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MAD Papers:

To view the published open abstracts, go to and enter the DOI.



Osorio-Murrilo, C., H. Savoy, and Y. Rubin. “Managing forward models in an inverse modeling framework through a master driver approach.” 2016. Submitted to Computers and Geosciences. 

Wang, C., H. Savoy, C. Osoirio-Murillo, X. Li, H. Zhu, and Y. Rubin. “Bayesian inverse modeling for probabilistic site characterization of shallow soil-tunneling system.” 2015. Submitted to Mathematical Geology




Heße, F., H. Savoy, C. A. Osorio-Murillo, J. Sege, S. Attinger, and Y. Rubin (2015), Characterizing the impact of roughness and connectivity features of aquifer conductivity using Bayesian inversion, Journal of Hydrology, 531, 73–87, doi:10.1016/j.jhydrol.2015.09.067.

The conductivity of an aquifer is usually difficult to represent in a precise manner due to having both a high degree of spatial variability combined with a scarcity of information. As a result, the conductivity is commonly modeled as a spatial random field, defined by its expected value and a covariance function. This covariance is usually parameterized by fitting a model function to experimental data derived from point measurements of the conductivity. In this study, we investigated the two features that are often difficult to discern when using such classic characterization schemes: roughness and connectivity. These two features both share the fact that, based on point measurements of the conductivity alone, they are difficult to discern. It therefore stands to rea- son that the use of additional data could alleviate this problem. To that end, we used the Method of Anchored Distributions (MAD), which is a novel Bayesian tool for the inverse characterization of spatial random fields. MAD is versatilewith respect to the used data, it has a modular structure and it does not assume any formal relationship between the target variable, i.e. the log hydraulic conductivity, and the data used for the inversion process, e.g. head measurements, draw-down from pumping tests or breakthrough curves. With respect to the characterization of the aforementioned features, we investigated the impact of several factors, such as different data types on the characterization process. Our findings suggest that both roughness and connectivity have only a limited impact on flow predictions, suggesting that the choice of such features is of lesser relevance if one is mainly interested in flow simulations. In case of roughness, this limited sensitivity was also seen for transport predictions if the solute had already travelled some distance. Connectivity however, can be a decisive factor for such simulations.


Over, M. W., U. Wollschlaeger, C. A. Osorio-Murillo, and Y. Rubin (2015), Bayesian inversion of Mualem-van Genuchten parameters in a multilayer soil profile: A data-driven assumption-free likelihood function, Water Resources Research, 51(2), 861–884, doi:10.1002/2013WR014956.

This paper introduces a hierarchical simulation and modeling framework that allows for infer- ence and validation of the likelihood function in Bayesian inversion of vadose zone hydraulic properties. The likelihood function or its analogs (objective functions and likelihood measures) are commonly assumed to be multivariate Gaussian in form; however, this assumption is not possible to verify without a hierarchical simulation and modeling framework. In this paper, we present the necessary statistical mechanisms for uti- lizing the hierarchical framework. We apply the hierarchical framework to the inversion of the vadose zone hydraulic properties within a multilayer soil profile conditioned on moisture content observations collected in the uppermost four layers. The key result of our work is that the goodness-of-fit validated likelihood func- tion form provides empirical justification for the assumption of multivariate Gaussian likelihood functions in past and future inversions at similar sites. As an alternative, the likelihood function need not be assumed to follow a parametric statistical distribution and can be computed directly using nonparametric methods. The nonparametric methods are considerably more computationally demanding, and to demonstrate this approach, we present a smaller dimension synthetic case study of evaporation from a soil column. The main drawback of our work is the increased computational expense of the inversion.


Osorio-Murillo, C. A., M. W. Over, H. Savoy, D. P. Ames, and Y. Rubin (2015), Software framework for inverse modeling and uncertainty characterization, Environmental Modelling & Software, 66, 98–109, doi:10.1016/j.envsoft.2015.01.002.

Estimation of spatial random fields (SRFs) is required for predicting groundwater flow, subsurface contaminant movement, and other areas of environmental and earth sciences modeling. This paper presents an inverse modeling framework called MAD# for characterizing SRFs, which is an imple- mentation of the Bayesian inverse modeling techniqueMethod of Anchored Distributions (MAD).MAD# allowsmodelers to “wrap” simulation models using an extensible driver architecture that exposes model parameters to the inversion engine. MAD# is implemented in an open source software package with the goal of lowering the barrier to using inverse modeling in education, research, and resource management. MAD# includes an intentionally simple user interface for simulation configuration, external software integration, spatial domain and model output visualization, and evaluation of model convergence. Four test cases are presented demonstrating the novel functionality of this framework to apply inversion in order to calibrate the model parameters characterizing a groundwater aquifer



Over, M. W., X. Chen, Y. Yang, and Y. Rubin (2013), A strategy for improved computational efficiency of the method of anchored distributions, Water Resour. Res., 49, 1-19, doi: 10.1002/wrcr.20182.

Y. Yang, M. Over, and Y. Rubin (2012) Strategic placement of localization devices (such as pilot points and anchors) in inverse modeling schemes, Water Resour. Res., 48, W08519, doi: 10.1029/2012WR011864.

X. Chen, H. Murakami, M. S. Hahn, G. E. Hammond, M. L. Rockhold, J. M. Zachara, and Y. Rubin (2012) Three-dimensional Bayesian geostatistical aquifer characterization at the Hanford 300 Area using tracer test data, Water Resour. Res., 48, W06501, doi: 10.1029/2011WR010675.

H. Murakami, X. Chen, M. S. Hahn, Y. Liu, M. L. Rockhold, V. R. Vermeul, J. M. Zachara, and Y. Rubin (2010) Bayesian approach for three-dimensional aquifer characterization at the Hanford 300 Area, Hydrol. Eart Syst. Sci., 14, doi: 10.5194/hess-14-1989-2010.

Rubin, Y., X. Chen, H. Murakami, and M. Hahn (2010), A Bayesian approach for inverse modeling, data assimilation,and conditional simulation of spatial random fields, Water Resour. Res., 46, W10523, doi:10.1029/2009WR008799.


MAD Abstracts:

MAD Presentation(s):

Archived Software Editions and Documentation:


WRR Paper #2012WR011957R Supporting Material

Last edited May 1, 2016 at 9:34 PM by frystacka, version 76


matthewoverUCB Sep 9, 2013 at 11:36 PM 
Heather, please maintain the supporting material WRR paper link even in a revised version.

matthewoverUCB Aug 25, 2011 at 11:22 PM 
Please track changes to any documents as we further the development, so there is a running record of what has been accomplished/revised/opened.