3D Joint hydrogeophysical inversion using similarity measures

Abstract

Parameter and state estimation for groundwater models within a coupled hydrogeophysical framework has become common in the last few years. It has been shown that such estimates are usually better than those from a single data inversion and different approaches have been suggested in literature to combine the essentially two different modalities in order to obtain a single estimate for the property of interest. However, most of the coupled hydrogeophysical frameworks rely on implementing some petrophysical relationships to couple the groundwater and geophysical variable. Such relationships are usually uncertain and hard to parametrize for a large region and can produce mass errors in the final estimates. Replacing the fixed petrophysical relationship by a more loose similarity constraint is therefore an appealing alternative for coupling the two different models.

In this work we further explore the potential of structure similarity measures for coupled inversion in 3D, specifically a version of cross-gradient field product. Furthermore, we propose an efficient computational approach for minimization of coupled objective function with multiple data misfits, which is applicable to large scale inverse problems. To test the applicability of the structure-coupled inversion we analyzed three different synthetic scenarios for solute tracer tests, estimating initial conditions or hydraulic conductivity.

Introduction

The idea of a structure-coupled inversion to image the subsurface using geophysical methods has been studied over the last two decades [1], [2], [3]. Typically, indirect data are collected by multiple methods to map some geological feature, hydrological state or conductivity. Since each method is based on a different physical phenomena, the independent inversion estimates often vary [4]. Moreover, each set of data might have a very different resolution and is contaminated by measurement errors. However, if all datasets are dependent on the same property, inverting them in some coupled fashion is desired in order to provide a single estimate, or a single confidence interval if stochastic methods are applied.

In the structure-coupled inversion the two or more different models to be estimated are not linked by an explicit petrophysical relationship but instead, some similarity is assumed, often implying that the two different models exhibit changes in the same area. This similarity is then imposed as a structural constraint in the minimization procedure leading to model estimates of different properties that share the same “structure”, where under the term structure we do not necessarily refer to geological structure (e.g. lithology). Examples of structure-coupled geophysical inversions can be found in [1], [5], [6], [7].

Geophysical imaging became also popular in hydrology to monitor hydrological states or to provide indirect data for parameters such as hydraulic conductivity, porosity or for example dispersion. Coupled approaches, i.e. inverting both the geophysical and hydrogeological data, have been shown to give better estimates compared to uncoupled ones, ensuring that geophysical methods provide realistic estimates with respect to groundwater flow processes (see in [8], [9] or [10]), the success of coupled approach is of course dependent on how correct is the conceptual hydrogeological model. Besides, the majority of coupled approaches is based on some petrophysical relationship, that links the geophysical and hydrological property; for example electrical conductivity with solute content or seismic velocity with porosity, fluid conductivity or geological facies location. The parameters for such petrophysical relationships have to be either calibrated in the field, estimated based on laboratory soil samples or become a part of the estimation procedure within the inverse problem, and thus increasing the amount of unknowns.

Petrophysical relationships represent an explicit way to impose a similar structure between the two different models and for environments with well mapped geology and well calibrated parameters it is clearly the best way to link groundwater and geophysical data, yet such information is generally not available or highly uncertain. Some authors question the validity of the petrophysical links (e.g. knowledge of porosity distribution) for their purpose of study, and in some cases they do calibrate it in the field, but often this petrophysical relation is accepted just by applying some fixed parameters for either synthetic or real datasets. In many cases the direct link might cause significant errors in the solute content estimates or enforce the coincidence of the two models, when there is none. This naturally leads to the idea of replacing the fixed petrophysical relationship by a more loose structural constraint, which will become a part of the objective function to be minimized when solving the inverse problem. The structural constraint has a preference for similarity between the two models (states or parameters), i.e. the spatial changes of the models occur at the same location; however, the two models (parameter fields) do not need to coincide. The consistency between the models ideally results from the data and not from assumed petrophysical relations.

