The test case considered in this work is similar to the laboratory experiment developed in Mboh et al. (2012), involving a falling-head infiltration phase followed by a drainage phase (Fig. 1). This experiment is representative of several laboratory SP experiments (Linde et al., 2007; Allègre et al., 2010; Jougnot and Linde, 2013, among others). Quartz sand is evenly packed in a plastic tube with an internal diameter of 5 cm to a height of Ls=117.5 cm. The column is initially saturated with a ponding of Lw=48 cm above the soil surface. Five sensors allowing SP measurements are installed 5, 29, 53, 77 and 101 cm from the surface, respectively. The column has a zero pressure head maintained at its bottom. At the top of the column, the boundary condition corresponds to a Dirichlet condition with a prescribed pressure head condition during the falling-head phase, followed by a Neumann condition with zero infiltration flux during the drainage phase. During the falling-head phase, the prescribed pressure head htop=48 cm has an exponential behavior driven by the saturated conductivity htop=Ls+Lwe-KsLst-Ls. The falling-head phase remains until the ponding vanishes at the critical time tc=-LsKsln⁡LsLs+Lw.

The total electrical current density j (Am-2) is determined from the generalized Ohm's law as follows: j=-σ∇φ+js, where φ (V) is the streaming potential, js (Am-2) is the streaming current density and σ (Sm-1) is the electrical conductivity distribution that is assumed to be isotropic.

Hence, the conservation equation ∇⋅j=0 is written as ∇⋅σ∇φ=∇⋅js. In addition, the electrical conductivity distribution can be estimated using the saturation Sw=θ/θs as follows (Mboh et al., 2012): σ=σsatSwna, where σsat (Sm-1) is the electric conductivity at saturation and na is the Archie saturation exponent (Archie, 1942).

The streaming current density js can be related to the Darcy velocity q(cmmin-1) by (Linde et al., 2007; Revil et al., 2007) js=-σsatρgKsCsatSwq, where Ks (cmmin-1) is the saturated hydraulic conductivity, ρ (kgmin-3) is the water density, g (ms-1) is the gravitational acceleration and Csat (VPa-1) is the voltage coupling coefficient at saturation.

Hence, the combination of the previous Eqs. (1)–(4) leads to the following partial differential equation governing the SP signals: ∇⋅Swna∇φ=∇⋅-ρgCsatSwKsq. However, the flow through an unsaturated soil column can be modeled by the one-dimensional Richard's equation: ∂θ∂t=ch+Ssθθs∂H∂t=-∇⋅q, where H (cm) and h (cm) are, respectively, the hydraulic and pressure head, such that H=h-z; z (cm) is the depth (downward positive); Ss (–) is the specific storage; θs (cm3cm-3) and θ (cm3cm-3) are the saturated and actual water contents, respectively; c(h) (cm-1) is the specific moisture capacity; and K(h) (cmmin-1) is the hydraulic conductivity.

The water velocity q is given from Darcy's law: q=-Kh∇H. In the current study, the standard models of Mualem (1976) and van Genuchten (1980) are used to relate pressure head, hydraulic conductivity and water content: Seh=θh-θrθs-θr=11+αhnmh<01h≥0KSe=KsSe1/21-1-Se1/mm2, where Se (–) is the effective saturation, θr (cm3cm-3) is the residual water content, Ks (cmmin-1) is the saturated hydraulic conductivity, m=1-1/n, and α  (cm-1) and n (–) are the Mualem–van Genuchten-shaped parameters.

