Comment on “Aquifer parameter identification using the extended Kalman filter” by C. H. Leng and H. D. Yeh

Abstract

[1] The paper by Leng and Yeh [2003] is a good example of how automatic computer analyses can yield highly precise but inaccurate parameter estimates. Leng and Yeh [2003] illustrate their extended Kalman filter (EKF) methodology by analyzing confined aquifer test data with the Theis [1935] model and unconfined aquifer test data with the Neuman [1972] model. Neither analysis demonstrates the superiority of EKF over existing methodologies. [2] For the confined aquifer the claim by Leng and Yeh [2003] that the proposed methodology achieves high accuracy when compared with graphical methods is specious, and the fact that they obtained stable parameter estimates with only 2.5 hours of drawdown data for the 4-hour test is inconsequential. The data used by Leng and Yeh [2003] for the confined aquifer, which are attributed to Todd [1980], derive from class notes handed out by John Ferris (formally of the U.S. Geological Survey, now deceased) for the purpose of illustrating Theis type curve methodology. The data were generated directly from the Theis equation for an assumed pumping rate, transmissivity, and storativity. With the selected theoretical data set, equally accurate parameter estimates can be obtained by any existing approach using only 5–10 min of drawdown data (see Todd [1980, Figure 4.10], and apply the Cooper-Jacob method). EKF methodology as implemented by Leng and Yeh [2003] does not improve upon this. [3] The data used by Leng and Yeh [2003] for the unconfined aquifer originally come from an aquifer test conducted at Saint Pardon de Conques, Gironde, France [Neuman, 1975; Batu, 1998]. To illustrate the “advantages” of using the EKF approach for unconfined aquifer parameter identification, Leng and Yeh [2003] selected data from a single observation well located at a distance of 10 m from the pumped well [Batu, 1998, Table 9–9]. Data from a second observation well located at a distance of 30 m from the pumped well are also available [see Batu, 1998, Table 9–10] but were not used. Because of the methodology used by Leng and Yeh [2003], which apparently allows for analysis of data from only one observation well at a time, and because actual aquifer test data are not error-free and are influenced by natural variations in aquifer hydraulic properties, they obtained estimates of specific yield that are unrealistically small for the aquifer in question. Neuman [1975] and Batu [1998] obtained similar, inaccurate results for the same reasons. Leng and Yeh [2003] state, however, that their reported values of mean error and standard error of estimate indicate the “good accuracy” of their results. [4] Analytical models [e.g., Theis, 1935; Neuman, 1972] nearly always assume that the aquifer is homogeneous. Because of field data imperfections it is always advisable for parameter estimation to use drawdown data from as many observation wells as are available and to analyze all observation well data simultaneously rather than each well independently. No matter the methodology used, a single observation well, unless unusually free of error, is generally not sufficient for accurate parameter estimates. For parameter estimation using type curves, Ferris et al. [1962] point out, and experienced groundwater practitioners know, that this is best accomplished by means of a composite plot (i.e., a plot of the logarithm of drawdown versus the logarithm of time over distance squared). By using this approach for an unconfined aquifer, piezometers located at varying distances from the pumped well extend the “late-time” range over which the measured data fall on the type curves of the relevant model, and piezometers located at varying depths below the water table and distances from the pumped well narrow down the range of possible values of the ratio of vertical to horizontal hydraulic conductivity. In this way the “composite plot” approach yields improved estimates of aquifer parameters when data contain measurement errors and the aquifer is not perfectly homogeneous. [5] Moench [1994] illustrates the use of composite plots in type curve analyses of three different aquifer tests, one of which is the test from Saint Pardon de Conques, Gironde, France. Table 1 displays the results obtained by Moench [1994] for the French test. Also shown in Table 1 are the results obtained by Heidari and Moench [1997] using nonlinear least squares and the simultaneous treatment of all available observation well data. Both analyses make use of the Neuman [1974] model, allowing for the small influence of the partially penetrating pumped well (which penetrated the lower 80% of the aquifer saturated thickness). The parameter estimates obtained by Neuman [1975] and Leng and Yeh [2003] are shown in Table 1 for comparison. In particular, note the differences in the values of specific yield and values of the ratios of vertical to horizontal hydraulic conductivity that result from analyses of data from only a single observation well [Neuman, 1975; Leng and Yeh, 2003] and from simultaneous analyses of data from two observation wells [Moench, 1994; Heidari and Moench, 1997]. In general, the accuracy of estimated aquifer parameters improves as additional observation wells are incorporated into an analysis. [6] Leng and Yeh [2003] maintain that their EKF approach can eliminate the need for long-term aquifer tests and still attain a high level of accuracy. For the unconfined aquifer they say that a 7-hour period may be sufficient for the 49-hour test when using EKF for parameter estimation. Actually, from the results presented in Table 5 (case 7), it would appear that “good,” but equally inaccurate, parameter estimates are obtained with only 15 min of drawdown data. One can, however, question the desirability of reducing the length of an aquifer test, as much is to be gained from a long-term test (e.g., definition of aquifer geometry). As pointed out in the preceding discussion, the accuracy of EKF methodology as described by Leng and Yeh [2003] is in doubt, even for the full 49-hour test. [7] Leng and Yeh [2003, paragraph 41] state, “Clearly, EKF has the advantage that it avoids erroneous estimation caused by human subjectivity during the curve fitting procedure.” The paper demonstrates to me that the exclusion of human judgment from the analysis of an aquifer test can lead to not only erroneous parameter estimates but also to poor aquifer test design and execution. Leng and Yeh [2003, paragraph 6] suggest that hydraulic parameters can be estimated in the field “on-line as the observations come in” so that the pumping test can be terminated once the parameter estimates stabilize. With the proposed EKF methodology and models used by Leng and Yeh [2003], I find this suggestion ill-advised.