## 1. Introduction

[2] Subsurface characterization for accurate groundwater flow and transport simulation is an important part of hydrogeologic studies such as groundwater resources planning and contamination remediation. Among subsurface properties of interest for the characterization are spatially distributed hydraulic conductivity, porosity, storage coefficients, and dispersion coefficients, as well as aquifer geometry and boundary conditions. To estimate the spatial structure of these subsurface properties, available observations are inverted through the relationship between observations and the subsurface properties of interest, which is formulated and treated as an “inverse problem.” Solutions to the inverse problem have been an active area of study in groundwater engineering [e.g., *Yeh*, 1986; *McLaughlin and Townley*, 1996; *Carrera et al*., 2005; *Oliver and Chen*, 2011]. A representative groundwater inverse problem is the estimation of hydraulic conductivity from hydrogeologic data such as hydraulic heads and direct soil/rock samples collected in boreholes [*Hoeksema and Kitanidis*, 1984; *Carrera and Neuman*, 1986; *Rubin and Dagan*, 1987]. Due to expensive and time-consuming hydrogeologic data collection process, near-surface geophysical techniques [*Rubin and Hubbard*, 2005; *Knight and Endres*, 2005] such as ground penetrating radar, electrical resistivity tomography and electromagnetic induction methods have also been introduced to provide additional information on the subsurface geology thus improving the estimation of unknown aquifer properties. With recent advances in computational capacity to handle large-scale data from various data collection techniques, inversion methods allow us to depict possible “images” of the subsurface for making predictions about subsurface phenomena and optimal decisions in engineering applications.

[3] However, subsurface characterization from hydrogeologic or geophysical measurements is fraught with challenges. Due to the limited number of measurements, which is usually much smaller than the number of spatially distributed parameters of subsurface properties, the weak sensitivity of measurements to unknowns, and the presence of noise in the observations, the typical inverse problem encountered in practice is ill-posed, i.e., there exist an infinite number of subsurface images consistent with measurements. In addition, the solution to an inverse problem must account for epistemic uncertainty introduced not just by sampling error in measurements but also by imperfect representation of the relationship between measurements and subsurface properties. Whereas information in the data is limited, inversion techniques utilize prior knowledge reflecting understanding of the subsurface to weigh possible solutions among those consistent with data. These solutions can be evaluated in a probabilistic way, which is commonly treated in the Bayesian framework by finding the probability density of feasible solutions that satisfy both model fitting and prior information [*Kaipio and Somersalo*, 2007; *Tarantola*, 2005; *Kitanidis*, 2012]. In the Bayesian geostatistical inversion approach [*Kitanidis and Vomvoris*, 1983; *Dagan*, 1985; *Kitanidis*, 1995], for example, prior information is described through the model that subsurface properties are distributed randomly with prescribed mean and spatial covariance functions, which is suitably casted into a probabilistic distribution function (pdf). Sometimes, one looks for the most probable solution that represents, in some sense, the best guess given data and prior information. In this case, the Bayesian inverse problem is reduced to the deterministic optimization problem of maximizing the posterior distribution. In the first-order Tikhonov regularization method [*Tikhonov and Arsenin*, 1977], the inverse problem is formulated as the optimization problem of seeking the “flattest” solution that reproduces the data. In fact, it is shown that the solution of the Tikhonov regularization method is equivalent to the best estimate obtained from the geostatistical inversion method with a certain class of generalized covariance functions [*Kitanidis*, 1999]. In summary, whether one uses deterministic or stochastic inversion methods, smoothness of the unknown subsurface structure is usually introduced as a desirable feature, which has the side effect of image blurring as a result of smoothing details that cannot be inferred from the incomplete and noisy information that measurements contain.

[4] However, when one strongly suspects that subsurface consists of a few relatively uniform geologic units with abrupt changes between units [*Fienen et al*., 2004], this approach seems inappropriate. Especially with a few available measurements, the data provides very weak sensitivity to the boundaries so that conventional techniques often perform poorly in the facies detection or zonation problem, defined as the challenging problem of finding regions of relatively uniform properties or, equivalently, identifying boundaries between such regions. Even if the subsurface is heterogeneous with earth properties varying at all scales, it may still be useful to categorize spatial parameters of interest as belonging to a few important discrete types or facies. For example, when one is interested in groundwater quality, it may be desirable to classify the subsurface into areas wherein groundwater meets potable, agricultural, or industrial use.

[5] The common approach to facies detection in groundwater engineering is the method of zonation in which the unknown field is parameterized into a few variables that describe the shape and location of zones or facies [*Jacquard*, 1965; *Yeh and Yoon*, 1981; *Tsai et al*., 2003]. During the inversion, the location and number of facies are optimized until the method achieves an acceptable data fit. Each facies in the method of zonation is typically assumed to have a constant property and then with identified facies distribution, variability within each facies can be simulated through geostatistical inversion methods. Other researchers [*Berre et al*., 2007; *Dorn and Villegas*, 2008; *Cardiff and Kitanidis*, 2009] identify zonal structures using the level-set method in which the shapes of facies are evolved during the inversion process to fit the data and thus one does not have to predetermine the shape. These approaches have shown satisfactory results in synthetic or practical applications, however, additional information such as the number and/or shape of zones needs to be defined before solving the inverse problem and the criteria for updating zonal parameters during the inversion may be ad hoc and depend largely on the choice of zonal structure parameterization.

