3.1. Laboratory Column Experiments
 The column experiments of Williams et al.  were designed to examine the geophysical response to microbe-induced ZnS and FeS precipitation during a biostimulation experiment performed using sulfate-reducing bacteria. The experimental columns were instrumented along their length with geophysical sensors, as well as with biogeochemical fluid sampling ports. The experiments were conducted under temperature-controlled conditions over a period of 78 days using five polycarbonate columns having inner diameters of 5.08 cm and lengths of 30.5 cm. Although different columns were used to collect seismic, complex resistivity and biogeochemical data sets and to serve as abiotic control columns, care was taken to ensure that the column packing, flow rates, and other experimental parameters were similar across the columns.
 Several pore volumes of lactate were flushed through the water-saturated, sand packed system before the experiment started, at which time the sulfate-reducing bacteria Desulfovibrio vulgaris were introduced into the middle and the nutrients were introduced into the bottom of the upward-flowing column. From the multilevel sampling ports, spaced 3.8 cm along column length, sulfate reduction was monitored over seven weeks, as indicated by decreasing substrate and metals concentrations, increasing biomass, and visually discernable regions of metal sulfide accumulation. The region of sulfide mineral precipitation showed a shift toward the influent (bottom) portion of the column over time as a result of microbial chemotaxis toward elevated substrate concentrations at the base of the column [Williams et al., 2005]. Upon termination, the fluid sampling and geophysical measurement columns were destructively evaluated; the sediment samples were collected to determine grain-affixed biomass, extractable metals, and to provide materials for electron microscopy.
 Williams et al.  showed that changes in seismic and complex resistivity measurements tracked the onset, spatial distribution, and aging of FeS and ZnS accumulation. In addition, the scanning electron microscope (SEM) images indicated that the biostimulation led to the aggregation of sulfide-encrusted bacterial cells. In this study, we extend this effort from a qualitative tracking of the system response using geophysical measurements to a quantitative estimation of the bioaggregated precipitate characteristics over time.
3.2. Geochemical Data and Evolution Model
 For the column experiments of Williams et al. , we can estimate the volume fraction of FeS and ZnS precipitates from the profiles of the measured total dissolved Fe2+ and Zn2+ concentrations using a mass balance method. For every mole of acetate produced, one half mole of sulfide is generated, which then results in the precipitation of sulfides according to the reactions Fe2+ + S2− FeSs, and Zn2+ + S2− ZnSs. For a column having a steady flow, the mass change of an aqueous species (after ignoring dispersion process) can be described by R = −ϕv() at steady state ( = 0), where ϕ is porosity, v is flow velocity, R is the precipitation rate of the sulfide mineral phase, C is the concentration of Fe or Zn in solution, and x is the distance along the column from the base. According to this equation, the loss rates of Fe(II) and Zn in the aqueous phase were computed by dividing their corresponding concentration differences by the distance between two consecutive sampling ports.
Here Rj−1/2 is the reaction rate defined in the interval between two discrete data points in space xj and xj−1, where the aqueous concentrations Cj and Cj−1 are measured [Steefel and Maher, 2009]. The FeS and ZnS accumulated during given time intervals were calculated by multiplying equation (6) by the time interval during the sampling process. The accumulated FeS and ZnS calculated using equation (6) matches well the amount of extractable FeS and ZnS measured at the end of the experiment (see Figure 3). The overall reaction stoichiometry outlined above is also supported by the measurements of other redox-active species (lactate, acetate, and sulfate) in the column, which are in the proper proportions for the electron balance. This further supports the validity of the use of the aqueous concentrations to calculate mineral precipitation rates.
