An iterative geostatistical inverse method for steady flow in the vadose zone

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Abstract. A linear estimator, cokriging, was applied to estimate hydraulic conductivity, using pressure head, solute concentration, and solute arrival time measurements in a hypothetical, heterogeneous vadose zone under steady state infiltrations at different degrees of saturation. Covariances and cross-covariances required by the estimator were determined by a first-order approximation in which sensitivity matrices were calculated using an adjoint state method, The effectiveness of the pressure, concentration, and arrival time measurements for the estimator were then evaluated using two statistical criteria, L • and L 2 norms, i.e., the average absolute error and the mean square error of the estimated conductivity field. Results of our analysis showed that pressure head measurements at steady state flow provided the best estimation of hydraulic conductivity among the three types of measurements. In addition, head measurements of flow near saturation were found more useful for estimating conductivity than those at low saturations. The arrival time measurements do not have any significant advantage over concentration. Factors such as variability, linearity, and ergodicity were discussed to explain advantage and limitation of each type of data set. Finally, to take advantage of different types of data set (e.g., head and concentration), a computationally efficient estimation approach was developed to combine them sequentially to estimate the hydraulic conductivity field. The conductivity field estimated by using this sequential approach proves to be better than all the previous estimates, using one type of data set alone. IntroductionDuring the past decades, the cokriging technique has been applied extensively to many studies of subsurface hydrology. timates of the heterogeneous conductivity field. On the other hand, Yeh et al. [1995, 1996] proposed iterative cokriging techniques for nonlinear systems in which the requirement of unbiasedness and minimum variance were imposed during each iteration. By incorporating the nonlinear relationship between head and conductivity in groundwater flow systems, the estimated conductivity field revealed more detailed heterogeneity than using the linear model and more closely resembled the true field. This iterative approach was further extended to unsaturated flow by Zhang and Yeh [1997] to estimate parameters for unsaturated hydraulic conductivity in the vadose zone. Similarly, a quasi-linear geostatistical approach was presented by Kitanidis [1995] in an attempt to incorporate the nonlinear relationship between the parameter and secondary information of the subsurface flow system.In spite of the advance in parameter estimation techniques, a practical question remains regarding what kind of measurements (e.g., head, concentration, or other variables) are most useful for estimating the hydrological parameters. Several attempts to address this issue were carried out in the past. For example, Yeh and Zhang [1996] found that pressure head measurements can improve estimates of s...

Section: Paper Number 1999wr900268mentioning

confidence: 99%

Cokriging estimation of the conductivity field under variably saturated flow conditions

Yeh²

1999

Self Cite

“…The approach requires a linearized relationship between log transmissivity and hydraulic head perturbations in order to develop a cross-covariance function, which is valid if the variance of log transmissivity is small. As a result, for highly heterogeneous cases, cokriging cannot take full advantage of hydraulic head information collected at sample locations [Yeh et al, 1996;Zhang and Yeh, 1997].…”

Section: Introductionmentioning

confidence: 99%

A full‐Bayesian approach to the groundwater inverse problem for steady state flow

Woodbury¹,

Ulrych²

2000

Abstract. A full-Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two-dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are "hyperparameters"; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (in(T)) field with hydraulic head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a in(T) field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the in(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the In(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when hydraulic head measurements are noisy and little or no information is specified on mean values of in(T), variance of in(T), and integral scales. It is common in hydrogeologic practice to utilize nonlinear regression methods for parameter estimation. There are currently several private and public domain software packages that are used for this purpose. In order to achieve unique solutions and avoid unstable behavior in the results, it is necessary to parameterize or zone the problem in such a way as to define a smaller (usually much smaller) number of parameters than data points. As stated above, aquifers are highly heterogeneous, and hydraulic properties can vary significantly over very short distances thus making conceptual models based on large zones questionable in our opinion. Another difficulty lies 2081

“…The resultant (cross) spectra,Ĉ ss (k), C sh (k),Ĉ sf (k),Ĉ sa (k) andĈ sy (k), are given [Zhang and Yeh, 1996;Russo, 1998] by:…”

mentioning

confidence: 99%

Block‐effective macrodispersion in variably saturated heterogeneous formations

Russo

2003

[1] Numerical models are frequently used in order to quantify the effect of the spatial heterogeneity in the formation hydraulic properties on field-scale solute transport. Because of practical constraints it is common to discretize the flow domain into relatively large numerical cells, with characteristic length scales, ' 1 , ' 2 , ' 3 , comparable with those of the formation heterogeneity, I 1 , I 2 , I 3 , leading to a loss in the velocity variability, and, concurrently, in solute spreading. We analyzed block-effective dispersion coefficients, D 0 ij (i, j = 1, 2, 3), required to compensate for this loss in three-dimensional, variably saturated heterogeneous formations under conditions of steady state, gravity-dominated, unsaturated flow. The results that the principal components of D 0 ij are controlled by the ratio ' 0 = ' 2 /I 2 = ' 3 /I 3 , that they may reach their ergodic limits if ' 0 is sufficiently large, and that their tendency to their asymptotic limits with increasing scaled travel time, t 0 , is inversely related to ' 0 , are in agreement with previous results for saturated flow. New findings suggest that under unsaturated flow conditions, both the time required to reach the asymptotic limits of D 0 ij and the length scale needed to reach the ergodic limits of D 0 ij depend on soil stratification, soil texture, and mean soil water saturation and differ for the longitudinal and the transverse components of D 0 ij .