In this work we further explore the potential of joint hydrogeophysical inversion when the parameters for petrophysical relationships are unknown or highly uncertain. We focus on coupling between soil bulk conductivity and solute content, which are often correlated in hydrological state and parameter estimation.

Different computational approaches exist to alleviate the need of the fixed petrophysical relationship. For example in the work of [11] authors assumed the parameter of interest is mainly affected by geological facies and used a level set method to minimize the coupled objective function. The same method could be expanded for joint inversion with groundwater and geophysical data using the same level set parametrization (geological facies distribution). The approach was tested with seismic and hydrological tomography data on a synthetic example. Due to the level set method, there was no need for a direct link between seismic velocity and hydraulic conductivity, however, in this application the structure similarity was enforced by the a-priori assumption, i.e. knowledge of the location of geological facies, and could be applied only due to the fact that both geophysical and hydrological data were directly dependent on the same geological property.

A different approach was tested in [12], where the authors replaced the petrophysical relationship by an assumption of a strong correlation between the fluid and soil bulk electrical conductivity. In particular, for their synthetic study with a solute tracer test, they used ERT time lapse sampling together with groundwater head and fluid conductivity data to determine hydraulic conductivity. In their approach they maximized the correlation between transient changes in the fluid and soil bulk conductivity and thus avoid solving the ERT inversion; the input for the ERT forward model were fluid conductivity changes alone. The correlation term was then added when solving the standard groundwater (GW) inverse problem to estimate hydraulic conductivity. The hydraulic head and fluid conductivity sensitivities were solved by perturbation method in parallel and hydraulic conductivity was parametrized by Pilot points method with linear interpolation. A high performance cluster was necessary to solve the inverse problem. The presented example showed that even indirect geophysical information lead to improvement in the estimates for hydraulic conductivity.

To our knowledge, the first work applying a structural constraint in hydrogeophysical inverse problem is the study of Lochbuhler et al. [13]. The authors used a structural constraint to combine two datasets, ground-penetrating radar data with hydraulic tomography or tracer mean arrival times to estimate hydraulic conductivity. A cross product of the spatial gradients of the geophysical and groundwater model was implemented as a structural constraint, creating a penalty in the coupled objective function. Both hydrological and geophysical data were inverted jointly in one coupled objective function, and the weight parameters adjusted at each iteration to account for different convergence behavior of each dataset. The weight for cross-gradient term was assigned a rather large value to enforce the cross-gradient term minimization. The authors tested the structural coupled approach on 2D synthetic case as well as field data from the Widen gravel aquifer in Switzerland. The recovered models do indeed display similar structure, though not following a simple linear petrophysical relationship. The estimates for hydraulic conductivity were in accordance with previous field studies.

Our work further expands on the joint inversion with a cross-gradient field product by [13]. Compared to their work we implemented a full 3D inversion and jointly inverted the geophysical and hydrological data that are not directly dependent on the property of interest. Furthermore, we replaced the cross product with an equivalent expression that can be more stably discretized, especially in 3D. Analytically derived sensitivities enabled us to develop a highly efficient computational scheme that can solve structurally coupled problems in 3D. We tested the scheme on synthetic examples representing solute tracer tests.

We solved the inverse problem for either initial solute states or hydraulic conductivity field. The groundwater data in this study are rather scarce making even the standard joint inversion (i.e. with the knowledge of the petrophysical relationship) a challenging ill-posed inverse problem. In our formulation, we minimize both groundwater and geophysical data misfits together with the similarity measure term to update both estimates of the groundwater and geophysical property field. We compared two computationally different methods to solve this coupled problem and also compared the obtained results with separate groundwater and geophysical inversions.

In the following section we briefly introduce our discretized groundwater and geophysical model, and derive the sensitivities necessary for the Gauss-Newton method. In Section 3 we formulate the coupled inverse problem with inexact constraint represented by the structure similarity term and introduce the minimization methods. In Section 4 we introduce different synthetic scenarios and show the results by different methods. Discussion and summary follows.