4.1. Parameter Sensitivities
 Some of the unknown parameters may not be identifiable due to low parameter sensitivity. Parameter sensitivity measures how sensitive an observation is to the changes of a parameter. The dimensionless scaled sensitivities (SS) are defined as:
where βj denotes the jth parameter. The overall sensitivity of a parameter is described by the composite scaled sensitivities (CSSs), which is the average of the SS values associated with the same parameter. The CSS for the jth parameter, CSSj, is calculated as [Hill, 1992; Anderman et al., 1996; Hill, 1998]
Given the same observation error, a lower CSS value indicates lower sensitivity to the changes in a parameter and a larger uncertainty of parameter estimate. For easy comparison of the CSS values associated with different parameters, Zhang et al. [2003a] defined a CSS ratio (γ) for the jth parameter as
where max(CSS) is the maximum of the CSS values for all the parameters. Hence the maximum value of γ is unity, which is associated with the parameter with the maximum CSS value. According to Hill , if one or more of the CSS ratios (γ) is less than about 0.01, the nonlinear regression is unlikely to converge, and these parameters are likely not identifiable using corresponding observations.
 Figure 4 shows the CSS ratios of the parameters of the reference material at initial parameter values. Parameters r, L, and α had the smallest CSS ratios, which are between 0.01 and 0.10; thus these parameters may or may not be identifiable using available information. The materials of the Hanford's Sisson and Lu site are relatively coarse, and none of the θr values exceeded 0.0411 m3 m−3; therefore r was set as a constant. To constrain the uncertainty in parameters α and L, their local-scale values were used as prior information. We assume that the uncertainty, expressed as the 95% confidence interval (CI), of the prior information is about a factor of 10 for α and ±5 for L. Note that a small change of the uncertainty of the prior information does not significantly affect the optimized values. We also used a local-scale value of θs as prior information with a 95% CI of ±0.20 m3 m−3 to constrain the value of θs within a physically meaningful range. The weight associated with prior information was set to be unity and that with the observation data set was 1/50. This means that the effect of each of the prior information in the objection is equivalent to 50 observations.
4.2. Upscaled Parameter Values
 After completion of parameter scaling, the number of unknowns was reduced by a factor equal to the number of material types (i.e., 5 in this case) from 35 to 7. Since r was set as a constant, six of the seven parameters were then estimated by solving equation (14) inversely using UCODE and STOMP, subject to the objective function equation (16).
 The uniqueness of the parameter estimates was evaluated by the correlation coefficients (R) between the parameters (Table 3) at the optimized parameter values. The absolute values of the correlation coefficients ∣R∣ ≥ 0.95 indicate that the parameter values cannot be uniquely estimated with the observations used in the regression [Hill, 1998]. A smaller value of ∣R∣ indicates the higher uniqueness. In our case, the largest value of ∣R∣ was 0.659, which is much smaller than the critical value of 0.95 and suggests that all the parameter estimates are unique. The sum of squared residual (SSR) decreased by 85.8% from 22.8 using the local-scale hydraulic parameters (Table 1) to 3.24 after CPSIT upscaling of these parameters.
Table 3. Correlation Coefficient Between the Field-Scale Reference Values of the Hydraulic Parameters of the Sisson and Lu Site
|sh||0.398|| || || || |
|sv||0.381||0.659|| || || |
|−0.045||0.567||0.019|| || |
 Table 4 lists the mean and the 95% linear confidence interval (LCI) of the optimized field-scale hydraulic parameters of the reference material. Since parameters s, , and were log-transformed when they were estimated, their 95% LCIs are expressed as the mean values multiplied or divided (×/÷) by a factor, which has the minimum value of unity. Comparing the FS values with the LS values (Table 4) shows that parameters and had the largest scale effects. Their FS values differ from the LS values by factors of 96.8 and 29.6, respectively. These findings are consistent with other research results which show that the saturated hydraulic conductivity tends to increase with spatial scale [e.g., Schulze-Makuch et al., 1999]. The FS values of and differed from their LS values by factors of 0.782 and 1.40, respectively, which suggests that and are weakly dependent on spatial scale. Parameters and had the smallest changes and varied by factors of 1.01 and 1.04, respectively, from local to field scale. This suggests that the LS values of and may be used to represent their FS values. Zhang et al. [2002, 2004] investigated FS hydraulic parameters of two layered soils and also found that the spatial-scale effects were largest for Ks, medium for α, and smallest for θs and n.