 The mass-balance-based estimation, however, is practically impossible under field conditions because many more processes are involved in the mass balance of Fe(II) and Zn, and it is typically challenging to decouple these different processes. For example, in addition to the process of FeS precipitation, minerals such as iron oxide can absorb Fe(II) on their surfaces. Since our ultimate goal of developing the estimation framework is to apply it to field data sets, we assume that the direct estimates of FeS and ZnS will not be available through this simple procedure, but that we may be able to approximate the expected amount and distribution of these mineral phases using more sophisticated geochemical models, normally multicomponent reactive transport models [Steefel and Maher, 2009]. The accuracy of the approximation might range from simple qualitative relationships to more sophisticated numerical reactive transport modeling platforms, such as CrunchFlow [Steefel, 2008] and TOUGH-React [Xu et al., 2003], depending on available information. As a result, for the purposes of this study, we use the results obtained from the above mass balance method as the ground truth for evaluating the applicability and effectiveness of our state-space estimation framework.
 For our application example, we use a simple qualitative relationship with a statistical model for describing possible uncertainty to represent the geochemical evolution. On the basis of the observation from the column experiments, we assume that the increment in volume fraction of metal precipitates is nonlinearly proportional to the concentrations of acetate. This assumption is not important and the relationship can be replaced with a more sophisticated numerical model as it becomes available. For now, this approach is sufficient for testing the developed framework. Let zt represent the increment of total acetate concentrations from time t − 1 to time t and let pt and pt−1 represent volume fraction of metal precipitates at time t and t − 1, respectively. The increment of volume fraction thus can be modeled using function B(zt, θ1, θ2) = θ1(1 − exp(−θ2zt)), where θ1 and θ2 are parameters associated with the model. This empirical model is intuitively plausible because it is consistent with the fact that the increment of volume fraction increases with increasing of acetate concentrations and the rate of increase in volume fraction decreases. Parameter θ1 is the limit of the increment of volume fraction, whereas parameter θ2 depends on the unit of acetate concentrations and the increasing speed of volume fraction. To account for uncertainty in the model, we assume that the two parameters are known within some ranges and the output of the model is subject to Gaussian relative random noise with standard deviation of β1. Consequently, we obtain the following statistical model that we use for this example to describe the evolution of the precipitate volume fraction from time t − 1 to time t.
3.3. Complex Resistivity Data and Petrophysical Model
 The complex resistivity data were collected from several locations along the length of the column and over time by using frequencies from 0.01 Hz to 1000 Hz. In this example, we focus only on the complex resistivity data collected between ports 1 and 2, which correspond to the length interval between 3.5 cm and 7.0 cm away from the column base. Theoretically based models for predicting spectral induced polarization (SIP) signatures in metal containing soils are lacking, despite recent advances in semitheoretical modeling of SIP signatures in nonmetallic soils [Leroy et al., 2008]. The one exception is the classic electrochemical model of Wong . He attributed the polarization in metallic soils when the metal is less than 10% of the soil volume to diffusion of redox active and inactive ions that are predominantly perpendicular to the metal surface under an applied electric field and to an electrochemical mechanism associated with the redox active ions that facilitate transport of charge between ionic and electronic conduction. In the model, he also assumed no interaction between the electric fields of the individual polarizable particles (i.e., the metallic minerals), a condition that Wong  stated was reasonable for metal concentrations up to 16%. However, since the theoretical model requires the definition of several (more than eight) electrochemical parameters that are typically poorly determined, no practical applications have been presented in the peer-reviewed literature.
 Given the lack of easily applied theoretical models to adequately describe the SIP response of soils containing metallic minerals, phenomenological formulations, such as the Cole-Cole relaxation model [Cole and Cole, 1941], are often invoked [Pelton et al., 1978, 1983; Binley et al., 2005; Slater et al., 2006]. Similar to those studies, the complex resistivity data are first inverted for Cole-Cole model parameters (e.g., chargeability and time constant) using the stochastic inversion method developed by Chen et al. . Figure 4 shows the real and imaginary components of the measured complex resistivity data after inoculation as well as their corresponding fits to Cole-Cole models. Figures 5 and 6 give the medians and 95% predictive intervals of the inverted chargeability normalized by zero-frequency resistivity (referred to as normalized chargeability) and time constant parameters from day 13 to day 48. We did not get reliable estimates of Cole-Cole parameters from the complex resistivity data collected on the date earlier than day 13. We speculate that under the conditions where geochemical (i.e., aqueous chemistry) conditions are changing rapidly, Cole-Cole parameters may not adequately capture changes in the complete spectral response.