Although several numerical techniques have been developed for the solution of the multidimensional Richards equation (e.g., Fahs et al., 2009; Belfort et al., 2009; Younes et al., 2013; Deng and Wang, 2017), the standard finite volume method is used here for the spatial discretization of the one-dimensional Richard's equation (Eq. 6). The integration of this equation over the finite volume (i) between (i-1/2) and (i+1/2) gives ∫i-1/2i+1/2ch+Ssθθs∂H∂tdz=qi-1/2-qi+1/2. Using expressions of the Darcy velocity at the element interfaces qi-1/2=-Ki-12Δz(Hi-Hi-1) and qi+1/2=-Ki+12Δz(Hi+1-Hi) in the case of a uniform spatial discretization with a spatial step, we obtain Δzci+Ssθiθs∂Hi∂t=Ki+1/2Hi+1-Hi-Ki-1/2Hi-Hi-1. Using τ=Swna and δ=ρgCsatSwKs, the integration of Eq. (5) over the finite volume (i) yields τi+1/2Δzφi+1-φi-τi-1/2Δzφi-φi-1-δi+1/2Ki+1/2Hi+1-Hi+δi-1/2Ki-1/2Hi-Hi-1=0, where the values at the interface τi±1/2, δi±1/2 and Ki±1/2 are calculated using the arithmetic mean between adjacent elements (for instance, τi+1/2=τi+τi+1/2).

Then, the temporal discretization of the obtained nonlinear ODE/DAE system (9–10) is performed with the method of lines (MOL) using the DASPK (Brown et al., 1994) time solver. The MOL is suitable for strongly nonlinear systems since it allows high-order temporal integration methods with formal error estimation and control (Miller et al., 1998; Younes et al., 2009; Fahs et al., 2009, 2011). In the current study, the relative and absolute local error tolerances are fixed to 10-6.

Numerical simulations are performed assuming typical MVG hydraulic parameters for the sandy soil with (according to Carsel and Parrish, 1988) Ks=0.495 cm min-1, θs=0.43 cm3 cm-3, θr=0.045 cm3 cm-3, α=0.145 cm-1 and n=2.68. The voltage coupling coefficient at saturation is Csat=-2.9×10-7V Pa-1 and the Archie saturation exponent is na=1.6.

Based on these hydraulic and geophysical parameters, a reference (mesh-independent) solution is obtained using a uniform mesh of 235 cells of 0.5 cm length. Data are generated by sampling the output SP signals every 10 min during 1800 min. Figure 2 shows that the SP signals have an almost linear behavior in the saturated falling-head phase. During the drainage phase, they have a nonlinear behavior and approach zero voltage for the dry conditions occurring toward the end of the experiment. The SP signals are independent Gaussian random noises with a standard deviation of 2.73 10-5 V. This noise level was obtained by Mboh et al. (2012) from laboratory measurements. The noised data (Fig. 2) are used as “observations” in the calibration exercise.

The aim of GSA is to assess the effect of the variation of parameters on the model output (Mara and Tarantola, 2008). Such knowledge is important for determining the most influential parameters as well as their regions and periods of influence (Fajraoui et al., 2011). The sensitivity of a model to its parameters can be assessed using variance-based sensitivity indices. These indices evaluate the contribution of each parameter to the variance of the model (Sobol', 2001). Polynomial chaos theory (Wiener, 1938) has been largely used to perform variance-based sensitivity analysis of computer models (see for instance, Sudret, 2008; Blatman and Sudret, 2010; Fajraoui et al., 2012; Younes et al., 2016; Shao et al., 2017; Mara et al., 2017). It can be stated that the PCE method is a surrogate-based approach. However, we argue that this method employs ANOVA (analysis Of variance) decomposition and hence can be considered as a spectral method (such as the Fourier amplitude sensitivity test; Cukier et al., 1973; Saltelli et al., 1999). Indeed, with this method, the sensitivity indices are directly obtained from the PCE coefficients without needing to run the surrogate model.

Let us consider a mathematical model with a random response f(ξ) which depends on (d) independent random parameters ξ=ξ1,ξ2,…,ξd. With PCE, f(ξ) is expanded using a set of orthonormal multivariate polynomials (up to a polynomial degree p): fξ≈∑α≤psαΨαξ, where α=α1…αd∈Rd is a dth-dimensional index. Sα represents the polynomial coefficients and Ψα represents the generalized polynomial chaos of degree α=∑i=1dαi, such as Hermite, Legendre and Jacobi polynomials, for instance. In this work, Legendre polynomials are employed because uniform distributions are considered for the parameters. The noninformative uniform distributions are used here to express the absence of prior information, which makes all possible values of the parameter equally likely.

