## 1. Introduction

[2] Reliable prediction of subsurface flow and transport plays an important role in effective resource development planning and implementation. In particular, proper representation of spatially distributed subsurface flow properties, and their connectivity, dominate the flow and transport behavior in geologic formations. To infer the connectivity patterns in subsurface flow properties, an inverse modeling framework is applied to combine flow and transport response measurements with a forward model that relates the observed quantities to the subsurface properties of interest. Formulation and solution of the resulting inverse problem have been widely discussed in the subsurface flow and transport literature [e.g., *Yeh*, 1986; *McLaughlin and Townley*, 1996; *Carrera et al*., 2005; *Oliver and Chen*, 2011; *Dai and Samper*, 2004].

[3] A major difficulty in solving subsurface characterization inverse problems is data limitation, which is caused by the inconvenience and high cost of data acquisition. In general, the number of parameters used to describe distributed subsurface properties with a high enough resolution overwhelmingly exceeds the number of available measurements, rendering the resulting inverse problems severely ill-posed. As a consequence, an infinite number of subsurface property maps can be conceived that reproduce the measurements but provide very distinct predictions. The solution set of such ill-posed inverse problems is typically constrained by incorporating prior information to discriminate against implausible outcomes. However, prior model comes with its associated uncertainty. A general approach to incorporate the uncertainties in all prior knowledge (as well as data and model errors) is the probabilistic approach [*Kaipio and Somersalo*, 2007; *Tarantola*, 2005a; *Kitanidis*, 2012; *Harp et al*., 2008; *Ye and Khaleel*, 2008; *Dai et al*., 2010].

[4] Deterministic formulations of ill-posed inverse problems, on the other hand, use prior knowledge to constrain the solution set, for example, by adding, either implicitly or explicitly, information about the attributes of the desired solution or by using the prior knowledge to reduce the number of parameters (i.e., parameterization). Classical regularization methods such as Tikhonov regularization [*Tikhonov and Arsenin*, 1977] of zeroth, first, and second order promote the shortest, smoothest, and flattest solutions, respectively. These classical methods are useful when strong evidence suggests that the properties of interest should exhibit gradual spatial variability. In formations where spatial variability is characterized by abrupt changes in the spatial variability (e.g., facies distribution), more sophisticated techniques must be employed. Parameterization methods such as zonation [*Jacquard*, 1965; *Yeh and Yoon*, 1981; *Tsai et al*., 2003], level-set [*Berre et al*., 2007; *Dorn and Villegas*, 2008; *Cardiff and Kitanidis*, 2009], sparse geological indicator [*Dai et al*., 2005], and truncated Gaussian/pluri-Gaussian [*Galli et al*., 1994] have been introduced for abruptly changing geologic formations (or facies types). However, these methods generally rely on knowledge about the number, location, and shape of the facies, and are known to be inflexible, and in most cases hard to interpret geologically. In recent years, multiple-point geostatistics has been proposed for generating subsurface facies maps from a conceptual model that experienced geologists can infer from outcrop surveys and process-based modeling [*Strebelle*, 2002; *Caers*, 2003; *Ronayne et al*., 2008].

[5] Prior training methods have been widely used in machine learning and computer vision. A popular example is the face recognition problem where a large training data sets of human faces is used to inform the face recognition inversion algorithm of the expected patterns in the solution, thereby constraining the solution feasible set. Similarly, reliable prior information can significantly enhance the plausibility and quality of the inversion solutions in subsurface imaging problems, in particular when complex geologic connectivity patterns are considered. For example, a training data set that characterizes the general shape, connectivity, and geometric attributes of the expected geologic patterns can be adopted to significantly constrain the connectivity in the solution. In a recent publication [*Khaninezhad et al*., 2012], sparse geologic dictionaries were introduced as a flexible approach for summarizing and incorporating prior geologic knowledge into the solution of the subsurface flow model calibration inverse problems. The sparse learning approach is motivated by the recent developments in sparse representation and approximation literature, formalized under the compressed sensing paradigm [*Donoho*, 2006; *Candès and Tao*, 2005; *Candès et al*., 2006; *Elad*, 2010]. In particular, learned sparse dictionaries [*Tošić and Frossard*, 2011; *Aharon et al*., 2006] have been introduced as an alternative to generic image compression bases, where specialized dictionaries are learned from application-specific prior data sets for a more effective representation of patterns similar to those in the prior data set.

[6] Several approaches have been introduced to implement sparse dictionary learning in image processing [*Aharon et al*., 2006; *Tošić and Frossard*, 2011; *Khaninezhad et al*., 2012]. In *Khaninezhad et al*. [2012], the formulation and advantages of learned sparse dictionaries for solving ill-posed subsurface flow inverse problems are discussed using the K-SVD algorithm [*Aharon et al*., 2006]. The computational cost of learning sparse dictionaries can, however, limit their application to large (full-size) images. To circumvent this problem, in image processing, sparse dictionaries are learned from smaller image segments/patches that are combined to form a full image. In data-deficient inverse modeling applications, such as subsurface model calibration, spatial image segmentation can lead to structural distortion and discontinuity at the boundaries of the image patches. Therefore, a more effective approach that honors the intrinsic and desired properties of the application of interest is more appropriate. For example, in data-scarce geoscience applications where the focus is primarily on identifying the overall shape and connectivity of the large-scale geologic features in the solution, an alternative approach to spatial segmentation should be sought. In this paper, we address this issue by learning sparse dictionaries from low-rank (low-dimensional) spectral representations of the training data set. To this end, we first consider the spectral representation of the prior data set in a linear compressive basis and discard the expansion dimensions that correspond to insignificant details to arrive at an effective low-rank representation of the full size image library. This step significantly reduces the dimension of the training data set without introducing a noticeable loss of quality in the prior model of facies connectivity. The subsequent sparse learning from the low-dimensional representation of the prior geologic connectivity can be performed efficiently. In addition to computational gain, working with the proposed low-rank data representations eliminates small-scale details from the solution and facilitates the estimation of large-scale connectivity features.