## 1. Introduction

[2] This paper presents a new approach for inverse modeling called method of anchored distributions (MAD). MAD is an inverse method focused on estimating the distributions of parameters in spatially variable fields. MAD addresses several of the main challenges facing inverse modeling. These challenges fall into two broad categories: data assimilation and modularity.

[3] Data assimilation in inverse modeling is the challenge of using multiple and complementary types of data as sources of information relevant to the target variable(s). In hydrogeological applications, for example, one may be interested in mapping the spatial distribution of the hydraulic conductivity *K* [cf. *Kitanidis and Vomvoris*, 1983; *Kitanidis*, 1986, 1991, 1995, 1997a, 1997b; *Carrera and Neuman*, 1986a, 1986b; *Hernandez et al.*, 2006] using measurements of the hydraulic head, measurements of concentrations and travel times obtained from solute transport experiments [*Bellin and Rubin*, 2004], and measurements of geophysical attributes obtained from geophysical surveys.

[4] Another example is the mapping of ocean circulation, which relies on a variety of data types (e.g., temperature, density, velocity vector components) obtained from ship surveys, moored instruments, buoys drifting freely on or floating below the ocean surface, and satellites. These data are measured over a wide range of scales and frequencies, and they need to be assimilated to yield accurate circulation models.

[5] In a third example, air quality management requires constructing maps of dry deposition pollution levels. Ideally, such maps would be based on a dense network of monitoring stations, but generally such networks do not exist. Alternative and related information must be used instead. For example, there are two main sources of information for dry deposition levels in the United States: one is pollution measurements at a sparse set of about 50 monitoring stations called CASTNet, with spacing between stations on the order of hundred of kilometers, and the other is the output of regional scale air quality models with grid resolution on the order of a few kilometers [*Fuentes and Raftery*, 2005].

[6] In all these cases, the observations can be related to the target variables by functions that relate measurement to target variables. The challenge in all these cases is to combine the multiple sources of data into a coherent map of the target variables without introducing external factors such as smoothing and weighting.

[7] To address such a wide range of problems we would need to address the challenge of modularity. Modularity means an inverse modeling approach that is not tied to particular models or data types and maintains the flexibility to accommodate a wide range of models and data types. Inverse methodology and the numerical simulation of data-generating processes have become very closely intertwined, in a way that makes them very limited in applications. (“Data-generating processes” in this document refers to the natural processes that result in a quantity being measured. These processes are usually simulated by numerical codes.) This can be attributed to the increasing complexity of the processes that are being analyzed and of the computational techniques needed for their analysis. For example, inverse modeling in hydrogeology evolved from Theis' type-curve matching into modern studies that include complex and specialized elements such as (1) adaptive and parallel computing techniques, (2) geophysical modeling of electromagnetic fields and of the propagation of seismic waves, and (3) complex multi component chemical reactions. The range of skills needed for implementing these elements forced researchers to build the inversion procedure around their own or favorite numerical codes. As a result, the potential for expanding the range of applications beyond the original application, for example, by changing the data types or the numerical models used, is limited. Modularity is a strategy for alleviating this difficulty by pursuing a model-independent inverse modeling framework.

[8] This paper, through presentation of the MAD concept, explores all these issues using a Bayesian framework. A theoretical approach is developed and demonstrated with two subsurface flow problems.