Equation (12) is similar to an ANOVA representation of the original model (Sobol' 1993), from which it is straightforward to express V[f(ξ)], the variance of f(ξ) as the sum of the partial contribution of the inputs, as follows: Vfξ=∑αsα2. The first-order sensitivity index Si and the total sensitivity index STi are defined by Si=VEfξξiVfξ∈0,1 STi=EVfξξ-iVfξ∈0,1, where ξ-i=ξξi, E(|) is the conditional expectation operator and V(|) the conditional variance. Si measures the amount of variance of f(ξ) due to ξi alone, while STi≥Si measures the amount of all contributions of ξi to the variance off(ξ), including its cooperative nonlinear contributions with the other parameters ξj. The input/output relationship is said to be additive when STi=Si,∀,i=1,…,d, and in this case ∑i=1dSi=1.

In the sequel, a PCE is constructed for each SP signal at each observable time. The number of coefficients for a full PCE representation is P=d+p!/(d!p!). The evaluation of the PCE coefficients requires at least P simulations of the nonlinear hydrogeophysical model. Note that P increases quickly with the order of the PCE and the number of parameters. Hence, several sparse PCE representations, in which only the significant coefficients are sought, have been proposed in the literature in order to reduce the computational cost of the estimation of the Sobol indices. For instance, Blatman and Sudret (2010) developed a sparse PCE representation using an iterative forward–backward approach based on nonintrusive regression. Fajraoui et al. (2012) developed a technique whereby only the sensitive coefficients (that affect significantly model variance) are retained in the PCE. Recently, Shao et al. (2017) developed an algorithm based on Bayesian model averaging (BMA) to select the best sparse PCE from a given data set using the KIC (Kayshap, 1982). The main idea of this algorithm is to increase the degree of an initial PCE progressively and compute the KIC until a satisfactory representation of the model responses is obtained. This algorithm is used hereafter to compute the sensitivity indices of the SP signals.

The SP responses are considered for uniformly distributed parameters over the large intervals shown in Table 1. These intervals include the reference values reported in Mboh et al. (2012). The sensitivity indices of the six input parameters (Ks,θrα,n,na,Csat) are estimated using an experimental design formed by N=212=4096 parameter sets. The order of the sparse PCE is automatically adapted for each observable time and location. For some observable times, the PCE is highly sparse; it reaches a degree of 31 but only contains 112 nonzero coefficients.

Figure 3 depicts the temporal distribution of the streaming potential variance, represented by the blue curve, and the relative contribution of the parameters, represented by the shaded area. This figure corresponds to the temporal ANOVA decomposition for sensor 1 (5 cm from the soil surface) and for sensor 4 (77 cm from the soil surface). Interactions between parameters are represented by the blank region between the variance curves and the shaded area. Note that because a Dirichlet boundary condition with zero SP is maintained at the outlet boundary, the variance of the SP signal is zero at the bottom and reaches its maximum value near the soil surface. Hence, the variance is higher for the first sensor, located 5 cm from the soil surface (Fig. 3a) than for sensor 4, located 77 cm from the soil surface (Fig. 3b).

The SP signals at different altitudes exhibit similar behavior (Fig. 3). In the following, we comment on the results of sensor 1 (Fig. 3a). Because Ks varies between 0.1 and 2 cm min-1, the saturated falling-head phase remains until the ponding vanishes at tc=-LsKsln⁡LsLs+Lw. Depending on the value of Ks (see Table 1), tc varies between t1=20 min and t2=403 min. Thus, in Fig. 3a, we can see that during the first time period (t≤t1), the SP signal is strongly influenced by the value of the parameter Csat. The first-order and total sensitivity indices at t=10 min (Table 2a) confirm that only the saturated parameters Ks and Csat are influential. Csat is about 17 times more influential than Ks. As expected, the remaining parameters have no influence during the first period. The total variance is 0.72 mv, and there is no interaction between the two parameters Ks and Csat since STi=Si for both of them and ∑i=1dSi=1.