 The upscaled FS hydraulic parameter values for the five materials are listed in Table 5. Table 5 also shows that the anisotropy coefficients (Ksh/Ksv) of the saturated hydraulic conductivity for the materials at the experiment site were between 4.0 and 24.7. These values are larger than those in Table 1, which were determined by taking the arithmetic and harmonic means using local-scale parameter values.
Table 5. Effective Parameters of Individual Materials Using the Combined Parameter Scaling and Inverse Technique (CPSIT)
|Materials||θs, m3 m−3||Ksh, m s−1||Ksv, m s−1||α, m−1||n||L||θr, m3 m−3||Ksh/Ksv|
 The classic inverse technique estimates all the parameters simultaneously and hence is referred to as simultaneous inversion (SI). The SI approach is often used to estimate the field-scale hydraulic parameters. It is well-known that simultaneously inverting a large number of parameters will encounter problems such as model divergence, extremely long simulation time, and nonuniqueness of results. In our case, there were 30 parameters, and attempts to simultaneously invert them, using guessed values or even parameter values measured in the laboratory, led to divergence of the problem and difficulty in obtaining results. Even if the problem had not been divergent, it would have taken about two months to complete the simulation. After a few tries, we found that the simultaneous inversion would only converge if (1) the parameter estimates obtained using CPSIT (Table 5) were used as the starting values and (2) prior information was applied to θs, α, n and L. Because these starting values were very close to the final values, the SI after CPSIT took only eight iterations to converge.
 Table 6 summarizes the optimized parameter values and the 95% LCIs. Compared with CPSIT alone, inclusion of simultaneous inversion decreased SSR by 26% from 3.24 to 2.40. A comparison of the optimized parameters using CPSIT and SI by calculating the ratio of the corresponding parameters is shown in Table 7. The ratios for parameters θs, α, n and L are between 0.791 and 1.629, indicative of good agreement. The ratios for parameters Ksh and Ksv of texture A, B and C varies from 0.276 to 3.012, which are not much more than the measurement error under well-controlled conditions. However, both Ksh and Ksv for D and E show relatively large discrepancy as indicated by ratios between 0.082 and 6.489. By checking the anisotropy of the two materials, we found SI gives anisotropy coefficients of 61.7 for D and 572.5 for E, while CPSIT gives 13.9 for D and 7.2 for E. These results suggest that the SI may have overestimated the degree of soil anisotropy in Ks. The 95% LCIs of Ksh and Ksv from SI (Table 6) for D and E have larger uncertainty compared with those from CPSIT (Table 5). The large uncertainty in Ksh and Ksv derived from SI is due to the fact that these two textures are at the largest depth in the soil profile where the temporal variation in water content after the introduction of the injected water was small (no more than 0.05 m3 m−3). With the CPSIT approach, the parameters of different textures were bound together by the scaling factors which force them to have the same uncertainty.
Table 6. Effective Parameters and 95% Linear Confidence Interval of Individual Materials Using Simultaneous Inversion (SI) With the Numbers in Table 5 as Start Valuesa
|Materials||θs, m3 m−3||Ksh, m s−1||Ksv, m s−1||α, m−1||n||L||θr, m3 m−3||Ksh/Ksv|
|A||0.324 ± 0.002||1.18E−03 ×/÷ 1.082||6.15E−04 ×/÷ 1.138||2.587 ×/÷ 1.019||2.675 ×/÷ 1.010||−0.257 ± 0.044||[0.039]||1.92|
|B||0.352 ± 0.002||8.65E−04 ×/÷ 1.101||9.18E−05 ×/÷ 1.063||4.094 ×/÷ 1.019||1.997 ×/÷ 1.011||0.115 ± 0.046||[0.027]||9.42|
|C||0.346 ± 0.002||5.08E−04 ×/÷ 1.192||4.50E−04 ×/÷ 1.157||3.460 ×/÷ 1.019||2.404 ×/÷ 1.025||1.211 ± 0.049||[0.040]||1.13|
|D||0.342 ± 0.002||2.22E−03 ×/÷ 1.142||3.60E−05 ×/÷ 1.136||3.253 ×/÷ 1.019||2.774 ×/÷ 1.038||0.898 ± 0.048||[0.045]||61.67|
|E||0.388 ± 0.002||2.25E−02 ×/÷ 2.198||3.93E−05 ×/÷ 3.548||3.457 ×/÷ 1.019||3.113 ×/÷ 1.200||0.782 ± 0.049||[0.031]||572.52|
Table 7. Ratio of the Effective Parameters Obtained Using CPSIT (Table 5) to Those Obtained Using SI (Table 6)
|Materials||θs, m3 m−3||Ksh, m s−1||Ksv, m s−1||α, m−1||n||L|
 This comparison of the parameter estimates using CPSIT and SI indicates that CPSIT treated a parameter of all the textures (e.g., Ksh for the five textures) as a whole while SI treated them as independent parameters. Consequently, some parameters (e.g., Ksh and Ksv for D and E) using SI may not be estimated well due to the lack of observations or very small changes within a material while they may be well estimated using CPSIT. Compared to simultaneous inversion, CPSIT also has the following advantages.
