven. The Marquardt method required somewhat less central memory than the quasi-linearization metilod for three of the four problems. For all four problems the quasi-Newton method required roughly two thirds to three quarters of the memory required by the Marquardt method, and the Fletcher-Reeves method required slightly less memory than the quasi-Newton method. Memory requirements were not excessive for any of the four methods.

Water Resources Research↗

USGS2007

Calculation of nonlinear confidence and prediction intervals for ground-water flow models

A method is derived to efficiently compute nonlinear confidence and prediction intervals on any function of parameters derived as output from a mathematical model of a physical system. The method is applied to the problem of obtaining confidence and prediction intervals for manually-calibrated ground-water flow models. To obtain confidence and prediction intervals resulting from uncertainties in parameters, the calibrated model and information on extreme ranges and ordering of the model parameters within one or more independent groups are required. If random errors in the dependent variable are present in addition to uncertainties in parameters, then calculation of prediction intervals also requires information on the extreme range of error expected. A simple Monte Carlo method is used to compute the quantiles necessary to establish probability levels for the confidence and prediction intervals. Application of the method to a hypothetical example showed that inclusion of random errors in the dependent variable in addition to uncertainties in parameters can considerably widen the prediction intervals.

Journal of the American Water Resources Associatio↗

USGS2006

User guide to the UNC process and three utility programs for computation of nonlinear confidence and prediction intervals using MODFLOW-2000

This report introduces and documents the Uncertainty (UNC) Process, a new Process in MODFLOW-2000 that calculates uncertainty measures for model parameters and for predictions produced by the model. Uncertainty measures can be computed by various methods, but when regression is applied to calibrate a model (for example when using the Parameter-Estimation Process of MODFLOW-2000) it is advantageous to also use regression-based methods to quantify uncertainty. For this reason the UNC Process computes (1) confidence intervals for parameters of the Parameter-Estimation Process and (2) confidence and prediction intervals for most types of functions that can be computed by a MODFLOW-2000 model calibrated by the Parameter-Estimation Process. The types of functions for which the Process works include hydraulic heads, hydraulic head differences, head-dependent flows computed by the head-dependent flow packages for drains (DRN6), rivers (RIV6), general-head boundaries (GHB6), streams (STR6), drain-return cells (DRT1), and constant-head boundaries (CHD), and for differences between flows computed by any of the mentioned flow packages. The UNC Process does not allow computation of intervals for the difference between flows computed by two different flow packages. The report also documents three programs, RESAN2-2k, BEALE2-2k, and CORFAC-2k, which are valuable for the evaluation of results from the Parameter-Estimation Process and for the preparation of input values for the UNC Process. RESAN2-2k and BEALE2-2k are significant updates of the residual analysis and modified Beale's measure programs first published by Cooley and Naff (1990) and later modified for use with MODFLOWP (Hill, 1994) and MODFLOW-2000 (Hill and others, 2000). CORFAC-2k is a new program that computes correction factors to be used by UNC.

Techniques and Methods↗

USGS2004

A theory for modeling ground-water flow in heterogeneous media

Construction of a ground-water model for a field area is not a straightforward process. Data are virtually never complete or detailed enough to allow substitution into the model equations and direct computation of the results of interest. Formal model calibration through optimization, statistical, and geostatistical methods is being applied to an increasing extent to deal with this problem and provide for quantitative evaluation and uncertainty analysis of the model. However, these approaches are hampered by two pervasive problems: 1) nonlinearity of the solution of the model equations with respect to some of the model (or hydrogeologic) input variables (termed in this report system characteristics) and 2) detailed and generally unknown spatial variability (heterogeneity) of some of the system characteristics such as log hydraulic conductivity, specific storage, recharge and discharge, and boundary conditions. A theory is developed in this report to address these problems. The theory allows construction and analysis of a ground-water model of flow (and, by extension, transport) in heterogeneous media using a small number of lumped or smoothed system characteristics (termed parameters). The theory fully addresses both nonlinearity and heterogeneity in such a way that the parameters are not assumed to be effective values. The ground-water flow system is assumed to be adequately characterized by a set of spatially and temporally distributed discrete values, ?, of the system characteristics. This set contains both small-scale variability that cannot be described in a model and large-scale variability that can. The spatial and temporal variability in ? are accounted for by imagining ? to be generated by a stochastic process wherein ? is normally distributed, although normality is not essential. Because ? has too large a dimension to be estimated using the data normally available, for modeling purposes ? is replaced by a smoothed or lumped approximation y?. (where y is a spatial and temporal interpolation matrix). Set y?. has the same form as the expected value of ?, y 'line' ? , where 'line' ? is the set of drift parameters of the stochastic process; ?. is a best-fit vector to ?. A model function f(?), such as a computed hydraulic head or flux, is assumed to accurately represent an actual field quantity, but the same function written using y?., f(y?.), contains error from lumping or smoothing of ? using y?.. Thus, the replacement of ? by y?. yields nonzero mean model errors of the form E(f(?)-f(y?.)) throughout the model and covariances between model errors at points throughout the model. These nonzero means and covariances are evaluated through third and fifth-order accuracy, respectively, using Taylor series expansions. They can have a significant effect on construction and interpretation of a model that is calibrated by estimating ?.. Vector ?.. is estimated as 'hat' ? using weighted nonlinear least squares techniques to fit a set of model functions f(y'hat' ?) to a. corresponding set of observations of f(?), Y. These observations are assumed to be corrupted by zero-mean, normally distributed observation errors, although, as for ?, normality is not essential. An analytical approximation of the nonlinear least squares solution is obtained using Taylor series expansions and perturbation techniques that assume model and observation errors to be small. This solution is used to evaluate biases and other results to second-order accuracy in the errors. The correct weight matrix to use in the analysis is shown to be the inverse of the second-moment matrix E(Y-f(y?.))(Y-f(y?.))', but the weight matrix is assumed to be arbitrary in most developments. The best diagonal approximation is the inverse of the matrix of diagonal elements of E(Y-f(y?.))(Y-f(y?.))', and a method of estimating this diagonal matrix when it is unknown is developed using a special objective function to compute 'hat' ?. When considered to be an estimate of f