[6] Another approach to delineate discrete geologic structures is the compactness or minimum support regularization method [*Last and Kubik*, 1983]. In this method, a subsurface image with the minimum area or volume of anomalies is selected thus geologic units with sharp contrasts can be recovered. The compactness regularization approach has been applied in gravity and electromagnetic inverse problems [*Portniaguine and Zhdanov*, 1999] and seismic travel-time tomography [*Ajo-Franklin et al*., 2007]. However, the compactness regularization term is nonconvex and discontinuous so that the minimization of the objective function is challenging and measurement noise may degrade the solution quality significantly. In addition, auxiliary parameters have to be assigned before or during the inversion.

[7] In a Bayesian framework, a natural solution to the zonation problem can be obtained by introducing appropriate information about the structure of the unknown function through the prior distribution. In other words, a prior pdf that captures the discretely changing structure of unknowns needs to be selected to produce multiple facies structures. For example, if one knows the exact number of facies, their threshold values for truncation and appropriate covariance models, the truncated Gaussian/pluri-Gaussian method [*Galli et al*., 1994] can be used to generate possible multiple-facies geologic images. Conceptual subsurface maps that professional geologists draw from outcrop survey can also be used to generate subsurface images using multiple-point geostatistics [*Strebelle*, 2002; *Hu and Chugunova*, 2008]. These prior images are then conditioned on measurement data to produce posterior realizations of the subsurface property field [*Caers*, 2003; *Liu and Oliver*, 2004; *Ronayne et al*., 2008; *Alcolea and Renard*, 2010; *Jafarpour and Khodabakhshi*, 2011].

[8] For the situation that one only has limited information such as the degree of the smoothness of unknowns, *Kitanidis* [2012] discusses the improper (generalized) prior in Bayesian inverse problems and how the choice of priors affects how sharply sharp boundaries may be identified. Resulting priors, for some reasonable choices of information, are identical to those obtained from Markov Random Field theory [*Li*, 2009], which states that a field characterized by joint probability distribution of the local properties under the Markov property is equivalent to a random field described by the global measure on it. Therefore, the improper prior can be a useful tool in modeling discrete geologic structure, as will be shown in the following sections.

[9] In this paper, we present a methodology of detecting zonal structure without any demanding assumptions about the location, shape, and number of zonal properties. For the zonal subsurface structure that contains sharp boundaries between units of distinct properties, it may be reasonable to assume there are a few large changes in geologic distribution of entire domain and small, perhaps negligible, variability within each zone. It will be shown that this prior information is translated suitably into a probability distribution that gives higher weight to solutions with a few sharp changes through the Laplace distribution of gradient, also known as total variation prior. The idea of using the Laplace distribution as a prior has long been recognized as a way to provide a “sparse” estimation whose elements contain many zeros (a zero, in this case, signifying no change between neighboring elements or nodes) and a few nonzeros (contrasts) [*Claerbout and Muir*, 1973; *Tibshirani*, 1996].

[10] Furthermore, recently developed compressed sensing theory [*Donoho*, 2006] explains that the optimization with *ℓ*_{1}-norm regularization [*Daubechies et al*., 2004], which attempts to find the mode of posterior pdf with a Laplace prior, can be equivalent to minimization of the number of nonzero elements in the solution leading to the sparsest solution; in our case, minimization of the number of the changes in the unknown field would be the most appropriate formulation for characterizing and classifying the subsurface structure with sharp contrasts. Several studies have implemented the *ℓ*_{1}-norm regularization approach based on compressed sensing theory such as subsurface reservoir permeability estimation using discrete cosine transform (DCT) [*Jafarpour et al*., 2009; *Li and Jafarpour*, 2010a] and seismic wavefield reconstruction using an extension of wavelets [*Lin and Herrmann*, 2007] under the assumption that a few nonzero coefficients in the discrete cosine or wavelet transform domain can represent the true solution effectively. *Li and Jafarpour* [2010b] extend DCT-based *ℓ*_{1}-norm regularization approach in a tractable Bayesian framework by replacing a Laplace prior with hierarchical models and account for the uncertainty on the cosine coefficients of estimated solutions.

[11] The remainder of this paper is organized as follows. In section 'Methodology', a Bayesian inversion approach with total variation prior is presented in detail. The structural parameters that account for uncertainty in prior information and likelihood function, measurements as well as forward model are estimated using an iterative Expectation-Maximization method. In section 'Applications', three numerical applications are presented. We briefly present a simple 1-D groundwater pumping rate identification example [*Kitanidis*, 2007] and how well uncertainty of the estimate is quantified compared to a classical geostatistical approach. Then we apply our method to a synthetic 2-D seismic borehole tomography problem [*Cardiff and Kitanidis*, 2009] and compare the result with those obtained by the geostatistical method and a level-set inversion method. Lastly, an example of the hydraulic conductivity estimation from sequential pumping tests is presented to illustrate a nonlinear inversion.