A statistical approach to the inverse problem of aquifer hydrology: 1. Theory

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Section: Parameterization and Inverse Approachmentioning

confidence: 99%

Three‐dimensional numerical inversion of pneumatic cross‐hole tests in unsaturated fractured tuff: 1. Methodology and borehole effects

Vesselinov¹,

Neuman²,

Illman³

2001

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“…One, however, encounters another problem in the Bayesian procedure: the observation data (i.e., the objective information) and the prior information (i.e., the subjective information) are fundamentally incommensurate; thus a simple superposition of these two kinds of information most of the time does not lead to an appropriate solution [Neuman and Yakowitz, 1979;Cooley, 1982]. Neuman and Yakowitz [1979] proposed multiplying an adjusting scaler (notation d 2 is used for this scaler throughout this paper) by the prior covariance matrix when imposing it over the observation data, which they termed the Extended Bayesian Method (EBM).…”

Section: Matching Of the Subjective And The Objective Informationmentioning

confidence: 99%

Matching objective and subjective information in groundwater inverse analysis by Akaike's Bayesian information criterion

Honjo¹,

Kashiwagi²

1999

Abstract. In order to overcome the illposedness of groundwater inverse analysis it is inevitable to introduce prior information of some form and thus Bayesian statistics. One of the essential problems in Bayesian inverse formulation is the optimum matching between the objective information (i.e., the observation) and the subjective information (i.e., the prior information). In this study, Akaike's Bayesian Information Criterion (ABIC) is introduced to overcome this problem. ABIC is also effective in the model identification problem, and this aspect is also emphasized. The effectiveness of the method is illustrated by analyses on an actual aquifer system. Both steady and transient state analyses are carried out. The paper also provides the background of ABIC in some detail. The method also recognizes the dour fact that whether the prior statistics are determined from actual field data or borrowed from other aquifers, they will seldom be known with sufficient accuracy to justify imposing them on the inverse solution without further modification; for this reason we do not require that the covariance matrices of the random vectors entering into the model be specified exactly, but to a scaler of multiplication. Model Selection and ParameterizationThe other important aspect of groundwater inverse analysis, besides the matching of the objective and subjective information, is the model selection or model identification problem. It is well recognized in the multiple regression analysis that if one increases the number of explanatory variables in a regression model, fit of the model to the given data improves, whereas reliability of estimated parameters decreases, and vice versa. There is a tradeoff between the parameter dimension and the reliability of estimation for estimated parameter values. In groundwater inverse analysis this problem typically appears in selecting number of zones in calculation model. This problem has been studied by many people; an excellent review on the topic is given by Yeh [1986]. Correra and Neuman [1986a, b, and c] solved this p•/oblem by using Akaike's Information Criterion (AIC). From the start of the development, AIC was aiming at solving this problem of model selection; it worked most conveniently for this purpose. Akaike's Bayesian Information Criterion (ABIC) introduced in this study is also developed on the same information theory principal as AIC; it can also be used to select the best model from several alternative models as is illustrated in the example in this paper.

“…This method was first proposed for using both hydraulic head and prior information on model parameters [Neuman and Yakowitz, 1979]. It has also been used with joint hydraulic head-thermal data [Woodbury and Smith, 1988 …”

Section: Analysis Of Data Residuals (Resid)mentioning

confidence: 99%

Parameter space methods in joint parameter estimation for groundwater flow models

Weiss¹,

Smith²

1998

Abstract. The joint estimation of model parameters using hydraulic head and tracer concentration data is examined using methods based on parameter space analysis. Response surfaces and confidence regions can indicate how multiple data sets interact to reduce parameter uncertainty. Each data set produces a unique response surface and confidence region. If these surfaces are oriented differently for the two data sets, parameter uncertainties are significantly reduced when the second data set is included in the inversion. If these surfaces are similar for the two data sets, the second data set will not reduce the parameter uncertainty significantly. The axes of the linearized confidence ellipsoid are analyzed to determine the difference in orientation of the confidence regions for the two data sets. The use of confidence regions can be extended to predict the value of a second data set in reducing the uncertainty of parameter estimates, before the data are collected. Parameter space approaches are also introduced for selecting the relative weights for the individual data sets in joint parameter estimation. The conventional method based on the analysis of data residuals is compared to three other methods: with the weights selected to maximize parameter stability, to minimize the volume of the confidence region, or to minimize the longest axis of the confidence region. Each criterion can lead to substantially different weights applied to each of the data sets. The application of these methods is demonstrated using hydraulic head measurements and •4Cconcentrations to calibrate a model of groundwater flow in the San Juan Basin, New Mexico.