 We develop a petrophysical model to link the inverted Cole-Cole parameters to the properties of metal precipitates based on the observations of the column experiments on the date after day 13. From Figures 5 and 6, we can see that the normalized chargeability, a nearly linear function of the surface area of sulfide minerals in contact with water, is decreasing through time while volume fraction of the precipitates suggested by the geochemical data (Figure 2) is increasing through time. These observations perhaps are different from the response of complex resistivity obtained at early time because at early time, a single cell has an increasing layer of sulfide on it and both surface area and volume fraction increase over time. To explain the observations at the later time, we develop a rock-physics model of cells aggregating into clusters, which provides the key geometric parameters involved in modeling both permeability and induced polarization (IP) responses of the sand column. In the following, we conceptually describe the petrophysical model and present the results that are directly related to the inverted Cole-Cole parameters (i.e., normalized chargeability and time constant). The detailed derivations are given in Appendix A. The developed model involves many parameters, some of which can be approximately determined from SEM images and some need to be estimated during the inversion, which are also explicitly given in the following description.
 We assume that the formation of metal precipitates includes two main phases based on our observations from the column experiments. Similar processes were also observed by Moreau et al.  under the natural conditions where the concentration of aqueous metals (e.g., zinc) was much lower. The early phase involves the coating of an individual cell, that is, the bacterial cells in the system produce sulfide mineral to the point that they become entirely covered in a sulfide layer and ultimately die [Williams et al., 2005]. The subsequent second phase involves the aggregation of individual coated biominerals, in which the dispersed individual coated cells form clusters. Since we only have data after 13 days of bioremediation, we assume the dominant process involved in this example is the cell aggregation. For ease of description, we assume that both cells and metal sulfides are spherical, and the effects of deviations between the actual shape and that of a sphere will be addressed by some coefficients. As shown in Figure 7, all cells with a sulfide coating are assumed to be initially dispersed (i.e., widely separated from one another). Over time, the dispersed cells gradually aggregate into clusters, in the present simple model, taking the form of spheres. We employ a face-centered sphere packing approach to represent the aggregation, as is described in Appendix A. These spherical clusters grow through the attachment of additional dispersed cells. Since an isolated coated cell has a larger mineral-fluid surface area than a cell attached to a cluster, the surface area of sulfide will decline as long as the rate of cells attaching to clusters is greater than the rate at which new dispersed cells are being formed. This is the case in the column experiments as shown by Williams et al. . Given the near complete consumption of lactate within the first 1.9 cm of the column by day 12, this loss of the primary electron donor (lactate) severely limits subsequent microbial growth and cell division, thereby minimizing the rate at which new dispersed cells are formed. To account for the observation that total sulfide volume in the pores increases over time, we also assume that the cells in a cluster have a thicker layer of sulfide on them than do the dispersed cells (i.e., hc ≫ hd in Figure 7). To describe the process, we define two key parameters: One is the volume fraction of metal precipitates (pt) and the other is the fraction of dispersed coated biominerals (wt). Both are functions of time and will be estimated in the inversion.
