## 1. Introduction

[2] Numerical inversion of time-lapse geophysical data has found widespread application to image temporal changes of geophysical properties caused by mass transfer through porous and fractured media, such as water movement in the vadose zone [e.g.,*Daily et al.*, 1992; *Binley et al.*, 2002; *Doetsch et al.*, 2010] and solute transport through aquifers [e.g., *Day-Lewis et al.*, 2003; *Singha and Gorelick*, 2005; *Day-Lewis and Singha*, 2008]. Various studies have shown that such hydrodynamic characterization of the subsurface using geophysical data can be much enhanced, compared to traditional geophysical analysis, if the inversion is directly constrained or coupled with hydrological properties, processes or insights. In this way, geophysical data are used to explore the permissible parameter range of a model describing hydrological properties or some aspect of flow or solute transport, using known or jointly estimated petrophysical relationships [e.g., *Kowalsky et al.*, 2005; *Hinnell et al.*, 2010; *Huisman et al.*, 2010; *Irving and Singha*, 2010; *Scholer et al.*, 2011].

[3] Commonly used smoothness-constrained deterministic inversions used to image plume targets typically exhibit difficulty to accurately resolve plume mass and spread [e.g.,*Singha and Gorelick*, 2005; *Day-Lewis et al.*, 2007; *Doetsch et al.*, 2010]. Such inverse problems can be reformulated to conserve mass and to yield recovered plumes that are more compact [e.g., *Ajo-Franklin et al.*, 2007; *Farquharson*, 2008], but they do not appropriately consider model nonlinearity during the inference of subsurface properties. Also, geophysical data are known to most frequently contain insufficient information to uniquely resolve spatially varying subsurface properties at a high spatial resolution, which means that multiple solutions that fit the data equally well are possible. Ideally, (hydro)geophysical inversion methods should consider this inherent nonuniqueness and provide an ensemble of model realizations that accurately span the range of possible (hydro)geophysical interpretations, with probabilistic properties that are consistent with the available data and prior information [e.g., *Mosegaard and Tarantola*, 1995; *Ramirez et al.*, 2005].

[4] We present a novel stochastic inversion method to recover three-dimensional tracer distributions from time-lapse geophysical data, and provide consistent estimates of model uncertainty within a Bayesian framework. This approach employs a lower-dimensional model parameterization related to the Legendre moments of the plume [*Teague*, 1980; *Day-Lewis et al.*, 2007]. During the inversion, the proposed moments are mapped into a tracer distribution (e.g., moisture content) which is subsequently transformed into a geophysical model (e.g., radar wave speed) using a petrophysical relationship. The geophysical model response (e.g., first-arrival travel times) is then simulated and compared with measurement data. To make sure that we accurately mimic expected plume mass and morphology, we do not work directly with the Legendre moments, but instead sample the null-space of the singular value decomposition (SVD) of a matrix containing predescribed mass and morphological constraints. This further reduces the dimensionality of the permissible model space thereby allowing for Markov chain Monte Carlo (MCMC) simulation to explore the posterior distribution of the plume geometry. We illustrate our method with two synthetic vadose zone water tracer application experiments involving increasingly complex plume shapes that are sampled using different amounts of cross-hole ground-penetrating radar (GPR) travel time data. We compare and evaluate our results against both classical least squares and compact mass conservative deterministic inversion results. To the best of our knowledge, this is the first mass conservative MCMC inversion of three-dimensional (3-D) tracer transport using time-lapse geophysical data.

[5] This paper is organized as follows. Section 2 presents the theoretical concepts underlying the proposed inversion approach. In section 3, we evaluate our method against state-of-the-art deterministic inversion techniques for a synthetic experiment involving a moisture plume with a relatively simple form similar to*Doetsch et al.* [2010]. This is followed in section 4 by an application to a much more heterogeneous moisture plume. Then, section 5 discusses important aspects that influence the performance of the method and suggests some further developments. Finally, section 6 draws conclusions about the presented work.