During the second period (t1≤t≤t2), the flow is either saturated or unsaturated depending on the value of Ks. Figure 3a shows that the variance of the SP signal exhibits its maximum value around 2.4 mv, with strong influences of the parameters Ks and Csat and weak interactions between them (small blank region between the variance curve and the shaded area). These results are confirmed by the sensitivity indices calculated at t=70 min and reported in Table 2a for sensor 1. Both first-order and total sensitivity indices indicate that Ks is the most influential parameter. The second influential parameter is Csat, which has a total sensitivity index about 12 times less than Ks. The parameter α is irrelevant since its total sensitivity index is 109 times less than Ks and its partial variance is Vi=Si×Vi=0.01 mv, which is less than the 95 % confidence interval associated with the SP measurement ±0.055mv. The total variance at t=70 min is calculated to be 2.17 mv, and the output–input relationship is close to being additive since ∑i=1dSi=0.94, which means that interactions between parameters exist but are not significant.

During the third period (t≥t2), the variance of the SP signal reduces to 0.3 mv (Fig. 3a) and significant interactions are observed between parameters (large blank region between the shaded area and the variance curve). Table 2a shows that for t=800 min, which corresponds to dry conditions, the total variance is 0.22. First-order sensitivity indices are very small, except for θr. The latter is highly influential since it has a significant first-order sensitivity index Si=0.27 and a more significant total-sensitivity index STi=0.74. The parameters Csat and Ks are irrelevant as they have very small first-order and total sensitivity indices. Further, strong interactions are observed between the parameters since the sum of the first-order indices is far from 1 ∑i=1dSi=0.47. The total sensitivity indices are significantly different from first-order sensitivity indices for almost all parameters. For instance, the ratio between these two indices is around 4 for α, 5 for na and 7 for n. The total sensitivity index of α remains small (0.065), whereas significant total sensitivity indices are obtained for nSTi=0.27 and naSTi=0.47, which indicates that these two parameters are influential (although their first-order sensitivity indices are small) because of the interaction between the parameters.

Figure 3b shows similar behavior for sensor 4 located 77 cm from the soil surface. The results in Table 2b indicate that the total variance observed at t=10, 70 and 800 min are around 8 times less than for sensor 1. For the first time period, the first and total sensitivity indices are identical to those observed for sensor 1 since saturated conditions occur inside the whole column and the same effect of Ks and Csat can be observed whatever the location inside the column. For the second time period, the sensitivity indices for sensor 4 (Table 2b) are similar to those observed for sensor 1. However, the results for the third time period show an improvement of the relevance of the parameter α, with an increase of both first and total sensitivity indices. Indeed, compared to the results of sensor 1, both first-order and total sensitivity indices tripled. Moreover, the total sensitivity index for αSTi=0.22 becomes close to that of nSTi=0.24.

In summary, the GSA applied to SP signals identifies the influential parameters and their periods of influence and shows that

the parameter Csat is highly influential during the first time period (t≤t1) during which no interactions are observed between parameters;

Calibration of computer models is an essential task since some parameters (like the Mualem–van Genuchten-shaped parameters α and n) cannot be directly measured. In such an exercise, the unknown model parameters are investigated by comparing the model responses to the observations. Recently, Mboh et al. (2012) showed that the inversion of SP signals can yield an accurate estimate of the saturated hydraulic conductivity Ks, the MVG fitting parameters α and n and the Archie saturation exponent na. Moreover, they showed that the quality of the estimation was comparable to that obtained from the calibration of pressure heads. In their study, Mboh et al. (2012) used the FOA approach with the shuffled complex evolution optimization algorithm SCE-UA (Duan et al., 1993).

As important as the determination of the optimal parameter sets are the associated 95 % confidence intervals (CIs) to quantify the uncertainty of the estimated values. The determination of CIs is not straightforward if the observed model responses are highly nonlinear functions of model parameters (Christensen and Cooley, 1999). In the sequel, the parameter estimation is performed using two approaches: the popular FOA approach and the Bayesian approach based on the MCMC sampler. Contrarily to FOA, the MCMC method is robust since no assumptions of model linearity or differentiability are required. Furthermore, prior information available for the parameters can be included. MCMC provides not only an optimal point estimate of the parameters but also a quantification of the entire parameter space. Several MCMC strategies have been developed for Bayesian sampling of the parameter space (Gallagher and Doherty, 2007; Vrugt, 2016). In a groundwater and vadose zone modeling context, the most widely used of these strategies is the Metropolis–Hastings algorithm (Metropolis et al., 1953; Hastings, 1970). It proceeds as follows (Gelman et al., 1996). i.