 We can obtain an analytical relationship between normalized chargeability (mt) and parameters pt, wt, and θ3, the latter of which is a coefficient that accounts for incomplete knowledge about the thickness of encrusted cells. Within the model and under certain assumptions, we can obtain the specific area St = G0(pt, wt, θ3) (see equation (A4)). Additionally, normalized chargeability (i.e., polarization magnitude) has been repeatedly shown to scale with St in laboratory studies conducted on both metallic soils [e.g., Slater et al., 2006] and nonmetallic soils [Scott and Barker, 2005; Slater et al., 2006]. Therefore we can assume that normalized chargeability is proportional to specific surface area, i.e., mt = θ4St = G1(pt, wt, θ3, θ4), where θ4 is a parameter that may partially account for disparity in the shapes between spheres and actual ones and partially explain the ratios between the specific area and chargeability. This is an empirical based model, which is critical for the success of our estimation because it links the IP responses to the physical properties of geochemical precipitation. To consider uncertainty in the model, we also assume the empirical relationship is subject to relative Gaussian random errors with the standard deviation of β2. This is a common assumption for likelihood functions because the Gaussian distribution is the most robust probability distribution for characterizing errors, even the errors are non-Gaussian [Stone, 1996]. Thus we obtain the following model.
 We can also obtain an analytical formula to link time constant (τt) to the fraction of dispersed biominerals (wt). Time constant, describing the length scale of the relaxation in IP responses, has been widely recognized as a function of the pore or grain size characteristics of soils [e.g., Olhoeft, 1985; Chelidze and Gueguen, 1999] and therefore can be linked to the mean radius of clusters formed from metal precipitates. Schwartz  showed that the function is consistent with electrochemical theory for colloidal suspensions, whereby we can tie time constant τt at time t to the mean radius of aggregated clusters (rt) using the following formula: τt = rt2/(2D), where D is referred to as the surface ionic diffusion parameter and its value is given by 3 × 10−9 m2/s as used by Tarasov and Titov  and Slater et al. . In addition, we can derive the mean radius as rt = θ5l0 (see equation (A8)), where l0 is the characteristic pore-throat radius of the system and has a value of 1.3 × 10−4 m as determined from Thompson et al.  permeability model prior to precipitation, and θ5 is a parameter that explains the effects of differences between the actual shape and the used sphere and the effects of uncertainty in the values of the surface ionic parameter and the characteristic pore-throat radius. This parameter will be determined in the inversion with a value between 0.2 and 0.9. By combining the above two relationships, we obtain τt = θ52l02(1 − wt)/(2D) = G2(wt, θ5). This is an important relation for the estimation because it provides a linkage between time constant and the fraction of dispersed cells. Again, to account for uncertainty in the model, we assume the empirical relationship is subject to relative Gaussian random errors with the standard deviation of β3. Thus we obtain the following model.
3.4. Bayesian Model
 We apply the estimation framework given in section 2 to the column experimental data described by Williams et al. . We consider volume fraction (p1, p2, …, pn) as state variables and time-lapse normalized chargeability (m1obs, m2obs, …, mnobs) and time constant (τ1obs, τ2obs, …, τnobs) as measurements with Gaussian relative random errors. We also consider the fraction of dispersed biominerals (w1, w2, …, wn) and five time-independent parameters (θ1, θ2, …, θ5) as unknowns. We jointly estimate those state variables and time- dependent and independent parameters by conditioning on the inverted Cole-Cole parameters.
 We can specify the general Bayesian framework given in equation (5) with the geochemical evolution model described in section 3.2, and the complex resistivity rock-physics model conceptually summarized in section 3.3 (and described in detail in Appendix A) to obtain the following specific Bayesian model for estimation of precipitate related parameters (see Appendix B).
 In equation 10, we assume p0 = 0 (i.e., no precipitates at time t0), β1 = 5%, β2 = 1%, and β3 = 10%. In this model, we only take account for random measurement errors, and systematic errors in data, model assumptions, and parameterization cannot be resolved. However, given the flexibility of our estimation framework, we can certainly combine them into the model if we know the structures of those systematic errors. To obtain samples from the joint posterior distribution given in equation (10), we first derive conditional distributions for unknown variables and then use the MCMC sampling methods to obtain many samples of the unknowns. Details about the MCMC sampling methods are provided by Chen et al.  and in Appendix C.