## 1. Introduction

[2] The inverse problem in hydrogeology aims to gain understanding about the characteristics of the subsurface, i.e., identification of model structure and corresponding parameters by integrating observed model responses such as hydraulic head and mass concentration data. Several inverse methods have been proposed to solve the inverse problem in the last several decades. At the early stages of inverse modeling, a single “best” estimate of hydraulic conductivities was pursued. Examples can be found in the works by *Kitanidis and Vomvoris* [1983] and *Hoeksema and Kitanidis* [1984, 1985], who proposed the geostatistical method to identify the parameters of the underlying variogram that describes the multi-Gaussian random function used to characterize the spatial heterogeneity of hydraulic conductivities; once these parameters were identified, the hydraulic conductivity map was obtained by cokriging using the conductivity and piezometric head data. Another example can be found in the work by *Carrera and Neuman* [1986], who treated the aquifer properties as piecewise homogeneous. These approaches produced maps of conductivity which were capable of reproducing the observed heads but which were too smooth to be used for transport predictions, since they lacked the short-scale variability observed in the field. It was, thus, realized that the aquifer should be characterized by heterogeneous distributions of the parameters (see *de Marsily et al.* [2005] for a historic perspective on the treatment of heterogeneity in aquifer modeling). There are already several inverse methods capable of dealing with this heterogeneity, e.g., the pilot point method [*RamaRao et al.*, 1995], the self-calibration method [*Gómez-Hernández et al.*, 1997; *Wen et al.*, 1999; *Hendricks Franssen et al.*, 2003], the ensemble Kalman filter [*Evensen*, 2003; *Chen and Zhang*, 2006; *Hendricks Franssen and Kinzelbach*, 2008; *Zhou et al.*, 2011, 2012], or the Markov chain Monte Carlo method [*Oliver et al.*, 1997; *Fu and Gómez-Hernández*, 2009].

[3] In the above referred inverse methods, the groundwater model structure is described by a variogram model, which basically measures the correlation between two spatial locations. This two-point variogram-based model is not able to characterize curvilinear features, e.g., cross-bedded structures in fluvial deposits or erosion fractures in karstic formations, while these curvilinear structures play a key role in flow and especially solute migration modeling [e.g., *Kerrou et al.*, 2008; *Li et al.*, 2011a]. A solution to address this issue is to use multiple-point geostatistics. A “training image,” which contains the types of features to be reproduced by the aquifer model, is introduced as a geological conceptual model [*Guardiano and Srivastava*, 1993]. This training image is used to derive experimental local conditional distributions that serve to propagate the curvilinear patterns onto the simulated aquifer. Several programs based on multiple-point geostatistics are available, e.g., SNESIM [*Strebelle*, 2002], FILTERSIM [*Zhang et al.*, 2006], SIMPAT [*Arpat and Caers*, 2007], and DS [*Mariethoz et al.*, 2010a], and a detailed review on multiple-point geostatistics is provided by *Hu and Chugunova* [2008]. The advantages of using multiple-point geostatistics for the characterization of hydraulic conductivity and for flow and transport prediction have been confirmed after comparison with variogram-based simulation methods, both in synthetic examples and in real aquifers [e.g., *Feyen and Caers*, 2006; *Huysmans and Dassargues*, 2009; *Journel and Zhang*, 2006].

[4] Most of the inverse methods construct an objective function to measure the deviation between the simulated and observed data. Then, through an optimization algorithm, the initial aquifer models are modified until the observed data are well reproduced by the model predictions. However, during the optimization process, the aquifer spatial structure may be modified with respect to the structure of the initial guesses and become geologically unrealistic [*Kitanidis*, 2007]. To prevent this departure, techniques such as including a regularization term or using a plausibility criterion are combined with the objective function to constrain the deviation of the updated model from the prior model [*Alcolea et al.*, 2006; *Emsellem and de Marsily*, 1971; *Neuman*, 1973]. But these methods have been challenged on their theoretical foundations [*RamaRao et al.*, 1995; *Rubin et al.*, 2010]. Some recent inverse methods use other avenues in an attempt to preserve the prior structure when perturbing the parameter values in the prior fields.

[5] Considering the limits of the conventional inverse methods and the advantages of multiple-point geostatistics, a reasonable solution is to use the multiple-point geostatistics to characterize the nonlinear structure and to try to preserve this structure when the model is updated using inverse methods. In this way, the curvilinear features are characterized properly and the model remains physically realistic during the inverse process. A few examples of such inverse methods include the gradual deformation method (GDM) [*Hu*, 2000; *Caers*, 2003], the probability perturbation method (PPM) [*Caers*, 2002; *Caers and Hoffman*, 2006] and the probability conditioning method (PCM) [*Jafarpour and Khodabakhshi*, 2011]. In all three methods, the prior model structure can be characterized by multiple-point statistics and the property realizations are updated in such a way that the prior model statistics are kept. The difference between the three methods resides in the way the observations are integrated and the way the realizations are updated. The main idea of the GDM is that the realizations are perturbed by modifying the random number used to draw from the conditional distribution functions inherent to the sequential simulation algorithm. This random number is chosen through optimizing a deformation parameter so that the mismatch between the simulated and observed dynamic data is reduced. The PPM is based on modifying the conditional probability functions themselves. For the case of PCM, the realizations are updated with a multiple-point simulation method under a soft constraint given by a probability map inferred from observed flow data. The probability map is built with the help of the ensemble Kalman filter.

[6] Alternatively to the inverse methods formulated in the framework of minimizing an objective function, the Markov chain Monte Carlo method provides another way to tackle the problem, namely, sampling from a posterior distribution that is already conditioned to observations. Two such examples that are capable of dealing with curvilinear structures are the blocking moving window algorithm [*Alcolea and Renard*, 2010] and the iterative spatial resampling [*Mariethoz et al.*, 2010b]. Another avenue is treating the inverse problem as a search problem, e.g., the distance-based inverse method proposed by *Suzuki and Caers* [2008]. A large number of multiple-point simulations are constructed, from which a search scheme is used to select those consistent with the observed dynamic data. The spatial structure of the parameters is not disturbed since no modification is performed, simply a selection is carried out. The updated model should be geologically realistic as long as the prior model is so.

[7] In this paper, we present a novel approach to constrain hydraulic conductivity realizations to dynamic flow data. The most distinct novelty of the proposed method is that we formulate the inverse problem on the basis of pattern search instead of minimizing an objective function or sampling the posterior distribution. We assume that the hydraulic conductivity to be simulated is related to the geologic structure and to the flow dynamics in its neighborhood. The value at each simulated cell is determined by searching for matches, through an ensemble of realizations, to the conditional pattern composed of simulated hydraulic conductivities and observed flow data. The proposed pixel-based method is not only convenient to condition to local data but it is also able to capture the geologic structure inherent to the initial seed realization. The pattern is searched through an ensemble of realizations, all of which are consistent with the geologic structure, so that the pattern search method ensures that the updated fields are physically realistic and the prior statistics are preserved.

[8] The rest of the paper is organized as follows. In section 2, the proposed method is presented in detail. In section 3, a synthetic example is described to assess the performance of the method. In section 4, the results of the synthetic experiment are presented and analyzed. In section 5, the method is further evaluated with another example to test the effect of the number of conditioning data and of the boundary conditions. In section 6, a few issues about the method are discussed. In section 7, some conclusions about the proposed method are given.