Ensemble Smoother with Multiple Data Assimilation as a Tool for Curve Fitting and Parameter Uncertainty Characterization: Example Applications to Fit Nonlinear Sorption Isotherms

Abstract

The ensemble smoother with multiple data assimilation (ES-MDA) coupled to a normal-score transformation is used to fit a Langmuir isotherm curve to estimate its parameters ($S_{m}$ and b) and their uncertainty. The highlights of this work are threefold: (1) the ES-MDA can be used as a curve fitting procedure, (2) the ES-MDA provides also a full uncertainty quantification about the fitted parameters, and (3) for the specific case of the Langmuir isotherm, parameter $S_m$ is well identified with little uncertainty, while parameter b is well identified with a larger uncertainty, indicating that solute concentrations are more sensitive to $S_m$ than to b. As a by-product, the number of samples required to characterize the joint uncertainty of Langmuir isotherm parameters is also investigated; it can be concluded that the minimum number of samples to use is six, with best results obtained with eight samples, a value larger than the number recommended in the literature.

Appendix: Additional Synthetic Examples

Fig. 7

The three isotherms analyzed in the paper. The Freundlich isotherm uses the left vertical axis, while the Langmuir isotherms use the right vertical axis

Fig. 8

From top to bottom, variation in the mean of parameter $S_m$, b, $K_f$, and $n^{-1}$ with the number of isotherm samples

In the following, two additional examples are included to support the claims in the main body of the text; the first one is for a synthetic soil with adsorption characteristics given by a Langmuir isotherm with parameters $S_{m}=7.2$ $\hbox {mg}\cdot \hbox {g}^{-1}$ and $b=0.174$ $\hbox {l}\cdot \hbox {mg}^{-1}$ (?), and the second one for a soil characterized by a Freundlich isotherm

$$\begin{aligned} S=K_f C_e^{1/n} \end{aligned}$$

(16)

with parameters $K_f = 1.5$ and $n^{-1} = 0.39$; S is given in $\hbox {mg}\cdot \hbox {g}^{-1}$ and $C_e$ in $\hbox {mg}\cdot \hbox {l}^{-1}$ (?). Figure 7 shows the three isotherms considered in the paper.

The procedure for fitting the curves is the same as the one used in the body of the text. The initial sets of realizations are drawn from the following bivariate uncorrelated uniform distributions: $(S_m, b) \in {U}[0,230]\times U[0,0.8]$, and $(K_f,n^{-1})\in U[0,30]\times U[0.001,0.99]$.

The evolution with the number of samples of the best estimate as given by the mean of the ensemble of updated parameters for the two cases can be seen in Fig. 8. The conclusions that can be drawn from the analysis of these figures are the same as from the analysis of the example in the main body. The estimated values are affected by the magnitude of the measurement errors: the larger the measurement errors, the larger the bias of the estimated value (as given by the mean of the ensemble results). When the error standard deviation is set to 1%, the estimates are quite close to the true value of the synthetic soil. For the Langmuir isotherm, the fluctuations of the mean $_m$ and mean b stabilize at about six samples, with stabilization improving as the number of realizations of the ensemble used increases. For the Freundlich isotherm, the estimation needs at least 13 samples and either 100 or 300 realizations to retrieve good estimates when the error standard deviation is above 1%; for smaller error, seven samples are necessary before the mean estimate stabilizes close to the reference value. It an be concluded that the Freundlich estimate may need a greater number of samples to ensure a good estimation of its parameters.