Choose an initial candidate x0=ξ0,σ0 formed by the initial estimate of the parameter set ξ0 and the hyperparameter σ0 and a proposal distribution q that depends on the previous accepted candidate.

Many improvements have been proposed in the literature to accelerate the MCMC convergence rate (e.g., Haario et al., 2006; ter Braak and Vrugt, 2008; Dostert et al., 2009, among others). Vrugt et al. (2009a, b) developed the DREAM MCMC sampler based on the differential evolution–Markov chain method of ter Braak (2006) to improve sampling efficiency. DREAM runs multiple Markov chains in parallel and uses subspace sampling and outlier chain correction to speed up MCMC convergence (Vrugt, 2016). Laloy and Vrugt (2012) developed the DREAM(ZS) MCMC sampler, in which a candidate for each chain is drawn from an archive of past states denoted Z, which plays the role of the generator q. Interested readers are referred to Vrugt (2016) for more details about the properties and implementation of DREAM and DREAM(ZS). In the current study, the DREAM(ZS) software is used for the MCMC estimation of the hydrogeophysical parameters. Note that because of the large number of model evaluations required, the MCMC method remains rarely used in practical applications compared to the FOA approach. Indeed, with FOA, the CIs are estimated once by assuming that the Jacobian remains constant within the CIs. This assumption was found to be reasonably accurate in nonlinear problems by Donaldson and Scnabel (1987). However, recently, several authors stated that parameter interdependences and model nonlinearities violate this assumption (see, for instance, Vrugt and Bouten, 2002; Vurgin et al. 2007; Gallagher and Doherty, 2007; Mertens et al., 2009; Kahl et al., 2015).

In the following, both MCMC and FOA approaches are employed for the inversion of the highly nonlinear hydrogeophysical problem using SP measurements.

Hydrogeophysical parameters are estimated using the DREAM(ZS) MCMC sampler (Laloy and Vrugt, 2012). Independent uniform distributions are considered for model parameter priors and likelihood hyperparameters (see Table 1). The parameter posterior distribution is written as pξ|ymes,σ∝σ-Nexp⁡-SSξ2σ2, where SSξ=∑k=1Nymesk-ymodkξ2 is the sum of the squared differences between the observed ymesk and modeled ymodk SP signals at time tk for N, the total number of SP observations.

The DREAM(ZS) software computes multiple sub-chains in parallel to thoroughly explore the parameter space. Taking the last 25 % of individuals (when the chains have converged) yields multiple sets used to estimate the updated parameter distributions and therefore the optimal parameter values and their CIs. In the sequel, the DREAM(ZS) MCMC sampler is used with three parallel chains.

We assume that the saturated water content has been initially measured with a fair degree of accuracy. However, instead of fixing its value (as in Kool et al. , 1987, van Dam et al., 1994, and Nützmann et al., 1998, among others), we assign a Gaussian distribution to θs to take associated uncertainty and its effect on the estimation of the rest of the parameters into account. It is assumed here that the saturated water content was accurately measured to be θs=0.43 cm3 cm-3 by weighing the saturated soil. The corresponding error measurements are independently and normally distributed with a zero mean and a standard deviation σθ=0.01 cm3 cm-3. Hence a Gaussian distribution is assigned to θs, with a mean value of 0.43 cm3 cm-3 and a 95 % CI [0.41-0.45] cm3 cm-3. The rest of the hydrogeophysical parameters have noninformative uniform distributions over the ranges reported in Table 1. The error (measurement) variance is also considered to be unknown and is simultaneously estimated with the physical parameters. Two scenarios are considered to check whether the FOA approach can provide a reliable estimation of parameters and associated uncertainties for the highly nonlinear hydrogeophysical problem investigated, both in the case of abundant data (small uncertainty regions) and in the case of scarcity of data (large uncertainty regions). In the first scenario, SP data collected from the sensors located at the five locations are taken into account for the calibration. In the second scenario, only the SP data from the first sensor located 5 cm from the soil surface serve as conditioning information for model calibration. Results of the MCMC sampler are compared to those of the FOA approach for both scenarios.