 1. The number of unknowns is Np × M for SI but is Np for CPSIT, which reduces the unknowns by a factor of M. For example, for a soil with 5 textures and 6 parameters needed to describe the hydraulic properties of each texture, there is a total of 30 parameters. If the SI method is used, all the 30 parameters must be estimated. With CPSIT, only 6 are estimated with the advantage of a fivefold reduction.
 2. The estimated number of runs of the application model is 2NpM(NpM + 1) for SI and 2Np(Np + 1) for CPSIT, which reduces the simulation time by a factor of approximately M(NpM + 1)/(Np + 1) (≈M2). Using the above example, if the SI method is used, the estimated number of runs is 1860. If the CPSIT method is used, the expected number of runs is 84, reduced by a factor of 22.
 3. To obtain unique estimates, the number of observations needed to estimate the unknowns using CPSIT is much less than that using SI. For a specific type of observation, generally when the ratio between the number of observations (Ny) and the number of parameters needs to be estimated becomes larger, and the results are more unique. For the same number of observations, the ratio is Ny/Np/M if the SI approach is used and is Ny/Np if the CPSIT method is used. The latter is M times larger than the former. Therefore CPSIT is much less susceptible to nonuniqueness problems than SI.
 4. The CPSIT approach uses both the field-scale and local-scale data while the SI method uses only the field-scale data. In cases where there are no local-scale parameter values available for the use of CPSIT, they may be determined using easily obtainable information, e.g., bulk density and particle size distribution. For the SI approach, neglecting local-scale data is a waste of valuable information.
 One limitation of using CPSIT is that any error in the local-scale parameter values is compounded through the upscaling process to the upscaled results. This is because the local scale parameter values are used to calculate scaling factors, and the values of the scaling factors are fixed during the upscaling process. This limitation may be minimized by taking more samples and using more advanced laboratory methods. Another limitation is that in a given simulation the hydraulic properties of all the EHMs must be described using the same type of hydraulic function, e.g., either the Brooks and Corey  or the van Genuchten  model but not mixed.
 Using the field-scale parameter values in Table 5 and the values of θr in Table 1, the injection experiment described in section 3.1 was simulated. As a comparison, flow was also simulated using the local-scale parameter values in Table 1. Note that the data for this validation were those associated with injections 2 (days 7 to 14), 6 (days 35 to 42), and 10 (days 63 to 123), which included 10583 observations, and were different from the data used in the model calibration. The simulation covered a period of 123 days. A comparison between the simulated and observed water contents is shown in Figure 5. Results show that the simulated water contents improved significantly after applying parameter scaling and inverse modeling. When the local-scale parameter values were used, the mean squared residual (MSR) of water content was 3.99 × 10−3 (Figure 5a). Water contents of most observations were significantly overestimated when the local-scale parameter values were used to simulate flow (Figure 5a). When the upscaled field-scale parameter values were used, the MSR value decreased by 83.2% to 6.71 ± 10−4 (Figure 5b). The same experiment was simulated by Sisson and Lu , Lu and Khaleel , Smoot and Lu , and Smoot and Williams  using different types of models. Although these investigations did not report the goodness of fit, in general, the prediction of flow was unsatisfactory.
Figure 5. Comparisons of simulated and observed water contents of the Sisson and Lu experiment during injections 2 (days 7–14), 6 (days 35–42), and 10 (days 63–123) using (a) the local-scale and (b) the field-scale values of hydraulic parameters. There were 10,583 observations.
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 Rockhold et al.  conducted simulations of the same experiment using a conditional simulation and upscaling method. They reported root-mean-square error (RMSE) values for water content of 0.0335 and 0.0280 for two simulations of flow in a single day. Simulations based on the proposed CPSIT and using the upscaled parameters produced an RMSE of 0.0259 for a period of 123 days. Thus CPSIT is clearly a robust and effective approach for upscaling hydraulic parameters from the local-scale to field-scale. The efficiency of the technique is further demonstrated through the need to invert only for the parameters of the reference material and in the M2 reduction in simulation time.