Professional Paper↗

USGS2000

An analysis of the pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields

An analysis of the pilot point method for automated calibration of an ensemble of conditionally simulated transmissivity fields was conducted on the basis of the simplifying assumption that the flow model is a linear function of log transmissivity. The analysis shows that the pilot point and conditional simulation method of model calibration and uncertainty analysis can produce accurate uncertainty measures if it can be assumed that errors of unknown origin in the differences between observed and model‐computed water pressures are small. When this assumption is not met, the method could yield significant errors from overparameterization and the neglect of potential sources of model inaccuracy. The conditional simulation part of the method is also shown to be a variant of the percentile bootstrap method, so that when applied to a nonlinear model, the method is subject to bootstrap errors. These sources of error must be considered when using the method.

Water Resources Research↗

USGS1999

Evaluation of prediction intervals for expressing uncertainties in groundwater flow model predictions

We tested the accuracy of 95% individual prediction intervals for hydraulic heads, streamflow gains, and effective transmissivities computed by groundwater models of two Danish aquifers. To compute the intervals, we assumed that each predicted value can be written as the sum of a computed dependent variable and a random error. Testing was accomplished by using a cross‐validation method and by using new field measurements of hydraulic heads and transmissivities that were not used to develop or calibrate the models. The tested null hypotheses are that the coverage probability of the prediction intervals is not significantly smaller than the assumed probability (95%) and that each tail probability is not significantly different from the assumed probability (2.5%). In all cases tested, these hypotheses were accepted at the 5% level of significance. We therefore conclude that for the groundwater models of two real aquifers the individual prediction intervals appear to be accurate.

Water Resources Research↗

USGS1999

Practical Scheffe‐type credibility intervals for variables of a groundwater model

Simultaneous Scheffé‐type credibility intervals (the Bayesian version of confidence intervals) for variables of a groundwater flow model calibrated using a Bayesian maximum a posteriori procedure were derived by Cooley [1993b]. It was assumed that variances reflecting the expected differences between observed and model‐computed quantities used to calibrate the model are known, whereas they would often be unknown for an actual model. In this study the variances are regarded as unknown, and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance. The credibility intervals are calculated from the posterior distribution, which was developed by considering each group variance to be a random variable about which nothing is known a priori, then eliminating it by integration. Numerical experiments using two test problems illustrate some characteristics of the credibility intervals. Nonlinearity of the statistical model greatly affected some of the credibility intervals, indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems. The parameter characterizing the probability level for the credibility intervals was, however, accurately computed using a linear model approximation, as compared with values calculated using second‐order and fully nonlinear formulations. This allows the credibility intervals to be computed very efficiently.

Water Resources Research↗

USGS1992

A modular finite-element model (MODFE) for areal and axisymmetric ground-water-flow problems, Part 2: Derivation of finite-element equations and comparisons with analytical solutions

MODFE, a modular finite-element model for simulating steady- or unsteady-state, area1 or axisymmetric flow of ground water in a heterogeneous anisotropic aquifer is documented in a three-part series of reports. In this report, part 2, the finite-element equations are derived by minimizing a functional of the difference between the true and approximate hydraulic head, which produces equations that are equivalent to those obtained by either classical variational or Galerkin techniques. Spatial finite elements are triangular with linear basis functions, and temporal finite elements are one dimensional with linear basis functions. Physical processes that can be represented by the model include (1) confined flow, unconfined flow (using the Dupuit approximation), or a combination of both; (2) leakage through either rigid or elastic confining units; (3) specified recharge or discharge at points, along lines, or areally; (4) flow across specified-flow, specified-head, or head-dependent boundaries; (5) decrease of aquifer thickness to zero under extreme water-table decline and increase of aquifer thickness from zero as the water table rises; and (6) head-dependent fluxes from springs, drainage wells, leakage across riverbeds or confining units combined with aquifer dewatering, and evapotranspiration. The matrix equations produced by the finite-element method are solved by the direct symmetric-Doolittle method or the iterative modified incomplete-Cholesky conjugate-gradient method. The direct method can be efficient for small- to medium-sized problems (less than about 500 nodes), and the iterative method is generally more efficient for larger-sized problems. Comparison of finite-element solutions with analytical solutions for five example problems demonstrates that the finite-element model can yield accurate solutions to ground-water flow problems.

Techniques of Water-Resources Investigations↗

USGS1985

CONCEPTUAL FRAMEWORK FOR REGRESSION MODELING OF GROUND-WATER FLOW.

The author examines the uses of ground-water flow models and which classes of use require treatment of stochastic components. He then compares traditional and stochastic procedures for modeling actual (as distinguished from hypothetical) systems. Finally, he examines the conceptual basis and characteristics of the regression approach to modeling ground-water flow.

Conference Paper↗